Proceedings of the SUNY Institute of Technology Conference on Theoretical High Energy Physics, June 6, 2002 [illustrated edition] 9780660190655, 0-660-19065-6

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Proceedings of the

SUNY Institute of Technology Conference on

Theoretical High Energy Physics

June 6, 2002

EDITORS: M.R. Ahmady

A.H. Fariborz

Department of Physics Mount Allison University Sackville, NB E4L 1E6 CANADA

Department of Mathematics/Science SUNY Institute of Technology Utica, New York 13504-3050 USA

NRC Research Press Ottawa 2003

© 2003 National Research Council of Canada All rights reserved. No part of this publication may be reproduced in a retrieval system, or transmitted by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the National Research Council of Canada, Ottawa, Ontario K1A 0R6, Canada. Printed in Canada on acid–free paper. ISBN 0–660–19065–6 Electronic ISBN 0-660-19257-8 NRC No. 46322 Publisher’s Note: This book has been prepared from camera ready copy provided by the editors.

National Library of Canada cataloguing in publication data SUNY Institute of Technology Conference on Theoretical High Energy Physics (2nd : 2003 : Utica, N.Y.) Proceedings of the SUNY Institute of Technology Conference on Theoretical High Energy Physics Issued by : National Research Council of Canada. Issued also on the Internet. Includes bibliographical references. ISBN 0-660-19065-6 Cat. no. NR15-68/2002E 1. Particles (Nuclear physics) - Congresses. 2. Nuclear physics - Congresses. I. Ahmady, Mohammad R., 1960- . II. Fariborz, A.H. (Amir Hossein), 1965- . III. National Research Council Canada. IV. Title. QC793.S86 2003

539.7'6

C2003-980174-8

Inquiries: Proceedings, NRC Research Press, National Research Council of Canada, Ottawa, Ontario K1A 0R6, Canada. Correct citation for this publication: Amhady, M.R. and Fariborz, A.H. (Editors). 2003. Proceedings of the SUNY Institute of Technology Conference on Theoretical High Energy Physics. NRC Research Press, National Research Council of Canada, Ottawa, ON K1A 0R6, Canada, 130 pages.

CONTENTS Preface · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

iv

Optimal RG-Improvement of Perturbative Calculations in QCD V. Elias · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Study of Scalar Mesons and Related Radiative Decays D. Black, M. Harada, and J. Schechter · · · · · · · · · · · · · · · · · · · · · · Gaussian Sum Rules, Scalar Gluonium, and Instantons T.G. Steele, D. Harnett, and G. Orlandini · · · · · · · · · · · · · · · · · · Possible Applications of Chiral Lagrangians in QCD Sum-Rules A.H. Fariborz · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Magnetic Catalysis and Anisotropic Confinement in QCD V.A. Miransky · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Supersymmetry in Anti de Sitter Space D.G.C. McKeon, C. Schubert, and T.N. Sherry · · · · · · · · · · · · · 0 ¯ 0 Mixing and CP-Violation in Charm D −D A. Petrov · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Extraction of |Vcb | via Inclusive Method F.A. Chishtie · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Recent QCD Related Issues in B-Physics M.R. Ahmady · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

117

Schedule · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · List of participants · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

127 129

iii

1

13

25

51

65 87 95 111

PREFACE This conference was held on June 6, 2002 at the SUNY Institute of Technology, and was the second high energy physics conference at this institute. It took place during a three-week meeting (June 2 June 22, ’02) of a group of collaborators from several universities in Canada, Mount Allison University, University of Western Ontario and the University of Saskatchewan, together with SUNYIT, Cornell University and the National University of Ireland. The conference also brought to SUNYIT speakers and participants from Syracuse University, SUNY at Albany, SUNY Morrisville, Wayne State University and Mohawk Valley Community College. We thank the School of Arts & Sciences at the SUNYIT, and the Office of Conference Services & Sponsored Research at the SUNYIT for their support. We also thank the office of Vice-President, Academic and Research, at Mount Allison University for the financial support to publish this proceedings.

M .R. Ahmady and A.H. Fariborz SUNY Institute of Technology

iv

————————– SUNY Institute of Technology Conference on Theoretical High Energy Physics June 6th, 2002

————————– Optimal RG-Improvement of Perturbative Calculations in QCD V. Elias 1 ,

2

Perimeter Institute for Theoretical Physics, 35 King Street North, Waterloo, Ontario N2J 2W9, Canada

————————————————– Abstract Using renormalization-group methods, differential equations can be obtained for the all-orders summation of leading and subsequent non-leading logarithmic corrections to QCD perturbative series for a number of processes and correlation functions. For a QCD perturbative series known to four orders, such as the e+ e− annihilation cross-section, explicit solutions to these equations are obtained for the summation to all orders in αs of the leading set and the subsequent two non-leading sets of logarithms. Such summations are shown for a number of processes to lead to a substantial reduction in sensitivity to the renormalization scale parameter. Surprisingly, such summations are also shown to lower the infrared singularity within the perturbative expression for the e+ e− annihilation cross-section to coincide with the Landau pole of the naive one-loop running QCD couplant.

