High Energy Physics and Cosmology 1998 - Proceedings of the Summer School (The Ictp Theoretical Physics) [1998 ed.] 9810238347, 9789810238346

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1998 SUMMER SCHOOL IN

HIGH ENERGY PHYSICS AND COSMOLOGY

the

abdus salam international centre for theoretical physics

united nations educational, scient~ic and cultural organization

international atomic energy agency

\.f:'\ 'J:jf§

THE ICTP SERIES IN THEORETICAL PHYSICS - VOLUME 15

1998 SUMMER SCHOOL IN

HIGH ENERGY PHYSICS AND COSMOLOGY IGTP, Trieste, Italy 29 June -17 July 1998

Editors

A. MASIERO SISSA, Trieste, Italy

G. SENJANOVIC A. SMIRNOV ICTp, Trieste, Italy

1I ·p

World Scientific 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

Library of Congress Cataloging-in-Publication Data Summer School in High Energy Physics and Cosmology (1998 : Trieste, Italy) 1998 Summer School in High Energy Physics and Cosmology: ICTP, Trieste, Italy, 29 June-17 July 1998/ editors, A. Masiero, G. Senjanovic, A. Smirnov. p. cm. -- (The ICTP series in theoretical physics; v. 15) At head of title: The Abdus Salam International Centre for Theoretical Physics, United Nations Educational, Scientific and Cultural Organization, International Atomic Energy Agency. ISBN 9810238347 (alk. paper) I. Nuclear astrophysics -- Congresses. 2. Particles (Nuclear physics) -- Congresses. 3. Cosmology -- Congresses. I. Masiero, A. (Antonio). II. Senjanovic, G. (Goran). III. Smirnov, A. (Alexei). IV. Unesco. V. Abdus Salam International Centre for Theoretical Physics. VI. International Atomic Energy Agency. VII. Title. VIII. Series. QB463.S86 1998 523.01'97--dc21 99-29818 CIP

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

Copyright © 1999 by The Abdus Salam International Centre for Theoretical Physics

Printed in Singapore by Uto-Print

v

PREFACE

The ICTP 1998 Summer School was devoted mostly to different aspects of astroparticle physics and cosmology. The series of introductory lectures on present day status of Standard Model of the electroweak interactions and Higgs physics, aspects of QCD and supersymmetry, provided particle physics background for astrophysical applications. The rest of the lectures focused on the in-depth coverage of inflation, high temperature field theory and phase transitions, baryogenesis, dark matter and structure formation. Special emphasis was placed on neutrino astrophysics in view of the super-Kamiokande results on atmospheric neutrinos. In the pages that follow you will find this beautifully described by our lecturers. We wish to thank them 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. Antonio Masiero, Goran Senjanovic and Alexei Smirnov

vii

CONTENTS

Preface

v

Introduction to Electroweak Symmetry Breaking S. Dawson

1

Introduction to QCD M. L. Mangano

84

Neutrino Astrophysics at the Cross Roads G. G. Raffelt

147

Finite Temperature Field Theory and Phase Transitions M. Quiros

187

An Introduction to Cosmological Inflation A. R. Liddle

260

The Large-Scale Structure of the Universe J. Silk

296

Theories of Baryogenesis A. Riotto

326

INTRODUCTION TO ELECTROWEAK SYMMETRY BREAKING

s.

Dawson

Physics Department, Brookhaven National Laboratory, Upton, NY 11973 An introduction to the physics of electroweak symmetry breaking is given. We discuss Higgs boson production in e+ e- and hadronic collisions and survey search techniques at present and future accelerators. Indirect limits on the Higgs boson mass from triviality arguments, vacuum stability, and precision electroweak measurements are presented. An effective Lagrangian, valid when there is no low mass Higgs boson, is used to discuss the physics of a strongly interacting electroweak symmetry breaking sector. Finally, the Higgs bosons of the minimal supersymmetric model are considered, along with the resulting differences in phenomenology from the Standard Model.

1

Introduction

The search for the Higgs boson has become a major focus of all particle accelerators. In the simplest version of the electroweak theory, the Higgs boson serves both to give the Wand Z bosons their masses and to give the fermions mass. It is thus a vital part of the theory. In these lectures, we will introduce the Higgs boson of the Standard Model of electroweak interactions.1 ,2 Section 2 contains a derivation of the Higgs mechanism, with particular attention to the choice of gauge. In Section 3 we discuss indirect limits on the Higgs boson mass coming from theoretical arguments and from precision measurements at the LEP and LEP2 colliders. The production of the Standard Model Higgs boson is then summarized in Sections 4 - 8, beginning with a discussion of the Higgs boson branching ratios in Section 4. Higgs production in e+ e- collisions at LEP and LEP2 and in hadronic collisions at the Tevatron and the LHC are discussed in Sections 5 and 6, with an emphasis on the potential for discovery in the different channels. Section 7 contains a derivation of the effective W approximation and a discussion of Higgs production through vector boson fusion at the LHC. The potential for a Higgs boson discovery at a very high energy e+ e- collider, (vis > 500 Ge V) is discussed in Section 8. Suppose the Higgs boson is not discovered in an e+ e- collider or at the LHC? Does this mean the Standard Model with a Higgs boson must be abandoned? In Section 9, we discuss the implications of a very heavy Higgs boson,

2

(Mh » 800 GeV). In this regime the Wand Z gauge bosons are strongly interacting and new techniques must be used. We present an effective Lagrangian valid for the case where Mh >> VS. Section 10 contains a list of some of the objections which many theorists have to the minimal Standard Model with a single Higgs boson. One of the most popular alternatives to the minimal Standard Model is to make the theory supersymmetric. The Higgs sector of the minimal supersymmetric model (MSSM) is surveyed in Section 11. We end with some conclusions in Section 12. 2

2.1

The Higgs Mechanism

Abelian Higgs Model

The central question of electroweak physics is : "Why are the Wand Z boson masses non-zero?" The measured values, Mw = 80 GeV and Mz = 91 GeV, are far from zero and cannot be considered as small effects. To see that this is a problem, we consider a U(I) gauge theory with a single gauge field, the photon. The Lagrangian is simpll

(1) where

(2) The statement of local U (1) gauge invariance is that the Lagrangian is invariant under the transformation:AI'(x) -t AI'(x) - 01'1](x) for any 1] and x. Suppose we now add a mass term for the photon to the Lagrangian, r _ J.- -

-41 FI'IIF I'll + 21 m 2 A I' AI' .

(3)

It is easy to see that the mass term violates the local gauge invariance. It is

thus the U(I) gauge invariance which requires the photon to be massless. We can extend the model by adding a single complex scalar field with charge -e which couples to the photon. The Lagrangian is now,

(4) where

01' - ieAI' 1if; 12 +.x(1 if;

{l2

12)2.

(5)

3

Figure 1: Scalar potential with J1.2

> o.

v (¢) is the most general renormalizable potential allowed by the U (1) gauge mvanance. This Lagrangian is invariant under global U(l) rotations, ¢ -+ ei8 ¢, and also under local gauge transformations: AJj(x)

¢(x)

-+ -+

AJj(x) - oJj1J(x) e- ie7J (x)¢(x).

(6) (7)

There are now two possibilities for the theory.a If J-l2 > 0 the potential has the shape shown in Fig. 1 and preserves the symmetries of the Lagrangian. The state of lowest energy is that with ¢ = 0, the vacuum state. The theory is simply quantum electrodynamics with a massless photon and a charged scalar field ¢ with mass J-l. The alternative scenario is more interesting. In this case J-l2 < 0 and the potential can be written as, 3,4

(8) which has the Mexican hat shape shown in Fig. 2. In this case the minimum energy state is not at ¢ = 0 but rather at

(¢) aWe assume .\ minimum energy.

> O.

If .\

< 0,

[JZi _

v

= V-v: = v12.

(9)

the potential is unbounded from below and has no state of

4

Figure 2: Scalar potential with 1-'2

< o.

( o.

> 0,

(47)

13

where A is some large scale where new physics enters in. Taking the reference scale Qo = v, and substituting Eq. 44 gives an approximate upper bound on the Higgs mass, (48) Requiring that there be no new physics before 10 16 GeV yields the approximate upper bound on the Higgs boson mass, Mh

< 160 GeV.

(49)

As the scale A becomes smaller, the limit on the Higgs mass becomes progressively weaker and for A '" 3 TeV, the bound is roughly Mh < 600 GeV. Of course, this picture is valid only if the one loop evolution equation of Eq. 45 is an accurate description of the theory at large oX. For large oX, however, higher order or non-perturbative corrections to the evolution equation must be included? Lattice gauge theory calculations have used similar techniques to obtain a bound on the Higgs mass. As above, they consider a purely scalar theory and require that the scalar self coupling oX remain finite for all scales less than some cutoff which is arbitrarily chosen to be 211" Mh. This gives a limit;lo Mh(lattice)

< 640 GeV.

(50)

The lattice results are relatively insensitive to the value of the cutoff chosen. Note that this bound is in rough agreement with that found above for A '"

3 TeV. Of course everything we have done so far is for a theory with only scalars. The physics changes dramatically when we couple the theory to fermions and gauge bosons. Since the Higgs coupling to fermions is proportional to the Higgs boson mass, the most relevant fermion is the top quark. Including the top quark and the gauge bosons, Eq. 45 becomeg. 1

2+9' 2)+_(2g4+(g2+9' doX = __ 1 [ 12oX 2+12oXg 2-12g4--oX(3g 3 3 _ 2)2) ] (51) dt

1611"2

t

t

2

gt

t == - v

where

For a heavy Higgs boson, oX running of oX are,

> gt, g, g',

16

M

(52)

and the dominant contributions to the

(53)

14

There is a critical value of the quartic coupling A which depends on the top quark mass,

(54) The evolution of the quartic coupling stops when A = Ac.12 If Mh > Mh == v2AcV then the quartic coupling becomes infinite at some scale and the theory is non-perturbative. If we require that the theory be perturbative (i.e., the Higgs quartic coupling be finite) at all energy scales below some unification scale (~ 10 16 Ge V) then an upper bound on the Higgs mass is obtained as a function of the top quark mass. To obtain a numerical value for the Higgs mass limit, the evolution of the gauge coupling constants and the Higgs Yukawa coupling must also be included. For M t = 175 Ge V this bound is Mh < 170 Ge V. 12 If a Higgs boson were found which was heavier than this bound, it would require that there be some new physics below the unification scale. The bound on the Higgs boson mass as a function of the cut-off scale from the requirement that the quartic coupling A(A) be finite is shown as the upper curve in Fig. 4.

3.2

Vacuum Stability

A bound on the Higgs mass can also be derived by the requirement that spontaneous symmetry breaking actually occursf3 that is,

V(v)

< V(O).

(55)

This bound is essentially equivalent to the requirement that A remain positive at all scales A. (If A becomes negative, the potential is unbounded from below and has no state of minimum energy.) For small A, Eq. 51 becomes,

dA dt

1 [ 4 = 161l'2 -12g t +

3 4 16 (2g

+

(g

2

+ 9I

2 2 ]

))

(56)

This is easily solved to find,

Requiring A(A)

> 0 gives

the bound on the Higgs boson mass,

(58)

15

A more careful analysis along the same lines as above 14 using the 2 loop renormalization group improved effective potentiaI' and the running of all couplings gives the requirement from vacuum stability if we require that the Standard Model be valid up to scales of order 10 16 GeV,9,15

Mh(GeV) > 130.5+ 2.1(Mt -174)

(59)

If the Standard Model is only valid to 1 TeV, then the limit of Eq. 59 becomes,

Mh(GeV) > 71 + .74(Mt

-

174)

(60)

We see that when>. is small (a light Higgs boson) radiative corrections from the top quark and gauge couplings become important and lead to a lower limit on the Higgs boson mass from the requirement of vacuum stability, >'(A) > O. If >. is large (a heavy Higgs boson) then triviality arguments, C}A) > 0), lead to an upper bound on the Higgs mass. The allowed region for the Higgs mass from these considerations is shown in Fig. 4 as a function of the scale of new physics, A. If the Standard Model is valid up to 10 16 Ge V, then the allowed region for the Higgs boson mass is restricted to be between about 130 GeV and 170 Ge V. A Higgs boson with a mass outside this region would be a signal for new physics. 3.3

Bounds from Electroweak Radiative Corrections

The Higgs boson enters into one loop radiative corrections in the Standard Model and so precision electroweak measurements can bound the Higgs boson mass. For example, the p parameter gets a contribution from the Higgs bosorr6g P

=1-

11g2 tan 2 Ow log ( Mh- ) . 9611" Mw

--2

(61)

Since the dependence on the Higgs boson mass is only logarithmic, the limits derived on the Higgs boson from this method are relatively weak. In contrast, the top quark contributes quadratically to many electroweak observables such as the p parameter. It is straightforward to demonstrate that at one loop all electroweak parameters have at most a logarithmic dependance on Mh .17 This fact has been 'The renormalization group improved effective potential sums all potentially large logarithms, log(Q2/v 2). 9This result is scheme dependent. Here p == Mtv/M~ cos2 Ow(Mw), where cos Ow is a running parameter calculated at an energy scale of Mw.

16 800.0

,.....,~---,-~---.,........----,------.

600.0

---> Q.)

CD 400.0

----

Landau pole

.r:

~

200.0

0.0 ' o - - - - - - - - c - - - - - , ; - - - - - - - , , ; - - - - - " 5 1 1 10 10

A (GeV) Figure 4: Theoretical limits on the Higgs boson mass. The allowed region is shaded. The region above the solid line (labelled Landau pole) is forbidden because the quartic coupling is infinite. The region below the dot-dash line is forbidden because the quartic coupling is negative, causing the potential to be unbounded from below.

glorified by the name of the "screening theorem" .18 In general, electroweak radiative corrections involving the Higgs boson take the form,

Mh 9 2 ( log--

Mw

+g 2 -Ml 2-'" Mw

)

.

(62)

That is, effects quadratic in the Higgs boson mass are always screened by an additional power of 9 relative to the lower order logarithmic effects and so radiative corrections involving the Higgs boson can never be large.19 From precision measurements at LEP and SLC of electroweak observables, the direct measurements of Mw and Mt at the Tevatron, and the measurements of v scattering experiments, there is the bound on the Higgs boson mass coming from the effect of radiative corrections,2°

Mh

< 280 GeV

(95% confidence level).

(63)

This bound does not include the Higgs boson direct search experiments and applies only in the minimal Standard Model. Since the bound of Eq. 63 arises

17

Mt=175.5 + 5.1 GeV 80.50

:>

MW=80.38_+.09 GeV

80.40

CD

CJ .......

i

80.30 -I

a (Mz)=128.99 a(Mz)=·118

80.20

80.10 """'::................--'''-______--'_......._--'_---o._---L_......._ ...... 200.0 150.0 160.0 170.0 180.0 190.0

M, (GeV) Figure 5: Limits on the Higgs boson mass from the Tevatron measurements of Mw and M t for various values of the Higgs boson mass.

from loop corrections, it can be circumvented by unknown new physics which contributes to the loop corrections. The relationship between Mw and M t arising from radiative corrections depends sensitively on the Higgs boson mass. The radiative corrections to Mw can be written as,

(64) where 8r is a function of M? and log(Mh)' The results using the Tevatron measurements of Mw and M t are shown in Fig. 5 and clearly prefer a relatively light Higgs boson, in agreement with the global fit of Eq. 63. While the Higgs boson mass remains a free parameter, the combination of limits from electroweak radiative corrections and the triviality bounds of the previous section suggest that the Higgs boson may be relatively light, in the few hundred GeV range.

18

4

Higgs Branching Ratios

In the Higgs sector, the Standard Model is extremely predictive, with all couplings, decay widths, and production cross sections given in terms of the unknown Higgs boson mass. The measurements of the various Higgs decay channels will serve to discriminate between the Standard Model and other models with more complicated Higgs sectors which may have different decay chains and Yukawa couplings. It is hence vital that we have reliable predictions for the branching ratios in order to verify the correctness of the Yukawa couplings of the Standard Model.21 In this section, we review the Higgs boson branching ratios in the Standard Model. 4.1

Decays to Fermion Pairs

The dominant decays of a Higgs boson with a mass below the W+ W- threshold are into fermion- antifermion pairs. In the Born approximation, the width into charged lepton pairs is (65)

Jl -

where /31 == 4M? / Ml is the velocity of the final state leptons. The Higgs boson decay into quarks is enhanced by the color factor Nc = 3 and also receives significant QeD corrections, (66)

where the QeD correction factor, !l~CD, can be found in Ref. 22. The Higgs boson clearly decays predominantly into the heaviest fermion kinematically allowed. A large portion of the QeD corrections can be absorbed by expressing the decay width in terms of a running quark mass, Mq(p), evaluated at the scale P = Mh. The QeD corrected decay width can then be written as,22

where 0:. (Ml) is defined in the MS scheme with 5 flavors and AlI:f:s = 150 GeV. The 0(0:;) corrections are also known in the limit Mh » M q . 3

19

For 10 GeV < Mh < 160 GeV, the most important fermion decay mode is h -+ bb. In leading log QeD, the running of the b quark mass is, (68) where Mb(M2) == M implies that the running mass at the position of the propagator pole is equal to the location of the pole. For Mb(M;) == 4.2 GeV, this yields an effective value Mb((Mh == 100 GeV)2) == 3 GeV. Inserting the QeD corrected mass into the expression for the width thus leads to a significantly smaller rate than that found using Mb == 4.2 GeV. For a Higgs boson in the 100 GeV range, the 0(0:.) corrections decrease the decay width for h -+ bb by about a factor of two. The electroweak radiative corrections to h -+ qq are not significant and amount to only a few percent correction.24 These can be neglected in comparison with the much larger QeD corrections. The branching ratios for the dominant decays to fermion- antifermion pairs are shown in Fig. 6.h The decrease in the h -+ If branching ratios at Mh '" 150 GeV is due to the turn-on of the WW* decay channel, where W* denotes a virtual W. For most of the region below the W+W- threshold, the Higgs decays almost entirely to bb pairs, although it is possible that the decays to r+r- will be useful in the experimental searches. The other fermionic Higgs boson decay channels are almost certainly too small to be separated from the backgrounds. Even including the QeD corrections, the rates roughly scale with the fermion masses and the color factor, N c ,

r(h -+ bb) r(h -+ r+r-)

(69)

and so a measurement of the branching ratios could serve to verify the correctness of the Standard Model couplings. The largest uncertainty is in the value of 0:., which affects the running b quark mass, as in Eq. 68 .

4.2

Decays to Gauge Boson Pairs

The Higgs boson can also decay to gauge boson pairs. At tree level, the decays h -+ W+W- and h -+ ZZ are possible, while at one-loop the decays h -+ gg, ii, iZ occur. h A convenient FORTRAN code for computing the QCD radiative corrections to the Higgs boson decays is HDECAY, which is documented in Ref. 25.

20

Higgs Branching Ratios to Fermion Pairs

bb +

't't

--"'-"'-"'-----"'- ... cc

-......,-,

-, \,

'.

.

\

\

ss

---------10"

50.0

100.0

150.0

Mh (GeV)

Figure 6: Branching ratios of the Standard Model Higgs boson to fermion-antifermion pairs, including QCD radiative corrections. The radiative corrections were computed using the program HDECAy.25

The decay widths of the Higgs boson to physical W+ W- or Z Z pairs are given by,

f(h --+ W+W-) = f(h --+ ZZ) =

(70)

where rv == 4M~ / M~. Below the W+W- and ZZ thresholds, the Higgs boson can also decay to vector boson pairs VV·, (V = W±, Z), with one of the gauge bosons off-shell. The widths, summed over all available channels for V· --+ II are: 26

f(h --+ ZZ·) f(h --+ WW·)

21

where Sw F(x)

== sin 2 Ow and == - l l - x

2 (~7x2_123 + :2) 1

2 4 -3(1-6X +4X ) Iln(x)

_l(3X2-1) 31-8x2+20x4 V4x 2 -1 2x

+

COS

1

(72)

3

These widths can be significant when the Higgs boson mass approaches the real W+W- and ZZ thresholds, as can be seen in Fig. 7. The WW· and ZZ· branching ratios grow rapidly with increasing Higgs mass and above the rate for h --+ W+ W- is close to 1. The decay width to Z Z· is roughly an order of magnitude smaller than the decay width to Ww· over most of the Higgs mass range due to the smallness of the neutral current couplings as compared to the charged current couplings. The decay of the Higgs boson to gluons arises through fermion loops,l

2Mw

fo(h --+ gg)

=

GFa 2 M3

L

64~7r3h 1

F l / 2 (Tq ) 12

(73)

q

where

Tq

==

4Mi/ M; and F l / 2 ( Tq) is defined to be, F l / 2 (Tq )

The function

!(Tq)

==

-2Tq

[1 + (1- Tq)!(Tq )]

(74)

is given by, if Tq

~ 1

(75) if Tq with x± = 1 ±

VI -

Tq .

< 1, (76)

In the limit in which the quark mass is much less than the Higgs boson mass, (the relevant limit for the b quark),

2Mi 2(Mq) Mh .

F l / 2 --+ M; log

(77)

A Higgs boson decaying to bb will therefore be extremely narrow. On the other hand, for a heavy quark, Tq --+ 00, and F l / 2 (Tq) approaches a constant, F l / 2 --+

4

-"3'

(78)

22

It is clear that the dominant contribution to the gluonic decay of the Higgs boson is from the top quark loop and from possible new generations of heavy fermions. A measurement of this rate would serve to count the number of heavy fermions since the effects of the heavy fermions do not decouple from the theory. The QCD radiative corrections from h -+ ggg and h -+ gqlj to the hadronic decay of the Higgs boson are large and they typically increase the width by more than 50%. The radiatively corrected width can be approximated by

r(h -+ ggX)

= ro(h -+ gg) [1 + cQ'~)]

(79)

where C = 2/25 - 26310g(J.l2 / MK), for Mh < 2Mt .27 The radiatively corrected branching ratio for h -+ ggX is the solid curve in Fig. 7. The decay h -+ Z I is not useful phenomenologically, so we will not discuss it here although the complete expression for the branching ratio can be found in Ref. 28. On the other hand, the decay h -+ II is an important mode for the Higgs search at the LHC. At lowest order, the branching ratio is, 29

(80) where the sum is over fermions and W± bosons with F 1 / 2 (Tq ) given in Eq. 23, and (81) Fw(rw) = 2 + 3rw[1 + (2 - rw )f(rw)]

rw = 4Mlv/M,;, NCi = 3 for quarks and 1 otherwise, and Qi is the electric charge in units of e. The function f(Tq) is given in Eq. 75. The h -+ II decay channel clearly probes the possible existence of heavy charged particles. (Charged scalars, such as those existing in supersymmetric models, would also contribute to the rate.f In the limit where the particle in the loop is much heavier than the Higgs boson, T -+ 00, F 1/ 2

-+

Fw

-+

4 3 7

(82)

The top quark contribution (F1 / 2 ) is therefore much smaller than that of the W loops (Fw) and so we expect the QeD corrections to be less important than is the case for the h -+ gg decay. In fact the QCD corrections to the total width for h -+ II are quite small~o The h -+ II branching ratio is the

23

Higgs Branching Ratios to Gauge Boson Pairs 10°

/

/

wvf ,/"

. ....

/

//

,, // / /

,"

,

"

,

/

,,

, ,,

zz

*

/,.

/'

,/

,I

",

"

"-"---.......~.

,I ,!

/ Zy

10.6 50.0

100.0

150.0

200.0

Mh (GeV) Figure 7: Branching ratios of the Standard Model Higgs boson to gauge boson pairs, including QCD radiative corrections. The rates to WW· and ZZ· must be multiplied by the appropriate branching ratios for W· and Z· decays to pairs. The radiative corrections were computed using the program HDECAy.25

lJ

dotted line in Fig. 7. For small Higgs masses it rises with increasing Mh and peaks at around 2 x 10- 3 for Mh ,...., 125 GeV. Above this mass, the WW· and Z Z· decay modes are increasing rapidly with increasing Higgs mass and the "f"f mode becomes further suppressed. The total Higgs boson width for a Higgs boson with mass less than Mh ,...., 200 GeV is shown in Fig. 8. As the Higgs boson becomes heavier twice the W boson mass, its width becomes extremely large (see Eq. 122). Below around Mh ,...., 150 GeV, the Higgs boson is quite narrow with rh < 10 MeV. As the WW· and Z Z· channels become accessible, the width increases rapidly with rh""" 1 GeV at Mh ,...., 200 GeV. Below the W+W- threshold, the Higgs boson width is too narrow to be resolved experimentally. The total width for the lightest neutral Higgs boson in the minimal supersymmetricmodel is typically much smaller than the Standard Model width for the same Higgs boson mass

24

Higgs Boson Decay Width

..-

>CD

CJ

10.3 50.0

100.0

150.0

200.0

Mh (GeV)

Figure 8: Total Higgs boson decay width in the Standard Model, including QeD radiative corrections. The turn-on of the W+ W- threshold at Mh '" 160 GeV is obvious.

and so a measurement of the total width could serve to discriminate between the two models.

5

Higgs Production at LEP and LEP2

Since the Higgs boson coupling to the electron is very small, '" me/v, the s-channel production mechanism, e+ e- --t h, is minute and the dominant Higgs boson production mechanism at LEP and LEP2 is the associated production with a Z, e+e- --t Z· --t Zh, as shown in Fig. 9. At LEP2, a physical Z boson can be produced and the cross section is~l

(83)

25

,,

z e+

,, h

,,

Figure 9: Higgs production through e+e-

-t

Zh.

where

(84)

>.V;:

The center of mass momentum of the produced Z is .JS/2 and the cross section is shown in Fig. 10 as a function of .JS for different values of the Higgs boson mass. The cross section peaks at .JS ~ Mz + 2Mh. From Fig. 10, it is apparent that the cross section increases rapidly with increasing energy and so the best limits on the Higgs boson mass will be obtained at the highest energy. The electroweak radiative corrections are quite small at LEP2 energies~2 Photon bremsstrahlung can be important, however, since it is enhanced by a large logarithm, log(slm;). The photon radiation can be accounted for by integrating the Born cross section of Eq. 83 with a radiator function F which includes virtual and soft photon effects, along with hard photon radiation~3 (J'

=~

!

(85)

ds' F(x, s)o-(s')

where x = 1- s'ls and the radiator function F(x, s) is known to O(a 2 ), along with the exponentiation of the infrared contribution,

F(x, s)

tx

t

2:

-

1

{

1+

~t} + {~ -

[log(~~) -1]

l}t + O(e) (86)

Photon radiation significantly reduces the Z h production rate from the Born cross section as shown in Fig. 11.

26

Mh=90 GeV Mh=100 GeV Mh=110 GeV

0.6

/""-,,

I

/

./

-- -- -----

./

I

0.2 I

I

, I I

0.0 L..o...............................o-.......................................o-..............................._ _..............................J 180.0 190.0 200.0 210.0 ECM

(GeV)

Figure 10: Born cross section for e+ e- -+ Zh as a fu~ction of center of mass energy.

A Higgs boson which can be produced at LEP or LEP2 will decay mostly to bb pairs, so the final state from e+ e- -+ Z h will have four fermions. The dominant background is Zbb production, which can be efficiently eliminated by b-tagging almost up to the kinematic limit for producing the Higgs boson. LEP2 studies estimate that with 200 GeV and .c 100 pb- 1 per experiment, a Higgs boson mass of 107 GeV could be observed at the 50" level~4 A higher energy e+e- machine (such as an NLC with Vs '" 500 GeV) could push the Higgs mass limit to around Mh '" .7VS. Currently the highest energy data at LEP2 is VS = 183 Ge V. The combined limit from the four LEP2 detectors iSla

vs =

Mh

> 89.8 GeV

=

at 95% c.l.

(87)

This limit includes both hadronic and leptonic decay modes of the Z. Note how close the result is to the kinematic boundary. The cross section for e+e- -+ Zh is s-wave and so has a very steep dependence on energy and on the Higgs boson mass at threshold, as is clear from

27

0.9

0.1 70.0

-------

ECM =192 ECtoA=192 EcM =175 ECM =175

80.0

GeV, GeV, GeV, GeV,

with ISR Born only with ISR Born only

90.0

100.0

Mh (GeV)

Figure 11: Effects of initial state radiation{ISR} on the process e+ e- -+ Zh. The curves labelled "Born only" are the results of Eq. 83, while those labelled "with ISR" include the photon radiation as in Eq. 85.

Fig. 10. This makes possible a precision measurement of the Higgs mass. By measuring the cross section at threshold and normalizing to a second measurement above threshold in order to minimize systematic uncertainties a high energy e+e- collider with VB = 500 GeV could obtain a 10" measurement of the mass 35 for Mh = 100 GeV

(88)

where L is the total integrated luminosity. The precision becomes worse for larger Mh because of the decrease in the signal cross section. (Note that the luminosity at LEP2 will not be high enough to perform this measurement.) The angular distribution of the Higgs boson from the e+ e- -+ Zh process is 1 dO" 2. 2 8M~ (89) - - - '" AZh sm () + - 0" dcos () s

28

- - - h

Figure 12: Higgs boson production through gluon fusion with a quark loop.

so that at high energy the distribution is that of a scalar particle, 1 du ---u dcos ()

~

. 2 sm ()

(90)

If the Higgs boson were CP odd, on the other hand, the angular distribution would be 1 + cos 2 (). Hence the angular distribution is sensitive to the spin-parity assignments of the Higgs boson~6 The angular distribution of this process is also quite sensitive to non-Standard Model ZZh couplings.37

6

Higgs Production in Hadronic Collisions

We turn now to the production of the Higgs boson in pp and pp collisions. 6.1

Gluon Fusion

Since the coupling of a Higgs boson to an up quark or a down quark is proportional to the quark mass, this coupling is very small. The primary production mechanism for a Higgs boson in hadronic collisions is through gluon fusion, gg -t h, which is shown in Fig. 12~8 The loop contains all quarks in the theory and is the dominant contribution to Higgs boson production at the LHC for all Mh < 1 Te V. (In extensions of the standard model, all massive colored particles run in the loop.) The lowest order cross section for gg -t h is,l·38 u(gg -t

h)

10~~v2 1L F1/2(Tq) 12 8( 1- ~l) q

UO(99-th)d(1-

~l)

(91)

29

where s is the gluon-gluon sub-process center of mass energy, v = 246 GeV, and F1/2(Tq) is defined in Eq. 74. In the heavy quark limit, (Mt/Mh -+ 00), the cross section becomes a constant,

(92) Just like the decay process, h -+ gg, this rate counts the number of heavy quarks and so could be a window into a possible fourth generation of quarks. The Higgs boson production cross section at a hadron collider can be found by integrating the parton cross section, (1o(pp -+ h), with the gluon structure functions, g( x),

(1o(pp-+h) =1 and 1/>2, interacting with the Standard Model Higgs boson, h, (the reason for

63

__t.'.!2.._

,",;1"

... ,

I

I I\

, '

' ,

,

' ....""

............

I

I'

-----------~----------h h

Figure 29: Scalar contributions to Higgs mass renormalization.

introducing 2 scalars is that with foresight we know that a supersymmetric theory associates 2 complex scalars with each massive fermion - we could just as easily make the argument given below with one additional scalar and slightly different couplings),

1OJJ{ + t·Tij~),

T ·).

