Proceedings of the 1999 Summer School in Particle Physics: Ictp, Trieste, Italy 21 June-9 July 1999 [1 ed.] 9810242387, 9789810242381

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PROCEEDINGS OF THE 1999 SUMMER SCHOOL IN

PARTICLE PHYSICS

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the

abdus salam

international centre for theoretical physics international atomic energy agency

united nations educational, scientific and cultural organization

(~) '-,.......

THE ICTP SERIES IN THEORETICAL PHYSICS - VOLUME 16

PROCEEDINGS OF THE 1999 SUMMER SCHOOL IN

PARTICLE PHYSICS ICTP, Trieste, Italy 21 June - 9 July 1999

Editors

G. SENJANOVIC A. Yu. SMIRNOV ICTp, Trieste, Italy

b World Scientific

III

'

Singapore· New Jersey· London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. POBox 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661

UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

PROCEEDINGS OF THE 1999 SUMMER SCHOOL IN PARTICLE PHYSICS Copyright © 2000 by The Abdus Salam International Centre for Theoretical Physics

ISBN 981-02-4238-7

This book is printed on acid and chlorine free paper. Printed in Singapore by World Scientific Printers

v

Preface

In this volume, precision tests of the Standard Model and a wide spectrum of physics beyond it, such as supersymmetry, grand unification and the fermion mass problem, are covered. The emphasis is on the areas where new experimental results will lead to significant progress: neutrino physics, CP violation and B physics. The articles written by top level experts in the fields, give a comprehensive view of the state-of-the-art of modern particle physics. We wish to thank the lecturers again for the great job they have done. We also wish to thank the students for their enthusiasm and interest, and all of our staff, for their invaluable help.

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vii

Contents

Preface

v

The Electroweak Standard Model W.Hollik Aspects of Quantum Chromodynamics A. Pich

53

Neutrino Physics E. Kh. Akhmedov

103

CP Violation In and Beyond the Standard Model Y. Nir

165

Introduction to B Physics M. Neubert

244

Supersymmetry Phenomenology H. Murayama

296

Unification and Supersymmetry R. N. Mohapatra

336

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THE ELECTROWEAK STANDARD MODEL W. HOLLIK Institut fur Theoretische Physik, Universitiit Karlsruhe, Kaiserstr. 12, D-76128 Karlsruhe, Germany E-mail: [email protected] In these lectures we discuss the structure of the electroweak Standard Model and its quantum effects. In the first part we give an introduction to the construction of the basic Lagrangian, followed by some details on the calculation of amplitudes at the one-loop level. It is shown how in higher-order calculations finite results for electroweak precison observables are obtained, with emphasis on the W, Z masses and the Z resonance observables. In the second part, the predictions of the Standard Model are compared with the recent experimental data, with implications for the Higgs boson mass range. A brief review on the status of the vector-boson selfinteractions and of the scalar sector of the Standard Model concludes the lectures.

1

Introduction

The electroweak Standard Model 1-3 is the commonly accepted theory of the fundamental electroweak interaction. It is a gauge invariant quantum field theory based on the the symmetry group SU(2)xU(1),which is spontaneously broken by the Higgs mechanism. The renormalizability of the Standard Model 4 allows us to make precise predictions for measureable quantities also in higher orders of the perturbative expansion, in terms of a few input parameters. The higher-order terms, radiative corrections or quantum corrections, contain the self-coupling of the vector bosons as well as their interactions with the Higgs field and the top quark, even for processes involving only light fermions. Their calculation provides the theoretical basis for electroweak precision tests. Assuming the validity of the Standard model, the presence of the top quark and the Higgs boson in the loop contributions to electroweak observables allows us to obtain indirectly significant bounds on their masses from precision measurements of these observables. The high precision experiments impose stringent tests on the Standard Model. Besides the impressive achievements in the determination of the Z boson parameters 5,6 and the W mass 7,6, the most important step has been the discovery of the top quark at the Tevatron 8, now at a mass mt = 174.3 ± 5.1 GeV 9, which coincides perfectly with the mass range obtained indirectly from the radiative corrections. The only unknown quantity at present is the Higgs boson. Its mass is nowadays getting more and more constrained as well, by a comparison of the Standard Model predictions with the experimental

2

data. In these lectures we give a brief discussion of the structure of the Standard Model and its quantum corrections for testing the electroweak theory at present and future colliders. The predictions for the vector boson masses, cross sections, and the Z resonance observables like the width of the Z resonance, partial widths, effective neutral current coupling constants and mixing angles at the Z peak, are presented in some detail. Moreover, we address the study of the vector-boson self-interaction and the constraints on the Higgs sector. Comparisons with the recent experimental data are shown and their implications for the present status of the Standard Model are discussed.

2 2.1

Formulation of the Standard Model The classical Lagrangian

The fundamental fermions, as families with left-handed doublets and righthanded singlets, appear as the fundamental representation of the group SU(2)xU(1),

(~) L' (~T) L'

(:e )

L'

(

~ ) L'

(

~ ) L'

G)

L'

UR,

eR,

dR,

MR,

TR

CR,···

They can be classified by the quantum numbers of the weak isospin I, h, and the weak hypercharge Y. The Gell-Mann-Nishijima relation establishes the relation of these basic quantum numbers to the electric charge Q:

Q= h

y

+ "2.

(1)

This structure can be incorporated in a gauge invariant field theory of the unified electromagnetic and weak interactions by interpreting SU(2)xU(1) as the group of gauge transformations under which the Lagrangian is invariant. This full symmetry has to be broken by the Higgs mechanism down to the electromagnetic gauge symmetry; otherwise the W±, Z bosons would be massless. The minimal formulation, the Standard Model, requires a single Higgs field which is a doublet under SU(2). According to the general principles of constructing a gauge invariant field theory with spontaneous symmetry breaking, the gauge, Higgs, and fermion parts of the electroweak Lagrangian (2)

3

are specified in the following way: Gauge fields. SU(2)xU(1) is a non-Abelian group which is generated by the isospin operators h, h, h and the hypercharge Y (the elements of the corresponding Lie algebra). Each of these generalized charges is associated with a vector field: a triplet of vector fields W~,2,3 with h,2,3 and a singlet field B JJ with Y. The isotriplet W;, a = 1,2,3, and the isosinglet B JJ lead to the field strength tensors

W:v = vI' W: - Vv W:

+ 92 fa be WiW~,

B JJv = vJJB v - vvB/J"

(3)

92 denotes the non-Abelian SU(2) gauge coupling constant and gl the Abelian

U(l) coupling. From the field tensors (3) the pure gauge field Lagrangian £

G

= _~ W a WJLv,a _ ~B BJLv 4 JJV 4 J1.V

(4)

is formed, which is invariant under non-Abelian gauge transformations. Fermion fields and fermion-gauge interactions. The left-handed fermion fields of each lepton and quark family

1/1; = (1/1{+) 1/1Jwith family index j are grouped into SU(2) doublets with component index a = ±, and the right-handed fields into singlets .I,R _ .I,R 'Pj

-

'Pju'

Each left- and right-handed multiplet is an eigenstate of the weak hypercharge Y such that the relation (1) is fulfilled. The covariant derivative DJ1. = vI' - i92 JaW:

.

Y

+ Z91"2 BJ1.

(5)

induces the fermion-gauge field interaction via the minimal substitution rule

£F

-~""" .T.L·

-

~ 'Pj Z"( j

I'D

.I,L

J1.'Pj

+

~ .T.R .

~ 'PjuZ"f

I'D

.I,R

J1.'Pju·

(6)

j,u

Higgs field and Higgs interactions. For spontaneous breaking of the SU(2)xU(1) symmetry leaving the electromagnetic gauge subgroup U(l)em unbroken, a single complex scalar doublet field with hypercharge Y = 1 (7)

4

is coupled to the gauge fields through LH = (DIL1f/2

(120)

dO. dO. '

Without QED corrections, photon-exchange and Z-photon interference terms, it is given by

0, _

3 AFB - 4" Ae A ,. - the longitudinal polarization of a final-state PTpoi -- A r,

(121)

T

lepton (122)

- the left--right asymmetry ALR

=

(1L - (1R . (1L (1R

+

~ Pe

(123)

with the cross section (1L(R) for longitudinally polarized, left (right)-handed, initial state electrons, with the degree of polarization Pe . Without QED corrections, photon-exchange and Z-photon interference terms, the on-resonance asymmetry is given by (124)

5.3

Z line-shape and Z widths

The integrated cross section (1(8) for e+e- -+ f! around the Z resonance with unpolarized beams is obtained from the formulae of the previous section in a straight forward way, expressed in terms of the effective vector and axial vector coupling constants. It is, however, convenient to rewrite (1(8) in terms of the Z width and the partial widths r e, r, in order to have a more model independent expression. We concentrate on the pure Z-resonance term. It differs from a BreitWigner line-shape by the 8-dependence of the width. Denoting by r z the total width, the pure Z-resonance part of the integrated cross section for e+e- -+ f! has the form 127r

rer,

(10= M2 .~.

Z

Z

(125)

32 The s-dependent width gives rise to a dislocation of the peak maximum by ~ -34 GeV 72,73 By means of the substitution 73 s - M'i

+ isfz/Mz

= (1

+ it)(s -

M'i

+ iMzI'z)

(126)

with Mz

= M z (l + ,/)-1/2,

I'z

= fz(l + "l)-1/2,

'Y

=

fz Mz

(127)

a genuine Breit-Wigner resonance shape is recovered: sI'2 ures(S) = Uo (s _ M'i)2:

M'ir~

(128)

Numerically one finds: Mz - Mz ~ -34 MeV, r z - fz ~ -1 MeV. Uo is not changed. Mz corresponds to the real part of the S-matrix pole of the Z-resonance 74. From line-shape measurements one obtains the parameters M z , fz, Uo or the partial widths, respectively. Mz is used as a precise input parameter, together with a and GIL; the width and partial widths allow comparisons with the predictions of the Standard Model. The total Z width f z can be calculated essentially as the sum over the fermionic partial decay widths. Expressed in terms of the effective coupling constants, they read up to second order in the fermion masses: ff = f0

M~ ) 1. (1 + Q2 3a) 471" + ~fQCD [I9v 12+ 19 12 ( 1 - 6m} f

f

f

f

A

(129)

with fo

= Nb y'2~;71"Ml,

Nb = 1 (leptons), = 3 (quarks).

The QCD correction for the light quarks with mq

~fbcD with

= fo

~

0 is given by

(lg~12 + 19~12) . KQCD

(130)

75

KQCD

= a. + 1.41 (a.) 71"

2

71"

_12.8 (as) 71"

3

Q2

_.-l..

aa. .

4 71"2

(131)

For b quarks the QCD corrections are different, because of finite b mass terms and to top-quark-dependent 2-loop diagrams for the axial part:

~f~CD = ~f~CD

+

fo

[19t1 2Rv +

19~12 RAJ.

(132)

33

The coefficients in the perturbative expansions

)2 + cv (as)3 + ... , s cAas + c A(a )2 + ...

Rv = C1vas -:; RA =

1 -:;

+ c2v

s (a -:;

2

-:;

3

-:;

depending on mb and mt, are calculated up to third order in as, except for the mb-dependent singlet terms, which are known to O(a;) 76,77. For a review of the QCD corrections to the Z width, see 78. The partial decay rate into b-quarks, in particular the ratio Rb = fb/f had , is an observable of special sensitivity to the top quark mass. Therefore, beyond the pure QCD corrections, also the 2-100p contributions of the mixed QCDelectroweak type, are important. The QCD corrections were first derived for the leading term of O(asGl'mF) 79 and were subsequently completed by the O(a s ) correction to the residual terms of O(aa s ) 80-82. In the same spirit, also the complete 2-100p O(aa s ) to the partial widths into the light quarks have been obtained, beyond those that are already contained in the factorized expression (130) with the electroweak 1-100p couplings 83. These "non-factorizable" corrections yield an extra negative contribution of -0.55(3) MeV to the total hadronic Z width (converted into a shift of the strong coupling constant, they correspond to c5a s = 0.001). In summary, the 2-100p corrections of O(aa s ) to the hadronic Z widths are by now completely under control. 5.4

Accuracy of the Standard Model predictions

For a discussion of the theoretical reliability of the Standard Model predictions, one has to consider the various sources contributing to their uncertainties: Parametric uncertainties result from the limited precision in the experimental values of the input parameters, essentially as = 0.119 ± 0.002 59, mt = 174.3 ± 5.1 GeV, mb = 4.7 ± 0.2 GeV, and the hadronic vacuum polarization as discussed in section 4.2. The conservative estimate of the error in Eq. (90) leads to c5Mw = 13 MeV in the W-mass prediction, and c5 sin 2 () = 0.00023 common to all of the mixing angles. The uncertainties from the QCD contributions can essentially be traced back to those in the top quark loops in the vector boson self-energies. The knowledge of the O(a;) corrections to the p-parameter and ~r yields a significant reduction; they are small, although not negligible (e.g. rv 3 . 10- 5 in s;).

34

The size of unknown higher-order contributions can be estimated by different treatments of non-leading terms of higher order in the implementation of radiative corrections in electroweak observables ('options') and by investigations of the scheme dependence. Explicit comparisons between the results of 5 different computer codes based on on-shell and M S calculations for the Z-resonance observables are documented in the "Electroweak Working Group Report" 84. The inclusion of the non-leading 2-100p corrections rv G~m; Mj reduce the uncertainty 85,67 in Mw below 6 MeV and in s~ below 10-4, typically to ±4 . 10- 5 • 6

Standard Model and precision data

We now confront the sm predictions for the discussed set of precision observabIes with the most recent sample of experimental data 5,6. In Table 1 the Standard Model predictions for Z-pole observables and the W mass are put together for the best-fit input data set, given in (135). The experimental results 6 on the Z observables are from LEP and the SLe, the W mass is from combined LEP and pP data; the top mass is from the Tevatron, and sin20w(vN) from deep-inelastic neutrino-nucleon scattering. The notation for the various quantities is that of section 5. In addition, O"had denotes the hadronic peak cross-section, and

R-~ C -

rhad

(133)

are ratios of partial widths, explained in section 5.3. The deviation from the Standard Model prediction in the quantity Rb has been reduced below one standard deviation by now. Other small deviations, at the level of one and two standard devaitions, are observed; the largest effect occurs in the forward-backward asymmetry for b quarks, A}B' which is 2.2 0" lower than the predicted value. Whereas the leptonic asymmetries, in particular ALR from SLD, favours a very light Higgs boson, the b quark asymmetry requires a heavy Higgs. Table 2 contains the combined LEP jSLD value for the leptonic form factor for the overall normalization, Pi, and the leptonic mixing angle s~; these quantities incorporate the quantum contributions to the leptonic neutral current couplings in Eq. (111). The experimental values are derived from partial widths and asymmetries under the assumption of lepton universality. Note that the experimental value for Pi points to the presence of genuine electroweak corrections by 3.5 standard deviations. In s~ the presence of purely bosonic radiative corrections is clearly established when the experi-

35 Table 1. Precision electroweak measurements and Standard Model predictions for the bestfit p,!-rameters.

Observable

Exp. Result

8M Prediction

Pull

LEP

91.1871 ± 0.0021 2.4944 ± 0.0024 41.544 ± 0.037 20.768 ± 0.024 0.01701 ± 0.00095

Mz (GeV) rz (GeV) O"had

(nb)

Rl AOl FB T

91.1869 2.4957 41.479 20.740 0.01625

0.08 -0.56 1.749 1.156 0.80

0.1425 ± 0.0044 0.1483 ± 0.0051

0.1472 0.1472

-1.07 0.21

0.2321 ± 0.0010

0.23150

0.60

0.23099 ± 0.00026

0.23150

-1.95

0.21642 ± 0.00073 0.1674 ± 0.0038 0.0988 ± 0.0020 0.0692 ± 0.0037 0.911 ± 0.025 0.630 ± 0.026

0.21583 0.1722 0.1032 0.0738 0.935 0.668

0.81 -1.27 -2.20 -1.23 -0.95 -1.46

80.350 ± 0.056

80.385

-0.62

80.448 ± 0.062 0.2254 ± 0.0021 174.3 ± 5.1

80.385 0.2229 173.1

1.02 1.13 0.23

polarization:

AT Ae qij charge asymmetry:

sin20f «

QFB

»

SLD sin 2 Of (ALR) LEP jSLD Heavy Flavor

Rb Rc AO,b FB AO,c FB Ab Ac LEP2 Mw (GeV) pP and vN Mw (GeV) sinOw (vN) mt (GeV)

mental result is compared with a theoretical value containing only the fermion loop corrections, an observation that has been persisting already for several years 86. The W mass prediction in Table 2 is obtained from Eq. (107) (including the higher-order terms) from Mz, Gp., a. and MH,mt. The experimental value for the W mass is from the combined LEP 2, UA2, CDF and DO results. The

36 Table 2. Combined experimental results and SM predictions for the best fit

Observable

Exp. Result

Pi

1.0041 ± 0.0012 0.23161 ± 0.00018 80.394 ± 0.042

2 8l

Mw (GeV)

SM Prediction 1.0053 0.23150 80.385

Standard Model prediction for Mw is shown in Figure 2 together with the results from the direct measurements and from the Standard Model fit to the Z observables and ~r 5.

_.. - LEP2,

80.5

pp Data

68% CL

:> Q)

(9

'---' 80.4

;: ~

80.3 Preliminary

80.2 -f--,,......:;.;~....;::.:::~...:...;::.;=r-...--...--.-...,.-.....-.--.-:~ 130 150 170 190 210

mt [GeV] Figure 2. The W mass versus mt. Standard Model prediction and direct and indirect determination.

The quantity sin20w(vN) is essentially the mass ratio Mw /Mz, measured indirectly in deep-inelastic neutrino-nucleon scattering. sin 2 Ow(vN) = 1 - M'fv/M;.

(134)

37

Table 1 contains the average from the experiments CCFR, CDHS, CHARM and NUTEV result 87, extracted for mt = 175 GeV and MH = 150 GeV The value corresponds to Mw = 80.25 ± 0.11 GeV and is hence consistent with the direct vector boson mass measurements and with the standard theory.

Stanford 1999 Measurement

Pull

mz [GeV]

91.1871 ± 0.0021

r z [GeV]

2.4944 ± 0.0024 41.544 ± 0.037 20.768 ± 0.024

.08 -.56 1.75 1.16

(J~adr [nb] Ae ~be

Ae At sin2a~:t mw[GeV]

Ab Ac ~bb ~bc

Ab Ac sin2a~:t 2 sin aw mw[GeV]

mt[GeV]

~~~d(mZ)

0.01701 ± 0.00095 .80 0.1483 ± 0.0051 .21 0.1425 ± 0.0044 -1.07 0.2321 ± 0.0010 80.350 ± 0.056

.60 -.62

0.21642 ± 0.00073

.81 -1.27 0.1674 ± 0.0038 0.0988 ± 0.0020 -2.20 -1.23 0.0692 ± 0.0037 0.911 ± 0.025 -.95 -1.46 0.630 ± 0.026 0.23099 ± 0.00026 -1.95 0.2255 ± 0.0021 80.448 ± 0.062

1.13 1.02

174.3±5.1 0.02804 ± 0.00065

.22 -.05 -3 -2 -1 0 1 2 3

Figure 3. Experimental results and pulls from a standard model fit obs(SM)/(exp.error).

5.

pull = obs( exp)-

Standard model global fits: The FORTRAN codes ZFITTER 88 and TOPAZO 89 have been updated by incorporating all the recent precision calculation results that were discussed in the previous section. Comparisons have shown good agreement between the predictions from the two independent programs 85. Global fits of the Standard Model parameters to the electroweak precision data done by the Electroweak Working Group 5 are based on these

38

4

2

o

EXcl!,1.ded.

Preliminary

10

Figure 4. Higgs mass dependence of X2 in the global fit to precision data 5,6. The shaded band displays the error from the theoretical uncertainties obtained from various options in the codes ZFITTER and TOPAZO.

recent versions. Including mt and M w from the direct measurements in the experimental data set, together with s~ from neutrino scattering, the Standard Model parameters for the best-fit result are 5,6 mt = 173.1

± 4.6GeV

MH = 92:!:~g GeV

as

= 0.119 ± 0.003.

(135)

The overall X2/d.o.f. = 22/15 for the overall fit has slightly increased within the last year. As can be seen from Figure 3 and Table 1, the deviations of the individual quantities from the Standard Model best-fit values are, with one exception, below two standard deviations. The upper limit on the Higgs mass at the 95% C.L. is MH < 245 GeV, where the theoretical uncertainty is included. Thereby the hadronic vacuum polarization in Eq. (90) has been used (solid line in Figure 4). Improvements of ~ahad lead to a small shift in the central value; the la upper bound on MH is influenced only marginally. The reason is that, simultaneously with

39

the shift in the central value to larger M H , the error is reduced (see Figure

4). Compared with previous years, one observes a significant decrease of the error on the Higgs mass, which besides the experimental improvements results from the reduction of the theoretical uncertainties of pure electroweak origin. The shaded band around the solid line in Figure 4 is the influence of the various 'options' (see section 5.4) in the codes ZFITTER and TOPAZO after the implementation of the 2-loop electroweak terms rv m; 85. It is thus the update of the error estimate described in the study group report 84. On the other hand, the remaining theoretical uncertainty associated with the Higgs mass bounds should be taken seriously. The effect of the inclusion of the next-to-Ieading term in the mrexpansion of the electroweak 2-loop corrections in the precision observables has shown to be sizeable, at the upper margin of the estimate given in 84. It is thus not guaranteed that the subsequent sub leading terms in the mt-expansion are indeed smaller in size. Also the variation of the MH-dependence at different stages of the calculation, as discussed in the previous subsections, indicate the necessity of more complete results at two-loop order. Having in mind also the variation of the Higgs mass bounds under the fluctuations of the experimental data 90, the limits for MH derived from the analysis of electroweak data in the frame of the Standard Model still carry a noticeable uncertainty. Nevertheless, as a central message, it can be concluded that the indirect determination of the Higgs mass range has shown that the Higgs boson is light, with its mass well below the non-perturbative regime. 7

The vector-boson self-interaction

The success of the Standard Model in the correct description of the electroweak precision observables is simultaneously an indirect confirmation of the Yang-Mills structure of the gauge boson self-interaction. For conclusive confirmations the direct experimental investigation is required. At LEP 2 (and higher energies), pair production of on-shell W bosons can be studied experimentally, allowing tests of the trilinear vector boson self-couplings and precise Mw measurements. For LEP 2, an error of about 40 MeV in Mw can be reached. Further improvements are expected from the Tevatron Run II and from the LHC, with 8Mw ~ 15 MeV. Pair production of W bosons in the Standard Model is described in Born approximation by the amplitude based on the Feynman graphs in Figure 5. Besides the t-channel v-exchange diagram, which involves only the Wfermion coupling, the s-channel diagrams contain the triple gauge interaction

40

Figure 5. Feynman graphs for e+e-

-t

W+W- in lowest order .

..[$"2'.189 GeV: preliminary

L

,......., ..0

a. ........

-----~i

20 ,

~

I

:

,,

,

,"

,

".

,;

~

~ ~

:~

:,' / '/

,i

:'

10

•

CD CD

+

Data

- - Standard t\;lodel

I:) ----

- - - no ZVVW vertex .h

.....," 0

160

170

-rs Figure 6. Cross-section for e+eprediction.

-t

••••• Ve

exchange

180

190

200

[GeV]

W+W-. measured at LEP, and the Standard Model

between the vector bosons. The gauge self-interactions of the vector bosons are essential for the high-energy behaviour of the production cross-section in accordance with the principle of unitarity. The self-interaction of the vector bosons is part of the Lagrangian (4); generalizing the triple couplings one

41

finds, with the notation Fl'v = ol'Av - ovAI' and ZI'V and w;tv analogously (replacing A -+ Z, W+): V .CWWI'/Z = e [(01' W,; - Ov W;) W-I' A

+'"I' W+WI' v Fl'v

+~ Mar W+PI' W-I' v FPv + h. c. 1 +ecotOw [(OI'W,; - ovW;) W-I' ZV +"'z W+WZI'V I' v

.Az w+ + Ma, PI' W-I' v zpv

h ] +. c.

(136)

In the Standard Model the coefficients have the values, dictated by gauge invariance according to section 2.1, (137) "'I' = "'z = 1, .AI' = .Az = O. Deviations from these values spoil the high-energy behaviour of the crosssections and would be visible at energies sufficiently above the production threshold. Measurements of the cross section for e+ e- -+ WW at LEP have confirmed the prediction of the Standard Model, as visualized in Figure 6 5,6. The non-observation of deviations from the Standard Model predictions can be converted into bounds on potential deviations of the trilinear coupling constants from their Standard Model values (denoted as anomalous couplings), which might be assigned to new physics beyond the minimal model. Those bounds are displayed in Figure 7 5,6. Experimental information on quartic gauge couplings via e+e- -+ W+W--y is also getting available in the meantime, again confirming the minimal model 6. Experimentally, WW final states are identified through their fermionic decay products. In view of the experimental precision, Standard Model calculations for the process e+e- -+ W+W- -+ 4f and the corresponding 4fermion background processes are mandatory at the accuracy level of at least 1%. This requires the understanding of the radiative corrections to the W boson production and decay processes, as well as a careful treatment of the finite-widths effects. The systematic treatment of the complete radiative corrections is a task of enormous complexity. For practical purposes, improved Born approximations are in use for both resonating and non-resonating processes, dressed by initial-state QED corrections. A status report can be found in ref 92.

42

ALEPH + DELPHI

+ OPAL

3 OJ)

~

2.5

2.5

,

2

2

1.5

1.5

1

1

0.5

0.5

o

o -0.4 -0.2

0

0.2 0.4

-0.2

-0.1

0

0.1

0.2

ilgf 3 OJ)

o

930 GeV (Figure 8 101). This result is confirmed by the calculation of the next-to-leading order correction in the liN expansion, where the Higgs sector is treated as an O(N) symmetric a-model 99. A similar increase of the 2-loop perturbative contribution with MH is observed for the fermionic decay 100 H -t f /, but with opposite sign leading to a cancellation of the one-loop correction for MH == 1100 GeV (Figure 8). The requirement of applicability of perturbation theory therefore puts a stringent upper limit on the Higgs mass. The indirect Higgs mass bounds obtained from the precision analysis show, however, that the Higgs boson is well below the mass range where the Higgs sector becomes non-perturbative. The lattice result 97 for the bosonic Higgs decay in Figure 8 for MH = 727 GeV is not far from the perturbative 2-loop result. The difference may at least partially be interpreted as missing higher-order terms. The behaviour of the quartic Higgs self-coupling ,x, as a function of a rising energy scale p" follows from the renormalization group equation with the /3function dominated by ,x and the top quark Yukawa coupling gt contributions: (140)

In order to avoid unphysical negative quartic couplings from the negative top quark contribution, a lower bound on the Higgs mass is derived. The requirement that the Higgs coupling remains finite and positive up to a scale A yields constraints on the Higgs mass M H , which have been evaluated at the 2-loop level 94,95. These bounds on MH are shown in Figure 9 95 as a

44

1.5 1.4

KV

M

... ~ ~

1.3

------- Kr

1.2 Lattice --)(

1.1