1

Electronic address: [email protected] Permanent address: Department of Applied Mathematics, The University of Western Ontario, London, Ontario N6A 5B7, Canada 2

1

Theoretical High Energy Physics

1

Optimal RG Improvement of Γ(B → Xuℓ−ν¯ℓ)

Optimal renormalization-group (RG) improvement of a perturbative series to a given order in the expansion couplant is the idea of including within that series all higher-order contributions that can be extracted by renormalization-group methods [1]. We call such terms, which involve leading and successive logarithms of the renormalization scale µ, RG-accessible. Techniques have been developed to obtain closed-form summations of such RG-accessible contributions to all orders in the perturbative expansion parameter, and such summations have been shown to lead to “optimally RGimproved” expressions for perturbative quantities that have significantly diminished dependence on µ [2, 3]. For example, leading and subleading perturbative QCD corrections to the inclusive semi-leptonic B → Xu ℓ− ν¯ℓ decay rate, which in tree-order is purely a charged-current weak interaction process, are given by a QCD series [4] αs (µ) µ2 S = 1 + 4.25360 + 5 log 2 mb (µ) π
  2 µ + 26.7848 + 36.9902 log m2b (µ)


+



415 µ2 log2 24 m2b (µ) 

 

such that

αs (µ) π

 

2



 3  α (µ) s  +O

π

(1)

G2F |Vub |2 5 mb (µ)S. (2) 192π 3 If one substitutes eq. (1) into eq. (2), one obtains a decay rate that decreases monotonically with increasing µ, raising the question as to which value of µ is most appropriate for comparing the calculation (2) to experiment. Clearly, such dependence on the unphysical parameter µ is an embarassment; indeed the renormalization group equation for the series S



Γ b → uℓ− ν¯ℓ =






 ∂ ∂ ∂ 0 = µ2 2 + β(g) 2 + mb γm (g) + 5γm (g) S µ2 , g 2 (µ), mb (µ) ∂µ ∂g ∂mb (3)

2

Elias

is nothing more than a chain rule expression for the requirement that the physically measurable decay rate be impervious to changes in the renormalization scale parameter µ, 0=

d − . Γ B → X ℓ ν ¯ u ℓ dµ2

(4)

The residual µ-dependence of the decay rate obtained from the series (1) is necessarily a consequence of the truncation of that series, as well as the relatively large value of the expansion constant αs (µ)/π. In fact, the series (1) may be expressed as a double summation over powers of logarithms and the expansion parameter, i.e., in the following form: n ∞ S [x, L] =



Tn,m xn Lm ,

(5)

n=0 m=0

where x ≡ αs (µ)/π,

L ≡ log(µ2 /m2b (µ)).

(6)

The first few constants of this series, i.e., the set {T0,0 (= 1), T1,0 , T1,1 , T2,0 , T2,1 , T2,2 }, are given by eq. (1). However, all higher-order constants of the form Tn,n , Tn,n−1 and Tn,n−2 can be obtained via eq. (3), and, hence, are RG-accessible. In terms of the new variables x and L, the RG-equation (3) may be expressed as




∂ ∂ 0 = (1 − 2γm (x)) + β(x) + 5γm (x) S[x, L]. ∂L ∂x

(7)

If we substitute the series (5) into the RG-equation (7), as well as the known QCD series expansions of the RG-functions β(x) = −

∞

n=0

βn xn+2 , γm (x) = −

∞

γn xn+1 ,

(8)

n=0

we find for any integer p that the aggregate coefficients of xp Lp−1 , xp Lp−2 and xp Lp−3 on the right hand side of eq. (7) necessarily vanish: xp Lp−1 : 0 = pTp,p − β0 Tp−1,p−1 (p − 1) − 5γ0 Tp−1,p−1

3

(9)

Theoretical High Energy Physics

xp Lp−2 : 0 = (p − 1)Tp,p−1 + 2γ0 (p − 1)Tp−1,p−1 − β0 (p − 1)Tp−1,p−2 − β1 (p − 2)Tp−2,p−2 − 5γ0 Tp−1,p−2 − 5γ1 Tp−2,p−2

(10)

xp Lp−3 : 0 = (p − 2)Tp,p−2 + 2γ0 (p − 2)Tp−1,p−2

+ 2γ1 (p − 2)Tp−2,p−2 − β0 (p − 1)Tp−1,p−3

− β1 (p − 2)Tp−2,p−3 − β2 (p − 3)Tp−3,p−3

− 5γ0 Tp−1,p−3 − 5γ1 Tp−2,p−3 − 5γ2 Tp−3,p−3 .

(11)

Given knowledge of T0,0 (= 1), one can calculate any coefficient Tp,p through successive applications of eq. (9). Indeed the eq. (1) values T1,1 = 5 and T2,2 = 415/24 follow from just two successive iterations of (9) using the nf = 5 QCD values γ0 = 1, β0 = 23/12. Similarly, knowledge of all Tp,p coefficients, as obtained via (9), plus knowledge of T1,0 = 4.25360 [eq. (1)] is sufficient via successive applications of (10) to determine all coefficients Tp,p−1 . Finally, knowledge of all coefficients Tp,p Tp,p−1 plus the single coefficient T2,0 = 26.7848 is sufficient via successive applications of (11) to determine all coefficients Tp,p−2 , since the set of M S RG-function coefficients β0 , β1 , β2 , γ0 (= 1), γ1 , and γ2 } have all been calculated [5]. Since we now see that all coefficients Tp,p , Tp,p−1 and Tp,p−2 are RGaccessible, it makes sense to restructure the double- summation series (4) in the form S[x, L] =

∞

Tp,p xp Lp +

p=0 ∞

+

∞

∞

Tp,p−1 xp Lp−1 +

Tp,p−2 xp Lp−2

p=2

p=1

Tp,p−3 xp Lp−3 + ... ,

(12)

p=3

since the first three terms above are completely determined by eqs. (9), (10) and (11). We express (12) in the more compact form S[x, L] = ≡

∞

n=0 ∞

x

n


∞

Tp,p−n (xL)

p=n

xn Sn (xL)

n=0

4

p−n



(13)

Elias

and note that S0 (xL), S1 (xL) and S2 (xL) all correspond to RG-accessible functions, based upon the information given in (1). Indeed, the program of optimal RG-improvement is nothing more than the explicit closed-form evaluation of these functions, and their subsequent incorporation into the calculated decay rate. To evaluate the summation S0 (u), as defined by (13), we simply multiply eq. (9) by up−1 and sum from p = 1 to infinity: 0 =

∞

p=1

− 5γ0

pTp,p up−1 − β0 ∞

∞

p=1

(p − 1)Tp−1,p−1 up−1

Tp−1,p−1 up−1

p=1

= (1 − β0 u)

dS0 − 5γ0 S0 . du

(14)

We note from the definition (13) of the series Sn (xL) that Sn (0) = Tn,0 .