2

I

(180)

The D terms then contribute to the scalar potential:

(181)

(182)

68

we find,

VD =

(183) The remainder of the scalar potential is given in terms of a function, W (the superpotential), which can be at most cubic in the scalar superfields. The SU(2)L x U(l)y gauge invariance allows only one interaction involving only the Higgs scalar fields,

(184) The supersymmetry algebra requires that W give a contribution to the scalar potential~4

(185) where Zi is a superfield. To obtain the interactions, we take the derivative of W with respect to Z and then evaluate the result in terms of the scalar component of z. The supersymmetric scalar potential is then,

v =1 JJ 12

(I 1 12 + 12 12) +

g2

~ gl2 (I 2 12 _ 11 12) 2 + g;

1;: . 2 12 . (186)

This potential has its minimum at (~) = (g) = 0, giving (V) = 0 and so represents a model with no electroweak symmetry breaking. It is difficult to break supersymmetry (and we know that it must be broken since there are no scalars degenerate in mass with the known fermions). The simplest solution is simply to add all possible soft supersymmetry breaking mass terms. In the Higgs sector, this amounts to adding masses for each doublet, along with an arbitrary mixing term! The scalar potential involving the Higgs bosons becomes,

(187) IThese soft supersymmetry breaking terms do not generate quadratic divergences.

69 The Higgs potential of the SUSY model can be seen to depend on 3 independent combinations of parameters,

1J.L 12 +mi, 1J.L 12 +m~, J.LB ,

(188)

where B is a new mass parameter. This is in contrast to the general 2 Higgs doublet model where there are 6 arbitrary coupling constants (and a phase) in the potential. From Eq. 187, it is clear that the quartic couplings are fixed in terms of the gauge couplings and so they are not free parameters. Note that VH automatically conserves CP since any complex phase in J.LB can be absorbed into the definitions of the Higgs fields. Clearly, if J.LB = 0 then all the terms in the potential are positive and the minimum of the potential occurs with V = 0 and (~) = (g) = O. Hence all 3 parameters must be non-zero in order for the electroweak symmetry to be broken. m The symmetry is broken when the neutral components of the Higgs doublets get vacuum expectation values, (189) By redefining the Higgs fields, we can always choose Vl and V2 positive. In the MSSM, the Higgs mechanism works in the same manner as in the Standard Model. When the electroweak symmetry is broken, the W gauge boson gets a mass which is fixed by Vl and V2,

2

Mw

g2 2 2 = 2(Vl + v2)

(190)

Before the symmetry was broken, the 2 complex SU(2)L Higgs doublets had 8 degrees of freedom. Three of these were absorbed to give the Wand Z gauge bosons their masses, leaving 5 physical degrees of freedom. There is now a charged Higgs boson, H±, a CP -odd neutral Higgs boson, AO, and 2 CP-even neutral Higgs bosons, hand H. It is a general prediction of supersymmetric models that there will be an expanded sector of physical Higgs bosons. After + v~ such that the W boson gets the correct mass, the Higgs sector fixing is then described by 2 additional parameters. The usual choice is

vr

(191) mWe assume that the parameters are arranged in such a way that the scalar partners of the quarks and leptons do not obtain vacuum expectation values. Such vacuum expectation values would spontaneously break the SU(3) color gauge symmetry or lepton number.

70 120

110

/'

-> -

I

100

/

Q)

(!J

90

.c

,I

~

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

80 I

,/

I

,

,

, /

/ I

I

,,'"

,""

,...~--

-------------

/

/

I

tan ~ = 2.5 tan ~ =5 tan ~ =40

70

120

160

200

240

MA (GeV) Figure 31: Mass of the neutral Higgs bosons as a function of the pseudoscalar mass, M A , and tan,B. This figure assumes a common scalar mass of 1 TeV, and neglects mixing effects, (Ai J.t 0).

= =

and M A , the mass of the pseudoscalar Higgs boson. Once these two parameters are given, then the masses of the remaining Higgs bosons can be calculated in terms of MA and tanJ3. Note that we can chose 0 ~ 13 ~ ~ since we have chosen VI, v2 > O. In the MSSM, the J.l parameter is a source of concern. It cannot be set to zero because then there would be no symmetry breaking. The Z-boson mass can be written in terms of the radiatively corrected neutral Higgs boson masses and J.l: 2 M2 = 2 [MK - Mk tan 13] _ 2 2 (192) Z tan2J3 _ 1 J.l . This requires a delicate cancellation between the Higgs masses and J.l. This is unattractive, since much of the motivation for supersymmetric theories is the desire to avoid unnatural cancellations. The J.l parameter can, however, be generated naturally in theories with additional Higgs singlets.

71

Maximum Higgs Mass (No mixing) 120.0

110.0

->

100.0

C)

90.0

--

~ = 50_----.:::::-...::::...-...::::...--== ... .... -_.;::..--..........::,..;.... .... tan

:.._-

--

--

tan

~ =10

Q)

)(

ttl

E

.s:

~

80.0

70.0

60.0 L.-_"""'--_-L.._--"_ _L.-_"""'--_-L.._---'"_----I 200.0 400.0 600.0 800.0 1000.0 msquark (GeV) Figure 32: Maximum value of the lightest Higgs boson mass as a function of the squark mass. This figure includes 2-loop radiative corrections and renormalization group improvements. (We have assumed degenerate squarks and set the mixing parameters Ai jJ, 0.)

= =

It is straightforward to find the physical Higgs bosons and their masses in terms of the parameters of Eq. 187. Details can be found in Ref. 1. The neutral Higgs masses are found by diagonalizing the 2 x 2 Higgs mass matrix and by convention, h is taken to be the lighter of the neutral Higgs. At tree level, the masses of the neutral Higgs bosons are given by,

The pseudoscalar mass is given by, M2 = A

21 p,B I, sin 2{3

(194)

and the charged scalar mass is,

(195) We see that at tree level, Eq. 187 gives important predictions about the relative

72

masses of the Higgs bosons,

Mw Mz

MA Mz

I cos 213 I

(196)

These relations yield the attractive prediction that the lightest neutral Higgs boson is lighter than the Z boson. However, loop corrections to the relations of Eq. 196 are large. In fact the corrections to M; grow like GpM( and receive contributions from loops with both top quarks and squarks. In a model with unbroken supersymmetry, these contributions would cancel. Since the supersymmetry has been broken by splitting the masses of the fermions and their scalar partners, the neutral Higgs boson masses become at one- 100p~6

where

fh

is the contribution of the one-loop corrections, (198)

We have assumed that all of the squarks have equal masses, m, and have neglected the smaller effects from the mixing parameters, Ai and Jl. In Fig. 31, we show the masses of the neutral Higgs bosons as a function of the pseudoscalar mass and for three values of tan 13. For tan 13 > 1, the mass eigenvalues increase monotonically with increasing MA and give an upper bound to the mass of the lightest Higgs boson, (199)

The corrections from fh are always positive and increase the mass of the lightest neutral Higgs boson with increasing top quark mass. From Fig. 31, we see that Mh obtains its maximal value for rather modest values of the pseudoscalar mass, MA > 300 GeV. The radiative corrections to the charged Higgs masssquared are proportional to M? and so are much smaller than the corrections to the neutral masses.

73

Higgs Couplings to b in SUSY

20.0

tan13 = 30

s:::

.0 .0

() 10.0

0.0 50.0

~ 250.0

tanf3 = 450.0

1.5

650.0

850.0

MA (GeV) Figure 33: Coupling of the lightest Higgs boson to charge -1/3 quarks. The value C bbh = 1 corresponds to the Standard Model coupling of the Higgs boson to charge -1/3 quarks.

There are many analyseSl 6 which include a variety of two-loop effects, renormalization group effects, etc., but the important point is that for given values of tan j3 and the squark masses, there is an upper bound on the lightest neutral Higgs boson mass. For large values of tan j3 the limit is relatively insensitive to the value of tan j3 and with a squark mass less than about 1 Te V, the upper limit on the Higgs mass is about 110 Ge V if mixing in the top squark sector is negligible (AT"'" 0). For large mixing, this limit is raised to around 130 GeV . • The minimal supersymmetric model predicts a neutral Higgs boson with a mass less than around 130 GeV. Such a mass scale may be accessible at LEP2, an upgraded Tevatron or the LHC and provides a definitive test of the MSSM. In a more complicated supersymmetric model with a richer Higgs structure, the upper bound on the lightest Higgs boson mass will be changed. However,

74 the requirement that the Higgs self coupling remain perturbative up to the Planck scale gives an upper bound on the lightest supersymmetric Higgs boson of around 150 Ge V in all models with only singlets and doublets of Higgs bosons.87 This is a very strong statement. It implies that either there is a relatively light Higgs boson or else there is some new physics between the weak scale and the Planck scale which causes the Higgs couplings to become non-perturbative. Another feature of the MSSM is that the fermion- Higgs couplings are no longer strictly proportional to mass. From the superpotential can be found both the scalar potential and the Yukawa interactions of the fermions with the scalars: " I~ oW I2 -2 1" ' [ 1/;iL - 7f7)-:1/;j 02W (200) = - 'L...J L...J + h.c. ] ,

.cw

i

Z,

ij

Z,

zJ

where Zi is a chiral superfield. This form of the Lagrangian is dictated by the supersymmetry and by the requirement that it be renormalizable. To obtain the interactions, we take the derivatives of W with respect to the superfields', Zi, and then evaluate the result in terms of the scalar component of Zi. The usual approach is to write the most general SU(3) x SU(2)L x U(I)y invariant superpotential with arbitrary coefficients for the interactions,

W

= (ijJ1.iI~iI~ + (ij [ALiI~LCj EC+ ADiI~Qj b c + AuiI~QiUC] +(ij [A1Li £i ftc

+ A2LiQi b C] + A3 U c b c b c ,

(201)

(where i,j are SU(2) indices). We have written the superpotential in terms of the fields of the first generation. In principle, the Ai could all be matrices which mix the interactions of the 3 generations. The terms in the square brackets proportional to AL, AD, and AU give the usual Yukawa interactions of the fermions with the Higgs bosons from the term _ ( 02W )

1/;; OZiOZj

tPj.

(202)

Hence these coefficients are determined in terms of the fermion masses and the vacuum expectation values of the neutral members of the scalar components of the Higgs doublets and are not free parameters at all. It is convenient to write the couplings for the neutral Higgs bosons to the fermions in terms of the Standard Model Higgs couplings,

75

Table 5: Higgs Boson Couplings to fermions

f

Gjjh

GjjH

G jjA

cos(\'

sinO! sin i3

cot (3

cos a

tan (3

U

d

sin i3 _sin~

cos

cos/f

where Gjjh is 1 for a Standard Model Higgs boson. The Gjji are given in Table 5 and plotted in Figs. 33 and 34 as a function of M A . We see that for small MA and large tan (3, the couplings of the neutral Higgs boson to fermions can be significantly different from the Standard Model couplings; the b-quark coupling becomes enhanced, while the t-quark coupling to the lightest Higgs boson is suppressed. When MA becomes large the Higgs-fermion couplings approach their standard model values, GJJh -+ 1. In fact even for MA ,...., 300 GeV, the Higgs-fermion couplings are very close to their Standard Model values. The Higgs boson couplings to gauge hosons are fixed hy the SU(2)L x U(l) gauge invariance. Some of the phenomenologically important couplings are: ZI'Z"h: ZI'Z"H:

igMz . ((3 -a )9 1''' ---sm cos8 w igMz - - cos((3 - a )gl''' cos8 w igMw sin((3 - a)gl''' igMw cos((3 - a)gl'''

Zl'h(p)A(p') : ZI' H(p)A(p') :

gcos((3 - a) ( ')1' '-----'------'- P + P 2cos8 w 9 sin((3 - a) (

-

2cos8 w

')1'

p+ P

(204)

We see that the couplings of the Higgs hosons to the gauge bosons all depend on the same angular factor, (3 - a. The pseudoscalar, A 0 , has no tree level coupling to pairs of gauge bosons. The angle (3 is a free parameter while the neutral Higgs mixing angle, a, which enters into many of the couplings, can he found at leading logarithm in terms of MA and (3: (205) With our conventions, -~ :::; a :::; O. It is clear that the couplings of the neutral scalars to vector bosons (V = W±, Z) are suppressed relative to those of the

76

Figure 34: Coupling of the lightest Higgs boson to charge 2/3 quarks. The value C tth yields the Standard Model coupling of the Higgs boson to charge 2/3 quarks.

= 1

standard model (206)

where gHVV is the coupling of the Higgs boson to vector bosons. Because of this sum rule, the WW scattering production mechanism tends not to be as important in supersymmetric models as in the Standard Model. A complete set of couplings for the Higgs bosons (including the charged and pseudoscalar Higgs) at tree level can be found in Ref. 1. These couplings completely determine the decay modes of the supersymmetric Higgs bosons and their experimental signatures. The important point is that (at lowest order) all of the couplings are completely determined in terms of MA and tan {3. When radiative corrections are included there is a dependence on the squark masses

77

and the mixing parameters in the mass matrices. This dependence is explored in detail in Ref. 34. It is an important feature of the MSSM that for large MA, the Higgs sector looks exactly like that of the Standard Model. As MA -+ 00, the masses of the charged Higgs bosons, H±, and the heavier neutral Higgs, H, become large leaving only the lighter Higgs boson, h, in the spectrum. In this limit, the couplings of the lighter Higgs boson, h, to fermions and gauge bosons take on their Standard Model values. We have, sin(,8 - a) cos(,8 - a)

(207)

From Eq. 204, we see that the heavier Higgs boson, H, decouples from the gauge bosons in the heavy MA limit, while the lighter Higgs boson, h, has Standard Model couplings. The Standard Model limit is also rapidly approached in the fermion-Higgs couplings for MA > 300 Ge V. In the limit of large M A , it will thus be exceedingly difficult to differentiate a supersymmetric Higgs sector from the Standard Model Higgs boson . • The supersymmetric Higgs sector with large MA looks like the Standard Model Higgs sector. In this case, it will be difficult to discover supersymmetry through the Higgs sector. Instead, it will be necessary to find some of the other supersymmetric partners of the observed particles. At a hadron collider, the neutral Higgs bosons of a supersymmetric model can be searched for using many of the same techniques as in the Standard Model~5.53 For most choices of the parameter space, gluon fusion is the dominant production mechanism. In the Standard Model, it was only the top quark contribution to gluon fusion which was important. In a supersymmetric model, however, the coupling to the b quark can be important for small values of cos,8, as can be seen from Table 5. Supersymmetric models have a rich particle spectrum in which to search for evidence of the Higgs mechanism. The various decays such as h, H -+ ''{''/, H+ -+ [+11, AO -+ r+r-, etc, are sensitive to different regions in the MA - tan,8 parameter space. It takes the combination of many decay channels in order to be able to cover the parameter space completely with out any holes. Discussions of the capabilities of the LHC detectors to experimentally observe evidence for the Higgs bosons of supersymmetric models can be found in the ATLASi3 and CMSi4 studies. An upgraded Tevatron will also have the capability to obtain meaningful limits on the symmetry breaking sector of a supersymmetric model!38

78 Table 6: Higgs Mass Reach of Future Accelerators Accelerator

Luminosity

LEP2 (192 GeV) Tevatron TEV-33

150 pb- 1 5 - 10 fb- 1 25 - 30 fb- 1 100 fb- 1 50 fb- 1

LHC NLC (500 GeV)

Higgs Mass Reach 95 GeV 80 - 100 GeV 120 GeV 800 GeV 350 GeV

Both the Tevatron and the LEP and LEP2 colliders have searched for the Higgs bosons and other new particles occuring in a SUSY model and have ruled out large portions of the tan{3- MA parameter space~9

12

Conclusions

Our current experimental knowledge of the Standard Model Higgs boson gives only the limits Mh > 90 Ge V and Mh < 280 Ge V from direct search experiments and precision measurements at the LEP and LEP2 experiments. From here, we must wait until the advent of an upgraded Tevatron and the LHC for further limits. Through the decay h -+ 'Y'Y and the production process pp -+ Z[+[-, the LHC will probe the mass region between 100 < Mh < 180 GeV, while the Tevatron is sensitive to Mh < 130 GeV with 30 fb- 1 through the Wh production process. For the higher mass region, 180 < Mh < 800 GeV, the LHC will be able to see the Higgs boson through the gold plated decay mode, h -+ Z Z -+ [+ [-[+ [-. The expected sensitivity of future colliders is summarized in Table 6.9° One of the important yardsticks for all current and future accelerators is their ability to discover (or to definitively exclude) the Higgs boson of the Standard Model. We hope that at the time of the LHC, we will be able to probe all mass scales up to Mh ,.., 800 GeV. Having found the Higgs boson, the next goal will be to determine if it is a Standard Model Higgs boson, or a Higgs boson of some more complicated theory such as the MSSM. If the Higgs boson is not found below this mass scale then we are in the regime where perturbative unitarity has broken down and we are led to the exciting conclusion that there must be new physics beyond the Standard Model waiting to be discovered.

79

Acknowledgements I am grateful to G. Senjanovic, A. Masiero, and D. Smirnov for organizing such a successful and enjoyable summer school. 1. J. Gunion, H. Haber, G. Kane, and S. Dawson, The Higgs Hunters Guide, (Addison-Wesley, Menlo Park, 1990). 2. Similar material can be found in J. Bagger, Physics Beyond the Standard Model, lectures given at the 1991 TASI summer school, Boulder, CO, 2-28 June, 1991, (World Scientific, Singapore, 1992); S. Dawson, Introduction to the Physics of Higgs Bosons, lectures given at the 1994 TASI summer school, Boulder, CO, 29 May -24 June, 1994, (World Scientific, Singapore, 1994), hep-ph/9411325; A. Djouadi, Int. J. Mod. Phys. AI0 (1995) 1; R. Chivukula, Models of Electroweak Symmetry Breaking: Course, lectures given at the Les Houches Center of Physics, Les Houches, France, 16-26 June, 1998, hep-ph/9803219; M. Spira and P. Zerwas, Electroweak Symmetry Breaking and Higgs Physics, lectures given at the International University School of Nuclear and Particle Physics, Schladming, Austria, 1-8 March 1997, hep-ph/9803257. 3. An introduction to the Standard Model can be found in C. Quigg, Gauge Theories of the Strong, Weak, and Electromagnetic Interactions, (Benjamin-Cummings, Reading, MA., 1983). 4. E. Abers and B. Lee, Phys. Rep. 9 (1975) 1. 5. P. W. Higgs, Phys. Rev. Lett. 13 (1964)508; Phys. Rev. 145 (1966) 1156; F. Englert and R. Brout, Phys. Rev. Lett. 13 (1964) 321; G. S. Guralnik, C. R. Hagen, and T. Kibble, Phys. Rev. Lett. 13 (1965) 585; T. Kibble, Phys. Rev. 155 (1967) 1554. 6. S. Glashow, Nucl. Phys. 22 (1961) 579; S. Weinberg, Phys. Rev. Lett. 19 (1976) 1264; A. Salam, in Elementary Particle Theory, ed. N. Svartholm (Almqvist and Wiksells, Stockholm, 1969), p. 367. 7. K. Wilson, Phys. Rev. B4 (1971) 3184; K. Wilson and J. Kogut, Phys. Rep. 12 (1974) 75; R. Dashen and H. Neuberger, Phys. Rev. Lett. 50 (1983) 1897; P. Hasenfratz and J. Nager, Z. Phys. C37 (1988); J. Kuti, L. Lin, and Y. Shen, Phys. Rev. Lett. 61 (1988) 678; M. Luscher and P. Weisz, Phys. Lett. B212 (1988) (472). 8. R. Chivukula and E. Simmons, Phys. Lett. B388 (1996) 788. 9. M. Quiros, Perspectives in Higgs Physics, Ed. G. Kane, (World Scientific, Singapore, 1997). 10. A. Hasenfratz, Quantum Fields on the Computer, Ed. M. Creutz, (World Scientific, Singapore, 1992),p. 125.

80 11. T. Cheng, E. Eichten and L. Li, Phys. Rev. 09 (1974) 2259; B. Pendleton and G. Ross, Phys. Lett. B98 (1981) 291; C. Hill, Phys. Rev. 024 (1981) 691; J. Bagger, S. Dimopoulos and E. Masso, Nucl. Phys. B253

12. 13.

14.

15. 16. 17.

18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

(1985) 397; M. Beg, C. Panagiotakopoulos, and A. Sirlin, Phys. Rev. Lett. 52 (1984) 883; M. Duncan, R. Philippe, and M. Sher, Phys. Lett. B153 (1985) 165; K. Babu and E. Ma, Phys. Rev. Lett. 55 (1985) 3005. N. Cabibbo et.al., Nucl. Phys. BI58 (1979) 295. A. Linde, JETP Lett 23 (1976) 64; Phys. Lett. B62 (1976) 435; S. Weinberg, Phys. Rev. Lett. 36 (1976) 294; S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 188. M. Lindner, M. Sher, and H. Zaglauer, Phys. Lett. B228 (1989) 139; C. Ford et. al., Nucl. Phys. B395 (1993) 62; M. Sher, Phys. Rep. 179 (1989)274; F. del Aguila, M Martinez, M. Quiros, Nucl. Phys. B381 (1992) 451; J. Casas, J. Espinosa, and M. Quiros, Phys. Lett. B324 (1995) 171; Phys. Lett. B382 (1996) 374; J. Espinosa and M. Quiros, Phys. Lett. B353 (1995) 257. M. Sher, Phys. Lett. B3I7 (1993)159; addendum, B331 (1994) 448. W. Marciano and A. Sirlin, Phys. Rev. Lett. 46 (1981) 163; W. Marciano, S. Sarantakos, and A. Sirlin, Nucl. Phys. B217 (1988) 84. T. Appelquist and C. Bernard, Phys. Rev. D22 (1980) 200; A. Longhitano, Nucl. Phys. B188 (1981) 118; T. Appelquist and G. Wu, Phys. Rev. 051 (1995) 240. M. Veltman, Acta. Phys. Pol. B8 (1977) 475. M. Einhorn and J. Wudka, Phys. Rev. D39 (1989) 2758. D. Karlen, results presented at XXIX International Conference on High Energy Physics, Vancouver, Canada, July 23-29, 1998. M. Spira, Fortsch. Phys. 46 (1998) 203. E. Braaten and J. Leveille, Phys. Rev. D22 (1980) 715; and M. Drees and K. Hikasa, Phys. Lett. B240 (1990) 455. K. Melnikov, Phys. Rev. D53 (1996) 5020. A. Djouadi and P. Gambino, Phys. Rev. Lett. 73 (1994) 2528. A. Djouadi, J. Kalinowski, and M. Spira, Comput. Phys. Comm. 108 (1998) 56. W.-Y. Keung and W. Marciano, Phys. Rev. D30 (1984) 248. T. Inami, T. Kubota, and Y. Okada, Z. Phys. CI8 (1983) 69; M. Spira, A. Djouadi, D. Graudenz, and P. Zerwas, Nucl. Phys. B453 (1995) 17. R. Cahn, M. Chanowitz, and N. Fleishon, Phys. Lett. B82 (1979)113; G. Gamberini, G. Giudice and G. Ridolfi, Nucl. Phys. B292 (1987) 237.

81

29. A. Vainshtein, M. Voloshin, V. Sakharov, and M. Shifman, Sov. J. Nucl. Phys. 30 (1979) 711. 30. S. Dawson and R. Kauffman, Phys. Rev. D49 (1993) 2298; A. Djouadi, M. Spira, and P. Zerwas, Phys. Lett. B311 (1993) 255. 31. B. Ioffe and V. Khoze, Sov. J. Part. Nucl. Phys. 9 (1978) 50. 32. B. Kniehl, Phys. Rep. 240C (1994) 211. 33. F. Berends, W. van Neerven, and G. Burgers, Nucl. Phys. B297 (1988) 429; erratum B304 (1988) 921. 34. M. Carena and P. Zerwas, Higgs Physics, CERN Yellow Report, CERN96-01, hep-ph/9602250. 35. V. Barger, M. Berger, J. Gunion, and T. Han, Phys. Rev. Lett. 78 (1997) 3991. 36. V. Barger, K. Cheung, A. Djouadi, B. Kniehl, and P. Zerwas, Phys. Rev. 049 (1994) 49. 37. W. Kilian, M. Kramer, and P. Zerwas, Phys. Lett. B381 (1996) 243. 38. F. Wilczek, Phys. Rev. Lett. 39 (1977) 1304; J. Ellis et.al., Phys. Lett. 83B (1979) 339; H. Georgi et. al., Phys. Rev. Lett. 40 (1978) 692; T. Rizzo, Phys. Rev. 022 (1980) 178. 39. M. Spira, A. Djouadi, D. Graudenz, and P. Zerwas, Nucl. Phys. B453 (1995) 17. 40. S. Dawson, Nucl. Phys. B359 (1991) 283. 41. A. Vainshtein et.al., Sov. J. Nucl. Phys.; M. Voloshin, Sov. 1. Nucl. Phys. B44 (1986) 478. 42. R. Kauffman, Phys. Rev. 045 (1992) 1512. 43. LO! of the ATLAS Collaboration, CERN/LHCC/92-4, Oct., 1992. 44. LO! of the CMS Collaboration, CERN/LHCC/92-3, Oct., 1992. 45. F. Paige, Supersymmetry Signatures at the Cern LHC, lectures given at the 1997 TAS7 summer school, Boulder, CO, 2-28 June, 1997, (World Scientific, Singapore, 1998). 46. P. Agrawal and S. Ellis, Phys. Lett. B229 (1989) 145. 47. J. Gunion, G. Kane, and J. Wudka, Nucl. Phys. B299 (1988) 231. 48. K. Ellis et.al., Nucl. Phys. B297(1988) 221. 49. D. Rainwater, D. Zeppenfeld, and K. Hagiwara, hep-ph/9808468. 50. W. Marciano, A. Stange, and S. Willenbrock, Phys. Rev. 049 (1994) 1354; Phys. Rev. D50 (1994) 4491. 51. T. Han and S. Willenbrock, Phys. Lett. B273 (1991) 167. 52. Future Electroweak Physics at the Fermilab Tevatron; Report of the Te V 2000 Study Group, Ed. D. Amidei and R. Brock, Fermilab-PUB-96-082, April, 1996. 53. S. Mrenna and C. P. Yuan, Phys. Lett. B416 (1998) 200.

82

54. S. Kim, S. Kuhlmann, and W. -M. Yao, in Proceedings of the 1996 DPF/DPB Summer Study on New Directions for High Energy Physics, 1996; W. - M. Yao, in Proceedings of the 1996 DPF/DPB Summer Study on New Directions for High Energy Physics, (Snowmass, Colorado, July, 1996. 55. S. Dawson, Nucl. Phys. B249 (1985) 42; G. Kane, W. Repko, and W. Rolnick, Phys. Lett B148 (1984) 367; M Chanowitz and M. Gaillard, Phys. Lett. B142 (1984) 85. 56. S. Brodsky, T. Kinoshita, and H. Terazawa, Phys. Rev. D4 (1971) 1532. 57. U. Baur and E. Glover, Nucl. Phys. B347 (1990) 12; Phys. Rev. D44 (1991) 99. 58. R. Cahn et. al. Phys. Rev. D35 (1987) 1626; V. Barger, T. Han, and R. Phillips, Phys. Rev. D37 (1988) 2005; J. Gunion and M. Soldate, Phys. Rev. D34 (1986) 826. 59. A Djouadi, D. Haidt, B. Kniehl, B. Mele, and P. Zerwas, Proceedings of e+ e- Collisions at 500 GeV: The Physics Potential, (Munich-AnnecyHamburg), ed. P.Zerwas, DESY 92-123A. 60. S. Dawson and J. Rosner, Phys. Lett., B148 (1984) 497. 61. A. Djouadi, J. Kalinowski, and P. Zerwas, Zeit. fur Phy. eM (1992) 255; K. Gaemers and G. Gounaris, Phys. Lett. B77 (1978) 379. 62. S. Dawson and L. Reina, hep-ph/9808443; S. Dittmaier et.al., hepph/9808433. 63. W. Marciano and S. Willenbrock, Phys. Rev. D37 (1988) 2509. 64. B. Lee, C. Quigg, and H. Thacker, Phys. Rev. D16 (1977) 1519; D. Dicus and V. Mathur, Phys. Rev. D7 (1973) 3111. 65. M. Duncan, G. Kane, and W. Repko, Nucl. Phys. B272 (1986) 517. 66. J. Cornwall, D. Levin, and G. Tiktopoulos, Phys. Rev. D10 (1974) 1145; Dll (1975) 972E; B. Lee, C. Quigg, and H. Thacker, Phys. Rev. D16 (1977) 1519; M. Chanowitz and M. Gaillard, Nucl. Phys. B261 (1985) 379; Y-P. Yao and C. Yuan, Phys. Rev. D38 (1988) 2237; J. Bagger and C. Schmidt, Phys. Rev. D41 (1990) 264; H. Veltman, Phys. Rev. D41 (1990) 2294. 67. S. Dawson and S. Willenbrock, Phys. Rev. D40 (1989); M. Veltman and F. Yndu rain, Nucl. Phys. B163 (1979) 402. 68. M. Chanowitz and M. Gaillard, Nucl. Phys. B261 (1985) 379. 69. M. Chanowitz, H. Georgi, and M. Golden, Phys. Rev. Lett. 57 (1986) 2344; Phys. Rev.D36 (1987) 1490. 70. J. Gasser and H. Leutwyler, Ann. Phys. 158 (1984) 142; Nucl. Phys. B250 (1985) 465. 71. S. Weinberg, Phys. Rev. Lett. 17 (1966) 616.

83

72. S. Dawson and G. Valencia, Nucl. Phys. B439 (1995) 3. 73. S. Alam, S. Dawson and R. Szalapski, Phys. Rev. D57 (1998) 1577. 74. R. Szalapski, Phys. Rev. D57 (1998) 5519; A. De Rujula, M. Gavela, P. Hernandez, and E. Masso, Nucl. Phys. B384 (1992)3. 75. M. Peskin and T. Takeuchi, Phys. Rev. Lett. 65 (1990) 964; D. Kennedy and B. Lynn, Nucl. Phys. B322 (1989) 1. 76. K. Hagiwara, S. Ishihara, R. Szalapski, and D. Zeppenfeld, Phys. Rev. D48 (1993) 2182. 77. A. Falk, M. Luke, and E. Simmons, Nucl. Phys. B365 (1991) 523. 78. J. Bagger, S. Dawson, and G. Valencia, Nucl. Phys. B399 (1993) 364. 79. J. Bagger et.al., Phys. Rev. D49 (1994) 1246; R. Chivukula, M. Dugan, and M. Golden, Ann. Rev. Nucl. Part. Sci. 45 (1995) 255. 80. J. Donoghue, C. Ramirez, and G. Valencia, Phys. Rev. D38 (1988) 2195; Phys. Rev. D39 (1989) 1947; M. Herrero and E. Morales, Nucl. Phys. B418 (1993) 364. 81. T. Appelquist and J. Carrazone, Phys. Rev. DU (1975) 2856. 82. See lectures in this school by N. Arkani-Hamed. 83. For a review of SUSY phenomenology, see G. Kane and H. Haber, Phys. Rep. U7e (1985) 75; H. Haber, TASI 1992 (World Scientific, Singapore, 1992). 84. J. Bagger and J. Wess, Supersymmelry and Supergravity (Princeton University Press, 1983). 85. G. Kane, H. Haber, and T. Stirling, Nucl. Phys. B161 (1979) 493. 86. H. Haber and R. Hempfling, Phys. Rev. Lett. 66 (1991) 1815; J. Ellis, G. Ridolfi and F. Zwirner, Phys. Lett. B257 (1991) 83; M. Berger, Phys. Rev. D41 (1990) 225; Y. Okada, M. Yamaguchi, and T. Yanagida, Prog. Theor. Phys. Lett. 85 (1991)1; M. Carena, M. Quiros, and C. Wagner, Nucl. Phys. B461 (1996) 407. 87. G. Kane, C. Kolda,M. Quiros, and J. Wells, Phys. Rev. Lett. 70 (1993) 2686. 88. M. Carena, S. Mrenna, and C. Wagner, ANL-HEP-PR-98-54, hepph/9808312. 89. A. Djouadi, report of the MSSM working group for the Workshop on GDR-Supersymmetry, hep-ph/9901246; Daniel Treille, results presented at XXIX International Conference on High Energy Physics, Vancouver, Canada, July 23-29, 1998. 90. H. Haber, et. aI., in Proceedings of the 1996 DPF/DPB Summer Study on New Directions for High Energy Physics, 1996.