~~~~~~~ 1-lp

~~~~

1.0 400

__ ..2-lp

---.--.--.-.---*-.-.-.--.--.--.-=.:::..-:.:-~.,,-...-.. tree level

600

BOO

~~

1000

1200

MH (GeV) Figure 8. Correction factors for the Higgs decay widths H in 1- and 2-1oop order.

-t

VV (V =

w, Z)

and H

-t

ff

function of the cut-off scale A up to which the standard Higgs sector can be extrapolated, for mt = 175 GeV and Qs(Mz) = 0.118. The allowed region is the area between the lower and the upper curves. The bands indicate the theoretical uncertainties associated with the solution of the renormalization group equations 95. It is interesting to note that the indirect determination of the Higgs mass range from electroweak precision data via radiative corrections is compatible with a value of MH where A can extend up to the Planck scale. 9

Conclusions

The electroweak Standard Model has developed into the quantum field theory of the electromagnetic and weak interactions. The experimental data for testing the electroweak theory have achieved an impressive accuracy. For the interpretation of the precision experiments radiative corrections, or quantum effects, play a crucial role. The calculation of radiative corrections is theoretically well established, and many contributions have become available over the past few years to improve and stabilize the Standard Model predictions.

45

,......., 600

>

mt = 175 GeV

Q)

~

L......J

::r: 400

::a

200

Figure 9. Theoretical limits on the Higgs boson mass from the absence of a Landau pole and from vacuum stability.

After taking the measured Z mass, besides Q: and G p , for completion of the input, each other precision observable provides a test of the electroweak theory. The theoretical predictions of the Standard Model depend on the mass of the recently discovered top quark and of the as yet experimentally unknown Higgs boson through the virtual presence of these particles in the loops. As a consequence, precision data can be used to pin down the allowed range of the mass parameters. The comparison of the theoretical predictions with experimental data has confirmed the validity of the Standard Model in a convincing way: - the description of the data is of high qualaity, with small deviations which might be considered as normal; - the quantum effects of the Standard Model have been established at the level of several a; - direct and indirect determinations of the top-quark mass are compatible with each other; - the Higgs-boson mass is meanwhile also being constrained within the perturbative mass regime with the possibility that the Standard Model may be extrapolated up to energies around the Planck scale.

46

In spite of this success, the conceptual situation with the Standard Model is unsatisfactory for quite a few deficiencies: - the smallness of the electroweak scale v....., 246GeV « M p1 (the so-called hierarchy problem); - the large number of free parameters (gauge couplings, vacuum expectation value, MH, fermion masses, CKM matrix elements), which are not predicted but have to be taken from experiments; - the pattern that occurs in the arrangement of the fermion masses; - the missing way to connect to gravity. Besides the list of conceptual theoretical problems, a new perspective arises through the recent experimental results on the atmospheric neutrino anomaly 102, which can most easily be explained by oscillations between different v species, associated with neutrino masses different from zero. In the strict-minimal model, neutrinos are massless and right-handed neutrino components are absent. The evidence for massive neutrinos requires a modification of the minimal model in order to accommodate neutrinos with mass. The straightforward way to introduce mass terms is the augmentation of the fermion sector by right-handed partners VR; together with the familiar IlL these allow the presence of Dirac mass terms....., myvv with v = IlL + VR and my as additional mass parameters, without altering the global architecture of the standard model and without spoiling the successful description of all the other electroweak phenomena. What appears unsatisfactory is the unexplained smallness of the neutrino Dirac masses, as enforced by the empirical situation. A commonly accepted elegant solution is given by the seasawmechanism where a lepton-number-violating Majorana mass term ....., M is introduced. Together with the Dirac mass, which is of the order of the usual charged lepton masses, a very light and a very heavy v component appear with the light one almost entirely left-handed, when M is of the order of the GUT scale. Candidates for specific models are Grand Unification scenarios such as the SO(lO)-GUT, where VR fits into the same 16-dimensional representation as the other fermions of a family. Hence, the appearance of small neutrino masses points towards a new high-mass scale beyond the minimal model, which may be associated with the concept of further unification of the fundamental forces.

47 References

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53

ASPECTS OF QUANTUM CHROMODYNAMICS Antonio Pich Departament de F{sica Teorica, IFIG, Universitat de Valencia Apt. Gorreus 2085, E-46071 Valencia, Spain E-mail: [email protected]

GSIG

These lectures provide an overview of Quantum Chromodynamics (QCD), the SU(3)c gauge theory of the strong interactions. The running of the strong coupling and the associated property of Asymptotic Freedom are analyzed. Some selected experimental tests and the present knowledge of a. and the quark masses are briefly summarized. A short description of the QCD flavour symmetries and the dynamical breaking of chiral symmetry is also given. A more detailed discussion can be found in standard textbooks l ,2,3,4 and recent reviews 5,6.

1

Quarks and Colour

A fast look into the Particle Data Tables 7 reveals the richness and variety of the hadronic spectrum. The large number of known mesonic and baryonic states clearly signals the existence of a deeper level of elementary constituents of matter: quarks 8 • In fact, the messy hadronic world can be easily understood in terms of a few constituent spin-! quark flavours:

Q=+~

u

c

t

Q_13 -

d

s

b

Assuming that mesons are M == qij states, while baryons have three quark constituents, B == qqq, one can nicely classify the entire hadronic spectrum. There is a one-to-one correspondence between the observed hadrons and the states predicted by this simple classification. Thus, the Quark Model appears to be a very useful Periodic Table of Hadrons. However, the quark picture faces a problem concerning the Fermi-Dirac statistics of the constituents. Since the fundamental state of a composite system is expected to have L = 0, the b. ++ baryon (J = with h = + corresponds to u t u t u t , with the three quark spins aligned into the same direction (S3 = +!) and all relative angular momenta equal to zero. The wave function is symmetric and, therefore, the b. ++ state obeys the wrong statistics. The problem can be solved assuming 9 the existence of a new quantum number, colour, such that each species of quark may have Nc = 3 different colours: qa, 0' = 1,2,3 (red, yellow, violet). Then, one can reinterpret this

¥)

¥

54

\

.

I

~

Figure 1: Two-jet event from the hadronic decay of a Z boson (DELPHI).

state as d++ '" f cx {3-y lu!u~u~) (notice that at least 3 colours are needed for making an antisymmetric state). In this picture, baryons and mesons are described by the colour-singlet combinations

(1.1 ) In order to avoid the existence of non-observed extra states with non-zero colour, one needs to further postulate that all asymptotic states are colourless, i.e. singlets under rotations in colour space. This assumption is known as the confinement hypothesis, because it implies the non-observability offree quarks: since quarks carry colour they are confined within colour-singlet bound states. The quark picture is not only a nice mathematical scheme to classify the hadronic world. We have strong experimental evidence of the existence of quarks. Fig. 1 shows a typical Z -+ hadrons event. Although there are many hadrons in the final state, they appear to be collimated in 2 jets of particles, as expected from a two-body decay Z -+ qij, where the qij pair has later hadronized.

1.1

Evidence of Colour

A direct test of the colour quantum number can be obtained from the ratio

- a-(e+e- -+ hadrons)

R e+e-

= a-(e+e- -+ /1-+ /1--) .

(1.2)

55

e-

q

e+

q

't - - - - - ( - ; :

-

Figure 2: e+ e- -+ hadrons.

-

v e ' vJl' Figure 3:

T

U

decay.

The hadronic production occurs through e+ e- -+ ,., Z· -+ qij -+ hadrons. Since quarks are assumed to be confined, the probability to hadronize is just one; therefore, summing over all possible quarks in the final state, we can estimate the inclusive cross-section into hadrons. At energies well below the Z peak, the cross-section is dominated by the ,-exchange amplitude; the ratio Re+e- is then given by the sum of the quark electric charges squared:

=

(NJ 3 : u,d,s) (NJ = 4 : u, d, s, c)

(1.3)

(NJ = 5 : u,d,s,c,b)

The measured ratio is shown in Fig. 4. Although the simple formula (1.3) cannot explain the complicated structure around the different quark thresholds, it gives the right average value of the cross-section (away from thresholds), provided that Ne is taken to be three. The agreement is better at larger energies. Notice that strong interactions have not been taken into account; only the confinement hypothesis has been used. Additional evidence for Ne = 3 is obtained from the hadronic decay of the T lepton, which proceeds through the W -emission diagram shown in Fig. 3. Since the W coupling to the charged current is of universal strength, there are (2 + Ne) equal contributions (if final masses and strong interactions are neglected) to the T decay width. Two of them correspond to the leptonic decay modes T- -+ II r e-Ve and T- -+ IIr /-l-V JJ , while the other Ne are associated with the possible colours ofthe quark-antiquark pair in the T- -+ IIr d eu decay mode (de == cos Oed + sin Oe s). Hence, the branching ratios for the different channels are expected to be approximately:

(1.4) (1.5)

56

5

I n2

4

+ { MARKIILGW I MEA

MARK!

R

3

, illlllllllllill 1

R

o AMY

3 T(nS) n=1234

• CELLO .. CLEO

o CRYSTAL BALL ;:CCUSB 6 DASP II

o JADE

+ LENA • MAC

-?-MARKJ 7MD-l "PLUTO

)O> A, as (J-l2) --+ 0, so that we recover asymptotic freedom. At lower energies the running coupling gets larger; for J-l --+ A, as (J-l2) --+ 00 and perturbation theory breaks down. The scale A indicates when the strong coupling blows up. Eq. (4.18) suggests that confinement at low energies is quite plausible in QeD; however, it does not provide a proof because perturbation theory is no longer valid when J-l --+ A.

73

4.4 Higher Orders Higher orders in perturbation theory are much more important in QeD than in QED, because the coupling is much larger (at ordinary energies). The 13 function is known to four loops; in the MS scheme, the computed higher-order coefficients take the values (3 = 1.202056903 ...)

22:

132

51 = -4 +

19 12 Nj ;

134

=.2 [(149753 35641") 128 6 +.,,3

133

= 641

[

5033 325 2] -2857 + -9- Nj - 27 N j

_ (1078361 6508 I" ) N 162 + 27.,,3 j

50065 6472 I" ) N 2 1093 N 3 ] + ( 162 + 81.,,3 j + 729 j

:s

.

(4.19)

:s

8, f32 < 0 (f33 < 0 for Nj 5, while f34 is always negative) which further reinforces the asymptotic freedom behaviour. The scale dependence of the running coupling at higher orders is given by: If Nj

5

5.1

Perturbative QeD Phenomenology

e+e-

--7

Hadrons

Figure 14: e+ e- -t -y", Z" -t hadrons.

The inclusive production of hadrons in e+e- annihilation is a good process for testing perturbative QeD predictions. The hadronic production occurs through the basic mechanism e+e- --7 , ' , Z" --7 qij, where the final q-ij pair interacts through the QeD forces; thus, the quarks exchange and emit gluons (and q' -ij' pairs) in all possible ways. At high energies, where O:s is small, we can use perturbative techniques to predict the different subprocesses: e+ e- --7 qij, qijG, qijGG, ... However, we do

74

e-

q

e-

e+

e+ q I

Figure 15: Diagrammatic relation between the total hadronic production cross-section and the two-point function IJI"'(q). The qij blob contains all possible QeD corrections. The dashed vertical line indicates that the blob is cut in all possible ways, so that the left and right sides correspond to the production amplitude T and its complex conjugate Tt, respectively, for a given intermediate state.

not have a good understanding of the way quarks and gluons hadronize. Qualitatively, quarks and gluons are created by the q-ij current at very short distances, x '" 1/.;s. Afterwards, they continue radiating additional soft gluons with smaller energies. At larger distances, x '" 1/A, the interaction becomes strong and the hadronization process occurs. Since we are lacking a rigorous description of confinement, we are unable to provide precise predictions of the different exclusive processes, such as e+ e- -+ 161['. However, we can make a quite accurate prediction for the total inclusive production of hadrons:

The details of the final hadronization are irrelevant for the inclusive sum, because the probability to hadronize is just one owing to our confinement assumption. Well below the Z peak, the hadronic production is dominated by the ,exchange contribution. Thus, we can compute the cross-sections of all subprocesses e+e- -+ -+ qij, qijG, ... (at a given order in as), and make the sum. Technically, it is much easier to compute the QeD T-product of two electromagnetic currents [J';m = 2::! Q! qnl' q! l:

,*

As shown in Fig. 15, the absorptive part of this object (i.e. the imaginary part, which results from cutting -putting on shell- the propagators of the intermediate exchanged quarks and gluons in all possible ways) just corresponds to the sum of the squared moduli of the different production amplitudes. The

75 exact relation with the total cross-section is:

Re+e-

_ O"(e+e- -+ hadrons)

=

0"

(+ +) e e- -+ 11 11-

= 12rr ImIIem(s).

(5.2)

Neglecting the small (away from thresholds) quark mass corrections, the ratio Re+e- is given by a perturbative series in powers of as(s):

The second expression, shows explicitly how the running coupling as (s) sums an infinite number of higher-order logarithmic terms. So far, the calculation has been performed to order a~, with the result (in the MS scheme) 23:

Fl

= 1,

F2

= 1.986 -

F3

= -6.637 -

1.200 Nj - 0.005 N j - 1.240 3 ~j

0.115 Nj, 2

(~jQjf QJ

(5.4)

The different charge dependence on the last term is due to the contribution from three intermediate gluons (with a separate quark trace attached to each electromagnetic current in Fig. 15). For N j = 5 flavours, one has:

The perturbative uncertainty of this prediction is of order a! and includes ambiguities related to the choice of renormalization scale and scheme. Although, the total sum of the perturbative series is independent of our renormalization conventions, different choices of scale and/or scheme lead to slightly

76

different numerical predictions for the truncated series. For instance, the perturbative series truncated at a finite order N has an explicit scale dependence of order a~+1. The numerical values of as and the Fn (n 2: 2) coefficients depend on our choice of scheme (also fin for n > 2). The prediction for Re+e- (s) above the b-b threshold is compared with the data 7 in Fig. 4, taking into account mass corrections and Z-exchange contributions. The two curves correspond to A~;=5) = 60 MeV (lower curve) and 250 MeV (upper curve). The rising at large energies is due to the tail of the Z peak. A global fit to all data between 20 and 65 GeV yields 24 a s (35 GeV) = 0.146 ± 0.030.

(5.5)

Z --+ Hadrons

5.2

The hadronic width of the Z boson can be analyzed in a similar way: R

= f(Z --+ hadrons) = f(Z --+ e+e-)

REW N.

Z -

Z

C

{I

' " F.

+ L...J

n>l

n

(as(M~))n + 0 IT

(mJ)} M~

The global factor (5.6) contains the underlying electroweak Z --+ Lf qfiiJ decay amplitude. Since both vector and axial-vector couplings are present, the QeD correction coefficients Fn are slightly different from Fn for n > 2. To determine as from Rz, one performs a global analysis of the LEP ISLe data, taking properly into account the higher-order electroweak corrections. The latest as value reported by the LEP Electroweak Working Group 25 is as(M~)

5.3

r- --+

l/r+

= 0.119± 0.003.

(5.7)

Hadrons

The calculation of QeD corrections to the inclusive hadronic decay of the r lepton 26 looks quite similar from a diagrammatic point of view. One puts all possible gluon (and qij) corrections to the basic decay diagram in Fig. 3, and computes the sum of all relevant par tonic subprocesses. As in the e+e- case, the calculation is more efficiently performed through the two-point function

IIi,I/(q) ==

if d xe 4

iqx

(OIT(V'(x)LI/(O)t) 10)

= (_gi'l/ q2 + ql'ql/) IIr) (q2) + ql'ql/ IIlO) (q2) ,

(5.8)

77 Im(s)

Figure 16: Integration contour in the complex s-plane used to compute R r .

which involves the T-product of two left-handed currents, V' = u,i-' (1- ,5)do• This object can be easily visualized through a diagram analogous to Fig. 15, where the photon is replaced by a W- line and one has a TV T pair in the external fermionic lines instead of the e+e- pair. The precise relation with the ratio R T , defined in (1.5), is:

1 !; (1- ~;) {(I +2~;) m

RT = 121r

;

2

ImII}.l)(s) + 1m II}.O) (s) }

The three-body character of the basic decay mechanism, T- -+ vTud o, shows here a crucial difference with e+e- annihilation. One needs to integrate over all possible neutrino energies or, equivalently, over all possible values of the total hadronic invariant-mass s. The spectral functions ImII}.o,l\s) contain the dynamical information on the invariant-mass distribution of the final hadrons. The lower integration limit corresponds to the threshold for hadronic production, i.e. m" (equal to zero for massless quarks). Clearly, this lies deep into the non-perturbative region where confinement is crucial. Thus, it is very difficult to make a reliable prediction for the s distribution. Fortunately, we have precious non-perturbative information on the dynamical functions II~,l)(S), which allows us to accurately predict R T • II}.O,l)(s) are analytic functions in the complex s-plane except for a cut in the positive real axis. The physics we are interested in lies in the singular region, where hadrons are produced. We need to know the integral along the physical cut of ImII}.o,l)(S)= -HII~,l)(s+i{) _II~,l)(s-i{)l. However, we can use Cauchy's theorem (close integrals of analytic functions are zero if there are no singularities within the integration contour) to express 26 RT as a contour integral in

78

the complex s-plane running counter-clockwise around the circle lsi = m;:

The advantage of this expression is that it requires dynamical information only for complex s of order which is significantly larger than the scale associated with non-perturbative effects in QCD. A perturbative calculation of Rr is then possible. Using the so-called Operator Product Expansion techniques it is possible to show 26 that non-perturbative contributions are very suppressed ['" (Ajm r )6]. Thus, Rr is a perfect observable for determining the strong coupling. In fact, T decay is probably the lowest energy process from which the running coupling constant can be extracted cleanly, without hopeless complications from nonperturbative effects. The T mass lies fortuitously in a compromise region where the coupling constant is large enough that Rr is very sensitive to its value, yet still small enough that the perturbative expansion still converges well. The explicit calculation gives 26:

m;,

as

where SEW = 1.0194 and JEw = 0.0010 are the leading and next-to-leading electroweak corrections 27, and J p contains the dominant perturbative QCD contribution:

Jp = a

s(;;;) + [F2 -

+

I

[

19

~!th] C~s(;;;)r 19

F3 - 12F2,61 - 24,62

= 0'0(;;;)

+5.2023

+

2] (as(m;))3 + 0(0',)4

265 288,61

(ao(;;;)r

+26.366

7r

(5.10)

(as~;)r +0(0':).

The remaining factor 8NP includes the small mass corrections and non-perturbative contributions. Owing to its high sensitivity 26 to a, the ratio Rr has been a subject of intensive study. Many different sources of possible perturbative and nonperturbative contributions have been analyzed in detail. Higher-order logarithmic corrections have been resummed 28, leading to very small renormalizationscheme dependences. The size of the non-perturbative contributions has been experimentally analyzed, through a study of the invariant-mass distribution of the final hadrons 29; the data imply30 that 8NP is smaller than 1%.

79 The present experimental measurements 30 give J p = 0.200 ± 0.013, which corresponds (in the MS scheme) to 10

as(m;)

= 0.345 ±

0.020,

(5.11)

significantly larger (110-) than the Z decay measurement (5.7). After evolution up to the scale Mz, the strong coupling constant in Eq. (5.11) decreases to as(M~) = 0.1208±0.0025, in excellent agreement with the direct measurement at the Z peak and with a similar error bar. The comparison of these two determinations of as in two extreme energy regimes, mr and M z , provides a beautiful test of the predicted running of the QeD coupling; i.e. a very significant experimental verification of asymptotic freedom.

e-

G

+

e+ Figure 17: Gluon bremsstrahlung corrections to e+ e- -+ qq.

At lowest order in as) the hadronic production in e+ e- collisions proceeds through e+ e- -+ qij. Thus, at high energies, the final hadronic states are predicted to have mainly a two-jet structure, which agrees with the empirical observations. At O(a s ), the emission of a hard gluon from a quark leg generates the e+e- -+ qijG transition, leading to 3-jet configurations. For massless quarks, the differential distribution of the 3-body final state is given by:

xi + x~ 3rr (1- xl)(1 - X2) ,

2a s

(5.12)

where

4;; Nc EQ; 2

,*

Nt

(5.13)

1=1

-+ qij cross-section. The kinematics is defined is the lowest-order e+ e- -+ through the invariants s == q2 and Sij == (Pi + pj)2 = (q - Pk)2 == s(1 - Xk) (i, j, k = 1,2,3), where PI, P2 and P3 are the quark, antiquark and gluon

80

momenta, respectively, and q is the total e+ e- momentum. For given s, there are only two independent kinematical variables since (5.14) In the centre-of-mass system, Xi = E;/ Ee = 2E;/..;s. Eq. (5.12) diverges as Xl or X2 tend to 1. This is a very different infinity from the ultraviolet ones encountered before in the loop integrals. In the present case, the tree amplitude itself is becoming singular in the phase-space boundary. The problem originates in the infrared behaviour of the intermediate quark propagators:

+ P3)2 = 2 (P2 . P3) (Pl + P3)2 = 2 (Pl· P3) (P2

-+ 0 ; -+ 0 .

(5.15)

There are two distinct kinematical configurations leading to infrared divergences: 1. Collinear gluon: The 4-momentum of the gluon is parallel to that of either the quark or the antiquark. This is also called a mass singularity, since the divergence would be absent if either the gluon or the quark had a mass (P311P2 implies S23 = 0 if p~ = P§ = 0).

2. Soft gluon: P3 -+

o.

In either case, the observed final hadrons will be detected as a 2-jet configuration, because the qG or ijG system cannot be resolved. Owing to the finite resolution of any detector, it is not possible (not even in principle) to separate those 2-jet events generated by the basic e+e- -+ qij process, from e+e- -+ qijG events with a collinear or soft gluon. In order to resolve a 3-jet event, the gluon should have an energy and opening angle (with respect to the quark or antiquark) larger than the detector resolution. The observable 3-jet cross-section will never include the problematic region Xl,2 -+ 1; thus, it will be finite, although its value will depend on the detector resolution and/or the precise definition of jet (i.e. a depends on the chosen integration limits). On the other side, the 2-jet configurations will include both e+e- -+ qij and e+e- -+ qijG with an unobserved gluon. The important question is then the infrared behaviour of the sum of both amplitudes. The exchange of virtual gluons among the quarks generate an O(a s ) correction to the e+e- -+ qij amplitude: (5.16)

81

Figure 18: I-loop gluonic corrections to e+e-

--t

qq.

where To is the lowest-order (tree-level) contribution, Tl the O(a s ) correction, and so on. The interference of To and Tl gives rise to an O( as) contribution to the e+e- ~ qij cross-section. We know already that loop diagrams have ultraviolet divergences which must be renormalized. In addition, they also have infrared divergences associated with collinear and soft configurations of the virtual gluon. One can explicitly check that the O(a s ) infrared divergence of a(e+e- ~ qij) exactly cancels the one in a{e+e- ~ qijG), so that the sum is well-defined:

(5.17) This is precisely the inclusive result discussed in Sect. 5.1. This remarkable cancellation of infrared divergences is actually a general result (Bloch-Nordsieck 31 and Kinoshita-Lee-Nauenberg 32 theorems): for inclusive enough cross-sections both the soft and collinear infrared divergences cancel.

Figure 19: 2-jet configuration.

While the total hadronic cross-section is unambiguously defined, we need a precise definition of jet in order to classify a given event as a 2-, 3-, ... , or n-jet configuration. Such a definition should be free of infrared singularities, and insensitive to the details of the non-perturbative fragmentation into hadrons. A popular example of jet definition is the so-called JADE algorithm 33, which

82 makes use of an invariant-mass cut y: 3 jet

Sij

== (Pi + Pj)Z > ys

(Vi, j

= 1,2,3) .

(5.18)

Clearly, both the theoretical predictions and the experimental measurements depend on the adopted jet definition. With the JADE algorithm, the fraction of 3-jet events is predicted to be: s R3 = -2a 31T

{( 3

+ 4Liz where

y) + 2 InZ( -1 -y) - + -5 - 6 y y 2

- 6y ) In ( - 1 - 2y

(_y_) _

2

1T

1- y

.

-9 Y2 2

(5.19)

},

3

LIZ(Z) == -

12

-d~-

o 1- ~

In~

.

(5.20)

The corresponding fraction of 2-jet events is given by R2 = 1 - R3. The fraction of 2- or 3-jet events obviously depends on the chosen cut y. The infrared singularities are manifest in the divergent behaviour of R3 for y --+ O. At higher orders in as one needs to define the different multi-jet fractions. For instance, one can easily generalize the JADE algorithm an classify a {Pll Pz, ... , Pn} event as a n-jet configuration provided that Sij > ys for all i, j = 1, ... , n. If a pair of momenta does not satisfy this constraint, they are combined into a single momentum and the event is considered as a (n - 1) jet configuration (if the constraint is satisfied by all other combinations of momenta). The general expression for the fraction of n-jet events takes the form:

(5.21) with

I:n Rn =

1. A few remarks are in order here:

• The jet fractions have a high sensitivity to a, [Rn '" a~-zl. Although the sensitivity increases with n, the number of events decreases with the jet multiplicity. • Higher-order as (J-! Z)j Ink (s / J-! 2 ) terms have been summed into as (s). However, the coefficients C;n)(y) still contain Ink(y) terms. At low values of y, the infrared divergence (y --+ 0) reappears and the perturbative series becomes unreliable. For large y, the jet fractions Rn with n ~ 3 are small.

83

• Experiments measure hadrons rather than partons. Therefore, since these observables are not fully inclusive, there is an unavoidable dependence on the non-perturbative fragmentation into hadrons. This is usually modelled through Monte Carlo analyses, and introduces theoretical uncertainties which need to be estimated. Different jet algorithms and jet variables (jet rates, event shapes, energy correlations, ... ) have been introduced to optimize the perturbative analysis. In some cases, a resummation of as (s)n lnm(y) contributions with m > n has been performed to improve the predictions at low y values 34. Many measurements of a., using different jet variables, have been performed. All results are in good agreement, providing a consistency test of the QCD predictions. An average of measurements at the Z peak, from LEP and SLC, using resummed O(a;) fits to a large set of shape variables, gives 7: as(M~)

= 0.122 ± 0.007.

(5.22)

A recent O(a;) DELPHI analysis 35 of 18 different event-shape observables finds an excellent fit to the data, by allowing the renormalization scale to be determined from the fit; this procedure gives a more precise value: as (M~) = 0.117 ± 0.003. Three-jet events can also be used to test the gluon spin. For a spin-O gluon, the differential distribution is still given by Eq. (5.12), but changing the xi+x~ factor in the numerator to 4. In general, one cannot readily be sure which hadronic jet emerges via fragmentation from a quark (or antiquark), and which from a gluon. Therefore, one adopts instead a jet ordering, Xl > X2 > X3, where Xl refers to the most energetic jet, 2 to the next and 3 to the least energetic one, which most likely would correspond to the gluon. When X2 -t 1 (Xl -t 1) the vector-gluon distribution is singular, but the corresponding scalar-gluon distribution is not because at that point X3 = (I-Xl) + (l-x2) -t O. The measured distribution agrees very well with the QCD predictions with a spin-l gluon; a scalar gluon is clearly excluded. The predictions for jet distributions and event shapes are functions of the colour group factors TF = 1/2, CF = (N't; - 1)/(2Nc) and C A = Nc. These quantities, defined in Eq. (A.5), result from the colour algebra associated with the different interaction vertices, and characterize the colour symmetry group. If the strong interactions were based on a different gauge group, the resulting predictions would differ in the values of these three factors. Since the vertices contribute in a different way to different observables, these colour factors can be measured by performing a combined fit. The data are in excellent agreement with the SU(3) values, and rule out the Abelian model and many classical Lie groups.

xV

84 6

Determinations of the Strong Coupling 0.5

n----::=::::::::::::::::::::::::::::;:::::::==::::I .~

.3 Deep fnelas.t!c Scattering et!f1A.1\ilibil~iiPp HlIafon CoIllSi";';s·· ~vy Qua~~Onia

;, . .,.', >i{'· ·.

III

A~,,,):'q;s(Mz)

() 123 QeD .,,::275 ,. Mev.·-.·.·:.;.·-.· ,.'.,..... .

0.3

220MeV-O.119 • .. 175 MeV -:;:::.~O.115

O(c:X1)

0.2

0.1 1

10

Q [GeV]

100

Figure 20: Compilation 36 of ()(. measurements as function of the energy scale.

In the massless quark limit, QCD has only one free parameter: the strong coupling as. Thus, all strong interaction phenomena should be described in terms of this single input. The measurements of as at different processes and at different mass scales provide then a crucial test of QCD. Obviously, the test should be restricted to those processes where perturbative techniques are reliable. Moreover, the same definition of a. should be taken everywhere; the MS scheme is usually adopted as the standard convention. Since the running coupling is a function of energy, one can either compare the different determinations at the different scales where they are measured, checking in this way the predicted Q2 dependence of the coupling, or use this prediction to bring all measurements to a common reference scale where they are compared. Nowadays, the Z mass scale is conventionally chosen for such a comparison. In order to assess the significance of the test, it is very important to have a good understanding of the uncertainties associated with the different measurements. This is not an easy question, because small non-perturbative effects can be present in many observables. In addition, some quantities have been computed to a very good perturbative accuracy (next-to-next-to-leading or-

85

DIS [pol. sIre!. fctn.] ~ DIS IBj-SR] ~ DIS [GLS-SRI ~ .-decays ILEPI .;.. I >--0-< F2 • F3 Iv -DIS I

0-- PT. In principle, this is not a problem because only PT has physical meaning; we should just sum over the physical transverse polarizations to get the right answer. However, the problem comes back at higher orders. The covariant gluon propagator contains the 4 polarization components; therefore higher-order graphs such as the ones in Fig. 23 get unphysical contributions from the longitudinal and scalar gluon polarizations propagating along the internal gluon lines. The absorptive part of these I-loop graphs (i.e. the imaginary part obtained putting on-shell the two gluon lines within the loop) is equal to IT(qij -t GG)I2. Thus, these loops suffer the same probability problem than the tree-level qij -t GG amplitude. The propagation of unphysical gluon components implies then a violation of unitarity (the two fake polarizations

98

a')

b')

c')

Figure 23: I-loop diagrams contributing to qq

-t

qq.

contribute a positive probability). In QED this problem does not appear because the gauge-fixing condition 01-' AI-' = 0 still leaves a residual gauge invariance under transformations satisfying DO = O. This guarantees that (even after adding the gauge-fixing term) the electromagnetic current is conserved, i.e. Ol-'Jtm ol-'(eQIit,I-''lt) O. If one considers the e+e- -+ " process, which proceeds through diagrams identical to a) and b) in Fig. 22, current conservation implies kl-'JI-'I-" k~,JI-'I-" 0, where kl-' and k~, are the momenta of the photons with polarizations A and X, respectively (remember that the interacting vertices contained in JI'I" are in fact the corresponding electromagnetic currents). As a consequence, the contributions from the scalar and longitudinal photon polarizations vanish and, therefore, Pc = PT' The reason why Pc =I PT in QeD stems from the third diagram in Fig. 22, involving a gluon self-interaction. Owing to the non-abelian character of the SU(3)c group, the gauge-fixing condition 0l-'G~ = 0 does not leave any residual invarianceb • Thus, kl'Jl'l" =I O.

=

=

=

=

qr"cp G

-f

"

~

q

'ip

d) Figure 24: Feynman diagrams involving the ghosts.

=

bTo maintain 81'(G~)' 0 after the gauge transformation (3.7), one would need OMa g.fabc81'(/i(h)G~, which is not possible because G~ is a quantum field.

=

99 Again, the problem could be solved adopting a non-covariant quantization where only the physical transverse polarizations propagate; but the resulting formalism would be awful and very inconvenient for performing practical calculations. A more clever solution consist 58 in adding additional unphysical fields, the so-called ghosts, with a coupling to the gluons such that exactly cancels all unphysical contributions from the scalar and longitudinal gluon polarizations. Since a positive fake probability has to be cancelled, one needs fields obeying the wrong statistics (i.e. of negative norm) and thus giving negative probabilities. The magic cancellation is achieved by adding to the Lagrangian the Faddeev-Popov term 59, (B.5) where ¢a, ¢a (a = 1, ... , N't: - 1) is a set of anticommuting (i.e. obeying the Fermi-Dirac statistics), massless, hermitian, scalar fields. The covariant derivative DI1¢a contains the needed coupling to the gluon field. One can easily check that diagrams d) and d') in Fig. 24 exactly cancel the unphysical contributions from diagrams c) and c') of Figs. 22 and 23, respectively; so that finally Pc = PT. Notice, that the Lagrangian (B.5) is necessarily not hermitian, because one needs to introduce an explicit violation of unitarity to cancel the unphysical probabilities and restore the unitarity of the final scattering amplitudes. The exact mechanism giving rise to the Lpp term can be understood in a simpler way using the more powerful path-integral formalism, which is beyond the scope of these lectures. The only point I would like to emphasize here, is that the addition of the gauge-fixing and Faddeev-Popov Lagrangians is just a mathematical trick, which allows to develop a simple covariant formalism, and therefore a set of simple Feynman rules, making easier to perform explicit calculations.

References

1. I.J.R. Aitchison and A.J.G. Hey, Gauge Theories in Particle Physics, Graduate Student Series in Physics (lOP Publishing Ltd, Bristol, 1989). 2. T. Muta, Foundations of Quantum Chromodynamics, Lecture Notes in Physics - Vol. 5 (World Scientific, Singapore, 1987). 3. P. Pascual and R. Tarrach, QCD: Renormalization for the Practitioner (Springer-Verlag, Berlin, 1984). 4. F.J. Yndur 7.5 MeV were detected, whereas the threshold used by Super-Kamiokande is at present 5.5 MeV. With these energy cuts, the Kamiokande and Super-Kamiokande detection rates are only sensitive to the 8B component of the solar neutrino flux. Reaction (86) has a very interesting feature - for neutrino energies E » me the angular distribution of the recoil electrons is forward-peaked, i.e. it points in the direction of the momentum of the incoming neutrino. The angular distributions of neutrinos detected in the Kamiokande and Super-Kamiokande experiments have a prominent peak at 1800 from the direction to the sun, which is a beautiful proof of the solar origin of these neutrinos. The origin of the neutrinos detected in the radiochemical experiments is unknown, and our belief that they come from the sun is based on the fact that no other sources of sufficient strength and energy are known. In all five solar neutrino experiments, fewer neutrinos than expected were detected, the degree of deficiency being different in the experiments of different types (fig. 9). This is the essence of the solar neutrino problem. The observed deficiency of solar neutrinos could be due to an insufficient knowledge of solar physics or an error in some input parameters (astrophysical solution), or due to experiment-related errors (such as miscalculated detection efficiency or cross section), or due to some unknown neutrino physics (particle physics solution of the solar neutrino problem). As we have already discussed, the standard solar models are well grounded, agree with each other and with helioseismological observations. All the solar neutrino experiments but one (Homestake) have been calibrated, and their experimental responses were found to be in a

137 1.5

...

•

III

"Ii'

"09t>.

•

Ii

• BP 118 OBP98 CIT + CDR 97 ~ RVCD 98 o GONG 118

•

• OS 118 eBP 95 • Proffitt 94 • KS 94 "CDr 94 *JCD 94 .. SSD 94 A cor 93 _TCL 93 D BPML 93 +BP 92

~

... 0.6

(£30~nlmum

0 0

0.6

• BP 92 "SBr90 +BU 88

If I

1.6

2

~'B)/BPII8

Figure 8: Predictions of standard solar models since 1988. The 7Be and BB fluxes are normalized to the predictions of the BP98 model. The rectangular box defines the 30- error range of the BP98 model. The best-fit 7Be flux is negative (from ref. 40). Total Rates: Standard Model vs. Experiment Bahcall-Pinsonneaull 98

O.54±O.07

67±8

0.47±O.02 2.56±O.23

SupcrK

CI

Kamloka

H.O

Theory iOOl 'Be aB

•

p-p. pep

•

CNO

SAGE

Ga Experiments •

Figure 9: Solar neutrino measurements and theoretical flux predictions. For Cl and Ga experiments, the units are SNU (1 SNU = 10- 36 captures per target atom per second); for H 2 0 experiments, the ratio data/expo is shown. From ref. 39.

138

very good agreement with expectations. The argon extraction efficiency of the Homestake detector was also checked by doping it with a known small number of radioactive argon atoms, but no calibration has been carried out since no artificial source of neutrinos with a suitable energy spectrum exists. The solar neutrino problem is not just the problem of the deficit of the observed neutrino flux: results of different experiments seem to be inconsistent with each other. In the absence of new neutrino physics, the energy spectra of the various components of the solar neutrino flux are given by the standard nuclear physics and well known, and only the total fluxes of these components may be different from those predicted by the standard solar models. One can then infer the flux of 8B neutrinos directly from the Kamiokande or SuperKamiokande data, and use this flux to find the corresponding contribution to the Homestake detection rate. This contribution turns out to be larger than the total detection rate, so the best-fit contribution of the 7Be neutrinos to the Homestake detection rate is negative! In fact, if one assumes that the solar neutrino spectra are undistorted, one can demonstrate the existence of the solar neutrino problem without using any information about solar physics except the solar luminosity constraint, or even without this constraint, i.e. in a solar-model independent way. Therefore the probability of the astrophysical solution of the solar neutrino problem is very low. Alternatively, one could assume that the observed deficit of solar neutrinos is due to some unknown experimental errors. However, in this case it is not sufficient to assume that one of the solar neutrino experiments is wrong; one would have to assume that the experiments of at least two different types (chlorine, gallium and water Cherenkov) are wrong, which is very unlikely. The remaining possibility is that the spectra of solar neutrinos are distorted, which requires new neutrino physics. There are several possible particle-physics solutions of the solar neutrino problem, the most natural one being neutrino oscillations (for an alternative solution related to nonzero neutrino magnetic moments, see e.g. 16). The neutrino oscillation solution has become even more plausible after the strong evidence for atmospheric neutrino oscillations was reported by the SuperKamiokande Collaboration. Neutrino oscillations can convert a fraction of solar Ve into vI-' or Vr (or their combination). Since the energy of solar neutrinos is smaller than the masses of muons and tauons, these vI-' or Vr cannot be detected in the CC reactions of the type (84) or (85) and therefore are invisible in the chlorine and gallium experiments. They can scatter on electrons through the NC interactions and therefore should contribute to the detection rate in water Cherenkov detectors. However, the cross section of the NC channel of reaction (86) is about a factor of 6 smaller than that of the CC

139

channel, and so the deficit of the neutrino flux observed in the Kamiokande and Super-Kamiokande experiments can be explained. The probabilities of neutrino oscillations depend on neutrino energy, and the distortion of the energy spectra of the experimentally detected solar neutrinos, which is necessary to reconcile the data of different experiments, is readily obtained.

10~

10--

~ ~

-

10'"

I

10-'7 ClAr + GALlBX + SAGE + SUperl(amiokande: rates only

ClAr + SAGE + GAUEX

+ SuperKamioimnd.

ssw:

Bahco.ll and

Pln~nnee.ult

199B

SSW: Bahcall and Pln.onneaull 1996

0.2

0.4

0.6 Bin'(28)

0.8

Figure 10: Left panel: Allowed regions (99 % c.l.) for the MSW solutions of the solar neutrino problem for 2-flavour oscillations of Ve into active neutrinos (only rates included). Right panel: The same for the VO solution. From ref. 41.

The oscillations of solar neutrinos inside the sun can be strongly enhanced due to the MSW effect which we discussed in sec. 4, and the solar data can be fitted even with a very small vacuum mixing angle. Solar matter can also influence neutrino oscillations if the vacuum mixing angle is not small. The allowed values of the neutrino oscillation parameters sin 2 200 and 60m 2 which fit the detection rates in the chlorine, gallium and water Cherenkov experiments in the 2-flavour scheme are shown in fig. 10 for matter enhanced (left panel) and vacuum (right panel) oscillations of Ve into active neutrinos. In the case of the matter enhanced oscillations, there are three allowed ranges of the parameters corresponding to the small mixing angle (SMA), large mixing angle (LMA) and low 60m 2 (LOW) MSW solutions 41,42. The LOW solution has a very low probability and only exists at 99% c.l.. The vacuum oscillation (VO) solution corresponds to very small values of 60m 2 , for which the neutrino oscillation length (23) for typical solar neutrino energies ("" a few MeV) is comparable

140

to the distance between the sun and the earth. This solution is also known as "just so" oscillation solution. For Ve --t Vsterile oscillations, there is only the SMA solution with the allowed region of parameters similar to that for oscillations into active neutrinos. Solar neutrinos detected during night travel some distance inside the earth on their way to the detector, and their oscillations can be affected by the matter of the earth. In particular, a fraction of VI-' or Vr produced as a result of the solar Ve oscillations can be reconverted into Ve by oscillations enhanced by the matter of the earth. This earth "regeneration" (or day/night) effect can be appreciable in the case of the LMA solution, but is expected to be very small in the case of the SMA solution. The day/night effects (and in general, the zenith angle dependence of the neutrino signal) can in principle be observed in real-time experiments, such as Super-Kamiokande. In the case of the MSW solutions of the solar neutrino problem the neutrino state arriving at the earth is an incoherent superposition of the mass eigenstates VI and V2. The probability PS E of finding a solar Ve after it traverses the earth depends on the average Ve survival probability in the sun Fs: PSE

=

Ps

+

1- 2Fs 2() cos 0

2

(P2e -

sin ()o).

(87)

Here P 2e = P(V2 --t v e ) is the probability of oscillations of the second mass eigenstate into electron neutrino inside the earth. It follows from this expression that the sign of the night-day asymmetry (N - D)/(N + D) depends on whether Fs is larger or smaller than 1/2. For the LMA solution, Fs < 1/2, and the night-day asymmetry is expected to be positive. For the SMA solution, Fs for the Super-Kamiokande experiment is close to 1/2, and therefore the expected day/night effect is small. It can in principle be of either sign. The value measured in the Super-Kamiokande experiment is 43 N D - 1 = 0.067 ± 0.033 (stat.) ± 0.013 (syst.) ,

(88)

i.e. shows an excess of the night-time flux at about 20'. The zenith angle event dependence measured by the Super-Kamiokande shows a rather flat distribution of the excess of events over different night-time zenith angle bins. This is rather typical for the LMA solution whereas for the SMA solution one normally expects the excess (or deficiency) to be concentrated in the vertical upward bin with zenith angles () in the range -1 < cos () < -0.8. Thus, the night-time zenith angle dependence and the overall night-day asymmetry (88) favour the LMA solution of the solar neutrino problem.

141

As we discussed before, oscillations of solar neutrinos in the 3-flavour scheme can be described through the effective two-flavour Ve survival probability [see eq. (36)). For a recent detailed analysis of the solar neutrino oscillations in the 3-flavour framework, see 44.

i

1

: SK SLE 524day + LE 825day 22.5kt ALL

ID

~0.9

j 5.5·20MeV

~ co 0.8

1

{If. ·

! I/stat.'.,ySl'

Cl

0.7 06

(Preliminary)

I stat. error

"-,

T~

.' , .. ~ .... 0.5

-,'-:L...

0.4

0.3 0.2

~in'2" = O.B. 6m' = 3.2X 10-' •..... kin'2" = 6.3Xl0·' , ~m2 = 5Xl0·' ........

~jnt2" = 0.79, 6m2 = 4.3X 10"0 = 0.83. 6m' = 7.1 X 10-"

_. - .•. rn'2"

(hASW. LhAS) (MSW, SMA)

(juslso, best fit)

(justso. typical)

0.1

Energy(MeV)

Figure 11: Recoil electron energy spectrum in the Super-Kamiokande experiment along with typical predictions of SMA, LMA and VO solutions (from ref. 43).

As we have already discussed, neutrino oscillations (both in matter and in vacuum) depend on neutrino energy and therefore should result in certain distortions of the spectrum of the detected solar neutrinos. One can obtain an information on the spectrum of incoming neutrinos by measuring the recoil electron spectrum in the reaction (86). This spectrum has been measured in the Super-Kamiokande experiment, and the results (along with the typical predictions ofthe LMA, SMA and VO solutions) are shown in fig. (11). In the absence of the neutrino spectrum distortion, the ratio of measured/expected electron spectrum presented in this figure should be a horizontal line. The characteristic feature of the measured spectrum is an excess of events in the region of high energies. The experimental errors are still too large to allow a clean discrimination between different solutions of the solar neutrino problem; however, the spectrum favours VO over the other solutions. Another possible explanation of the excess of the high energy events is that the flux of the highest-energy hep neutrinos has been underestimated by about a factor of 20.