(15)

The solution of the differential equation (14) with initial condition S0 (0) = T0,0 = 1 is S0 (u) = (1 − β0 u)−5γ0 /β0 . (16) A similar procedure is employed to find S1 , and S2 . If we multiply eq. (10) by up−2 and then sum from p = 2 to ∞, we find after a little algebra that (1 − β0 u)

dS1 dS0 . − (β0 + 5γ0 )S1 = 5γ1 S0 − (2γ0 − β1 u) du du

(17)

Substituting the solution (16) into the right hand side of (17) and noting that S1 (0) = T1,0 , we find that S1 (u) = +

5(γ0 β1 /β0 −γ1 )/β0 (1−β0 u)5γ0 /β0

T1,0 − 5(γ0 β1 /β0 − γ1 )/β0 + [5γ0 (2γ0 − β1 /β0 )/β0 ] log(1 − β0 u) . (1 − β0 u)(β0 +5γ0 )/β0 (18) 5

Theoretical High Energy Physics

Similarly, we can multiply eq. (11) by up−3 and then sum from p = 3 to ∞ to obtain the differential equation (1 − β0 u)

dS2 − (2β0 + 5γ0 )S2 du dS1 + (β1 + 5γ1 )S1 = (β1 u − 2γ0 ) du dS0 +(β2 u − 2γ1 ) + 5γ2 S0 du

(19)

whose solution is given by eq. (2.28) of ref. [2].

2

Order-by-Order Elimination of Renormalization Scale Dependence

If one substitutes solutions for S0 (xL), S1 (xL) and S2 (xL) into eq. (13), one obtains the following optimally RG-improved version of the series S [2]: S ∼ = S0 (xL) + xS1 (xL) + x2 S2 (xL)   18655 1020 = w−60/23 + x − w + 10.1310 + log w w−83/23 3174 529    3171350 2 2 + x 13.2231w − 47.4897 + log w w 279841   719610 2 log w w−106/23 (20) 61.0515 + 25.5973 log w + + 279841 with w ≡ 1 − β0 xL

(21)

and with x and L given by eq. (6). When multiplied by m5b (µ), this expression has the remarkable property of being almost entirely independent of µ. Figure 1 of ref. [2] displays a head to head comparison of the µ dependence of eq. (20), and the same expression with S given by the known terms of eq. (1). For the latter case, [mb (µ)]5 S is seen to decrease from ≈ 2500 GeV5 to ≈ 1500 GeV5 as µ decreases from 1.5 GeV to 9.0 GeV. For eq. (20), however, the quantity [mp (µ)]5 S is seen to be 1816 ± 6 GeV5 over the same range of 6

Elias

µ, effectively removing all µ-dependence from the optimally RG-improved two-loop calculation. Such elimination of renormalization-scale dependence via optimal RGimprovement is also upheld for a number of perturbative expressions, including QCD corrections to the inclusive semileptonic decay of B-mesons to charmed states (B → Xc ℓ− ν¯ℓ ), QCD corrections to Higgs boson decays, the perturbative portion of the QCD static potential function, the (Standard-Model) Higgs-mediated W W → ZZ cross section at very high energies, and QCD sum-rule scalar- and vector-current correlation functions [2]. This last example is of particular relevance for QCD corrections to the benchmark electromagnetic cross-section ratio R(s) = σ(e+ e− → hadrons)/σ(e+ e− → µ+ µ− ). Such QCD corrections are proportional to the imaginary part of the vector-current correlation function series, a series which is fully known to three subleading orders in αs [6]. For five active flavours, we have S ≡ 3R(s)/11

23 = 1 + x + 1.40924 + L x2 12   529 2 3 L x + ... + −12.8046 + 7.81875L + 144 



(22)

where x = αs (µ)/π, as before, and where L is now the logarithm L ≡ log(µ2 /s).

(23)

Note that dependence on the physical scale s resides entirely in the logarithm, and that the all-orders series (22) for S, a measurable quantity, is necessarily impervious to changes in µ. However, progressive truncations of (22) introduce progressively larger amounts of renormalization scale dependence. For example, if the series S is truncated after all its known √ terms, as listed in eq. (22), we find for s = 15 GeV that to order x3 , S increases modestly from 1.0525 to 1.0540 as µ increases from 7.5 to 30 GeV. 1 Had we truncated the series (22) following its O(x2 ) term, we 1

In all estimates presented here, αs (µ) is assumed to evolve via its known 4-loop order β-function from αs (Mz ) = 0.118 [7].

7

Theoretical High Energy Physics

find that such a truncation of S now decreases from 1.056 to 1.053 over the same range of µ, doubling the magnitude of µ-dependence evident over this range. Finally, if we consider only the lowest order correction to unity (S = 1 + x(µ)), we find that S decreases from 1.061 to 1.045 as µ increases from 7.5 GeV to 30 GeV. Optimal RG-improvement of the known terms of the series (22) has been shown by the same methods delineated above to lead to the following expression [2]: S = 1 + x/w + x2 [1.49024 − 1.26087 log w] /w2 + x3 [0.115003w − 12.9196 − 5.14353 log w

+ 1.58979 log2 w /w3 + ...