84

INTRODUCTION TO QCD

M.L. MANGANO

CERN, TH Division, 1211 Geneva 23, Switzerland I review in this series of lectures the basics of perturbative quantum chromodynamics and some simple applications to the physics of high-energy collisions.

1

Introduction

Quantum Chromodynamics (QCD) is the theory of strong interactions. It is formulated in terms of elementary fields (quarks and gluons), whose interactions obey the principles of a relativistic QFT, with a non-abelian gauge invariance SU(3). The emergence of QCD as theory of strong interactions could be reviewed historically, analyzing the various experimental data and the theoretical ideas available in the years 1960-1973 (see e.g. refs. [17,18]). To do this accurately and usefully would require more time than I have available. I therefore prefer to introduce QCD right away, and to use my time in exploring some of its consequences and applications. I will therefore assume that you all know more or less what QCD is! I assume you know that hadrons are made of quarks, that quarks are spin-1/2, colour-triplet fermions, interacting via the exchange of an octet of spin-1 gluons. I assume you know the concept of running couplings, asymptotic freedom and of confinement. I shall finally assume that you have some familiarity with the fundamental ideas and formalism of QED: Feynman rules, renormalization, gauge invariance. If you go through lecture series on QCD, you will hardly ever find the same item twice. This is because QCD today covers a huge set of subjects and each of us has his own concept of what to do with QCD and of what are the "fundamental" notions of QCD and its "fundamental" applications. As a result, you will find lecture series centred around non-perturbative applications, (lattice QCD, sum rules, chiral perturbation theory, heavy quark effective theory), around formal properties of the perturbative expansion (asymptotic behaviour, renormalons), techniques to evaluate complex classes of Feynman diagrams, or phenomenological applicati,ons of QCD to possibly very different sets of experimental data (structure functions, deep-inelastic scattering (DIS) sum rules,

85

polarized DIS, small x physics (including hard pomerons, diffraction), LEP physics, pp collisions, etc. I can anticipate that I will not be able to cover or to simply mention all of this. After introducing some basic material, I will focus on some elementary applications of QeD in high-energy e+ e-, ep and pp collisions. The outline of these lectures is the following: 1. Gauge invariance and Feynman rules for QeD.

2. Renormalization, running coupling, renormalization group invariance. 3. QeD in e+e- collisions: from quarks and gluons to hadrons, jets, shape variables. 4. QeD in lepton-hadron collisions: DIS, parton densities, parton evolution. 5. QeD in hadron-hadron collisions: formalism, WjZ production, jet production. Given the large number of papers which contributed to the development of the field, it is impossible to provide a fair bibliography. I therefore limit my list of references to some excellent books and review articles covering the material presented here, and more. Papers on specific items can be easily found by consulting the standard hep-th and hep-ph preprint archives.

2 QeD Feynman rules There is no free lunch, so before starting with the applications, we need to spend some time developing the formalism and the necessary theoretical ideas. I will dedicate to this purpose the first two lectures. Today, I concentrate on Feynman rules. I will use an approach which is not canonical, namely it does not follow the standard path of the construction of a gauge invariant Lagrangian and the derivation of Feynman rules from it. I will rather start from QED, and empirically construct the extension to a non-Abelian theory by enforcing the desired symmetries directly on some specific scattering amplitudes. Hopefully, this will lead to a better insight into the relation between gauge invariance and Feynman rules. It will also provide you with a way of easily recalling or checking your rules when books are not around!

86

2.1

Summary of QED Feynman rules

We start by summarizing the familiar Feynman rules for Quantum Electrodynamics (QED). They are obtained from the Lagrangian:

(1) where 'l/J is the electron field, of mass m and coupling constant e, and F/-LV is the electromagnetic field strength.

(2) The resulting Feynman rules are summarized in the following table: ~----~ =Z~~~---

p+m p2 - m 2 + if

(3)

-i {/-Lv.

(Feynman gauge)

(4)

p

i

P-

p

.

m

+ if

+Zf

(5)

where in the last equation Q = -1 for the electron, Q = 2/3 for the u-quark, etc. Let us start by considering a simple QED process, e+e- ----; 'Y'Y (for simplicity we shall always assume m = 0): q

q

q

q

The total amplitude M"( is given by: i l l e 2 M"( == Dl +D2 = v(ii)!-2 g _ ~l!-l u(q) + v(ii)!-l g _ ~2!-2 n(q)

== M/-LVfif2 (7)

Gauge invariance demands that

(8)

87

MIJ == M lJv f.2 is in fact the current that couples to the photon k1 . Charge conservation requires 8IJMIJ = 0:

!J

M Od 3 x =

8IJMIJ, = 0 ::::}

JV . M

d3 x =

J

8oMo d 3 x

r

is

M . di5 =

0

(9)

-+ co

In momentum space, this means

(10)

kiMIJ, = O.

Another way of saying this is that the theory is invariant if f.1J,(k) ---- f.1J(k) + f(k) kw This is the standard Abelian gauge invariance associated to the vector potential transformations: (11)

Let us verify that M-y is indeed gauge invariant. Using gu(q) from the Dirac equation, we can rewrite ki MIJ, as: V(ij)t2

1

g _ ~l (~l -

-v(ij)t 2u(q)

g)u(q)

+ V(ij)(~l

+ v(ij)t 2u(q) = 0

= v(ij), = 0

1

- ij) kl _ /2 U(q)

(12)

Notice that the two diagrams are not individually gauge invariant, only the sum is. Notice also that the cancellation takes place independently of the choice of f.2. The amplitude is therefore gauge invariant even in the case of emission of non-transverse photons. Let us try now to generalize our QED example to a theory where the "electrons" carry a non-Abelian charge, i.e., they transform under a non-trivial representation R of a non-Abelian group G (which, for the sake of simplicity, we shall always assume to be of the SU(N) type. Likewise, we shall refer to the non-abelian charge as "colour"). The standard current operator belongs to the product R ® R. The only representation that belongs to R ® R for any R is the adjoint representation. Therefore the field that couples to the colour current must transform as the adjoint representation of the group G. So the only generalization of the photon field to the case of a non-Abelian symmetry is a set of vector fields transforming under the adjoint of G, and the simplest generalization of the coupling to fermions takes the form: a,a

A

ion

k.m

(13)

88

where the matrices )..a represent the algebra of the group on the representation R. By definition, they satisfy the algebra: (14) for a fixed set of structure constants f abc , which uniquely characterize the algebra. We shall call quarks (q) the fermion fields in Rand gluons (g) the vector fields which coule to the quark colour current. The non-abelian generalization of the e+e- -+ II process if the qij -+ gg annihilation. Its amplitude can be evaluated by including the >. matrices in Eq. (6): (15) with (a, b) colour labels (i.e. group indices) of gluons 1 and 2, (i, j) colour labels of ij, q, respectively. Using Eq. (14), we can rewrite (15) as: (16) If we want the charge associated with the group G to be conserved, we still need to demand kJJ.1f "2MJJ." = fJJ.k" = 0. (17) 12 MJJ." 9 9

Substituting

fi -+ ki

in (16) we get instead, using (12): k 1 JJ. M 9JJ. --

-

9 2fabc

\C

-.(-)

/\ij V.

q

J

.()

"2 U.

q

(18)

The gauge cancellation taking place in QED between the two diagrams is spoiled by the non-Abelian nature of the coupling of quarks to gluons (Le., )..a and ).b do not commute, and bc f. 0).

r

The only possible way to solve this problem is to include additional diagrams. That new interactions should exist is by itself a reasonable fact, since gluons are charged (Le., they transform under the symmetry group) and might want to interact among themselves. If we rewrite (18) as follows: (19)

we can recognize in the second factor the structure of the qijg vertex. The first factor has the appropriate colour structure to describe a triple-gluon vertex,

89 with a, b, c the colour labels of the three gluons:

(20) c

a

Equation (19) therefore suggests the existence of a coupling like (20), with a Lorentz structure VILIIL21L3 to be specified, giving rise to the following contribution to qij ~ gg: q pJ.L

q

_ig 2 D3 = (ig)..fj)v(ijkylLu(q)j (;:)

9

r bc V

ILVP (

-p, kl' k 2 ) fr(k 1 ) f~(k2)

(21)

We now need to find VILIIL21L3(Pl,P2,P3) and to verify that the contribution of the new diagram to kl . Mg cancels that of the first two diagrams. We will now show that the constraints of Lorentz invariance, Bose symmetry and dimensional analysis uniquely fix V, up to an overall constant factor. Dimensional analysis fixes the coupling to be linear in the gluon momenta. This is because each vector field carries dimension 1, there are three of them, and the interaction must have total dimension equal to 4. So at most one derivative (Le. one power of momentum) can appear at the vertex. In priciple, if some mass parameter were available, higher derivatives could be included, with the appropriate powers of the mass parameter appearing in the denominator. This is however not the case. It is important to remark that the absence of interactions with higher number of derivatives is also crucial for the renormalizability of the interaction. Lorentz invariance requires then that V be built out of terms of the form Bose symmetry requires V to be fully antisymmetric under the exchange of any pair (J-Li, Pi) f-t (J-Lj, Pj) since the colour structure jabc is totally antisymmetric. As a result, for example, a term like g1LI1L2P~3 vanishes under antisymmetrization, while gILIIL2Pi 3 doesn't. Starting from this last term, we can easily add the pieces required to obtain the full antisymmetry in all three indices. The result is unique, up to an overall factor: gILIIL2P/L3.

90

To test the gauge variation of the contribution D 3, we set /13 = /1, tl = kl and k3 = -(kl + k 2) in eq. (21), and we get: kr't~2

VIl l/L21l (kI, k2' k3) = Vo {-(kl

+ k2)Il(kl . (2) + 2(kl . k2)t~ -

(k2 . (2)ki} (23)

The gauge variation is therefore:

The first term cancels the gauge variation of DI + D2 provided Vo = 1, the second term vanishes for a physical gluon k2' since in this case k2 . t2 = O. DI + D2 + D3 is therefore gauge invariant but, contrary to the case of QED, only for physical external on-shell gluons. Having introduced a three-gluon coupling, we can induce processes involving only gluons, such as 99 ---- 99:

(25)

Once more it is necessary to verify the gauge invariance of this amplitude. It turns out that one more diagram is required, induced by a four-gluon vertex. Lorentz invariance, Bose symmetry and dimensional analysis uniquely determine once again the structure of this vertex. The overall factor is fixed by gauge invariance. The resulting Feynman rule for the 4-gluon vertex is given in fig. 1. You van verify that the 3- and 4-gluon vertices we introduced above are exactly those which arise from the Yang-Mills Lagrangian: r '-'YM -_

1"

-4: ~ FaIlV Fall v

'th

WI

C Fa '" A av[ - 9 fabcA b[Il A v] IlV -- U[1l

(26)

a

It can be shown that the 3 and 4-gluon vertices we generated are all is needed to guarantee gauge invariance even for processes more complicated than those studied in the previous simple examples. In other words, no extra 5- or more gluon vertices have to be introduced to achieve the gauge invariance of higherorder amplitudes. At the tree level this is the consequence of dimensional analysis and of the locality of the couplings (no inverse powers of the momenta can appear in the Lagrangian). At the loop level, these conditions are supplemented by the renormalizability of the theory [3,7].

91

p

a,~p

a

p

b

i"ab

u

i,n

k,m

(Feynman gauge)

i_ oab _ _

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

p

-i gof3 p2 + if p2

Oik

+ if t

p- m+if

Imn

b,p

C,-y

a,a

_ig 2fxae fXbd (go(3 9 'YO _ gOO gf3'Y) _ig 2fxad f xbe (go(3 9'YO _ g0'Y g(30) _ig 2rab red (g0'Y g(3o _ gOO gf3'Y)

b

i,n

a,a

C

k,m

Figure 1: Feynman rules for QeD. The solid lines represent the femions, the curly lines the gluons, and the dotted lines represent the ghosts.

92 Before one can start calculating cross-sections, a technical subtlety that arises in QeD when squaring the amplitudes and summing over the polarization of external states needs to be discussed. Let us again start from the QED example. Let us focus, for example, on the sum over polarizations of photon k 1:

2: IMI2 (2: fif r*) =

E1

MI-£M:

(27)

El

The two independent physical polarizations of a photon with momentum k = (ko; 0, 0, ko) are given by fi R = {O; 1, ±i, O)jJ2. They satisfy the standard normalization properties: '

We can write the sum over physical polarizations in a convenient form by introducing the vector k = (k o;0, 0, -ko):

(28)

We could have written the sum over physical polarizations using any other momentum £1-£' provided k· £ -:f. O. This would be equivalent to a gauge transformation (prove it as an exercise). In QED the second term in eq. (28) can be safely dropped, since kl-£MI-£ = o. As a cross check, notice that kl-£MI-£ = 0 implies Mo = M 3 , and therefore:

i=L,R

(29) Therefore, the production of the longitudinal and time-like components of the photon cancel each other. This is true regardless of whether additional external photons are physical or not, since the gauge invariance kl . M = 0 shown in Eq. (12) holds regardless of the choice for f2, as already remarked. In particular,

(30) (for n photons, ki 1 k~2 ... k~n Ml-£l ... lJ.n = 0) and the production of any number of unphysical photons vanishes. The situation in the case of gluon emission is different, since kl . M ex f2 • k2' which vanishes only for a physical f2. This implies that the production of one physical and one non-physical gluons is

93

equal to 0, but the production of a pair of non-physical gluons is allowed! If €2 . k2 ¥- 0, then Mo is not equal to M 3 , and eq. (28) is not equivalent to L €J.I.€~ = -gJ.l.v· Exercise: show that

L

l€i€2 MJ.l.vI 2 =

non-physical

li

g2 rbc)..c

2k1 k v(q)ki

U(Q)1

2

(31)

1 2

In the case of non-Abelian theories, it is therefore important to restrict the sum over polarizations and (because of unitarity) the off-shell propagators to physical degrees of freedom with the choice of physical gauges. Alternatively, one has to undertake a study of the implications of gauge-fixing in non-physical gauges for the quantization of the theory (see refs. [3,7]). The outcome of this analysis is the appearance of two colour-octet scalar degrees of freedom (called ghosts) whose role is to enforce unitarity in non-physical gauges. They will appear in internal closed loops, or will be pair-produced in final states. They only couple to gluons. Their Feynman rules are supplemented by the prescription that each closed loop should come with a -1 sign, as if they obeyed Fermi statistics. Being scalars, this prescription breaks the spin-statistics relation, and leads as a result to the possibility that production probabilities be negative. This is precisely what is required to cancel the contributions of nontransverse degrees of freedom appearing in non-physical gauges. Adding the ghosts contribution to qq -> gg decays (using the Feynman rules from fig. 1) gives in fact 2

(32)

which exactly cancels the contribution of non-transverse gluons in the nonphysical gauge L €J.I.€~ = -gJ.l.v, given in in eq. 31. The detailed derivation of the need for and properties of ghosts (including their Feynman rules and the "-I" prescription for loops) can be found in the suggested textbooks. I will not derive these results here since we will not need them for our applications (we will use physical gauges or will consider processes not involving the 3g vertex). The full set of Feynman rules for the QeD Lagrangian is given in fig. 1.

94

2.2

Some Useful Results in Colour Algebra

The presence of colour factors in the Feynman rules makes it necessary to develop some technology to evaluate the colour coefficients which multiply our Feynman diagrams. To be specific, we shall assume the gauge group is SU(N). The fundamental relation of the algebra is (33)

r

with bc totally antisymmetric. This relation implies that all A matrices are traceless. For practical calculations, since we will always sum over initial, final, and intermediate state colours, we will never need the explicit values of bc • All of the results can be expressed in terms of group invariants (a.k.a. Casimirs), some of which we will now introduce. The first such invariant (TF) is chosen to fix the normalization of the matrices A:

r

(34)

where by convention TF = 1/2 for the fundamental representation. Should you change this convention, you would need to change the definition (Le. the numerical value) of the coupling constant g, since 9 Aa appears in the Lagrangian and in the Feynman rules. Exercise: Show that tr(Aa Ab ) is indeed a group invariant. Hint: write the action on Aa of a general group transformation with infinitesimal parameters fb as follows: abc C bAa = fbf A (35)

L

b,c

The definition of TF allows to evaluate the colour factor for an interesting diagram, Le. the quark self-energy:

(36)

The value of GF can be obtained by tracing the relation above: GFN

=

t r '"' \a \a = ~ A A a

2

>:ab

U

T FUab >: = N 2- 1

(37)

95

where we used the fact that of gluons) for SU(N).

oab Oab

= N2 -

1, the number of matrices Aa (and

There are some useful graphical tricks (which I learned from P. Nason, ref. [9]) which can be used to evaluate complicated expressions. The starting point is the following representation for the quark and gluon propagators, and for the qijg and ggg interaction vertices:

E

~( JL ~ ~ )

fermion

(38)

gluon

(39)

Fermion-GIuon Vertex (t a ) (40)

~(A-A)

3-Gluon Vertex

(41)

(tabe)

Contraction over colour indices is obtained by connecting the respective colour (or anticolour) lines. A closed loop of a colour line gives rise to a factor N, since the closed loop is equivalent to the trace of the unit matrix. So the above representation of the qijg vertex embodies the idea of "colour conservation", whereby the colour-anticolour quantum numbers carried by the qij pair are transferred to the gluon. The piece proportional to 1/N in the qqg vertex appears only when the colour of the quark and of the antiquark are the same. It ensures that Aa is traceless, as it should. This can be easily checked as an exercise. The factor 1/ J2 is related to the chosen normalization of TF. As a first example of applications, let us reevaluate GF:

i

.AaOA~

~(JL J2

j

1 N

li

"

)

1

x-

J2

(JL

1 N

li

"

)

96

~(~-~N ~-~~+ N 2

1

~

+ N2 - - - - - - - - -

As an exercise, you can calculate the colour factor for qij show that:

(42)

--+

qij scattering, and

a

(43) This result can be used to evaluate the one-loop colour factors for the interaction vertex with a photon:

(44)

For the interaction with a gluon we have instead:

97

Notice that in the case of the coupling to the photon the qij pair is in a coloursinglet state. The gluon exchange effect in this case has a positive sign (=} attraction). In the case of the coupling to the gluon the qij pair is in a colouroctet state, and the gluon-exchange correction has a negative sign relative to the Born interaction. The force between a qij pair is therefore attractive if the pair is in a colour-singlet, while it is repulsive if it is in a colour-octet state! This gives a qualitative argument for why no colour-octet qij bound state exists. The remaining important relation that one needs is the following:

2:r r bc

bd

= CA 8

cd

with C A = N .

(46)

a,b

You can easily prove it by using the graphical representation given in eq. (41), or by using eq. (43) and bc = -2i tr([).a, ).b] ).C).

r

3

Renormalization, or: INFINITIES !"

"THEORISTS ARE NOT AFRAID OF

QeD calculations are extremely demanding. Although perturbative, the size of the coupling constant even at rather large values of the exchanged momentum,

98 Q2, is such that the convergence of the perturbative expansion is slow. Several orders of perturbation theory (PT) are required in order to obtain a good accuracy. The complexity of the calculations grows dramatically with the order of the approximation. As an additional complication, the evaluation of a large class of higher-order diagrams gives rise to results which are a priori ill-defined, namely to infinities. A typical example of what is known as an ultraviolet divergence, appears when considering the corrections to the quark self-energy. Using the Feynman rules presented in the previous lecture, one can obtain: p

where simple manipulations lead to the following expression for

~(p)

= iCF

d4£ 1 (27r)4 £2(p + £)2 '

J

~(p):

(48)

which is logarithmically divergent in the ultraviolet (1£1 ----> 00) region. In this lecture we will discuss how to deal with these infinities. To start with, we study a simple example taken from standard electrostatics.

3.1

The potential of an infinite line of charge

Let us consider a wire of infinite length, carrying a constant charge density>.. By definition, the dimensions of>. are [length]-l. Our goal is to evaluate the electric potential, and eventually the electric field, in a point P at distance R from the wire. There is no need to do any calculation to anticipate that the evaluation of the electric potential will cause some problem. Using the fact that the potential should be linear in the charge density>., we write V(R) = >.f(R). Since the potential itself has the dimensions of [lengtht 1, we clearly see that there is no room for f(R) to have any non-trivial functional dependence on R. The problem is made explicit if we try to evaluate V(R) using the standard EM formulas: V(R)

=

J

>'(r) dx r

=

>.

j+oo VR2dx+ x 2

(49)

-00

where the integral runs over the position x on the wire. This integral is logarithmically divergent, and the potential is ill-defined. We know however that

99

this is not a serious issue, since the potential itself is not a physical observable, only the electric field is measurable. Since the electric field is obtained by taking the gradient of the scalar potential, it will be proportional to

j

V'(R) rv,X

dx

+oo

(R2

-00

+ X2)3/2

(50)

'

which is perfectly convergent. It is however interesting to explore the possibiity of providing a useful operative meaning to the definition of the scalar potential. To do that, we start by regularizing the integral in eq. (49). This can be done by introducing the regularized V(R) defined as: VA(R) = jA,X -A

dx

VR2 + x 2

= 'xlo

+ R2 + A] VA 2 + R2 - A

[VA2 g

(51)

We can then define the electric field as E(R) =

(52)

lim [-VVA(R)]

A_oo

It is easy to check that this prescription leads to the right result: E(R)

=

lim A-oo

R 2,X R

L

V£2 + R2

___ 2'x R

R

(53)

Notice that in this process we had to introduce a new variable A with the dimension of a length. This allows us to solve the puzzle first pointed out at the beginning. At the end, however, the dependence of the physical observable (Le. the electric field) on this extra parameter disappears. Notice also that the object: (54)

is well defined. This suggests a way of defining the potential which is meaninfgul even in the A --- 00 limit. We can renormalize the potential, by subtracting V(R) at some fixed value of R = Ro, and taking the A --- 00 limit: V(R) --- V(R) - V(Ro)

= ,X log

(~O

(55)

The non-physical infinities present in V(R) and V(Ro) cancel each other, leaving a finite result, with a non-trivial R-dependence. Once again, this is possible because a dimensionful parameter (in this case Ro) has been introduced. This example suggests a strategy for dealing with divergencies:

100

i) Identify an appropriate way to regularize infinite integrals

ii) Absorb the divergent terms into a redefinition of fields or parameters, e.g, via subtractions. This step is usually called renormalization.

iii) Make sure the procedure is consistent, by checking that the physical results do not depend on the regularization prescription. In the rest of this lecture I will explain how this strategy is applied to the case of ultraviolet divergencies encountered in perturbation theory.

3.2

Dimensional regularization

The typical expressions we have to deal with have the form:

(56) You can easily show that the integral encountered in the quark self-energy diagram can be rewritten as: 1

1

r

£2 (£ _ p)2 = io

l

dx (£2

1.

+ M2)2'

2

With L = £ - xp, M = x(l - x)p

2

(57)

The most straighforward extension of the ideas presented above in the case of the infinite charged wire is to regularize the integral using a momentun cutoff, and to renormalize it with a subtraction (for example I(M2) - I(MJ)). Experience has shown, however, that the best way to regularize I(M2) is to take the analytic continuation of the integral in the number of space-time dimensions. In fact (58)

is finite VD < 4. If we could assign a formal meaning to ID(M2) for continuous values of D away from D = 4, we could then perform all our manipulations in D =I- 4, regulate the divergences, renormalize fields and couplings, and then go back to D = 4. To proceed, one defines (for Euclidean metrics): (59)

101

with dD.D-l the differential solid angle in D dimensions. D.D-l is the surface of a D-dimensional sphere. It can be obtained by using the following formal identity:

J

dDfe- P

==

[J

dfe- ]D =1r D / 2 l2

(60)

The integral can also be evaluated, using eq. (59), as

J

dDf e- P

D.D

1

D.D

~

00

2

l2

fD-l e- df

=

D.D

~

1

roo dx e-Xx¥ == D.D2 f

10

00

df2(f2)

D22

(D)

e-e

2

(61)

2

Comparing eqs. (60) and (61) we get: ID(M2)

(41r;D/2

1 (41r)D/2

f(~/2) 10

00

dx x¥(x

f(2 - D12) (M2)~-2 f(2)

+ M2)2 (62)

Defining D = 4 - 2t (with the understanding that t will be taken to 0 at the end of the day), and using the small-t expansion:

1 f(t) = - - 1< t

+ OCt)

(63)

we finally obtain: (64)

The divergent part of the integral is then regularized as a pole in (D - 4). The M -dependent part of the integral behaves logarithmically, as expected because the integral itself was dimensionless in D = 4. The Ill' pole can be removed by a subtraction: (65)

where the subtraction scale J.L2 is usually referred to as the "renormalization scale" . One can prove (and you will find this in the quoted textbooks) that other divergent integrals which appear in other loop diagrams can be regularized in a similar fashion, with the appearance of 1 It poles. Explicit calculations and more details on this techinque can be found in the literature quoted at the end.

102

3.3 Renormalization Let us come back now to our quark self-energy diagram, eq. (47). After regulating the divergence using dimensional regularization, we can eliminate it by adding a counterterm to the Lagrangian: £

---+ £,

+ 'E(p)7f;ifJ'l/J =

[1

+ 'E(p)]7f;ifJ'l/J + '"

(66)

In this way, the corrections at 0(g2) to the inverse propagator are finite:

.0.

= -iP'E(p) + ip'E(p)

= 0

(67)

The inclusion of this counterterm can be interpreted as a renormalization of the quark wave function. To see this, it is sufficient to define: (68)

and verify that the kinetic part of the Lagrangian written in terms of 'l/J R takes again the canonical form. It may seem that this regularization/renormalization procedure can always be carried out, with all possible infinities being removed by ad hoc counterterms. This is not true. That these subtractions can be performed consistently for any possible type of divergence which develops in PT is a highly non-trivial fact. To convince you of this, consider the following example.

Let us study the QeD corrections to the interaction of quarks with a photon:

(69)

-

p

p

_

,2

- (-zg) CF

J

. d40(. pZ 'J1. r" (27T)4 'Y P+ (-z e'Y ) [

~

t

,

Z

p+ t

1 (~2i)

P

'Y

103

It is easily recognized that V(q2) is divergent. The divergence can be removed by adding a counterterm to the bare Lagrangian: -e AIJ.'¢'IJ.'l/J

-[1

--+

+ V(q2)] e

-eAIJ.'¢'IJ.'l/J - eV(q2)AIJ.'¢,IJ.'l/J

AIJ.'¢'IJ.'l/J

(70)

If we take into account the counterterm that was introduced to renormalize the quark self-energy, the part of the quark Lagrangian describing the interaction with photons is now:

Defining a renormalized charge by: (72)

we are left with the renormalized Lagrangian: (73)

Can we blindly accept this result, regardless of the values of the counterterms V(p2) and I:(q2)? The answer to this question is NO! Charge conservation, in fact, requires eR = e. The electric charge carried by a quark cannot be affected by the QCD corrections, and cannot be affected by the renormalization of QCD-induced divergencies. There are many ways to see that if eR # e the electric charge would not be conserved in strong interactions. The simplest way is to consider the process e+lIe --+ W+ --+ ud. The electric charge of the initial state is +1 in units of e. After including QCD corrections (which in the case of the interaction with a Ware the same as those for the interaction of quarks with a photon), the charge of the final state is +1 in units of eR. Unless eR = e, the total electric charge would not be conserved in this process! The non-renormalization of the electric charge in presence of strong interactions is the fact that makes the charge of the proton to equal the sum of the charges of its constituent quarks, in spite of the complex QCD dynamics that holds the quarks together.

104

As a result, the renormalization procedure is consistent with charge conservation if and only if (74)

This identity should hold at all orders of perturbation theory. It represents a fundamental constraint on the consistency of the theory, and shows that the removal of infinities, by itself, is not a trivial trick which can be applied to arbitrary theories. Fortunately, the previous identity can be shown to hold. You can prove it explicitly at the one-loop order by explicitly evaluating the integrals defining V(q) and E(p). To carry out the renormalization program for QeD at I-loop order, several other diagrams in addition to the quark self-energy need to be evaluated. One needs the corrections to the gluon self-energy, to the coupling of a quark pair to a gluon, and to the 3-gluon coupling. Each of these corrections gives rise to infinities, which can be regulated in dimensional regularization. For the purposes of renormalization, it is useful to apply the concept of D dimensions not only to the evaluation of the infinite integrals, but to the full theory as well. In other words we should consider the Lagrangian as describing the interactions of fields in D-dimensions. Nothing changes in its form, but the canonical dimensions of fields and couplings will be shifted. This is because the action (defined as the integral over space-time of the lagrangian), is a dimensionless quantity. As a result, the canonical dimensions of the fields, and of the coupling constants, have to depend on D:

o =} [£] =

[ / dDx £(x)]

[81'1>81'1>] = D [~~7/J] = D [~47/Jg]

=D

1-

D = 4-

=}

[1>]

=

=}

[7/J]

= 3/2 - 10

=}

[g]

= 10

210

10

The gauge coupling constant acquires dimensions! This is a prelude to the non-trivial behaviour of the renormalized coupling constant as a function of the energy scale ("running"). But before we come to this, let us go back to the calculation of the counter-terms and the construction of the renormalized Lagrangian.