142

Since the hep flux is very poorly known, this is still a possibility. At present, the situation with the solar neutrino data is somewhat confusing: different pieces of the data favour different solutions of the solar neutrino problem. The total rates are better fitted within the SMA solution, the night/day ratio prefers the LMA solution, and the electron recoil spectrum in Super-Kamiokande favours the VO solution. Clearly, more data are needed to clear the situation up. Fortunately, two new experiments which can potentially resolve the problem are now under way or will soon be put into operation. The SNO (Sudbury Neutrino Observatory) experiment has started taking data, and its first results are expected some time during 2000. The Borexino experiment is scheduled to start data taking in 2001. The SN 0 detector consists of 1000 tons of heavy water, and it detects solar neutrinos in three different channels:

+ d --t p + P + eVa + d --t p + n + Va

Ve

(CC),

Emin

= 1.44 MeV,

(89)

(NC),

Emin

= 2.23 MeV,

(90)

and Vae scattering process (86) which can proceed through both CC and NC channels. The CC reaction (89) is very well suited for measuring the solar neutrino spectrum. Unlike in the case of Vae scattering (86) in which the energy of incoming neutrino is shared between two light particles in the final state, the final state of the reaction (89) contains only one light particle electron, and a heavy 2p system whose kinetic energy is very small. Therefore by measuring the electron energy one can directly measure the spectrum of the solar neutrinos. Detecting the solar neutrino spectrum would be of great importance since any deviation from the spectrum predicted by the nuclear beta decay would be a "smoking gun" signature of new neutrino physics. The cross section of the NC reaction (90) is the same for neutrinos of all three flavours, and therefore oscillations between Ve and vJl or V T would not change the NC detection rate in the SNO experiment. On the other hand, these oscillations would deplete the solar Ve flux, reducing the CC event rate. Therefore the CC/NC ratio is a sensitive probe of neutrino flavour oscillations. If solar Ve oscillate into electroweak singlet (sterile) neutrinos, both CC and NC event rates will be suppressed. The Borexino experiment will detect solar neutrinos through the Vae scattering with a very low energy threshold, and will be able to detect the 7Be neutrino line. Different solutions of the solar neutrino problem predict different degree of suppression of 7Be neutrinos, and their detection could help discriminate between these solutions. Observation of the 7Be neutrino line would be especially important in the case of the VO solution. Due to the eccentricity of the earth's orbit the distance between the sun and the earth varies

143

by about 3.5% during the year, and this should lead to varying oscillation phase (and therefore varying solar neutrino signal) in the case of vacuum neutrino oscillations. This seasonal variation can in principle he separated from the trivial 7% variation due to the 1/ L2 law which is not related to neutrino oscillations. However, the oscillation phase depends on neutrino energy [see (21)], and integration over significant energy intervals may make it difficult to observe the seasonal variations of the solar neutrino flux due to the VO. The 7Be neutrinos are monochromatic, which should facilitate the observation of the seasonal variations. The oscillation phase in (21) depends on the ratio L/ E, therefore for neutrinos with continuous spectrum VO should lead to seasonal variations of the spectrum distortion which are correlated with the seasonal variations of the flux 45. Such variations would be a clear signature of the VO solution. Although up to now the sensitivities of the solar neutrino experiments have been insufficient to allow a clear-cut discrimination between different solutions of the solar neutrino problem, one can hope that the combined data of the currently operating and forthcoming experiments will allow to finally resolve the solar neutrino problem. 6

Atmospheric neutrinos

Atmospheric neutrinos are electron and muon neutrinos and their anti neutrinos which are produced in the hadronic showers induced by primary cosmic rays in the earth's atmosphere. The main mechanism of production of the atmospheric neutrinos is given by the following chain of reactions: p(a, ... )

+ Air ~ 7f±(K±) +

X 7f±(K±) ~ f-l± + v,.,(v,.,) f-l± ~ e± + ve(ve) + V,., (v,.,)

(91)

Atmospheric neutrinos can be observed directly in large mass underground detectors predominantly by means of their CC interactions:

+ A ~ e-(e+) + X , V,., (v,.,) + A ~ f-l-(f-l+) + X . Ve(V e )

(92)

Naively, from the reaction chain (91) one would expect to have two atmospheric muon neutrinos or antineutrinos for every electron neutrino or antineutrino. In reality, the situation is more complicated: one should take into account the differences in the lifetimes of 7f±, K± and f-l± as well as the differences in their spectra. Also, although the reaction chain (91) is the dominant source of

144

atmospheric neutrinos, it is not the only one. Calculations of the atmospheric neutrino fluxes predict the v/-,/ve ratio that depends on neutrino energy and the zenith angle of neutrino trajectory, approaching 2 for low energy neutrinos and horizontal trajectories but exceeding this value for higher energy neutrinos and for trajectories close to vertical. Accurate calculation of the atmospheric neutrino fluxes is a difficult job which includes such ingredients as spectra and chemical composition of cosmic rays (including geomagnetic effects and solar activity), cross sections of 7r and K production off the nuclear targets, Monte Carlo simulation of hadronic cascades in the atmosphere and the calculation of neutrino spectra including muon polarization effects. Each step introduces some uncertainty in the calculation. The overall uncertainty of the calculated atmospheric neutrino fluxes is rather large, and the total fluxes calculated by different authors differ by as much as 20 - 30%. At the same time, the ratio of the muon to electron neutrino fluxes is fairly insensitive to the above uncertainties, and different calculations yield the ratios of muon-like to electron-like contained events which agree to about 5%. This ratio has been measured in a number of experiments, and the Kamiokande and IMB Collaborations reported smaller than expected ratio in their contained events, with the double ratio R(p/e) == [(v/-, + v/-,)/(ve + veldata/[(v/-, + v/-,)/(ve + ve)lMc := 0.6 where MC stands for Monte Carlo simulations. The discrepancy between the observed and predicted atmospheric neutrino fluxes was called the atmospheric neutrino anomaly. The existence of this anomaly was subsequently confirmed by Soudan 2, MACRO and Super-Kamiokande experiments. Most remarkably, the SuperKamiokande (SK) Collaboration obtained a very convincing evidence for the up-down asymmetry and zenith-angle dependent deficiency of the flux of muon neutrinos, which has been interpreted as an evidence for neutrino oscillations. We shall now discuss the SK data and their interpretation. The SK detector is a 50 kt water Cherenkov detector (22.5 kt fiducial volume) which is overseen by more than 13,000 photomultiplier tubes. The charged leptons born in the CC interactions of neutrinos produce the rings of the Cherenkov light in the detector which are observed by the phototubes. Muons can be distinguished from electrons since their Cherenkov rings are sharp whereas those produced by electrons are diffuse. The SK Collaboration subdivided their atmospheric neutrino events into several groups, depending on the energy of the charged leptons produced. Fully contained (FC) events are those for which the neutrino interaction vertex is located inside the detector and all final state particles do not get out of it. FC events are further subdivided into sub-GeV (visible energy < 1.33 GeV) and multi-GeV (visible energy > 1.33 Ge V) events. Partially contained (PC) events are those for which the

145

produced muon exits the inner detector volume (only muons are penetrating enough). The average energy of a neutrino producing a PC event in SK is '" 15 GeV. Muon neutrinos can also be detected indirectly by observing the muons that they have produced in the material surrounding the· detector. To reduce the background from atmospheric muons, only upward-going neutrino-induced muons are usually considered. A rough estimate of the energy spectrum of the upward-going muons has been obtained dividing them in two categories, passing (or through-going) and stopping muons. The latter, which stop inside the detector, correspond to the average parent neutrino energy '" 10 GeV, whereas for the through-going muons the average neutrino energy is'" 100 GeV. The measurements of the double ratio R(J-t/e) for contained events at SK (848 live days) give 3

= 0.68 ± 0.02 (stat.) ± 0.05 (syst.) R = 0.68 ± 0.04 (stat.) ± 0.08 (syst.)

R

(sub-GeV) , (multi-GeV) .

(93)

The value of R for sub-GeV events is different from unity (to which it should be equal in no-oscillation case) by 5.90-.

600 sub-GeVe-like

sUb-GeV II-like

multi-GeV e-like

mUlti-GeV II-like + PC

400

200

0 7:-':2 "'?

200

o

-0.8 -0.4

0 0.4 0.8 casEl

-0.8 -0.4

0 0.4 0.8 casEl

Figure 12: Zenith angle distributions for sub-GeY and multi-GeY e-like and Jt-like events at SK. The bars show the (no-oscillations) Monte Carlo predictions; the lines show the predictions for VI' t-+ v.,. oscillations with the best-fit parameters ~m2 = 3.5 x 10- 3 ey2, sin 2 29 = 1.0. From 46.

146

We shall now discuss the zenith angle distributions of the atmospheric neutrino events. It should be remembered that the zenith angle distributions of the charged leptons which are experimentally measured do not coincide with those of their parent neutrinos: for multi-GeV neutrinos the average angle between the momenta of neutrinos and charged leptons is about 17°, whereas for sub-GeV neutrinos it is close to 60°. This is properly taken into account in MC simulations. For PC events and upward going muons the correlation between the directions of momenta of muons and parent neutrinos is much better. The distances L traveled by neutrinos before they reach the detector vary in a wide range: for vertically downward going neutrinos (neutrino zenith angle Ov = 0) L rv 15 km; for horizontal neutrino trajectories (Ov = 90°) L rv 500 km; the vertically up-going neutrinos (Ov = 180°) cross the earth along its diameter and for them L rv 13,000 km. In fig. 12 the zenith angle distributions of the SK e-like and {l-like events are shown separately for sub-GeV and multi-GeV contained events. The data correspond to 736 live days. (By the time these lecture notes were being written, the results for 848 live days of data taking had been reported, but no high-quality figures for zenith angle distributions were publicly available). One can see that for e-like events, the measured zenith angle distributions agree very well with the MC predictions (shown by bars), both in the sub-GeV and multi-GeV samples, while for {l-like events both samples show zenith-angle dependent deficiency of event numbers compared to expectations. The deficit of muon neutrinos is stronger for upward going neutrinos which have larger pathlengths. In the multi-GeV sample, there is practically no deficit of events caused by muon neutrinos coming from the upper hemisphere (cosO> 0), whereas in the sub-GeV sample, all {l-like events exhibit a deficit which decreases with cos f). This pattern is perfectly consistent with oscillations vI-' H Vr or vI-' H Vs where Vs is a sterile neutrino. Muon neutrinos responsible for the multi-GeV sample are depleted by the oscillations when their pathlength is large enough; the depletion becomes less pronounced as the pathlength decreases (cos f) increases); for neutrinos coming from the upper hemisphere, the pathlengths are too short and there are practically no oscillations. Neutrinos responsible for the sub-GeV {l-like events have smaller energies, and so their oscillation lengths are smaller; therefore even neutrinos coming from the upper hemisphere experience sizeable depletion due to the oscillations. For up-going sub-GeV neutrinos the oscillation length is much smaller than the pathlength and they experience averaged oscillations. The solid line in fig. 12 obtained with the vI-' H Vr oscillation parameters in the 2-flavour scheme ~m2 = 3.5 x 10- 3 eV 2 , sin 2 20 = 1.0 gives an excellent fit of the data. An informative parameter characterizing the distortion of the zenith angle

147

~~: ~~I'~=~~~:~t~c:::J f

>: o

2-

10"'

10

11-1ike

0.5

o

FC ~----">----+--1

I(r'

I

~

10"

r-----r--~---r--~___"

VIl-V t Super Kamiokande Preliminary 848 days

10'

PC

•..••.~. ""."••.~,."",=.,.= ,' .. ""

-0.5

~

'>

10

Momentum (GeV/c)

68%C.L 9O%CL --- 99% C.L 0.6

0.8

sirr 2e

I

Figure 13: Left panel: up-down asymmetry vs event momentum for single ring e-like and f.t-like events in SK. Right panel: SK allowed regions for oscillations parameters for VI' f-t V T channel in 2-flavour scheme (FC + PC single ring events). From 3.

distribution is the up-down asymmetry A, where up corresponds to the events with cos e < -0.2 and down to those with cos e > 0.2. The flux of atmospheric neutrinos is expected to be nearly up-down symmetric for neutrino energies E ~ 1 GeV, with minor deviations coming from geomagnetic effects which are well understood and can be accurately taken into account. In particular, at the geographical location of the SK detector small positive asymmetry is expected. Any significant deviation of the up-down asymmetry of neutrino induced events from the asymmetry due to the geomagnetic effects is an indication of neutrino oscillations or some other new neutrino physics. The asymmetry measured for the SK multi-GeV JL-like events is 3

U-D +D

A = -U-- = -0.32 ± 0.04 (stat.) ± 0.01 (syst.) ,

(94)

i.e. is negative and differs from zero by almost 8a! The dependence of the asymmetries for e-like FC and JL-like FC+PC events on the event momentum is shown in fig. 13 (left panel). One can see that for e-like events A ~ 0 for all momenta. At the same time, for JL-like events the asymmetry is close to zero at low momenta and decreases with momentum. This is easily understood in terms of the vI-' oscillations. For very small momenta, the oscillation length is small and both up-going and down-going neutrino fluxes are depleted by oscillations to about the same extent; in addition, loose correlation between the directions of the momenta of the charged lepton and of its patent neutrino tends to smear out the asymmetry at low energies. With increasing momentum

148

the oscillation length increases, and the pathlength of down-going neutrinos becomes too small for oscillations to develop. The right panel of fig. 13 gives the allowed values of the oscillation parameters following from the SK FC and PC event data. The best fit corresponds to ~m2 = 3.1 x 10- 3 eV2, sin2 2fJ = 0.99 and has a very good X2/d.o.f. = 55/67. In this analysis the poorly known overall normalization factor of the neutrino flux was considered as a free parameter, and the best fit was achieved with 5% upward renormalization of the flux of Honda et al. 47. The above value of X2 should be compared with that of the no-oscillation hypothesis: X2 / d.o.f. = 177/69, which is a very poor fit.

VIl ~ V't

SuperK

, ,,

2

X (Osc. Best Fit) = 67.5/82 d.o.f ".

X2(NuU Osc.) = 214/84 d.o.f .'

,'.

A

_

..

"

_~.~--,-----"I

I

r. ,I

'

,

,,

'

.

'. - --------Kamiokande 90% "' ...

······68% C.L. - 9O%C.L. ---- 99 % C.L.

....

I ,

---- ........ . - .

-

I

•

a=O.06

•

.. --- ---

I

\

sin2 26 = 1.0 run2 = 3.5 x 10.3 [eV2]

_

~"--

,

SuperK9O% MACR09O% MACRO Sens. Soudan 68% Soudan 90%

",

"

la5~~~~~~~~u

0.6

0.8

1

sin' 28

1 sin' 28

Figure 14: Left panel: SK allowed regions of oscillation parameters for vI-' foot V T channel in 2-fl.avour scheme (FC + PC single ring events + upward through-going + upward stopping events). Right panel: comparison of allowed values of oscillation parameters obtained in different experiments. From 3.

The SK data show evidence for neutrino oscillations not only in their FC and PC JL-like events: upward stopping and upward through-going events also demonstrate zenith angle dependent deficiency of muon neutrinos consistent with neutrino oscillations, although the statistics for up-going muons is lower than that for contained events. Fig. 14 (left panel) summarizes the allowed region for 2-flavour neutrino oscillations obtained from the combined analysis of the SK FC and PC single ring events (848 live days), upward through-

149

going (923 live days) and upward stopping events (902 live days). The best fit (X2/d.o.f = 67.5/82) was obtained for 6m 2 = 3.5x 10-3 eV 2 , sin2 20 = 1.0 and the upward 6% renormalization of the flux of Honda et at. The no-oscillation fit gives X2 /d.o.f. = 214/84. The right panel gives the comparison of the allowed regions obtained in different atmospheric neutrino experiments. The evidence for neutrino oscillations is very impressive. Are neutrino oscillations that are responsible for the depletion of the vp. flux Vp. ++ V T or vp. ++ vs? For contained events, the oscillation probabilities in these two channels are nearly the same and the data can be fitted equally well in both cases, with very similar allowed ranges of the oscillation parameters. However, for higher energy upward going events there are important differences between these two cases. In the 2-flavour scheme, vp. ++ V T oscillations are not affected by matter because the interactions of Vp. and V T with matter are identical. However, sterile neutrinos do not interact with matter at all, and therefore the vp. ++ Vs oscillations are affected by the matter-induced potential Vp. - Vs = Vp.- At low energies, the kinetic energy difference 6m 2 /2E dominates over Vp., and the earth's matter effects are unimportant. They become important at higher energies, when 6m 2 /2E '" Vp.; at very high energies, when 6m 2 /2E « Vp., matter strongly suppresses neutrino oscillations both in vp. ++ Vs and vp. ++ Vs channels (see eq. (56) where Ne has to be replaced by =r:Nn/2 for these oscillation channels). Therefore the oscillations of high energy neutrinos travelling significant distances in the earth should be strongly suppressed in this case. Fig. 15 (upper left panel) shows the zenith angle distribution of the upward through-going events ((E) '" 100 Ge V) along with the predictions of vp. ++ V T and vp. ++ Vs scenarios with parameters obtained from the fits of contained events. It can be seen that the vp. ++ V T oscillation channel describes the "vertical" (cosO < -0.4) events better than the vp. ++ Vs channel. For horizontal and nearly horizontal events (-0.4 < cos 0 < 0) both scenarios give similar predictions since the neutrinos do not travel inside the earth or travel only small distances there. This is further illustrated by the left bottom panel of fig. 15 in which the Vertical/Horizontal event ratio is plotted. The excluded ranges of the oscillation parameters for various oscillation channels are shown in the right panel of fig. 15. One can conclude that the vI-' ++ Vs oscillations are currently disfavoured at the 20- level, although not yet completely ruled out as a possible solution of the atmospheric neutrino problem. Another way to discriminate between Vp. ++ V T and vI-' ++ Vs as the two possible main channels of the atmospheric neutrino oscillations is through the detection of single pions produced in the NC reactions vN --+ vNno. Sterile neutrinos do not participate in these reactions and therefore vp. ++ Vs oscil-

150 •

Upward

,ii' o

"vertkal"

..! ~ "horizontal" ~

!r-r-

I

Through-going Muons

I

SuperKamiokande (very preliminary)

!

ITO:::: I J j I'

~

2

L._.i- -

VJ.1--+Vs

~

,-----j--

+.-----~

:

I~t~-

.1.m2 = 10-2.5 eV2

I sin226 = 1

VJ.l --+ Vt ,0 _I

.~

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

1.-_ _ _ _ _ _ _ _ _""co"-;'9

e: :.-_

t ----:F~:~~': ~-~-~·~-~·~·~·~~·~Y~~t~·,·~·~

0.8

~~ 0.7

0.8

0.9

I

sin 2 29

0.7

:Jtjli(

-3

·2.8

-2.6

-2.4 -2.2 -2 loglO(Am 2)

C,L

95'!c., c.L. :::. 20- exclusion 99% C.L. ::::.. 30' exclusion

Figure 15: Left top panel: Zenith angle distribution of upward through-going events at SK The dashed and short-dashed histograms show the vI' f-t Vr and vI' f-t V. solutions respectively" Left bottom: Vertical/Horizontal ratio for upward throughgoing muons along with the predictions of VI' f-t Vr and VI' f-t Vs oscillation scenarios" Right panel: Exclusion regions obtained from Up/Down ratio for energetic PC sample (Evis > 5 GeV, (E) ~ 25 GeV) and Vertical/Horizontal ratio for upward throughgoing events" From 3"

lations would deplete the number of the observed 'lr°'s. Unfortunately, cross sections of these NC reactions are only known with large uncertainties, and at the moment a clear discrimination between the vp. f-t Vr and vp. f-t Vs oscillations of atmospheric neutrinos through the detection of single neutral pions does not seem to be possible. The situation may change in the near future, however, as the K2K Collaboration is planning to perform high statistics measurements of neutrino interactions in their near detector. Can vp. ++ Ve oscillations be responsible for the observed anomalies in the atmospheric neutrino data? The answer is no, at least not as the dominant channel. Explaining the data requires oscillations with large mixing equal or close to the maximal one; vp. ++ Ve oscillations would then certainly lead to a significant distortion of the zenith angle distributions of the e-like contained events, contrary to observations. In addition, for 11m 2 in the range'" 10- 3 eV 2 which is required by the atmospheric neutrino data, vp. ++ Ve oscillations

151

are severely restricted by the CHOOZ reactor antineutrino experiment, which excludes these oscillations as the main channel of the atmospheric neutrino oscillations (see sec. 7). However, vI' t-t Ve and Ve t-t V T can be present as sub dominant channels of the oscillations of atmospheric neutrinos. This may lead to interesting matter effects on oscillations of neutrinos crossing the earth on their way to the detector. Matter can strongly affect Ve t-t VJ.L,T and ve t-t VJ.L,T oscillations leading to an enhancement of the oscillation probabilities for neutrinos and suppression for antineutrinos or vice versa, depending on the sign of the corresponding mass squared difference. For certain neutrino energies, one can expect an enhancement due to the MSW effect in the mantle or in the core of the earth. In addition, there can be a specific enhancement effect, different from the MSW one, due to a cumulative build-up of the transition probability for neutrinos crossing the earth's mantle, core and then again mantle (parametric resonance 35,36,37,30). The matter enhancement effects are expected to take place in a relatively narrow range of neutrino energies, and it is rather difficult to observe them experimentally. Yet the possibility of the observation of matter effects on oscillations of atmospheric neutrinos is very exciting. Since three neutrino flavours are known to exist, oscillations of atmospheric neutrinos should in general be considered in the full 3-flavour framework (assuming that no sterile neutrinos take part in these oscillations). As follows from the analyses of solar and atmospheric neutrino data, there should be two distinct mass squared difference scales in the three neutrino framework, ~m~tm ,...., 1O-3 eV 2 and ~m~ 1O-4 eV 2 . The hierarchy ~m~tm » ~m~ means that one of the three neutrino mass eigenstates (which we denote V3) is separated by the larger mass gap from the other two. The neutrino mass squared differences thus satisfy ~m~1 ~ ~m~2 » ~m~I' We know already that vI' t-t V T should be the main channel of oscillations whereas Ve t-t VJ.L,T oscillations can only be present as the subdominant channels. In the 3-flavour framework this means that the element jUe31 of the lepton mixing matrix (26) is small. The smallness of jUe31 means that the vI' t-t V T and vI' t-t Ve oscillations approximately decouple, and the 2-flavour analysis gives a good first approximation. In terms of the parametrization (27) of the lepton mixing matrix, the mixing angle describing the main channel of the atmospheric neutrino oscillations is 023 , and its best-fit value following from the SK data is 45°. This fact and the smallness of IUd = 813 mean that the mass eigenstate V3 mainly consists of the flavour eigenstates vI' and V T with approximately equal weights, while the admixture of Ve to this state is small or zero. Can one observe true 3-flavour effects in the atmospheric neutrino oscillations? In principle, yes. Although in the 2-flavour scheme vI' t-t V T oscillations

:s

152

are not affected by matter, in the full 3-flavour approach these oscillations are modified by matter because of the mixing with Ve. These matter effects are, however, rather small because of the smallness of IUe3 1. Three-flavour effects can also lead to interesting phenomena in the subdominant vI' t+ Ve and Ve t+ V T oscillations. To see this, let us consider the fluxes FI-' and Fe of atmospheric vI' and Ve in the presence of oscillations. As we discussed in sec. 4.4, in the case of the hierarchy Llm~l ~ Llm~2 » Llm~l the oscillation probabilities in the 3-flavour scheme simplify significantly and can be expressed through the evolution matrix of the effective 2-flavour problem. Using eqs. (78) - (82) one can express FI-' and Fe through the corresponding no-oscillation fluxes F~ and F~ as Fe FI'

= F~

[1 + P2(r8~3

-

1)] ,

= ~ [1 - 8~3 (1 - ~) P2 + 28~3~3 [Z cos cI> + W3 sin cI> -

(95) 11], (96)

where r(E,O,,) = ~(E,O,,)/F~(E,O,,). It is interesting that the sub dominant t+ VI-',T oscillations may lead either to an excess or to a deficiency of e-like events depending on the values of rand 823. Indeed, the effect of oscillations on the electron neutrino flux is proportional to the factor (r8~3 - 1). If one assumes r = 2, there will be an excess of e-like events for (}23 > 45° and a deficiency for (}23 < 45°. The SK best fit is (}23 = 45°; for this value of (}23 there would be no deviation from the prediction for r = 2. However, for upward going neutrinos in the multi-GeV range r is typically 3 - 3.5 rather than 2, so there should be an excess of e-like events for (}23 ::: 33° and a deficiency 33°. Thus, the distortions of the zenith angle distributions of the for 023 e-like events due to the subdominant Ve t+ VI',T oscillations should depend on the value of the mixing angle (}23 governing the dominant vI-' t+ Vr oscillation channel. However, these distortions are expected to be small because of the smallness of iUe31 (notice that the probability P2 in (95) and (96) vanishes in the limit Ue3 -t 0). For recent analyses of the SK atmospheric neutrino data in the 3-flavour framework, see 48,30. If the solution of the solar neutrino problem is the LMA MSW effect, Llm~l can be as large as rv 10-4 eV 2 and its influence of the atmospheric neutrino oscillations cannot be neglected. In that case eqs. (95) and (96) have to be modified. It was shown in 49 that in this case the sub dominant oscillations can lead to a negative up-down asymmetry of e-like events at relatively large momenta. If the trend of slightly decreasing with momentum asymmetry of the e-like events which can be seen in the left top panel of fig. 13 is of physical origin and not just due to statistical fluctuations, it can be explained by relatively large Llm~l. Ve

:s

153

Are the standard neutrino oscillations the sole possible explanation of the observed atmospheric neutrino anomalies? In principle, other explanations are possible. Those include exotic types of neutrino oscillations - matter-induced oscillations due to flavour-changing interactions of neutrinos with medium 50, oscillations due to small violations of the Lorentz or CPT invariance or of the gravitational equivalence principle 51, and also neutrino decay 52. Exotic oscillations lead to periodic variations of the vI' survival probability with the oscillation lengths losc ()( E-n where n = 0 in the case of flavour-changing neutrino interactions or violation of CPT invariance, and n = 1 for oscillations due to the violations of the Lorentz invariance or equivalence principle. This has to be contrasted with n = -1 in the case of the standard neutrino oscillations. The energy dependence of the oscillation length can be tested in the atmospheric neutrino experiments as the energies of detected neutrinos span more than 3 orders of magnitude. The analysis was performed in 53, and the authors found that the fit of the atmospheric neutrino data assuming oscillations with losc ()( E-n gives n = -0.9 ± 0.4 at 90% c.l.. This clearly favours the standard oscillations over the exotic ones. In contrast to this, the neutrino decay mechanism fits the SK data quite well, the quality of the fit being as good as the one for the standard neutrino oscillations 54. However, the assumed neutrino decay mode (into a sterile neutrino and a Majoron) requires particle physics models which look less appealing than those that provide neutrino masses required for the standard neutrino oscillations. 7

Reactor and accelerator neutrino experiments

In reactor neutrino experiments oscillations of electron antineutrinos into another neutrino species are searched for by studying possible depletion of the ve flux beyond the usual geometrical one 55,56. These are the disappearance experiments, because the energies of the reactor Ve's ((E) == 3 MeV) are too small to allow the detection of muon or tauon antineutrinos in CC experiments. Small ve energy makes the reactor neutrino experiments sensitive to oscillations with rather small values of 6m 2 • Up to now, no evidence for neutrino oscillations has been found in the reactor neutrino experiments, which allowed to exclude certain regions in the neutrino parameter space. The best constraints were obtained by the CHOOZ experiment in France 57 (see fig. 16). As follows from eq. (34), in the case of the hierarchy 6m~1 == 6m~2 » 6m~1' the survival probability of Ve (which by CPT invariance is equal to that of ve) in the 3-flavour scheme has a simple form which coincides with that in the two-flavour framework. Therefore the excluded regions obtained in the 2-flavour analysis apply to the 3-flavour case

154 ~ 'r-~----~~~--------,

~

121 90% CL Kamiokande (multi-GcV) ~ 90% CL Kamiokande (sub+multi-GcV)

v'" oscillations. The region to the right of the curves are excluded. Shaded areas correspond to the Kamiokande allowed region for atmospheric VI'- -t Vc oscillations 57. Right panel: LSND allowed region for vI'- -t Vc oscillations (shaded areas) along with KARMEN, BNL776 and Bugey exclusion regions 56. Figure 16: Left panel: CHOOZ exclusion region for Vc -t

as well, with the identification 6.m 2 = 6.m~l ~ 6.m~2' sin 2 2() = sin 2 2()13' For the values of 6.m51 == 6.m~t7n in the Super-Kamiokande allowed region (2 - 6) . 10- 3 ey2, the CHOOZ results give the following constraint on the element Ue3 of the lepton mixing matrix: IUd 2 (1 - lUe312) < 0.045 - 0.02, i.e. lUe31 is either small or close to unity. The latter possibility is excluded by solar and atmospheric neutrino observations, and one finally obtains sin 2 ()13 ==

IUd 2

~ (0.047 - 0.02)

for

6.m~l = (2 - 6) . 10- 3 ey2.

(97)

This is the most stringent constraint on lUe31 to date. Presently, a long baseline reactor experiment KamLAND is under construction in Japan. This will be a large liquid scintillator detector experiment using the foriner Kamiokande site. KamLAND will detect electron antineutrinos coming from several Japanese power reactors at an average distance of about 180 km. KamLAND is scheduled to start taking data in 2001 and will

155

be sensitive to values of D.m 2 as low as 4 x 10- 6 eV 2 , i.e. in the range relevant for the solar neutrino oscillations! It is expected to be able to probe the LMA solution of the solar neutrino problem (see fig. 17). It may also be able to directly detect solar 8B and 7Be neutrinos after its liquid scintillator has been purified to ultra high purity level by recirculating through purification 58.

K2K

10 '

MINOS

~

~i

Low E ~

,

-2

10

-3

10

10"

10-5

-

Sensitivities

~~-,

solar SMA 10

Figure 17: Left panel: results of the present and sensitivities of future Ve f-+ VI' oscillations searches (90% c.l.) 58. Right panel: sensitivities of MINOS and K2K long baseline experiments 59,

In the 3-flavour framework, the oscillations due to the large D.m~tm 3 x 10- 3 eV 2 should be in the averaging regime in the KamLAND experiment and therefore the ve survival probability should be described by eqs. (36) and (37) rather than by (34). In the case of SMA, LMA and VO solutions of the solar neutrino problem the ve survival probability at KamLAND is then just P(ve -+ Ve) '== Ct3' i.e. the mixing angle ()13 is directly measured. The CHOOZ data allow Ct3 ~ 0.90 for D.m~tm in the SK range. In the case of the LMA solution, P(ve -+ vehll '== Ct3P(Ve -+ vehll where P(ve -+ vehll is the usual 2-flavour survival probability (37). Thus, in this case the 2-flavour analysis is modified by a non-vanishing ()13. There have been a number of accelerator experiments looking for neutrino oscillations. In all but one no evidence for oscillations was found and constraints on oscillation parameters were obtained 56,59. The LSND Collabo-

156

ration have obtained an evidence for iiI-' -+ iie and vI-' -+ Ve oscillations 4. The LSND result is the only indication for neutrino oscillations that is a signal and not a deficiency. The KARMEN experiment is looking for neutrino oscillations in iiI-' -+ ve channel. No evidence for oscillations has been obtained, and part of the LSND allowed region has been excluded. In fig. 16 (right panel) the results from LSND and KARMEN experiments are shown along with the relevant constraints from the BNL 776 and Bugey experiments. One can see that the only domain of the LSND allowed region which is presently not excluded is a narrow strip with sin 2 2() ~ 1.5 x 10- 3 - 4 X 10- 2 and 6.m 2 ~ 0.2 - 2 ey2. The existing neutrino anomalies (solar neutrino problem, atmospheric neutrino anomaly and the LSND result), if all interpreted in terms of neutrino oscillations, require three different scales of mass squared differences: 6.m~ ;S 10- 4 ey2, 6.m~tm ,...., 10- 3 ey2 and 6.m~SND ~ 0.2 ey2. This is only possible with four (or more) light neutrino species. The fourth light neutrino cannot be just the 4th generation neutrino similar to V e , vI-' and V T because this would be in conflict with the experimentally measured width of ZO boson [see eq. (2)]. It can only be an electroweak singlet (sterile) neutrino. Therefore the LSND result, if correct, would imply the existence of a light sterile neutrino. Out of all experimental evidences for neutrino oscillations, the LSND result is the only one that has not yet been confirmed by other experiments. It is therefore very important to have it independently checked. This will be done by the MiniBooNE (first phase of BooNE) experiment at Fermilab. MiniBooNE will be capable of observing both vI-' -+ Ve appearance and VI-' disappearance. If the LSND signal is due to vI-' -+ Ve oscillations, MiniBooNE is expected to detect an excess of ,....,1500 Ve events during its first year of operation, establishing the oscillation signal at ,...., 8a level (see fig. 17). If this happens, the second detector will be installed, with the goal to accurately measure the oscillation parameters. MiniBooNE will begin taking data in 2002. A number of long baseline accelerator neutrino experiments have been proposed to date. They are designed to independently test the oscillation interpretation of the results of the atmospheric neutrino experiments, accurately measure the oscillation parameters and to (possibly) identify the oscillation channel. The first of these experiments, K2K (KEK to Super-Kamiokande) already started taking data in 1999. It has a baseline of 250 km and is looking for vI-' disappearance. K2K should be able to test practically the whole region of oscillation parameters allowed by the SK atmospheric neutrino data except perhaps the lowest-6.m 2 part of it (see fig. 17). Two other long baseline projects, NuMI - MINOS (Fermilab to Soudan mine in the US) and CNGS (CERN to Gran Sasso in Europe), each with the baseline of about 730 km, will be sensitive to smaller values of 6.m 2 and should be able to test the whole

157

allowed region of SK (fig. 17). MINOS will look for vI-' disappearance and spectrum distortions due to vI-' -t v'" oscillations. It may run in three different energy regimes - high, medium and low energy. MINOS is scheduled to start data taking in the end of 2002. CERN to Gran Sasso will be an appearance experiment looking specifically for vI-' -t Vr oscillations. It will also probe vI-' -t Ve appearance. At the moment, two detectors have been approved for participation in the experiment - OPERA and ICANOE. The whole project was approved in December of 1999 and the data taking is planned to begin on May 15, 2005 5 . Among widely discussed now future projects are neutrino factories - muon storage rings producing intense beams of high energy neutrinos. In addition to high statistics studies of neutrino interactions, experiments at neutrino factories should be capable of measuring neutrino oscillation parameters with high precision and probing the subdominant neutrino oscillation channels, matter effects and CP violation effects in neutrino oscillations 60 • 8

Phenomenological neutrino mass matrices

The information on neutrino masses and lepton mixing obtained in solar and atmospheric the neutrino experiments can be summarized as follows: SMA:

~m~ :::'

LMA:

~m~

yO: Atm:

(4 - 10) . 10- 6 ey2 ,

:::' (2 - 20) . 10- 5 ey2,

sin22B0 :::' (0.1 - 1.0) . 10- 2 sin2 2B0 :::' 0.65 - 0.97

~m~ :::' (0.5 - 5) . 10- 10 ey2 ,

sin22B0 :::' 0.6 - 1.0

~m~tm :::' (2 - 6) .

sin22Batm :::' 0.82 - 1.0 (98)

10- 3

ey2,

In the 3-flavour framework (neglecting the LSND result), ~m0 has to be identified with ~m~1' ~matm with Am~1 :::' ~m~2' B0 with B12 , and Batm with B23 • The 2-flavour analysis gives a good first approximation to the results of the 3-flavour studies 44,48 because the mixing angle B13 is constrained to be rather small by the CHOOZ data (97). A very concise and illuminating way of summarizing the neutrino data is to present the neutrino mass eigenstates graphically as rectangles of unit area, the position of which on the vertical scale reflects their mass whereas the itreas of differently marked parts are equal to lUai/ 2 , i.e. to the weights with which the ath flavor eigenstate is present in the ith mass eigenstate 61. In the 3-flavour framework, the present-day data allow three main possibilities corresponding to the three main solutions of the solar neutrino problem. They are shown in fig. 18 and in fig. 19 (left panel). The schemes have different contributions of V e , vI-' and Vr in the two low-lying mass eigenstates. The weight of Ve in the

158 10° , - - - - - - - - - - - - ,

10° ~ Ve

10 -I

_V~ _ V

~

~

t

10 -2

:;;

~

E 10-3 10-4

~

~

10 -5 VI

v

2

v3

Figure 18: 3-flavour schemes of neutrino masses and mixing_ Solutions of the solar

neutrino problem are SMA (left panel) and LMA right panel) 61

mass eigenstate V3 which is separated from VI and V2 by a large mass gap is IUe3 12 ; it is either small or zero_ The weights of vI-' and Vr within each mass eigenstate are approximately equal to each other as a consequence of 823 :::: 45° and lUe312 = SI3 « L This, in particular, means that the solar neutrinos oscillate into an almost equal superposition of vI-' and Vn independently of the type of the solution of the solar neutrino problem (SMA, LMA, LOW or VO)_ The scheme of the left panel of fig. 18 corresponds to single large mixing, whereas those of the right panel of fig. 18 and left panel of fig. 19 describe bi-Iarge or bi-maximal (in the case Ue 3 = 0) mixing. The 3-flavour schemes of figs 18 and 19 correspond to the so-called normal neutrino mass hierarchy, ml « m2 «m3. In addition, there are schemes (not shown in the figures) in which the position of the mass eigenstate V3 and of the pair (VI, V2) are interchanged (inverted mass, hierarchy, m3 « mi :::: m2)- The present-day data do not discriminate between the normal and inverted hierarchies; such a discrimination may become possible in future if the earth's matter effects in atmospheric or long baseline Ve t+ vI-' or Ve t+ Vr oscillations are observed. The normal and inverted neutrino mass hierarchies discussed above do not exhaust all possible schemes of 3-flavour neutrino mass schemes. Since neutrino oscillation experiments only allow the determination of neutrino mass squared differences and not of the masses themselves, it is also possible that neutrinos are quasi-degenerate in mass, and only their mass squared differences satisfy the hierarchies shown in figs. 18 and 19. Direct neutrino mass measurements allow the average neutrino mass as large as a few eV (provided that the 2j30v constraint is satisfied). In that case neutrinos could constitute a significant fraction of the dark matter of the universe (hot dark matter)_ If the LSND result is correct, there should be at least four light neutrino species. Four-flavour neutrino schemes can be analyzed similarly to

159

-

m,eV e;:;J ve

10-2

_ _

V~

v,

10-3

10

1

DvS

10- 1

_

v,

e- 10-2

~~}

~ 2vHDM

ATM

LSND

~~

10- 3

-6

--

c:=-

""'" Ve

~

10-4 10- 5

10

,I ·'1

10-1

10-4 vI

v,

'3

vI

v2

v 4

v3

Figure 19: Schemes of neutrino masses and mixing_ Left panel: 3-flavour scheme with the VO solution of the solar neutrino problem_ Right panel: SMA solution in the 4-flavour scheme 61 _

the 3-flavour case_ The data allow essentially two schemes_ In each of these schemes there are two pairs of nearly degenerate mass eigenstates separated by large 6.mls N D- The mass splittings between the components of the quasidegenerate pairs are 6.m~ and 6.m~tm _ An example of a 4-flavour scheme is shown in fig_ 19 (right panel)_ More detailed discussions of the 4-flavour schemes can be found in 62 . The experimental information on the neutrino masses and lepton mixing angles allows one to reconstruct the phenomenologically allowed forms of the neutrino mass matrix. This can be done by inverting the relation (mL)diag = diag(ml' m2, m3) = UTmLU. The detailed structure of the neutrino mass matrix mL is not yet known since some of the neutrino paramet\rs are still rather uncertain. However, one can derive the zeroth-approximation structures (textures) of the neutrino mass matrices just using the already known gross features of neutrino spectrum and mixing angles. In particular, in the zeroth approximation one can assume ml = m2, 813 = 0, (}23 = 45°. Some of the neutrino mass matrix textures that can be obtained in this way are summarized in Table 2 taken from 63. Notice that these textures are defined only up to the trivial sign changes due to the rephasing of the neutrino fields. In realistic models, zeros in neutrino mass textures must be filled with small elements and in addition the large entries can be perturbed by small corrections. The small entries of the neutrino mass matrices should satisfy certain constraints; one of them follows from the requirement that for the LMA and SMA MSW solutions of the solar neutrino problem the lower-lying of the mass eigenstates VI and V2 have a larger Ve component. As was discussed in sec. 4, if this condition is not satisfied, the MSW resonance is only possible for antineutrinos and not for neutrinos. This requirement on the small entries of

160

Table 2: Zeroth order form of the neutrino mass matrix for double and single maximal mixing. A - normal mass hierarchy, B - inverted hierarchy, C - quasi-degenerate neutrinos. From 63. double maximal mixing

mdiag

A

Diag[O,O,l]

Bl

Diag[I,-I,O]

B2

Diag[I,I,O]

CO

Diag[I,I,I]

Cl

Diag[-I,I,I]

C2

Diag[I,-I,I]

[~

o 1/2 -1/2 o o

[~

0 1/2 1/2

[1o 01 o 0

Diag[I,I,-I]

0 0

1~2] 1/2

~]

[~

0 0 1

-~/2] 1/2

[~

0 -1/2 -1/2

-1/2

[1o 1/2 0 o 1/2

-~/2]

1~2]

1/2

[~ ~]

[0 1/,f2 1M] 1/-/2 1/2 -1/2 -1/2

[~

0 1/2 -1/2

0 1 0

-1/-/2 -1/-/2] 1/2 -1/2 -1/2 1/2

1/-/2

C3

0] -1/2 1/2

1/,f2 1/,f2]

[1/~ 1/-/2

[0 -1/-/2 -1/-/2

single maximal mixing

1/2

n

[-1

o o

[~ [~

0 1 0

0 0 -1 0 0 1

~] ~1]

n

161

the neutrino mass matrices disfavours the inverted mass hierarchy and quasidegenerate neutrinos and favours the normal mass hierarchy 64. The neutrino mass matrix textures can provide us with a hint of the symmetries or dynamics underlying the theory of neutrino mass. With the forthcoming data from future neutrino experiments, it may eventually become possible to unravel the mechanism of the neutrino mass generation, which may hold the clue to the general fermion mass problem. We live in a very fascinating time for neutrino physics. It is very likely that in a few years from now new solar and long baseline neutrino experiments will bring us an important knowledge about neutrinos allowing to answer many questions and solving the solar neutrino problem. Neutrinos may also bring us new surprises, as they did many times in the past. Acknowledgements

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162

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163

28. 29. 30. 31. 32.

33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51.

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61.

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63. 64.

Phys. Rev. D39 (1989) 3606. V. Barger et at., Phys. Rev. Lett. 82 (1999) 2640. G.L. Fogli et al., Phys.Rev. D60 (1999) 053006. V. Barger et at., Phys. Lett. B462 (1999) 109. G. Gratta, hep-exj9905011. L. DiLella, hep-exj991201O. CHOOZ Collaboration, M. Apollonio et at., hep-exj9907037. A. Suzuki, Talk given at 8th International Workshop on Neutrino Telescopes, Venice, Italy, February 23 - 26, 1999. J. Conrad, hep-exj9811009. For discussions of neutrino oscillation experiments at muon storage rings see, e.g., A. De Rujula, M. B. Gavela and P. Hernandez, ref. 32 j M. Campanelli, A. Bueno and A. Rubbia, hep-phj9905240j V. Barger, S. Geer and K. Whisnant, hep-phj9906487j O. Yasuda, hep-phj9910428j I. Mocioiu and R. Shrock, hep-phj9910554; V. Barger, S. Geer, R. Raja and K. Whisnant, hep-phj9911524. M. Freund et al., hep-phj9912457. A.Yu. Smirnov, Invited talk at 34th Rencontres de Moriond: Electroweak Interactions and Unified Theories, Les Arcs, France, March 1320, 1999j hep-phj9907296. For recent discussions of neutrino mass and mixing schemes and data analyses in the 4-flavour framework, see e.g. C. Giunti, hep-phj9909395j D. Dooling et at., hep-phj9908513j C. Giunti, M.C. Gonzalez-Garcia, C. Peiia-Garay, hep-phjOOOll01. G. Altarelli, F. Feruglio, hep-phj9905536. E.Kh. Akhmedov, Phys. Lett. B467 (1999) 95.

165