(24)

where w is given by the definition (21), but with L now given by eq. (23). Eq. (24) is, of course, really a restructured version of the same infinite series as eq. (21), and similarly must be independent of µ when taken to all orders. However, for the series (24) such imperviousness to changes in renormalization scale is now evident on an order-by-order basis. Truncation of the series (24) after its first nonleading term (i.e., S = 1 + x/w) still provides an expression that exhibits less variation with µ than all four known terms of the series (22). As µ increases from 7.5 to 30 GeV, we find for √ s = 15 GeV that 1 + x/w decreases from 1.0524 to 1.0516. Similarly, truncation of the series (24) after its third term leads to a slow decrease from 1.0557 to 1.0553, and retention of all four known terms leads to an almost flat value (1.05372 ± 0.00004) over the same 7.5 GeV - 30 GeV spread in µ. Consequently, we see that the program of optimal RG-improvement, as described above, is seen to yield order-by-order perturbation-theory predictions which are almost entirely decoupled from the particular choice of renormalization scale.

3

Lowering the Infrared Bound on Perturbative Approximations to R(s)

The optimally RG-improved series (24) is term-by-term singular when w = [1 − β0 x(µ) log(µ2 /s)] is zero. Since s is the external momentum scale 8

Elias

characterising the physical e+ e− annihilation process, we see that the use of (24) is possible only if [3] 



π s > µ exp − . β0 αs (µ) 2

(25)

It is particularly curious that this bound on s corresponds to the infrared bound on the naive one-loop (1L) running couplant (x1L = (αs (µ))1L /π) µ2

dx1L = −β0 x21L dµ2

whose solution αs (µ) =

(26)

π β0 log(µ2 /Λ21L )

(27)

can be inverted as follows to express the 1L infrared cut-off in terms of some reference value of αs (µ): Λ21L





π = µ exp − . β0 αs (µ) 2

(28)

Consequently, for a given choice of µ for which αs (µ) is known (e.g. the value αs (mτ ) extracted from τ -decay experiments), we see that each of the terms in the optimally RG-improved series (24) diverges as s approaches Λ21L from above. The idea that QCD perturbative series break down in the infrared is hardly new, but the location of this breakdown is usually identified with an IR-divergence in the all-available-orders evolution of αs (µ), not the naive 1L Landau pole of eq. (28). To consider the IR boundary of QCD corrections to e+ e− annihilation, we find for three active flavours that QCD corrections to the e+ e− annihilation cross-section are, as before, obtained from the perturbative series within the imaginary part of the QCD vector current correlation function [6]: 9 R(s)/2 = 1 + x + 1.63982 + L x2 4   81 2 3 + −10.2839 + 11.3792L + L x 16 + ... 



9

(29)

Theoretical High Energy Physics

where x = αs (µ) and where L = log(µ2 /s). The standard phenomenological approach to this series is to first recognize its all-orders invariance under changes in µ, and then to assume such invariance applies to truncation of the series after its four known terms. This (seldom stated) assumption [8] motivates the choice µ2 = s (i.e. L = 0) leading to the usual nf = 3 expression [7],  √ √ √ R(s) = 2 1 + x( s) + 1.63982 x2 ( 2) − 10.2839 x3 ( s) + ... . (30)

√ Such an expansion necessarily falls apart in the infrared when x( s) be√ comes large. Indeed, the large coefficient of x3 ( s) in (30) manifests itself √ in a sharp drop in R(s) at s ∼ = 650 MeV [3]. It is interesting to compare the known terms in eq. (30) to the optimally RG-improved version of the known terms in eq. (29) [2, 3]: R(s) = s [1 + x(µ)/w(µ, s)   16 + x2 (µ) 1.63982 − log (w(µ, s)) /w2 (µ, s) 9 + x3 (µ) (−1.31057w(µ, s) − 8.97333 − 8.99096 log (w(µ, s)) +





3.16049 log2 (w(µ, s)) /w3 (µ, s)

where

(31)

9 w(µ, s) = 1 − x(µ) log(µ2 /s). (32) 4 √ We first note that x(µ) and x( s) occurring in eqs. (29), (30), (31) and (32) are evolved through all known orders of the β-function (8). Consequently, we are free to assign to eq. (31) an nf = 3 empirical value for µ (µ = mτ or, alternatively, µ = 1 GeV) safely outside the infrared region. √ To facilitate comparison of eqs. (30) and (31), we will assume that x( s) devolves via the full β-function from this same initial choice of µ until, √ for a sufficiently small value of s, x( s) becomes infinite. The point here is that for nf = 3, the first four known terms of the β-function are all same-sign: (β0 x2 + β1 x3 + β2 x4 + β3 x5 ) > β0 x2 . Thus for a given value of x, the full β-function is more negative than the one-loop β-function of eq. (26). Since the evolution of both equations is referenced to the same √ initial value µ, the all-orders couplant x( s) will diverge at a value of s 10

Elias

that is larger than Λ21L , the Landau pole of eq. (26). Hence the series (31) will probe more deeply into the infrared than the series (30) for R(s). √ As an example, consider the running couplant x( s) obtained via a two-loop β-function β(x) = −β0 x2 − β1 x3 . Solution of the differential equation s dx/ds = β(x) with the initial value x(µ) yields the exact constraint   
√ 1 β1 µ2 x(µ)[x( s) + β0 /β1 ] 1 √ − √ + = . (33) log β0 log s x(µ) x( s) β0 x( s)[x(µ) + β0 /β1 ] √ The two-loop Landau pole s2L occurs when x( s2L ) → ∞, i.e., when s2L



1 = µ exp − β0 x(µ) 2

=

Λ21L




β0 1+ β1 x(µ)




β0 1+ β1 x(µ)

β1 /β 2

β1 /β 2 0

0

.