105

Replace the bare fields and couplings with renormalized oneS': 'lj;bare = Zi/2 'Ij; R - Z1/2 AI' bare3 R

AI'

gbare = Zg /-(gR We explicitly extracted the dimensions out of gbare, introducing the dimensional parameter f.l (renormalization scale). In this way the renormalized coupling gR is dimensionless (as it should be once we go back to 4-dimensions). The Lagrangian, written in terms of renormalized quantities, becomes:

.c =

-

1

1/2

Z2'1j;if!'Ij; - 4Z3F:.f/:v + Z9Z2Z3

-

f.l' g'lj;Jf..'Ij; + (gauge fixing, ghosts, ... ) (75)

It is customary to define

Z1

= Zg Z2 Zi/2

(76)

= 1 + 8n , we then obtain:

If we set Zn

.c

i{; i f!'Ij; -

~F:vFJLVa + f.l' gi{;Jf..'Ij; + [ghosts,

GM]

+ 82 i{; if)'Ij; - ~83F:vFJLVa + 81f.l'gi{;Jf..'Ij;

(77)

The counter-terms 8i are fixed by requiring the I-loop Green functions to be finite. The explicit evaluation, which you can find carried out in detail, for example, in refs. [7,3], gives:

1)

quark self-energy

-=> 82 = -CF C~s 47r f

gluon self-energy

=> 83 =

qijg vertex corrections

(78)

(~CA - ~nfTF)

1

=> 81 = - (C A + CF ) -as 47r f

as 47r

1

--

(79)

f

(80)

As usual we introduced the notation as = g2/47r. The strong-coupling renormalization constant Zg can be obtained using these results and eq. (76):

Z1

---::-= =

z2 zi/ 2

1 +8 1 - 82

-

1 as -1 [11 2 ] -8 --CA + -nFTF 3 = 1+ 2 47r f 6 3

a For the sake of simplicity, here and in the following we shall assume the quarks as massless. The inclusion of the mass terms does not add any interesting new feature in what follows.

106

(81) Notice the cancellation of the terms proportional to CF, between the quark self-energy (Z2) and the abelian part of the vertex correction (Zl). This is the same as in the case of the QCD non-renormalization of the electric coupling, discussed at the beginning of the lecture. The non-abelian part' of the vertex correction contributes viceversa to the QCD coupling renormalization. This is a consequence of gauge invariance. The separation of the non-abelian contributions to the self-energy and to the vertex is not gauge-invariant, only their sum is. Notice also that the consistency of the renormalization procedure requires that the renormalized strong coupling 9 defining the strength of the interaction of quarks and gluons should be the same as that defining the interaction of gluons among themselves. If this didn't happen, the gauge invariance of the qij - t gg process so painfully achieved in the first lecture by fixing the coefficient of the 3-gluon coupling would not hold anymore at I-loop! Once again, this additional constraint can be shown to hold through an explicit calculation.

3.4

Running of as

The running of as is a consequence of the renormalization-scale independence of the renormalization process. The bare coupling gbare knows nothing about our choice of J-t. The parameter J-t is an artifact of the regularization prescription, introduced to define the dimensionful coupling in D dimensions, and should not enter in measurable quantities. As a result:

dgbare = 0 dJ-t

(82)

Using the definition of g: gbare = J-t€ Zg g, we then get f,//2€ t""

where

Z2g a S

+ 112€ t""

2€ Z2 das - 0 a S 2Zg dZg dt+J-t gdt-

d 2 d dt = J-t dJ-t2

d dlogJ-t2

(83)

(84)

Zg depends upon J-t only via the presence of as. If we define (3(a s )

=

das dt

(85)

107

we then get: j3(a s )

+ 2Zas9

dZ9 j3(a ) -d s as

=

(86)

-Ea.

Using eq. (81) and expanding in powers of as, we get: (87)

1+2Q.... §

Zg da s

and finally: j3(a s ) = -boa~

with

bo = -1 (11 -CA 2rr 6

-

2 nf TF ) 3

-

= - 1 (33 -

N=3

12rr

2nf)

(88) We can now solve eq. (85), assuming bo > 0 (which is true provided the number of quark flavours is less than 16) and get the famous running of as: (89)

The parameter A describes the boundary condition of the first order differential equation defining the running of as, and corresponds to the scale at which the coupling becomes infinity.

3.5

Renormalization group in variance

The fact that the coupling constant as depends on the unphysical renormalizaiton scale p, should not be a source of worry. This is because the coupling constant itself is not an observable. What we observe are decay rates, spectra, or cross sections. These are given by the product of the coupling constant times some matrix element, which in general will acquire a non-trivial renormalization-scale dependence through the renormalization procedure. We therefore just need to check that the scale dependence of the coupling constant and of the matrix elements cancel each other, leaving results which do not depend on p,. Consider now a physical observable, for example the ratio R = a( e+ e- -+ hadrons)/a(e+e- -+ p,+p,-). R can be calculated in perturbation theory within QCD, giving rise to an expansion in the renormalized coupling as (p,): 00

R[a s ,s/p,2]

=

1 +as h(t) +a~ h(t)

+ ... =

La~ JCn)(t) n=O

(90)

108

where t = s / p,2 (and we omitted a trivial overall factor 3 L f Q}). R depends on p, explicitly via the functions J(n)(t) and implicitly through as. Since R is an observable, it should be independent of p" and the functions J(n)(t) cannot be totally arbitrary. In particular, one should have: (91)

Before we give the general, formal solution to this differential equation, it is instructive to work out directly its form within perturbation theory. dR p, 2 dp,2

= 0 =

dlt j3(as ) It () t +as p, 2 dp,2 +2as j3 (as)

dh h () t +a 22 +... (92) s p, dp,2

At order as (remember that j3 is of order a~) we get

dlt

dp,2

= 0 => It = constant ==

(93)

al

This is by itself a non-trivial result! It says that the evaluation of R at one-loop is finite, all UV infinities must cancel without charge renormalization. If they didn't cancel, It would depend explicitly on p,. As we saw at the beginning, this is a consequence of the non-renormalization of the electric charge. At order a~ we have:

o => h

2

=

bo al log t:... +a2 (integration constant)

s

(94)

So up to order a~ we have: R

= 1+

al as

'"-v-'" one-loop

+ al bo a~

log p,2 / S ..

+ a2a~ + ... '

(95)

two-loops

Notice that the requirement of renormalization group invariance allows us to know the coefficient of the logarithimc term at 2-loops without having to carry out the explicit 2-loop calculation! It is also important to notice that in the limit of high energy, S -> 00, the logarithmic term of the two-loop contribution becomes very large, and this piece becomes numerically of order as as soon as log s/ p,2 ~ l/boas. You can easily check that renormalization scale invariance requires the presence of such logs at all orders of PT. In particular:

(96)

109

We can collect all these logs as follows:

R

=

J.L2 1 + alas [1 + asbo log -; l+al

as(J.L)

1 + as (J.L)b o log ~

1 + alas(s)

J.L2 2 + (asb o log -;) +... ] + a2as2 + ...

+ a 2+ a2 s ...

+ a2a~ +...

(97) (98)

In fact: (99) RG invariance constrains the form of higher-order corrections. All of the higher-order logarithmic terms are determined in terms of lower-order finite coefficients. They can be resummed by simply setting the scale of as to s. You can check by yourself that this will work also for the higher-order terms, such as those proportional to a2. So the final result has the form: (100)

Of course al, a2, ... have to be determined by an explicit calculation. However, the truncation of the series at order n has now an accuracy which is truly of order a~+l, contrary to before when higher-order terms were as large as lower-order ones. The explicit calculation has been carried out up to the a3 coefficient. In particular, (101) The formal proof of the previous equation can be obtained by showing that the general form of the equation (102)

is given by

R(as(s), 1), with {

das s = (3(a s ) og J]I

dl

(103)

110

4

QeD in e+ e- collisions

e+e- collisions provide one of the cleanest environments in which to study applications of QCD at high energy. This is the place where theoretical calculations have today reached their best accuracy, and where experimental data are the most precise, especially thanks to the huge statistics accumulated by LEP, LEP2 and SLC. The key process is the annihilation of the e+e- pair into a virtual photon or ZO boson, which will subsequently decay to a qij pair. e+ecollisions have therefore the big advantage of providing an almost point-like source of quark pairs, so that, contrary to the case of interactions involving hadrons in the initial state, we at least know very precisely the state of the quarks at the beginning of the interaction process. Nevertheless, it is by no means obvious that this information is sufficient to predict the properties of the hadronic final state. We all know that this final state is clearly not simply a qij pair, but some high-multiplicity set of hadrons. It is therefore not obvious that a calculation done using the simple picture e+ e- -+ qij will have anything to do with reality. For example, one may wonder why don't we need to calculate a( e+ e- -+ qijg . .. g ... ) for all possible gluon multiplicities to get an accurate estimate of a( e+ e- -+ hadrons). And since in any case the final state is not made of q's and g's, but of 7r'S, K's, p's, etc., why would a(e+e- -+ qijg ... g) be enough? The solution to this puzzle lies both in a question of time and energy scales, and in the dynamics of QCD. When the qij pair is produced, the force binding q and ij is proportional to Cts(s) (y'S being the e+e- centre-of-mass energy). Therefore it is weak, and q and ij behave to good approximation like free particles. The radiation emitted in the first instants after the pair creation is also perturbative, and it will stay so until a time after creation of the order of (1 GeV)-l, when radiation with wavelengths ~ (1 GeV)-l starts being emitted. At this scale the coupling constant is large, non-perturbative phenomena and hadronization start playing a role. However, as we will show, colour emission during the perturbative evolution organizes itself in such a way as to form colour-neutral, low mass, parton clusters highly localized in phase-space. As a result, the complete colour-neutralization (i.e., the hadronization) does not involve long-range interactions between partons far away in phase-space. This is very important, because the forces acting among coloured objects at this time scale would be huge. If the perturbative evolution were to separate far apart colour-singlet qij pairs, the final-state interactions taking place during the hadronization phase would totally upset the structure of the final state. As an

111

additional result of this "pre-confining" evolution, memory of where the local colour-neutral clusters came from is totally lost. So we expect the properties of hadronization to be universal: a model that describes hadronization at a given energy will work equally well at some other energy. Furthermore, so much time has passed since the original qij creation, that the hadronization phase cannot significantly affect the total hadron production rate. Perturbative corrections due to the emission of the first hard partons should be calculable in PT, providing a finite, meaningful cross-section. The nature of non-perturbative corrections to this picture can be explored. One can prove for example that the leading correction to the total rate Re+eis of order F / s2, where F ex (OIQsF!vFl-'va 10) is the so-called gluon condensate. Since F rv 0(1 GeV4), these NP corrections are usually very small. For example, they are of 0(10- 8 ) at the ZO peak! Corrections scaling like A2 / s or AI VB can nevertheless appear in other less inclusive quantities, such as event shapes or fragmentation functions. We now come back to the perturbative evolution, and will devote the first part of this lecture to justifying the picture given above. In the second half we shall discuss jet cross-sections and shape variables.

4.1

Soft Gluon Emission

Emission of soft gluons plays a fundamental role in the evolution of the final state [6,15]. Soft gluons are emitted with large probability, since the emission spectrum behaves like dE/E, typical of bremstrahlung even in QED. They provide the seed for the bulk of the final-state multiplicity of hadrons. The study of soft-gluon emission is simplified by the simplicity of their couplings. Being soft (Le., long wavelength) they are insensitive to the details of the veryshort-distance dynamics: they cannot distinguish features of the interactions which take place on time scales shorter than their wavelength. They are also insensitive to the spin of the partons: the only feature they are sensitive to is the colour charge. To prove this let us consider soft-gluon emission in the qij decay of an off-shell photon:

p. j

p. j

k.a

p. i

(104)

112

Asoft

u(p)f(k)(ig)

-i

P+ ~ r lt v(p)

[2: k u(p )f( k)(p +

i

+ u(p) r lt p + ~ (ig)f(k) v(P)

)..fj

~)rlt v(p)

2;'

-

k u(p) r lt (p +

)..fj

~)f( k) V(p)] )..ij

I used the generic symbol r It to describe the interaction vertex with the photon to stress the fact that the following manipulations are independent of the specific form of r w In particular, r It can represent an arbitrarily complicated vertex form factor. Neglecting the factors of ~ in the numerators (since k « p,p, by definition of soft) and using the Dirac equations, we get:

(::~ -

Asoft = g)..fj

;f

k

) A Born

(105)

We then conclude that soft-gluon emission factorizes into the product of an emission factor, times the Born-level amplitude. From this exercise, one can extract general Feynman rules for soft-gluon emission:

p.

Exercise: Derive the 9

j~ ~ ----+

a. f-L

p. i

(106)

gg soft-emission rules:

a. f-L

c~ b. P

=

i9r bc 2plt

(107)

gYp

Example: Consider the "decay" of a virtual gluon into a quark pair. One more diagram should be added to those considered in the case of the electroweak decay. The fact that the quark pair is not in a colour-singlet state anymore makes things a bit more interesting:

k.a

p. j

p. j

a

p.

p. i

p. j

(108)

113

k-+O

b [ -Qf. - -Pf. ] g ().. B )..).. 'J Qk pk

+g

B ().. b )..)

[pf. -

pk

'J

- -Qf. ] Qk

(109)

The two factors correspond to the two possible ways colour can flow in this process:

(110)

In the first case the antiquark (colour label j) is colour connected to the soft gluon (colour label b), and the quark (colour label i) is connected to the decaying gluon (colour label a). In the second case, the order is reversed. The two emission factors correspond to the emission of the soft gluon from the antiquark, and from the quark line, respectively. When squaring the total amplitude, and summing over initial and final-state colours, the interference between the two pieces is suppressed by 1/N 2 relative to the individual squares:

L a,b,i,j

L B,b,i,j

I()..B)..b)ij 12 =

2

Ltr a,b

()..B)..b)ij[()..b)..B)ij]* =

L

()..B)..b)..b)..B) =

N -1 CF = O(N3) 2 2

tr()..B)..b)..B)..b) =

B,b

N -1 (C F 2

-

~A)

(111)

= O(N)

~

(112) As a result, the emission of a soft gluon can be described, to the leading order in 1/N 2 , as the incoherent sum of the emission form the two colour currents. 4.2

Angular ordering for soft-gluon emission

The results presented above have important consequences for the perturbative evolution of the quarks. A key property of the soft-gluon emission is the so-

114

called angular ordering. This phenomenon consists in the continuous reduction of the opening angle at which successive soft gluons are emitted by the evolving quark. As a result, this radiation is confined within smaller and smaller cones around the quark direction, and the final state will look like a collimated jet of partons. In addition, the structure of the colour flow during the jet evolution forces the qij pairs which are in a colour-singlet state to be close in phase-space, thereby achieving the pre-confinement of colour-singlet clusters alluded to at the beginning of the lecture. Let us start first by proving the property of colour ordering. Consider the qij pair produced by the decay of a rapidly moving virtual photon. The amplitude for the emission of a soft gluon was given in eq. (105). Squaring, summing over colours and including the gluon phase-space we get the following result:

L

d3 k IAsoltl (271-)32ko 2

L IAol

2

_2pIJ'jjV 2 (pk)(pk) g

d3 k

L €I'€~ (27r)32kO

2(pjj) 2 (d¢) kOdkO dao (pk )(pk) g C I 271" 871"2 d cos () CXsCF dko dao - - 71" kO

d¢ 1 - cos ()ij d () cos 271" (1 - cos ()ik)(l - cos ()jk)

(113)

where ()Ot/3 = ()Ot - ()/3, and i, j, k refer to the q, ij and gluon directions, respectively. We can write the following identity: 1 - cos ()ij (1 - cos ()ik)(l - cos ()jk)

1 [ cos ()jk - cos ()ij 2 (1 - cos ()ik)(l - cos ()jk) 1

+ "2[i j] == W(i) + W(j)

+

1 1-

]

COS()ik

(114)

We would like to interpret the two functions W(i) and W(j) as radiation probabilities from the quark and antiquark lines. Each of them is in fact only singular in the limit of gluon emission parallel to the respective quark: --->

finite if k II j

(cos ()jk

--->

1)

(115)

--->

finite if k II i (cos ()ik

--->

1)

(116)

The intepretation as probabilities is however limited by the fact that neither W(i) nor W(j) are positive definite. However, you can easily prove that (117) otherwise

115

where the integral is the azimuthal average around the q direction. A similar result holds for W(j):

(118) otherwise

As a result, the emission of soft gluons outside the two cones obtained by rotating the antiquark direction around the quark's, and viceversa, averages to o. Inside the two cones, one can consider the radiation from the emitters as being uncorrelated. In other words, the two colour lines defined by the quark and antiquark currents act as independent emitters, and the quantum coherence (Le. the effects of interference between the two graphs contributing to the gluon-emission amplitude) is accounted for by constraining the emission to take place within those fixed cones. If one repeats now the exercise for emission of one additional gluon, one will find the same angular constraint, but this time applied to the colour lines defined by the previously established antenna. As shown in the previous subsection, the qijg state can be decomposed at the leading order in liN into two independent emitters, one given by the colour line flowing from the gluon to the quark, the other given by the colour line flowing from the antiquark to the gluon. So the emission of the additional gluon will be constrained to take place either within the cone formed by the quark and the gluon, or within the cone formed by the gluon and the antiquark. Either way, the emission angle will be smaller than the angle of the first gluon emission. This leads to the concept of angular ordering, with successive emission of soft gluons taking place within cones which get smaller and smaller.

The fact that colour always flows directly from the emitting parton to the emitted one, the collimation of the jet, and the softening of the radiation emitted at later stages, ensure that partons forming a colour-singlet cluster are close in phase-space. As a result, hadronization (the non-perturbative process that will bind together colour-singlet parton pairs) takes place locally inside the jet and is not a collective process: only pairs of nearby partons are involved. The inclusive properties of jets (e.g. the particle multiplicity, jet mass, jet broadening, etc.) are independent of the hadronization model, up to corrections of order (A/.js)n (for some integer power n, which depends on the observable), with A ~ 1 GeV.

116

4.3

Jet rates

We now present explicit calculations of interesting observables. For simplicity, we will work with the soft-gluon approximation for the matrix elements and the phase-space. As a result, the correction to the differential e+ e- ~ qij cross-section from one-gluon emission becomes: da

g

= ao

2a s CF dko dcosB 1f ko 1 - cos2 B

with ao

=

Born amplitude.

(119)

In this equation we used the fact that in the soft-g limit the q and ij are back-to-back, and q. ij = 2qoijo,

q. k

= qoko(l -

cos B),

ijk

= ijok o (1 + cos B)

(120)

Notice the presence in da g of soft and collinear singularities. They will have to cancel in the total cross-section which, as we saw in the previous lecture, is finite. They do indeed cancel against the contribution to the total crosssection coming from the virtual correction diagram, where a gluon is exchanged between the two quarks. In the total cross-section (and for other sufficiently inclusive observables) the final states produced by the virtual diagrams and by the real emission diagrams in the soft or collinear limit are the same, and both contribute. In order for the total cross-section to be finite, the virtual contribution will need to take the following form: 2a s

-aD

dkodcosB

-CF 1f

1# o11 dk

0

-k' 0

-1

1

(1

28(ko) [8(1 - cos B)

d cos B' -

cos 2B' x

+ 8(1 + cos B)]

(121)

plus finite corrections. In this way:

{ # dk o

Jo

11

-1

2

dcosB

[

d ag dkodcosB

+

2

d a v ] = finite dkod cos B

(122)

With the form of the virtual corrections available (at least in this simplified soft-gluon-dominated approximation) we can proceed and calculate other quantities. Jets are usually defined as clusters of particles close-by in phase-space. A typical jet definition distributes particles in sets of invariant mass smaller than a given parameter M, requiring that one particle only belongs to one jet, and

117

that no other particles (or jets) can be added to a given jet without its mass exceeding M. In the case of a three-particle final state, such as the one we are studying, we get three-jet events if (q + k)2, (ij + k)2 and (q + ij)2 are all larger than M2. We will have two-jet events when at least one of these quantities gets smaller than M2. For example emission of a gluon near the direction of the quark, with 2qk = 2qok O(1 - cos ()) < M2, defines a two-jet event, one jet being given by the ij, the other by the system q + k. One usually introduces the parameter y = M2 / s, and studies the jet multiplicity as a function of y. Let us calculate the two- and three-jet rates at order as. The phase-space domain for two-jet events is given by two regions. The first one is defined by 2qk = 2qoko(1 - cos (}) < ys. This region consists of two parts: (I)a :

ko < YVs { 0

1

.!. N

E 10

-1

(3)

VS,

where the upper sign refers to neutrinos, the lower sign to antineutrinos, G F is the Fermi constant, nB the baryon density, Yn the neutron and Ye the electron number per baryon (both about 1/2 in normal matter). Numerically we have

Fr;.:

G = 1.9 2v2

X

10- 14 eV

p

g cm- 3

·

(4)

The dispersion relation is Ev = Vweak + Jp~ + m~ so that Vweak should be compared with m~/2pv. For ~m~ around 10- 3 eV 2 , PI' of a few GeV, and p of a few g cm- 3 , the energy difference between vI-' and Vs arising from Vweak is about the same as that from ~m~/2pv. The resulting modification of the oscillation pattern can cause rather peculiar zenith-angle distributions 21-23, but the current data do not allow one to exclude the Vs channel.

151

While the Vr is quasi-sterile in the detector because of the large mass of the T-Iepton, there is still an important difference to a Vs because the Vr produces pions in neutral-current collisions such as vN -t NV7f o which can be seen by 7f0 -t 21'. With better statistics and a dedicated analysis one may be able to distinguish the Vr and Vs oscillation channels 24-26. The evidence for atmospheric neutrino oscillations is very compelling, yet an independent confirmation is urgently needed. Hopefully it will come from one of the long-baseline experiments where an accelerator neutrino beam is directed toward a distant detector. The most advanced project is the K2K experiment 27 between KEK and Kamioka with a baseline of 250 km. Other projects include detectors in the Soudan mine at a distance of 730 km from Fermilab 28,29, or in the Gran Sasso Laboratory at 732 km from CERN 30-32. 2.2

Solar Neutrinos

The Sun, like other hydrogen-burning stars, liberates nuclear binding energy by the effective fusion reaction 4p + 2e- -t 4He + 2ve + 26.73 MeV so that its luminosity implies a Ve flux at Earth of 6.6 x 1010 cm- 2 S-I. In detail, the production of helium involves primarily the pp-chains-the CNO cycle is important in stars more massive than the Sun. The expected solar neutrino flux is shown in Fig. 3, where solid lines are for the three contributions which ~~~~'" Superkar

Kamiokal Homestal

z. SAGE, GAl

Figure 3: Solar neutrino flux at Earth. Continuum spectra in cm- 2 S-I MeV-I, line spectra in cm- 2 S-I. Solid lines are the sources of dominant experimental significance.

152

are most important for the measurements, pp: Beryllium: Boron:

p + p -+ 2H + e+ + Ve e- + 7Be -+ 7Li + Ve p + 7Be -+ 8B -+ 8Be*

+ e+ + Ve

(Ev < 0.420 MeV), (Ev = 0.862 MeV), (Ev ;S 15 MeV).

(5)

A crucial feature of these reactions is that the beryllium and boron neutrinos both arise from 7Be which may either capture a proton or an electron so that their relative fluxes depend on the branching ratio between the two reactions. The solar neutrino flux has been measured in five different experiments with three different spectral response characteristics; the relevant energy range is indicated by the hatched bars above Fig. 3. The radiochemical gallium experiments GALLEX 33,34 and SAGE 35 reach to the lowest energies and pick up fluxes from all source reactions. The Homestake chlorine experiment 36 picks up beryllium and boron neutrinos, while the Kamiokande 37 and SuperKamiokande 38 ,39 water Cherenkov detectors see only the upper part of the Total Rates: Standard Model vs. Experiment Bahcall-Pinsonneault 98

O.54±O.07

67±B

0.47±O.02 2.56±O.23

SuperK

Kamioka H20

Cl

Theory II1II 'Be BB

II1II p-p. pep II1II CNO

SAGE Ga

Experiments II1II

Figure 4: Solar neutrino fluxes measured in five experiments vs. theoretical predictions from a standard solar model 40. (Figure courtesy of J. Bahcall.)

153

boron flux. All of the experiments see a flux deficit relative to standard-solar model predictions as summarized in Fig. 4 and in a recent overview 41 . It has been widely discussed that there is no possibility to account for the measured fluxes by any apparent astrophysical or nuclear-physics modification of the standard solar models so that an explanation in terms of neutrino oscillations is difficult to avoid 41,42. Moreover, at something like the 99.8% CL one cannot account for the measurements by an energy-independent global suppression factor 41. Therefore, one cannot appeal to neutrino oscillations with an arbitrary D.m~ and a large mixing angle. One viable possibility are vacuum oscillations with a large mixing angle and a D.m~ around 10- 10 eV2, providing an oscillations length of order the Sun-Earth distance and thus an energy-dependent suppression factor. Second, one can have solutions with D.m~ in the neighborhood of 10- 5 eV2 where the mass difference between the oscillating flavors (energy of order 1 Me V) can be canceled by the neutrino refractive effect in the Sun, leading to resonant or MSW oscillations 43,44, again with an energy-dependent suppression factor of the Ve flux. In this case one may have a nearly maximal mixing angle, or a small one as shown in Table 1. The large-angle MSW region does not provide a credible fit for Ve -+ Vs oscillations while the other solutions are possible for the Ve -+ VI-',T or Ve -+ Vs channels, of course with somewhat different contours of preferred mixing parameters 41. It is noteworthy that the spectral distortion of the spectrum of recoil electrons measured at SuperKamiokande seems to single out the vacuum case as the preferred solution 39, although this must be considered a rather preliminary conclusion at present.

2.3

LSND

The LSND (Liquid Scintillation Neutrino Detector) experiment is the only case of a pure laboratory experiment which shows indications for neutrino oscillations 45 . It utilizes a proton beam at the Los Alamos National Laboratory in the US. The protons are directed at a target where neutrinos arise from the same basic mechanism Eq. (1), upper line, that produces them in the atmosphere. From 7r+ decay-in-flight one obtains a vI-' beam of up to 180 MeV while the subsequent decay-at-rest of stopped /1+'s provides a iiI-' beam of less than 53 MeV. The beam should not contain any iie's; they can be detected by iieP -+ ne+ in coincidence with np -+ d,(2.2 MeV). For energies above 36 MeV, the 1993-95 data included 22 such events above an expected background of 4.6 ± 0.6; this excess is interpreted as evidence for iiI-' -+ iie oscillations. The LSND data favor a large range of ve-vl-'-mixing parameters. After taking the exclusion regions of other experiments into account, one is left with

154

a sliver of mixing parameters in the range indicated in Table 1. The KARMEN experiment is also sensitive in this range, but has not seen any events 46. This lack of confirmation, however, does not exclude the LSND evidence as the non-observation of only a few expected events is not a statistically persuasive conflict. Moreover, if one excludes the background-infested 20-36 MeV data in LSND one finds a much broader range of allowed mixing parameters than could have been probed by KARMEN 48. Within 2-3 years all of the LSND area will be covered with high sensitivity by MiniBooNE 47, a new experiment at Fermilab, which will settle this case.

2.4

Global Interpretation

In Table 1 we summarize the neutrino oscillation channels and mixing parameters indicated by the atmospheric and solar neutrino anomalies and the LSND experiment. Clearly there is no straightforward interpretation because there are too many indications! If only three different mass eigenstates mi, i = 1,2,3, exist, the mass splittings must satisfy

L

6m~

= (m~ - mD + (m~ - mI) + (m~ - mD = 0,

(6)

Splitting.

a trivial condition which is not met by the independent 6m~ from Table l. Some of the experiments may not be due to a single 6m~ but rather to nontrivial three-flavor oscillation patterns 49-52. Even then it appears that one must ignore some of the experimental evidence or stretch the errors beyond plausible limits to accommodate all experiments in a three-flavor scheme. If one has to throw out one of the indications, LSND is usually taken as the natural victim because there is no independent confirmation, and because

Table 1: Experimentally favored neutrino mass differences and mixing angles.

Experiment

Favored Channel

LSND Atmospheric

vI-' vI-' vI-'

Solar Vacuum MSW (small angle) MSW (large angle)

Ve Ve Ve

-+ -+ -+

ve Vr Va

-+ anything -+ anything -+ vI-' or Vr

6m 2 [eV2]

sin 2 2e

0.2-10 (1-8) X 10- 3 (2-7) X 10- 3

(0.2-3) x 10- 2 0.85-1 0.85-1

(0.5-8) x 10- 10 (0.4-1) x 10- 5 (3-30) X 10- 5

0.5-1 10- 3 -10- 2 0.6-1

155

the other cases simply look too strong to be struck from the list. Once LSND has been disposed of, a typical mass and mixing scheme may be as shown in Fig. 5 where the small-angle MSW solution has been taken for solar neutrinos. 100 ~ve

10 -I

_

vJl

_v't 10 -2

>.,

.

e

10-3 10-4

~

A

10 -5 VI

V 2

V3

Figure 5: Hierarchical mass and mixing scheme to account for solar and atmospheric neutrinos, the former by the small-angle MSW solution. The flavor content of each mass eigenstate is indicated by the fill-patterns. (Figure 53 reproduced with kind permission of A. Smirnov.)

However, the large mixing angle which is needed to account for the atmospheric neutrino anomaly suggests that more than one mixing angle may be large. Moreover, the spectral distortion observed in SuperKamiokande suggests that the solar vacuum solution may be preferred 39. Of course, the vastly different values for Llm~ implied by atmospheric neutrinos and the solar vacuum solution looks unnatural. Shrugging off this objection, there are several workable schemes involving more than one large mixing angle, for example bi-maximal mixing or threefold maximal mixing 53 . It is also conceivable that the mass differences are not representative of the masses themselves, i.e. that all three flavors have, say, an eV-mass with small splittings as implied by solar and atmospheric neutrinos (degenerate mass pattern). Of course, such a scheme is very different from the hierarchical patterns that we know in the quark and charged-lepton sectors, but the large mixing angle or angles look very unfamiliar, too. If the neutrino masses are all Majorana, one may still evade bounds on the effective Ve Majorana mass (m~Jeff relevant for neutrinoless (3(3 decay. For example, in the bi-maximal mixing case there is an exact cancellation so that (m~e )eff = 0 in the limit where the mass differences can be neglected relative to the common mass scale. At the present time there is no objective reason to ignore LSND. As a consequence, a very radical conclusion follows: there must be four independent

156

mass eigenstates, i.e. at least one low-mass neutrino degree of freedom beyond the three sequential flavors. This fourth flavor Vs would have to be sterile with regard to the standard weak interactions. Probably the most natural mass and mixing pattern is one like Fig. 6, but there are also other possibilities 53,54. 10

1

OVs

HDM

_vI! _v

:..---ATM

~ve 10- 1

t

;>

" S- 10-2 W- 3

-

LSND

Vs~

10-4

v4

VI

Figure 6: Representative four-flavor mass and mixing scheme to account for all experimental evidence_ (Figure 53 reproduced with kind permission of A. Smirnov.)

Of course, it would be an extremely radical and unexpected finding if the oscillation experiments had not only turned up evidence for neutrino masses, but for an additional, previously unsuspected low-mass sterile neutrino. A confirmation of LSND by MiniBooNE 47 would make this conclusion difficult to avoid so that this new experiment is perhaps the most urgent current effort in experimental neutrino physics. 3

3.1

Cosmology

Big-Bang Nucleosynthesis

Massive neutrinos and the existence of sterile neutrinos can have a variety of important cosmological consequences. One immediately wonders if a fourth neutrino flavor is not in conflict with the well-known big-bang nucleosynthesis (BBN) limit on the effective number of thermally excited primordial neutrino degrees of freedom 55-57. However, there are several questions. The first and most obvious one is whether the observationally inferred light-element abundances strictly exclude a fourth flavor at the epoch of BBN. The unfortunate answer is that, while a fourth flavor clearly would make a very significant difference, BBN is not in a position to exclude this possibility with the sort of confidence that would be required to dismiss the sterile-neutrino hypothesis 58 .