CP VIOLATION IN AND BEYOND THE STANDARD MODEL YOSEF NIR School of Natural Sciences, Institute for Advanced Study, Princeton NJ 08540, USA E-mail: [email protected] and Department of Particle Physics, Weizmann Institute of Science, Rehovot 76100, Israel The special features of CP violation in the Standard Model are presented. The significance of measuring CP violation in B, K and D decays is explained. The predictions of the Standard Model for CP asymmetries in B decays are analyzed in detail. Then, four frameworks of new physics are reviewed: (i) Supersymmetry provides an excellent demonstration of the power of CP violation as a probe of new physics. (ii) Left-right symmetric models are discussed as an example of an extension of the gauge sector. CP violation suggests that the scale of LRS breaking is low. (iii) The variety of extensions of the scalar sector are presented and their unique CP violating signatures are emphasized. (iv) Vector-like down quarks are presented as an example of an extension of the fermion sector. Their implications for CP asymmetries in B decays are highly interesting.

1

Introduction

CP violation arises naturally in the three generation Standard Model. The CP violation that has been measured in neutral K-meson decays (cK and CK) is accommodated in the Standard Model in a simple way 1. Yet, CP violation is one of the least tested aspects of the Standard Model. The value of the· C K parameter 2 as well as bounds on other CP violating parameters (most noticeably, the electric dipole moments of the neutron, dN, and of the electron, de) can be accounted for in models where CP violation has features that are very different from the Standard Model ones. It is unlikely that the Standard Model provides the complete description of CP violation in nature. First, it is quite clear that there exists New Physics beyond the Standard Model. Almost any extension of the Standard Model has additional sources of CP violating effects (or effects that change the relationship of the measurable quantitites to the CP violating parameters of the Standard Model). In addition there is a great puzzle in cosmology that relates to CP violation, and that is the baryon asymmetry of the universe 3. Theories that explain the observed asymmetry must include new sources of CP violation 4: the Standard Model cannot generate a large enough matterantimatter imbalance to produce the baryon number to entropy ratio observed

166

in the universe today 5-7. In the near future, significant new information on CP violation will be provided by various experiments. The main source of information will be measurements of CP violation in various B decays, particularly neutral B decays into final CP eigenstates 8 -10. First attempts have already been reported 11-13 and interpreted in the framework of various models of new physics 14 -15. Another piece of valuable information might come from a measurement of the KL -t n:ovii decay 16-19. For the first time, the pattern of CP violation that is predicted by the Standard Model will be tested. Basic questions such as whether CP is an approximate symmetry in nature will be answered. It could be that the scale where new CP violating sources appear is too high above the Standard Model scale (e.g. the GUT scale) to give any observable deviations from the Standard Model predictions. In such a case, the outcome of the experiments will be a (frustratingly) successful test of the Standard Model and a significant improvement in our knowledge of the CKM matrix. A much more interesting situation will arise if the new sources of CP violation appear at a scale that is not too high above the electroweak scale. Then they might be discovered in the forthcoming experiments. Once enough independent observations of CP violating effects are made, we will find that there is no single choice of CKM parameters that is consistent with all measurements. There may even be enough information in the pattern of the inconsistencies to tell us something about the nature of the new physics contributions 20-22. The aim of these lectures is to explain the theoretical tools with which we will analyze new information about CP violation. The first part, chapters 2-6, deal with the Standard Model while the second, chapters 7-10, discuss physics beyond the Standard Model. In chapter 2, we present the Standard Model picture of CP violation. We emphasize the features that are unique to the Standard Model. In chapter 3, we give a brief, model-independent discussion of CP violating observables in B meson decays. In chapter 4, we discuss CP violation in the K system (particularly, the CK and c~ parameters) in a model independent way and in the framework of the Standard Model. We also describe CP violation in K -t 7rvii. In chapter 5, we discuss CP violation in D -t K 7r decays. In chapter 6, we present in detail the Standard Model predictions for CP asymmetries in B decays. In chapter 7, the power of CP violation as a probe of new physics is explained. Then, we discuss specific frameworks of new physics: Supersymmetry (chapter 8), Left-Right symmetry as an example of extensions of the gauge sector (chapter 9), extensions of the scalar sector (chapter 10), and extra down singlet-quarks as an example of

167

extensions of the fermion sector (chapter 11). Finally, we summarize our main points in chapter 12.

2

Theory of CP Violation in the Standard Model

2.1

Yukawa Interactions Are the Source of CP Violation

A model of the basic interactions between elementary particles is defined by the following three ingredients: (i) The symmetries of the Lagrangian; (ii) The representations of fermions and scalars; (iii) The pattern of spontaneous symmetry breaking. The Standard Model (SM) is defined as follows: (i) The gauge symmetry is

GS M = SU(3)c

X

SU(2)L x U(I)y.

(2.1)

(ii) There are three fermion generations, each consisting of five representations:

QL(3, 2)+1/6, u~i(3, 1)+2/3, d~i(3, 1)-1/3, LL(1, 2)-1/2, £~i(l, 1)-1. (2.2) Our notations mean that, for example, the left-handed quarks, Qi, are in a triplet (3) of the SU(3)c group, a doublet (2) of SU(2)L and carryhypercharge Y = QEM - T3 = +1/6. The index I denotes interaction eigenstates. The index i = 1,2,3 is the flavor (or generation) index. There is a single scalar multiplet: (1,2)+1/ 2'

(2.3)

(iii) The scalar assumes a VEV,

("/b requires that there exists CP violation in I::1F = 1 processes (¢'a - ¢'b =I- 0), namely direct CP violation. Experimentally, both direct and indirect CP violation have been established. Below we will see that SK signifies indirect CP violation while s'x signifies direct CP violation. Theoretically, most models of CP violation (including the Standard Model) have predicted that both types of CP violation exist. There is, however, one class of models, that is superweak models, that predict only indirect CP violation. The measurement of s'x =I- 0 has excluded this class of models. 4·2

The

SK

and s'x Parameters

Historically, a different language from the one used by us has been employed to describe CP violation in K --+ 7f7f and K --+ 7f£V decays. In this section we 'translate' the language of SK and s'x to our notations. Doing so will make it easy to understand which type of CP violation is related to each quantity. The two CP violating quantities measured in neutral K decays are (7f07f0 11i IK L) 1100 = (7fo7f ol1iIKs)'

(7f+7f-I1iIKL) 11+- = (7f+7f-I1iIKs)

Define, for (ij) = (00) or (+-), Aij = (7fi7f j l1iIK O),

(4.2)

\. _('l)

A'J -

(4.1)

P

Aij K

Aij

•

(4.3)

Then 1 - >"00 1100= l+Aoo'

(4.4)

184

The 1700 and 17+- parameters get contributions from CP violation in mixing (l(q/p)IK ;j:. 1) and from the interference of decays with and without mixing (Im>"ij ;j:. 0) at 0(10- 3 ) and from CP violation in decay (IAij/Aijl ;j:. 1) at

0(10- 6 ). There are two isospin channels in K -+ 1m leading to final (211"h=0 and

(211"h=2 states:

(4.5)

The fact that there are two strong phases allows for CP violation in decay. The possible effects are, however, small (on top of the smallness of the relevant CP violating phases) because the final I = 0 state is dominant (this is the t:J..I = 1/2 rule). Defining

AI = ((1I"1I"hIHIK o),

AI = ((1I"1I"hIHlkO),

(4.6)

we have, experimentally, IA2/Aol >:::! 1/20. Instead of 1700 and 17+- we may define two combinations, CK and c~, in such a way that the possible effects of CP violation in decay (mixing) are isolated into c~ (cK). The experimental definition of the CK parameter is CK

=1

3(1700

+ 217+-)·

(4.7)

To zeroth order in A 2 /A o, we have 1700 = 17+- = CK. However, the specific combination (4.7) is chosen in such a way that the following relation holds to first order in A2/Ao: CK

1- >"0 = 1 + >"0'

(4.8)

where (4.9)

Since, by definition, only one strong channel contributes to >"0, there is indeed no CP violation in decay in (4.8). It is simple to show that Re CK ;j:. 0 is a manifestation of CP violation in mixing while Im CK ;j:. 0 is a manifestation of CP violation in the interference between decays with and without mixing. Since experimentally argcK >:::! 11"/4, the two contributions are comparable. It is also clear that CK ;j:. 0 is a manifestation of indirect CP violation: it could be described entirely in terms of a CP violating phase in the M12 amplitude.

185

The experimental value of CK is given by ICKI

26

= (2.280 ± 0.013) x 10- 3 .

The experimental definition of the CK

cK

(4.10)

parameter is

=~(77+- 7700). 3

(4.11)

The theoretical expression is

c~ ~ ~().,oo -

(4.12)

).,+-).

Obviously, any type of CP violation which is independent of the final state does not contribute to cK' Consequently, there is no contribution from CP violation in mixing to (4.12). It is simple to show that Re CK :/; 0 is a manifestation of CP violation in decay while Im cK :/; 0 is a manifestation of CP violation in the interference between decays with and without mixing. Following our explanations in the previous section, we learn that cK :/; 0 is a manifestation of direct OP violation: it requires .(B -t 1jJK s)

= - (~bvt:) (Vc: Vc:) (Vc'd Vc:) vtb ~d Vcb v"s Vcd Vcs

===}

Im>'t/JKs

= sin(2,B).

(6.18) We have seen in the previous section that, for b -t ccs decays, we have a very small CP violation in decay, 1-IA/AI rv >.2rpT . Consequently, eq. (6.18) is clean ofhadronic uncertainties to better than 0(10- 2 ). This means that the measuremnet of at/JKs can give the theoretically cleanest determination of a CKM parameter, even cleaner than the determination of !Yusl from K -t 7rev. (If BR(KL -t 7rviJ) is measured, it will give a comparably clean determination of'T/.) A second example of a theoretically clean mode in class (i) is B -t ¢Ks. We showed in the previous section that, for b -t sss decays, we have small CP violation in decay, 1 - lA/AI = 0(>.2) = 0(0.05). We can neglect this effect. The analysis is similar to the 1jJKs case, and the asymmetry is proportional to sin(2,B). The same quark subprocesses give theoretically clean CP asymmetries also in Bs decays. These asymmetries are, however, very small since the relative phase between the mixing amplitude and the decay amplitudes (,Bs defined in (2.37)) is very small. The best known example of class (ii) is B -t 7r7r. The quark subprocess is b -t uud which is dominated by the W-mediated tree diagram. Neglecting for the moment the second, pure penguin, term we find (6.19) The CP eigenvalue for two pions is +1. Combining eqs. (6.5) and (6.19), we

201

get >"(B

--+ 1f+1f-) =

(~bVi:) (V:d V: b) Vib ~d

Vud Vub

==}

Im>"n = sin(2a).

(6.20)

The pure penguin term in A(uud) in eq. (6.8) has a weak phase, arg(~dVib), different from the term with the tree contribution, so it modifies both Im>"n and (if there are non-trivial strong phases) I>"nl. The recent CLEO results mentioned above suggest that the penguin contribution to B --+ 1f1f channel is significant, probably 10% or more. This then introduces CP violation in decay, unless the strong phases cancel (or are zero, as suggested by factorization arguments). The resulting hadronic uncertainty can be eliminated using isospin analysis 86. This requires a measurement of the rates for the isospinrelated channels B+ --+ 1f+1f0 and BO --+ .1f01f0 as well as the corresponding CP-conjugate processes. The rate for 1fo1f o is expected to be small and the measurement is difficult, but even an upper bound on this rate can be used to limit the magnitude of hadronic uncertainties 87. Related but slightly more complicated channels with the same underlying quark structure are B --+ p01fo and B --+ a~1fo. Again an analysis involving the isospin-related channels can be used to help eliminate hadronic uncertainties from CP violations in the decays 88,89. Channels such as pp and al P could in principle also be studied, using angular analysis to determine the mixture of CP-even and CP-odd contributions. The analysis of B --+ D+ D- proceeds along very similar lines. The quark subprocess here is b --+ ccd, and so the tree contribution gives >"(B--+D+D-)=rJD+D-

(~:~~) (i:~~)

==}

Im>"DD=-sin(2/3),

(6.21) where we used rJD+ D- = + 1. Again, there are hadronic uncertainties due to the pure penguin term in (6.8), but they are estimated to be small. A summary of our results for CP violation in the interference of decays with and without mixing in Bq --+ fcp is given in table III. For each mode, we give the asymmetry that would arise if the dominat contribution were the only contribution. In all cases the above discussions have neglected the distinction between strong penguins and electroweak penguins. The CKM phase structure of both types of penguins is the same. The only place where this distinction becomes important is when an isospin argument is used to remove hadronic uncertainties due to penguin contributions. These arguments are based on the fact that gluons have isospin zero, and hence strong penguin processes have definite b.I. Photons and Z-bosons on the other hand contribute to

202 Table III. ImA(Bq ---+

Quark process

Iep).

Sample Bd mode

ImABr+fcp

Sample Bs mode

ImABs-+fcp

DsDs

sin 2{Js

'ljJKs

sin 2{J

b ---+ sss

Ks

sin 2{J

rl'

0

b ---+ uud

7r7r

sin 20

7r°Ks

sin 2"(

b ---+ cm

D+D-

sin 2{J

'ljJKs

sin 2{Js

sin 2{J

7r

0

0

Ks

b ---+

cCs

b ---+ uus b ---+ ssd

7r°Ks 7r

0

sin 2{J

more than one 111 transition and hence cannot be separated from tree terms by isospin analysis. In most cases electroweak penguins are small, typically no more than ten percent of the corresponding strong penguins and so their effects can safely be neglected. However in cases (iii) and (iv), where tree contributions are small or absent, their effects may need to be considered. (A full review of the role of electroweak penguins in B decays has been given in ref. 90.) 7

CP Violation Can Probe New Physics

We have argued that the Standard Model picture of CP violation is rather unique and highly predictive. We have also stated that reasonable extensions of the Standard Model have a very different picture of CP violation. Experimental results are too few to decide between the various possibilities. But in the near future, we expect many new measurements of CP violating observabIes. Our discussion of CP violation in the presence of new physics is aimed to demonstrate that, indeed, models of new physics can significantly modify the Standard Model predictions and that the near future measurements will therefore have a strong impact on the theoretical understanding of CP violation. To understand how the Standard Model predictions could be modified by New Physics, we focus on CP violation in the interference between decays with and without mixing. As explained above, this type of CP violation may give, due to its theoretical cleanliness, unambiguous evidence for New Physics most easily.

203

Let us consider five specific CP violating observables. (i) Im)..",Ks' the CP asymmetry in B --+ 'l/JKs. This measurement will cleanly determine the relative phase between the B - B mixing amplitude and the b --+ ccs decay amplitude (sin 2(3 in the Standard Model). The b --+ ccs decay has Standard Model tree contributions and therefore is very unlikely to be significantly affected by new physics. On the other hand, the mixing amplitude can be easily modified by new physics. We parametrize such a modification by a phase (}d:

2(}d = arg(M12 /Mfr)

====}

Im).."'Ks = sin[2((3 + (}d)].

(7.1)

(ii) Im)..q,Ks, the CP asymmetry in B --+ ¢Ks. This measurement will cleanly determine the relative phase between the B - B mixing amplitude and the b --+ sss decay amplitude. The b --+ sss decay has only Standard Model penguin contributions and therefore is sensitive to new physics. We parametrize the modification of the decay amplitude by a phase () A 91: (}A

-

-8M

=arg(Aq,Ks/Aq,Ks)

====}

Im)..q,Ks =sin[2((3+(}d+(}A)].

(7.2)

(iii) a"vli, the CP violating ratio of K --+ 7rviJ decays, defined in (4.38). This measurement will cleanly determine the relative phase between the K - k mixing amplitude and the s --+ dviJ decay amplitude. The experimentally measured small value of eK requires that the phase of the K - k mixing amplitude is not modified from the Standard Model prediction. (More precisely, it requires that the phase in the mixing amplitude is very close to the phase in the s --+ duu decay amplitude.) On the other hand, the decay, which in the Standard Model is a loop process with small mixing angles, can be easily modified by new physics. (iv) aD-+Krr, the CP violating quantity in D --+ K±7r~ decays (see (5.5) and (5.11)): _ Im()..K-rr+) - Im()..K~rr-) aD-+Krr I'/\K-rr+ 1

(7.3) It depends on the relative phase between the D - jj mixing amplitude and the c --+ dsu and c --+ sdu decay amplitudes. The two decay channels are tree

level and therefore unlikely to be affected by new physics 68. On the other hand, the mixing amplitude can be easily modified by new physics 69. (V) dN, the electric dipole moment of the neutron. We did not discuss this quantity so far because, unlike CP violation in meson decays, flavor changing couplings are not necessary for dN. In other words, the CP violation that induces dN is flavor diagonal. It does in general get contributions from flavor

204 Table IV. Features of various CP violating observables. Process

Observable

Mixing

B --+ 1jJKs

Im),,pKs

B --+ Ks

Im),q,Ks

K --+ 7rvii

D --+ K7r

Decay

SM

B-B

b --+ ccs

sin 2(3

B-B

b --+ sss

sin 2(3

a 7rVV

K-K

s --+ dVii

aD-+K-rr

D-D

c --+ dsu

dN

~

sin 2 (3

NP

+ Od) + Od + OA)

sin 2((3 sin 2((3

sin 2 OK

0

~sinK-rr

,$ 1O- 6 d';;P

FD phases

changing physics, but it could be induced by sectors that are flavor blind. Within the Standard Model (and ignoring the strong CP angle 0QCD), the contribution from 8KM arises at the three loop level and is at least six orders of magnitude below the experimental bound. We denote the present 90% C.L upper bound on dN by d";p. It is given by 92

d";P = 6.3

X

10- 26 e cm.

(7.4)

The main features of the observables that we chose are summarized in Table IV. The various CP violating observables discussed above are sensitive then to new physics in the mixing amplitudes for the B - jj and D - j) systems, in the decay amplitudes for b -+ sss and s -+ dvv channels and to flavor diagonal CP violation. If information about all these processes becomes available and deviations from the Standard Model predictions are found, we can ask rather detailed questions about the nature of the new physics that is responsible to these deviations: (i) Is the new physics related to the down sector? the up sector? both?

(ii) Is the new physics related to 6.B = 1 processes? 6.B = 2? both? (iii) Is the new physics related to the third generation? to all generations? (iv) Are the new sources of CP violation flavor changing? flavor diagonal? both? It is no wonder then that with such rich information, flavor and CP violation provide an excellent probe of new physics.

205 8

Supersymmetry

A generic sl1persymmetric extension of the Standard Model contains a host of new flavor and CP violating parameters. (For recent reviews on supersymmetry see refs. 93 - 97. The following chapter is based on 98.) It is an amusing exercise to count the number of parameters. The supersymmetric part of the Lagrangian depends, in addition to the three gauge couplings of GSM, on the parameters of the superpotential W, which can be written as a function of the scalar matter fields:

W =

L

(Y;jhuiiL(URj

+ y;1 hdiiLi dRj + Y;~hdLL;lRj) + J.Lhuhd'

(8.1)

i,j

In addition, we have to add soft supersymmetry breaking terms:

.csoft

= - (aijhuiiLi'URj

+ atjhdiiLidRj + aLhdLLJRj + bhuhd + h.c.) (8.2)

The three Yukawa matrices Y f depend on 27 real and 27 imaginary parameters. Similarly, the three a f -matrices depend on 27 real and 27 imaginary parameters. The five m S2 hermitian 3 x 3 mass-squared matrices for sfermions (5 = Q, dR, UR, L, eR) have 30 real parameters and 15 phases. The gauge and Higgs sectors depend on (8.3)

that is 11 real and 5 imaginary parameters. Summing over all sectors, we get 95 real and 74 imaginary parameters. If we switch off all the above parameters but the gauge couplings, we gain global symmetries: G~J;;~~(Yf,J.L,af,b,m2,m = 0) = U(3)5 x U(l)pQ

X

U(l)R'

(8.4)

where the U(l)pQ x U(l)R charge assignments are: U(l)PQ U(l)R

hu

hd

Qu Qd

Ll

1 1

1 1

-1 1

-l. 1

-1 1

(8.5)

Consequently, we can remove at most 15 real and 32 imaginary parameters. But even when all the couplings are switched on, there is a global symmtery, that is (8.6)

206

so that 2 of the 32 imaginary parameters cannot be removed. We are left then with 80 real . 124 = { 44 . . physIcal parameters. Imagmary

(8.7)

In particular, there are 43 new CP violating phases! In addition to the single Kobayashi-Maskawa of the SM, we can put 3 phases in M I , M 2 , /L (we used the U(l)pQ and U(l)R to remove the phases from /LB· and M 3 , respectively) and the other 40 phases appear in the mixing matrices of the fermion-sfermiongaugino couplings. (Of the 80 real parameters, there are 11 absolute values of the parameters in (8.3), 9 fermion masses, 21 sfermion masses, 3 CKM angles and 36 SCKM angles.) Supersymmetry provides a nice example to our statement that reasonable extensions of the Standard Model may have more than one source of CP violation. The requirement of consistency with experimental data provides strong constraints on many of these parameters. For this reason, the physics of flavor and CP violation has had a profound impact on supersymmetric model building. A discussion of CP violation in this context can hardly avoid addressing the flavor problem itself. Indeed, many of the supersymmetric models that we analyze below were originally aimed at solving flavor problems. As concerns CP violation, one can distinguish two classes of experimental constraints. First, bounds on nuclear and atomic electric dipole moments determine what is usually called the supersymmetric CP problem. Second, the physics of neutral mesons and, most importantly, the small experimental value of 10K pose the supersymmetric 10K problem. In the next two subsections we describe the two problems. Then we describe various super symmetric flavor models and the ways in which they address the supersymmetric CP problem. Before turning to a detailed discussion, we define two scales that play an important role in supersymmetry: As, where the soft supersymmetry breaking terms are generated, and A F , where flavor dynamics takes place. When AF » As, it is possible that there are no genuinely new sources of flavor and CP violation. This leads to models with exact universality, which we discuss in section 8.3. When AF ;$ As, we do not expect, in general, that flavor and CP violation are limited to the Yukawa matrices. One way to suppress CP violation would be to assume that, similarly to the Standard Model, CP violating phases are large, but their effects are screened, possibly by the same physics that explains the various flavor puzzles. Such models, with Abelian or non-Abelian horizontal symmetries, are described in section 8.4. It is also possible that CP violating effects are suppressed because squarks are heavy.

207

This scenario is also discussed in section 8.4. Another option is to assume that CP is an approximate symmetry of the full theory (namely, CP violating phases are all small). We discuss this scenario in section 8.5. A brief discussion of the implications of c' / c is included in this subsection. Some concluding comments regarding CP violation as a probe of supersymmetric flavor models are given in section 8.6.

8.1

The Supersymmetric CP Problem

One aspect of super symmetric CP violation involves effects that are flavor preserving. Then, for simplicity, we describe this aspect in a supersymmetric model without additional flavor mixings, i.e. the minimal supersymmetric standard model (MSSM) with universal sfermion masses and with the trilinear SUSY-breaking scalar couplings proportional to the corresponding Yukawa couplings. (The generalization to the case of non-universal soft terms is straightforward.) In such a constrained framework, there are four new phases beyond the two phases of the Standard Model (8KM and 8QCD). One arises in the bilinear Jl-term of the superpotential (8.1), while the other three arise in the soft supersymmetry breaking parameters of (8.2): in (the gaugino mass), a (the trilinear scalar coupling) and b (the bilinear scalar coupling). Only two combinations of the four phases are physical 99,100. To see this, note that one could treat the various dimensionful parameters in (8.1) and (8.2) as spurions which break the U(l)PQ x U(l)R symmetry, thus deriving selection rules:

in

o

A 0

-2

-2

b -2 -2

Jl -2 0

(8.8)

(where we defined A through at = Ayt). Physical observables can only depend on combinations of the dimensionful parameters that are neutral under both U (1) 'so There are three such independent combinations: inJlb*, AJlb* and A*in. However, only two of their phases are independent, say

=(:~ :~).

(9.7)

The spontaneous symmetry breaking occurs in two stages, GLRS

x DLRS

-+

GSM

-+ SU(3)c

X

(9.8)

U(l)EM'

due to the VEVs of the neutral members of the scalar fields:

(In general, all four VEVs are complex. There is, however, freedom of rotations by U(l)B-L for 6.L and 6.R and by U(l)r3L x U(1)r3R for ¢l and ¢2, so that only two phases are physical.) These VEVs are assumed to satisfy the hierarchy (9.10)

The first stage of symmetry breaking in (9.8) takes place at the scale the second at k = ylk; + k~. 9.2

VR

and

Flavor Parameters

The quark Yukawa couplings have the following form: CYuk

= fQL«I>QR

+ hQL~QR + h.c.,

(9.11)

where ~ = T2«I>*T2. As a consequence of DLRS, the Yukawa matrices f and h are symmetric and real: P requires that they are hermitian, C requires that they are symmetric, and CP requires that they are real. The resulting mass matrices,

+ hk2 e- iOl , hkl + fk 2 e iOl ,

Mu = fk 1 Md =

(9.12)

are complex symmetric matrices. How many independent physical flavor parameters (and, in particular, phases) does this model have? We have two symmetric and complex mass matrices, that is twelve real and twelve imaginary Yukawa parameters. If we set h = f = 0, we gain a global U(3) symmetry (DLRs does not allow independent U(3)L and U(3)R rotations). This means that we can remove three real and six imaginary parameters. When h and f are different from

217

zero, there is no global symmetry (U(l)B-L is part of the gauge symmetry). We conclude that there are nine real and six imaginary flavor parameters. Six of the real parameters are the six quark masses. To identify the other flavor parameters, note that the symmetric mass matrices can be diagonalized by a unitary transformation of the form (9.13) Consequently, the mixing matrices VL and VR describing, respectively, the WL and WR interactions,

Cee =

~

(WtJlULVL'i.£dL + Wit,URVRrJldR) +h.c.,

(9.14)

are related: VL = pUVii pd ,

(9.15)

where pu and pd are diagonal phase matrices. The three real parameters are then the three mixing angles, which are equal in VL and VR. The six phases can be arranged in various ways. A convenient choice is to have a single phase in VL, which is then just 8KM of VCKM of the Standard Model, and five phases in VR. (It is also possible to have VL = Vii with six phases in each.) 9.3

What is the Low Energy Effective Theory of the LRS Model?

It is interesting to ask what is the low enrgy effective theory below the scale It is straightforward to show that the light fields are precisely those of the Standard Model: the fermions are chiral under SU(2)L except for the right-handed neutrinos in LRi which acquire Majorana masses at the scale VR due to their coupling to /:1R. (The left-handed neutrinos acquire very light masses from both the see-saw mechanism and their direct coupling to /:1L') Only the one Higgs doublet related to GS M breaking, that is kl4>l + k 2 e- ia 4>2, remains light. The question is then whether the left-right symmetry constrains Standard Model parameters. To answer this question, we first argue that phenomenological constraints forbid r == k2/kl = 0(1). (More precisely, it is r sin 0: which is constrained to be very small.) Consider eqs. (9.12). They lead to the following equations:

VR.

Mure ia

-

Md = k 1 h(r2 - 1),

Mu - Mdre- ia = kd(l - r2).

(9.16)

The right hand side of these equations is real. Then, the imaginary part of the left-hand side should vanish. Let us put all quark masses to zero, except

218

for mt and mb. We take then (Muh3 rmt sin(Ot

= mteillt

+ 0:) -

and (Mdh3

mb sin Ob

= mbeillb.

= 0,

mt sin Ot - mb sin(Ob - 0:) = O.

The second equation implies that ()t gives 173

:5 mb/mt.

We get:

(9.17)

Then, the first equation (9.18)

(Note that the model is symmetric under r --t 1/r and

0:

--t

-0:.

Therefore,

r sin 0: 2: mt/mb is acceptable and physically equivalent.)

The only source of CP violation in the quark mass matrices is the phase (The phase f3 in (~R> does not affect quark masses, though it may affect neutrino masses.) Moreover, if one of the (¢?> vanished, then again there would be no CP violation in the quark mass matrices. As a consequence of these two facts, all CP violating phases in the mixing matrices VL and VR are proportional to rsino:. Hence the importance of (9.18). In particular, for the Kobayashi-Maskawa phase, one finds 173

0:.

OKM .......

r sin o:(mclm.) :::; 0(0.1).

(9.19)

We learn that the low energy effective theory ofthe left-right symmetric model is the Standard Model with a small value for OKM. Phenomenologically, it is difficult, though not impossible, to account for cK with just the Standard Model contribution and a small KM phase. There are then two possibilities: (i) The left-right symmetry is broken at a very high scale. The low energy theory is to a good approximation just the Standard Model. CP is, however, an approximate symmetry in the kaon sector. The hadronic parameters playing a role in the calculation of c K have to assume rather extreme values. (ii) The left-right symmetry is broken at low enough scale so that there are significant new contributions to various rare processes. In particular, box diagrams with intermediate W wboson and tree diagrams with a heavy neutral Higgs dominate c K. This sets up an upper bound on the scale VR, of order tens of TeV. 9.4

Phenomenology of CP Violation

The smallness of r sin 0: does not necessarily mean that CP is an approximate symmetry in the quark sector; the phases in the mixing matrices depend, in

219

addition to r sin a, on quark mass ratios, some of which are large. An explicit calculation shows that the six phases actually divide to two groups; the KM phase and the three phases that appear in VR in a two generation model (usually denoted by (h, 62 and,) are all small 173, while the two extra phases that appear in the three generation VR (denoted by a1, a2) are not 174;

6,6 1 ,62 "

IX

rsina(mc/ms)

:s: 0(0.1), (9.20)

In C K, it is mainly 61 and 62 which playa role. (We here assume that the hadronic parameters are close to their present theoretical estimate and therefore cK cannot be explained in this framework by the Standard Model contribution alone.) Assuming that the W L - WR box diagram gives the dominant contribution, one is led to conclude that 173

(9.21) is favored. Note that CP conserving processes provide a lower bound

175,176,

(9.22) The favored range for M(WR ) is then very constrained in this framework. Taking into account this upper bound and the fact that the ai phases are enhanced by a factor of about 10 compared to the 6i phases, one finds that the left-right symmetric contributions compete with or even dominate over the Standard Model contributions to B - B mixing and to Bs - Bs mixing 174,177-·179. This means that CP asymmetries in B or Bs decays into final CP eigenstates could be substantially different from the Standard Model prediction. Moreover, the phases in the left-right symmetric contributions to B - Band Bs - Bs mixing are closely related, predicting correlations between the deviations. The CP asymmetry in semileptonic B decays could also be significantly enhanced 180. The recent measurement of a,pKs gives first constraints on a1 leading to new bounds on aSL 14. Finally, LRS models could enhance the electric dipole moments of the neutron and of the electron 181-184. 10

Multi-Scalar Models

The Standard Model has a single scalar field, (1,2h/2, that is responsible for the spontaneous symmetry breaking, SU(2)L x U(I)y --+ U(I)EM. Within the framework of the Standard Model, the complex Yukawa couplings of the scalar doublet to fermions are the only source of flavor physics and of CP

220 violation. However, in the mass basis, the interactions of the Higgs particle are flavor diagonal and CP conserving. There are several good reasons for the interest in multi-scalar models in the context of flavor and CP violation: a. If there exist additional scalars and, in particular, SU(2)L-doublets, then not only there are new sources of CP violation, but also the Yukawa interactions in the mass basis as well as the scalar self-interactions may violate CPo b. CP violation in scalar interactions has very different features from the W -mediated CP violation of the Standard Model. For example, it could lead to observable flavor diagonal CP violation in top physics or in electric dipole moments, or it could induce transverse lepton polarization in semileptonic meson decays. c. With more than a single scalar doublet, CP violation could be spontaneous. Indeed, there is no good reason to assume that the Standard Model doublet is the only scalar in Nature. Most extensions of the Standard Model predict that there exist additional scalars. For example, models with an extended gauge symmetry (such as GUTs and left-right symmetric models) need extra scalars to break the symmetry down to GSM; Supersymmetry requires that there exists a scalar partner to each Standard Model fermion. However, scalar masses are generically not protected by a symmetry. Consequently, in models where the low energy effective theory is the Standard Model, we expect in general that the only light scalar is the Standard Model doublet. The study of multi-scalar models is then best motivated in the following cases: (i) The scale of new physics is not very high above the electroweak scale. One has to remember, however, that in such cases there is more to the new physics than just extending the scalar sector. (ii) The scalar is related to the spontaneous breaking of a global symmetry. In some cases, a discrete symmetry is enough to make a scalar light. We will discuss scalar SU(2)L-doublets and -singlets only. There are two reasons for that. First, the VEVs of higher multiplets need to be very small in order to avoid large deviations from the experimentally successful relation p = 1. Second, higher multiplets do not couple to the known fermions. (The only exception is an SU(2)L-triplet that can couple to the left-handed leptons.)

221

10.1

Multi Higgs Doublet Models

The most popular extension of the Higgs sector is the multi Higgs doublet model (MHDM) and, in particular, the two Higgs doublet model (2HDM). These models have, in addition to the Kobayashi-Maskawa phase of the quark mixing matrix, several new sources of CP violation 185: (i) A complex mixing matrix for charged scalars

186.

(ii) Mixing of CP-even and CP-odd neutral scalars

187.

(iii) CP-odd Yukawa couplings (in the quark mass basis). (iv) Complex quartic scalar couplings. The CP violation that is relevant to near future experiments always involves fermions. Therefore, we will only discuss the new sources (i), (ii) and (iii). A generic MHDM, with all dimensionful parameters at the electroweak scale and all dimensionless parameters of order one, leads to severe phenomenological problems. In particular, some of the physical scalars have flavor changing (and CP violating) couplings at tree level, violating bounds on rare processes such as !:l.mK and cK by several orders of magnitude. There are three possible solutions to these problems: (I) Natural flavor conservation (NFC) 188: only a single scalar doublet couples to each fermion sector. 2HDM where the same (a different) scalar doublet couples to the up and the down quarks are called type I (II). The absence of flavor changing and/or CP violating Yukawa interactions in this case is based on the same mechanism as within the Standard Model. (II) Approximate flavor symmetries (AFS) 189: it is quite likely that the smallness and hierarchy in the fermion masses and mixing angles are related to an approximate flavor symmetry, broken by a small parameter. If so, then it is unavoidable that the Yukawa couplings of all scalar doublets are affected by the selection rules related to the flavor symmetry. In such a case, couplings to the light generations and, in particular, off-diagonal couplings, are suppressed. (III) Heavy scalars 190: all dimensionful parameters that are not constrained by the requirement that the spontaneous breaking of GS M occurs at the electroweak scale are actually higher than this scale, ANP » AEW . Then all the new sources of flavor and CP violation in the scalar sector are suppressed by O(A~w/A~p). In table VII we summarize the implications·· of the various multi-scalar models for CP violation. Note that, if we impose NFC, spontaneous CP violation (SCPV) 187 is impossible in 2HDM 186 and (since the combination

222 Table VII. Multi Higgs Doublet Models. Framework

Model (Example)

SCPV

(i)

(ii)

NFC

MSSM

Excluded

Yes·

Yes·

No

AFS

Horizontal Sym.

Yes

Yes·

Yes

O(mq/mz)

Heavy

LRS

Yes

O(~)· A

O(~~W)

O(~~W )

NP

(iii)

NP

NP

of SCPV and NFC leads to 8KM = 0 191) is phenomenologically excluded in MHDM 192,193. Explicit models of spontaneous CP violation have been constructed within the frameworks of approximate NFC 194, approximate flavor symmetries 139,154 and heavy scalars 184. In the supersymmetric framework, one has to add at least two scalar singlets to allow for spontaneous CP violation 195. Entries marked with '*' mean that the number of scalar doublets should be larger than 2 (that is, the answer is 'No' in 2HDM).

10.2

(i) Charged Scalar Exchange

We investigate a multi Higgs doublet model (with n 2: 3 doublets) with NFC and assume that a different doublet couples to each of the the down, up and lepton sectors: - £y

= - -¢>i UViCKM Mdiagp d R ~

D + -¢>t UMdiagVi P D- ¢>tv-Ml PR f-o+h .c., u CKM L ~

~

(10.1) where PL,R = (1 =f "(5)/2. We denote the physical charged scalars by Ht (i = 1,2, ... , n - 1), and the would-be Goldstone boson (eaten by the W+) by H-::. We define K to be the matrix that rotates the charged scalars from the interaction- to the mass-eigenbasis. Then the Yukawa Lagrangian in the mass basis (for both fermions and scalars) is C1/2 n-1

£y

=

2{;4 2)HtU[Y;M~iagVcKMPL ,=1

+ X;VCKMM~iagpRlD + (10.2)

223

where X. - _ Kt1 •K*' n1

(10.3)

CP violation in the charged scalar sector comes from phases in K. CP violating effects are largest when the lightest charged scalar is much lighter than the heavier ones 196,197. Here we assume that all but the lightest charged scalar (Hi) effectively decouple from the fermions. Then, CP violating observables depend on three parameters:

~Im(X;Yn

~ ~ i=l

2 mHo

'

(10.4)

~

~ Im(YiZn ~ i=l

2

•

m Hi

Im(XY*) induces CP violation in the quarks sector, while Im(X Z*) and Im(Y Z*) give CP violation that is observable in semi-Ieptonic processes. There is an interesting question of whether charged scalar exchange could be the only source of CP violation. In other words, we would like to know whether a model of extended scalar sector with spontaneous CP violation and NFC is viable. In these models, 8KM = 0 and EK has to be accounted for by charged Higgs exchange. This requires very large long distance contributions. The CP violating coupling should fulfill 198,199 Im(XY*) 2: 0(40). However, the upper bounds on dN

192

and on BR(b -+ 8,)

Im(XY*)

~

0(1).

(10.5) 193

require (10.6)

We conclude that models of SCPV and NFC are excluded. It is, of course, still a viable possibility that CP is explicitly broken, in which case both quark and Higgs mixings provide CP violation. The bound (10.6) implies that the charged Higgs contribution to B - 13 mixing is numerically small and would modify the Standard Model predictions for CP asymmetries in B decays by no more than 0(0.02) 193. On the other hand, the contribution to d N can still be close to the experimental upper bound.

224

The lepton transverse polarization cannot be generated by vector or axialvector interactions only 200,201, so it is particularly suited for searching for CP violating scalar contributions. As triple-vector correlation is odd under time-reversal, the experimental observation of such correlation would signal T and - assuming CPT symmetry - CP violation. (It is possible to get nonvanishing T-odd observables even without CP violation (see e.g. 202). Such "fake" asymmetries can arise from final state interactions (FSI). They can be removed by comparing the measurements in two CP conjugate channels.) The muon transverse polarization in K -+ 1f J-t v decays and the tau transverse polarization in B -+ X TV are examples of such observables. The lepton transverse polarization, P l.., in semileptonic decays is defined as the lepton polarization component along the normal vector of the decay plane,

P _ Si· (Pi l.. -

X

PX)

Ipt xpxl

'

(10.7)

where Si is the lepton spin three-vector and Pi (Px) is the three-momentum of the lepton (hadron). Experimentally, it is useful to define the integrated CP violating asymmetry

r+ -r-

acp

== (Pl..) = r+ +r-'

(10.8)

where r+ (r-) is the rate of finding the lepton spin parallel (anti-parallel) to the normal vector of the decay plane. A non-zero acp can arise in our model from the interference between the W-mediated and the H+ -mediated tree diagrams. For example, in the semi-taonic bottom quark decay, the asymmetry is given by acp = Cps Im(~Z') and could be as large as 0.3 (see mH e.g. 203-205). 10.3

(ii) Effects of CP-even and CP-odd Scalar Mixing in Top Physics

It is possible that the neutral scalars are mixtures of CP-even and CP-odd scalar fields 187,206-209,196-197. Such a scalar couples to both scalar and pseudoscalar currents: (10.9)

where Hi is the physical Higgs boson and a{ ,b{ are functions of mixing angles in the matrix that diagonalizes the neutral scalar mass matrix. (Specifically, they are proportional to the components of, respectively, Reu and Imu in Hd CP violation in processes involving fermions is proportional to a{b{*. The natural place to look for manifestations of this type of CP violation is

225

top physics, where the large Yukawa couplings allow large asymmetries (see e.g. 210). Note that unlike our discussion above, the asymmetries here have nothing to do with FCNC processes. Actually, in models with NFC (even if softly broken 207), the effects discussed here contribute negligibly to CK and to CP asymmetries in B decays. On the other hand, two loop diagrams with intermediate neutral scalar and top quark can induce a CP violating three gluon operator 211,212 that would give d N close to the experimental bound 212-215.

10.4

(iii) Flavor Changing Neutral Scalar Exchange

Natural flavor conservation needs not be exact in models of extended scalar sector 194,207,216,217. In particular, it is quite likely that the existence of the additional scalars is related to flavor symmetries that explain the smallness and hierarchy in the Yukawa couplings. In this case, the new flavor changing couplings of these scalars are suppressed by the same selection rules as those that are responsible to the smallness of fermion masses and mixing, and there is no need to impose NFC 189,218-222,185. An explicit framework, with Abelian horizontal symmetries, was presented in 223,137,98. (For another related study, see 224.) We explain the general idea using these models. We emphasize that in this example the scalar with flavor changing couplings is a Standard Model singlet, and not an extra doublet, but the idea that these couplings are suppressed by approximate horizontal symmetries works in the same way. The simplest model of ref. 223 extends the SM by supersymmetry and by an Abelian horizontal symmetry 11 = U(l) (or ZN). The symmetry 11 is broken by a VEV of a single scalar S that is a singlet of the SM gauge group. Consequently, Yukawa couplings that violate 11 arise only from nonrenormalizable terms and are therefore suppressed. Explicitly, the quark Yukawa terms have the form (10.10) where M is some high energy scale and n1j is the horizontal charge of the combination Qiiij¢>q (in units of the charge of S). The terms (10.10) lead to quark masses and mixing as well as to flavor changing couplings, ZD' for the scalar S. The magnitude of the latter is then related to that of the effective Yukawa couplings ~j: (10.11)

226

Since the order of magnitude of each entry in the quark mass matrices is fixed in these models in terms of quark masses and mixing, the Z~ couplings can be estimated in terms of these physical parameters and the scale (8). For example, these couplings contribute to K - k mixing proportionally to (10.12)

With arbitrary phase factors in the various Z~ couplings, the contributions to neutral meson mixing are, in general, CP violating. In particular, there will be a contribution to CK from Im(Zt2Z~i). Requiring that the 8mediated tree level contribution does not exceed the experimental value of CK gives, for 0(1) phases, (10.13)

We learn that (for Ms ~ (8)) the mass of the 8-scalar could be as low as 1.5 TeV, some 4 orders of magnitude below the bound corresponding to 0(1) flavor changing couplings. The flavor changing couplings of the 8-scalar lead also to a tree level contribution to B - B mixing proportional to d d. Z13 Z 31

mdmb

~ (8)2 .

(10.14)

This means that, for phases of order 1, the neutral scalar exchange accounts for at most a few percent of B - fJ mixing. This cannot be signaled in 6.mB (because of the hadronic uncertainties in the calculation) but could be signalled (if (8) is at the lower bound) in CP asymmetries in BO decays. Finally, the contribution to D - fJ mixing, proportional to zu zu* 12 21

~

mumc (8)2 '

(1O.15)

is below a percent of the current experimental bound. This is unlikely to be discovered in near-future experiments, even if the new phases maximize the interference effects in the DO --* K-1f+ decay. To summarize, models with horizontal symmetries naturally suppress flavor changing couplings of extra scalars. There is no need to invoke NFC even for new scalars at the TeV scale. Furthermore, the magnitude of the flavor changing couplings is related to the observed fermion parameters. Typically, contributions from neutral scalars with flavor changing couplings could dominate CK. If they do, then a signal at the few percent level in CP asymmetries in neutral B decays is quite likely.

227

10.5

The Superweak Scenario