(34)

Since β0 , β1 and x(µ) are all positive, s2L > Λ21L . Note that β2 and β3 [9] persist in being positive. Consequently, for a given initial value x(µ), the singularity in eq. (31) that occurs at w(µ, s) = 0 (i.e., at s = Λ21L ), √ continues to precede the Landau singularity of x( s) characterizing eq. (30). Thus, the optimally RG-improved eq. (31) extends the applicability of perturbative QCD to lower values of s than in the conventional eq. (30) approach to R(s), as is explicitly shown in Fig. 2 of ref. [3]. Finally, we note that one must distinguish between the infrared limitations on the domain of perturbative approximations to R(s), and any such limitations on R(s) itself. For example, each term within the toy series ∞ 2 n 2 2 n=0 [−x/(s − Λ )] diverges at s = Λ , but the function (s − Λ )/(s − 2 Λ + x) from which this series is extracted, is clearly finite at s = Λ2 . Similarly, the all-orders function R(s), as opposed to truncations of its series representations, may indeed proceed smoothly to a finite limit as s → 0 [10]. If such is the case, the best one can hope for in a perturbative series representation of R(s) is the deepest possible penetration of that series into the low-s region.

Acknowledgments: I am grateful to the Natural Sciences and Engineering Council of Canada for financial support, and to M. R. Ahmady, F. A. Chishtie, A. H. Fariborz, 11

Theoretical High Energy Physics

N. Fattahi, D. G. C. McKeon, T. N. Sherry, A. Squires and T. G. Steele for their contributions to much of the research summarized in this talk.

References [1] C. J. Maxwell, Nucl. Phys. B (Proc. Suppl.) 86, 74 (2000). [2] M.R. Ahmady et al., Phys. Rev. D66, 014010 (2002). [3] M. R. Ahmady et al., hep-ph/0208025. [4] T. van Ritbergen, Phys. Lett. B454, 353 (1999). [5] D. J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973); H. D. Politzer, Phys. Rev. Lett. 30, 1346 (1973); W. E. Caswell, Phys. Rev. Lett. 33, 244 (1974); D.R.T. Jones, Nucl. Phys. B75, 531 (1974); E. S. Egorian and O. V. Tarasov, Theor. Mat. Fiz. 41, 26 (1979); O. V. Tarasov, A. A. Vladimirov and A. Yu. Zharkov, Phys. Lett. B93, 429 (1980); S. A. Larin and J. A. M. Vermaseren, Phys. Lett. B303, 334 (1993). [6] S. G. Gorishny, A. L. Kataev and S. A. Larin, Phys. Lett. B259, 144 (1991); L. R. Surguladze and M. A. Samuel, Phys. Rev. Lett. 66, 560 (1991) and 2416 (E); K. G. Chetyrkin, Phys. Lett. B391, 402 (1997); F. Chishtie, V. Elias and T. G. Steele, Phys. Rev. D59 105013 (1999). [7] Particle Data Group, D. E. Groom et al., Eur. Phys. J. C15, 1 (2000). [8] F. J. Yndurain, Quantum Chromodynamics (Springer, New York, 1983) 58-62. [9] T. van Ritbergen, J. A. M. Vermaseren and S. A. Larin, Phys. Lett. B400, 379 (1997). [10] D. M. Howe and C. J. Maxwell, hep-ph/0204036.

12

————————– SUNY Institute of Technology Conference on Theoretical High Energy Physics June 6th, 2002

————————– Study of Scalar Mesons and Related Radiative Decays Deirdre Black a, 1 , Masayasu Harada Joseph Schechter c, 3 a

b, 2

,

Theory Group, Jefferson Laboratory, 12000 Jefferson Avenue, Newport News, Virginia, 23606, USA b Department of Physics, Nagoya University, Nagoya 464-8602, Japan c Physics Department, Syracuse University, Syracuse, NY 13244-1130, USA

————————————————– Abstract After a brief review of the puzzling light scalar meson sector of QCD, a brief summary will be given of a paper concerning radiative decays involving the light scalars. There, a simple vector meson dominance model is constructed in an initial attempt to relate a large number of the radiative decays involving a putative scalar nonet to each other. As an application it is illustrated why a0 (980) − f0 (980) mixing is not expected to greatly alter the f0 /a0 production ratio for radiative φ decays.

1

Electronic address: [email protected] Electronic address: [email protected] 3 Electronic address : [email protected]

2

13

Theoretical High Energy Physics

1

Introduction

Why might the subject of light scalar mesons be of interest to physicists now that QCD is known to be the correct theory of Strong Interactions and the burning issue is to extend the Standard Model to higher energies? Simply put, another goal of Physics is to produce results from Theory which can be compared with Experiment. At very large energy scales, the asymptotic freedom of QCD guarantees that a controlled perturbation expansion is a practical tool, once the relevant ”low energy stuff” is suitably parameterized. At very low energy scales, for example close to the threshold of ππ scattering. the running QCD coupling constant is expected to be large and perturbation theory is not expected to work. Fortunately, a controlled expansion based on an effective theory with the correct symmetry structure- Chiral Perturbation Theory[1]- seems to work reasonably well. The new information about Strong Interactions which this approach reveals is closely related to the spectrum and flavor ”family” properties of the lowest lying pseudoscalar meson multiplet and was, in fact, essentially known before QCD. Clearly it is important to understand how far in energy above threshold the Chiral Perturbation Theory program will take us. To get a rough estimate consider the experimental data for the real part of the I = J = 0 ππ scattering amplitude, R00 displayed in Fig. 1. The chiral perturbation series should essentially give a polynomial fit to this shape, which up to about 1 GeV is crudely reminiscent of one cycle of a sine curve. Now consider polynomial approximations to one cycle of the sine curve with various numbers of terms. These are illustrated in Fig. 2. Note that each succesive term departs from the true sine curve right after the preceding one. It is clear that something like eight terms are required for a decent fit. This would correspond to seven loop order of chiral perturbation theory and seems presently impractical.