157

Second, a sterile neutrino need not attain thermal equilibrium in the first place. It is excited by oscillations in conjunction with collisions so that its contribution to the cosmic energy density at the BBN epoch depends on the mass difference and mixing angle with an active flavor 59-65 . If the atmospheric neutrino anomaly is due to v/-, --+ v. oscillations, the large mixing angle and large ~m~ imply that the sterile neutrino would be fully excited at the time of BBN. On the other hand, for the small-angle MSW solution or the vacuum solution of the solar neutrino problem, it is barely excited so that the additional energy density is negligible. Therefore, of the different four-flavor patterns BBN favors those where Ve-Vs oscillations solve the solar neutrino problem over those where v/-,-vs oscillations explain the atmospheric neutrino anomaly. Even this conclusion can be avoided if a lepton asymmetry of order 10- 5 exists at the time of the primordial v/-, --+ v. oscillations 66. It may be possible to create such asymmetries among the active neutrinos by oscillations between, say, V (i/ and sterile states 67 ,68, although the exact requirements on the mass and mixing parameters are controversial in some cases 69-73. Be that as it may, a sterile neutrino provides for a rich oscillation phenomenology in the early universe, but at the same time BBN is not quite enough of a precision tool to distinguish seriously between different four-flavor patterns. As it stands, BBN would benefit more from pinning down the neutrino mass and mixing pattern experimentally than the other way round. T

3.2

T )

Dark Matter

Irrespective of the possible existence of a sterile neutrino, it has become difficult to dispute that neutrinos have masses. Therefore, they could play an important role for the cosmological dark matter. Standard calculations in the framework of the big-bang cosmology reveal that the present-day universe contains about 100 em -3 neutrinos and antineutrinos per active flavor 74, leading to a cosmological mass fraction of

(7) where h is the Hubble constant in units of 100 km S-1 Mpc- 1 . The observed age of the universe together with the measured expansion rate reveals that nh 2 ;S 0.4, leading to the most restrictive limit on the masses of all neutrino flavors 78,79. Once we believe the current indications for oscillations, the mass differences are so small that this limit reads mv ;S 13 eV for the common mass scale of all flavors, roughly identical with the world-averaged tritium endpoint limit on mv. of about 81 15 eV.

158

If the neutrino masses were in this range they could be the cosmic dark matter as first pointed out more than 25 years ago 80. However, it was quickly recognized that neutrinos do not make for a good universal dark matter candidate. The simplest counter-argument ("Tremaine-Gunn-limit") arises from the phase space of spiral galaxies which cannot accommodate enough neutrinos to account for their dark matter unless the neutrino mass obeys a lower limit 82,83. For typical spiral galaxies it is 84 m" 2: 20 e V, for dwarf galaxies even m" 2: 100-200 eV, difficult to reconcile with the cosmological upper limit.

3.3

Large-Scale Structure

The Tremaine-Gunn-limit is only the tip of the iceberg of evidence against neutrino dark matter. The most powerful argument arises from cosmic structure formation. At early times the universe was extremely smooth as demonstrated by the tiny amplitude of the temperature fluctuations of the cosmic microwave background radiation across the sky. The present-day distribution of matter, on the other hand, is very clumpy. There are stars, galaxies, clusters of galaxies, and large-scale coherent structures on scales up to about 100 Mpc. A perfectly homogeneous expanding universe stays that way forever. The standard theory 74-77 for the formation of structure has it that the universe was initially almost, but not quite, perfectly homogeneous, with a tiny modulation of its density field. The action of gravity enhances the density contrast as time goes on, leading to the observed structures. The outcome of this evolution depends on the initial spectrum of density fluctuations which is usually taken to be approximately flat, i.e. of the "Harrison-Zeldovich-type," corresponding to the power-law-index n = 1. However, the effective spectrum relevant for structure formation is the processed spectrum which obtains at the epoch when the universe becomes matter dominated. As the matter which makes up the cosmic fluid can diffuse around, the smallest-scale density fluctuations will be wiped out. This effect is particularly important for weakly interacting particles which can diffuse far while they are relativistic. Low-mass particles stay relativistic for a long time and thus wipe out the primordial fluctuations up to large scales. Massive particles stay put earlier and thus have this effect only on small scales. One speaks of "hot dark matter" (HDM) if the particle masses are small enough that all fluctuations are wiped out beyond scales which later correspond to a galaxy. Conversely, "cold dark matter" (CDM) has this effect only on sub-galactic scales. One way of presenting the results of calculations of structure formation is to show the expected power-spectrum of the present-day matter distribution (Fig. 7) which can be compared to the observed galaxy distribution. The theory

159

of structure formation then predicts the form, but not the amplitude of the spectrum which can be fit either on large scales to the observed temperature fluctuations of the cosmic microwave background radiation as observed by the CO BE satellite, or else on small scales to the observed galaxy distribution. Figure 7 illustrates that HDM (neutrinos) suppresses essentially all small-scale structure below a cut-off corresponding to a supercluster scale and thus does not seem to be able to account for the observations. While cold dark matter works impressively well, it has the problem of producing too much clustering on small scales. Ways out include a primordial power spectrum which is not strictly flat (tilted dark matter), a mix of cold and hot dark matter, or the assumption of a cosmological constant. Currently there is a broad consensus that some variant of a CDM cosmology where structure forms by gravitational instability from a primordial density fluctuations of approximately the Harrison-Zeldovich type is probably how our universe works. Thus, while it is widely accepted that neutrinos are not the main darkmatter component, quite conceivably they contribute something like 20%, giving rise to a hot plus cold dark matter (HCDM) scenario which avoids the overproduction of small-scale structure of a pure CDM cosmology 86-89. A HDM fraction exceeding about 20% is inconsistent with the size of voids in d (h· 1 Mpc)

10 5

1000 Microwave Background

100

10

Superclusters

Clusters

Galaxies

10 4

"?

.s::.

.>
c) logU( 4>c) / J-L}

(105)

~(t(4>c»4>c

Using (105) we can write the effective potential and its derivatives (102) as 4>c-functions, (106) and

Using Eq. (104) one can easily prove that 14, (108)

208

Fixing the scale is a matter of convention. Fixing the scale, as we have just described, as a function of ¢c (i.e. giving different scales for different values of the field) is usually done to optimize the validity of the perturbative expansion, i.e. minimizing the value of radiative corrections to the effective potential around the minimum of the field. A very interesting result obtained in Ref. 13 is: The RGE improved effective potential exact up to (next-to-leading)L log order c is obtained using the L-Ioop effective potential and the (L+1)-100p RGE (J-functions. 2

Field Theory at Finite TeIDperature

The formalism used in conventional quantum field theory is suitable to describe observables (e.g. cross-sections) measured in empty space-time, as particle interactions in an accelerator. However, in the early stages of the universe, at high temperature, the environment had a non-negligible matter and radiation density, making the hypotheses of conventional field theories impracticable. For that reason, under those circumstances, the methods of conventional field theories are no longer in use, and should be replaced by others, closer to thermodynamics, where the background state is a thermal bath. This field has been called field theory at finite temperature and it is extremely useful to study all phenomena which happened in the early universe: phase transitions, inflationary cosmology, ... Excellent articles 15,16, review articles 17,18,19 and text books 20 exist which discuss different aspects of these issues. In this section we will review the main methods which will be useful for the theory of phase transitions at finite temperature. 2.1

Grand-canonical ensemble

In this section we shall give some definitions borrowed from thermodynamics and statistical mechanics. The IDicrocanonical enseIDble is used to describe an isolated system with fixed energy E, particle number N and volume V. The canonical enseIDble describes a system in contact with a heat reservoir at temperature T: the energy can be exchanged between them and T, N and V are fixed. Finally, in the grand canonical enseIDble the system can exchange energy and particles with the reservoir: T, V and the chemical potentials are fixed. Consider now a dynamical system characterized by a hamiltonian d Hand a set of conserved (mutually commuting) charges QA. The equilibrium state of CThe convention is (next-to-leading)O ""leading, i.e. L = O. For L to next-to-leading log. d All operators will be considered in the Heisenberg picture.

= 1 the potential is exact

209 the system at rest in the large volume V is described by the grand-canonical density operator p = exp( -cI» exp { -

~ aAQA -

{JH}

(109)

where cI> == log Tr exp { - L:A aA Q A - {JH} is called the Massieu function (Legendre transform of the entropy), aA and {J are Lagrange multipliers given by {J = T- 1, aA = -(J/-LA, T is the temperature and /-LA are the chemical potentials. Using (109) one defines the grand canonical average of an arbitrary operator 0, as (110) (0) == Tr(Op) satisfying the property (1) = l. In the following of this section we will always consider the case of zero chemical potential. It will be re-introduced when necessary.

2.2

Generating functionals

We will start considering the case of a real scalar field ¢J( x), carrying no charges (/-LA = 0), with hamiltonian H, i.e.

(111)

¢J(x) = eitH¢J(O,x)e-itH

where the time Xo = t is analytically continued to the complex plane. We define the thermal Green function as the grand canonical average of the ordered product of the n field operators (112)

where the Te ordering means that fields should be ordered along the path C in the complex t-plane. For instance the product of two fields is defined as,

Te¢J(x)¢J(y) = Oc(xo - yO)¢J(x)¢J(y)

+ Oc(yo -

XO)¢J(y)¢J(x)

(113)

If we parameterize C as t = z( T), where T is a real parameter, Te ordering means standard ordering along T. Therefore the step and delta functions can be given as Oc(t) = OCT), Je(t) = (ozjOT)-l J(T). The rules of the functional formalism can be applied as usual, with the prescription Jj(y)jJj(x) = Jc( x O_yO)J(3) (x-iJ), and the generating functional Z,6[j] for the full Green functions,

Z,6[j] =

L 00

.n [

~

n=O n. e

d4Xl ... d4x n j(xd·· .j(xn)G(e)(Xl, ... ,xn )

(114)

210

can also be written as,

zt3[j] =

(TC exp {i Latxj(x)¢(x) } )

(115)

which is normalized to z!3[O] = (1) = 1, as in (110), and where the integral along t is supposed to follow the path C in the complex plane. Similarly, the generating functional for connected Green functions W t3 [j] is defined as Zt3 [j] == exp{ iWt3 (j]), and the generating functional for IPI Green functions r t3 [4)], by the Legendre transformation,

r t3 [4)] =

Wt3(j] -

L

atx

~)

(295) (296)

251

are the D-term contributions. The field-dependent stop masses are then

rnZ

rnZ (¢) + rnZ (¢)

(¢) =

tL

h,2

2

tR

±

rnZ [

tL

(¢) 2

rnZ

(¢)]2

+ [m~(¢)12.

tR

(297)

The corresponding effective T-dependent masses, M! (¢, T), are given by tl,2

expressions identical to (297), apart from the replacement

The IItL,R (T) are the leading parts of the T-dependent self-energies of 'h,R, (299)

IItL (T)

IItR (T)

=

(300)

where gs is the strong gauge coupling constant. Only loops of gauge bosons, Higgs bosons and third generation squarks have been included, implicitly assuming that all remaining supersymmetric particles are heavy and decouple. We shall work in the limit in which the left handed stop is heavy, mQ ~ 500 GeV. In this limit, the supersymmetric corrections to the precision electroweak parameter 6.p become small and hence, this allows a good fit to the electroweak precision data coming from LEP and SLD. Lower values of mQ make the phase transition stronger and we are hence taking a conservative assumption from the point of view of defining the region consistent with electroweak baryogenesis. The left handed stop decouples at finite temperature, but, at zero temperature, it sets the scale of the Higgs masses as a function of tan jJ. For right-handed stop masses below, or of order of, the top quark mass, and for large values of the CP-odd Higgs mass, mA » Mz, the one-loop improved Higgs effective potential admits a high temperature expansion,

VO+V1

=-m

2

(T) ¢2_T

2

ESM

[

¢3

+ (2N

)

(rnZ+ IIt

127r

c

where Nc = 3 is the number of colours and in the Standard Model case.

(T)

tR

ESM

)3/2] + A(T) ¢4+ ... 8

(301) is the cubic term coefficient

252 Within our approximation, the lightest stop mass is approximately given by

~t '"- m 2 + U

0

2 cos 2(3 + m 2

. 15MZ

t

(1- .4;)

(302)

m2

Q

where At At - pi tan (3 is the stop mixing parameter. As was observed in Ref. 68, the phase transition strength is maximized for values of the soft breaking parameter m~ = -TIiR(T), for which the coefficient of the cubic term in the effective potential, 3

E ~ ESM

_

(

h~ sin (3 1 - Ai 1mb

+

In

4v27f

)3/2 (303)

'

can be one order of magnitude larger than ESM 68. In principle, the above would allow a sufficiently strong first order phase transition for Higgs masses as large as 100 GeV. However, it was also noticed that such large negative values of m~ may induce the presence of color breaking minima at zero or finite temperature 68,69. Demanding the absence of such dangerous minima, the one loop analysis leads to an upper bound on the lightest CP-even Higgs mass of order 80 GeV. This bound was obtained for values of m~ = -m~ of order (80 GeV)2. The most important two loop corrections are of the form ¢2 log( ¢) and, as said above, are induced by the Standard Model weak gauge bosons as well as by the stop and gluon loops 70. It was recently noticed that the coefficient of these terms can be efficiently obtained by the study of the three dimensional running mass of the scalar top and Higgs fields in the dimensionally reduced theory at high temperatures 71. Equivalently, in a four dimensional computation of the MSSM Higgs effective potential with a heavy left-handed stop, we obtain 68 AH

~T ¢2T2 327f 2

~

( mbA;- )]2 +

51 2 2. 2 16 9 - 3 h t sm(3 1-

[

[

2 2 . 2 89s h t sm (3

(_mbA; )] 1-

where the first term comes from the Standard Model gauge boson-loop contributions, while the second and third terms come from the light supersymmetric particle loop contributions. The scale AH depends on the finite corrections, which may be obtained by the expressions given in 72. As mentioned above, the two-loop corrections are very important and, as °has been shown in Ref. 70, they

253

can make the phase transition strongly first order even for mu ~ 0 72. Concerning the validity of the perturbative expansion, the .B-parameter, similarly to .BSM, (283), can be shown to be given by (305)

which leads to a reliable perturbative expansion for a value of the Higgs mass enhanced, with respect to its Standard Model value, by a factor'" (mt!mh)2. An analogous situation occurs in the U-direction (U == tR). The one-loop expression is approximately given by

Vo (U)

+ VI ( U,T )

=

- 2 ( -mu

+"(uT 2) U 2 -TEuU 3

AU 4 , + TU

(306)

where "(u and Eu were given in 72 . Analogous to the case of the field , the two loop corrections to the Upotential are dominated by gluon and stop loops and are approximately given by

U [100 g9s - 2h 2

T2 V2 (U, T) = 167r 2

4

2. 2

t

sm.B

(

1-

A; )] log (Au) mb U

(307)

Au

where, as in the Higgs case, the scale may only be obtained after the finite corrections to the effective potential are computed 72. Once the effective potential in the and U directions are computed, one can study the strength of the electroweak phase transition, as well as the presence of potential color breaking minima. At one-loop, it was observed that requiring the stability of the physical vacuum at zero temperature was enough to assure the absolute stability of the potential at finite temperature. As has been first noticed in Ref. 71 , once two loop corrections are included, the situation is more complicated 72. At zero temperature the minimization of the effective potential for the fields and U shows that the true minima are located for vanishing values of one of the two fields. The two set of minima are connected through a family of saddle points for which both fields acquire non-vanishing values. Due to the nature of the high temperature corrections, we do not expect a modification of this conclusion at finite temperature. Two parameters control the presence of color breaking minima: m defined as the smallest value of for which a color breaking minimum deeper than the electroweak breaking minimum is present at T = 0, and Tcu , the critical temperature for the transition into a color breaking minimum in the

mu

u,

254

U-direction. The value of mb may be obtained by analysing the effective potential for the field U at zero temperature, and it is approximately given by 68 (308) Defining the critical temperature as that at which the potential at the symmetry preserving and broken minima are degenerate, four situations can happen in the comparison of the critical temperatures along the ¢ (Tc) and U (TcU ) transitions: a) TP < Tc; mu < mij; b) TcU < Tc; mu > mij; c) TcU > Te; mu < mb; d) Tcu > Te; mu > mb· In case a), as the universe cools down, a phase transition into a color preserving minimum occurs, which remains stable until T = O. This situation, of absolute stability of the physical vacuum, is the most conservative requirement to obtain electroweak baryogenesis. In case b), at T = 0 the color breaking minimum is deeper than the physical one implying that the color preserving minimum becomes unstable for finite values of the temperature, with T < T e . A physically acceptable situation may only occur if the lifetime of the physical vacuum is smaller than the age of the universe. We shall denote this situation as "metastability". In case c), as the universe cools down, a color breaking minimum develops which, however, becomes metastable as the temperature approaches zero. A physically acceptable situation can only take place if a two step phase transition occurs, that is if the color breaking minimum has a lifetime lower than the age of the universe at some temperature T < Te 71. Finally, in case d) the color breaking minimum is absolutely stable and hence, the situation becomes physically unacceptable. Figure 9 shows the region of parameter space consistent with a sufficiently strong phase transition for At = 0,200,300 GeV. For low values of the mixing, At ;S 200 GeV, case a) or c) may occur but, contrary to what happens at one-loop, case b) is not realized. For the case of no mixing, this result is in agreement with the analysis of 71. The region of absolute stability of the physical vacuum for At ~ 0 is bounded to values of the Higgs mass of order 95 GeV. There is a small region at the right of the solid line, in which a two-step phase transition may take place, for values of the parameters which would lead to v/T < 1 for T = Te, but may evolve to larger values at some T < Te at which the second of the two step phase transition into the physical vacuum takes place. This region disappears for larger values of the stop mixing mass parameter. For values of the mixing parameter At between 200 GeV and 300 GeV, both situations, cases b) and c) may occur, depending on the value of tanfj. For large values of the stop mixing, At > 300 GeV, a two-step phase

255

105

.•

\ \

\

\ \

\ \ \

rna= 1 TeV

\

\\ 95

\ \

\\ \

\ \\

. U \ \\

\\ \\• \\

85

\\

\\

75 115

125

135

mr Figure 9: Values of mh, m t for which v(Tc)/Tc

145

155

165

(GeV) = 1 (solid

line), TP

= Tc

(dashed line),

mu = mt (short-dashed line), for mQ = 1 TeV and At = 0,200,300 GeV. The region on the left of the solid line is consistent with a strongly first order phase transition. A two step phase transition may occur in the regions on the left of the dashed line, while on the left of the short-dashed line, the physical vacuum at T = 0 becomes metastable. The region on the left of both the dashed and short-dashed lines leads to a stable color breaking vacuum state at zero temperature and is hence physically unacceptable.

transition does not take place. All together, and even demanding absolute stability of the physical vacuum, electroweak baryogenesis seems to work for a wide region of Higgs and stop mass values. Higgs masses between the present experimental limit, of about 90 GeV, and around 105 GeV are consistent with this scenario. Similarly, the running stop mass may vary from values of order 165 GeV (of the same order as the top quark mass one) and 100 GeV. Observe that, due to the influence of the D-terms, values of m t := 165 GeV, At := 0 and mh := 75 GeV, are achieved for small positive values of mu. Also observe that for lower values of mQ the phase transition may become more strongly first order and slightly larger values of the stop masses may be obtained. Remind that these results are

256

based on the two-loop improved effective potential. Recent non-perturbative calculations 73 confirm the validity of these perturbative results.

References

1. W. Heisenberg and H. Euler, Z. Phys. 98 (1936) 714; J. Schwinger, Phys. Rev. 82 (1951) 664. 2. J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127 (1962) 965; G. Jona-Lasinio, Nuovo Cimento 34 (1964) 1790. 3. S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888. 4. R. Jackiw, Phys. Rev. D9 (1974) 1686. 5. J. Iliopoulos, C. Itzykson and A. Martin, Rev. Mod. Phys. 47 (1975) 165. 6. J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics (McGrawHill, 1964); Relativistic Quantum Fields (McGraw-Hill, 1965). 7. 1.S. Gradshteyn and 1.M. Ryzhik, Table of Integrals Series and Products (Academic Press, 1965). 8. G. t'Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189; C.G. Bollini and J.J. Giambiagi, Phys. Lett. B40 (1972) 366; J.F. Ashmore, Nuovo Cimento Letters 4 (1972) 289. 9. G. t'Hooft and M. Veltman, Nucl. Phys. B61 (1973) 455; W.A. Bardeen, A.J. Buras, D.W. Duke and T. Muta, Phys. Rev. D18 (1978) 3998. 10. See, e.g.: J. Collins, Renormalization (Cambridge University Press, 1984). 11. W. Siegel, Phys. Lett. B84 (1979) 193. 12. G.W. Anderson and L.J. Hall, Phys. Rev. D45 (1992) 2685. 13. B. Kastening, Phys. Lett. B283 (1992) 287; M. Banda, T. Kugo, N. Maekawa and H. Nakano, Phys. Lett. B301 (1993) 83; C. Ford, D.R.T. Jones, P.W. Stephenson and M.B. Einhorn, Nucl. Phys. B395 (1993) 17. 14. A. Casas, J.R. Espinosa, M. Quiros and A. Riotto, Nucl. Phys. B436 (1995) 3. 15. L. Dolan and R. Jackiw, Phys. Rev. D9 (1974) 3320. 16. S. Weinberg, Phys. Rev. D9 (1974) 3357. 17. R.H. Brandenberger, Rev. Mod. Phys. 57 (1985) 1. 18. N.P. Landsman and Ch.G. van Weert, Phys. Rep. 145 (1987) 141. 19. M. Quiros, Helv. Phys. Acta 67 (1994) 451. 20. J.1. Kapusta, Finite-temperature field theory (Cambridge University Press, 1989). 21. R. Kubo, J. Phys. Soc. Japan 12 (1957) 570; P.C. Martin and J.

257

Schwinger, Phys. Rev. 115 (1959) 1342. 22. T. Matsubara, Prog. Theor. Phys. 14 (1955) 351. 23. H. Matsumoto, Y. Nakano, H. Umezawa, F. Mancini and M. Marinaro, Prog. Theor. Phys. 70 (1983) 599; H. Matsumoto, Y. Nakano and H. Umezawa, J. Math. Phys. 25 (1984) 3076. 24. L.V. Keldish, Sov. Phys. JETP 20 (1964) 1018. 25. T.S. Evans, Phys. Rev. D47 (1993) R4196; T. Altherr, CERN preprint, CERN-TH.6942/93. 26. R. Kobes, Phys. Rev. D42 (1990) 562 and Phys. Rev. Lett. 67 (1991) 1384; T.S. Evans, Phys. Lett. B249 (1990) 286, Phys. Lett. B252 (1990) 108, Nucl. Phys. B371 (1992) 340; P. Aurenche and T. Becherrawy, Nucl. Phys. B379 (1992) 259; M.A. van Eijck and Ch. G. van Weert, Phys. Lett. B278 (1992) 305. 27. D.A. Kirzhnits, JETP Lett. 15 (1972) 529. 28. D.A. Kirzhnits and A.D. Linde, Phys. Lett. 42B (1972) 471; D.A. Kirzhnits and A.D. Linde, JETP 40 (1974) 628; D.A. Kirzhnits and A.D. Linde, Ann. Phys. 101 (1976) 195. 29. A.D. Linde, Rep. Prog. Phys. 42 (1979) 389; A.D. Linde, Phys. Lett. 99B (1981) 391; A.D. Linde, Phys. Lett. 99B (1981) 391; A.D. Linde, Particle Physics and Inflationary Cosmology (Harwood, Chur, Switzerland, 1990). 30. A.D. Linde, Phys. Lett. BI08 (1982) 389; A. Albrecht and P.J. Steinhardt, Phys. Rev. Lett. 48 (1082) 1226. 31. M. Dine, R.G. Leigh, P. Huet, A. Linde and D. Linde, Phys. Lett. B283 (1992) 319; Phys. Rev. D46 (1992) 550. 32. S. Coleman, Phys. Rev. D15 (1977) 2929. 33. C.G. Callan and S. Coleman, Phys. Rev. D16 (1977) 1762. 34. S. Coleman and F. De Luccia, Phys. Rev. D21 (1980) 3305. 35. A.D. Linde, Phys. Lett. 70B (1977) 306; 100B (1981) 37; Nucl. Phys. B216 (1983) 421. 36. 1. McLerran, M. Shaposhnikov, N. Turok and M. Voloshin, Phys. Lett. B256 (1991) 451. 37. M. Dine, P. Huet and R. Singleton Jr., Nucl. Phys. B375 (1992) 625. 38. J.M. Moreno, M. Quiros and M. Seco, Nucl. Phys. B526 (1998) 489. 39. E.W. Kolb and M.S. Turner, The Early Universe (Addison-Wesley, 1990). 40. A.D. Sakharov, Zh. Eksp. Teor. Fiz. Pis'ma 5 (1967) 32; JETP Lett. 91B (1967) 24. 41. Particle Data Group, Review of Particle Properties, Eur. Phys. J. C3 (1998) 1. 42. E.W. Kolb and S. Wolfram, Nucl. Phys. B172 (1980) 224; Phys. Lett.

258

B91 (1980) 217. 43. V.A. Kuzmin, V.A. Rubakov and M.E. Shaposhnikov, Phys. Lett. BI55 (1985) 36j Phys. Lett. BI91 (1987) 171. 44. A.G. Cohen, D.B. Kaplan and A.E. Nelson, Annu. Rev. Nucl. Part. Sci. 43 (1993) 27. 45. G. t' Hooft, Phys. Rev. Lett. 37 (1976) 8j Phys. Rev. DI4 (1976) 3432. 46. M. Kobayashi and M. Maskawa, Prog. Theor. Phys. 49 (1973) 652. 47. N.S. Manton, Phys. Rev. D28 (1983) 2019; F.R Klinkhamer and N.S. Manton, Phys. Rev. D30 (1984) 2212. 48. J. Kunz, B. Kleihaus and Y. Brihaye, Phys. Rev. D46 (1992) 3587. 49. Y. Brihaye and J. Kunz, Phys. Rev. D48 (1993) 3884. 50. J.M. Moreno, D.H. Oaknin and M. Quiros, [hep-phj9605387j, Nucl. Phys. B483 (1997) 267. 51. P. Arnold and L. McLerran, Phys. Rev. D36 (1987) 581; S.Yu. Khlebnikov and M.E. Shaposhnikov, Nucl. Phys. B308 (1988) 885. 52. J. Ambjorn, M. Laursen and M. Shaposhnikov, Phys. Lett. BI97 (1989)49j J. Ambjorn, T. Askaard, H. Porter and M. Shaposhnikov, Phys. Lett. B244 (1990) 479; Nucl. Phys. B353 (1991) 346j J. Ambjorn and K. Farakos, Phys. Lett. B294 (1992) 248j J. Ambjorn and A. Krasnitz, Phys. Lett. B362 (1995) 97. 53. P. Arnold, D. Son and L.G. Yaife,Phys. Rev. D55 (1997) 6264; hepphj9810216j hep-phj9810217j G.D. Moore, Phys. Rev. D59 (1999) 014503; hep-phj9810313j D. Bodeker, Phys. Lett. B426 (1998) 351; hep-phj9810265. 54. L. Carson, Xu Li, L. McLerran and R-T. Wang, Phys. Rev. D42 (1990) 2127. 55. M.E. Shaposhnikov, Nucl. Phys. B287 (1987) 757; Nucl. Phys. B299 (1988) 797j A.1. Bochkarev and M.E. Shaposhnikov, Mod. Phys. Lett A2 (1987) 417. 56. A.1. Bochkarev, S.V. Kuzmin and M.E. Shaposhnikov, Phys. Rev. D43 (1991) 369. 57. P. Fendley, Phys. Lett. BI96 (1987) 175. 58. J.R Espinosa, M. Quiros and F. Zwirner, Phys. Lett. B29I (1992) 115. 59. D.J. Gross, RD. Pisarski and L.G. Yaife, Rev. Mod. Phys. 53 (1981) 43. 60. RR Parwani, Phys. Rev. D45 (1992) 4695. 61. P. Arnold and O. Espinosa, Phys. Rev. D47 (1993) 3546. 62. M.E. Carrington, Phys. Rev. D45 (1992) 2933. 63. K. Jansen, Nucl. Phys. (Proc. Supl.) B47 (1996) 196 [hep-Iatj9509018]. 64. A.G. Cohen and A.E. Nelson, Phys. Lett. B297 (1992) 111.

259 65. J.R. Espinosa, M. Quiros and F. Zwirner, Phys. Lett. B307 (1993) 106. 66. A. Brignole, J.R. Espinosa, M. Quiros and F. Zwirner, Phys. Lett. B324 (1994) 181. 67. G.F. Giudice, Phys. Rev. D45 (1992) 3177; S. Myint, Phys. Lett. B287 (1992) 325. 68. M. Carena, M. Quiros and C.E.M. Wagner, Phys. Lett. B380 (1996) 81 69. M. Carena and C.E.M. Wagner, Nucl. Phys. B452 (1995) 45 70. J.R. Espinosa, Nucl. Phys. B475 (1996) 273; B. de Carlos and J.R. Espinosa, [hep-ph/9703212], Nucl. Phys. B503 (1997) 24. 71. D. Bodeker, P. John, M. Laine and M.G. Schmidt, Nucl. Phys. B497 (1997) 387 72. M. Carena, M. Quiros and C.E.M. Wagner, [hep-ph/9710401] Nucl. Phys. B524 (1998) 3. 73. M. Laine and K. Rummukainen, Phys. Rev. Lett. 80 (1998) 5259; Nucl. Phys. B535 (1998) 423.

260 AN INTRODUCTION TO COSMOLOGICAL INFLATION ANDREW R. LIDDLE Astronomy Centre, University of Sussex, Brighton BN1 9QJ, U. K. and Astrophysics Group, The Blackett Laboratory, Imperial College, London SW7 2BZ, U. K. (present address) An introductory account is given of the inflationary cosmology, which postulates a period of accelerated expansion during the Universe's earliest stages. The historical motivation is briefly outlined, and the modelling of the inflationary epoch explained. The most important aspect of inflation is that it provides a possible model for the origin of structure in the Universe, and key results are reviewed, along with a discussion of the current observational situation and outlook.

1

Overview

One of the central planks of modern cosmology is the idea of inflation. Originally introduced by Guth 1 in order to explain the initial conditions for the hot big bang model, it has subsequently been given a much more important role as the currently-favoured candidate for the origin of structure in the Universe, such as galaxies, galaxy clusters and cosmic microwave background anisotropies. This article seeks to give an introductory account of the inflationary cosmology, with the focus aimed towards inflation as a model for the origin of structure. It begins with a quick review of the big bang cosmology, and the problems with it which led to the introduction of inflation. The modelling of the inflationary epoch using scalar fields is described, and then results giving the form of perturbations produced by inflation are quoted. Finally, the current observational situation is briefly sketched. 2

Big bang problems and the idea of inflation

The standard hot big bang theory is an extremely successful one, passing some crucial observational tests of which I'd highlight five. • The expansion of the Universe. • The existence and spectrum of the cosmic microwave background radiation. • The abundances of light elements in the Universe (nucleosynthesis).