The original superweak scenario 225 stated that CP violation appears in a new f).S = 2 interaction while there is no CP violation in the SM f).S = 1 transitions. Consequently, the only large observable CP violating effect is eK, while e'/e ,. . ., 10- 8 and EDMs are negligibly small. At present, the term "superweak CP violation" has been used for many different types of models. There are several reasons for this situation: (i) The work of ref. 225 was concerned only with CP violation in K decays. In extending the idea to other mesons, one may interpret the idea in various ways. On one side, it is possible that the superweak interaction is significant only in K - K mixing and (apart from the relaxation of the e K- bounds on the CKM parameters) has no effect on mixing of heavier mesons. On the other extreme, one may take the superweak scenario to imply that CP violation comes from f).F = 2 processes only for all mesons. The most common use of the term 'superweak' refers to the latter option, namely that there is no direct CP violation. (ii) The scenario proposed in 225 did not employ any specific model. It was actually proposed even before the formulation of the Standard Model. To extend the idea to, for example, the neutral B system, a model is required. Various models give very different predictions for CP asymmetries in B decays. If one extends the superweak scenario to the B system by assuming that there is CP violation in f).B = 2 but not in f).B = 1 transitions, the prediction for CP asymmetries in B decays into final CP eigenstates is that they are equal for all final states 226 - 228. Whether these asymmetries are all small or could be large is model dependent. In addition, the asymmetries in charged B decays vanish. CP violation via neutral scalar exchange is the most commonly studied realization of the superweak idea. In particular, if the complex couplings presented in the previous section were the only source of CP violation, then this model would be superweak. The smallness of the couplings would make the contribution from neutral Higgs mediated diagrams negligible compared to the Standard Model diagrams in f).S = 1 processes, but the fact that they contribute to mixing at tree level would allow them to dominate the f).S = 2 processes. Various models (or scenarios) that realize the main features of the superweak idea can be found in refs. 229-230,194,216. As mentioned above, there is a considerable variation in their predictions for e'/e, dN and other quantities. If we take the term 'superweak CP violation' to imply that there is only indirect CP violation, or at least that there is no direct CP violation in K decays, then e' Ie :j:. 0 is inconsistent with this scenario which

Z'0

Z'0

228

is therefore excluded.

11

Extensions of the Fermion Sector: Down Singlet Quarks

The fermion sector of the Standard Model is described in eq. (2.2). It can be extended by either a fourth, sequential generation or by non-sequential fermions, namely 'exotic' representations, different from those of (2.2). (The four generation model can only be viable if it is further extended to evade bounds related to the neutrino sector 231 and to electroweak precision data 26.) Our discussion in this chapter is focused on non-sequential fermions and their implications on CP asymmetries in neutral B decays and on KL -+ 7fviJ.

11.1

The Theoretical Framework

We consider a model with extra quarks in vector-like representations of the Standard Model gauge group, (11.1) Such (three pairs of) quark representations appear, for example, in E6 GUTs. The mass matrix in the down sector, M d , is now 4 x 4. (Note that the Mti entries do not violate GS M and are, therefore, bare mass terms.) How many independent CP violating parameters are there in Md and MU? Since Md (MU) is 4 x 4 (3 x 3) and complex, there are 25 real and 25 imaginary parameters in these matrices. If we switch off the mass matrices, there is a global symmetry added to the model, (11.2) One can remove, at most, 12 real and 23 imaginary parameters. However, the model with the quark mass matrices switched on has still a global symmetry of U(l)B' so one of the imaginary parameters cannot be removed. We conclude that there are 16 flavor parameters: 13 real ones, that is seven masses and six mixing angles, and 3 phases. These three phases are independent sources of CP violation.

11.2 Z-Mediated FGNG The most important feature of this model for our purposes is. that it allows CP violating Z-mediated Flavor Changing Neutral Currents (FCNC). To understand how these FCNC arise, it is convenient to work in a basis where the up sector interaction eigenstates are identified with the mass eigenstates. The

229 down sector interaction eigenstates are then related to the mass eigenstates by a 4 x 4 unitary matrix K. Charged current interactions are described by

£~

~(W:VijUiL"/LdjL + h.c.).

=

(11.3)

The charged current mixing matrix V is a 3 x 4 sub-matrix of K:

Vij = Kij for i = 1,2,3; j = 1,2,3,4.

(11.4)

The V matrix is parameterized, as anticipated above, by six real angles and three phases, instead of three angles and one phase in the original CKM matrix. All three phases may affect CP asymmetries in BO decays. Neutral current interactions are described by

rZ _

Lint -

9

--()-

cos

W

ZIL (J1L3

-

. 2 () W JIL ) sm EM , (11.5)

The neutral current mixing matrix for the down sector is U = vtV. As V is not unitary, U::j:. 1. In particular, its non-diagonal elements do not vanish:

Upq =

-K;pK4q

for p::j:. q.

( 11.6)

The three elements which are most relevant to our study are

= V:dVus + Vc'dVcs + l't'dVis, Udb = V:dVub + Vc'dVcb + l't'dVib,

Uds

(11. 7)

The fact that, in contrast to the Standard Model, the various Upq do not necessarily vanish, allows FCNC at tree level. This may substantially modify the predictions for CP asymmetries. The flavor changing couplings of the Z contribute to various FCNC processes. Relevant constraints arise from semileptonic FCNC B decays:

r(B -+ f+f- X)z _ [(1/2 _ . 2 ())2 reB -+ f+vX) sm w

. 4 ()

+ sm w

]

\Udb \2 + \Usb \2 (11.8) IVub\2 + FpsIVcbl2'

where Fps '" 0.5 is a phase space factor. The experimental upper bound on

reB -+ f+ f- X) gives

I~:: I ~ 0.04, I~:: I ~ 0.04.

(11.9)

230

Additional constraints come from neutral B mixing: (

Using v'BBIB

A

umB

)

Z

:G 0.16 GeV,

db 12 .

=V2GFBBI'JJmB'fJBIU

3

(11.10)

we get

IUdbl

:5 9 x 10- 4 .

(11.11)

As concerns ~mB.' only lower bounds exist and consequently there is no analog bound on IUsbl. Bounds on Uds can be derived from the measurements of BR(KL -t J.L+J.L-) , BR(K+ -t 7T+YV), eK and e'/e yielding, respectively (for recent derivations , see 160,54) ,

IRe(Uds)1

:5

10- 5,

5 IUdsl < 3 x 10- ,

IRe(Uds) Im(Uds) I

:5

1.3 x 10- 9,

IIm(Uds )I

:5

10- 5.

(11.12)

(Note that the combination of bounds from BR(KL -t J.L+J.L-) and from the recently improved e'/e is stronger than the bounds from BR(K+ -t 7T+YV) and eK 54. The latter bounds are, however, subject to smaller hadronic uncertainties. )

11.3

CP Asymmetries in B Decays

The most interesting effects in this model concern CP asymmetries in neutral B decays into final CP eigenstates232-239,15. We describe these effects in detail as they illustrate the type of new ingredients that are likely to affect CP asymmetries in neutral B decays and the way in which the SM predictions might be modified. (If there exist light up quarks in exotic representations, they may introduce similar, interesting effects in neutral D decays 69.) If the Uqb elements are not much smaller than the bounds (11.9) and (11.11), they will affect several aspects of physics related to CP asymmetries in B decays. (i) Neutral B mixing: The experimentally measured value of ~mBd (and the lower bound on ~mB.) can be explained by Standard Model processes, namely box diagrams with intermediate top quarks. Still, the uncertainties in the theoretical calculations, such as the values of IB and Vid (and the absence of an upper bound on ~mBJ allow a situation where 8M processes do not

231

give the dominant contributions to either or both of Cl.mBd and Cl.mBs 14. The ratio between the Z-mediated tree diagram and the Standard Model box diagram is given by (q = d, s)

(Cl.mB.}tree (Cl.mBq)box

=

2

2V27r 1~12 GFmrvSo(Xt) Vtq~~

~ 150 1~12 < Vtq~~

'"

{5 0.25

q =d . q= s

(11.13) (The last inequality is derived under the assumption that the violation of CKM unitarity is not strong. The bound on (Cl.mBJtree/(Cl.mBJbox is higher if IVtd~~1 < 0.005 holds.) From (11.9) and (11.13) we learn that the Zmediated tree diagram could give the dominant contribution to Cl.mBd but at most 25% of Cl.mB s . (ii) Unitarity of the 3 x 3 CKM matrix: Within the SM, unitarity of the three generation CKM matrix gives:

+ Vcd Vcs + ~d Vts = 0, == V':d Vub + Vcd Vcb + l't'd Vtb = 0, == V':s Vub + Vc: Vcb + l't: vtb = O.

Uds == V':dVus Udb Usb

(11.14)

Eq. (11.7), however, implies that now (11.14) is replaced by (11.15) A measure of the violation of (11.14) is given by

Udb * I :5 0.18, Ivtd~b

Usb

IVts~~

I :5

0.04.

(11.16)

The bound on IUdb/(Vtd~~)1 is even weaker if IVtdl is lower than the three generation unitarity bound. We learn that the first of the SM relations in (11.14) is practically maintained, while the third can be violated by at most 4%. However, the Udb = 0 constraint may be violated by 0(0.2) effects. The Standard Model unitarity triangle should be replaced by a unitarity quadrangle. After the recent measurement of a,pKs 13, not only the magnitude of Udb but also the phases Q and j3, Q

== arg (

VUdV*b)

U;: '

_ arg (U;b) {J- = V V* ,

(11.17)

cd cb

are constrained 15, but the constraints are not very strong. (iii) Z-mediated B decays: Our main interest in this chapter is in hadronic BO decays to CP eigenstates, where the quark sub-process is b -+ UiUidj,

232 with Ui = U, c and dj = d, s. These decays get new contributions from Zmediated tree diagrams, in addition to the standard W -mediated ones. The ratio between the amplitudes is

Az - = [ -1 - -2. sm 2 Ow ] Aw 2 3

I-Uib - *I' V;jV:b

(11.18)

We find that the Z contributions can be safely neglected in b -t ccs ( ,$ 0.013) and b -t ccd ( ,$ 0.03). On the other hand, it may be significant in b -t uud ( ,$ 0.12), and in processes with no SM tree contributions, e.g. b -t sss, that may have comparable contributions from penguin and Z-mediated tree diagrams. (iv) New contributions to r12(Bq): The difference in width comes from modes that are common to Bq and Bq. As discussed above, there are new contributions to such modes from Z-mediated FCNC. However, while the new contributions to M12 are from tree level diagrams, i.e. 0(g2), those to r 12 are still coming from a box-diagram, i.e. 0(g4). Consequently, no significant enhancement of the SM value of r 12 is expected, and the relation r 12 « M12 is maintained. (The new contribution could significantly modify the leptonic asymmetry in neutral B decays 240,15 though the asymmetry remains small.) The fact that M 12 (BO) could be dominated by the Z-mediated FCNC together with the fact that this new amplitude depends on new CP violating phases means that large deviations from the Standard Model predictions for CP asymmetries are possible. As r 12 « M12 is maintained, future measurements of certain modes will still be subject to a clean theoretical interpretation in terms of the extended electroweak sector parameters. Let us assume that, indeed, M12 is dominated by the new physics. (Generalization to the case that the new contribution is comparable to (but not necessarily dominant over) the Standard Model one is straightforward 234,238.) Then

(qp)

B

~

Udb Udb'

(11.19)

We argued above that b -t ccs is still dominated by the W mediated diagram. Furthermore, the first unitarity constraint in (11.14) is practically maintained. Then it is straightforward to evaluate the CP asymmetry in B -t 'ljJKs. We find that it simply measures an angle of the unitarity quadrangle: acp(B -t 'ljJKs) = -sin2,B.

(11.20)

The new contribution to b -t ccd is 0(3%), which is somewhat smaller than the SM penguins. So we still have, to a good approximation, (taking into

233

account CP-parities) acp(B -t 'ljJKs)

~

-acp(B -t DD).

(11.21)

Care has to be taken regarding b -t uud decays. Here, direct CP violation may be large 237 and prevent a clean theoretical interpretation of the asymmetry. Only if the asymmetry is large, so that the shift from the Z-mediated contribution to the decay is small, we get acp(B -t 1m) = - sin2a.

(11.22)

The important point about the modification of the SM predictions is then not that the angles a, (J and 'Y may have very different values from those predicted by the SM, but rather that the CP asymmetries do not measure these angles anymore. As the experimental constraints on a and fJ are still rather weak 15, a large range is possible for each of the asymmetries. This model demonstrates that there exist extensions of the SM where dramatic deviations from its predictions for CP asymmetries in B decays are not unlikely. Another interesting point concerns Bs decays. If Bs - Bs mixing as well as the b -t ccs decay are dominated by the SM diagrams, we have, similar to the 8M, (11.23) As shown in ref. 20, this is a sufficient condition for the angles extracted from B -t 'ljJKs, B -t 7r7r and the relative phase between the Bs - Bs mixing amplitude and the b -t uud decay amplitude (if it can be deduced from experiment) to sum up to 7r (up to possible effects of direct CP violation). This happens in spite of the fact that the first two asymmetries do not correspond to (J and a of the unitarity triangle. 11.4

The K L -t 7rvil Decay

In chapter 7 we argued that the only potentially significant new contribution to a1rVV can come from the decay amplitude. Z-mediated FCNC provide an explicit example of New Physics that may modify the 8M prediction for a 1rVV of eq. (4.38). Assuming that the Z-mediated tree diagram dominates K -t 7rvil, we get 232 (11.24) Bounds on the relevant couplings were given in eq. (11.12) above. We learn that large effects are possible. When IRe(Uds)1 and IIm(Uds)1 are close to their upper bounds, the branching ratios BR(K+ -t 7r+viI) and

234 a1l"1IV = 0(1). Furthermore, as in this case the SM contribution is small, the measurement of BR(K+ --+ 7r+viJ) approximately determines IUdsl, and with the additional measurement of BR(KL --+ 7r°viJ), arg(Uds) is approximately determined as well.

BR(KL --+ 7r°viJ) are both 0(10- 10 ) and

12

Conclusions

Experiments have not yet probed in a significant way the mechanism of CP violation. There is a large number of open questions concerning CP violation. Here are some examples: • Why are the measured parameters, CK and c~, small? The answer in the Standard Model is that CP violation is screened in processes that are dominated by the first two quark generations by small mixing angles. We have seen examples of new physics, that is supersymmetry with approximate CP, where the reason is the smallness of all CP violating phases. Observing CP asymmetries of order one, as expected in processes that involve the first and third generations such as B --+ 'ljJKs, K --+ 7rviJ or even in charged B decays, B± --+ K±7r°, will exclude the approximate CP scenario. • What is the number of independent CP violating phases? The answer in the Standard Model is one, the Kobayashi-Maskawa phase. We have encountered models with a larger number, e.g. forty four in the supersymmetric standard model. If the pattern predicted by the Standard Model, e.g. small CP asymmetries in B s --+ 'IjJ¢ and in D --+ K 7r, a strong correlation between CP violation in B --+ 'ljJKs and in KL --+ 7rviJ, equal asymmetries in B --+ 'ljJKs and in B --+ ¢Ks, etc., is inconsistent with measurements, then probably there are several independent phases. • Why is CP violated? The answer in the Standard Model is explicit breaking by complex Yukawa couplings. In left-right-symmetric models, the Lagrangian can be CP symmetric and the breaking is spontaneous. It will be difficult to answer this question by experimental measurements, unless the correlations predicted by a specific model of spontaneous CP violation will be experimentally confirmed. • Is CP violation restricted to flavor changing interactions? This is indeed the case in the Standard Model. But in many of its extensions, such as supersymmetry, there is flavor diagonal CP violation. Observation of an electric dipole moment or of CP violation in tt production will provide strong hints for flavor diagonal CP violation.

235

• Is CP violation restricted to quark interactions? This is the case in the Standard Model but not if neutrinos have masses. Observation of CP asymmetries in neutrino oscillation experiments will be a direct evidence of CP violation in the lepton sector . • Is CP violation restricted to the weak interactions? In the Standard Model, CP violation appears in charged current (that is, W-mediated) weak interactions only. In multi-scalar models, it appears in scalar interactions. In supersymmetry, it appears in strong interactions. Observation of transverse lepton polarization in meson decays will provide evidence for CP violation in interactions that are not mediated by vector bosons. There are more questions that we can ask and answers that we will learn in the near future. But the list above is enough to demonstrate how unique the Standard Model picture of CP violation is, how sensitive is CP violation to new physics, and how important are present and future experiments that will search for CP violation. Acknowledgements I thank Gabriela Barenboim, Sven Bergmann, Galit Eyal, Yuval Grossman, Stephane Plaszczynski, Helen Quinn, Marie-Helene Schune and Joao Silva for useful discussions. Y.N. is supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities, and by the Minerva Foundation (Munich).

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244

INTRODUCTION TO B PHYSICS MATTHIAS NEUBERT Newman Laboratory of Nuclear Studies Cornell University, Ithaca, NY 14853, USA These lectures provide an introduction to various topics in heavy-flavor physics. We review the theory and phenomenology of heavy-quark symmetry, exclusive weak decays of B mesons, inclusive decay rates, and some rare B decays.

1

Introduction

The rich phenomenology of weak decays has always been a source of information about the nature of elementary particle interactions. A long time ago, {3and JL-decay experiments revealed the structure of the effective flavor-changing interactions at low momentum transfer. Today, weak decays of hadrons containing heavy quarks are employed for tests of the Standard Model and measurements of its parameters. In particular, they offer the most direct way to determine the weak mixing angles, to test the unitarity of the CabibboKobayashi-Maskawa (CKM) matrix, and to explore the physics of CP violation. Hopefully, this will provide some hints about New Physics beyond the Standard Model. On the other hand, hadronic weak decays also serve as a probe of that part of strong-interaction phenomenology which is least understood: the confinement of quarks and gluons inside hadrons. The structure of weak interactions in the Standard Model is rather simple. Flavor-changing decays are mediated by the coupling of the charged current Jbe to the W-boson field:

Lee = - ~ Jbe W~

+ h.c.,

(1)

where

J~c = (v

e , v"

v,) Y'

(

~) + (UL, "",[Lh' V

CKM

Gn

(2)

contains the left-handed lepton and quark fields, and (3)

245

is the CKM matrix. At low energies, the charged-current interaction gives rise to local four-fermion couplings of the form (4)

where

g2

GF -

- 4v'2Mfv

= 1.16639(2) GeV- 2

(5)

is the Fermi constant.

B_[> ___ and the heavy-quark effective theory 7 -17, which provide the modern theoretical framework for the description of the properties and decays of hadrons containing a heavy quark. For a more detailed description of this ,subject, the reader is referred to the review articles in Refs. 18-24.

2.1

The Physical Picture

There are several reasons why the strong interactions of hadrons containing heavy quarks are easier to understand than those of hadrons containing only light quarks. The first is asymptotic freedom, the fact that the effective coupling constant of QCD becomes weak in processes with a large momentum transfer, corresponding to interactions at short distance scales 25,26. At large distances, on the other hand, the coupling becomes strong, leading to nonperturbative phenomena such as the confinement of quarks and gluons on a length scale Rhad ""' 1/ A QCD ""' 1 fm, which determines the size of hadrons. Roughly speaking, AQCD ""' 0.2 GeV is the energy scale that separates the regions of large and small coupling constant. When the mass of a quark Q is much larger than this scale, mQ » AQCD, it is called a heavy quark. The

248

quarks of the Standard Model fall naturally into two classes: up, down and strange are light quarks, whereas charm, bottom and top are heavy quarks~ For heavy quarks, the effective coupling constant O:s(mQ) is small, implying that on length scales comparable to the Compton wavelength AQ '" l/mQ the strong interactions are perturbative and much like the electromagnetic interactions. In fact, the quarkonium systems (QQ), whose size is of order AQ/O:s(mQ) « Rhad, are very much hydrogen-like. Systems composed of a heavy quark and other light constituents are more complicated. The size of such systems is determined by Rhad, and the typical momenta exchanged between the heavy and light constituents are of order AQCD. The heavy quark is surrounded by a complicated, strongly interacting cloud of light quarks, antiquarks and gluons. In this case it is the fact that AQ « Rhad, i.e. that the Compton wavelength of the heavy quark is much smaller than the size of the hadron, which leads to simplifications. To resolve the quantum numbers of the heavy quark would require a hard probe; the soft gluons exchanged between the heavy quark and the light constituents can only resolve distances much larger than AQ. Therefore, the light degrees of freedom are blind to the flavor (mass) and spin orientation of the heavy quark. They experience only its color field, which extends over large distances because of confinement. In the rest frame of the heavy quark, it is in fact only the electric color field that is important; relativistic effects such as color magnetism vanish as mQ -+ 00. Since the heavy-quark spin participates in interactions only through such relativistic effects, it decouples. It follows that, in the limit mQ -+ 00, hadronic systems which differ only in the flavor or spin quantum numbers of the heavy quark have the same configuration of their light degrees of freedom 1-{;. Although this observation still does not allow us to calculate what this configuration is, it provides relations between the properties of such particles as the heavy mesons B, D, B* and D* , or the heavy baryons Ab and Ac (to the extent that corrections to the infinite quark-mass limit are small in these systems). These relations result from some approximate symmetries of the effective strong interactions of heavy quarks at low energies. The configuration of light degrees of freedom in a hadron containing a single heavy quark with velocity v does not change if this quark is replaced by another heavy quark with different flavor or spin, but with the same velocity. Both heavy quarks lead to the same static color field. For Nh heavy-quark flavors, there is thus an SU(2Nh ) spin-flavor symmetry group, under which the effective strong interactions are invariant. These symmetries are in close correspondence to familiar properties of atoms. The flavor symmetry a Ironically, the top quark is of no relevance to our discussion here, since it is too heavy to form hadronic bound states before it decays.

249

is analogous to the fact that different isotopes have the same chemistry, since to good approximation the wave function of the electrons is independent of the mass of the nucleus. The electrons only see the total nuclear charge. The spin symmetry is analogous to the fact that the hyperfine levels in atoms are nearly degenerate. The nuclear spin decouples in the limit me/mN -t O. Heavy-quark symmetry is an approximate symmetry, and corrections arise since the quark masses are not infinite. In many respects, it is complementary to chiral symmetry, which arises in the opposite limit of small quark masses. There is an important distinction, however. Whereas chiral symmetry is a symmetry of the QeD Lagrangian in the limit of vanishing quark masses, heavy-quark symmetry is not a symmetry of the Lagrangian (not even an approximate one), but rather a symmetry of an effective theory that is a good approximation to QeD in a certain kinematic region. It is realized only in systems in which a heavy quark interacts predominantly by the exchange of soft gluons. In such systems the heavy quark is almost on-shell; its momentum fluctuates around the mass shell by an amount of order AQCD . The corresponding fluctuations in the velocity of the heavy quark vanish as AQco/mQ -t O. The velocity becomes a conserved quantity and is no longer a dynamical degree of freedom 14. Nevertheless, results derived on the basis of heavy-quark symmetry are model-independent consequences of QeD in a well-defined limit. The symmetry-breaking corrections can be studied in a systematic way. To this end, it is however necessary to cast the QeD Lagrangian for a heavy quark, (6)

into a form suitable for taking the limit mQ -t

00.

2.2 Heavy-Quark Effective Theory The effects of a very heavy particle often become irrelevant at low energies. It is then useful to construct a low-energy effective theory, in which this heavy particle no longer appears. Eventually, this effective theory will be easier to deal with than the full theory. A familiar example is Fermi's theory of the weak interactions. For the description of the weak decays of hadrons, the weak interactions can be approximated by point-like four-fermion couplings, governed by a dimensionful coupling constant GF [cf. (4)]. The effects of the intermediate W bosons can only be resolved at energies much larger than the hadron masses. The process of removing the degrees of freedom of a heavy particle involves the following steps 27 -29: one first identifies the heavy-particle fields and "integrates them out" in the generating functional of the Green functions

250 of the theory. This is possible since at low energies the heavy particle does not appear as an external state. However, whereas the action of the full theory is usually a local one, what results after this first step is a non-local effective action. The non-locality is related to the fact that in the full theory the heavy particle with mass M can appear in virtual processes and propagate over a short but finite distance ~x '" 1/M. Thus, a second step is required to obtain a local effective Lagrangian: the non-local effective action is rewritten as an infinite series of local terms in an Operator Product Expansion (OPE) 30,31. Roughly speaking, this corresponds to an expansion in powers of 1/M. It is in this step that the short- and long-distance physics is disentangled. The long-distance physics corresponds to interactions at low energies and is the same in the full and the effective theory. But short-distance effects arising from quantum corrections involving large virtual momenta (of order M) are not described correctly in the effective theory once the heavy particle has been integrated out. In a third step, they have to be added in a perturbative way using renormalization-group techniques. These short-distance effects lead to a renormalization of the coefficients of the local operators in the effective Lagrangian. An example is the effective Lagrangian for non-Ieptonic weak decays, in which radiative corrections from hard gluons with virtual momenta in the range between mw and some low renormalization scale J.L give rise to Wilson coefficients, which renormalize the local four-fermion interactions 32-34. The heavy-quark effective theory (HQET) is constructed to provide a simplified description of processes where a heavy quark interacts with light degrees of freedom predominantly by the exchange of soft gluons 7 -17. Clearly, mQ is the high-energy scale in this case, and AQCD is the scale of the hadronic physics we are interested in. The situation is illustrated in Fig. 3. At short distances, i.e. for energy scales larger than the heavy-quark mass, the physics is perturbative and described by conventional QCD. For mass scales much below the heavy-quark mass, the physics is complicated and non-perturbative because of confinement. Our goal is to obtain a simplified description in this region using an effective field theory. To separate short- and long-distance effects, we introduce a separation scale J.L such that A QCD « J.L «mQ. The HQET will be constructed in such a way that it is equivalent to QCD in the long-distance region, i.e. for scales below J.L. In the short-distance region, the effective theory is incomplete, since some high-momentum modes have been integrated out from the full theory. The fact that the physics must be independent of the arbitrary scale J.L allows us to derive renormalization-group equations, which can be employed to deal with the short-distance effects in an efficient way. Compared with most effective theories, in which the degrees of freedom of a heavy particle are removed completely from the low-energy theory, the

251

EWtheory

short-distance physics

QeD

perturbation theory + RGE

!= I"

few

x

AQCD long-distance physics

HQET A QCD

J

non-perturbaril'e techniques

Figure 3: Philosophy of the heavy-quark effective theory.

HQET is special in that its purpose is to describe the properties and decays of hadrons which do contain a heavy quark. Hence, it is not possible to remove the heavy quark completely from the effective theory. What is possible is to integrate out the "small components" in the full heavy-quark spinor, which describe the fluctuations around the mass shell. The starting point in the construction of the HQET is the observation that a heavy quark bound inside a hadron moves more or less with the hadron's velocity v and is almost on-shell. Its momentum can be written as (7)

where the components of the so-called residual momentum k are much smaller than mQ. Note that v is a four-velocity, so that v 2 = 1. Interactions of the heavy quark with light degrees of freedom change the residual momentum by an amount of order 6.k '" AQCD , but the corresponding changes in the heavyquark velocity vanish as AQCD/mQ -+ O. In this situation, it is appropriate to introduce large- and small-component fields, hv and H v , by

(8)

252 where P+ and P_ are projection operators defined as

(9) It follows that

(10) Because of the projection operators, the new fields satisfy phv = hv and pH v = -Hv. In the rest frame, i.e. for vI-' = (1,0,0,0), hv corresponds to the upper two components of Q, while Hv corresponds to the lower ones. Whereas hv annihilates a heavy quark with velocity v, Hv creates a heavy antiquark with velocity v. In terms of the new fields, the QeD Lagrangian (6) for a heavy quark takes the form

where Di = DI-' - vI-' V· D is orthogonal to the heavy-quark velocity: V· D..L = 0. In the rest frame, Di = (0, jj) contains the spatial components of the covariant derivative. From (11), it is apparent that hv describes massless degrees of freedom, whereas Hv corresponds to fluctuations with twice the heavy-quark mass. These are the heavy degrees of freedom that will be eliminated in the construction of the effective theory. The fields are mixed by the presence of the third and fourth terms, which describe pair creation or annihilation of heavy quarks and antiquarks. As shown in the first diagram in Fig. 4, in a virtual process, a heavy quark propagating forward in time can turn into an antiquark propagating backward in time, and then turn back into a quark. The energy of the intermediate quantum state hhH is larger than the energy of the incoming heavy quark by at least 2mQ. Because of this large energy gap, the virtual quantum fluctuation can only propagate over a short distance A.x '" l/mQ. On hadronic scales set by Rhad = 1/ AQCD, the process essentially looks like a local interaction of the form kv iJfJ..L - 12 iJfJ ..Lhv , mQ

(12)

where we have simply replaced the propagator for Hv by 1/2mQ. A more correct treatment is to integrate out the small-component field H v , thereby deriving a non-local effective action for the large-component field hv, which can then be expanded in terms of local operators. Before doing this, let us mention a second type of virtual corrections involving pair creation, namely heavy-quark loops. An example is shown in the second diagram in Fig. 4.

253

Heavy-quark loops cannot be described in terms of the effective fields hv and H v , since the quark velocities inside a loop are not conserved and are in no way related to hadron velocities. However, such short-distance processes are proportional to the small coupling constant O:s (mQ) and can be calculated in perturbation theory. They lead to corrections that are added onto the lowenergy effective theory in the renormalization procedure.

z Figure 4: Virtual fluctuations involving pair creation of heavy quarks. Time flows to the right.

On a classical level, the heavy degrees of freedom represented by Hv can be eliminated using the equation of motion. Taking the variation of the Lagrangian with respect to the field H v , we obtain

(iv·D

+ 2mQ) Hv = iIjJ.Lhv.

(13)

This equation can formally be solved to give

Hv = 2

mQ

~.W· D iIjJ .Lhv ,

(14)

showing that the small-component field Hv is indeed of order 1/mQ. We can now insert this solution into (11) to obtain the "non-local effective Lagrangian" Ceff = hv iv·D hv

-

+ hv iIjJ.L

1 2 . D iIjJ .Lhv· mQ+w·

(15)

Clearly, the second term corresponds to the first class of virtual processes shown in Fig. 4. It is possible to derive this Lagrangian in a more elegant way by manipulating the generating functional for QCD Green functions containing heavy-quark fields 17. To this end, one starts from the field redefinition (10) and couples the large-component fields hv to external sources Pv. Green functions with an arbitrary number of hv fields can be constructed by taking derivatives with respect to Pv. No sources are needed for the heavy degrees of freedom represented by Hv. The functional integral over these fields is Gaussian and can be performed explicitly, leading to the effective action

(16)

254

with Ceff as given in (15). The appearance of the logarithm of the determinant

~ = exp (~Tr In [2mQ + iv·D -

(17)

i1J])

is a quantum effect not present in the classical derivation presented above. However, in this case the determinant can be regulated in a gauge-invariant way, and by choosing the gauge v . A = 0 one can show that In ~ is just an irrelevant constant 17,35. Because of the phase factor in (10), the x dependence of the effective heavyquark field hv is weak. In momentum space, derivatives acting on hv produce powers of the residual momentum k, which is much smaller than mQ. Hence, the non-local effective Lagrangian (15) allows for a derivative expansion:

Ceff = ltv iv·D hv

+ - 12 mQ

f: ltv il/h (- iv· D) n iIfJ.l.hv.

(18)

2mQ

n=O

Taking into account that hv contains a P+ projection operator, and using the identity

P+ il/h il/hP+ = P+ [(iD.l.)2

+ ~CTp,v GlJv]

P+,

(19)

where i[DIJ, DV] = 9s GIJV is the gluon field-strength tensor, one finds that 12,16

In the limit mQ -+

00,

only the first term remains:

(21) This is the effective Lagrangian of the HQET. It gives rise to the Feynman rules shown in Fig. 5. Let us take a moment to study the symmetries of this Lagrangian 14. Since there appear no Dirac matrices, interactions of the heavy quark with gluons leave its spin unchanged. Associated with this is an SU(2) symmetry group, under which Coc> is invariant. The action of this symmetry on the heavy-quark fields becomes most transparent in the rest frame, where the generators Si of SU(2) can be chosen as (22)

255

j

_i_l+Fo .. v· k

2

JZ

Figure 5: Feynman rules of the HQET (i,j and a are color indices). A heavy quark with velocity v is represented by a double line. The residual momentum k is defined in (7).

Here a i are the Pauli matrices. An infinitesimal SU(2) transformation hv -+ (1 + if· 5) hv leaves the Lagrangian invariant:

6.c

00

= hv [iv·D, if'·

5] hv =

o.

(23)

Another symmetry of the HQET arises since the mass of the heavy quark does not appear in the effective Lagrangian. For N h heavy quarks moving at the same velocity, eq. (21) can be extended by writing Nh

.coo =

L h~ iv·D h~.

(24)

i=l

This is invariant under rotations in flavor space. When combined with the spin symmetry, the symmetry group is promoted to SU(2Nh). This is the heavy-quark spin-flavor symmetry 6,14. Its physical content is that, in the limit mQ -+ 00, the strong interactions of a heavy quark become independent of its mass and spin. Consider now the operators appearing at order l/mQ in the effective Lagrangian (20). They are easiest to identify in the rest frame. The first operator, 1 - . 2 1 .~2 Okin = -2- hv (zDJ...) hv -+ - - - hv (zD) hv, mQ 2mQ

(25)

is the gauge-covariant extension of the kinetic energy arising from the residual motion of the heavy quark. The second operator is the non-Abelian analogue of the Pauli interaction, which describes the color-magnetic coupling of the heavy-quark spin to the gluon field:

gs v gs - ~ ~ Omag = -4- hv ap'v Cp. hv -+ - - - hv S·Bc hv· mQ

mQ

(26)

256

Here S is the spin operator defined in (22), and B~ = _!€ijkGjk are the components of the color-magnetic field. The chromo-magnetic interaction is a relativistic effect, which scales like 1/mQ. This is the origin of the heavy-quark spin symmetry.

2.3

The Residual Mass Term and the Definition of the Heavy-Quark Mass

The choice of the expansion parameter in the HQET, i.e. the definition of the heavy-quark mass mQ, deserves some comments. In the derivation presented earlier in this section, we chose mQ to be the "mass in the Lagrangian" , and using this parameter in the phase redefinition in (10) we obtained the effective Lagrangian (21), in which the heavy-quark mass no longer appears. However, this treatment has its subtleties. The symmetries of the HQET allow a "residual mass" 8m for the heavy quark, provided that 8m is of order AQCD and is the same for all heavy-quark flavors. Even if we arrange that such a mass term is not present at the tree level, it will in general be induced by quantum corrections. (This is unavoidable if the theory is regulated with a dimensionful cutoff.) Therefore, instead of (21) we should write the effective Lagrangian in the more general form 36 (27)

+ Ll.m, the residual mass changes in the opposite way: 8m --+ 8m - Ll.m. This implies that there is a unique choice of the expansion parameter mQ such that 8m = O. Requiring 8m = 0, as it is usually done implicitly in the HQET, defines a heavyquark mass, which in perturbation theory coincides with the pole mass 37 . This, in turn, defines for each heavy hadron HQ a parameter A (sometimes called the "binding energy") through

If we redefine the expansion parameter according to mQ --+ mQ

(28) If one prefers to work with another choice of the expansion parameter, the

values of non-perturbative parameters such as A change, but at the same time one has to include the residual mass term in the HQET Lagrangian. It can be shown that the various parameters depending on the definition of mQ enter the predictions for physical quantities in such a way that the results are independent of the particular choice adopted 36 • There is one more subtlety hidden in the above discussion. The quantities mQ, A and 8m are non-perturbative parameters of the HQET, which have a similar status as the vacuum condensates in QeD phenomenology 38. These

257

parameters cannot be defined unambiguously in perturbation theory. The reason lies in the divergent behavior of perturbative expansions in large orders, which is associated with the existence of singularities along the real axis in the Borel plane, the so-called renormalons 39-47. For instance, the perturbation series which relates the pole mass mQ of a heavy quark to its bare mass, mQ

= m~are {I + Cl as(mQ) + C2 a;(mQ) + ... + Cn a~(mQ) + ... },

(29)

contains numerical coefficients C n that grow as n! for large n, rendering the series divergent and not Borel summable 48,49. The best one can achieve is to truncate the perturbation series at its minimal term, but this leads to an unavoidable arbitrariness of order 6.mQ '" AQCD (the size of the minimal term) in the value of the pole mass. This observation, which at first sight seems a serious problem for QeD phenomenology, should not come as a surprise. We know that because of confinement quarks do not appear as physical states in nature. Hence, there is no unique way to define their on-shell properties such as a pole mass. Remarkably, QeD perturbation theory "knows" about its incompleteness and indicates, through the appearance of renormalon singularities, the presence of non-perturbative effects. One must first specify a scheme how to truncate the QeD perturbation series before non-perturbative statements such as 15m = 0 become meaningful, and hence before non-perturbative parameters such as mQ and A become well-defined quantities. The actual values of these parameters will depend on this scheme. We stress that the "renormalon ambiguities" are not a conceptual problem for the heavy-quark expansion. In fact, it can be shown quite generally that these ambiguities cancel in all predictions for physical observables 50--52. The way the cancellations occur is intricate, however. The generic structure of the heavy-quark expansion for an observable is of the form: Observable", C[as(mQ)]

(1 +

:Q

+ ... ) ,

(30)

where C[as(mQ)] represents a perturbative coefficient function, and A is a dimensionful non-perturbative parameter. The truncation of the perturbation series defining the coefficient function leads to an arbitrariness of order AQCD/mQ, which cancels against a corresponding arbitrariness of order A QCD in the definition of the non-perturbative parameter A. The renormalon problem poses itself when one imagines to apply perturbation theory to very high orders. In practice, the perturbative coefficients are known to finite order in as (typically to one- or two-loop accuracy), and to be consistent one should use them in connection with the pole mass (and A etc.) defined to the same order.

258

2.4

Spectroscopic Implications

The spin-flavor symmetry leads to many interesting relations between the properties of hadrons containing a heavy quark. The most direct consequences concern the spectroscopy of such states 53,54. In the limit mQ -t 00, the spin of the heavy quark and the total angular momentum j of the light degrees of freedom are separately conserved by the strong interactions. Because of heavy-quark symmetry, the dynamics is independent of the spin and mass of the heavy quark. Hadronic states can thus be classified by the quantum numbers (flavor, spin, parity, etc.) of their light degrees of freedom 55. The spin symmetry predicts that, for fixed j ¥- 0, there is a doublet of degenerate states with total spin J = j ±!. The flavor symmetry relates the properties of states with different heavy-quark flavor. In general, the mass of a hadron HQ containing a heavy quark Q obeys an expansion of the form (31)

The parameter A represents contributions arising from terms in the Lagrangian that are independent of the heavy-quark mass 36, whereas the quantity ~m2 originates from the terms of order I/mQ in the effective Lagrangian of the HQET. For the ground-state pseudoscalar and vector mesons, one can parametrize the contributions from the kinetic energy and the chromo-magnetic interaction in terms of two quantities Al and A2, in such a way that 56 (32)

The hadronic parameters A, Al and A2 are independent of mQ. They characterize the properties of the light constituents. Consider, as a first example, the SU(3) mass splittings for heavy mesons. The heavy-quark expansion predicts that mBs - mBd =

As - Ad + O(l/mb) , (33)

where we have indicated that the value of the parameter A depends on the flavor of the light quark. Thus, to the extent that the charm and bottom quarks can both be considered sufficiently heavy, the mass splittings should be similar in the two systems. This prediction is confirmed experimentally, since

(90 ± 3) MeV, (99 ± 1) MeV.

(34)

259

As a second example, consider the spin splittings between the ground-state pseudoscalar (J = 0) and vector (J = 1) mesons, which are the members of the spin-doublet with j = ~. From (31) and (32), it follows that

m1. - m1 = 4.\2 + O(I/mb) , mb. - mb = 4.\2

+ O(I/me) .

(35)

The data are compatible with this:

m1. - m1

~ 0.49 GeV 2

mb. - mb ~ 0.55 GeV

2

, •

(36)

Assuming that the B system is close to the heavy-quark limit, we obtain the value (37) for one of the hadronic parameters in (32). This quantity plays an important role in the phenomenology of inclusive decays of heavy hadrons. A third example is provided by the mass splittings between the groundstate mesons and baryons containing a heavy quark. The HQET predicts that mAb - mB =

+ O(I/mb) , Arneson + O(I/me) .

Abaryon - Arneson

mAc - mD = Abaryon -

(38)

This is again consistent with the experimental results mAb -mB

= (345±9) MeV,

mAc - mD = (416

± 1) MeV,

(39)

although in this case the data indicate sizeable symmetry-breaking corrections. The dominant correction to the relations (38) comes from the contribution of the chromo-magnetic interaction to the masses of the heavy mesons,b which adds a term 3.\2/2mQ on the right-hand side. Including this term, we obtain the refined prediction that the two quantities 3.\2

mAb - mB - -2= (311 mB

3.\2 2mD

mA - mD - - c

± 9) MeV,

= (320± 1) MeV

(40)

bBecause of spin symmetry, there is no such contribution to the masses of AQ baryons.

260

should be close to each other. This is clearly satisfied by the data. The mass formula (31) can also be used to derive information on the heavy-quark masses from the observed hadron masses. Introducing the "spinaveraged" meson masses mB = (mB + 3mB-) ~ 5.31 GeV and mD = t (mD + 3mD*) ~ 1.97 GeV, we find that

t

Using theoretical estimates for the parameter AI, which lie in the range 57--66 Al = -(0.3 ± 0.2) GeV 2

,

(42)

this relation leads to

mb - me = (3.39 ± 0.03 ± 0.03) GeV,

(43)

where the first error reflects the uncertainty in the value of Al, and the second one takes into account unknown higher-order corrections. The fact that the difference of the pole masses, mb - me, is known rather precisely is important for the analysis of inclusive decays of heavy hadrons. 3

Exclusive Semi-Leptonic Decays

Semi-Ieptonic decays of B mesons have received a lot of attention in recent years. The decay channel f3 -t D* l ii has the largest branching fraction of all B-meson decay modes. From a theoretical point of view, semi-Ieptonic decays are simple enough to allow for a reliable, quantitative description. The analysis of these decays provides much information about the strong forces that bind the quarks and gluons into hadrons. Schematically, a semi-Ieptonic decay process is shown in Fig. 6. The strength of the b -t c transition vertex is governed by the element Veb of the CKM matrix. The parameters of this matrix are fundamental parameters of the Standard Model. A primary goal of the study of semi-Ieptonic decays of B mesons is to extract with high precision the values of lVeb 1and lVub I. We will now discuss the theoretical basis of such analyses.

3.1

Weak Decay Form Factors

Heavy-quark symmetry implies relations between the weak decay form factors of heavy mesons, which are of particular interest. These relations have been

261

v b

13

[

-

c~~~~

~L: c

]

..-----.q

D, D*, D**, ...

q

Figure 6: Semi-leptonic decays of B mesons.