2

Need for light scalar mesons

Thus an alternative approach is indicated for going beyond threshold of pi pi scattering up to about 1 GeV. The data itself suggests the presence of s-wave resonances, the lowest of which is denoted the ”sigma”. Physically, 14

Black, Harada, and Schechter

R

o o

s

(GeV)

Figure 1: Illustration of the real part of the pi pi scattering amplitude extracted from experimental data. one then expects the practical range of chiral perturbation theory to be up to about 450-500 MeV, just before the location of this lowest resonance. In the last few years there have been studies [2] by many authors which advance this picture. All of them are ”model dependent” but this is probably inevitable for the strongly coupled regime of QCD. For example [3], in a framework where the amplitude is computed from a non linear chiral Lagrangian containing explicit scalars as well as vectors and pseudoscalars, the fit shown in Fig. 1 emerges as a sum of four pieces: i. the current algebra ”contact” term, ii. the ρ exchange diagram iii. a non Breit Wigner σ(560) pole diagram and exchange, iv. an f0 (980) pole in the background produced by the other three. It is not just a simple sum of Born graphs but includes the approximate unitarization features of the non Breit Wigner shape of the sigma and a Ramsauer Townsend mechanism which reverses the sign of the f0 (980). Also note that i. and ii. provide very substantial background to the sigma pole, partially explaining why the sigma does not ”jump right out” of various experimental studies. Qualitative agreement with this approach is obtained by K-matrix unitarization of the two flavor 15

Theoretical High Energy Physics

approximate_sine_curves

1.5

1 y 0.5

0

1

2

3

4

5

6

x –0.5

–1

–1.5

Figure 2: Polynomial approximations to one cycle of the sine curve. linear sigma model [4] and three flavor linear sigma model [5] amplitudes. Workers on scalar mesons entertain the hope that, after the revelations about the vacuum structure of QCD confirmed by the broken chiral symmetric treatment of the pseudoscalars, an understanding of the next layer of the ”strong interaction onion” will be provided by studying the light scalars. An initial question is whether the light scalars belong to a flavor SU (3) multiplet as the underlying quark structure might suggest. Apart from the σ(560), the f0 (980) and the isovector a0 (980) are fairly well established. This leaves a gap concerning the four strange- so called kappastates. This question is more controversial than that of the sigma state . In the unitarized non linear chiral Lagrangian framework one must thus consider π − K scattering. In this case the low energy amplitude is taken[6] to correspond to the sum of a current algebra contact diagram, vector ρ and K ∗ exchange diagrams and scalar σ(560), f0 (980) and κ(900) exchange diagrams. The situation in the interesting I = 1/2 s-wave channel turns 16

Black, Harada, and Schechter

out to be very analogous to the I = 0 channel of s-wave ππ scattering. Now a non Breit Wigner κ is required to restore unitarity; it plays the role of the σ(560) in the ππ case. It was found that a satisfactory description of the 1-1.5 GeV s-wave region is also obtained by including the well known K0∗ (1430) scalar resonance, which plays the role of the f0 (980) in the ππ calculation. As in the case of the sigma, the light kappa seems hidden by background and does not jump right out of the initial analysis of the experimental data. Thus the nine states associated with the σ(560), κ(900), f0 (980) and a0 (980) seem to be required in order to fit experiment in this chiral framework. What would their masses and coupling constants suggest about their quark substructure if they were assumed to comprise an SU(3) nonet [7]? Clearly the mass ordering of the various states is inverted compared to the ”ideal mixing”[8] scenario which approximately holds for most meson nonets. This means that a quark structure for the putative scalar nonet of the form Nab ∼ qa q¯b is unlikely since the mass ordering just corresponds to counting the number of heavier strange quarks. Then the nearly degenerate f0 (980) and a0 (980) which must have the structure N11 ± N22 would be lightest rather than heaviest. However the inverted ordering will agree with this counting if we assume that the scalar mesons are schematically constructed as Nab ∼ Ta T¯b where Ta ∼ ǫacd q¯c q¯d is a ”dual” quark (or anti diquark). This interpretation is strengthened by consideration [7] of the scalars’ coupling constants to two pseudoscalars. Those couplings √ depend 3 1 2 on the value of a mixing angle, θs between N3 and (N1 −N2 )/ 2). Fitting the coupling constants to the treatments of ππ and Kπ scattering gives a mixing angle such that σ ∼ N33 + ”small”; σ(560) is thus a predominantly non-strange particle in this picture. Furthermore the states N11 ± N22 now would each predominantly contain two extra strange quarks and would be expected to be heaviest. Four quark pictures of various types have been sugggested as arising from spin-spin interactions in the MIT bag model[9], unitarized quark models[10] and meson-meson interaction models[11]. There seems to be another interesting twist to the story of the light scalars. The success of the phenomenological quark model suggests that there exists, in addition, a nonet of “conventional” p-wave q q¯ scalars in the energy region above 1 GeV. The experimental candidates for these states are a0 (1450)(I = 1), K0∗ (1430)(I = 1/2) and for I = 0, f0 (1370), 17

Theoretical High Energy Physics

f0 (1500) and f0 (1710). These are enough for a full nonet plus a glueball. However it is puzzling that the strange K0∗ (1430) isn’t noticeably heavier than the non strange a0 (1450) and that they are not lighter than the corresponding spin 2 states. These and another puzzle may be solved in a natural way[12] if the heavier p-wave scalar nonet mixes with a lighter qq q¯q¯ nonet of the type mentioned above. The mixing mechanism makes essential use of the ”bare” lighter nonet having an inverted mass ordering while the heavier ”bare” nonet has the normal ordering. A rather rich structure involving the light scalars seems to be emerging. At lower energies one may consider as a first approximation, ”integrating out” the heavier nonet and retaining just the lighter one.