261

• That the predicted age of the Universe is comparable to direct age measurements of objects within the Universe . • That given the irregularities seen in the microwave background by COBE, there exists a reasonable explanation for the development of structure in the Universe, through gravitational collapse. In combination, these are extremely compelling. However, the standard hot big bang theory is limited to those epochs where the Universe is cool enough that the underlying physical processes are well established and understood through terrestrial experiment. It does not attempt to address the state of the Universe at earlier, hotter, times. Furthermore, the hot big bang theory leaves a range of crucial questions unanswered, for it turns out that it can successfully proceed only if the initial conditions are very carefully chosen. The assumption of early Universe studies is that the mysteries of the conditions under which the big bang theory operates may be explained through the physics occurring in its distant, unexplored past. If so, accurate observations of the present state of the Universe may highlight the types of process occurring during these early stages, and perhaps even shed light on the nature of physical laws at energies which it would be inconceivable to explore by other means.

2.1

A hot big bang reminder

To get us started, I'll give a quick review of the big bang cosmology. More detailed accounts can be found in any of a number of cosmological textbooks. One of my aims in this section is to set down the notation for the rest of the article.

2.2

Equations of motion

The hot big bang theory is based on the cosmological principle, which states that the Universe should look the same to all observers. That tells us that the Universe must be homogeneous and isotropic, which in turn tells us which metric must be used to describe it. It is the Robertson-Walker metric

(1) Here t is the time variable, and r-()-t/J are (polar) coordinates. The constant k measures the spatial curvature, with k negative, zero and positive corresponding to open, flat and closed Universes respectively. If k is zero or negative, then the range of r is from zero to infinity and the Universe is infinite, while

262 if k is positive then r goes from zero to I/Jk. Usually the coordinates are rescaled to make k equal to -1,0 or +1. The quantity a(t) is the scale-factor of the Universe, which measures its physical size. The form of a(t) depends on the properties of the material within the Universe, as we'll see. If no external forces are acting, then a particle at rest at a given set of coordinates (r, (J, ¢) will remain there. Such coordinates are said to be comoving with the expansion. One swaps between physical (ie actual) and co moving distances via physical distance

= a(t) x comoving distance.

(2)

The expansion of the Universe is governed by the properties of material within it. This can be specified a by the energy density p(t) and the pressure p(t). These are often related by an equation of state, which gives p as a function of p; the classic examples are P

p

Radiation,

3

o

p

Non-relativistic matter.

(3) (4)

In general though there need not be a simple equation of state; for example there may be more than one type of material, such as a combination of radiation and non-relati vistic matter, and certain types of material, such as a scalar field (a type of material we'll encounter later which is crucial for modelling inflation) , cannot be described by an equation of state at all. The crucial equations describing the expansion of the Universe are 2

87r

k

3m~l

a

H = - - p - -2

Friedmann equation

(5)

p+3H(p+p)=O

Fluid equation

(6)

where overdots are time derivatives and H = a/a is the Hubble parameter. The terms in the fluid equation contributing to p have a simple interpretation; the term 3H p is the reduction in density due to the increase in volume, and the term 3H p is the reduction in energy caused by the thermodynamic work done by the pressure when this expansion occurs. These can also be combined to form a new equation 47r -Iia = - 3m - 2 (p+ 3p)

Acceleration equation

(7)

pl

aI follow standard cosmological practice of setting the fundamental constants c and 1i equal to one. This makes the energy density and mass density interchangeable (since the former is c2 times the latter). I shall also normally use the Planck mass mpl rather than the gravitational constant Gj with the convention just mentioned they are related by G == mpl2.

263 in which k does not appear explicitly.

2.3

Standard cosmological solutions

When k = 0 the Friedmann and fluid equations can readily be solved for the equations of state given earlier, leading to the classic cosmological solutions Matter Domination Radiation Domination

p

= 0:

p

= p/3:

P ex: a- 3

P ex: a-

4

a(t) ex: t 2 / 3 a(t) ex: t 1 / 2

(8) (9)

In both cases the density falls as r2. When k = 0 we have the freedom to rescale a and it is normally chosen to be unity at the present, making physical and comoving scales coincide. The proportionality constants are then fixed by setting the density to be Pa at time ta, where here and throughout the subscript zero indicates present value. A more intriguing solution appears for the case of a so-called cosmological constant, which corresponds to an equation of state p = -po The fluid equation then gives p = 0 and hence p = Pa, leading to

a(t) ex: exp (Ht)

(10)

More complicated solutions can also be found for mixtures of components. For example, if there is both matter and radiation the Friedmann equation can be solved be using conformal time T = f dt/a, while if there is matter and a non-zero curvature term the solution can be given either in parametric form using normal time t, or in closed form with conformal time.

2.4

Critical density and the density parameter

The spatial geometry is flat if k = O. For a given H, this requires that the density equals the critical density

(11) Densities are often measured as fractions of Pc:

n(t) ==

!!... . pc

(12)

The quantity n is known as the density parameter, and can be applied to individual types of material as well as the total density.

264

The present value of the Hubble parameter is still not that well known, and is normally parametrized as

Ho

h = 100h kms- 1 Mpc- 1 = 3000 Mpc- 1 ,

(13)

where h is normally assumed to lie in the range 0.5 ::; h ::; O.B. The present critical density is

2.5

Characteristic scales and horizons

The big bang Universe has two characteristic scales • The Hubble time (or length) H- 1 . • The curvature scale

alkl- 1 / 2 .

The first of these gives the characteristic timescale of evolution of a(t), and the second gives the distance up to which space can be taken as having a flat (Euclidean) geometry. As written above they are both physical scales; to obtain the corresponding comoving scale one should divide by a(t). The ratio of these scales actually gives a measure of 0; from the Friedmann equation we find

(15) A crucial property of the big bang Universe is that it possesses horizons; even light can only have travelled a finite distance since the start of the Universe t., given by t dt (16) dH(t) = a(t) t. a(t) .

i

For example, matter domination gives dH(t) = 3t = 2H- 1 . In a big bang Universe, dH(to) is a good approximation to the distance to the surface of last scattering (the origin of the observed microwave background, at a time known as 'decoupling'), since to » tdec. 2.6

Redshift and temperature

The redshift measures the expansion of the Universe via the stretching of light

1+ z

=

a(to) a (temission)

.

(17)

265 Redshift can be used to describe both time and distance. As a time, it simply refers to the time at which light would have to be emitted to have a present redshift z. As a distance, it refers to the present distance to an object from which light is received with a redshift z. Note that this distance is not necessarily the time multiplied by the speed of light, since the Universe is expanding as the light travels across it. As the Universe expands, it cools according to the law 1 T ex - . (18) a

In its earliest stages the Universe may have been arbitrarily hot and dense. 2.7

The history of the Universe

Presently the Universe is dominated by non-relativistic matter, but because radiation reduces more quickly with the expansion, this implies that at earlier times the Universe was radiation dominated. During the radiation era temperature and time are related by t - '" (1010K)2 -1 sec T-

(19)

The highest energies accessible to terrestrial experiment, generated in particle accelerators, correspond to a temperature of about 10 15 K, which was attained when the Universe was about 10- 10 sec old. Before that, we have no direct evidence of the applicable physical laws and must use extrapolation based on current particle physics model building. After that time there is a fairly clear picture of how the Universe evolved to reach the present, with the key events being as follows:

• 10- 4 seconds: Quarks condense to form protons and neutrons. • 1 second: The Universe has cooled sufficiently that light nuclei are able to form, via a process known as nucleosynthesis. • 10 4 years: The radiation density drops to the level of the matter density, the epoch being known as matter-radiation equality. Subsequently the Universe is matter dominated. • 10 5 years: Decoupling of radiation from matter leads to the formation of the microwave background. This is more or less coincident with recombination, when the up-to-now free electrons combine with the nuclei to form atoms. • 10 10 years: The present.

266

3

Problems with the Big Bang

In this section I shall quickly review the original motivation for the inflationary cosmology. These problems were largely ones of initial conditions. While historically these problems were very important, they are now somewhat marginalized as focus is instead concentrated on inflation as a theory for the origin of cosmic structure. 3.1

The flatness problem

Taking advantage of the definition of the density parameter, and ignoring a possible cosmological constant contribution, the Friedmann equation can be written in the form

In - 11 = _lk_1 . 2

(20)

a H2

During standard big bang evolution, a 2 H2 is decreasing, and so from one, for example Matter domination: Radiation domination:

n moves away

In - 11 ex: t 2 / 3 In - 11 ex: t

(21) (22)

where the solutions apply provided n is close to one. So n = 1 is an unstable critical point. Since we know that today n is certainly within an order of magnitude of one, it must have been much closer in the past. Inserting the appropriate behaviours for the matter and radiation eras (or if you like just assuming radiation domination all the way to the present) gives nucleosynthesis (t

~

electro-weak scale (t'"

1 sec) :

10- 11

sec) :

In - 11 < 0(10- 16 ) In - 11 < 0(10- 27 )

(23)

(24)

That is, hardly any choices of the initial density lead to a Universe like our own. Typically, the Universe will either swiftly recollapse, or will rapidly expand and cool below 3K within its first second of existence. 3.2

The horizon problem

Microwave photons emitted from opposite sides of the sky appear to be in thermal equilibrium at almost the same temperature. The most natural explanation for this is that the Universe has indeed reached a state of thermal equilibrium, through interactions between the different regions. But unfortunately in the big bang theory this is not possible. There was no time for

267 those regions to interact before the photons were emitted, because of the finite horizon size, td dt dt

I

t.

· V

--2 m p1 4>;

V' dj

and a

ajexp

= aj

at t

[1¥

7f m

---

3

mpi

",. _ m mpi ~t ( 'l'lt

y487f

2)] ,

(51)

= 0) and the total amount of inflation is

N tot

_

-

27f

II'>r

-2- m p1

1

-2 .

(52)

This last equation can be obtained from the solution for a, but in fact is more easily obtained directly by integrating Eq. (47), for which one needn't bother to solve the equations of motion. In order for classical physics to be valid we require V « m~l' but it is still easy to get enough inflation provided m is small enough. As we shall later see, m is in fact required to be small from observational limits on the size of density perturbations produced, and we can easily get far more than the minimum amount of inflation required to solve the various cosmological problems we originally set out to solve.

5.6

Reheating after inflation

During inflation, all matter except the scalar field (usually called the inflaton) is redshifted to extremely low densities. Reheating is the process whereby the inflaton's energy density is converted back into conventional matter after inflation, re-entering the standard big bang theory. Once the slow-roll conditions break down, the scalar field switches from being overdamped to being underdamped and begins to move rapidly on the Hubble timescale, oscillating at the bottom of the potential. As it does so, it decays into conventional matter. The details of reheating are an important area of research in inflationary cosmology at the moment for several reasons, but are not important for the generation and evolution of density perturbations

278 which is the main focus of the remainder of this article. Consequently, I'll just note that recently there has been quite a dramatic change of view as to how reheating takes place. Traditional treatments (e.g. as given in Kolb & Turner 7) added a phenomenological decay term; this was constrained to be very small and hence reheating was viewed as being very inefficient. This allowed su bstantial red shifting to take place after the end of inflation and before the Universe returned to thermal equilibrium; hence the reheat temperature would be lower, by several orders of magnitude, than suggested by the energy density at the end of inflation. This picture is radically revised in work by Kofman, Linde & Starobinsky 8 (see also Ref. 9), who suggest that the decay can undergo broad parametric resonance, with extremely efficient transfer of energy from the coherent oscillations of the inflaton field. This initial transfer has been dubbed preheating. With such an efficient start to the reheating process, it now appears possible that the reheating epoch may be very short indeed and hence that most of the energy density in the inflaton field at the end of inflation may be available for conversion into thermalized form. 5.7

The range of inflation models

Over the last fifteen years or so a great number of inflationary models have been devised, both with and without reference to specific underlying particle theories. Here I will discuss a very small subset of the models which have been introduced, just to give you a flavour of the variety. At the moment particle physics model building of inflation is undergoing a renaissance, and a detailed snapshot of the current situation can be found in the review of Lyth & Riotto.lO However, as we shall be discussing in the next section, observations have great prospects for distinguishing between the different inflationary models. By far the best type of observation for this purpose appears to be high resolution satellite microwave background anisotropy observations, and we are fortunate that two proposals have been approved - NASA has funded the MAP satellite 11 for launch around 2000, and ESA has approved the PLANCK satellite 12 for launch some later. These satellites should offer very strong discrimination between the inflation models I shall now discuss. Indeed, it may even be possible to attempt a more challenging type of observation - one which is independent of the particular inflationary model and hence begins to test the idea of inflation itself.

Chaotic inflation models This is the standard type of inflation mode1.6 The ingredients are

279 • A single scalar field, rolling in ... • A potential V (¢», which in some regions satisfies the slow-roll conditions, while also possessing a minimum with zero potential in which inflation is to end. • Initial conditions well up the potential, due to large fluctuations at the Planck era. There are a large number of models of this type. Some are Polynomial chaotic inflation Power-law inflation 'Natural'inflation Intermediate inflation

V(¢» V(¢» V(¢» V(¢» V (¢»

= ~m2¢>2 = >..¢>4 = Voexp(j¥-~) = Vorl

+ cos J]

ex: ¢> - {J

Some of these actually do not satisfy the condition of a minimum in which inflation ends; they permit inflation to continue forever. However, we shall see power-law inflation arising in a more satisfactory context shortly.

Multi-field theories A recent trend in inflationary model building has been the exploration of models with more than one scalar field. The classic example is the hybrid inflation model,13 which seems particularly promising for particle physics model building. The simplest version has a potential with two fields ¢> and '1jJ of the form

(53) which is illustrated in Figure 4. When ¢>2 is large, the minimum of the potential in the '1jJ-direction is at '1jJ = O. The field rolls down this 'channel' until it reaches ¢>rnst = >..M2 / >..', at which point '1jJ = 0 becomes unstable and the field rolls into one of the true minima at ¢> = 0 and '1jJ = ±M. While in the 'channel', which is where all the interesting behaviour takes place, this is just like a single field model with an effective potential for ¢> of the form

(54) This is a fairly standard form, the unusual thing being the constant term, which would not normally be allowed as it would give a present-day cosmological

280

Figure 4: The potential for the hybrid inflation model. The field rolls down the channel at 'IjJ = 0 until it reaches the critical ¢ value, then falls off the side to the true minimum at ¢ 0 and 'IjJ ±M.

=

=

constant. The most interesting regime is where that constant dominates, and it gives quite an unusual phenomenology. In particular, the energy density during inflation can be much lower than normal while still giving suitably large density perturbations, and secondly the field ¢ can be rolling extremely slowly which is of benefit to particle physics model building. Within the more general class of two and multi-field inflation models, it is quite common for only one field to be dynamically important, as in the hybrid inflation model - this effectively reduces the situation back to the single field case of the previous subsection. However, it may also be possible to have more than one important dynamical degree of freedom. In that case there is no attractor behaviour giving a unique route into the potential minimum, as in the single field case; for example, if the potential is of the form of an asymmetric bowl one could roll into the base down any direction. In that situation, the model loses some of its predictive power, because the late-time behaviour is not independent of the initial conditions.e

eOf course, there is no requirement that the 'true' physical theory does have predictive power, but it would be unfortunate for us if it does not.

281

Beyond general relativity Rather than introduce an explicit scalar field to drive inflation, some theories modify the gravitational sector of the theory into something more complicated than general relativity.14 Examples are • Higher derivative gravity (R + R2

+ .. J

• Jordan-Brans-Dicke theory. • Scalar-tensor gravity. The last two are theories where the gravitational constant may vary (indeed Jordan-Brans-Dicke theory is a special case of scalar-tensor gravity). However, a clever trick, known as the conformal transformation,15 allows such theories to be rewritten as general relativity plus one or more scalar fields with some potential. Often, only one of those fields is dynamical which returns us once more to the original chaotic inflation scenario! The most famous example is extended inflation.16 In its original form, it transforms precisely into the power-law inflation model that we've already discussed, with the added bonus that it includes a proper method of ending inflation. Unfortunately though, this model is now ruled out by observations~ Indeed, models of inflation based on altering gravity are much more constrained than other types, since we know a lot about gravity and how well general relativity works,14 and many models of this kind are very vulnerable to observations.

Open inflation

In the early 1990s, in the face of ever increasing evidence of a sub-critical matter density in the Universe, interest was refocussed on an idea which defies the original inflationary motivation and gives rise to a homogeneous but open Universe from inflation! Often in the past it has been declared that this is either impossible or contrived; however, it can be readily achieved in models with quantum tunnelling from a false vacuum (a metastable state) followed by a second inflationary stage.17 The tunnelling creates a bubble, and, incredibly, the region inside the expanding bubble looks just like an open Universe, with the bubble wall corresponding to the initial (coordinate) singularity. These models are normally referred to as 'open inflation' or 'single-bubble' models. So far it has turned out that such models are not all that easy to construct. 'That is, a genuinely open Universe with hyperbolic geometry and no cosmological constant.

282

These models are already very different from traditional inflation models, and subsequently an even bolder idea has been proposed,18 that an open Universe can be created via 'tunnelling from nothing' rather than from a preexisting inflationary phase. As I write this remains controversial. While both these types of open inflation models remain viable, they are considerably more complex than the standard inflation models, and at the moment not that well motivated as although observations continue to favour a low matter density, they also favour spatial flatness reintroduced by a cosmological constant. Therefore from now on I will restrict discussion to the single-field chaotic inflation models. 5.8

Recap

The main points of this long section were the following. • Cosmological scalar fields, which were introduced long before inflation was thought of, provide a natural framework for inflation. • Despite a wide range of motivations, most inflationary models are dynamically equivalent to general relativity plus a single scalar field with some potential V ( 5 or a massive galaxy cluster at z > 2 this would be another indication that the current library of cosmological models is deficient. New data sets such as SDSS and 2DF are urgently needed to verify whether the shape discrepancies in P( k) will persist.

319

6.4

Large-Scale Peculiar Velocities

Large-scale bulk flows may bypass the uncertainty in using luminous galaxies as tracers of the density fluctuations on 10 - 100 Mpc scales. Surveys which count galaxies are unavoidably biased, depending on the selection criterion. One does not know a priori what the best selection is for an optically-selected sample. Large-scale peculiar velocities offer an intriguing alternative. One can directly infer bpI p. There are assumptions, of course. The Tully-Fisher relation must be assumed to provide a universal calibration of peculiar velocities. This yields the radial component of peculiar velocity. If the flows are assumed to be correlated, one can reconstruct the three-dimensional velocity potential and density distribution. An ideal example is the Mark III catalog of galaxy peculiar velocities, generated by A. Dekel and collaborators, from which an unbiased map of the mass distribution over 12 - 80h- 1 Mpc has been produced. The corresponding CMB map prediction of the appearance of the precursor of our local region to an observer located on some appropriate space-like hypersurface is shown in figure 2 30. The combination of intrinsic potential and out-of-phase Doppler components results in a differential filter that varies the map structure as a function of geodesic curvature. Identical features are dramatically reduced in scale for an open model as !lm is lowered, because of the angular-size redshift relation for a low !lm model. The result is a sensitive function of background curvature. With surveys such as 2DF or especially SDSS, one will have thousands of independent cubes 100 Mpc on the side either in redshift or in real space that one can compare with the comparable numbers of CMB maps of similar resolution, namely over 0.1- 1 degree, that will be produced by future experiments, both long duration balloon and satellite. This will provide a novel way of probing geodesic curvature. The CMB fluctuation maps are unbiased, whereas the galaxy surveys yield biased maps. Moreover, non-Gaussianity may be present in the CMB maps that is not apparent in the large-scale structure maps since the latter are a highly processed version of the former. By cross-correlating the two types of maps, one should be able to study bias and the possible development of non-Gaussianity over 10 - 100 Mpc scales. How typical is our local patch of the universe?

7

Galaxy Formation

The stable LSP is a natural motivation for cold dark matter-dominated cosmology. Freeze-out occurred early at kT > 1 GeV. By matter-domination,

320 at kT rv 10eV, the particle velocity dispersion is negligible relative to that of any virialization velocity on scales of interest for structure formation. Hence bottom-up formation occurs. The primordial fluctuation spectrum is expected to be nearly scale-invariant, as a consequence of inflation-amplified quantum fluctuations. Cold dark matter allows growth of fluctuations on sub-horizon scales at a logarithmic rate during radiation-domination, and at a rate proportional to (1 + z)-l during matter-domination. The theory of galaxy formation commences with inflationary fluctuations, follows their linear evolution, and simulates the full non-linear development of structure formation. The aim is to account for the detailed properties of young galaxies in the early universe, as well as those of their massive counterparts in our vicinity. Ab initio galaxy formation theory has had mixed results: both spectacular successes and persistent failures. The distribution of galaxy luminosities is described by a luminosity function ¢(L) that has three parameters: number density normalization ¢., characteristic luminosity L., and asymptotic slope a at luminosity L ~ L. so that ¢(L) = ¢.(L./ L)Oi exp( -L/ L.). Analytic theory provides the mass function of the first nonlinear objects to form by gravitational instability of a local Gaussian density fluctuation field. The resulting Press-Schechter mass function has an exponential cut-off at M. and a ~ 2. The mass M. corresponds to that of objects that are currently entering the nonlinear regime. These are groups of galaxies. To derive the mass function of galaxies, one requires the baryons to have undergone cooling. This effectively sets a redshift threshold, and objects that satisfy the cooling condition have masses on the order of those of the most massive galaxies. However continued accretion means that there is not a sharp cut-off in galaxy mass. Tidal torquing between adjacent, quasilinear fluctuations results in acquisition of a dimensionless angular momentum per galaxy halo that is comparable to what is inferred from observations of disks with simple assumptions about baryonic collapse and angular momentum conservation within the dark halo. Excellent agreement with the general form of rotation curves for massive galaxies is founcfll . On larger scales, structure formation ab initio accounts for galaxy clustering as described by the correlations in the galaxy distributiorr 2 , and for the abundance and structure of galaxy clusterg33. A notable accomplishment has been the explanation of the distribution of intergalactic gas as measured in Lyman alpha cloudg34 in the redshift range z = 2 - 4. However there are some less well understood issues. The superficial success in deriving a characteristic cold gas mass, and inferred luminosity, that corresponds to L. in the Schechter function for the distribution of galaxy lumi-

321

nosities has not held up. The deeper the cold dark matter potential well, the greater is the mass of cooled gas. Cooling must be terminated. The predicted slope of a = 2 is steeper than the observed slope of a ~ 1 for red-selected galaxies. Interestingly, blue-selected galaxy samples have an upturn (a ~ -1.5) at MB ~ - 16. Infall of all or most of the diffuse matter surrounding forming galaxies results in a ratio of mass-to-Iuminosity that is excessive, relative to the mass-luminosity ratios inferred for spirals or ellipticals from the normalization of the Tully-Fisher and fundamental plane correlations, respectively. Appeal must be sought to feedback from star formation and death that effectively makes star formation progressively less efficient in lower mass galaxies. Various feedback self-regulation models have been explored with differing degrees of success. Star formation prescriptions have been applied that account for the observed luminosity function, but at the price of anomalously high mass-to-Iuminosity ratios for bright galaxies. Conversely, models that satisfy the observed mass-to-Iuminosity ratios cannot easily be reconciled with the bright galaxy normalisation (and even low mass tail) of the galaxy luminosity functiorr 5 • Another, potentially more serious, difficulty with galaxy formation models has arisen with the derivation of the size of galaxy disks in hydro dynamical simulation$36 • Clumpy collapse and baryonic cooling result in effective dynamical friction on the dense baryonic clumps and efficient angular momentum transfer. The resulting disk size is found from simulations to be about 20 percent of that predicted for a homogenous collapse of baryons in an initially slowly rotating dark halo, with initial rotation characterized by the dimensionless angular momentum parameter A predicted from tidal torque theory to be about 0.07. A typical disk half-mass radius is found to be ~ 0.2A R;, where Ri is the dark halo radius of ~ 50 - 100 kpc, and is far smaller than the ~ 5 - 10 kpc measured for typical disks. Presumably feedback associated with star formation will help resolve the disk crisis, but a plausible and generic prescription for this remains to be developed. For almost all of these unresolved issues, baryons appear to be the source both of the problem and of the potential solution, provided that a suitable prescription for star formation and associated feedback can be implemented. The problem of course is that a fundamental theory of star formation is lacking. We do not understand the details of star formation processes in the nearest region of massive star formation, Orion, let alone in the early universe or in the extreme conditions encountered for merging galaxies. The nature of star formation in protogalaxies is especially challenging. One cannot choose between the following alternatives in current star formation. Stellar masses are limited by feedback from protostellar energy release.

322

This drives an outflow that ulimately limits accretion. The mass of the parent gas clump determines the stellar mass: it provides the gas reservoir, and protostellar jets and outflows do not terminate the accretion. Perhaps the truth combines the two effects: molecular clouds are turbulent, while the turbulence is driven by outflows and also determines the spectrum of clump massses. One might hope that is the probable absence of magenetic fields and even heavy elements, formation of the first stars would have been a simpler process. But this is not obviously the case. Transfer of angular momentum must still be accomplished, and one has to appeal to disk instabilities to drive this. The physics is possibly less secure than in the situation of magnetised gas clumps. The first stars could have had a mass distribution very different from the present day initial mass function. One could argue equally convincingly that the primordial initial mass function was biased towards massive stars, or to low mass stars, or indeed that the first stars formed in highly exotic conditions in the vicinity of supermassive black holes. One confronts similarly large uncertainties in large-scale formation. Did spheroids form monolithically or in dribs and drabs? The efficiency of early star formation is unknown, and one cannot distinguish these possibilities from theoretical considerations alone. CDM theory certainly prefers a hierarchical approach in which stars form early in small galaxies that merge at late epochs to form the massive spheroids. However one measures high star formation rates at high redshift in for example the Lyman break galaxies at z = 3 4. Star formation rates in excess of 100 solar masses per year for relatively common objects are most easily reconcilable with the monolithic approach to protogalactic star formation. In the absence of any overriding theoretical guidance, a practical approach to galaxy formation appears to be to implement a theoretical prescription for star formation that is closely coupled to observational data on nearby starforming galaxies, an approach that in essence is semi-phenomenological. Of course this is also essentially the philosophy that one encounters in the ab initio approach. There however the cosmological issues are dominant and the calculations are sufficiently complex that, at least hitherto, only a token acknowledgement has been paid to the notion of incorporating star formation phenomenology. By building this in at the outset, one may hope to develop a more robust approach to the early universe. Acknowledgements I am grateful to my collaborators E. Gawiser, L. Amandola and S. Zaroubi for permission to use our joint results. I acknowledge support from NSF.

323 References 1. W. Freedman et al., in IAU Symposium 183, Cosmological Parameters and the Evolution of the Universe, in press (1998). 2. B. Chaboyer, Physics Reports, in press (1998). 3. D. Schramm and M. Turner, Rev. Mod. Phys. 70, 303 (1998). 4. S. Perlmutter et al., preprint (1998) 5. B. Schmidt et al., AJ, in press (1998) 6. K. Kuijken and G. Gilmore, ApJ 367, L9 (1991) 7. A. Gould, J. Bahcall and C. Flynn, ApJ 465, 759 (1996) 8. G. Worthey and D. Ottaviani, ApJS 111, 377 (1997) 9. N. A. Bahcall and R. Y. Cen, ApJ 407, L49 (1997) 10. A. Evrard, MNRAS 292, 289 (1997) 11. V. Eke, S. Cole and C. Frenk, MNRAS 282, 263 (1996) 12. Y. Sigad et al., ApJ 495, 516 (1998) 13. M. Davis, A. Nusser and J. Willick, ApJ 473,22 (1996) 14. T. Kolatt and A. Dekel, ApJ 479, 592 (1997) 15. S. BurIes and D. Tytler, ApJ 507, 744 (1998) 16. R. Cen and J. Ostriker, astro-ph/9806281 (1998) 17. A. Kravtsov et al., ApJ 502,48 (1998) 18. B. Moore et al., astroph/9711259 )(1997) 19. A. Burkert and J. Silk, ApJ 488,55 (1998) 20. B. D. Fields, G. A. Mathews and D. S. Schramm, ApJ 483, 625 (1997) 21. J. Ellis, Falk, T., K. Olive and M. Schmitt, Phys. Lett.B B388, 355 (1997) 22. J. Ellis, Falk, T. and K. Olive, preprint hep-ph/9810360 (1998) 23. B. Metcalf and J. Silk, ApJ 492L, 1 (1998) 24. S. Hancock et al., MNRAS 294L, 1 (1998) 25. E. Gawiser and J. Silk, Science 280, 1405 (1998) 26. N. A. Bahcall, X. H. Fan and R. Y. Cen, ApJ 485L, 53 (1997) 27. A. Blanchard, J. Bartlett and R. Sadat preprint astro-ph/9809182 (1998) 28. M. Balogh, A. Babul and D. R. Patton, preprint astro-ph/9809159 (1998) 29. L. Amendola et al., Phys. Lett.D 54, 7199 (1966) 30. S. Zaroubi et al., ApJ 490, 473 (1997) 31. J. Navarro, C. S. Frenk and S. D. M. White, ApJ 490, 493 (1997) 32. A. Jenkins et al., ApJ 499, 201 (1998) 33. P. Thomas et al., MNRAS 296, 106 (1998) 34. M. Rauch et al., ApJ 489, 7 (1997) 35. M. Steinmetz and J. Navarro, preprint astro-ph/9808076 (1998) 36. M. Steinmetz and E. Muller, MNRAS 276,549 (1995)

324

10 5

10 4

102

10 1

1 10- 3

10- 2 10- 1 k (h Mpc- 1 )

1

Figure 1: Constraints from LSS and CMB on ACDM model with inflationary bubbles added at k = O.06hMpc- 1 to primordial power spectrum.

325

....



"" -§ ~

8I

= .... I

-40

-20

0

20

40

Arcl.Yl.in.

a)

:; M1 > M2 > .. ·Mw,G J G(l) J G(2) J" ·SU(3)c0SU(2)L0U(1)y.

(94) What is the main motivation for invoking GUTs? Gauge couplings (couplings to gauge fields) are charactherized by a dimensionless constant g, or equivalently by a = g2/41r. (For electromagnetism, 9 is the electron charge and a evaluated at low energy is the fine structure constant aem = 1/137.) Gauge couplings are not supposed to be extremely small, and one should take 9 ~ 1 for crude order of magnitude estimates (making a one or two orders of magnitude below 1). Assuming small couplings, the perturbative effects usually dominate, and we focus on them for the moment. With perturbative quantum effects included, the effective masses and couplings depend on the relevant energy scale Q. The dependence on Q (called 'running') can be calculated through the renormalization group equations (RGE's), and is logarithmic. In the context of collider physics, Q can be taken to be the collision energy, if there are no bigger relevant scales (particle masses). For the Standard Model there are three gauge couplings, ai where i = 3,2,1, corresponding for respectively to the strong interaction (colour SU(3)c) the non-abelian electroweak interaction (SU(2)L) and electroweak hypercharge (U(l)y). (The electromagnetic gauge coupling is given by a- 1 = all + a 21 .) In the one-loop approximation,

354

ignoring the Higgs field, their running is given (at one loop) by do; b; 2 dln(Q2) = 411"0; .