derived by Isgur and Wise 6, generalizing ideas developed by Nussinov and Wetzel 3, and by Voloshin and Shifman 4,5. Consider the elastic scattering of a B meson, B(v) -t B(v ' ), induced by a vector current coupled to the b quark. Before the action of the current, the light degrees of freedom inside the B meson orbit around the heavy quark, which acts as a static source of color. On average, the b quark and the B meson have the same velocity v. The action of the current is to replace instantaneously (at time t = to) the color source by one moving at a velocity v', as indicated in Fig. 7. If v = v', nothing happens; the light degrees of freedom do not realize that there was a current acting on the heavy quark. If the velocities are different, however, the light constituents suddenly find themselves interacting with a moving color source. Soft gluons have to be exchanged to rearrange them so as to form a B meson moving at velocity v'. This rearrangement leads to a form-factor suppression, reflecting the fact that, as the velocities become more and more different, the probability for an elastic transition decreases. The important observation is that, in the limit mb -t 00, the form factor can only depend on the Lorentz boost , = v . v' connecting the rest frames of the initial- and final-state mesons. Thus, in this limit a dimensionless probability function ~(v· v') describes the transition. It is called the Isgur-Wise function 6. In the HQET, which provides the appropriate framework for taking the limit mb -t 00, the hadronic matrix element describing the scattering process can thus be written as _1_ (B(v')1 mB

bv,,"'bv IB(v))

=

~(v . v') (v + v'),.. .

(44)

Here bv and bv ' are the velocity-dependent heavy-quark fields of the HQET. It is important that the function ~(v . v') does not depend on mb. The factor 11mB on the left-hand side compensates for a trivial dependence on the heavymeson mass caused by the relativistic normalization of meson states, which is conventionally taken to be

(45)

262 Note that there is no term proportional to (v - v')/' in (44). This can be seen by contracting the matrix element with (v - v')/" which must give zero since pbv = bv and bv ' p' = bv "

t > to

t = to

t < to

Figure 7: Elastic transition induced by an external heavy-quark current.

It is more conventional to write the above matrix element in terms of an elastic form factor Fel(q2) depending on the momentum transfer q2 = (p_p')2: (B(v')1 bl'/'b IB(v)) = Fel(q2) (p

+ p')/' ,

(46)

where pe,) = mBv(1). Comparing this with (44), we find that

Fel(q2) = ~(v . v'),

q2 = -2m1(v· v' - 1).

(47)

Because of current conservation, the elastic form factor is normalized to unity at q2 = O. This condition implies the normalization of the Isgur-Wise function at the kinematic point V· v' = 1, i.e. for v = v': ~(1)

= 1.

(48)

It is in accordance with the intuitive argument that the probability for an elastic transition is unity if there is no velocity change. Since for v = v' the final-state meson is at rest in the rest frame of the initial meson, the point v . v' = 1 is referred to as the zero-recoil limit. The heavy-quark flavor symmetry can be used to replace the b quark in the final-state meson by a c quark, thereby turning the B meson into a D meson. Then the scattering process turns into a weak decay process. In the infinite-mass limit, the replacement bv ' --+ cv , is a symmetry transformation, under which the effective Lagrangian is invariant. Hence, the matrix element 1 (D(v')1 cv 'l'/'b v IB(v)) = JmBmD

~(v . v') (v + v')l-'

(49)

is still determined by the same function ~(v . v'). This is interesting, since in general the matrix element of a flavor-changing current between two pseudoscalar mesons is described by two form factors:

(D( v') I c I'/'b IB( v))

= J+(q2) (p + p')/' - f _(q2) (p -

p')/' .

(50)

263

Comparing the above two equations, we find that J±(q2) = mB ± mD 2JmBmD q2 = m~

~(v. v'),

+ m'b -

(51)

2mBmD v . v' .

Thus, the heavy-quark flavor symmetry relates two a priori independent form factors to one and the same function. Moreover, the normalization of the Isgur-Wise function at v . Vi = 1 now implies a non-trivial normalization of the form factors f±(q2) at the point of maximum momentum transfer, q~ax = (mB - mD)2:

(52) The heavy-quark spin symmetry leads to additional relations among weak decay form factors. It can be used to relate matrix elements involving vector mesons to those involving pseudoscalar mesons. A vector meson with longitudinal polarization is related to a pseudoscalar meson by a rotation of the heavy-quark spin. Hence, the spin-symmetry transformation C~, -+ C~, relates B -+ D with B -+ D* transitions. The result of this transformation is 6

[c*1-' (v· v'

+ 1) -

V'I-'

c*·

v]~(v. Vi), (53)

where c denotes the polarization vector of the D* meson. Once again, the matrix elements are completely described in terms of the Isgur-Wise function. Now this is even more remarkable, since in general four form factors, V(q2) for the vector current, and Ai(q2), i = 0,1,2, for the axial current, are required to parameterize these matrix elements. In the heavy-quark limit, they obey the relations 67 mB +mD· I 2 2 2 2 ~(v· v) = V(q ) = Ao(q ) = A 1(q ) JmBmD·

2 2 I q2 = mB+mD. - 2mBmD·V·V.

(54)

264

Equations (51) and (54) summarize the relations imposed by heavy-quark symmetry on the weak decay form factors describing the semi-Ieptonic decay processes B -t D£v and B -t D*£v. These relations are model-independent consequences of QCD in the limit where mb, me» AQco. They playa crucial role in the determination of the CKM matrix element iVeb I. In terms of the recoil variable w = V· v', the differential semi-leptonic decay rates in the heavyquark limit become 68

dr(B -t D £ v) dw dr(B -t D* £ v) dw

a2

48:3 rv"bl 2 (mB + mD)2 mb (w 2 -

4~:3 iVebl 2 (mB x

[1 + ~ w+

1

mD*)2

1)3/2 e(w) ,

mb* Vw2=-i (w + 1)2

m1- 2w mBmD* + m (mB - mD*)2

b*] e(w).

(55)

These expressions receive symmetry-breaking corrections, since the masses of the heavy quarks are not infinitely large. Perturbative corrections of order a:(mQ) can be calculated order by order in perturbation theory. A more difficult task is to control the non-perturbative power corrections of order (AQco/mQ)n. The HQET provides a systematic framework for analyzing these corrections. For the case of weak-decay form factors the analysis of the 1I mQ corrections was performed by Luke 69. Later, Falk and the present author have analyzed the structure of 1/m~ corrections for both meson and baryon weak decay form factors 56. We shall not discuss these rather technical issues in detail, but only mention the most important result of Luke's analysis. It concerns the zero-recoil limit, where an analogue of the Ademollo-Gatto theorem 70 can be proved. This is Luke's theorem 69, which states that the matrix elements describing the leading 1/mQ corrections to weak decay amplitudes vanish at zero recoil. This theorem is valid to all orders in perturbation theory 56,71,72. Most importantly, it protects the B -t D*£v decay rate from receiving first-order 1/mQ corrections at zero recoil 68 . [A similar statement is not true for the decay B -t D £ v. The reason is simple but somewhat subtle. Luke's theorem protects only those form factors not multiplied by kinematic factors that vanish for v = v'. By angular momentum conservation, the two pseudoscalar mesons in the decay B -t D £ v must be in a relative p wave, and hence the amplitude is proportional to the velocity IVD 1of the D meson in the B-meson rest frame. This leads to a factor (w 2 -1) in the decay rate. In such a situation, kinematically suppressed form factors can contribute 67.J

265

3.2 Short-Distance Corrections In Sec. 2, we have discussed the first two steps in the construction of the HQET. Integrating out the small components in the heavy-quark fields, a nonlocal effective action was derived, which was then expanded in a series of local operators. The effective Lagrangian obtained that way correctly reproduces the long-distance physics of the full theory (see Fig. 3). It does not contain the short-distance physics correctly, however. The reason is obvious: a heavy quark participates in strong interactions through its coupling to gluons. These gluons can be soft or hard, i.e. their virtual momenta can be small, of the order of the confinement scale, or large, of the order of the heavy-quark mass. But hard gluons can resolve the spin and flavor quantum numbers of a heavy quark. Their effects lead to a renormalization of the coefficients of the operators in the HQET. A new feature of such short-distance corrections is that through the running coupling constant they induce a logarithmic dependence on the heavy-quark mass 4. Since O:s(mQ) is small, these effects can be calculated in perturbation theory. Consider, as an example, the matrix elements of the vector current V = ij ,IJ.Q. In QCD this current is partially conserved and needs no renormalization. Its matrix elements are free of ultraviolet divergences. Still, these matrix elements have a logarithmic dependence on mQ from the exchange of hard gluons with virtual momenta of the order of the heavy-quark mass. If one goes over to the effective theory by taking the limit mQ -+ 00, these logarithms diverge. Consequently, the vector current in the effective theory does require a renormalization 11. Its matrix elements depend on an arbitrary renormalization scale p, which separates the regions of short- and long-distance physics. If p is chosen such that AQCD « p « mQ, the effective coupling constant in the region between p and mQ is small, and perturbation theory can be used to compute the short-distance corrections. These corrections have to be added to the matrix elements of the effective theory, which contain the long-distance physics below the scale p. Schematically, then, the relation between matrix elements in the full and in the effective theory is (V(mQ))QCD = Co(mQ,p) (VO(p))HQET

+

C 1 (mQ,p) mQ

(VI (p))HQET

+ ... ,

(56) where we have indicated that matrix elements in the full theory depend on mQ, whereas matrix elements in the effective theory are mass-independent, but do depend on the renormalization scale. The Wilson coefficients Ci (mQ, p) are defined by this relation. Order by order in perturbation theory, they can be computed from a comparison of the matrix elements in the two theories.

266

Since the effective theory is constructed to reproduce correctly the low-energy behavior of the full theory, this "matching" procedure is independent of any long-distance physics, such as infrared singularities, non-perturbative effects, and the nature of the external states used in the matrix elements. The calculation of the coefficient functions in perturbation theory uses the powerful methods of the renormalization group. It is in principle straightforward, yet in practice rather tedious. A comprehensive discussion of most of the existing calculations of short-distance corrections in the HQET can be found in Ref. 18. 3.3

Model-Independent Determination of Web I

We will now discuss the most important application of the formalism described above in the context of semi-Ieptonic decays of B mesons. A modelindependent determination of the CKM matrix element IVeb I based on heavyquark symmetry can be obtained by measuring the recoil spectrum of D* mesons produced in fJ -+ D* decays 68. In the heavy-quark limit, the differential decay rate for this process has been given in (55). In order to allow for corrections to that limit, we write

e;;

df dw

4~:3 (mB X

[1

mD_)2 m1- Vw 2 - 1 (w

+ 1)2

+ ~ m1- 2wmBmD- + m 1-]WebI 2 F2(W) , w +1 (mB - mD_)2

(57)

where the hadronic form factor F(w) coincides with the Isgur-Wise function up to symmetry-breaking corrections of order D:s(mQ) and AQcn/mQ. The idea is to measure the product Web IF(w) as a function of w, and to extract Web I from an extrapolation of the data to the zero-recoil point w = 1, where the B and the D* mesons have a common rest frame. At this kinematic point, heavy-quark symmetry helps us to calculate the normalization F(I) with small and controlled theoretical errors. Since the range of w values accessible in this decay is rather small (1 < w < 1.5), the extrapolation can be done using an expansion around w = 1: F(w) = F(I) [1 -

,p (w -

1) + c(w - 1)2 ... ] .

(58)

The slope if and the curvature c, and indeed more generally the complete shape of the form factor, are tightly constrained by analyticity and unitarity requirements 73,74. In the long run, the statistics of the experimental results

267

close to zero recoil will be such that these theoretical constraints will not be crucial to get a precision measurement of Webl. They will, however, enable strong consistency checks. 0.05 . . . - - - - - - - - - - - - - - - - - - , 0.04

~0.Q3 -:Q

;:,.." 0.02 0.01

1.1

1.2

1.3

1.4

1.5

w Figure 8: CLEO data for the product IVcbl.r(w), as extracted from the recoil spectrum in B -+ D* £ j) decays 75. The line shows a linear fit to the data.

Measurements of the recoil spectrum have been performed by several experimental groups. Figure 8 shows, as an example, the data reported some time ago by the CLEO Collaboration. The weighted average of the experimental results is 76 (59) Web I F(I) = (35.2 ± 2.6) x 10- 3 . Heavy-quark symmetry implies that the general structure of the symmetrybreaking corrections to the form factor at zero recoil is 68

AQCD A~CD) _ F(I) = 'TlA ( 1 + 0 x - - + const x - 2 - +... = 'TlA (1 mQ mQ

+ 61/ m 2),

(60)

where 'TlA is a short-distance correction arising from the finite renormalization of the flavor-changing axial current at zero recoil, and 611m 2 parameterizes second-order (and higher) power corrections. The absence of first-order power corrections at zero recoil is a consequence of Luke's theorem 69. The one-loop expression for 'TlA has been known for a long time 2,5,77:

+ me In mb - ~) ~ 0.96. mb - me me 3

'TlA = 1 + c¥s(M) (mb 7r

(61)

268

The scale M in the running coupling constant can be fixed by adopting the prescription of Brodsky, Lepage and Mackenzie (BLM) 78, where it is identified with the average virtuality of the gluon in the one-loop diagrams that contribute to 'T]A. If as(M) is defined in the MS scheme, the result is 79 M :::::: 0.51v'memb' Several estimates of higher-order corrections to 1JA have been discussed. A renormalization-group resummation of logarithms of the type (as In mb/me)n, as(asln mb/me)n and me/mb(as In mb/me)n leads to 11,80-83 1JA :::::: 0.985. On the other hand, a resummation of "renormalon-chain" contributions of the form 13~-la~, where 130 = 11- ~nf is the first coefficient of the QCD ;1-function, gives 84 'T]A :::::: 0.945. Using these partial resummations to estimate the uncertainty gives 1JA = 0.965 ± 0.020. Recently, Czarnecki has improved this estimate by calculating 1JA at two-loop order 85. His result, 1JA = 0.960 ± 0.007, is in excellent agreement with the BLM-improved oneloop expression (61). Here the error is taken to be the size of the two-loop correction. The analysis of the power corrections is more difficult, since it cannot rely on perturbation theory. Three approaches have been discussed: in the "exclusive approach" , all1/m~ operators in the HQET are classified and their matrix elements estimated, leading to 56,86 ~1/m2 = -(3 ± 2)%; the "inclusive approach" has been used to derive the bound ~1/m2 < -3%, and to estimate that 87 ~1/m2 = -(7 ± 3)%; the "hybrid approach" combines the virtues of the former two to obtain a more restrictive lower bound on ~1/m2. This leads to 88 ~1/m2 = -0.055 ± 0.025. Combining the above results, adding the theoretical errors linearly to be conservative, gives .1'(1) = 0.91 ± 0.03 (62) for the normalization of the hadronic form factor at zero recoil. Thus, the corrections to the heavy-quark limit amount to a moderate decrease of the form factor of about 10%. This can be used to extract from the experimental result (59) the model-independent value

iVebl = (38.7 ± 2.8exp ± 1.3t h) 3.4

x 10-

3

.

(63)

Measurements of B -+ D*ei} and B -+ Dei} Form Factors and Tests of Heavy-Quark Symmetry

We have discussed earlier in this section that heavy-quark symmetry implies relations between the semi-Ieptonic form factors of heavy mesons. They receive symmetry-breaking corrections, which can be estimated using the HQET. The extent to which these relations hold can be tested experimentally by comparing

269

the different form factors describing the decays fJ -t DC *) £ v at the same value ofw.

When the lepton mass is neglected, the differential decay distributions in fJ -t D* £ v decays can be parameterized by three helicity amplitudes, or equivalently by three independent combinations of form factors. It has been suggested that a good choice for three such quantities should be inspired by the heavy-quark limit 18,89. One thus defines a form factor hA1(W), which up to symmetry-breaking corrections coincides with the Isgur-Wise function, and two form-factor ratios

R 2(w) = [

q2 ] A2(q2) (mB+mD*)2 A 1(q2)·

1_

(64)

The relation between wand q2 has been given in (54). This definition is such that in the heavy-quark limit R1(W) = R2(W) = 1 independently of w. To extract the functions hAl (w), R1 (w) and R2 (w) from experimental data is a complicated task. However, HQET-based calculations suggest that the w dependence of the form-factor ratios, which is induced by symmetry-breaking effects, is rather mild 89. Moreover, the form factor hAl (w) is expected to have a nearly linear shape over the accessible w range. This motivates to introduce three parameters O~l' R1 and R2 by

(65)

where F(I) = 0.91 ± 0.03 from (62). The CLEO Collaboration has extracted these three parameters from an analysis of the angular distributions in fJ -t D*£ v decays 90. The results are O~l = 0.91 ± 0.16,

R1

= 1.18 ± 0.32,

R2

= 0.71 ± 0.23.

(66)

Using the HQET, one obtains an essentially model-independent prediction for the symmetry-breaking corrections to R 1, whereas the corrections to R2 are somewhat model dependent. To good approximation 18 R1 ~ 1 + R2

~

1-

4Q s (me)

3

me

1f

A

K,

-2me

A + -2-

~

~

0.8 ± 0.2 ,

1.3 ± 0.1, (67)

270

with '" ~ 1 from QCD sum rules 89. Here 1\ is the "binding energy" as defined in (28). Theoretical calculations 91 ,92 as well as phenomenological analyses 62 ,63 suggest that A ~ 0.45-0.65 GeV is the appropriate value to be used in one-loop calculations. A quark-model calculation of Rl and R2 gives results similar to the HQET predictions 93: R1 ~ 1.15 and R2 ~ 0.91. The experimental data confirm the theoretical prediction that R1 > 1 and R2 < 1, although the errors are still large. Heavy-quark symmetry has also been tested by comparing the form factor F( w) in B -t D* '- iJ decays with the corresponding form factor 9 (w) governing B -t D '- iJ decays. The theoretical prediction 74,89

9(1) F(l) = 1.08 ± 0.06

(68)

compares well with the experimental results for this ratio: 0.99±0.19 reported by the CLEO Collaboration 94, and 0.87±0.30 reported by the ALEPH Collaboration 95. In these analyses, it has also been tested that within experimental errors the shape of the two form factors agrees over the entire range of w values. The results of the analyses described above are very encouraging. Within errors, the experiments confirm the HQET predictions, starting to test them at the level of symmetry-breaking corrections. 4

Inclusive Decay Rates

Inclusive decay rates determine the probability of the decay of a particle into the sum of all possible final states with a given set of global quantum numbers. An example is provided by the inclusive semi-Ieptonic decay rate of the B meson, r(B - t X f. iJ), where the final state consists of a lepton-neutrino pair accompanied by any number of hadrons. Here we shall discuss the theoretical description of inclusive decays of hadrons containing a heavy quark 96-105. From a theoretical point of view such decays have two advantages: first, boundstate effects related to the initial state, such as the "Fermi motion" of the heavy quark inside the hadron 103,104, can be accounted for in a systematic way using the heavy-quark expansion; secondly, the fact that the final state consists of a sum over many hadronic channels eliminates bound-state effects related to the properties of individual hadrons. This second feature is based on the hypothesis of quark-hadron duality, which is an important concept in QCD phenomenology. The assumption of duality is that cross sections and decay rates, which are defined in the physical region (i.e. the region of time-like momenta), are calculable in QCD after a "smearing" or "averaging" procedure has been applied 106. In semi-Ieptonic decays, it is the integration over the

271

lepton and neutrino phase space that provides a smearing over the invariant hadronic mass of the final state (so-called global duality). For non-Ieptonic decays, on the other hand, the total hadronic mass is fixed, and it is only the fact that one sums over many hadronic states that provides an averaging (so-called local duality). Clearly, local duality is a stronger assumption than global duality. It is important to stress that quark-hadron duality cannot yet be derived from first principles; still, it is a necessary assumption for many applications of QCD. The validity of global duality has been tested experimentally using data on hadronic T decays 107. Using the optical theorem, the inclusive decay width of a hadron Hb containing a b quark can be written in the form (69)

where the transition operator T is given by (70)

Inserting a complete set of states inside the time-ordered product, we recover the standard expression

for the decay rate. For the case of semi-Ieptonic and non-Ieptonic decays, .ceff is the effective weak Lagrangian given in (4), which in practice is corrected for short-distance effects 32,33,108-110 arising from the exchange of gluons with virtualities between mw and mb. If some quantum numbers of the final states X are specified, the sum over intermediate states is to be restricted appropriately. In the case of the inclusive semi-Ieptonic decay rate, for instance, the sum would include only those states X containing a lepton-neutrino pair. In perturbation theory, some contributions to the transition operator are given by the two-loop diagrams shown on the left-hand side in Fig. 9. Because of the large mass of the b quark, the momenta flowing through the internal propagator lines are large. It is thus possible to construct an OPE for the transition operator, in which T is represented as a series of local operators containing the heavy-quark fields. The operator with the lowest dimension, d = 3, is bb. It arises by contracting the internal lines of the first diagram. The only gauge-invariant operator with dimension 4 is bilfJ bj however, the equations of motion imply that between physical states this operator can be

272

-

bb

•

b

9

Figure 9: Perturbative contributions to the transition operator T (left), and the corresponding operators in the OPE (right). The open squares represent a four-fermion interaction of the effective Lagrangian .ceff, while the black circles represent local operators in the OPE.

replaced by mbbb. The first operator that is different from bb has dimension 5 and contains the gluon field. It is given by bgsa I'IIGI'II b. This operator arises from diagrams in which a gluon is emitted from one of the internal lines, such as the second diagram shown in Fig. 9. For dimensional reasons, the matrix elements of such higher-dimensional operators are suppressed by inverse powers of the heavy-quark mass. Thus, any inclusive decay rate of a hadron Hb can be written as 97-99 (72)

where the prefactor arises naturally from the loop integrations, c! are calculable coefficient functions (which also contain the relevant CKM matrix elements) depending on the quantum numbers f of the final state, and (0) H are the (normalized) forward matrix elements of local operators, for which we use the short-hand notation (73)

In the next step, these matrix elements are systematically expanded in powers of 11mb, using the technology of the HQET. The result is 56,97,99

(74)

273 where we have defined the HQET matrix elements

M;(Hb)

= - 12 (Hb(V)1 bv (iD? bv IHb(V)) , mHb

(75)

Here (iD)2 = (iv· D)2 - (iD)2; in the rest frame, this is the square of the operator for the spatial momentum of the heavy quark. Inserting these results into (72) yields

r(Hb -+ X ) f

= G}mg 19211"3

{cf 3

(1 _M~(Hb) - M'i;(Hb)) + 2m2

b

2cf M'i;(Hb) 5 m2b

+ ... } . (76)

It is instructive to understand the appearance of the "kinetic energy" contribution M~, which is the gauge-covariant extension of the square of the b-quark

momentum inside the heavy hadron. This contribution is the field-theory analogue of the Lorentz factor (1 - Vb2 )1/2 ~ 1 - f2 /2m~, in accordance with the fact that the lifetime, T = l/r, for a moving particle increases due to time dilation. The main result of the heavy-quark expansion for inclusive decay rates is the observation that the free quark decay (i.e. the parton model) provides the first term in a systematic l/mb expansion 96. For dimensional reasons, the corresponding rate is proportional to the fifth power of the b-quark mass. The non-perturbative corrections, which arise from bound-state effects inside the B meson, are suppressed by at least two powers of the heavy-quark mass, i.e. they are of relative order (AQCD/mb)2. Note that the absence of firstorder power corrections is a consequence of the equations of motion, as there is no independent gauge-invariant operator of dimension 4 that could appear in the OPE. The fact that bound-state effects in inclusive decays are strongly suppressed explains a posteriori the success of the parton model in describing such processes 111,112. The hadronic matrix elements appearing in the heavy-quark expansion (76) can be determined to some extent from the known masses of heavy hadron states. For the B meson, one finds that

M;(B)

= -AI = (0.3 ± 0.2) Gey2,

M'i;(B) = 3A2 ~ 0.36 Gey2,

(77)

where Al and A2 are the parameters appearing in the mass formula (32). For the ground-state baryon A b , in which the light constituents have total spin

274 zero, it follows that

J.l&(A b) = 0,

(7S)

while the matrix element J.l; (Ab) obeys the relation

(mAb -

mAJ -

(mB - mD) = [J.l;(B) - J.l;(Ab)]

(2~e - 2~J + O(l/mb),

(79) where mB and mD denote the spin-averaged masses introduced in connection with (41). The above relation implies

(SO) What remains to be calculated, then, is the coefficient functions c~ for a given inclusive decay channel. To illustrate this general formalism, we discuss as an example the determination of iVebl from inclusive semi-Ieptonic B decays. In this case the short-distance coefficients in the general expression (76) are given by 97-99

iVebl 2 [1 - Sx 2

+ Sx 6

-

x8

-

12x 4 lnx 2

+ 0(0:

8 )] ,

-6iVebI 2 (1 - x 2 )4.

(Sl)

Here x = me/mb, and mb and me are the masses of the band c quarks, defined to a given order in perturbation theory 37. The O( O:s) terms in c~L are known exactly 113, and reliable estimates exist for the 0 (0:;) corrections 114. The theoretical uncertainties in this determination of IVeb I are quite different from those entering the analysis of exclusive decays. The main sources are the dependence on the heavy-quark masses, higher-order perturbative corrections, and above all the assumption of global quark-hadron duality. A conservative estimate of the total theoretical error on the extracted value of iVebl yields 115

[

6] B ] [1---:;:;;SL

iVebl = (0.040±0.003) 10.5%

1/2

.

pS 1/2

= (40±l exp ±3t h) x 10

-3

. (S2)

The value of iVebl extracted from the inclusive semi-Ieptonic width is in excellent agreement with the value in (63) obtained from the analysis of the exclusive decay 13 -+ D* £ ii. This agreement is gratifying given the differences of the methods used, and it provides an indirect test of global quark-hadron duality. Combining the two measurements gives the final result

iVebl = 0.039 ± 0.002 . After

Vud

and

V us ,

(S3)

this is the third-best known entry in the CKM matrix.

275 5

Rare B Decays and Determination of the Weak Phase 'Y

The main objectives of the B factories are to explore the physics of CP violation, to determine the flavor parameters of the electroweak theory, and to probe for physics beyond the Standard Model. This will test the CKM mechanism, which predicts that all CP violation results from a single complex phase in the quark mixing matrix. Facing the announcement of evidence for a CP asymmetry in the decays B -t J /1/J Ks by the CDF Collaboration 116, the confirmation of direct CP violation in K -t 1r1r decays by the KTeV and NA48 groups 117,118, and the successful start of the B factories at SLAC and KEK, the year 1999 has been an important step towards achieving this goal.

Figure 10: The rescaled unitarity triangle representing the relation 1 +

J)~Ud + ~t}~td = o. cb

cd

cb

cd

The apex is determined by the Wolfenstein parameters (p, ii). The area of the triangle is proportional to the strength of CP violation in the Standard Model.

The determination of the sides and angles of the "unitarity triangle" V:bVud+ Vcb Vcd+ ~bvtd = 0 depicted in Fig. 10 plays a central role in the B factory program. Adopting the standard phase conventions for the CKM matrix, only the two smallest elements in this relation, V:b and vtd, have non-vanishing imaginary parts (to an excellent approximation). In the Standard Model the angle fJ = -arg(vtd) can be determined in a theoretically clean way by measuring the mixing-induced CP asymmetry in the decays B -t J/1/JKs. The preliminary CDF result implies 116 sin2fJ = 0.79~g::!. The angle / = arg(V:b)' or equivalently the combination 0: = 180 0 - fJ - /, is much harder to determine 115. Recently, there has been significant progress in the theoretical understanding of the hadronic decays B -t 1r K, and methods have been developed to extract information on / from rate measurements for these processes. Here we discuss the charged modes B± -t 1r K, which from a theoretical perspective are particularly clean.

276

In the Standard Model, the main contributions to the decay amplitudes for the rare processes B ---t 7f K are due to the penguin-induced flavor-changing neutral current (FCNC) transitions b ---t sqij, which exceed a small, Cabibbosuppressed b ---t uus contribution from W -boson exchange. The weak phase 'Y enters through the interference of these two ("penguin" and "tree") contributions. Because of a fortunate interplay of isospin, Fierz and flavor symmetries, the theoretical description of the charged modes B± ---t 7f K is very clean despite the fact that these are exclusive non-Ieptonic decays 119-121. Without any dynamical assumption, the hadronic uncertainties in the description of the interference terms relevant to the determination of 'Yare of relative magnitude O(,X2) or O(f-su(3)/Ne), where ,X = sinOc ~ 0.22 is a measure of Cabibbo suppression, fSU(3) '" 20% is the typical size ofSU(3) breaking, and the factor liNe indicates that the corresponding terms vanish in the factorization approximation. Factorizable SU(3) breaking can be accounted for in a straightforward way. Recently, the accuracy of this description has been further improved when it was shown that non-Ieptonic B decays into two light mesons, such as B ---t 7f K and B ---t 7f7f, admit a systematic heavy-quark expansion 122. To leading order in 11mb, but to all orders in perturbation theory, the decay amplitudes for these processes can be calculated from first principles without recourse to phenomenological models. The QCD factorization theorem proved in Ref. 122 improves upon the phenomenological approach of "generalized factorization" 123, which emerges as the leading term in the heavy-quark limit. With the help of this theorem, the irreducible theoretical uncertainties in the description of the B± ---t 7f K decay amplitudes can be reduced by an extra factor of O(l/mb), rendering their analysis essentially model independent. As a consequence of this fact, and because they are dominated by FCNC transitions, the decays B± ---t 7f K offer a sensitive probe to physics beyond the Standard Model 121,124-127 , much in the same way as the "classical" FCNC processes B ---t Xs'Y or B ---t Xs e+e-. 5.1

Theory of B± ---t 7fK Decays

The hadronic decays B ---t 7fK are mediated by a low-energy effective weak Hamiltonian 128, whose operators allow for three different classes of flavor topologies: QCD penguins, trees, and electroweak penguins. In the Standard Model the weak couplings associated with these topologies are known. From the measured branching ratios one can deduce that QCD penguins dominate the B ---t 7f K decay amplitudes 129 , whereas trees and electroweak penguins are subleading and of a similar strength 130. The theoretical description of the two

277

charged modes B± ---+ 11"± KO and B± ---+ 11"0 K± exploits the fact that the amplitudes for these processes differ in a pure isospin amplitude, A 3 / 2 , defined as the matrix element of the isovector part of the effective Hamiltonian between a B meson and the 11" K isospin eigenstate with I = ~. In the Standard Model the parameters of this amplitude are determined, up to an overall strong phase ¢, in the limit of SU(3) flavor symmetry 119 . Using the QCD factorization theorem the SU(3)-breaking corrections can be calculated in a model-independent way up to non-factorizable terms that are power-suppressed in 11mb and vanish in the heavy-quark limit. A convenient parameterization of the non-Ieptonic decay amplitudes A+o A(B+ ---+ 11"+ KO) and Ao+ -v'2 A(B+ ---+ 11"0 K+) is 121

=

=

= P (1 -

Ca

ei'r ei '1) ,

Ao+ = P [1 -

ea

ei'r e i '1 -

A+o

e3/2 ei + €3/28EW

cacos77sin2, cos ¢> + €3/28EW

+,

(95)

where we have set cos¢> = 1 in the numerator of the O(ca) term. Using the QCD factorization theorem one finds that Ca cos 77 ~ -0.02 in the heavy-quark limit 137, and we assign a 100% uncertainty to this estimate. In evaluating the result (95) we scan the parameters in the ranges 0.15 :::; €3/2 :::; 0.27, 0.55 :::; 8EW :::; 0.73, -25° :::; ¢> :::; 25°, and -0.04 :::; Ca cos 77 sin 2, :::; O. Figure 13 shows the allowed regions in the (p, ij) plane for the representative values XR = 0.25, 0.75, and 1.25 (from right to left). We stress that with this method a useful constraint on the Wolfenstein parameters is obtained for any value of XR.

283

B.I. mixing constraint

\ 0.6 0.5

semilept nic decays

0.4 I~

0.3

XR = 1.25/

", ,I

/

I

0.2

,

I I I

0.1

\

"

\ "< I "',J \ I o~~--~~--~~~·--~~~~k-~

- 0.6

- 0.4

- 0.2

0

0.2

\ \

__~____

L.

0.4

0.6

p Figure 13: Allowed regions in the (p,ij) plane for fixed values of XR, obtained by varying all theoretical parameters inside their respective ranges of uncertainty, as specified in the text. The sign of ij is not determined.

Model-independent determination: It is important that, once more precise data on B± ~ 7r K decays will become available, it will be possible to test the prediction of a small strong phase ¢ experimentally. To this end, one must determine the CP asymmetry A defined in (90) in addition to the ratio R*. From (91) it follows that for fixed values of E3/2 and 8EW the quantities R* and A define contours in the (t, ¢) plane, whose intersections determine the two phases up to possible discrete ambiguities 120,121. Figure 14 shows these contours for some representative values, assuming E3/2 = 0.21, 8EW = 0.64, and ea = O. In practice, including the uncertainties in the values of these parameters changes the contour lines into contour bands. Typically, the spread of the bands induces an error in the determination of , of about 121 10°. In the most general case there are up to eight discrete solutions for the two phases, four of which are related to the other four by a sign change (t, ¢) ~ (-" -¢). However, for typical values of R* it turns out that often only four solutions exist, two of which are related to the other two by a sign change. The theoretical prediction that ¢ is small implies that solutions should exist where the contours intersect close to the lower portion in the plot. Other solutions with large ¢ are strongly disfavored. Note that according to (91) the sign of the CP asymmetry A fixes the relative sign between the two phases , and ¢. If we trust the theoretical prediction that ¢ is negative 137, it follows that in most cases there remains only a unique solution for " i.e. the CP-violating phase, can be determined without any

284 175. 150. 125. 100. ~

75. 50. 25. 25. 50. 75. 100. 125. 150. 175.

111 Figure 14: Contours of constant R. ("hyperbolas") and constant IAI ("circles") in the (hi, l.pi) plane. The sign of the asymmetry A determines the sign of the product sin'Y sin.p. The contours for R. refer to values from 0.6 to 1.0 in steps of 0.1, those for the asymmetry correspond to 5%, 15%, and 25%, as indicated.

discrete ambiguity. Consider, as an example, the hypothetical case where R* = 0.8 and A = -15%. Figure 14 then allows the four solutions where (t,¢) ~ (±82°,=r=21°) or (±158°, =r=78°). The second pair of solutions is strongly disfavored because of the large values of the strong phase ¢. From the first pair of solutions, the one with ¢ ~ - 21 0 is closest to our theoretical expectation that ¢ ~ -11 0 , hence leaving 'Y ~ 82 0 as the unique solution.

6

Sensitivity to New Physics

In the presence of New Physics the theoretical description of B± -+ 1f K decays becomes more complicated. In particular, new CP-violating contributions to the decay amplitudes may be induced. A detailed analysis of such effects has been presented in 127. A convenient and completely general parameterization of the two amplitudes in (84) is obtained by replacing

p-+p',

(96)

285

where p, a, b are real hadronic parameters, and tPP' tPa, tPb are strong phases. The terms ip and ib change sign under a CP transformation. New Physics effects parameterized by pI and pare isospin conserving, while those described by a and b violate isospin symmetry. Note that the parameter pI cancels in all ratios of branching ratios and thus does not affect the quantities R. and XR as well as any CP asymmetry. Because the ratio R. in (89) would be 1 in the limit of isospin symmetry, it is particularly sensitive to isospin-violating New Physics contributions. New Physics can affect the bound on "( derived from (92) as well as the extraction of "( using the strategies discussed above. We will discuss these two possibilities in turn.

6.1

Effects on the Bound on "(

The upper bound on R;l in (92) and the corresponding bound on XR shown in Fig. 11 are model-independent results valid in the Standard Model. Note that the extremal value of R;l is such that IXRI :::; (1 + 8EW ) irrespective of "(. A value of IXRI exceeding this bound would be a clear signal for New Physics 121,124,127. Consider first the case where New Physics may induce arbitrary CPviolating contributions to the B -+ 7f K decay amplitudes, while preserving isospin symmetry. Then the only change with respect to the Standard Model is that the parameter p may no longer be as small as O(ca). Varying the strong phases tP and cPP independently, and allowing for an arbitrarily large New Physics contribution to p, one can derive the bound 127

(97) The extremal value is the same as in the Standard Model, i.e. isospin-conserving New Physics effects cannot lead to a value of IXRI exceeding (1 +8EW). For intermediate values of"( the Standard Model bound on XR is weakened; but even for large p = 0(1), corresponding to a significant New Physics contribution to the decay amplitudes, the effect is small. If both isospin-violating and isospin-conserving New Physics contributions are present and involve new CP-violating phases, the analysis becomes more complicated. Still, it is possible to derive model-independent bounds on XR. Allowing for arbitrary values of p and all strong phases, one obtains 127

IXRI < v(lal + I cos "(1)2 + (Ibl + Isin "(1)2 (98)

286 where the last inequality is relevant only in cases where J a2 + b2 » 1. The important point to note is that with isospin-violating New Physics contributions the value of IXRI can exceed the upper bound in the Standard Model by a potentially large amount. For instance, if J a 2 + b2 is twice as large as in the Standard Model, corresponding to a New Physics contribution to the decay amplitudes of only 10-15%, then IXRI could be as large as 2.6 as compared with the maximal value 1.8 allowed (for arbitrary 'Y) in the Standard Model. Also, in the most general case where b and p are non-zero, the maximal value IXRI can take is no longer restricted to occur at the endpoints 'Y = 0° or 180°, which are disfavored by the global analysis of the unitarity triangle 115. Rather, IXRI would take its maximal value if I tan 'YI = Ipi = Ib/ al· The present experimental value of X R in (93) has too large an error to determine whether there is any deviation from the Standard Model. If XR turns out to be larger than 1 (i.e. at least one third of a standard deviation above its current central value), then an interpretation of this result in the Standard Model would require a large value I'YI > 91° (see Fig. 11), which would be difficult to accommodate in view of the upper bound implied by the experimental constraint on Bs-Bs mixing, thus providing evidence for New Physics. If X R > 1.3, one could go a step further and conclude that the New Physics must necessarily violate isospin 127.

6.2

Effects on the Determination of 'Y

A value of the observable R* violating the bound (92) would be an exciting hint for New Physics. However, even if a future precise measurement will give a value that is consistent with the Standard Model bound, B± -+ 7r K decays provide an excellent testing ground for physics beyond the Standard Model. This is so because New Physics may cause a significant shift in the value of 'Y extracted using the strategies discussed earlier, leading to inconsistencies when this value is compared with other determinations of 'Y. A global fit of the unitarity triangle combining information from semileptonic B decays, B-B mixing, CP violation in the kaon system, and mixinginduced CP violation in B -+ J /'I/J Ks decays provides information on 'Y which in a few years will determine its value within a rather narrow range 115. Such an indirect determination could be complemented by direct measurements of 'Y using, e.g., B -+ DK(*) decays, or using the triangle relation 'Y = 180° - (}; - f3 combined with a measurement of (};. We will assume that a discrepancy of more than 25° between the "true" 'Y = arg(V:b ) and the value 'YrrK extracted in B± -+ 7r K decays will be observable after a few years of operation at the B factories. This sets the benchmark for sensitivity to New Physics effects.

287

2. 1.5 1. ro

0.5 0 -0.5 -1.

0

25. 50. 75. 100. 125. 150. 175.

Y Figure 15: Contours of constant XR versus I and the parameter a, assuming I horizontal band shows the value of a in the Standard Model.

> O.

The

In order to illustrate how big an effect New Physics could have on the extracted value of 'Y, we consider the simplest case where there are no new CP-violating couplings. Then all New Physics contributions in (96) are parameterized by the single parameter aNP == a - JEW. A more general discussion can be found in Ref. 127. We also assume for simplicity that the strong phase

O. Obviously, even a moderate New Physics contribution to the parameter a can induce a large shift in 'Y. Note that the present central value of X R ~ 0.7 is such that values of a less than the Standard Model result a ~ 0.64 are disfavored, since they would require values of 'Y exceeding 100°, in conflict with the global analysis of the unitarity triangle 115.

6.3

Survey of New Physics models

In Ref. 127, we have explored how New Physics could affect purely hadronic FCNC transitions of the type b -+ sqq focusing, in particular, on isospin violation. Unlike in the Standard Model, where isospin-violating effects in these

288 Table 1: Maximal contributions to aNP and shifts in 'Y in extensions of the Standard Model. For the case of supersymmetric models with R-parity the first (second) row corresponds to maximal right-handed (left-handed) strange-bottom squark mixing. For the two-Higgsdoublet models we take mH+ > lOOGeV and tanf3 > 1.

Model

laNPI

1'Y1TK -

FCN C Z exchange extra Z' boson SUSY without R-parity

2.0 14 14

180 1800 1800

SUSY with R-parity

0.4 1.3 0.15 0.3

25 0 1800

two-Higgs-doublet models anomalous gauge-boson couplings

1'1

0

100 20 0

processes are suppressed by electroweak gauge couplings or small CKM matrix elements, in many New Physics scenarios these effects are not parametrically suppressed relative to isospin-conserving FCNC processes. In the language of effective weak Hamiltonians this implies that the Wilson coefficients of QCD and electroweak penguin operators are of a similar magnitude. For a large class of New Physics models we found that the coefficients of the electroweak penguin operators are, in fact, due to "trojan" penguins, which are neither related to penguin diagrams nor of electroweak origin. Specifically, we have considered: (a) models with tree-level FCNC couplings of the Z boson, extended gauge models with an extra Z' boson, supersymmetric models with broken R-parityj (b) supersymmetric models with R-parity conservation; (c) two-Higgs-doublet models, and models with anomalous gauge-boson couplings. Some of these models have also been investigated in Refs. 125 and 126. In case (a), the electroweak penguin coefficients can be much larger than in the Standard Model because they are due to tree-level processes. In case (b), these coefficients can compete with the ones ofthe Standard Model because they arise from strong-interaction box diagrams, which scale relative to the Standard Model like (a:8/a:)(m~/m§USY)' In models (c), on the other hand, isospin-violating New Physics effects are not parametrically enhanced and are generally smaller than in the Standard Model. For each New Physics model we have explored which region of parameter space can be probed by the B± -+ 7r K observables, and how big a departure from the Standard Model predictions one can expect under realistic

289

circumstances, taking into account all constraints on the model parameters implied by other processes. Table 1 summarizes our estimates of the maximal isospin-violating contributions to the decay amplitudes, as parameterized by laNPI. They are the potentially most important source of New Physics effects in B± -+ 7r K decays. For comparison, we recall that in the Standard Model a ~ 0.64. Also shown are the corresponding maximal values of the difference 111rK - 11· As noted above, in models with tree-level FCNC couplings New Physics effects can be dramatic, whereas in supersymmetric models with Rparity conservation isospin-violating loop effects can be competitive with the Standard Model. In the case of supersymmetric models with R-parity violation the bound (98) implies interesting limits on certain combinations of the trilinear couplings A~jk and A~jk' as discussed in Ref. 127. 7

Concluding Relllarks

We have presented an introduction to recent developments in the theory and phenomenology of B physics, focusing on heavy-quark symmetry, exclusive and inclusive weak decays of B mesons, and rare B decays that are sensitive to CP-violating weak phases of the Standard Model. The theoretical tools that allow us to perform quantitative calculations are various forms of heavy-quark expansions, i.e. expansions in logarithms and inverse powers of the large scale provided by the heavy-quark mass, mb » AQcn. Heavy-flavor physics is a rich and diverse area of current research, which is characterized by a fruitful interplay between theory and experiments. This has led to many significant discoveries and developments. B physics has the potential to determine many important parameters of the electroweak theory and to test the Standard Model at low energies. At the same time, through the study of CP violation it provides a window to physics beyond the Standard Model. Indeed, there is a fair chance that such New Physics will first be seen at the B factories, before it can be explored in future collider experiments at the Tevatron and the Large Hadron Collider. Acknowledgements: It is a great pleasure to thank the Organizers of the Trieste Summer School in Particle Physics for the invitation to present these lectures and for providing a stimulating and relaxing atmosphere, which helped to initiate many physics discussions. In particular, I wish to express my gratitude to Gia Dvali, Antonio Masiero, Goran Senjanovic and Alexei Smirnov for their great hospitality and their many efforts to make my stay in Trieste a memorable one. Last but not least, I wish to thank the students of the school for their lively interest in these lectures.