3

Radiative decays involving light scalars

In the last few years, a lot of experimental activity[13] at the e+ e− machines (Novosibirsk, DAΦNE and Jefferson Lab) has resulted in definitive measurements of the interesting reactions: φ(1020) → f0 (980) + γ → π 0 π 0 + γ, φ(1020) → a0 (980) + γ → π 0 η + γ.

(1) (2)

These measurements have been awaited by theorists for a number of years as proposed tests [14] of the nature of the f0 (980) and a0 (980) scalars. The theoretical models used for these tests were based on the ¯ observation that the vector meson, φ(1020) mainly decays into K + K so a virtual K loop diagram can reasonably be expected to dominate the decay mechanism. In this framework Achasov [15] has argued that the data are most consistent with a compact four quark structure for the f0 and a0 (as opposed to a two quark structure or a loosely bound meson meson ”molecule” structure). This situation makes it interesting to study in detail the extension of the picture to a full nonet (or two?) of scalar mesons as well as to further solidify the technical analysis of the K- loop class of diagrams. In addition, there is perhaps (depending on the exact masses and widths of the a0 and f0 mesons) a problem in that the experimentally derived ratio 18

Black, Harada, and Schechter

Γ(φ → f0 γ)/Γ(φ → a0 γ) is in the range 3-4 while theoretical estimates are mostly clustered around unity. We are presently working on K-loop type models but decided to start for ourselves with a much simpler preliminary picture. The goal of this model [16] is to try to correlate many different radiative processes involving the members of a full scalar nonet by using flavor symmetry. The model has the following features: 1. It is based on a chiral symmetric Lagrangian containing complete nonets of pseudoscalar, vector as well as (the putative) scalar fields. 2.Vector meson dominance for photon vertices is automatic in the formulation. 3. An effective flavor invariant SVV (scalar-vector-vector) vertex is postulated which has three relevant parameters. These are treated as the only a priori unfixed parameters of the model. Our framework is that of a standard non-linear chiral Lagrangian containing, in addition to the pseudoscalar nonet matrix field φ, the vector meson nonet matrix ρµ and a scalar nonet matrix field denoted N . Under chiral unitary transformations of the three light quarks; qL,R → UL,R qL,R , the chiral matrix U = exp(2iφ/Fπ ), where Fπ ≃ 0.131 GeV, transforms as U → UL U UR† . The convenient matrix K(UL , UR , φ) is defined by the following transformation property of ξ (U = ξ 2 ): ξ → UL ξK † = KξUR† , and specifies the transformations of “constituent-type” objects. The fields we need transform as N → KN K † ,

i ρµ → Kρµ K † + K∂µ K † , g˜ g [ρµ , ρν ] → KFµν K † , Fµν (ρ) = ∂µ ρν − ∂ν ρµ − i˜

(3)

where the coupling constant g˜ is about 4.04. The strong trilinear scalarvector-vector terms in the effective Lagrangian are: LSV V = βA ǫabc ǫa b c [Fµν (ρ)]aa′ [Fµν (ρ)]bb′ Ncc′ ′ ′ ′

+ βB Tr [N ] Tr [Fµν (ρ)Fµν (ρ)] + βC Tr [N Fµν (ρ)] Tr [Fµν (ρ)] + βD Tr [N ] Tr [Fµν (ρ)] Tr [Fµν (ρ)] .

(4)

Chiral invariance is evident from (3) and the four flavor-invariants are needed for generality. (A term ∼ Tr(F F N ) is linearly dependent on the 19

Theoretical High Energy Physics

four shown). Actually the βD term will not contribute in our model so there are only three relevant parameters βA , βB and βC . Equation (4) is analogous to the P V V interaction which was originally introduced as a πρω coupling a long time ago [17]. It is intended to be the simplest description of the production mechanism which contains the full symmetries of the problem. Elsewhere we will discuss modifications due to the effect of K-loops. One can now compute the amplitudes for S → γγ and V → Sγ according to the diagrams of Fig. 3. γ ( k 1, ε 1 ) 0

ρ ,ω,φ

V(p, εv)

ρ0,ω,φ

γ( k,ε)

S ρ0,ω,φ

S

γ(k , ε ) 2 2

(a)

(b)

Figure 3: Feynman diagrams for (a) S → γγ and (b) V → Sγ. Altogether there are many processes of these types. For the two photon decays one may consider the initial scalar to be any of σ(560), f0 (980) or a00 (980). With an initial vector state we have, in addition to φ → f0 , a00 +γ, the possibilities φ → σ + γ, ω → σ + γ and ρ0 → σ + γ. Furthermore for the cases when the scalar may be heavier than the vector, the same diagram allows one to compute the five modes f0 , a00 → ω, ρ0 + γ as well as κ0 → K ∗0 + γ. These are not all measured yet but an initial predicted correlation, is shown in [16]. This model can also be used to study a recent conjecture[18] which attempts to produce a large value for the ratio Γ(φ → f0 γ)/Γ(φ → a0 γ) by invoking the iso spin violating a0 (980)−f0 (980) mixing. Actually, a detailed refutation of this conjecture has already been presented[19]. However the calculation may illustrate our approach. One may simply introduce the mixing by a term in the effective Lagrangian: Laf = Aaf a00 f0 . A recent calculation [20] for the purpose of finding the effect of the scalar mesons in the η → 3π process obtained the value Aaf = −4.66 × 10−3 GeV2 . It 20

Black, Harada, and Schechter

is convenient to treat this term as a perturbation. Then the amplitude 0 factor for φ → f0 γ includes √ a correction term consisting of the φ → a0 γ a0 amplitude factor Cφ = 2 (βC − 2βA ) multiplied by Aaf and by the a0 propagator. The φ → a00 γ amplitude factor has a similar correction. The desired ratio is then, Cφf + Aaf Cφa /Da (m2f ) amp(φ → f0 γ) , = a amp(φ → a00 γ) Cφ + Aaf Cφf /Df (m2a )