(95)

The coefficients b; depend on the number of particles with mass « Q. Including all particles in the minimal supersymmetric standard model gives b1 = 11, b2 1 and b3 -3. Using the values of 0; measured by collider experiments at a scale Q ~ 100 MeV, one finds that all three couplings become equal at a scale 2,39,82:1 Q = MGUT, where

=

=

(96) The unified value is 0GUT

~

1/25.

(97)

One explanation of this remarkable experimental result may be that there is a GUT, involving a higher symmetry with a single gauge coupling, which is unbroken above the scale MGUT. Another might be that field theory becomes invalid above the unification scale, to be replaced by something like weakly coupled string theory or M-theory 124 which is the source of unification. At the time of writing there is no consensus about which explanation is correct, but in this section we will focus on Grand Unified Theories and their relevance for baryogenesis. It is a general property of GUTs that the same representation may contain both quarks and leptons and therefore there exist gauge bosons which mediate gauge interactions among fermions having different baryon number. This is not enough -though- to conclude that in GUTs the baryon number is violated, because it might be possible to assign a baryonic charge to the gauge bosons in such a way that each vertex boson-fermion-fermion the baryon number is

conserved. Let us discuss this crucial point in more detail. The fundamental fermions of the SM are

(1,2, -1/2), (3,2, 1/6), (1,1,1), (3,1, -2/3), (3,1,1/3),

fL QL

e'L C

UL

d'L =

=

=

(98)

aTo be precise, fOIl 012 013 QGUT, the factor 5/3 arising because the historical definition of OIl is not very sensible. In passing we note that the unification fails by many standard deviations in the absence of supersymmetry, which may be construed as evidence for supersymmetry and anyhow highlights the remarkable accuracy of the experiments leading to this result.

355 where in parenthesis we have written then SU(3)c ®SU(2)L ®U(I)y quantum numbers and all the spinors are left-handed. Given two spinors '!/JL and XL, it is possible to define a renormalizable coupling to a gauge boson VI' by

i '!/JilT!' XL V!,

+ h.c.,

(99)

where lTl' = (1, if) and if are the Pauli matrices. At this point one may try to write down all the couplings of the form (99) starting from the spinors of the SM and identify all the possible gauge bosons which may be present in a GUT having the same spinors of the SM. Of course, the same gauge boson may be coupled to more than one pair of spinors. If all the spinor pairs have the same baryon number B, then it suffices to assign a baryon number - B to the gauge boson and obtain a baryon number conserving theory. If there exist gauge bosons which couple to spinor pairs having different baryon number, one may write down baryon number violating interactions. These bosons are given in Table 1, where we have indicated, for every gauge boson, all the possible interactions and the corresponding baryon numbers B and baryon minus lepton numbers B - L Gauge boson

V

1

V2

= (3,2, -5/6) = (3,2,1/6)

Table 1 spinors u~TQL

QtLeCL £tLdCL £iui d~tQL

B

B-L

2/3 -1/3 -1/3 -1/3 2/3

2/3 2/3 2/3 2/3 2/3

Of course, every gauge boson listed in Table 1 has the corresponding antiboson. One can repeat the same procedure to identify the scalar bosons S which may mediate baryon number violation interactions via fermions. The generic coupling reads

(100) If we consider all the spinor pairs Xl '!/JL, even belonging to different families, we get the following possibilities

356

Table 2 Scalar boson

Sl

= (3,1, -1/3)

spmors

B

B-L

£tQl

-1/3 2/3 -1/3 2/3 -1/3 2/3 -1/3 2/3

2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3

QLQL

S-.!

= (3,1, -4/3)

S3

= (3,3, -1/3)

e1 u1 dct ct Lu L

e'}. d'}. u~tu~t

£lQl QLQL

Out of all possible scalar and gauge bosons which may couple to the fermions of the SM, only the five that we have listed may give rise to interactions which violate the baryon number. A crucial point for what we will be discussing in the following is that each of these bosons have the same combination B - L, which means that this combination may be not violated in any vertex boson-fermion-fermion. This is quite a striking result and originates only from having required the invariance under the SM gauge group and that the only fermions of the theory are those of the SM. The extension of the fermionic content of the theory may allow the presence of more heavy bosons which will possibly violate B and even B - L. In the Grand Unified Theories based on SO(10) - for instance - there is another fermion which is a singlet under the SM gauge group and is identified with the antineutrino N'i = (1,1,0). It carries lepton number equal to L = -1. It is possible to introduce a new scalar field S4 which may couple to N'iu1 and d~t d~t, thus violating the baryon number. It is remarkable that the choice for the lepton number of Nf leads to no new gauge boson which violates B - L. These considerations do not apply to supersymmetric models though (for a review see 56). Indeed, for every fermionic degree of freedom there exist a superpartner (squark or slepton) which does have the same quantum number. Furthermore, in the MinimalSupersymmetric Standard Model (MSSM) one has to introduce two Higgs doublets H1 = (1,2, -1/2) and H2 = (1,2,1/2) and the corresponding fermionic superpartners, the so-called higgsinos ii 1,2. Finally, every gauge boson has its own superparner, the gaugino. In this large zoo of new particles, one can easily find couplings that violate Band B - L. For instance, the higgsino ii1 may couple to the quark doublet QL and to the scalars Sl and S3 of the Table 2. The pair iii has baryon number B = -1/3 and B-L = -1/3 and both quantum numbers are not conserved. Nevertheless, in the supersymmetric models which are phenomenologically acceptable, even without considering the presence of superheavy particles, it is necessary to suppress some supersymmetric couplings which would lead at the weak scale

Q1

357 to a proton decay at a rate which is too fast for being in agreement with the tight experimental constraints. One commonly accepted solution is to introduce a discrete symmetry Z2, called R-parity, under which all the fields of the SM are even and all the superpartners are odd. The scalar component of any chiral supermultiplet has the following R-parity number

R

= (_1)3(B-L),

(101)

while the corresponding fermion has the same number multiplied by -1. If we impose that R-parity is exact, then it is easy to check that, besides suppressing the fast proton decay at the weak scale, one avoids the presence Band L violating couplings of heavy fields with the light fermionic fields of the MSSM. Indeed, all the heavy bosons of Tables 1 and 2 have R-parity R = 1, while for instance - the pair has parity R = -1. Similar considerations apply to other fermionic pairs. We conclude that in the GUTs, both supersymmetric and non-supersymmetric, no B - L asymmetry may be generated through the out-ofequilibrium decay of gauge boson fields. We will mention in the following though - that the generation of such an asymmetry is possible in the framework of (supersymmetric) SO(10) via the out-of-equilibrium decay of the right-handed (s)neutrino, i.e. via the decay of a superheavy fermion (scalar). After having learned that GUTs are the perfect arena for baryon number violating interactions, we will illustrate now some features of the out-ofequilibrium decay scenario within some specific GUTs, like SU(5) and SO(10).

iii Qt

The case of SU(5) The gauge group SU(5) is the smallest group containing the SM gauge group and as such it represents the most appealing candidate to build up a Grand Unified Theory. The non-supersymmetric version of SU(5) is - however already ruled out by its prediction of the proton lifetime Tp ' " 1030 years, which is in disagreement with the experimental lower bound Tp ;::. 1032 years 9. Recent precise measurements of coupling constants at LEP suggest that the supersymmetric extension of SU(5) gives a consistent picture of coupling unification 2,39,82 and is a viable possibility. The fermionic content of SU(5) is the same as the one in the SM. Therefore, as we explained in Section 4.3, it is not possible to create any asymmetry in B - L. Fermions are assigned to the reducible representation 5J EEl 10 J as (102)

358

and (103)

There are 24 gauge bosons which belong to the adjoint representation 24v and may couple to the fermions through the couplings (104)

Among the 24 gauge bosons there are the bosons XY = VI = (3,2, -5/6) (and their CP-conjugate) which may decay violating the baryon number: XY -+ QL, QQ, where Q and L denotes an arbitrary quark and lepton, respectively. They have electric charges Qx = -1/3 and Qy = -4/3. The mass and the couplings of these bosons are determined by the gauge coupling unification Mxy Mxy

5 x 10 14 GeV, 10

16

GeV,

O'GUT

O'GUT

~ 1/45, non - supersymmetric SU(5),

~ 1/24, supersymmetric SU(5).

(105)

While in the gauge sector the structure is uniquely determined by the gauge group, in the Higgs sector the results depend upon the choice of the representation. The Higgs fields which couple to the fermions may be in the representation 5H or in the representations 10H, 15H, 45H and 50H. If we consider the minimal choice 5H, we obtain (106)

where hU,D are matrices in the flavor space. The representation 5H contains the Higgs doublet of the SM, (1,2,1/2) and the triplet S1 = (3,1, -1/3) which is B-violating. Unfortunately, this minimal choice of the Higgs sector does not suffice to explain the baryon number of the Universe. The CP violation is due to the complex phases which cannot be reabsorbed by field redifinition (they are physical) in the Yukawa sector. At the tree-level these phases do not give any contribution to the baryon asymmetry and at the one-loop level the asymmetry is proportional to (107)

where the trace is over generation indices. This is because the Higgs on the external and internal legs of the one-loop interference diagrams is the same. A net baryon number only appears at three-loop, resulting in a baryon asymmetry '" 10- 16 which is far too small to explain the observed one. The same problem in present in the supersymmetric version of SU(5) where one has to introduce two Higgs supeerfields 5H and 5]r 55.

359 The problem of too tiny CP violation in SU(5) may be solved by complicating further the Higgs sector. One may introduce an extra scalar 5~ with the same quantum numbers of 5H, but with a different mass and/or lifetime 101. In that case one-loop diagrams with exchange of 5~ instead of 5H can give rise to a net baryon number proportional (108) where hU,D are the couplings of 5~ to QL and QQ, respectively. A second alternative is to introduce a different second Higgs representation. For example, adding a Higgs in the 45 representation of SU(5) an adequate baryon asymmetry may be producedb for a wide range of the parameters 57. The case of SO(10)

In the GUT based on SO(10) the spontaneous breaking down to the SM gauge group is generally obtained through different steps (for a general review, see 117). The main two channels are SO(lO)

M~T G 224 ~ G 214 ~ G 2113 M!!..t L SU(3)c ® SU(2)L ® U(l)y,

SO(10)

M~T G 224 ~

G2213

~ G 2113

M!!..t

L

SU(3)c ® SU(2)L ® U(l)y,

(109)

where G 224

SU(2)L ® SU(2)R ® SU(4),

G 214

SU(2)L ® U(1)I3R ® SU(4),

G2113 G2213

SU(2)L ® U(1)r3R ® U(l)B-L ® SU(3)c, SU(2)L ® SU(2)R ® U(l)B-L ® SU(3)c,

(110)

where the four intermediate scales have not to be necessarily different from each other. We notice that, - if we are interested in the generation of an asymmetry in B - L, the relevant scale is the scale at which the abelian group U(l)B-L breaks down, i.e. MB-L, and not the Grand Unification scale MGUT; - it is usually not possible to generate any baryon asymmetry at the scale MGUT. Indeed, the fermionic content of SO(10) is the one of the SM plus a right-handed neutrino N'L = (1,1,0). All the fermions belonging to the same generation are contained in the spino rial representation 16f. Differently from what happens for the case of SU(5), now all the fermions posses the corresponding antifermions and it is possible to define a conjugation operator

360 of the charge C starting from the operators of SO(10), in such a way that, if SO(10) is not broken, then C is conserved 96. In the simplest mechanism for the breaking of S0(10), the one into G 224 , a crucial role is played by the 54 H . In such a case there is a symmetry SU(2)L (59 SU(2)R 104,93,94 with equal coupling constants 9L and gR and consequently, C is still a symmetry of the theory. It is possible to see that, with this choice of the Higgs representation, C is not broken until U(I)B-L is broken, i.e. at the scale M B - L b. At this scale, the right-handed neutrino acquires a Majorana mass MN = O(MB_L) and its out-of-equilibrium decays may generate a nonvanishing B - L asymmetry 47. We will return to this point later. With a more complicated choice of the Higgs representation it is possible to break C at the scale MR where SU(2)R is broken and in such a case baryogenesis may take place at that scale. 5

The out-of-equilibrium decay scenario and the thermal history of the Universe

The out-of-equilibrium scenario that we have depicted in the previous section is operative only if a nonequilibrium number density of X heavy bosons was present in the early Universe. Usually massive particles are in equilibrium at at high temperatures, T » Mx and their number density exceeds the equilibrium one when T becomes of the same order of the mass Mx. We have seen that, if the decay rate is small enough around T ~ M x, see Eq. (58), then departure from equilibrium is attained and the subsequent decays of X and X particles may produce the observed baryon number asymmetry. The basic assumption - however - of this picture is that the superheavy bosons were as abundant as photons at very high temperatures T ;:: Mx. If the X particles are gauge or Higgs bosons of Grand Unification, the situation is somewhat more complicated because they might have never been in thermal equilibrium at the very early stages of the evolution of the Universe. Even if the temperature of the primeval plasma was higher than the Grand Unified scale MGUT ~ 10 16 GeV, the rate of production of superheavy particles would be smaller than the expansion rate of the Universe and the number density of superheavy bosons could always be smaller than the equilibrium one. Secondly, the temperature of the Universe might be always smaller than MGUT and correspondingly the thermally produced X bosons might be never as abundant as photons, making their role in baryogensis negligible. All these considerations depend crucially upon the thermal history of the Universe and deserve a closer look. bIn fact, if one uses the 210H to break 80(10), one can maintain the gauge part of the left-right symmetry but not C. Th eimplications for baryogenesis are discussed in 23.

361

5.1

Inflation and reheating: the old days

The flatness and the horizon problems of the standard big bang cosmology are elegantly solved if during the evolution of the early Universe the energy density happened to be dominated by some form of vacuum energy and comoving scales grow quasi-exponentially 54. An inflationary stage is also required to dilute any undesirable topological defects left as remnants after some phase transition taking place at early epochs. The vacuum energy driving inflation is generally assumed to be associated to the potential V(¢) of some scalar field ¢, the inflaton, which is initially displaced from the minimum of its potential. As a by-product, quantum fluctuations of the inflaton field may be the seeds for the generation of structure and the fluctuations observed in the cosmic microwave background radiation, 8T/T ~ 10- 583 ,90,84. Inflation ended when the potential energy associated with the inflaton field became smaller than the kinetic energy of the field. By that time, any preinflation entropy in the Universe had been inflated away, and the energy of the universe was entirely in the form of coherent oscillations of the inflaton condensate around the minimum of its potential. The Universe may be said to be frozen after the end of inflation. We know that somehow the low-entropy cold Universe dominated by the energy of coherent motion of the ¢ field must be transformed into a high-entropy hot Universe dominated by radiation. The process by which the energy of the inflaton field is transferred from the inflaton field to radiation has been dubbed reheating. In the old theory of reheating 35 ,1, the simplest way to envision this process is if the comoving energy density in the zero mode of the inflaton decays into normal particles, which then scatter and thermalize to form a thermal background. It is usually assumed that the decay width of this process is the same as the decay width of a free inflaton field. Of particular interest is a quantity known as the reheat temperature, denoted as TRH. The reheat temperature is calculated by assuming an instantaneous conversion of the energy density in the inflaton field into radiation when the decay width of the inflaton energy, r if> , is equal to H, the expansion rate of the universe. The reheat temperature is calculated quite easily. After inflation the inflaton field executes coherent oscillations about the minimum of the potential. Averaged over several oscillations, the coherent oscillation energy density redshifts as matter: P¢> ex a- 3 , where a is the Robertson-Walker scale factor. If we denote as PI and aJ the total inflaton energy density and the scale factor at the initiation of coherent oscillations, then the Hubble expansion rate as a

362 function of a is

H2(a) = 811" PI (a I )3 3 M~ a

(111)

Equating H(a) and f,,, leads to an expression for aI/a. Now if we assume that all available coherent energy density is instantaneously converted into radiation at this value of aI/a, we can find the reheat temperature by setting the coherent energy density, p

2 M2 g2 _ _ ~ g2 c 2 -E ~ g2 c 2

4MJ

MJ

X

10 12

»

1

(128)

and the resonance is broad. For certain values of the parameters (A, q) there are exact solutions Xk and the corresponding number density nk that grow exponentially with time because they belong to an instability band of the Mathieu equation (for a recent comprehensive review on preheating after chaotic inflation 70 and references therein)

(129) where the parameter J-lk depends upon the instability band and, in the broad resonance case, q » 1, it is ~ 0.2. These instabilities can be interpreted as coherent "particle" production with large occupancy numbers. One way of understanding this phenomenon is to consider the energy of these modes as that of a harmonic oscillator, Ek IXk 12 /2 + w~ IXk 12 /2 Wk (nk + 1/2). The occupancy number of level k can grow exponentially fast, nk ~ exp(2J-lkM",t) » 1, and these modes soon behave like classical waves. The parameter q during preheating determines the strength of the resonance. It is possible that the model parameters are such that parametric resonance does not occur, and then the usual perturbative approach would follow, with decay rate r ",. In fact, as the Universe expands, the growth of the scale factor and the decrease of the amplitude of inflaton oscillations shifts the values of (A, q) along the stability/instability chart of the Mathieu equation, going from broad resonance, for q » 1, to narrow resonance, q « 1, and finally to the perturbative decay of the inflaton.

=

=

368

It is important to notice that, after the short period of preheating, the Universe is likely to enter a long period of matter domination where the biggest contribution to the energy density of the Universe is provided by the residual small amplitude oscillations of the classical inflaton field and/or by the inflaton quanta produced during the back-reaction processes. This period will end when the age of the Universe becomes of the order of the perturbative lifetime of the inflaton field, t '" r;l. At this point, the Universe will be reheated up to a temperature TRH given in (112) obtained applying the old theory of reheating described in the previous section.

5.4

GUT baryogenesis and preheating

A crucial observation for baryogenesis is that even particles with mass larger than that of the inflaton may be produced during preheating. To see how this might work, let us assume that the interaction term between the superheavy bosons and the inflaton field is of the type g2¢>2IXI2. During preheating, quantum fluctuations of the X field with momentum k approximately obey the Mathieu equation where now

A{k)

=

k 2 +M2

M2 x

+ 2q.

(130)

cP

Particle production occurs above the line A = 2q. The width of the instability strip scales as q1/2 for large q, independent of the X mass. The condition for broad resonance 68,74 (131) becomes k

2

+ Ml < M2

",gM'

(132)

cP cP which yields for the typical energy of X bosons produced in preheating

Ei =

k

2

+ Ml

~gif>McP'

(133)

By the time the resonance develops to the full strength, if>2 '" 10- 5 M~. The resulting estimate for the typical energy of particles at the end of the broad resonance regime for McP '" 10 13 GeV is (134) Supermassive X bosons can be produced by the broad parametric resonance for Ex > Mx, which leads to the estimate that X production will be possible if Mx < gl/21015 GeV.

369

For g2 '" 1 one would have copious production of X particles (in this regime the problem is non-linear from the beginning and therefore g2 = 1 has to be understood as a rough estimate of the limiting case) as heavy as 10 15 GeV, i.e., 100 times greater than the inflaton mass. The only problem here is that for large coupling g, radiative corrections to the effective potential of the inflaton field may modify its shape at

0.2, and this critical value of r does not depend significantly upon mx or q 75. The most relevant case with q = 108 , where X-bosons as massive as ten times the inflaton mass can be created, is shown in Fig. 6 in the Hartree approximation. Note, that two lower curves which correspond to r equal to 0.08 and 0.12 never reach the limiting value (X2)max ,...., 1O-10M~, which is imposed by rescattering 65 , and the Hartree approximation ought to be reliable in this cases. As outlined above, one may consider a three part reheating process, with initial conditions corresponding to the frozen universe at the end of inflation. The first stage is explosive particle production, where a fraction 8 of the energy density at the end of preheating is transferred to X bosons, with (1- 8) of the initial energy remaining in ¢ coherent oscillation energy. We assume that this stage occurs within a few Hubble times of the end of inflation. The second stage is the X decay and subsequent thermalization of the decay products. We assume that decay of an X - X pair produces a net baryon number {, as well as entropy. Reheating is brought to a close in the third phase when the remaining energy density in ¢ oscillations is transferred to radiation. The final baryon asymmetry depends linearly upon the ratio 8 between the energy stored in the X particles at the end of the preheating stage and the energy stored in the inflaton field at the beginning of the preheating era 74. The description simplifies if we assume zero initial kinetic energy of the X s. One may also a assume that there are fast interactions that thermalize the massless decay products of the X. Then in a co-moving volume a3 , the total number of X bosons, Nx = nxa3, the total baryon number, NB = nBa 3, and the dimensionless radiation energy, R = PRa4, evolve according to Nx

.

.

R= -aMxNx;

374

10- 7 q=10 8 m x=10

10- 8 10- 9 10- 10 NO::

::g 10- 11

~

............. N

X

............

10- 12 10- 13 10- 14 10- 15 10- 16

0

2

4

6

8

10

12

14

16

18

T

Figure 6: The time dependence of the variance of X in the Hartree approximation with model parameters q = 108 , mx = 10 and for three values of r, from top to bottom: 0.04, 0.08,0.12.

375

101~~~~~~~~~~~~~~~~~~~~~

1 ............ :::::::.::.:.:.:............................ ······· ..........~~.a-3)/(p~aI3) 10- 1 10- 2

10- 3 10-4 10- 5

10-8 10-7 10-8 10- 9

..-

___ ------r--" : ":

~

"-

'--~-----

Figure 7: The evolution of the baryon number, the X number density, the energy density in oscillations, and the gravitino-to-entropy ratio as a function of the scale factor a.

376 (140) N~Q is the total number of X s in thermal equilibrium at temperature T (i), t) of the Heisenberg picture field operator J(i, t)

J(i, t)I4>(i), t)

= 4>(i)I4>(i), t).

(189)

Then the partition function may be written as "summation" over the eigenstates (190) z = 2:)4>(i), t = 0Ie- 1Jlf l4>(i), t = 0). 4>(x)

We can now make the analogy with the zero temperature case where in the language of field theory /

(4)''(i), t = Ole- iH (t"-t )I4>'(i), t = 0)

(4)''(i), t"I4>'(i), t') 0
Mq are partially reflected and partially trasmitted. In the reflection processes quarks and antiquarks acquire different probabilities of penetrating the bubble wall. In such a way, it might be possible to produce a net baryon number flux from outside to inside the bubble wall. For instance, considering momenta between M d and M s , then all the strange quarks might be reflected off, while down quarks have a nonvanishing probability of being transmitted. However, this effect is largely suppressed by the fact that fermions, when they propagate in the plasma, acquire a damping rate I '" 0.1 T » Ms and the quark energy and momenta cannot be defined exactly, but have a spread of the order of I » (Ms - Md). In other words, the lifetime of the quantum packet is much shorter than the typical reflection time from the bubble wall ('" 1/ Ms): C P violation, which is based on coherence and needs at least a time'" 1/Ms to be built up, cannot be efficient 49,59. Therefore, the common wisdom is that electroweak baryogenesis is not possible within the SM.

7.2

Electroweak baryogenesis in the MSSM

The most promising and well-motivated framework for electroweak baryogenesis beyond the SM seems to be supersymmetry (SUSY) 56,4. Let us remind the reader only a few notions about the MSSM that will turn out to be useful in the following. Let us consider the MSSM superpotential

where we have omitted the generation indices. The Higg sector contains the two Higgs doublets (240) The lepton Yukawa matrix he can be always taken real and diagonal while h U and h d contain the KM phase.

403

What is relevant for baryogenesis is to identify possible new sources of C P violation. They emerge from the operators which break softly supersymmetry i) Trilinear couplings: r

u

Hiju c+ r d HJ,JJc + r e H1Le" + h.c.,

(241)

where we have defined r(u,d,e)

== m3/2

A(u,d,e) . h(u,d,e).

(242)

Generally, in supergravity models the matrices A(u,d,e) are assumed to be proportional to the identiy matrix (243) where the A parameter can be complex. ii) bilinear couplings: (244)

iii) Majorana gaugino masses:

1

2 (M1A1Al + M2A2A2 + M3 A3A3) + h.c.

(245)

At the GUT scale it is usually assumed that (246)

ivy Scalar soft masses: (247) The new contributions to explicit violation of C P are given in the phases of the complex parameters A, B, Mi (i 1,2,3) and by the parameter p, in the superpotential (239). Two phases may be removed by redifining the phase of the superfield iI 2 in such a way that the phase of p, is opposite to that of B. The product p,B in (244) is therefore real. It is also possible to remove the phase of the gaugino mass M by an R symmetry transformation. The latter leaves all the other supersymmeric couplings invariant and only modifies the trilinear ones, which get multiplied by exp( -

0 is H{z)

+

(268)

where

..\± =

Vw±JV~+4rD

.

2D Imposing the continuity of H and H' at the boundaries, we find

(269)

From the form of the above equations one can see that C P violating densities are non zero for a time t '" D/v~ and the assumptions leading to the analytical

413

form of H (z) are valid provided that the interaction rates r t and r ss are larger than v3/ D 102,19. The equation governing the baryon asymmetry nB is given by 102 (271) where nL is the total number density of left-handed weak doublet fermions and we have assumed that the baryon asymmetry gets produced only in the symmetric phase. Expressing nL(z) in terms of the Higgs number density nL

= 9kq kT -

8kbkT - 5 k bk q H kH(kb + 9k q + 9kT )

(272)

and making use of Eqs. (267)-(271), we find that nB _ _ S

-

9

(k.)ADrsp I

2 VwS

'

(273)

where g(k;) is a numerical coefficient depending upon the light degrees of freedom present in the thermal bath. Eq. (273) summarizes all the ingredients we need to produce a baryon asymmetry in electroweak baryogenesis: 1) (the integral of) a CP violating source A, 2) baryon number violation provided by the sphaleron transitions with rate r sp and 3) out-of-equilibrium conditions provided by the expanding bubble wall. Out-of-equilibrium field theory with a broad brush

The next step in the computation of the baryon asymmetry is the evaluation of the C P violating sources for the right-handed stop number and the higgsino number. Non-equilibrium Quantum Field Theory provides us with the necessary tools to write down a set of quantum Boltzmann equations (QBE's) describing the local particle densities and automatically incorporating the C P violating sources. The most appropriate extension of the field theory to deal with these issues is to generalize the time contour of integration to a closed time-path (CTP). The CTP formalism is a powerful Green's function formulation for describing non-equilibrium phenomena in field theory, it leads to a complete non-equilibrium quantum kinetic theory approach and to a rigorous computation of the C P violating sources for the stop and the Higgsino numbers 109,110,111. What is more relevant, though, is that the CP violating sourcesand more generally the particle number changing interactions- built up from

414

the CTP formalism are characterized by "memory" effects which are typical of the quantum transport theory 32,58. C P violating sources are built up when right-handed stops and Higgsinos scatter off the advancing Higgs bubble wall and CP is violated at the vertices of interactions. In the classical kinetic theory the "scattering term" does not include any integral over the past history of the system. This is equivalent to assuming that any collision in the plasma does not depend upon the previous ones. On the contrary, the quantum approach reveals that the CP violating source is manifestly non-Markovian. We will now briefly present some of the basic features of the non-equilibrium quantum field theory based on the Schwinger-Keldysh formulation 115,61. The interested reader is referred to the excellent review by Chou et al. 22 for a more comprehensive discussion. Since we need the temporal evolution of the particle asymmetries with definite initial conditions and not simply the transition amplitude of particle reactions, the ordinary equilibrium quantum field theory at finite temperature is not the appropriate tool. The most appropriate extension of the field theory to deal with nonequilibrium phenomena amounts to generalize the time contour of integration to a closed-time path. More precisely, the time integration contour is deformed to run from - .p - G .p,

y)

8(x,y)G~(x,y)

(276)

G~(x,y) = G~ - G~ = G~ -G~.

For equilibrium phenomena, the brackets (- . -) imply a thermodynamic average over all the possible states of the system. While for homogeneous systems in equilibrium, the Green functions depend only upon the difference of their arguments (x, y) = (x - y) and there is no dependence upon (x + y), for systems out of equilibrium, the definitions (275) and (276) have a different meaning. The concept of thermodynamic averaging is now ill-defined. Instead, the bracket means the need to average over all the available states of the system for the non-equilibrium distributions. Furthermore, the arguments of the Green functions (x, y) are not usually given as the difference (x - y). For example, non-equilibrium could be caused by transients which make the Green functions depend upon (tx,ty) rather than (t x -ty). For interacting systems whether in equilibrium or not, one must define and calculate self-energy functions. Again, there are six of them: 1;t, ~f, 1;, E r and Ea. The same relationships exist among them as for the Green functions in (275) and (276), such as Er

= Et -

= E> -

E


= E< -

Ef.

(277)

The self-energies are incorporated into the Green functions through the use of Dyson's equations. A useful notation may be introduced which expresses four of the six Green functions as the elements of two-by-two matrices 31

- (Et - (GG> ±G t

G=

(278)

where the upper signs refer to bosonic case and the lower signs to fermionic case. For systems either in equilibrium or non-equilibrium, Dyson's equation is most easily expressed by using the matrix notation

where the superscript "0" on the Green functions means to use those for noninteracting system. This equation appears quite formidable; however, some simple expressions may be obtained for the respective Green functions. It is useful to notice that qyson's equation can be written in an alternate form, instead of (279), with GO on the right in the interaction terms,

Equations. (279) and (280) are the starting points to derive the quantum Boltzmann equations describing the temporal evolution of the CP violating particle density asymmetries. The quantum Boltzmann equations Our goal now is to find the QBE for the generic bosonic C P violating current (281) The zero-component of this current n¢> represents the number density of particles minus the number density of antiparticles and is therefore the quantity which enters the diffusion equations of supersymmetric electroweak baryogeneSlS.

418

Since the C P violating current can be expressed in terms of the Green function C~ (x, y) as (282) the problem is reduced to find the QBE for the interacting Green function c~ (x, y) when the system is not in equilibrium. This equation can be found

(ox

+m2) on both sides of the equation. Here from (279) by operating by m represents the bare mass term of the field cP. On the right-hand side, this operator acts only on G~

where I is the identity matrix. It is useful to also have an equation of motion for the other variable y. This is obtained from (280) by operating by +m2) on both sides of the equation. We obtain

(Oy

The two equations (283) and (284) are the starting point for the derivation of the QBE for the particle asymmetries. Let us extract from (283) and (284) the equations of motions for the Green function C~ (x, y)

(ox +m2) C~(x, y) C~ (x, y) (Oy +m2)

=

=

J J

d4X 3 [~~(X,X3)G~(X3'Y) -

~~(X,X3)G~(X3'Y)J, (285)

d4X3 [C¢(x, X3)~;(X3' y) - C;(x, X3)~~(X3' y)].

(286) If we now substract the two equations and make the identification x = y, the left-hand side is given by

I

.(X)

=-

f

d3 X3

IT

-00

-

> h.

G~E~]}.