290

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296

SUPERSYMMETRY PHENOMENOLOGY HITOSHI MURAYAMA Department of Physics, University of California, Berkeley, CA 94720, USA

Lawrence Berkeley National Laboratory, Berkeley, CA 94720 E-mail: murayamaOlbl.gov This is a very pedagogical review of supersymmetry phenomenology, given at ICTP Summer School in 1999, aimed mostly at people who had never studied supersymmetry before. It starts with an analogy that the reason why supersymmetry is needed is similar to the reason why the positron exists. It introduces the construction of supersymmetric Lagrangians in a practical way. The low-energy constraints, renormalization-group analyses, and collider phenomenology are briefly discussed.

1

1.1

Motivation

Problems in the Standard Model

The Standard Model of particle physics, albeit extremely successful phenomenologically, has been regarded only as a low-energy effective theory of the yet-more-fundamental theory. One can list many reasons why we think this way, but a few are named below. First of all, the quantum number assignments of the fermions under the standard SU(3)c x SU(2)L x U(I)y gauge group (Table 1) appear utterly

Table 1. The fermionic particle content of the Standard Model. Here we've put primes on the neutrinos in the same spirit of putting primes on the down-quarks in the quark doublets, indicating that the mass eigenstates are rotated by the MNS and CKM matrices, respectively. The subscripts g, r, b refer to colors.

297 Table 2. The bosonic particle content of the Standard Model.

Wl, W 2 ,H+,HW3,B,Irn(HO)

~

~

W+,W"(,Z

~

H

gx8

ReHO

bizarre. Probably the hypercharges are the weirdest of all. These assignments, however, are crucial to guarantee the cancellation of anomalies which could jeopardiz~ the gauge invariance at the quantum level, rendering the theory inconsistent. Another related puzzle is why the hypercharges are quantized in the unit of 1/6. In principle, the hypercharges can be any numbers, even irrational. However, the quantized hypercharges are responsible for neutrality of bulk matter Q(e) + 2Q(u) + Q(d) = Q(u) + 2Q(d) = 0 at a precision of 10- 21 . 1

The gauge group itself poses a question as well. Why are there seemingly unrelated three independent gauge groups, which somehow conspire together to have anomaly-free particle content in a non-trivial way? Why is "the strong interaction" strong and "the weak interaction" weaker? The essential ingredient in the Standard Model which appears the ugliest to most people is the electroweak symmetry breaking. In the list of bosons in the Standard Model Table 2, the gauge multiplets are necessary consequences of the gauge theories, and they appear natural. They of course all carry spin 1. However, there is only one spinless multiplet in the Standard Model: the Higgs doublet (1)

which condenses in the vacuum due to the Mexican-hat potential. It is introduced just for the purpose of breaking the electroweak symmetry SU(2)L x U(I)y -+ U(I)QED. The potential has to be arranged in a way to break the symmetry without any microscopic explanations. Why is there a seemingly unnecessary three-fold repetition of "generations"? Even the second generation led the Nobel Laureate 1.1. Rabi to ask "who ordered muon?" Now we face even more puzzling question of having three generations. And why do the fermions have a mass spectrum which stretches over almost six orders of magnitude between the electron and the top quark? This question becomes even more serious once we consider the recent evidence for neutrino oscillations which suggest the mass of the thirdgeneration neutrino v~ of about 0.05 eV. 2 This makes the mass spectrum

298

stretching over thirteen orders of magnitude. We have no concrete understanding of the mass spectrum nor the mixing patterns.

1.2 Drive to go to Shorter Distances All the puzzles raised in the previous section (and more) cry out for a more fundamental theory underlying the Standard Model. What history suggests is that the fundamental theory lies always at shorter distances than the distance scale of the problem. For instance, the equation of state of the ideal gas was found to be a simple consequence of the statistical mechanics of free molecules. The van der Waals equation, which describes the deviation from the ideal one, was the consequence of the finite size of molecules and their interactions. Mendeleev's periodic table of chemical elements was understood in terms of the bound electronic states, Pauli exclusion principle and spin. The existence of varieties of nuclide was due to the composite nature of nuclei made of protons and neutrons. The list would go on and on. Indeed, seeking answers at more and more fundamental level is the heart of the physical science, namely the reductionist approach. The distance scale of the Standard Model is given by the size of the Higgs boson condensate v = 250 Ge V. In natural units, it gives the distance scale of d = hc/v = 0.8 X 10- 16 cm. We therefore would like to study physics at distance scales shorter than this eventually, and try to answer puzzles whose partial list was given in the previous section. Then the idea must be that we imagine the Standard Model to be valid down to a distance scale shorter than d, and then new physics will appear which will take over the Standard Model. But applying the Standard Model to a distance scale shorter than d poses a serious theoretical problem. In order to make this point clear, we first describe a related problem in the classical electromagnetism, and then discuss the case of the Standard Model later along the same line. 3

1.3 Positron Analogue In the classical electromagnetism, the only dynamical degrees of freedom are electrons, electric fields, and magnetic fields. When an electron is present in the vacuum, there is a Coulomb electric field around it, which has the energy of ~ECoulomb

1

e2

= -47rco - -. re

(2)

299

Here, r e is the "size" of the electron introduced to cutoff the divergent Coulomb self-energy. Since this Coulomb self-energy is there for every electron, it has to be considered to be a part of the electron rest energy. Therefore, the mass of the electron receives an additional contribution due to the Coulomb self-energy: (3)

Experimentally, we know that the "size" of the electron is small, r e ~ 10- 17 cm. This implies that the self-energy AE is greater than 10 GeV or so, and hence the "bare" electron mass must be negative to obtain the observed mass of the electron, with a fine cancellation like 0.511 = -9999.489 + 10000.000MeV.

(4)

Even setting a conceptual problem with a negative mass electron aside, such a fine-cancellation between the "bare" mass of the electron and the Coulomb self-energy appears ridiculous. In order for such a cancellation to be absent, we conclude that the classical electromagnetism cannot be applied to distance scales shorter than e 2 j(47rcomeC2) = 2.8 x 10- 13 cm. This is a long distance in the present-day particle physics' standard. The resolution to the problem came from the discovery of the anti-particle of the electron, the positron, or in other words by doubling the degrees of freedom in the theory. The Coulomb self-energy discussed above can be depicted by a diagram where the electron emits the Coulomb field (a virtual photon) which is absorbed later by the electron (the electron "feels" its own Coulomb field). But now that the positron exists (thanks to Anderson back in 1932), and we also know that the world is quantum mechanical, one should think about the fluctuation of the "vacuum" where the vacuum produces a pair of an electron and a positron out of nothing together with a photon, within the time allowed by the energy-time uncertainty principle At rv hjAE rv hj(2meC2). This is a new phenomenon which didn't exist in the classical electrodynamics, and modifies physics below the distance scale d rv cAt rv hcj(2mec2) = 200 x 10- 13 cm. Therefore, the classical electrodynamics actually did have a finite applicability only down to this distance scale, much earlier than 2.8 x 10- 13 cm as exhibited by the problem of the fine cancellation above. Given this vacuum fluctuation process, one should also consider a process where the electTOn sitting in the vacuum by chance annihilates with the positron and the photon in the vacuum fluctuation, and the electron which used to be a part of the fluctuation remains instead as a real electron. V. Weisskopfl calculated this contribution to the electron selfenergy for the first time, and found that it is negative and cancels the leading

300

piece in the Coulomb self-energy exactly: 1 e2 tlEpair = --4--· 1I"€o

(5)

re

After the linearly divergent piece l/r e is canceled, the leading contribution in the r e -+ 0 limit is given by

tlE

30: 2 h = tlECoulomb + tlEpair = -4 me C log - - . 11" mecr e

(6)

There are two important things to be said about this formula. First, the correction tlE is proportional to the electron mass and hence the total mass is proportional to the "bare" mass of the electron, (me c2 )obs = (m ec 2hare [1

3

+ 40: log 11"

_h_] .

mecre

A7)

Therefore, we are talking about the "percentage" of the correction, rather than a huge additive constant. Second, the correction depends only logarithmically on the "size" of the electron. As a result, the correction is only a 9% increase in the mass even for an electron as small as the Planck distance re = l/Mpl = 1.6 X 10- 33 cm. The fact that the correction is proportional to the "bare" mass is a consequence of a new symmetry present in the theory with the antiparticle (the positron): the chiral symmetry. In the limit of the exact chiral symmetry, the electron is massless and the symmetry protects the electron from acquiring a mass from self-energy corrections. The finite mass of the electron breaks the chiral symmetry explicitly, and because the self-energy correction should vanish in the chiral symmetric limit (zero mass electron), the correction is proportional to the electron mass. Therefore, the doubling of the degrees of freedom and the cancellation of the power divergences lead to a sensible theory of electron applicable to very short distance scales. 1.4

Supersymmetry

In the Standard Model, the Higgs potential is given by V = p?IHI2

+ 'xIHI4,

(8)

where v 2 = (H)2 = -p? /2,X = (176 GeV)2. Because perturbative unitarity requires that ,X ;S 1, _/-L2 is of the order of (100 GeV)2. However, the mass squared parameter /-L 2 of the Higgs doublet receives a quadratically divergent contribution from its self-energy corrections. For instance, the process where

301

the Higgs doublets splits into a pair of top quarks and come back to the Higgs boson gives the self-energy correction 2

!:l.Jltop

1 = - 64h;-22' TH

( ) 9

7r

where THis the "size" of the Higgs boson, and h t :=:;j 1 is the top quark Yukawa coupling. Based on the same argument in the previous section, this makes the Standard Model not applicable below the distance scale of 10- 17 cm. The motivation for supersymmetry is to make the Standard Model applicable to much shorter distances so that we can hope that answers to many of the puzzles in the Standard Model can be given by physics at shorter distance scales. 5 In order to do so, supersymmetry repeats what history did with the positron: doubling the degrees of freedom with an explicitly broken new symmetry. Then the top quark would have a superpartner, stop, a whose loop diagram gives another contribution to the Higgs boson self energy 2

!:l.Jlstop

h;

1

= +64 22' TH

(10)

7r

The leading pieces in 11TH cancel between the top and stop contributions, and one obtains the correction to be 2

!:l.Jltop

2 + !:l.Jltop

h;

2 2) 1 = -62 (mi - m t log -2--2' 4 7r THm i

(11)

One important difference from the positron case, however, is that the mass of the stop, mi, is unknown. In order for the !:l.Jl2 to be of the same order of magnitude as the tree-level value Jl2 = _2-Xv2, we need to be not too far above the electroweak scale. Similar arguments apply to masses of other superpartners that couple directly to the Higgs doublet. This is the so-called naturalness constraint on the superparticle masses (for more quantitative discussions, see papers6).

mf

1.5

Other Directions

Of course, supersymmetry is not the only solution discussed in the literature to avoid miraculously fine cancellations in the Higgs boson mass-squared term. Technicolor (see a review 7 ) is a beautiful idea which replaces the Higgs doublet by a composite techni-quark condensate. Then TH '" 1 TeV is a truly aThis is a terrible name, which was originally meant to be "scalar top." If supersymmetry will be discovered by the next generation collider experiments, we should seriously look for better names for the superparticles.

302

physical size of the Higgs doublet and there is no need for fine cancellations. Despite the beauty of the idea, this direction has had problems with generating fermion masses, especially the top quark mass, in a way consistent with the constraints from the flavor-changing neutral currents. The difficulties in the model building, however, do not necessarily mean that the idea itself is wrong; indeed still efforts are being devoted to construct realistic models. Another recent idea is to lower the Planck scale down to the Te V scale by employing large extra spatial dimensions. 8 This is a new direction which has just started, and there is an intensive activity to find constraints on the idea as well as on model building. Since the field is still new, there is no "standard" framework one can discuss at this point, but this is no surprise given the fact that supersymmetry is still evolving even after almost two decades of intense research. One important remark about all these ideas is that they inevitably predict interesting signals at Te V-scale collider experiments. While we only discuss supersymmetry in this lecture, it is likely that nature has a surprise ready for us; maybe none of the ideas discussed so far is right. Still we know that there is something out there to be uncovered at TeV scale energies. 2

Supersymmetric Lagrangian

We do not go into full-fledged formalism of supersymmetric Lagrangians in this lecture but rather confine ourselves to a practical introduction of how to write down Lagrangians with explicitly broken supersymmetry which still fulfill the motivation for supersymmetry discussed in the previous section. One can find useful discussions as well as an extensive list of references in a nice review by Steve Martin. 9

2.1

Supermultiplets

Supersymmetry is a symmetry between bosons and fermions, and hence necessarily relates particles with different spins. All particles in supersymmetric theories fall into supermultiplets, which have both bosonic and fermionic components. There are two types of supermultiplets which appear in renormalizable field theories: chiral and vector supermultiplets. Chiral supermultiplets are often denoted by the symbol -htQTHu-h~IQI2IHuI2 -h~ITI2IHuI2 -mblQl2 -m}ITI2 -htAtQTHu , (26) where mb, m}, and At are soft parameters. Note that the fields Q, Tare spinor and Q, T, Hu are scalar components of the chiral supermultiplets (an unfortunate but common notation in the literature). This explicit Lagrangian allows us to easily work out the one-loop self-energy diagrams for the Higgs doublet H u , after shifting the field Hu around its vacuum expectation value (this also generates mass terms for the top quark and the scalars which have to be consistently included). The diagram with top quark loop from the first term in Eq. (26) is quadratically divergent (negative). The contractions of Q or T in the next two terms also generate (positive) contributions to the Higgs self-energy. In the absence of soft parameters mb = m} = 0, these two contributions precisely cancel with each other, consistent with the non-renormalization theorem which states that no mass terms (superpotential terms) can be generated by renormalizations. However, the explicit breaking terms mb, m} make the cancellation inexact. With a simplifying assumption d As will be explained in the next section, the right-handed spinors all need to be chargedconjugated to the left-handed ones in order to be part of the chiral supermultiplets. Therefore the chiral supermultiplet T actually contains the left-handed Weyl spinor (tR)c. The Higgs multiplet here will be denoted Hu in later sections. ·We dropped terms which do not contribute to the Higgs boson self-energy at the one-loop level.

307 m~

= m} = m,2, we find (27)

Here, A is the ultraviolet cutoff of the one-loop diagrams. Therefore, these mass-squared parameters are indeed "soft" in the sense that they do not produce power divergences. Similarly, the diagrams with two htA t couplings with scalar top loop produce only a logarithmic divergent contribution. 4

The Minimal Supersymmetric Standard Model

Encouraged by the discussion in the previous section that the supersymmetry can be explicitly broken while retaining the absence of power divergences, we now try to promote the Standard Model to a supersymmetric theory. The Minimal Supersymmetric Standard Model (MSSM) is a supersymmetric version of the Standard Model with the minimal particle content.

4.1

Particle Content

The first task is to promote all fields in the Standard Model to appropriate supermultiplets. This is obvious for the gauge bosons: they all become vector multiplets. For the quarks and leptons, we normally have left-handed and right-handed fields in the Standard Model. In order to promote them to chiral supermultiplets, however, we need to make all fields left-handed Weyl spinors. This can be done by charge-conjugating all right-handed fields. Therefore, when we refer to supermultiplets of the right-handed down quark, say, we are actually talking about chiral supermultiplets whose left-handed spinor component is the left-handed anti-down quark field. As for the Higgs boson, the field Eq. (1) in the Standard Model can be embedded into a chiral supermultiplet Hu. It can couple to the up-type quarks and generate their masses upon the symmetry breaking. In order to generate down-type quark masses, however, we normally use (28)

Unfortunately, this trick does not work in a supersymmetric fashion because the superpotential W must be a holomorphic function of the chiral supermultiplets and one is not allowed to take a complex conjugation of this sort.

308 Table 3. The chiral supermultiplets in the Minimal Supersymmetric Standard Model.. The numbers in the bold face refer to SU(3)c, SU(2)L representations. The superscripts are hypercharges.

L 1(1,2)-1/2 El(I,I)+1 Ql (3,2)1/6 U1(3,1)-2/3 Dl(3,1)+1/ 3

L 2(1,2)-1/2 E 2 (1, 1)+1 Q2(3,2)1/6 U2(3,1)-2/3 D2(3,1)+1/3 Hu(I,2)+1/2 H d (I,2)-1/2

L 3(1,2)-1/2 E3(1, 1)+1 Q3(3,2)1/6 U3(3,1)-2/3 D3(3,1)+1/ 3

Therefore, we need to introduce another chiral supermultiplet Hd which has the same gauge quantum numbers of ia2H* above'! In all, the chiral supermultiplets in the Minimal Supersymmetric Standard Model are listed in Table 3. The particles in the MSSM are referred to as follows. 9 First of all, all quarks, leptons are called just in the same way as in the Standard Model, namely electron, electron-neutrino, muon, muon-neutrino, tau, tau-neutrino, up, down, strange, charm, bottom, top. Their superpartners, which have spin O,are named with "s" at the beginning, which stand for "scalar." They are denoted by the same symbols as their fermionic counterpart with the tilde. Therefore, the superpartner of the electron is called "selectron," and is written as e. All these names are funny, but probably the worst one of all is the "sstrange" (8), which I cannot pronounce at all. Superpartners of quarks are "squarks," and those of leptons are "sleptons." Sometimes all of them are called together as "sfermions," which does not make sense at all because they are bosons. The Higgs doublets are denoted by capital H, but as we will see later, their physical degrees of freedom are hO, HO, AO and H±. Their superpartners are called "higgsinos," written as iie, ii;+;, iii, ii~. In general, fermionic superpartners of boson in the Standard Model have "ino" at the end of the name. Spin 1/2 superpartners of the gauge bosons are "gauginos" as mentioned in the previous section, and for each gauge groups: f Another reason to need both Hu and Hd chiral supermultiplets is to cancel the gauge anomalies arising from their spinor components. 9When I first learned supersymmetry, I didn't believe it at all. Doubling the degrees of freedom looked too much to me, until I came up with my own argument at the beginning of the lecture. The funny names for the particles were yet another reason not to believe in it. It doesn't sound scientific. Once supersymmetry will be discovered, we definitely need better sounding names!

309

gluino for gluon, wino for W, bino for U(l)y gauge boson B. As a result of the electroweak symmetry breaking, all neutral "inos", namely two neutral higgsinos, the neutral wino W3 and the bino iJ mix with each other to form four Majorana fermions. They are called "neutralinos" X? for i = 1,2,3,4. Similarly, the charged higgsinos iIt, iIi, W-, W+ mix and form two massive for i = 1,2. All particles with tilde do not exist Dirac fermions "charginos" in the non-supersymmetric Standard Model. Once we introduce R-parity in a later section, the particles with tilde have odd R-parity.

xt

4.2 Superpotential The SU(3)c x SU(2)L x U(l)y gauge invariance allows the following terms in the superpotential

(29) The first three terms correspond to the Yukawa couplings in the Standard Model (with exactly the same number of parameters). The subscripts i, j, k are generation indices. The parameter p, has mass dimension one and gives a supersymmetric mass to both fermionic and bosonic components of the chiral supermultiplets Hu and Hd. The terms in the second line of Eq. (29) are in general problematic as they break the baryon (B) or lepton (L) numbers. If the superpotential contains both B- and L-violating terms, such as A~12UIDID2 and A121QID2Ll, one can exchange D2 = s to generate a fourfermion operator (30)

where the spinor indices are contracted in each parentheses and the color indices by the epsilon tensor. Such an operator would contribute to the proton decay process p -+ e+7fo at a rate of r '" A'4m~/m~, and hence the partial lifetime of the order of Tp '"

6 x 10

-13

(ma)4

sec 1 TeV

1

N4 .

(31)

Recall that the experimental limit on the proton partial lifetime in this mode is Tp > 1.6 X 1033 years. 17 Unless the coupling constants are extremely small, this is clearly a disaster.

310

4.3 R-parity To avoid this problem of too-rapid proton decay, a common assumption is a discrete symmetry called R-parity18 (or matter parity19). The Z2 discrete charge is given by

(32) where s is the spin of the particle. Under R p , all standard model particles, namely quarks, leptons, gauge bosons, and Higgs bosons, carry even parity, while their superpartners odd due to the (_1)28 factor. Once this discrete symmetry is imposed, all terms in the second line of Eq. (29) will be forbidden, and we do not generate a dangerous operator such as that in Eq. (30). Indeed, B- and L-numbers are now accidental symmetries of the MSSM Lagrangian as a consequence of the supersymmetry, gauge invariance, renormalizability and R-parity conservation. One immediate consequence of the conserved R-parity is that the lightest particle with odd R-parity, i.e., the Lightest Supersymmetric Particle (LSP), is stable. Another consequence is that one can produce (or annihilate) superparticles only pairwise. These two points have important implications on the collider phenomenology and cosmology. Since the LSP is stable, its cosmological relic is a good (and arguably the best) candidate for the Cold Dark Matter particles (see, e.g., a review 20 on this subject). If so, we do not want it to be electrically charged and/or strongly interacting; otherwise we should have detected them already. Then the LSP should be a superpartner of Z, " or neutral Higgs bosons or their linear combination (called neutralino).h On the other hand, the superparticles can be produced only in pairs and they decay eventually into the LSP, which escapes detection. This is why the typical signature of supersymmetry at collider experiments is the missing energy /momentum. The phenomenology of R-parity breaking models has been also studied. If either B-violating or L-violating terms exist in Eq. (29), but not both, they would not induce proton decay.23 However they can still produce n-n oscillation and a plethora of flavor-changing phenomena. We refer to a recent compilation of phenomenological constraints 24 for further details.

h A sneutrino can in principle be the LSP, 12, but it cannot be the CDM to avoid constraints from the direct detection experiment for the CDM particies. 21 It becomes a viable candidate again if there is a large lepton number violation. 22

311

4.4

Soft Supersymmetry Breaking Terms

In addition to the interactions that arise from the superpotential Eq. (29), we should add soft supersymmetry breaking terms to the Lagrangian as we have not seen any of the superpartners of the Standard Model particles. Following the general classifications in Eq. (23), and assuming R-parity conservation, they are given by (33)

(34) £2 = -A~ >..~ii£JjHu - A1 >..1 iiJJj Hd - A;j >..~iiJjjHd

+ BpBuHd + c.c. (35)

The mass-squared parameters for scalar quarks (squarks) and scalar leptons (sleptons) are all three-by-three hermitian matrices, while the trilinear couplings Aij and the bilinear coupling B of mass dimension one are general complex numbers. i

4.5

Higgs Sector

It is of considerable interest to look closely at the Higgs sector of the MSSM.

Following the general form of the super symmetric Lagrangians Eqs. (17,15,18) with the superpotential W = J1HuHd in Eq. (29) as well as the soft parameters in Eq. (34), the potential for the Higgs bosons is given as V

12 (1 HLH

= L2

2 2 +J1 (1HuI

u2

u

1)2 +!L2(~ Ht~H

+Ht=-Hd d 2

2

u2

+ IHdI2) + mku IH ul + mkdlHdl2 2

u

~)2 +Ht~Hd d2

- (BJ1Hu H d + c.c.) (36)

It turns out that it is always possible to gauge-rotate the Higgs bosons such

that (37)

in the vacuum. Since only electrically neutral components have vacuum expectation values, the vacuum necessarily conserves U(l)QED.i Writing the iIt is unfortunate that the notation A is used both for the scalar components of chiral supermultiplets and the trilinear couplings. Hopefully one can tell them apart from the context. iThis is not necessarily true in general two-doublet Higgs Models. Consult a review. 25

312

potential (36) down using the expectation values (37), we find (38)

where g~ = g2 + g,2. In order for the Higgs bosons to acquire the vacuum expectation values, the determinant of the mass matrix at the origin must be negative, (39)

However, there is a danger that the direction Vu = Vd, which makes the quartic term in the potential identically vanish, may be unbounded from below. For this not to occur, we need 1'2

+ mh.. + 1'2 + mhd > 21'B.

(40)

In order to reproduce the mass of the Z-boson correctly, we need

Vu

= ~ sin/1,

Vd

= ~ cos/1,

v

= 250 GeV.

(41)

The vacuum minimization conditions are given by 8V/8v u = 8V/8Vd = 0 from the potential Eq. (38). Using Eq. (41), we obtain 1'2

2

2

m + m Hd = _---K 2

m 2H.. tan 2 /1 , tan 2 /1 - 1 -

(42)

and (43)

Because there are two Higgs doublets, each of which with four real scalar fields, the number of degrees of freedom is eight before the symmetry breaking. However three of them are eaten by W+, W- and Z bosons, and we are left with five physics scalar particles. There are two CP-even scalars hO, HO, one CP-odd scalar AO, and two charged scalars H+ and H-. Their masses can be worked out from the potential (38): 2

2

2

m H ± =mW+mA'

(44)

and

m~o, mho = ~ ( m~ + m~ ± J(m~ + m~)2 - 4m1m~ cos2 2/1) .

(45)

A very interesting consequence of the formula Eq. (45) is that the lighter CPeven Higgs mass m~o is maximized when cos 2 2/1 = 1: m~o = (m~ + m1 -

313

=

Im~ - m~I)/2. When mA < mz, we obtain m~o m~ mA > mz, m~o m~. Therefore in any case we find

=


H+ with H+

~ Ti'

10 9

• 7

";:.1.4

_._.\\\\,

LEP2 :",'VLdt=150pb-' " \,fs=192GeV .,

I

HlA ....., tt

i 250

JOO

J50

400

450

500

m. (GeV) Figure 2. Expected coverage of the MSSM Higgs sector parameter space by the LHC experiments, after three years of high-luminosity running.

-mZ S W c j3 mzcWcj3

o

-f.-t

(48)

315

Here, Sw = sin Ow, Cw = cos Ow, s(3 = sin,8, and c(3 = cos,8. Once Ml, M 2, J.L exceed mz, which is preferred given the current experimental limits, one can regard components proportional to mz as small perturbations. Then the neutralinos are close to their weak eigenstates, bino, wino, and higgsinos. But the higgsinos in this limit are mixed to form symmetric and anti-symmetric linear combinations iI~ = (iI~ + iI~)/.j2 and iI~ = (iI~ - iI~)/.j2. Similarly two positively charged inos: iI;t; and W+, and two negatively charged inos: iIi and W- mix. The mass matrix is given by

1:-::) -(W- iI-) ( d

M2

.j2mw c(3

.j2m W S(3) J.L

(~+) +C.C. Ht

(49)

Again once M 2 , J.L 2: mw, the chargino states are close to the weak eigenstates winos and higgsinos.

4.7 Squarks, Sleptons The mass terms of squarks and sleptons are also modified after the electroweak symmetry breaking. There are four different contributions. One is the supersymmetric piece coming from the 18W/84>d 2 terms in Eq. (15) with 4>i = Q, U, D, L, E. These terms add m} where mf is the mass of the quarks and leptons from their Yukawa couplings to the Higgs boson. Next one is combing from the 18W/84>iI 2 terms in Eq. (15) with 4>i = Hu or Hd in the superpotential Eq. (29). Because of the J.L term,

8W

8Ho

- = -J.LHd0+ " )..';1 QiUj,

(50)

u

(51) Taking the absolute square of these two expressions pick the cross terms together with (H~) = vcos/3/.j2, (H~) = vsin/3/.j2 and we obtain mixing between Q and U, Q and fl, and L and E. Similarly, the vacuum expectation values of the Higgs bosons in the trilinear couplings Eq. (35) also generate similar mixing terms. Finally, the D-term potential after eliminating the auxiliary field D Eq. (18) also give contributions to the scalar masses m~(h - Q sin 2 Ow) cos 2/3. Therefore, the mass matrix of stop, for instance, is given as

316

mt(At-ILcot{3) ) ( mb3+m;+mM!-~s~)C2{J mt(At -ILcot{3) mb3+m;+m~(-~s~)C2{J

(h) tR' (52)

with C2{J = cos 2{3. Here, h is the up component of Q3, and tR = t*. For first and second generation particles, the off-diagonal terms are negligible for most purposes. They may, however, be important when their loops in flavorchanging processes are considered.

4.8

What We Gained in the MSSM

It is useful to review here what we have gained in the MSSM over what we had in the Standard Model. The main advantage of the MSSM is of course what motivated the supersymmetry to begin with: the absence of the quadratic divergences as seen in Eq. (27). This fact allows us to apply the MSSM down to distance scales much shorter than the electroweak scale, and hence we can at least hope that many of the puzzles discussed at the beginning of the lecture to be solved by physics at the short distance scales. There are a few amusing and welcome by-products of supersymmetry beyond this very motivation. First of all, the Higgs doublet in the Standard Model appears so unnatural partly because it is the only scalar field introduced just for the sake of the electroweak symmetry breaking. In the MSSM, however, there are so many scalar fields: 15 complex scalar fields for each generation and two in each Higgs doublet. Therefore, the Higgs bosons are just "one of them." Then the question about the electroweak symmetry breaking is addressed in a completely different fashion: why is it only the Higgs bosons that condense? In fact, one can even partially answer this question in the renormalization group analysis in the next sections where "typically" (we will explain what we mean by this) it is only the Higgs bosons which acquire negative mass squared (39) while the masses-squared of all the other scalars "naturally" remain positive. Finally, the absolute upper bound on the lightest CP-even Higgs boson is falsifiable by experiments. However, life is not as good as we wish. We will see that there are very stringent low-energy constraints on the MSSM in the next section. 5

Low-Energy Constraints

Despite the fact that we are interested in superparticles in the 100-1000 GeV range, which we are just starting to explore in collider searches, there are many amazingly stringent low-energy constraints on superparticles.

317

9

9

Figure 3. A Feynman diagram which gives rise to bomK and

CK.

One of the most stringent constraints comes from the KO_j(o miXIng parameters D.mK and CK. The main reason for the stringent constraints is that the scalar masses-squared in the MSSM Lagrangian Eq. (34) can violate flavor, i.e., the scalar masses-squared matrices are not necessarily diagonal in the basis where the corresponding quark mass matrices are diagonal. To simplify the discussion, let us concentrate only on the first and the second generations (ignore the third). We also go to the basis where the down-type Yukawa matrix A;J is diagonal, such that \ij

_

Ad Vd -

(md0 ms0 )

(53)

.

Therefore the states K O = (ds) , k O = (sd) are well-defined in this basis. In the same basis, however, the squark masses-squared can have off-diagonal elements in general, m 2"lJ

-

D -

(m~dR m'i> ' 12) m2* D,12

m2

SR

.

(54)

Since their off-diagonal elements will be required to be small (as we will see later), it is convenient to treat them as small perturbation. We insert the off-diagonal elements as two-point Feynman vertices which change the squark flavor dL,R f-t h,R in the diagrams. To simplify the discussion further, we assume that all squarks and gluino the are comparable in their masses m. Then the relevant quantities are given in terms of the ratio (8f2)LL == mb,12/m2 (and similarly (8f2)RR = m'i> 121m2), as depicted in Fig. 3. The operator from this Feynman diagram is' estimated approximately as

f

2 2 ) 0005 . as (8 -2)1,L(d-L'Y J.L SL )(d-L'YJ.LSL· m

(55)

318

This operator is further sandwiched between KO and kO states, and we find Llmk '" 0.005fimko:;(8t2)iL _\ m

= 1.2 x 10-

12 Gey2

C6:~eY ) 2 (;~f (8t2)iL < 3.5

X

10-

15 Gey2, (56)

where the last inequality is the phenomenological constraint in the absence of accidental cancellations. This requires

~eY )

(8f2)LL ;S 0.05 (500

(57)

and hence the off-diagonal element m~,12 must be small. It turns out that the product (8t2)LL(8t2)RR is more stringently constrained, especially its imaginary part from CK. Much more careful and detailed analysis than the above order-of-magnitude estimate gives 29

There are many other low-energy observables, such as electron -and neutron electric dipole moments (EDM) , /L -+ which place important constraints on the supersymmetry parameters. 30 There are various ways to avoid such low-energy constraints on supersymmetry. The first one is called "universality" of soft parameters. 31 It is simply assumed that the scalar masses-squared matrices are proportional to identity matrices, i.e., m~, m'tJ, mb ex: 1. Then no matter what rotation is made in order to go to the basis where the quark masses are diagonal, the identity matrices stay the same, and hence the off-diagonal elements are never produced. There has been many proposals to generate universal scalar masses either by the mediation mechanism of the supersymmetry breaking such as the gauge mediated (see reviews 32 ), anomaly mediated 33 , or gaugino mediated 34 supersymmetry breaking, or by non-Abelian flavor symmetries. 35 The second possibility is called "alignment," where certain flavor symmetries should be responsible for "aligning" the quark and squark mass matrices such that the squark masses are almost diagonal in the same basis where the down-quark masses are diagonal. 36 Because of the CKM matrix it is impossible to do this both for down-quark and up-quark masses. Since the phenomenological constraints in the up-quark sector are much weaker than in the down-quark sector, this choice would alleviate many of the low-energy constraints (except for flavor-diagonal CP-violation such as EDMs). Finally there is a possibility

e"

319

called "decoupling," which assumes first- and second-generation superpartners much heavier than TeV while keeping the third-generation superpartners as well as gauginos in the 100 GeV range to keep the Higgs self-energy small enough. 37 Even though this idea suffers from a fine-tuning problem in general,38 many models had been constructed to achieve such a split mass spectrum recently. 39 In short, the low-energy constraints are indeed very stringent, but there are many ideas to avoid such constraints naturally within certain model frameworks. Especially given the fact that we still do not know any of the superparticle masses experimentally, one cannot make the discussions more clear-cut at this stage. On the other hand, important low-energy effects of supersymmetry are still being discovered in the literature, such as muon 9 - 2,40 and direct CP-violationY They may be even more possible low-energy manifestations of supersymmetry which have been missed so far. 6

Renormalization Group Analyses

Once supersymmetry protects the Higgs self-energy against corrections from the short distance scales, or equivalently, the high energy cutoff scales, it becomes important to connect physics at the electroweak scale where we can do measurements to the fundamental parameters defined at high energy scales. This can be done by studying the renormalization-group evolution of parameters. It also becomes a natural expectation that the supersymmetry breaking itself originates at some high energy scale. If this is the case, the soft supersymmetry breaking parameters should also be studied using the renormalization-group equations. We study the renormalizationgroup evolution of various parameters in the softly-broken supersymmetric Lagrangian at the one-loop level. l If supersymmetry indeed turns out to be the choice of nature, the renormalization-group analysis will be crucial in probing physics at high energy scales using the observables at the TeV-scale collider experiments. 44 6.1

Gauge Coupling Constants

The first parameters to be studied are naturally the coupling constants in the Standard Model. The running of the gauge couplings constants are described in term of the beta functions, and their one-loop solutions in non'Recently, there have been developments in obtaining and understanding all-order beta functions for gauge coupling constants 42 and soft parameters. 43

320

supersymmetric theories are given by 1 g2 (IL)

bo

1

IL

= g2 (IL') + 8rr2 log IL"

(59)

with 11 2 1 bo = -C2 (G) - -Sf - -Sb 3 3 3· This formula is for Weyl fermions factors are defined by

(60)

f and complex scalars b. The group theory

r

8ad C2 (G) = bc fdbc 8ab Sf ,b = TrTaT b

(61) (62)

and C2 (G) = Nc for SU(Nc) groups and Sf,b = 1/2 for their fundamental representations. In supersymmetric theories, there is always the gaugino multiplet in the adjoint representation of the gauge group. They contribute to Eq. (60) with Sf = C2 (G), and therefore the total contribution of the vector supermultiplet is 3C2 (G). On the other hand, the chiral supermultiplets have a Weyl spinor and a complex scalar, and the last two terms in Eq. (60) are always added together to Sf = Sb. Therefore, the beta function coefficients simplify to

(63) Given the beta functions, it is easy to work out how the gauge coupling constants measured accurately at LEP ISLe evolve to higher energies. One interesting possibility is that the gauge groups in the Standard Model SU(3)c x SU(2)L x U(1)y may be embedded into a simple group, such as 8U(5) or 80(10), at some high energy scale, called "grand unification." The gauge coupling constants at IL '" mz are approximately a-I = 129, sin 2 Ow ::: 0.232, and a;1 = 0.119. In the SU(5) normalization, the U(1) coupling constant is given by al = ~a' = ~a/ cos 2 Ow. It turns out that the gauge coupling constants become equal at IL ::: 2 x 1016 GeV given the MSSM particle content (Fig. 4). On the other hand, the three gauge coupling constants miss each other quite badly with the non-supersymmetric Standard Model particle content. This observation suggests the possibility of supersymmetric grand unification. 6.2

Yukawa Coupling Constants

Since first- and second-generation Yukawa couplings are so small, let us ignore them and concentrate on the third-generation ones. Their renormalization-

321

.., ~-

60

60

50

50

40

..,

30

~-

20

MSSM

40 30 20

Standard Model

10 0102

10'

10"

10"

10'0 10 12 1014 10'6 10'·

Jl (GeV)

10

°lo''''IO'''"'·'i(fioa-·~'iOi4'"'iQi6'lo'· Jl (GeV)

Figure 4. Running of gauge coupling constants in the Standard Model and in the MSSM.

group equations are given as (64) (65) (66)

The important aspect of these equations is that the gauge coupling constants push down the Yukawa coupling constants at higher energies, while the Yukawa couplings push them up. This interplay, together with a large top Yukawa coupling, allows the possibility that the Yukawa couplings may also unify at the same energy scale where the gauge coupling constants appear to unify (Fig. 5). It turned out that the actual situation is much more relaxed than what this plot suggests. This is because there is a significant correction to mb at tan J3 2: 10 when the superparticles are integrated out 45.

6.3

Soft Parameters

Since we do not know any of the soft parameters at this point, we cannot use the renormalization-group equations to probe physics at high energy scales. On the other hand, we can use the renormalization-group equations from boundary conditions at high energy scales suggested by models to obtain useful information on the "typical" superparticle mass spectrum.

322 100 '"b= 3.1---.,

...Iiii

---------------------------~

co..

c:

«S

10 Msusy= mt

o.s(M z)=O.118

A.b = A..

1

100

120

140

160

180

200

mt(mt) (GeV) Figure 5. The regions on (mt, tan {3) plane where hb = hT at the GUT-scale. 46

First of all, the gaugino mass parameters have very simple behavior that

d Mi

J-t- - 2 =0. d J-t 9i

(67)

Therefore, the ratios Mij g'f are constants at all energies. If the grand unification is true, both the gauge coupling constants and the gaugino mass parameters must unify at the GUT-scale and hence the ratios are all the same at the GUT-scale. Since the ratios do not run, the ratios are all the same at any energy scales, and hence the low-energy gaugino mass ratios are predicted to be

(68) at the TeV scale. We see the tendency that the colored particle (gluino in this case) is much heavier than uncolored particle (wino and bino in this case). This turns out to be a relatively model-independent conclusion. The running of scalar masses is given by simple equations when all Yukawa

323

couplings other than that of the top quark are neglected. We find 6 2 2 2 d 2 22 1611" J.t dJ.t mHu = 3Xt - 692 M 2 - 591 M1 , d

1611" J.t dJ.t mHd

22 = -692M2 -

2 d 2 1611" J.tdJ.tmQ3

=Xt -

2

2

d

2

32

6

2

()

69

2

()

70

591 M 1 ,

2

2

2

2

2

2

2

393M3 -692M2 - 1591M1' 32

2

2

2

32

2

2

1611" J.t dJ.t mus = 2Xt - 393M3 - 1591M1'

(

71

)

(72)

Here, X t = 2h;(mhu + m~3 + mf;s) and the trilinear couplings are also neglected. Even within this simplified assumptions, one learns interesting lessons. First of all, the gauge interactions push the scalar masses up at lower energies due to the gaugino mass squared contributions. Colored particles are pushed up even more than uncolored ones, and the right-handed sleptons would be the least pushed up. On the other hand, Yukawa couplings push the scalar masses down at lower energies. The coefficients of X t in the Eqs. (69, 71, 72) are simply the multiplicity factors which correspond to 3 of SU(3)c, 2 of SU(2)y and 1 of U(l)y. It is extremely amusing that the mhu is pushed down the most because of the factor of three as well as is pushed up the least because of the absence of the gluino mass contribution. Therefore, the fact that the Higgs mass squared is negative at the electroweak scale may well be just a simple consequence of the renormalization-group equations! Since the Higgs boson is just "one of them" in the MSSM, the renormalization-group equations provide a very compelling reason why it is only the Higgs boson whose mass-squared goes negative and condenses. One can view this as an explanation for the electroweak symmetry breaking. 6.4

Minimal Supergravity

Of course, nothing quantitative can be said unless one makes some specific assumptions for the boundary conditions of the renormalization-group equations. One common choice called "Minimal Supergravity" is the following set of assumptions: 2ij _ 2ij _ 2ij _ 2ij _ 2ij _ 2 >:ij mQ - mu - m D - m L - mE - mou , 2

2

2

mHu = mHd = mo, - Aij - A O ij - -Aij Au d-l-

M1 = M2 = M3 = M 1/ 2

324

at the GUT-scale. The parameter mo is called the universal scalar mass, Ao the universal trilinear coupling, and M 1 / 2 the universal gaugino mass. Once this assumption is made, there are only five parameters at the GUT-scale, (mo, M 1 / 2 , Ao , B, J1.). This assumption also avoids most of the low-energy constraints easily because the scalar mass-squared matrices are proportional to the identity matrices and hence there is no flavor violation. Of course this is probably an oversimplification of the parameter space, but it still provides useful starting point in discussing phenomenology. Especially most of the search limits from collider experiments have been reported using this assumption. In general, this choice of the boundary conditions, which actually have not much to do with supergravity itself, lead to acceptable and interesting phenomenology including the collider signatures, low-energy constraints as well as cosmology. 7

Collider Phenomenology

We do not go into much details of the collider phenomenology of supersymmetry in this lecture notes and we refer to reviews. 47 Here, we give only a very brief summary of collider phenomenology. Supersymmetry is an ideal target for current and new future collider searches. As long as they are within the mass scale expected by the argument given at the beginning of the lecture, we expect supersymmetric particles to be discovered at LEP-II (even though the phase space left is quite limited by now), Tevatron Run-II, or the LHC. The next two figures Figs. 6, 7 show the discovery reach of supersymmetry at LEP-II, Tevatron Run II, LHC. It is fair to say that the mass range of superparticles relevant to solve the problem of fine cancellation in the Higgs boson self-energy described at the beginning of the lecture is covered by these experiments. A future e+e- linear collider would playa fantastic role in proving that new particles are indeed superpartners of the known Standard Model particles and in determining their parameters. 50 Once such studies will be done, we will exploit renormalization-group analyses trying to connect physics at TeV scale to yet-more-fundamental physics at higher energy scales. Example of such possible studies are shown in Fig. 9. The measurements of gaugino masses were simulated. At the LHC, the measurements are basically on the gluino mass and the LSP mass which is assumed to be the bino state, and their mass difference can be measured quite well. By assuming a value of the LSP mass, one can extract the gluino mass. At the e+e- linear colliders, one can even disentangle the mixing in neutralino and chargino states employing expected high beam polarizations and determine Ml and M2 in a model-independent

325 250 554

200

436> Q) ~

----

-------------w;iOOio

r

- - - '3Q(2aolb)

~~~~~-r~179

50

o

314

100

200

300

400

500

ma(GeV)

200

~150

439> Q)

~

~

50

W,(47)

17&

~~~ 200 300 400 500

100

ma (GeV) Baer0116GM

Figure 6. Regions in the mo vs.ml/2 plane explorable by Tevatron and LEP II experiments. 4 7

matter. Combination of both types of experiments determine all three gaugino masses, which would provide a non-trivial test of the grand unification. 8

Mediation Mechanisms of Supersymmetry Breaking

One of the most important questions in the super symmetry phenomenology is how supersymmetry is broken and how the particles in the MSSM learn the effect of supersymmetry breaking. The first one is the issue of dynamical supersymmetry breaking, and the second one is the issue of the "mediation"

326

800 600 400 200

800

800

;;.,

600

600

::;'"

400

400

200

200

~

S

0

0

500

1000

1500

rno (GeV)

2000

500

1000

1500

2000

rno (GeV)

Figure 7. Regions in the rno VS.rnl/2 plane explorable by LHC experiments with lO fb- 1 of integrated luminosity.48 Different curves correspond to different search modes: 11 (single lepton), ItT (missing transverse ~ne!gy), SS (same sign dilepton), 31 (trilepton), OS (opposite sign dilepton), I (slepton), W 1 Z2 (charged wino, neutral wino associated production).

mechanism. The problem of the supersymmetry breaking itself has gone through a dramatic progress in the last few years thanks to works on the dynamics of supersymmetric gauge theories by Seiberg. lO The original idea by Witten 5 was that the dynamical supersymmetry breaking is ideal to explain the hierarchy. Because of the non-renormalization theorem, if supersymmetry is unbroken at the tree-level, it remains unbroken at all orders in perturbation theory.

327

1 rol

800

800

I~

I;;'

600

1

10

1

1

~

W\(500)

--~--~-------------

600

ok

~

§

400

400

'-1

W,(250)

NLC500

200

: NLC500

a)

,

200

tanp~2

a)

tanp~IO

JL , mh increases above the M z . However, it is now well established that in a large class of supersymmetric models (which do not differ too much from the MSSM), the Higgs mass is less than 150 Ge V or so. Another very interesting property of the MSSM is that electroweak symmetry breaking can be induced by radiative corrections. As we will see below, in all the schemes for generating soft supersymmetry breaking terms via a hidden sector, one generally gets positive (mass)2's for all scalar fields at the scale of SUSY breaking as well as equal mass-squares. In order to study the theory at the weak scale, one must extrapolate all these parameters using the renormalization group equations. The degree of extrapolation will of course depend on the strength of the gauge and the Yukawa couplings of the various fields. In particular, the mku will have a strong extrapolation proportional to l~r2 since Hu couples to the top quark. Since ht c:: 1, this can make (Mz) < 0, leading to spontaneous breakdown of the electroweak symmetry. An approximate

mk

340

solution of the renormalization group equations gives

(7) This is a very attractive feature of supersymmetric theories. 1.2

Why go beyond the MSSM ?

Even though the MSSM solves two outstanding peoblems of the standard model, i.e. the stabilization of the Higgs mass and the breaking of the electroweak symmetry, it brings in a lot of undesirable consequences. They are: (a) Presence of arbitrary baryon and lepton number violating couplings i.e. the A, A' and A" couplings described above. In fact a combination of A' and A" couplings lead to proton decay. Present lower limits on the proton lifetime then imply that A' A" ::; 10-25 for squark masses of order of a TeV. Recall that a very attractive feature of the standard model is the automatic conservation of baryon and lepton number. The presence of R-parity breaking termS' also makes it impossible to use the LSP as the Cold Dark Matter of the universe since it is not stable and will therefore decay away in the very early moments of the universe. We will see that as we proceed to discuss the various grand unified theories, keeping the R-parity violating terms under control it will provide a major constraint on model building. (b) The different mixing matrices in the quark and squark sector leads to arb.itrary amount of flavor violation manifesting in such phenomena as K L - K s mass difference etc. Using present experimental information and the fact that the standard model more or less accounts for the observed magnitude of these processes implies that there must be strong constraints on the mass splittings among squarks. Detailed calculations indicatE! that one must have ~mUm~ 10- 3 or so. Again recall that this undoes another nice feature of the standard model. (c) The presence of new couplings involving the super partners allows for the existence of extra CP phases. In particular the presence of the phase in the gluino mass leads to a large electric dipole moment of the neutron unless this phase is assumed to be suppressed by two to three orders of magnitud~ . This is generally referred to in the literature as the SUSY CP rpoblem. In addition, there is of course the famous strong CP problem which neither the standard model nor the MSSM provide a solution to. In order to cure these problems as well to understand the origin of the soft SUSY breaking terms, one must seek new physics beyond the MSSM. Below, we pursue two kinds of directions for new physics: one which analyses schemes

::;

341

that generate soft breaking terms and a second one which leads to automatic Band L conservation as well as solves the SUSY CP problem. The second model also provides a solution to the strong CP problem without the need for an axion under certain circumstances.

1.3 Supersymmetric Left-Right model One of the attractive features of the super symmetric models is its ability to provide a candidate for the cold dark matter of the universe. This however relies on the theory obeying R-parity conservation (with R == (_1)3(B-£)+2s). It is easy to check that particles of the standard model are even under R whereas their superpartners are odd. The lightest superpartner is then absolutely stable and can become the dark matter of the universe. In the MSSM, R-parity symmetry is not automatic and is achieved by imposing global baryon and lepton number conservation on the theory as additional requirements. First of all, this takes us one step back from the non-supersymmetric standard model where the conservation Band L arise automatically from the gauge symmetry and the field content of the model. Secondly, there is a prevalent lore supported by some calculations that in the presence of nonperturbative gravitational effects such as black holes or worm holes, any externally imposed global symmetry must be violated by Planck suppressed operators 10. In this case, the R-parity violating effects again become strong enough to cause rapid decay of the lightest R-odd neutralino so that there is no dark matter particle in the minimal supersymmetric standard model. It is therefore desirable to seek supersymmetric theories where, like the standard model, R-parity conservation (hence Baryon and Lepton number conservation) becomes automatic i.e. guaranteed by the field content and gauge symmetry. It was realized in mid-80's 11 that such is the case in the supersymmetric version of the left-right model that implements the see-saw mechanism for neutrino masses. We briefly discuss this model in the section. The gauge group for this model is SU(2)£ x SU(2)R X U(I)B_£ x SU(3)c. The chiral superfields denoting left-handed and right-handed quark superfields are denoted by Q == (u, d) and QC == (dC, _u C) respectively and similarly the lepton superfields are given by L == (v, e) and U == (e C , _v C ). The Q and L transform as left-handed doublets with the obvious values for the B - Land the QC and LC transform as the right-handed doublets with opposite B - L values. The symmetry breaking is achieved by the following set of Higgs superfields: # 0 ---+ Gstd

X

SUSY

(53)

To sudy this we have to use W G and calculate the relevant F -terms and set them to zero to maintain supersymmetry down to the weak scale. F~,{3

= z8$ + 2x3 + 3y~; = 0

(54)

356

-h

Taking < Trif! >= 0 implies that z = < Trif!2 >. If we assume that Diag < if! >= (al,a2,a3,a4,a5), then one has the following equations:

z

~iai

= 0

+ 2xai + 3ya;

= 0

(55)

with i = 1, ... 5. Thus we have five equations and two parameters. There are therefore three different choices for the ai's that can solve the above equations and they are: Case (A):

< if! >= 0

(56)

In this case, SU(5) symmetry remains unbroken. Case (B):

Diag < if! >= (a, a, a, a, -4a)

(57)

In this case, SU(5) symmetry breaks down to SU(4) x U(l) and one can find a = ~~. Case (C):

Diag

< «I> >= (b , b"b -~b -~b) 2' 2

(58)

This is the desired vacuum since SU(5) in this case breaks down to SU(3)c x SU(2)L x U(l)y gauge group of the standard model. The value of b = ~~ and we choose the parameters x to be order of Mu. In the supersymmetric limit all vacua are degenerate.

3.2

Low energy spectrum and doublet-triplet splitting

Let us next discuss whether the MSSM arises below the GUT scale in this model. So far we have only obtained the gauge group. The matter content of the MSSM is also already built into the P and T multiplets. The only remaining question is that of the two Higgs superfields Hu and Hd of MSSM. They must come out of the H and the fI multiplets. Writing H ==

fI ==

(~d).

From W G substituting the < «I>

(k:)

and

> for case (C), we obtain, (59)

357

If we choose 3/2b = M, then the massless standard model doublets remain and every other particle of the SU(5) model gets large mass. The uncomfortable aspect of this procedure is that the adjustment of the parameters is done by hand does not emerge in a natural manner. This procedure of splitting of the color triplets (u,d from SU(2)L doublets Hu,d is called doublet-triplet splitting and is a generic issue in all GUT models. An advantage of SUSY GUT's is that once the fine tuning is done at the tree level, the nonrenormalization theorem of the SUSY models preserves this to all orders in perturbation theory. This is one step ahead of the corresponding situation in non- SUSY GUT's, where the cancellation between band M has to be done in each order of perturbation theory. A more satisfactory situation would be where the doublet-triplet splitting emerges naturally due to requirements of group theory or underlying dynamics.

3.3

Fermion masses and Proton decay

Effective superpotential for matter sector at low energies then looks like:

Note that hd and hi arise from the T F H coupling and this satisfy the relation hd = hi. Similarly, hu arises from the TT H coupling and therefore obeys the constraint hu = h~. (None of these constraints are present in the MSSM). The second relation will be recognized by the reader as a partial Yukawa unification relation and we can therefore use the discussion of Section 2 to predict m T in terms of mb. The relation between the Yuakawa couplings however hold for each generation and therefor imply the undesirable relations among the fermion masses such as md/ms = me/mil" This relation is independent of the mass scale and therefore holds also at the weak scale. It is in disagreement with observations by almost a factor of 15 or so. This a major difficulty for minimal SU(5) model. This problem does not reflect any fundamental difficulty with the idea of grand unification but rather with this particular realization. In fact by including additional multiplets such as 45 in the theory, one can avoid this problem. Another way is to add higher dimensional operators to the theory such as TFif?H/MpI, which can be of order of a 0.1 GeV or so and could be used to fix the muon mass prediction from SU(5). The presence of both quarks and leptons in the same multiplet of SU(5) model leads to proton decay. For detailed discussions of this classic feature of GUTs, see for instance 2. In non-SUSY SU(5), there are two classes of Feynman diagrams that lead to proton decay in this model: (i) the exchange of

358

gauge bosons familiar from non-SUSY SU(5) where effective operators of type e+t udct u are generated; and (ii) exchange of Higgs fields. In the supersymmetric case theer is an additional source for proton decay coming from the exchange of Higgsinos, where QQH and QLfI via H fI mixing generate the effective operator QQQL/MH that leads to proton decay. In fact, this turns out to give the dominant contribution. The gauge boson exchange diagram leads to p -+ e+7r° with an amplitude M p ..... e+ 1ro ::: 4~C;u. This leads to a prediction for the proton lifetime of: u

T = 4.5 p

X

1029±.7 (

2.1

Mu

X

1014 GeV

)4

(61)

For Mu ::: 2 x 1016 GeV, one gets Tp = 4.5 X 1037±.7 yrs. This far beyond the capability of SuperKamiokande experiment, whose ultimate limit is ,...., 1034 years. Turning now to the Higgsino exchange diagram, we see that the amplitude for this case is given by:

(62) In this formula there is only one heavy mass suppression. Although there are other suppression factors, they are not as potent as in the gauge boson exchange case. As result this dominates. A second aspect of this process is that the final state is vK+ rather than e+7r°. This can be seen by studying the effective operator that arises from the exchange of the color triplet fields in the 5 + 5 i.e. 0 ilB=1 = QQQ L where Q and L are all superfields and are therefore bosonic operators. In terms of the isospin and color components, this looks like EijkuiUjdke- or EijkuidjdkV. It is then clear that unless the two u's or the d's in the above expressions belong to two different generations, the operators vanishes due to color antisymmetry. Since the charm particles are heavier than the protons, the only contribution comes from the second operators and the strange quark has to be present (i.e. the operator is EijkuidjSkVw Hence the new final state. Detailed calculations shoW36 that for this decay lifetime to be consistent with present observations, one must have MH > Mu by almost a factor of 10. This is somewhat unpleasant since it would require that some coupling in the superpotential has to be much larger than one.

359

3.4

Other aspects of SU(5)

There are several other interesting implications of SU(5) grand unification that makes this model attractive and testable. The model has very few parameters and hence is very predictive. The MSSM has got more than a hundred free parameters, that makes such models expertimentally quite fearsome and of course hard to test. On the other hand, once the model is embedded into SUSY SU(5) with Polonyi type supergravity, the number of parameters reduces to just five: they are the A, B, m3/2 which parameterize the effects of supergravity discussed in section I, J-L parameter which is the HuHd mixing term in the superpotential also present in the superpotential and mA, the universal gaugino mass. This reduction in the number of parameters has the following implications: (i) Gaugino unification:

At the GUT scale, we have the three gaugino masses equal (i.e. mAl = m A2 = m A3 • There value at the weak scale can be predicted by using the RG running as follows: dmAi

----;u- =

bi 21f (}:imA,

(63)

Solving these equations, one finds that at the weak scale, we have (64)

Thus discovery of gaugino's will test this formula and therefore SU(5) grand unification. (ii) Prediction for squark and slepton masses

At the supersymmetry breaking scale, all scalar masses in the simple supergravity schemes are equal. Again, one can predict their weak scale values by the RGE extrapolation. One finds the following formulae'l7: (65) k ;) 1 · 2() 1 l ' 2() d f k -- t(2-b h Q uz -- 2 were - "32szn w an d QZ d - -2 + "3 szn w an l+bk t an d bk are the coefficients of the RGE's for coupling constant evolutions given earlier. A very obvious formula for the sleptons can be written down. It omits the strong coupling factor. A rough estimate gives that my :::' m~/2 and m~ :::' m~/2 + 4mL. This could therefore serve as independent tests of the SUSY SU(5).

360

3.5

Problems and prospects for SUSY SU(5)

While the simple SUSY SU(5) model exemplifies the power and utility of the idea of SUSY GUTs, it also brings to the surface some of the problems one must solve if the idea eventually has to be useful. Let us enumerate them one by one and also discuss the various ideas proposed to overcome them.

(i) R-parity breaking: There are renormalizable terms in the superpotential that break baryon and lepton number: (66)

When written in terms of the component fields, this leads to R-parity breaking terms of the MSSM such as LaLbe~, QLdc as well as uCdcdc etc. The new point that results from grand unification is that there is only one coupling parameter that describes all three types of terms and also the coupling A satisfies the antisymmetry in the two generation indices b, c. This total number of parameters that break R-parity are nine instead of 45 in the MSSM. There are also nonrenormalizable terms of the form T F F( if! / Mpe)n38, which are significant for n = 1,2,3,4 and can add different complexion to the R-parity violation. Thus, the SUSY SU(5) model does not lead to an LSP that is naturally stable to lead to a CDM candidate. As we will see in the next section, the SO(lO) model provides a natural solution to this problem if only certain Higgs superfields are chosen. (ii) Doublet-triplet splitting problem:

We saw earlier that to generate the light doublets of the MSSM, one needs a fine tuning between the two parameters 3/2Ab and M in the superpotential. However once SUSY breaking is implemented via the hidden sector mechanism one gets a SUSY breaking Lagrangian of the form: LSB

= AAHif!H + BM H H

+ h.c.

(67)

where the symbols in this equation are only the scalar components of the superfields. In general supergravity scenarios, Ai-B. As a result, when the Higgsinos are fine tuned to have mass in the weak scale range, the same fine tuning does not leave the scalar doublets at the weak scale. There are two possible ways out of this problem: we discuss them below.

361

(iiA) Sliding singlet The first way out of this is to introduce a singlet field S and choose the superpotential of the form: WDT

= 2HipH

+ SHH

(68)

The supersymmetric minimum of this theory is given by: FH = Hu( -3b+

= 3b which is precisely the condition that keeps the doublets light. Thus the doublets remain naturally of the weak scale without any need for fine tuning. This is called the sliding singlet mechanism. In this case the supersymmetry breaking at the tree level maintains the massless of the MSSM doublets for both the fermion as well as the bosonic components. There is however a problem that arises once one loop corrections are included- because they lead to corrections for the < S > vev of order 16~2 m3/2Mu which then produces a mismatch in the cancellation of the bosonic Higgs masses. One is back to square one!

(iiB) Missing partner mechanism: A second mechanism that works better than the previous one is the so called missing partner mechanism where one chooses to break the GUT symmetry by a multiplet that has coupling to the Hand H and other multiplets in such a way that once SU(5) symmetry is broken, only the color triplets in them have multiplets in the field it couples to pair up with but not weak doublets. As a result, the doiublet naturally light. An example is provided by adding the 50, 5-0 (denoted by e~fj d'j = x!IO >j Ui = xh~x~IO >j e+ = t t t 10 > etc. X1X2X3 Other representations such as 10 are given simply by the r a, 45 by [r a, rbJ etc. In other words, they can be denoted by vectors with totally antisymmetric indices: The tensor representations that will be necessary in our discussion are 10 == Haj 45 == Aab, 120 == Aabe, 210 == ~abed and 126 == tl.abede. (All indices here are totally antisymmetric). One needs a charge conjugation operator to write Yukawa couplings such as I{! I{! H where H == 10. It is given by C == II i r 2i - 1 with i = 1, ... 5. The generators of SU(4) and SU(2)£ x SU(2)R can be written down in terms of the X's. The fact that SU(4) is isomorphic to SO(6) implies that the generators of SU(4) will involve only Xi and its hermitean conjugate for i = 1,2,3 whereas the SU(2)£ x SU(2)R involves only XP (and its h.c.) for p = 4,5. The SU(2)L generators are: It = X1x5 and Ii and h,£ can be found from it. Similarly, Iii = and the other right handed generators can be found from it. For instance hR = HhR - LRJ etc. We also have

xhl

1

B - L = --~·xtX· 3 ' " _ 1

t

+ ~ p xtx p p

(76)

t

Q - 3~iXiXi - X4X4 This formulation is one of many ways one can deal with the group theory of SO(2Nf1. An advantage of of the spinor basis is that calculations such as

365

those for 16.10.16 need only manipulations of the anticommutation relations among the xi's and bypass any matrix multiplication. As an example, suppose we want to evaluate up and down quark masses induced by the weak scale vev's from the 10 higgs. We have to evaluate lJIcr alJI Ha. To see which components of H corresponds to electroweak doublets, let us note that SO(lO) -t SO(6) x SO(4); denote a = 1, .. 6 as the 80(6) indices and p = 7.. 10 as the 80(4) indices. Now 80(6) is isomorphic to 8U(4) which we identify as 8U(4) color with lepton number as fourth color 3 and 80(4) is isomorphic to SU(2)L x SU(2)R group. To evaluate the above matrix element, we need to give vev to H 9 ,lO since all other elements have electric charge. This can be seen from the 8U(5) basis, where X5 , corresponding to the neutrino has zero charge whereas all the other X's have electric charge as can be seen from the formula for electric charge in terms of X's given above. Thus all one needs to evaluate is typically a matrix element of the type < 0lXlr9cx~xlxll0 >. In this matrix element, only terms X5 from r9 and X2X3X4X!X! will contribute and yield a value one.

4.2

Symmetry breaking and fermion masses

Let us now proceed to discuss the breaking of 80(10) down to the standard model. 80(10) contains the maximal subgroups SU(5) x U(l) and SU(4)c x SU(2)L x SU(2)R X Z2 where the Z2 group corresponds to charge conjugation. The SU(4)c group contains the subgroup SU(3)c x U(l)B-L. Before discussing the symmetry breaking, let us digress to discuss the Z2 subgroup and its implications. The discrete subgroup Z2 is often called D-parity in literature"2. Under D-parity, u -t u C ; e -t e C etc. In general the D-parity symmetry and the SU(2)R symmetry can be broken separately from each other. This has several interesting physical implications. For example if D-parity breaks at a scale (Mp) higher than SU(2)R (MR) (i.e. Mp > MR), then the Higgs boson spectrum gets asymmetrized and as a result, the two gauge couplings evolve in a different manner. At M R , one has gL =F gR. The 80(10) operator that implements the D-parity operation is given by D == r 2r 3 r 6 r 7 . The presence of D-parity group below the GUT scale can lead to formation of domain walls bounded by strings 43 . This can be cosmological disaster if Mp = MR43 whereas this problem can be avoided if'2 Mp > MR. Another way to avoid such problem will be to invoke inflation with a reheating temperature TR ::; MR. There are therefore many ways to break 80(10) down to the standard model. Below we list a few of the interesting breaking chains along with the

366 SO(1O) multiplets whose vev's lead to that pattern.

(A) SO(1O) -+ SU(5) -+

G STD

The Higgs multiplet responsible for the breaking at the first stage is a 16 dimensional multiplet (to be denoted 'ljJ H) which has a field with the quantum number of yC which is an SU(5) singlet but with non-zero B - L quantum number. The second stage can be achieved by

(77) The breaking of the SU(5) group down to the standard model is implemented by the 45-dimensional multiplet which contains the 24 dim. representation of SU(5), which as we saw in the previous section contains a singlet of the standard model group. In the matrix notation, we can write breaking by 45 as < A >= iT2 x Diag(a, a, a, b, b,) where a i- 0 whereas we could have b = 0 or nonzero. A second symmetry breaking chain of physical interest is: (B) SO(lO) -+

G 224 D

-+

GSTD

where we have denoted G 224D == SU(2)L X SU(2)R x SU(4)c X Z2. We will use this obvoius shorthand for the different subgroups. This breakingis achieved by the Higgs multiplet 54 = (1,1,1) + (3,3,1) + (1,1,20') + (2,2,6)

(78)

The second stage of the breaking of G 224D down to G ST D is achieved in one of two ways and the physics in both cases are very different as we will see later: (i) 16+1-6 or (ii) 126+ 126. For clarity, let us give the G 224D decomposition of the 16 and 126. 16 = (2,1,4) + (1,2,4) 126 = (3,1,10) + (1,3,10) + (2,2,15) + (1,1,6)

(79)

In matrix notation, we have

< 54 > = Diag(2a, 2a, 2a, 2a, 2a, 2a, -3a, -3a, -3a, -3a)

(80)

and for the 126 case it is the yCyC component that has nonzero vev. It is important to point out that since the super symmetry has to be maintained down to the electroweak scale, we must consider the Higgs bosons that

367 reduce the rank of the group in pairs (such as 16+ 16). Then the D-terms will cancel among themselves. However, such a requirement does not apply if a particular Higgs boson vev does not reduce the rank.

(C) SO(lO) -t

G 2231 -t GSTD

This breaking is achieved by a combination of 54 and 45 dimensional Higgs representations. Note the absence of the Z2 symmetry after the first stage of breaking. This is because the (1,1,15) (under G 224 ) submultiplet that breaks the SO(lO) symmetry is odd under the D-parity. The second stage breaking is as in the case (B).

(D) S0(10) -t G 224 -t G STD Note the absence of the D-parity in the second stage. This is achieved by the Higgs multiplet 210 which decomposes under G224 as follows: 210 = (1,1,15) + (1,1,1) + (2,2,10) +(2,2,10) + (1,3,15) + (3,1,15) + (2,2,6)

(81)

The component that acquires vev is < ~78910 >=1 O. It is important to point out that since the supersymmetry has to be maintained down to the electroweak scale, we must consider the Higgs bo~ons that reduce the rank of the group in pairs (such as 16+ 16). Then the D-terms will cancel among themselves. However, such a requirement does not apply if a particular Higgs boson vev does not reduce the rank. Let us now proceed to the discussion of fermion masses. As in all gauge models, they will arise out of the Yukawa couplings after spontaneous symmetry breaking. To obtain the Yukawa couplings, we first note that 16 x 16 = 10 + 120 + 126. Therefore the gauge invariant couplings are of the form 16.16.10 == IJiTC- 1 falJiHa; 16.16.120 == IJifafbfclJiAabc and 16.16.126 == IJifafbfcfdfelJi6.abcde. We have suppressed the generation indices. Treating the Yukawa couplings as matrices in the generation space, one gets the following symmetry properties for them: hlO = hio; h 120 = -hi20 and h 126 = hi26 where the subscripts denote the Yukawa couplings of the spinors with the respective Higgs fields. To obtain fermion masses after electroweak symmetry breaking, one has to give vevs to the following components of the fields in different cases: < H 9 ,10 >=1 0; A 789 ,7810 =I 0 or A 129 = A 349 = A569 =I 0 (or with 9 replaced by 10) and similarly ~12789 = ~34789 = ~56789 =I 0 etc. Several important constraints on fermion masses implied in the SO(lO) model are:

368

(i) If there is only one 10 Higgs responsible for the masses, then only

< HlO >¥

o ane has the relation Mu = Md = Me = MvD; where the MF denote the mass matrix for the F-type fermion. (ii) If there are two 10's, then one has Md = Me and Mu = MvD. (iii) If the fermion masses are generated by a 126, then we have the mass relation following from SU(4) symmetry i.e. 3Md = -Me and 3Mu = -MvD. It is then clear that, if we have only 10's generating fermion masses we have the bad mass relations for the first wo generations in the down-electron sector. On the other hand it provides the good b - T relation. One way to cure it would be to bring in contributions from the 126, which split the quark masses from the lepton masses- since in the G 224 language, it contains (2,2, 15) component which gives the mass relation me = -3md. This combined with the 10 contribution can perhaps provide phenomenologically viable fermion masses. With this in mind, we note the suggestion of Georgi and Jarlskogt4 who proposed that one should have the Md and Me of the following forms to avoid the bad mass relations among the first generations while keeping b - T unification: (82)

oao) °b

(Obc

Mu =

(83)

a

These mass matrices lead to mb = m at the GUT scale and .!!!:.# 0 leads to R-parity breaking terms of the form LHu discussed in the sec. 1. When one goes to the nonrenormalizable operators many other examples arise: e.g. lJIlJIlJIlJIH/Mpl after symmetry breaking lead to QLdc,LLe c as well as uCdcdc type terms.

4.4 Doublet-triplet splitting (D-T-S): As we noted in sec.3, splitting the weak doublets from the color triplets appearing in the same multiplet of the GUT group is a very generic problem of all grand unification theories. Since in the SO(10) models, the fermion masses are sensitive to the GUT multiplets which lead to the low energy doublets, the problem of D-T-S acquires an added complexity. What we mean is the following: as noted earlier, if there are only 10 Higgses giving fermion masses, then we have the bad relation me/m/-L = md/ms that contradicts observations. One way to cure this is to have either an 126 which leaves a doublet from it in the low energy MSSM in conjunction with the doublet from the 10's or to

372

have only 10's and have non-renormalizable operators give an effective operator which transforms like 126. This means that the process of doublet triplet splitting must be done in a way that accomplishes this goal. One of the simplest ways to implement D-T-S is to employ the missing vev mechanis'H, where one takes two 10's (denoted by H l ,2) and couple them to the 45 as AHl H 2. If one then gives vev to A as < A >= iT2 x Diag(a, a, a, 0, 0), then it is easy to varify that the doublets (four of them) remain light. This model without further ado does not lead to MSSM. So one must somehow make two of the four doublets heavy. This was discussed in great detail by Babu and Ban5l . A second problem also tackled by Babu and Barr is the question that once the SO(lO) model is made realistic by the addition of say 16 + 1-6 , then new couplings of the form 16.1-6.45 exist in the theory that give nonzero entries at the missing vev position thus destroying the whole suggestion. There are however solutions to this problem by increasing the number of 45's. Another more practical problem with this method is the following. As mentioned before, the low energy doublets in this method are coming from 10's only and is problematic for fermion mass relations. This problem was tackled in two papen,s2,53. In the first paper, it was shown how one can mix in a doublet from the 126 so that the bad fermion mass relation can be corrected. To show the bare essentials of this techniques, let consider a model with a single H, single pair ~ + 6. and a A and S == 54 and write the following superpotential: (88)

After symmetry breaking this leads to a matrix of the following form among the three pairs of weak doublets in the theory i.e. Hu,lO, Hu,t!., Hu,li and thew corresponding Hd's. In the basis where the column is given by(10, 126, 126) and similarly for the row, we have the Doublet matrix: MD =

(

0

< A >2 1M

o

< A >2 1M 0 M

0)

M

(89)

0

where the direct Hu,lOHd,lO mass term is fine tuned to zero. This kind of a three by three mass matrix leaves the low energy doublets to have components from both the 10 and 126 and thus avoid the mass relations. It is easy to check that the triplet mass matrix in this case makes all of them heavy. There is another way to achieve the similar result without resorting to fine tuning as we did here by using 16 Higgses. Suppose there are two 10's, one pair of 16 and 1-6 (denoted by 'II H, IjI H ). Let us write the following superpotential: Wbm

= WHWHH l

+ IjIHIjI HH 2 + AHlH2 + W2W2AA'H2

(90)

373

°

If we now give vev's to < V C >::/:- and < ,;c > ::/:- 0, then the three by three doublet matrix involving the Hu's from Hi and \I!H and Hd'S from the Hi's and \II H form the three by three matrix which has the same as in the above equation. As a result, the light MSSM doublets are admixtures of doublets from 10's and 16's. This in conjunction with the last term in the above superpotential gives precisely the GJ mass matrices without effecting the form of the up quark mass matrix. Another way to implement the doublet triplet splitting in SO(lO) models without using the Dimopoulos-Wilczek ansatz was recently proposed in Ref.54 • The basic idea is to use a vev pattern for the 45 that is orthogonal to that used by Dimopoulos and Wilczek i.e. < A >= iT2 x Diag(O, 0, 0, b, b). Clearly, one immediate advantage is that this vev pattern is not destabilized by the inclusion of 16+1-6. Then with the addition of a pair of 16+1-6 (denoted below by P,F,CandC) and an additional 45 denoted by A', one finds that the light doublets are the stanadrd model doublets in the P and F. They can then be quite easily mixed with the doublets from 10's to generate the MSSM doublets. The particular superpotential that does the job is given by

W CM = P AF + CA' F + CA' P + M A2 + M' A'2

(91)

It is then assumed that the Higgs fields C and C have vev's along the SU(5) singlet (or v C ) direction. Then it easy to see that the A vev makes all the fields which are SU(2)R doublets become superheavy leaving only the SU(2)L fields light. The color triplet fields which are SU(2h are part of the SU(5) 10 multiplet. They are made heavy by the last two terms in the W C M since the the SU(5) singlet field in C and C give mass to the SU(5) 10 and 1-0 pair from the P and P and the 45 A'. The SU(5) 24 in A and A' pick up direct mass from their mass terms. This leaves only the MSSM doublets in the P and P as the light doublets. It is easy to mix them with MSSM doublets from the 10 fields (denoted by H) by using the couplings of type CHP + CHF. One practical advantage of this way of splitting doublets from triplets is that one can preferentially have the Hu to contain a doublet from 10 while leaving the Hd in the 16. A consequence of this is that the top quark mass then comes from the renormalizable operators whereas the bottom quark mass comes only from higher dimensional operators. This explains why the bottom quark mass is so much smaller than the top quark mass. Thus it is possible to have D-T -S along with phenomenologically viable mass matrices for fermions.

374

4.5

Final comments on SO(10)

The 80(10) model dearly has a number of attractive properties over the 8U(5) model e.g. the possibility to have automatic R-parity conservation, small nonzero neutrino masses, interesting fermion mass relations etc. There is another aspect of the model that makes it attractive from the cosmological point of view. This has to do with a simple mechanism for baryogenesis. It was suggested by Fukugita and Yanagida"5 that in the 80(10) type models, one could first generate a lepton asymmetry at a scale of about 1011 GeV or so when the right handed Majorana neutrinos have mass and generate the desired lepton asymmetry via their decay. This lepton aymmetry in the presence of sphaleron processes can be converted to baryons. This model has been studied quantitatively in many papers and found to provide a good explanation of the observed B /

n n-r .

5

Other grand unifcation groups

While the 8U(5) and 80(10) are the two simplest grand unification groups, other interesting unfication models motivated for different reasons are those based on E 6 , SU(6), SU(5) x U(l) and SU(5) x SU(5). We discuss them very briefly in this final section of the lectures.

5.1

E6 grand unification

These unification models were considerecP 7 in the late seventies and their popularity increased in the late eighties after it was demonstrated that the CalabiYau compactification of the superstring models lead to the gauge group E6 in the visible sector and predict the representations for the matter and Higgs multiplets that can be used to build realistic modelS'8. To start the discussion of E6 model building, let us first note that E6 contains the subgroups (i) SO(lO) x U(l); (ii) SU(3)L x SU(3)R x SU(3)c and (iii) SU(6) x SU(2). The [SU(3)j3 subgroup shows that the E6 unification is also left right symmetric. The basic representation of the E6 group is 27 dimensional and for model building purposes it is useful to give its decomposition interms of the first two subgroups: SO(lO) x U(l) :: 27 [SU(3W :: 27 = (3,1,3)

= 16 1 + 10_ 2 + 14

+ (1,3,3) + (3,3,1)

(92)

375

The fermion assignment can be given in the [SU(3)]3 basis as follows:

(3,1,3)

~

0) ;

(1,3,3)

(3,3,1) =

~

0:,) ;

(93)

(J:I ~f ::) e

v no

We see that there are eleven extra fermion fields than the SO(10) model. Thus the model is non minimal in the matter sector. Important to note that all the new fermions are vector like. This is important from the low energy point of view since the present electroweak data5 9 (i.e. the precision measurement of radiative parameters S, T and U put severe restrictions on extra fermions only if they are not vectorlike. Also the vectorlike nature of the new fermions keeps the anomaly cancellation of the standard model. Turning now to symmetry breaking, we will consider two interesing chainsalthough E6 being a group of rank six, there are many possible ways to arrive at the standard model. One chain is: (94)

The first stage of the breaking can be achieved by a 650 dimensional Higgs field which is the lowest dim. representation that has a singlet under this group. In the case of string models this stage is generally achieved by the Wilson loops involving the gauge fields along the compactified direction. The second stage is achieved by means of the no field in the 27 dimensional Higgs boson. The final stage can be achieved in one of two ways depending on whether one wants to maintain the R-parity symmetry after symmetry breaking. If one does not care about breaking R-parity, the V C field in 27-Higgs can be used to arrive at the standard model On the other hand if one wants to keep R-parity conserved, the smallest dimensional Higgs field would be 351' is needed to arrive at the standard model. Another interesting chain of symmetry breaking is: E6

-+ SO(lO) x U(l) -+

G2213

-+

GSTD

(95)

The first stage of this chain is achieved by a 78 dim. rep. and the rest can be achieved by the 27 Higgs as in the previous case. The fermion masses in this model arise from 27 higgs since 27rn27 rn27H is E6 invariant and it contains the MSSM doublets (the Hi fields in the 27

376

given above). The [27]3 interaction in terms of the components can be written as [27P -+ QQD + QCQc DC + QQc H + LL cH +H2n o + DDcno + QLD c + QCLcD

(96)

Form this we see that in addition to the usual assignments of B-L to known fermions, if we assign B - L for D as -2/3 and DC as +2/3, then all the above terms conserve R-parity prior to symmetry breaking. However when < v C >=/:. 0, d C a D mix leading to breakdown of R-parity. They can for instance generate a uCdCdC term with strength ~~:;: . This can lead to the fj,B = 2 processes such as neutron-antineutron oscillation.

5.2

SU(5) x SU(5) unification

The SU(5) x SU(5) model that we will discuss here was motivated by the goal of maintaining automatic R-parity conservation as well as the simple see-saw mechanism for neutrino masses in the context of superstring compactification. The reason was the failure of the string models at any level to yield the 126 dim. rep. in the case of 80(10) yielding fermionic compactifications. Although no work has been done on higher level string compactifications with SU(5) x SU(5) as the GUT group, the model described here involves simple enough representaions that it may not be unrealistic to expect them to come out of a -consistent compactification scheme. In any case for pure SU(5) at level II all representations used here come out. Let us now see some details of the model. The matter fields in this case belong to left-right symmetric representations such as (5, 1) + (1,5) + (10,1) + (1,1-0) as follows: (denoted by F L , FR, TL, TR)'

(97)

377

This left-right symmetric fermion assignment was first considered in Ref. 60. But the R-parity conserving version of the model was considered in Ref~l. Crucial to R-parity conservation is the nature of the Higgs multiplets in the theory. We choose the higgses belonging to (5,5), (15, 1) + (1,1-5). The SU(5) xSU(5) group is first broken down to SU(3)c x SU(2)L x SU(2)R X U(l)B-L by the (5,5) acquiring vev's along Diag(a, a, a, 0, 0). This also makes the new vectorlike particles U, D, E superheavy. The left-right group is then broken down to the G ST D by the 15-dimensional Higgs acquiring a vev in its right handed multiplet along the yCyC direction. This component has B - L = 2 and therefore R-parity remains an exact symmetry. The light fermion masses and the electroweak symmetry breaking arise via the vev of a second (5,5) multiplet acquiring vev along the direction Diag(O, 0, 0, b, b). A new feature of these models is that due to the presence of new fermions, the normalization of the hypercharge and color are different from the standard 8U(5) or 80(10) unification models. In fact in this case, Iy

=

{1;(Y/2) and as result, at the GUT scale sin (}w = The GUT scale in this case is therefore much lower than the standard scenarios discussed prior to this. 2

5.3

3 1 6.

Flipped SU(5)

This model was suggested in Ref~2 and have been extensively studied as a model that emerges from string compactification. It is based on the gauge group SU(5) x U(l) and as such is not a strict grand unification model. Nevertheless it has several interesting features that we mention here. The matter fields are assigned to representations 5-3 (F), 1+5 (8) and 10+1 (T). The deatailed particle assignments are as follows:

(98)

378 The electric charge formula for this group is given by: 1

1

v'I5

5

Q = I3L - -A24 +-X

(99)

where Aa ( a= 1...24) denote the SU(5) generators and X is the U(I) generator with I3L == A3' The Higgs fields are assigned to representations ~(10+d + f; and H(5_2) + iI. The first stage of the symmetry breaking in this model is accomplished by ~45 -:j; O. This leaves the standard model group as the unbroken group. Another point is that since the ~45 has B - L = 1, this model breaks R-parity (via the nonrenormalizable interactiions). Hand iI contain the MSSM doublets. An interesting point about the model is the natural way in which doublet-triplet splitting occurs. To see this note the most general superpotential for the model involving the Higgs fields: (100)

On setting ~45 = Mu, the first term gives fijk~ij Hk which therefore pairs up the triplet in H with the triplet in 10 to make it superheavy and since there is no color singlet weak doublet in 10, the doublet remains light. This provides a neat realization of the missing partner mechanism for D-T-S. The fermion masses in this model are generated by the following superpotential: (101) It is clear that this model has no b -

T mass unification; thus we lose one very successful prediction of the SUSY GUTs. There is also no simple see-saw mechanism. And furthermore the model does not conserve R-parity automatically as already noted. For instance there are higher dim. terms of the form TT~F/Mp!, FF~S/Mpl that after symmetry breaking lead to R-parity breaking terms like QLdc and LLe c . Thus they erase the baryon asymmetry in the model.

5.4

SU(6) GUT and naturally light MSSM doublets

In this section, we discuss an SU(6) GUT model which has the novel feature that under certain assumptions the MSSM Higgs doublets arise as pseudoGoldstone multiplets in the process of symmetry breaking without any need for fine tuning. This idea was suggested by Berezhiani and Dvalp3 and has been pursued in several subsequent paper~4.

379

We will only discuss the Higgs sector of the model since our primary goal is to illustrate the new mechanism to understand the D-T -So Consider the Higgs fields belonging to the 35 (denoted by ~), and to 6 and 6 (denoted by H, fI respectively). Then demand that the superpotential of the model has the following structure: W = W~

+ W(H,fI)

(102)

i.e. set terms such as H~fI to zero. This is a rather adhoc assumption but it has very interesting consequences. Let the fields have the following pattern of vev's. 1

o < H >=< fI >=

o o ; < ~ >= Diag(l, 1, 1, 1, -2, -2) o o

(103)

Note that ~ breaks the SU(6) group down to SU(4) x SU(2) x U(l) whereas H field breaks the group down to SU(5) x U(l). Note that the Goldstone bosons for the breaking to SU(5) x U(l) are in 5 + 5 + 1 i.e. under the standard model group they tranform as : (SU(3) x SU(2) x U(l))

GB' s = (3,1) + (1,2) + (3,1) + (1,2) + (1,1)

(104)

whereas the Goldstone bosons generated by the breaking of SU(6) -+ SU(4) x SU(2) x U(l) by the ~ are: (3,2)

+ (3,2) + (1,2) + (1,2)

(105)

Since both the vev's break SU(6) -+ SU(3) x SU(2) x U(l), the massless states that are eaten up in the process of Higgs mechanism are (3,1)

+ (3,1) + (1,2) + (1,2) + (3,2) + (3,2) + (1,1)

(106)

We then see that the only Goldstones that are not eaten up are the two weak doublets (1,2)+ (1,2). These can be identified with the MSSM doublets. This model can be made realistic by adding matter fermiona to two 6's and a 15 per generation to make the model anomaly free. We do not discuss this here.

380

5.5 IL -+ e + "I as a test of supersymmetric gmnd unification One of the most precise limits on lepton flavor violation is for the process + "Ij the latest limits on the branching ratio B(IL -+ e + "I) from the MEGA experiment at Los Alamos is 1.2 x 10- 11 . Since the vanishing neutrino mass implied by the standard model leads to exact flavor conservation in the lepton sector, this process is a very sensitive barometer of new physics. However, it turns out that in most nonsupersymmetric extensions of the standard model, the branching ratio for IL -+ e + , is proportional to the (!];f) 4 in the most optimistic cases, the present limit on the new physics scale in the range of few TeV's. One exceptin to this rule is the supersymmetric models and in particular supersymmetric grand unification. What happens there is that since the super-partners of fermions are expected to be in the few hundred Ge V range and furthermore since they can support lepton flavor mixing terms even in the absence of neutrino mass, they can apriori lead to large lepton flavor violation. In fact this puts a severe constraints on the parameters of the MSSM. A way to satisfy these constraints is to assume specific forms for the supersymmetry breaking terms- in particular the assumption that helps is the universality of scalar masses that arise in supergravity models. Once one assumes this universality at the Planck scale, departures from this including flavor mixing terms can arise at the weak scale due to the flavor violation intrinsic to the Yukawa couplings. The amount of flavor violation is generically h2 has the magnitude ,...., 16 ". 21n MM PI • For typical Yukawa couplings this proSUSy duces flavor violating effects such as the IL -+ e+, process at the level of 10- 16 or so, which is beyond the reach of any currently contemplated experiment. It has been noted'5 that situation may be very different in grand unified theories where quarks and leptons are unified into one multiplet. In such cases (say for instance in SU(5) or SO(lO) model), the third generation coupling is the top quark Yukawa coupling which is of order one. Therefore as we extrapolate from the Mpl scale to the GUT scale, the third generation slepton masses split away by significant amount from the first and second generation ones. Now at the weak scale, when one does the weak rotation to come to the mass basis for quarks, it induces a mixing between j1, and e of order 812 ,...., 2 2 m V31 V23 mr - 2 i'. This mixing can be large and has been found to yield IL -+ mo e + , of order 10- 12 _10- 13 . This is within reach of the next generation of

IL -+ e

experiments being discussed'6. Several limitations on this result must be noted since it specifically depends on two assumptions: (i) Quark lepton unificationj (ii) universal scalar masses at Planck scale. One class of models that violate assumption (i) while being consistent with all known supersymmetry phenomenology are the super-Left-

381

right models where, the prediction for J.L -+ e + / branching ratio is in the range of 10- 15 or set°. Another class is the GMSB models where the partial universality of scalar masses holds at at a scale of order'" 100 TeV by which time any trace of quark lepton unification which may be present at the GUT scale is absent. Finally, even given the idea of grand unification, not all GUT models would lead to large J.L -+ e+/. An example is the SU(5) x SU(5) model discussed in this section since in the model, quarks unify with with superheavy leptons and vice versa. Thus no immediate room for enhancement of lepton flavor violation.

6

String theories, extra dimensions and grand unification

In this section, we would like to look beyond grand unification not only to see what possible scenarios exist at that level but to see if those possibilities impose any restrictions on the GUT scenarios discussed in the text. Two main and related areas we will explore are the string theories, both weakly and strongly coupled and the possibility that there may be extra large hidden dimensions in nature. Recent developements in string theories have such a discuusion more substantial. Let us start with a brief overview of why string theories are being taken so seriously by theoists. Recall that the point particle based local field theories have been largely responsible for whatever understanding (and it is considerable) we have of the nature of particles and forces. Most spectacular has been the success of spontaneously broken gauge theories in providing a remarkable description of all known low energy phenomena. Why then do we look for theories whose starting point is to abandon this successful recipe and consider nonlocal models which posit that the fundamental entities of nature are not points but strings? The answer to this question is that despite the success of gauge theories, they do not incorporate gravity and they are plagued with divergences. The latter is directly related to the point nature of the vertices that describe particle interactions. In string theories on the other hand there are no point vertices and therefore, not surprisingly no infinities. Much more interesting is the fact that closed string theories in fact lead to gravity theory. Thus if we can get other forces and particles of nature from a string model, we would then have a complete theory of all forces and matter.The enormous popularity of string theories rests on the fact that this indeed appears to be the case. To be more concrete, in string theories, the vibrational modes are identified

382

with particles; in particular the massless models are to taken as the fields of the standard model if they have the right quantum numbers. The excited states are spaced in mass spectra by an amount given by the string scale M str or the square root of the string tension. It turns out that the lowest state of the closed string has spin two and can be identified with the graviton. This feature is common to all string theories. What is nontrivial to see the emergence of gauge symmetries and quarks and leptons etc from these models. How they emerge can roughly be seen as follows. It turns out that conformal invariance of string theory demand that strings exist only in 10 or 26 dimensional space time (10 if the string is supersymmetric and 26 if it is purely bosonic). In order to come down to observed 3+1 space-time, we musy compactify the extra space dimensions. This is similar in concept to the Kaluza-Klein theories, where it is well known that compactifying the extra space dimensions turns the corresponding conserved momenta to gauge charges (hence the appearance of gauge symmetries) and one higher dimensional matter multiplet can lead to many matter fields in 3+ 1 dimensions. Here the role of supersymmetry becomes important.

To see how supersymmetry emerges from string theories, note that if we consider only bosonic string theories, it will not have any fermions. One must therefore incorporate fermionic string degrees of freedom. That with certain other stringy consistency conditions leads to the emergence of supersymmetry in the vibrational (particle) spectra. Once we have supersymmetry and gauge symmetry, in higher dimensions, the gauge multiplet will be accompanied by a gaugino multiplet. When the gaugino multiplet is reduced to 3+1 dimensions, quarks and leptons emerge from the strings. This is a simplistic overview of how standard model like features can emerge from string theories. There are five kinds of string theories: type I, type IIA and IIB, heterotic SO(32) and Es x E~. It was believed for a long time that of these only the heterotic string theory can be useful in providing the standard model at low energies- the reason being that in the other cases either the gauge group was not adequate or the matter content. This has changed in recent years due to the realization that previous conclusion was derived only in the weak coupling limit of the string theories but once the strongly coupled strings are considered, there emerge duality relations that makes all string theories equivalent. For instance in type IIB string theories, emergence of D-brane type solutions, the gauge group could be bigger to accomodate the standard model gauge group. In the subsequent sections, we would consider the constraints imposed by the different type of string theories ( weakly coupled or strongly coupled) on the nature of grand unification.

383

6.1

Weakly coupled heterotic string, mass scales and gauge coupling unification

Let us first address the question of mass scales in these theories. As a typical theory let us consider the Calabi-Yau compactification which begins with the 10 dimensional super Yang-Mills theory coupled to supergravity based on the gauge group Es x E~. The Lagrangian for the massless states of the theory is fixed by the above symmetry requirement and writing only the first two bosonic terms, we have 4 S -- (0/)3

JdlO xe -2 [Ra' + 4IF/-'vF

/-,1.'

+ ..... ]

(107)

Compactifying to 4-dimensions, it is easy to derive the relations: 0.,4 e 2

GN

= 647rV(6) j au

0.,3 e 2

= 167rV(6)

(108)

leading to the relation G N = la' au. Thus for typical values of the unified gauge coupling (say ~ 1/24), M str ~ O.IMpf i.e. they are of the same order and the string scale is larger than the GUT scale by a factor roughly of 20. How does one view this? One possible attitude is to say that at the GUT scale a grand unified group emerges so that between Mu and M str , the coupling evolves and presumably remains perturbative. This makes a heavy demand on the string theory that one must search for a vacuum which has a GUT group (say 80(10)) with three generations and the appropriate Higgs fields. Another way to look at this is that we have a puzzle and new idea is needed to understand this scale discrepancy. We will see that there exist some pome very interesting possibilities in strongly coupled string theories. Regardless of whether there is a grand unifying gauge group present at the the scale Mu or not, string models do have gauge coupling unification as is apparent from the above equation. In case where there are different gauge groups below the string scale, one has the more general reiatiOIf8 (109) where k i are the Kac-Moody level of the theory. For the simple heterotic construction with Calabi-Yau compactification, all ki's are unity (in the proper normalization for the hypercharge. In more general theories however, ki's could be different from one and a more general unification occurs.

384

6.2 Spectrum constraints String theories impose constraints on the allowed spectrum of the grand unified theories. This considerably narrow the field of allowed GUT models which is a welcome feature. To see the basic reason for the emergence of this constraint, let us again focus on the weakly coupled heterotic models. Note that the bosonic sector of the model exists only in 26 dimensions of which 22 extra dimensions must be compactified. As is familiar from Kaluza-Klein models, since the momentum corresponding to each extra space leads to a conserved gauge charge, the maximum number of commuting generators with 22 extra dimensions is clearly 22 i.e. the maximum rank of any gauge group that emerges from a string model must be 22. Now if we look at the zero modes of the heterotic string, the supersymmetry of the theory implies that the gauginos (which supply the fermions of the low energy gauge group) must belong to the adjoint representation of the gauge group. Therefore only those representations that are present in the adjoint of this gauge group will appear in the low energy spectrum. Is this really a restriction on the spectrum ? The answer is that it is a severe restriction. Let us look at some examples below. For the case of D=IO, Es x Es super Yang-Mills theory that arises in the heterotic string models maintaining N = 1 supersymmetry at low energies requires that gauge and the spin connections be identified'9. This reduces the gauge group to E6 x Es. The adjoint of the Es group is 248 dimensional and decomposes under E6 x SU(3) to 78, 1+ 27, 3+2-7, bar3 +1, 8. Since the SU(3) group is identified with the spin connection, it becomes "part" of the complex manifold and we only see the 27+27 representations of E6 group at low energies. At fir sight it might appear disastrous since there is no possible way to break the E6 group down to the standard model with only 27's. Luckily, in this case the singularities of the complex manifold provide a way via the so called Wilson loop mechanism to break the E6 group down in a manner that 78 of E6 would have done i.e. we would have a low energy group such as [SU(3)P or SU(3) x SU(2)L x SU(2)R X U{l)B-L x U{l) etc. The rest of the breaking down to the standard model can be achieved by the 27. But the main point that needs emphasizing here is that we have a very retsricted set of representations and many other representations of E6 which normally prove useful such as 351 etc are simply not allowed in this class of string models. Similar situation occurs when other compactifications are chosen such as fermionic ones and the low energy group for instance is only SO{IO). In this case we have only 16+1-6 and 10 representations. This is not adequate enough to build realistic models. The above discussions apply only to the level I compactifications and luck-

385

ily it is possible to obtain some other representations by considering higher level compactificatiow,A8,47. The detailed string theoretic construCtions of higher level compactifications are much too technical for such a review. But there is a simpler way to understand the basic points. To illustrate this let us consider the case of SU(5) model. At level one, the only representations that appear are the 5+10. If on the other hand we start with the group SU(5)x SU(5) at level I, we will have the same representations for each group. If we now break the group down and consider only the diagonal subgroup SU(5), then the resulting representations must be products of the original ones and it is easy to see from group theory that the resulting SU(5) representations that emerge are: 5+10, 15+24+40+45. This is what happens when one has a level II compactification. For the case of SO(lO), one not only has the spinor and the vector representations present for the case of level I compactification but also new ones such as 45+54 (but no more). Unfortunately, it appears that useful representations such as 126 can not be obtained at higher levels. It must however be noted that these representations are adequate for realistic model building and many models based on them have been constructed. Thus the bottom line message of this subsection is that the string models due to the presence of higher symmetries provide further restrictions on GUT models and therefore is a step farther towards unification. It nust be cautioned however that no realistic model with three generations and right representation content to yield complete symmtery breaking has yet appeared.

6.3

Strongly coupled strings, large extra dimensions and low string string scales

So far we were discussing weakly coupled strings and we found that only one possible hierarchy of scales is admissible where M str ::: Mcomp ::: O.IMpl ::: 20Mu. It has been realized in the past three years that once one goes to the strong soupling limit of string theories, many new possibilities emerge. To have an overall picture of how this happens, let us recall the basic relation between string coupling and the observable couplings such as Newton's constant and the gauge couplings: GN -

0/ 4

. k·o· -

V(6) 641T01O '

,

0/ 3 ---,-=-:-----

, - V(6)161T01O

kioi

(110)

= Ostr

where 010 and Ostr are the 10 and 4-dimensional string couplings respectively. If we identify V(6) IsimM.rl~, we can derive the following relations among the

386 couplings: (111) Clearly, if we now want to identify the astr and M str with au and Mu, the smallness of G N can be understood only if alO is very large i.e. we are in the strong coupling limit of strings. This indicates that the profile of mass scales can be considerably different in the strongly couped string theorieio. The first realization of these ideas in a concrete string model was presentaed by Horava and Witten in the context of the M-theory whose low energy limit is given by an 11-dimensional supergravity compactified on an S1 x Z2 orbifold. In this case the picture is that of a 11-dimensional bulk bounded by lO-dimensional walls where in resides the gauge groups with one Es on each wall. The separation between the two walls (or the compactification radius in the 11-th dimension) Rll is related in these modsels to the string coupling as Rll rv (a')1/2a~~3. This means that in the weak coupling limit the two walls sit on top of each other and we have the usual weakly coupled picture described in the previous subsection of this section. On the other hand as the string becomes larger, the two walls separate. The effective Lagrangian in this case is given bi 1 : (112) Upon compactification, we get the following relations between the four dimensional couplings: (113) It is now clear that adjusting R ll , we can get the correct string scale while

keeping both the string scale and the compactification scales at the Mu. One gets Rll rv (10 12 GeV)-1. Thus in strongly coupled string theories, the string scale can be lower. This idea was carried another step further by Lykkerl 2 •73 in the proposal that perhaps the string scale can be as low as a TeV. In such a scenario, one has the following general relation between the various mass scales: (114) It is then clear that one of the compactification radii could be quite largi 4

and indeed it was suggested in Ref. 74 that this would change Newton's laww at submillimeter distances. As it turns out validity of Newtons inverse square

387

law has only been checked only above millimeter distances. This is exciting for experimentalists. Sinilarly, the fact that the string scale can be as low as a TeV implies that string states could be excited in colliders.

6.4

Effect of extra dimensions on gauge coupling unification

Once it is accepted that the compactification scales as well as the string scales could be arbitrary, it is clear that the presence of Kaluza-Klein excitations will effect the evolution of couplings in gauge theories and may alter the whole picture of unification of couplings. This question was first studied by Dienes, Dudas and Gherghetta!5. The formula for this evolution above the compactification scale /-Lo was derived by Dienes, Dudas and Gherghetta (DDG) 75 on the base of an effective (4-dimensional) theory approach and the general result at one-loop level is given by

ail (/-Lo) = ail (A)

+ b 2~ b In (~) + :~ i

i

{

1=~2 ~: ~3 (7r~2 )

r,

(115)

with A as the ultraviolet cut-off, 8 the number of extra dimensions and R the compactification radius identified as 1/ /-Lo. The Jacobi theta function (116) n=-oo

reflects the sum over the complete (infinite) Kaluza Klein (KK) tower. In Eq. (115) bi are the beta functions of the theory below the /-Lo scale, and bi are the contribution to the beta functions of the KK states at each excitation level. Besides, the numerical factor r in the former integral could not be deduced purely from this approach. Indeed, it is obtained assuming that A » /-Lo and comparing the limit with the usual renormalization group analysis, decoupling all the excited states with masses above A, and al assuming that the number of KK states below certain energy /-L between /-Lo and A is well approximated by the volume of a 8-dimensional sphere of radius /-L/ /-Lo N(/-L,/-Lo) = Xs

(~) s

j

(117)

7r 2

with Xs = s/ /r(1 + 8/2). The result is a power law behaviour of the gauge coupling constants given by (118)

388

It however turns out that for MSSM the energy range between J.Lo and A identified as the unification scale-- is relatively small due to the steep behaviour in the evolution of the couplings. For instance, for a single extra dimension the ratio AI J.Lo has an upper limit of the order of 30, which substantially decreases for higher 8 to be less than 6. It is therefore clear whether the power law approximation is a good description of the coupling evolution. This question has been recently examined in Re£!6. In general, the mass of each KK mode is well approximated by Ii 2,",

2

2

J.L n = J.Lo L..J n i

(119)

•

i=1

Therefore, at each mass level J.Ln there are as many modes as solutions to Eq. (119). It means, for instance, that in one extra dimension each KK level will have 2 KK states that match each other, with the exception of the zero modes which are not degenerate and correspond to (some of) the particles in the original (4-dimensional) theory manifest below the J.Lo scale. In this particular case, the mass levels are separated by units of J.Lo. In higher extra dimensions the KK levels are not regularly spaced any more. Indeed, as it follows from Eq. (119), the Combining all these equations together is straightforward to get

which explicitly shows a logarithmic behaviour just corrected by the appearance of the n thresholds below J.L. Using the Stirling's formula n! ~ nn e-nv'27rn valid for large n, the last expression takes the form of the power law running

a·-1 (J.L) •

bi bi In (J.L) -= a·-1 ( J.Lo ) - •

27r

J.Lo

i

- -b . 2

27r

[(. -J.L ) J.Lo

rn-]

- In v 27r .

(121)

In the DDG paper, it was concluded that for MSSM, unification can essentially occur for arbitrary values of the Mu starting all the way from a TeV to 1016 GeV if one puts the gauge bosons in the bulk but leaves the chiral fermions in the brane; however, the value of astrong increases as Mu is lowered. This can be corrected in many wayf1 7,76. There are however certain immediate issues that come up in models with GUT scale as low as a Te V. The two main issues are that of proton decay and neutrino masses. It has been conjectured that in strongly coupled theories

389

there are U(l) symmetries that will help to stabilize the proton. One must therefore show that string vacua exist with such properties. As far as neutrino mass goes, there is no way to implement the seesaw mechanism now unless one generates the neutrino Dirac masses radiatively. So completely new approaches have been trie~B where the postulated bulk neutrinos form Dirac masses with the known neutrinos. These ideas will work only if the string scale is low. There is a new way to circumvent the constraint of low string scale if one considers the left-right symmetric models in the bulk'9. An important implication of this approach to neutrino masses is that the string scale has to be at least lOB Ge V. In such a case, one can have only one large extra dimension (i.e. in the millimeter range) without conflict with smallness of the Newton's constant. These problems becomes moot if one considers high string scale models but with large extra dimensions so that the interesting gravity remains. One particular result of interest in this connection is the way that high scale seesaw mechanism emerges from higher dimensional unification. Recall that the minimal susy left-right model with the seesaw mechanism resisted grand unification with the minimal particle content. It was noted in Ref!6 that in the presence of higher dimension, if all the gauge bosons are in the bulk and matter in the brane, then the left-right model unifies with a left-right seesaw scale around 10 13 GeV and the KK scale for one dimension slight above it. Reflections This set of lectures is meant to be a pedagogical overview of the vast (and still expanding) field of supersymmetric grand unification- recently re-energised by ideas from strongly coupled string theories that bring in many new concepts and possibilities such as large extra dimensions, low string scales, bulk neutrinos etc. There are many unsolved problems not just of technical nature but of fundamental nature. The most glaring of the fundamental ones relating to string theories is of course how to stabilize the dilaton that is at the heart of the strong coupling discussion as well the discussion of unification and compactification. Among the thechnical ones are: construction of explicit string models that embody the "fantasies" scattered throughout the literature, before they faint away and evaporate in the glare of some other new ideas. It will involve a better understanding of internal string dynamics, a really challenging task since we dont seem to have a string field theory. Only after this huddle is surpassed, can we hope to bridge the big "disconnect" between string theories and the experiments that still makes many people uncomfortable to accept them as the final stage for the ultimate drama of physics. Meanwhile for those who wish to stay away from the hardships of string life have plenty to exercise their imagination- anything from finding a better solution to the doublet-triplet splitting to other phenomenological studies that can discrimi-

390

nate between different models. Finally, a customary apology: the body of literature in this field is large and only a very selective sample has been given and this means that many important papers have not been cited. The ones cited should be consulted for additional references. It is hoped that this overview is of help in inspiring the reader to push the frontier in this extremely exciting field a bit further. Clearly, as it is stressed often here, there is an enormous amount that remains to be done and we are unlikely to see the final theory of everything anytime soon (a good thing too!), although progress in the last two decades has been enormous. Acknowledgement The author would like to thank G. Senjanovic and A. Smirnov for organizing a pleasant summer school and creating an environment that promotes interaction between the students and the lecturers in such an effective manner and ICTP for support. He would also like to thank many students for their comments. This work has been supported by the National Science Foundation grant no. PHY-9802551. References

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