(291)

Rearranging these terms

[>

dt3 E.p (X, X3)G.p< (X3, X)

-G~(X, x3)E~(X3' X) +Gi(X, x3)E~(X3' X) - Ei(X, X3)G~(X3' X)]. (292) This equation is the QBE for the particle density asymmetry and it can be explicitly checked that, in the particular case in which interactions conserve the number of particles and the latter are neither created nor destroyed, the number asymmetry n.p is conserved and obeys the equation of continuity on.p/oT+ V -i.p = O. During the production of the baryon asymmetry, however, particle asymmetries are not conserved. This occurs because the interactions themselves do not conserve the particle number asymmetries and there is some source of CP violation in the system. The right-hand side of -+

-

420 Eq. (292), through the general form of the self-energy Et{>, contains all the information necessary to describe the temporal evolution of the particle density asymmetries: particle number changing reactions and C P violating source terms, which will pop out from the corresponding self-energy Ecp. If the interactions of the system do not violate C P, there will be no C P violating sources and the final baryon asymmetry produced during supersymmetric baryogenesis will be vanishing. The kinetic Eq. (292) has an obvious interpretation in terms of gain and loss processes. What is unusual, however, is the presence of the integral over the time: the equation is manifestly non-Markovian. Only the assumption that the relaxation time scale of the particle asymmetry is much longer than the time scale of the non-local kernels leads to a Markovian description. A further approximation, i.e. taking the upper limit ofthe time integral to T -+ 00, leads to the familiar Boltzmann equation. The physical interpretation of the integral over the past history of the system is straightforward: it leads to the typical "memory" effects which are observed in quantum transport theory 32,58. In the classical kinetic theory the "scattering term" does not include any integral over the past history of the system which is equivalent to assume that any collision in the plasma does not depend upon the previous ones. On the contrary, quantum distributions posses strong memory effects and the thermalization rate obtained from the quantum transport theory may be substantially longer than the one obtained from the classical kinetic theory. As shown in 109,110,111 , memory effects play a fundamental role in the determination of the C P violating sources which fuel baryogenesis when transport properties allow the C P violating charges to diffuse in front of the bubble wall separating the broken from the unbroken phase at the electroweak phase transition. Notice that so far we have not made any approximation and the computation is therefore valid for all shapes and sizes of the bubble wall expanding in the thermal bath during a first-order electroweak phase transition. Let us now focus on the generic fermionic C P violating current. It reads (293)

,/1

represent the usual Dirac matrices. where 1/J indicates a Dirac fermion and Again, the zero-component of this current n", represents the number density of particles minus the number density of antiparticles and is therefore the relevant quantity for the diffusion equations of supersymmetric electroweak baryogenesis. We want to find a couple of equations of motion for the interacting fermionic Green function C",(x, y) when the system is not in equilibrium. Such equa-

421

tions may be found by applying the operators

(i fix -M)

and

(i /Jy +M)

on both sides of Eqs. (279) and (280), respectively. Here M represents the bare mass term of the fermion"p. We find

(i fix -M) C",(x,y) C",(x, y) (i f;y +M)

=

J(4)(x,y)4

+

= -8(4)(X, Y)4 -

Jd J

4x E",(x,X3)C",(X3,y),(294) 3

d4X3C",(X, x3)E",(X3, y). (295)

We can now take the trace over the spino rial indeces of both sides of the equations, sum up the two equations above and finally extract the equation of motion for the Green function G~

Tr{ [i ftx +i Iq G~(x,y)}

J

= d4x3Tr[1:~(X,X3)G~(X3'Y) - 1:~(X,X3)G~(X3'Y) -

G~ (x, X3)1:~(X3' y) + G~(x, X3)1:~ (X3, y)].

(296)

Making use of the centre-of-mass coordinate system, we can work out the lefthand side of Eq. (296)

Tr [i fix G~(T, X, t, r) + G~ (T, X, t, r)i f;y] It=r=o + 8~) i(1,b,I""p)lt=r=o 8 - 8Xp. ('I/J(XhP."p(X) i (8~

8 p. - 8Xp. J",.

(

297)

The next step is to employ the definitions in (276) to express the time-ordered functions G~, G~, 1:~, and 1:~ in terms of G~, G~, 1:~ and G~. The computation goes along the same lines as the analysis made in the previous section and we get 109,110,111

8n",(X) -+ 8T + V' -j",(X)

=

f

3

d i3

jT

-00

[

dt3Tr 1:~(X,X3)G~(X3'X)

-G~(X, X3)1:~(X3' X)+G~(X, X3)1:~(X3' X) -1:~(X, X3)G~(X3' X)]. (298) This is the "diffusion" equation describing the temporal evolution of a generic fermionic number asymmetry n",. As for the bosonic case, all the information regarding particle number violating interactions and C P violating sources are stored in the self-energy 1:",.

422

The C P violating source for higgsinos and the final baryon asymmetry As we mentioned, a strongly first order electroweak phase transition can be achieved in the presence of a top squark lighter than the top quark. In order to naturally suppress its contribution to the parameter I:!:..p and hence preserve a good agreement with the precision measurements at LEP, it should be mainly right-handed. This can be achieved if the left-handed stop soft supersymmetry breaking mass mQ is much larger than Mz. Under this assumption, however, the right-handed stop contribution to the baryon asymmetry results to be negligible. We will concentrate, therefore, only on the CP violating source for the Higgsino. The Higgs fermion current associated with neutral and charged Higgsinos can be written as J~ = fi,p. fi (299) H

where fi is the Dirac spinor

(300) and H2 fig (fit), HI = iff (fii) for neutral (charged) Higgsinos. The processes in the plasma which change the Higgsino number are the ones induced by the top Yukawa coupling and by interactions with the Higgs profile. The interactions among the charginos and the charged Higgsinos which are responsible for the C P violating source in the diffusion equation for the Higgs fermion number read

where Op. is the phase of the JL-parameter and we have indicated (HP(x)} by Vi(X), i = 1,2. Analogously, the interactions among the Bino, the W3-ino and the neutral Higgsinos are

To compute the source for the Higgs fermion number 'ii we perform a "Higgs insertion expansion" around the symmetric phase. At the lowest level of

423 perturbation, the interactions of the charged Higgsino induce a contribution to the self-energy of the form (and analogously for the other com ponent 8E~P, > ) H

8E~P,«x, y) H

= g~p(x, y)hCD.:.«x, y)PL + g~p(x, y)PRcD.:.«x, y)PR , W W

(303)

where g~p(x, y) g~p(x, y)

(304)

We have approximated the exact Green function of winos C w by the equilibrium Green function in the unbroken phase C~. This is because any departure W from thermal equilibrium distribution functions is caused at a given point by the passage of the wall and, therefore, is CJ(vw ). Since we will show that the source is already linear in v w , working with thermal equilibrium Green functions in the unbroken phase amounts to ignoring terms of higher order in vw . This is accurate as long as the bubble wall is moving slowly in the plasma. Similar formulae hold for the neutral Higgsinos. The dispersion relations of charginos and neutralinos are changed by high temperature corrections 123. Even though fermionic dispersion relations are highly nontrivial, especially when dealing with Majorana fermions 107, relatively simple expressions for the equilibrium fermionic spectral functions may be given in the limit in which the damping rate is smaller than the typical self-energy of the fermionic excitation 58. If we now insert the expressions (303) and (304) into the QBE (298), we get the CP violating source 109,110,111 I'ii

=-

Jd3X31~ dt3Tr[bE~P'>(X'X3)C~«X3'X)

- C~>(X x )bE~P,«x X) H ,3 H 3,

+ C~«X x )bEc:..P,>(x X) H' 3 H 3,

- bE~P,«X, X3)C~>(X3' X)],

(305)

which contains in the integrand the following function

g~p(X, X3) + g~p(X, X3) - g~P(X3' X) - g~P(X3' X) = 2i sin OIL [V2 (X)V1 (X3) - V1 (X)V2 (X3)],

(306)

which vanishes if Im(J-l) = 0 and if the tan (3( x) is a constant along the Higgs profile. In order to deal with analytic expressions, we can work out the thick wall limit and simplify the expressions obtained above by performing a derivative expansion

.(

)_

~~

an

.() (IL lL)n x3 - X .

V, X3 - ~ n! a(XIL)n v, X

(307)

424

The term with no derivatives vanishes in the expansion (307), V2(X)Vl (X) Vl(X)V2(X) = 0, which means that the static term in the derivative expansion does not contribute to the source. For a smooth Higgs profile, the derivatives with respect to the time coordinate and n > 1 are associated with higher powers of Vw I Lw, where Vw and Lw are the velocity and the width of the bubble wall, respectively. Since the typical time scale of the processes giving rise to the source is given by the thermalization time of the higgsinos l/rH, the approximation is good for values of Lwriilvw » 1. In other words, this expansion is valid only when the mean free path of the higgsinos in the plasma is smaller than the scale of variation of the Higgs background determined by the wall thickness, L w , and the wall velocity V w • The term corresponding to n = 1 in the expansion (307) gives a contribution to the source proportional to the function

which should vanish smoothly for values of X outside the bubble wall. Here we have denoted v2 == vi + v~. Since the variation of the Higgs fields is due to the expansion of the bubble wall through the thermal bath, the source 'ii will be linear in V w . The corresponding contribution tot he C P violating source reads

where

rW = H

roo dk_::-k_

J0 X

2 __

27r 2 w j[ww

[(1- 2Re(fW))I(wj[,fj[,ww ,fw ) + (1- 2Re(fR))

x I(ww,fw,wj[,fj[)

+ 2(Im(f~) + Im(jW))G(wj[,fj[,ww , f w )] (310)

andw~(W) = k 2 +1J.l1 2 (Mi) whilef~(W) = II [exp (wH(w/T+irii(w/T) + 1]. The functions I and G are given by

c] 1 1 . [ c] "2 [(a - c)2 + (b + d)2] sm 2arctan b + d ' 1

I(a,b,c,d)

1

. [

"2 [(a + c)2 + (b + d)2] sm

+

a

+

2arctan b + d a-

425

G(a, b, c, d)

=

1

1

[ a + c] 2arctan b + d

1

[ a - c] 2arctan b + d .

2 [(a + c)2 + (b + d)2] cos 1

2 [(a -

c)2

+ (b + d)2] cos

(311)

Notice that the function G(Wjj, rjj,W w ' rw) has a peak for Wjj '" w w . This resonant behaviour is associated to the fact that the Higgs background is carrying a very low momentum (of order of the inverse of the bubble wall width Lw) and to the possibility of absorption or emission of Higgs quanta by the propagating supersymmetric particles. The resonance can only take place when the higgsino and the wino do not differ too much in mass. By using the Uncertainty Principle, it is easy to understand that the width of this resonance is expected to be proportional to the thermalization rate of the particles giving rise to the baryon asymmetry. The damping rate of charged and neutral Higgsinos is expected to be of the order of 5 x 1O- 2 T. The Bino contribution may be obtained from the above expressions by replacing M2 by MI. The CP violating source for the Higgs fermion number is enhanced if M 2 , MI '" J.l and low momentum particles are transmitted over the distance Lw. This means that the classical approximation is not entirely adequate to describe the quantum interference nature of C P violation and only a quantum approach is suitable for the computation of the building up of the C P violating sources. Notice that the source is built up integrating over all the history of the system. This leads to "memory effect" that are responsible for some enhancement of the final baryon asymmetry. These memory effects lead to "relaxation" times for the C P violating sources which are typically longer than the ones dictated by the thermalization rates of the particles in the thermal bath. In fact, this observation is valid for all the processes described by the "scattering" term in the right-handed side of the quantum diffusion equations. The slowdown of the relaxation processes may help to keep the system out of equilibrium for longer times and therefore enhance the final baryon asymmetry. There are two more reasons why one should expect quantum relaxation times to be longer than the ones predicted by the classical approach. First, the decay ofthe Green's functions as functions of the difference of the time arguments: an exponential decay is found in thermal equilibrium when one ignore the frequency dependence of self-energies in the spectral functions, e.g. IG> (k, t, t')1 '" IG> (k)1 x exp [-r(k, w) It - t'I]. The decay of the Green's functions restrict the range of the time integration for the scattering term, reduces the integrals and, therefore, the change of the local particle number densities as a function of time. The second effect is the rather different oscillatory behaviour of the functions G> and G< for a given

426 momentum, as functions of the time argument difference. As we have previously mentioned, the final baryon asymmetry (273) depends sensitively on the parameter A. The parameter A computed from the higgsino source is

A

ex:

I

2!(k i )fii I DA+ '

roo du v2(u) df3(u) e-,),+" ~ roo du v2(u) d{3(u) ,

10

du

10

du

(312)

where !(k;) is a coefficient depending upon the number of degrees of freedom present in the thermal bath. The integral I has been computed including twoloop effects in ref. 106 and results to be I ~ 10- 2 for rnA = 150-200 GeV. The final baryon asymmetry turns out to be 110 nB '" S

-

(lsin( and radiation energy density, PRo We will assume that the decay rate of the inflaton field energy density is r 4>. We will also assume that the light degrees of freedom are in local thermodynamic equilibrium. With the above assumptions, the Boltzmann equations describing the redshift and interchange in the energy density among the different components IS

pq, + 3H P4> + r 4>P4> = 0 PR + 4H PR - r 4>P4> = 0,

(314)

where dot denotes time derivative. It is useful to introduce the dimensionless constant, of r 4> as r 4> = (}:4>M4>' For a reheat temperature much smaller than M4>,

r 4>

(}:

defined in terms

must be small.

(315)

428 It is also convenient to work with dimensionless quantities that can absorb the effect of expansion of the universe. This may be accomplished with the definitions ;we,. M-1 3. (316) 'I' = p¢ ¢ a , It is also convenient to use the scale factor, rather than time, for the independent variable, so we define a variable x aM¢. With this choice the system of equations can be written as (prime denotes d/dx)

=

~'

R' The constant

C1

(317)

is given by (318)

It is straightforward to solve the system of equations in Eq. (317) with initial conditions at x = XI of R(XI) = 0 and ~(xI) = ~I. It is convenient to express p¢(x = xI) in terms of the expansion rate at XI, which leads to

(319) The numerical value of XI is irrelevant. Before solving the system of equations, it is useful to consider the earlytime solution for R. Here, by early time, we mean H » f 1, in the early-time regime T scales as a- 3 / B . So entropy is created in the early-time regime. So if one is producing a massive particle during reheating it is necessary to take into account the fact that the maximum temperature is greater than TRH, and during the early-time evolution T ex a- 3 / B .

=

3) The euclidean action is given by (324) where R is the radius of the bubble. If we now indicate by 8R the thickness of the bubble wall and ~V = V«h) - V(4)>d < 0, we get

S ,...,2 R28R(84)>)28R 3 7r 8R + where 84» = 4»2 In such a case

- 4»1.

47rR3~V 3

'

(325)

Suppose now that the bubbles are thick, that is 8R ,..., R.

S3"'" 27rR(84)>)2 +

47rR3~V 3 .

(326)

The critical radius Rc is obtained as the maximum of the action (326)

R,..., c

84»

v'-2~V·

(327)

430

4) We introduce a chemical potential for any particle which takes part to fast processes, and then reduce the number of linearly independent chemical potentials by solving the corresponding system of equations. Finally, we can express the abundances of any particle in equilibrium in terms of the remaining linear independent chemical potentials, corresponding to the conserved charges of the system. Since strong interactions are in equilibrium inside the bubble wall, we can chose the same chemical potential for quarks of the same flavour but different color, and set to zero the chemical potential for gluons. Moreover, since inside the bubble wall 5U(2)L x U(1)y is broken, the chemical potential for the neutral Higgs scalars vanisheS'. The other fast processes, and the corresponding chemical potential equations are: i) top Yukawa: tL

+ Hg

h + H+

f-t tR f-t tR

+ g, + g,

Pt R)' PbL + PH+),

(328)

ii) 5U(2)L flavour diagonal: (Pv'L (Pu~ (PH+ (PH-

= = = =

Pe~ Pd'

L

+ Pw+)' + Pw+),

(i=1,2,3).

Pw+ ), -Pw+ ),

(329)

Neutral current gauge interactions are also in equilibrium, so we have zero chemical potential for the photon and the Z boson. Imposing the above constraints, we can reduce the number of independent chemical potentials to four, Pw+, PtL' PUL == 1/2 2::7=1 pu~, and PeL == 1/3 2::~=1 Pe~· These quantities correspond to the four linearly independent conserved charges of the system. Choosing the basis Q, (B - L), (B + L), and BP == B3 - 1/2(B 1 + B 2 ), where the primes indicate that only particles in equilibrium contribute to the various charges, and introducing the respective chemical potentials, we can go to the new basis using the relations PQ P(B-L) {

P(B+L)

PBP

3PtL + 2pUL - 3peL + llpw+, 3PtL + 4pUL - 6peL - 6pw+, 3PtL + 4pUL + 6peL' 3PtL - 2pUL·

CThis is true if chirality flip interactions, or processes like Z

~

(330)

Z· h, are sufficiently fast.

431

If sphaleron transitions were fast, then we could eliminate a further chemical potential through the constraint 3

3

3 LJlu~

3

+ 3 LJld~ + LJle~ = o.

;=1

i=l

(331)

i=l

+ L) would be determined by that of the other three charges according to the relation

In this case, the value of (B

(B

3

7

19 40

+ L)EQ = -Q + -BP 80 20

-(B - L).

(332)

The above result should not come as a surprise, since we already know that a non zero value for B - L gives rise to a non zero (B + L) at equilibrium. Stated in other words, sphaleron transitions erase the baryon asymmetry only if any conserved charge of the system has vanishing thermal average, otherwise the equilibrium point lies at (B + L)EQ f. O. At high temperature (Jl; « T) the free energy of the system is given by F

= I;

[3Jl~L + 3Jl~L + 6Jl~L + 3Jl;L + 3Jl;R + 6Jl~L + 3Jlt +6Jli:v+

+ 2Jlk+ + 2Jl~o + 2Jlko] 1

2

.

(333)

Using (328), (329) and (330) to express the chemical potentials in terms ofthe four conserved charges in (330) we obtain the free energy as a function of the density of (B + L) = JlB+£T2/6,

F [(B

+ L)] = 0.46

[(B

+ L) -

(B T2

+ L)EQ]2

+

constant terms,

(334)

where the "constant terms" depend on Q, (B-L), and BP but not on (B+L), and (B + L)EQ is given by (332). From the above expression, we can see that dnB+L -d-t-

rsp

IX

of

-T o(B + L)

rsp

IX

-T [(B + L) -

(B

+ L)EQ].

(335)

5) The contribution to the E parameter of the potential (231) is given generically by, see Eq. (229),

T """' -12rr L.. ni [2 m;{¢)

,

+ IIi(T) ]3/2 ,

(336)

432

for a generic bosonic particle i with plasma mass II;(T). In the case of the right-handed stop, we get (337) where Nc = 3 is the number of color and m~t is given in (251). The upper bound on the contribution to the E parameter from the right-handed stops is = mb + IIR(T) ~ O. This gives obtained when mb < 0 and meJ! t (338) Using now the fact that mt we get Eq. (253).

= htv

and that (¢(Tc))/Tc

= 2E/>. ~ 4v 2 E/m~,

References 1. L. F. Abbott, E. Fahri and M. Wise, Phys. Lett. B117, 29 (1982). 2. U. Amaldi, W. de Boer and H. Furstenau, Phys. Lett. B260, 447 (1991). 3. G.W. Anderson, A. Linde, A. Riotto, Phys. Rev. Lett. 77,3716 (1996), hep-ph/9606416. 4. N. Arkani-Hamed, this series of lectures. 5. P. Arnold, D. Son and L.G. Yaffe, Phys. Rev. D55, 6264 (1997). 6. P. Arnold, Phys. Rev. D55, 7781 (1997); ibidem hep-ph/9706305. 7. R. Battiston, Astroparticle physics with the Alpha Magnetic Spectrometer (AMS) in Vulcano 1996, Frontier Objects in Astrophysics and Particle Physics, pag. 543, Vulcano, Italy, 27 May - 1 Jun 1996. 8. J. Bernstein, Kinetic Theory in the Expanding Universe, Cambridge Monographs on Mathematical Physics, Cambridge, 1988. 9. See, for instance, G. Blewitt, Phys. Rev. Lett. 55, 2144 (1985). 10. A.1. Bocharev and M.E. Shaposhnikiv, Mod. Phys. Lett. A2, 417 (1987) . 11. P. Bock et al., CERN-EP-98-046. 12. D. Bodeker, P. John, M. Laine, M.G. Schmidt, Nucl. Phys. B497, 387 (1997). 13. D. Bodeker, Phys. Lett. B426, 351 (1998). 14. E. Braaten and R. Pisarski, Nucl. Phys. B337, 569 (1990); Phys. Rev. D45, 1827 (1992). 15. A. Brignole, J.R. Espinosa, M. Quiros and F. Zwirner, Phys. Lett. B324, 181 (1994).

433

16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46.

Y. Brihaye and J. Kunz, Phys. Rev. D46, 3587 (1992). W. Buchmuller and M. Plumacher, Phys. Lett. B389, 73 (1996). M. Carena, M. Quiros and C.E.M. Wagner, Phys. Lett. B380, 81 (1996). M. Carena, M. Quiros, A. Riotto, I. Vilja and C.E.M. Wagner, Nucl. Phys. B503, 387 (1997), [hep-ph/9702409]. M. Carena, M. Quiros and C.E.M. Wagner, CERN-TH/97-190, hepph/9710401. 1. Carson et al., Phys. Rev. D42, 2127 (1990). K. Chou, Z. Su, B. Hao and L. Yu, Phys. Rep. 118 (1985), 1 and references therein. D. Chang, R.N. Mohapatra and M.K. Parida, Phys. Lett. B142, 55 (1984). D.J.H. Chung, E.W. Kolb and A. Riotto, hep-ph/9802238. D.J.H. Chung, E.W. Kolb and A. Riotto, hep-ph/9805473. D.J.H. Chung, E.W. Kolb and A. Riotto, to appear. J.M. Cline and K. Kainulainen, Nucl. Phys. B482, 73 (1996). J.M. Cline, M. Joyce and K. Kainulainen, hep-ph/9708393. A.G. Cohen, D.B. Kaplan and A.E. Nelson, Annu. Rev. Nucl. Part. Sci. 43,27 (1993); L. Covi, E. Roulet and F. Vissani, Phys. Lett. B384, 169 (1996); Phys. Lett. B399, 113 (1997); Phys. Lett. B424, 101 (1998). R.A. Craig, J. Math. Phys. 9,605 (1968). P. Danielewicz, Ann. of Phys., 152, 239 (1984), ibidem 152, 305 (1984) D. Delepine, J.M. Gerard, R. Gonzalez Felipe and J. Weyers, Phys. Lett. B386, 183 (1996). S. Dimopoulos and L. Susskind, Phys. Rev. D18, 4500 (1978). A. D. Dolgov and A. D. Linde, Phys. Lett. B116, 329 (1982). A. De Rujula, astro-ph/9705045. M. Dine, P. Huet and R. Singleton Jr., Nucl. Phys. B375, 625 (1992). A.D. Dolgov, Phys. Rept. 222, 309 (1992). J. Ellis, S. Kelley and D.V. Nanopoulos, Phys. Lett. B249, 441 (1990); B260, 131 (1991). J. Ellis et al .. Phys. Lett. B118, 335 (1982). J. Ellis, J. Kim and D.V. Nanopoulos, Phys. Lett. B145, 181 (1984). J.R. Espinosa, M. Quiros and F. Zwirner, Phys. Lett. B307, 106 (1993). J.R. Espinosa, Nucl. Phys. B475, 273 (1996). B. de Carlos and J.R. Espinosa, hep-ph/9703317. G.R. Farrar and M.E. Shaposhnikov, Phys. Rev. Lett. 70, 2833 (1983) and Erratum) 71,210 (1993); Phys. Rev. D50, 774 (1994). J.N. Fry, K.A. Olive and M.S. Turner, Phys. Rev. D22, 2953 (1980);

434

Phys. Rev. D22, 2977 (1980); Phys. Rev. Lett. 45,2074 (1980). 47. M. Fukugita and T. Yanagida, Phys. Lett. 174,45 (1986). 48. K. Fujikawa, Phys. Rev. D21, 2848 (1980); Phys. Rev. D29, 285 (1984). 49. M.B. Gavela. P. Hernandez, J. Orloff, O. Pene and C. Quimbay, Mod. Phys. Lett. A9, 795 (1994); Nucl. Phys. B430, 382 (1994). 50. M. Gell-Mann, P. Ramond and R. Slanski, in Supergravity, ed. P. Van Nieuwenhuizen and D.Z. Freedman, North Holland (1979). 51. G.F. Giudice, Phys. Rev. D45, 3177 (1992). 52. G.F. Giudice and M. Shaposhnikov, Phys. Lett. B326, 118 (1994). 53. M. Grisaru, W. Siegel and M. Rocek, Nucl. Phys. B159, 429 (1979). 54. Guth, A. H., Phys. Rev. D23, 347 (1981). 55. H.E. Haber, Phys. Rev. D26, 1317 (1982). 56. H.E. Haber and G.L. Kane, Phys. Rept. 117, 75 (1985). 57. J.A. Harvey, E.W. Kolb, D.B. Reiss and S. Wolfram, Nucl. Phys. B201, 16 (1982). 58. P.A. Henning, Phys. Rep. 253, 235 (1995). 59. P. Huet and E. Sather, Phys. Rev. D51, 379 (1995). 60. M. Kawasaki and T. Moroi, Prog. Theor. Phys. 93, 879 (1995). 61. L.V. Keldysh, JETP 20 (1965), 1018. 62. S. Yu. Khlebnikov and M.E. Shaposhnikov Nucl. Phys. B308, 885 (1988) . 63. S.Yu. Khlebnikov and 1.1. Tkachev, Phys. Rev. Lett. 77,219 (1996). 64. S.Yu. Khlebnikov and I. Tkachev, Phys. Lett. B390, 80 (1997). 65. S.Yu. Khlebnikov and I. Tkachev, Phys. Rev. Lett. 79, 1607 (1997). 66. S.Yu. Khlebnikov and I. Tkachev, Phys. Rev. D56, 653 (1997). 67. R.F. Klinkhamer and N.S. Manton, Phys. Rev. D30, 2212 (1984). 68. L. A. Kofman, A. D. Linde and A. A. Starobinsky, Phys. Rev. Lett. 73, 3195 (1994); 69. L. Kofman, A.D. Linde and A.A. Starobinsky, Phys. Rev. Lett. 76, 1011 (1994). 70. L. Kofman, astro-ph/9802285. 71. E.W. Kolb and S. Wolfram, Nucl. Phys. B172, 224 (1980); Phys. Lett. B91, 217 (1980). 72. E.W. Kolb and M.S. Turner, Ann. Rev. Nucl. Part. Sci. 33,645 (1983). 73. See, for instance, E.W. Kolb and M.S. Turner, The Early Universe, Addison-Wesley (1990). 74. E.W. Kolb, A.D. Linde and A. Riotto, Phys. Rev. Lett. 77,4290 (1996), hep-ph/9606260. 75. Ed.W. Kolb, A. Riotto and 1.1. Tkachev, Phys. Lett. B423, 348 (1998),

435

hep-ph/9801306. 76. V.A. Kuzmin, V.A. Rubakov and M.E. Shaposhnikov, Phys. Lett. B155, 36 (1985). 77. M. Laine, Nucl. Phys. 481,43 (1996). 78. M. Laine and K. Rummukainen, Phys. Rev. Lett. 80,5259 (1998). 79. M. Laine and K. Rummukainen, hep-lat/9804019. 80. P. Langacker, Phys. Rept. 72, 185 (1981). 81. P. Langacker, R.D. Peccei and T. Yanagida, Mod. Phys. Lett. AI, 541 (1986) . 82. P. Langacker and M.-X. Luo, Phys. Rev. D44, 817 (1991). 83. A. R. Liddle and D. H. Lyth, Phys. Rep. 231, 1 (1993) 84. A.R. Liddle, this series of Lectures. 85. A.D. Linde, Nucl. Phys. B216, 421 (1983). 86. A. D. Linde, Particle Physics and Inflationary Cosmology, Harwood Academic, Switzerland (1990). 87. M. Losada, Phys. Rev. D56, 2893 (1997). 88. G.R. Farrar and M. Losada, Phys. Lett. B406, 60 (1997). 89. M. Losada, hep-ph/9806519 .. 90. D.H. Lyth and A. Riotto, Particle physics models of inflation and the cosmological density perturbation, hep-ph/9807278, accepted for publication in Phys. Rept. 91. M. A. Luty, Phys. Rev. D45, 455 (1992). 92. L. Mc Lerran, E. Mottola and M. Shaposhnikov, Phys. Rev. D43, 2027 (1991) . 93. R.N. Mohapatra, J.C. Pati, Phys. Rev. D11, 566 (1975). 94. R.N. Mohapatra and G. Senjanovic, Phys. Rev. D12, 1502 (1975). 95. R.N. Mohapatra and X. Zhang, Phys. Rev. D45, 2699 (1992). 96. See, for instance, A. Masiero in Grand Unification with and without supersymmetry and cosmological implications, World Scientific Edition (1984). 97. S.P. Mikheyev and A.Y. Smirnov, Nuovo Cim. 9C, 17 (1986). 98. G.D. Moore, C. Hu and B. Muller, hep-ph/9710436. 99. J.M. Moreno, M. Quiros and M. Seco, IEM-FT-168/98 preprint, [hepph/9801272]. 100. S. Myint, Phys. Lett. B287, 325 (1992). 101. D.V. Nanopoulos and S. Weinberg, Phys. Rev. D20, 2484 (1979). 102. P. Huet and A.E. Nelson, Phys. Rev. D53, 4578 (1996). 103. H. P. Nilles, Phys. Rep. 110, 1 (1984). 104. J.e. Pati and A. Salam, Phys. Rev. DI0, 275 (1974). 105. M. Plumacher, Z. Phys. C74, 549 (1997).

436

106. 107. 108. 109. 110. 111. 112. 113.

M. Quiros, this series of lectures. A. Riotto and I. Vilja, Phys. Lett. B402, 314 (1997); hep-ph/9612428. A. Riotto, OUTP-97-43-P preprint, hep-ph/9709286. A. Riotto, Nucl. Phys. B518, 339 (1998), hep-ph/9712221. A. Riotto, hep-ph/9803357. A. Riotto, hep-ph/9802240. A. Riotto and 1.1. Tkachev, Phys. Lett. B385, 57 (1996), V.A. Rubakov and M.E. Shaposhnikov, Usp. Fiz. Nauk 166, 493 (1996), Phys. Usp. 39,461 (1996). hep-ph/9604444. 114. A.D. Sakharov, Zh. Eksp. Teor. Fiz. Pis'ma 5, 32 (1967); JETP Lett. 91B, 24 (1967). 115. J. Schwinger, J. Math. Phys. 2 (1961),407. 116. M. Shaposhnikov, JETP Lett. 44, 465 (1986); Nucl. Phys. 287, 757 (1987); ibidem B299, 797 (1988). 117. R. Slanski, Phys. Rept. 79, 128 (1981). 118. F.W. Stecker, Nucl. Phys. B252, 25 (1985). 119. G. Steigman, Ann. Rev. Astron. Astrophys. 14, 336 (1976). 120. G. 't Hooft, Phys. Rev. Lett. 37, 37 (1976); Phys. Rev. D14, 3432 (1976). 121. 1.1. Tkachev, Phys. Lett. B376, 35 (1996). 122. M. Trodden, hep-ph/9803479. 123. H.A. Weldon, Phys. Rev. D26 (1982), 1394. 124. E. Witten, Nucl. Phys. B471, 135 (1996) 125. L. Wolfenstein, Phys. Rev. D17, 2369 (1978).