(5)

where Da (m2f ) = −m2f +m2a −ima Γa and Df (m2a ) = −m2a +m2f −imf Γf . In this approach the propagators are diagonal in the isospin basis. The numerical values of these resonance widths and masses are, according to the Review of Particle Physics [21] ma0 = (984.7 ± 1.3) MeV, Γa0 = 50– 100 MeV, mf0 = 980 ± 10 MeV and Γf0 = 40–100 MeV. For definiteness, from column 1 of Table II in Ref. [3] we take mf0 = 987 MeV and Γf0 = 65 MeV while in Eq. (4.2) of Ref. [22] we take Γa0 = 70 MeV. In fact the main conclusion does not depend on these precise values. It is easy to see that the mixing factors are approximately given by Aaf Aaf iAaf ≈ ≈ −0.07i. ≈ 2 2 Da (mf ) Df (ma ) ma Γa

(6)

Noting that Cφf /Cφa ≈ 0.75 in the present model, the ratio in Eq.(5) is roughly (0.75 − 0.07i)/(1 − 0.05i). Clearly, the correction to Γ(φ → f0 γ)/Γ(φ → a0 γ) due to a00 -f0 mixing only amounts to a few per cent, nowhere near the huge effect suggested in[18].

Acknowledgments: We are happy to thank N. N. Achasov for important communications and A. Abdel-Rehim, A. H. Fariborz and F. Sannino for very helpful discussions. Prof. Fariborz also is to be thanked for excellently organizing this stimulating conference. D.B. wishes to acknowledge support from the Thomas Jefferson National Accelerator Facility operated by the Southeastern Universities Research Association (SURA) under DOE contract number DE-AC05-84ER40150. The work of M.H. is supported in part by Grantin-Aid for Scientific Research (A)#12740144 and USDOE Grant Number DE-FG02-88ER40388. The work of J.S. is supported in part by DOE contract DE-FG-02-85ER40231. 21

Theoretical High Energy Physics

References [1] J. Gasser and H. Leutwyler, Ann. Phys. 158, 142(1984); Nucl. Phys. B250, 465 (1985). [2] See the dedicated conference proceedings, S. Ishida et al ”Possible existence of the sigma meson and its implication to hadron physics”, KEK Proceedings 2000-4, Soryyushiron Kenkyu 102, No.5 2001. Additional points of view are expressed in the proceedings, D. Amelin and A. M. Zaitsev ”Hadron spectroscopy”, Ninth international conference on hadron spectroscopy, Protvino, Russia (2001). [3] M. Harada, F. Sannino and J. Schechter, Phys. Rev. D 54, 1991 (1996). [4] N. N. Achasov and G. N. Shestakov, Phys. Rev. D 49,5779 (1984). See also S. Ishida, M. Y. Ishida, H. Takahashi, T. Ishida, K. Takamatsu and T. Tsuru, Prog. Theor. Phys. 95, 745 (1966). [5] See for example D. Black, A. H. Fariborz, S. Moussa, S. Nasri and J. Schechter, Phys. Rev. D 64, 014031 (2001). A more complete list of references is given here. [6] D. Black, A. H. Fariborz, F. Sannino and J. Schechter, Phys. Rev. D 58, 054012 (1998). [7] D. Black, A. H. Fariborz, F. Sannino and J. Schechter, Phys. Rev. D59, 074026 (1999). [8] S. Okubo, Phys. Lett. 5, 165 (1963). [9] R. L. Jaffe, Phys. Rev. D 15, 267 (1977). [10] N. A. Tornqvist, Z. Phys. C 68, 647 (1995); E. van Beveren and G. Rupp, Eur. Phys. J. C 10, 469 (1999). [11] J. D. Weinstein and N. Isgur, Phys. Rev. Lett. 48, 659 (1982). [12] D. Black, A. H. Fariborz and J. Schechter, Phys. Rev. D 61, 074001 (2000); D. Black et al,[5] above; T. Teshima, I. Kitamura and N. Morisita, J. Phys. G 28, 1391 (2002); F. Close and N. Tornqvist, arXiv:hep-ph/0204205. 22

Black, Harada, and Schechter

[13] M. N. Achasov et al. [SND Collaboration], Phys. Lett. B 479, 53 (2000); R. R. Akhmetshin et al. [CMD-2 Collaboration], Phys. Lett. B 462, 380 (1999); A. Aloisio et al. [KLOE Collaboration], arXiv:hepex/0107024. [14] N. N. Achasov and V. N. Ivanchenko, Nucl. Phys. B 315, 465 (1989); F. E. Close, N. Isgur and S. Kumano, Nucl. Phys. B 389, 513 (1993); N. N. Achasov and V. V. Gubin, Phys. Rev. D 56, 4084 (1997); Phys. Rev. D 57, 1987 (1998); N. N. Achasov, V. V. Gubin and V. I. Shevchenko, Phys. Rev. D 56, 203 (1997); J. L. Lucio Martinez and M. Napsuciale, Phys. Lett. B 454, 365 (1999); arXiv:hepph/0001136; A. Bramon, R. Escribano, J. L. Lucio M., M. Napsuciale and G. Pancheri, Phys. Lett. B 494, 221 (2000). [15] N. N. Achasov, page 112 of ”Hadron Spectroscopy” [2] above. [16] D. Black, M. Harada and J. Schechter, Phys. Rev. Letts. 88, 181603 (2002). [17] M. Gell-Mann, D. Sharp and W. G. Wagner, Phys. Rev. Lett. 8, 261 (1962). [18] F. E. Close and A. Kirk, Phys. Lett. B 489, 24 (2000); 515, 13 (2001). [19] N.N. Achasov and A. V. Kiselev, Phys. Lett. B 534, 83 (2002). [20] A. Abdel-Rehim, D. Black, A. H. Fariborz and J. Schechter, Phys. Rev. D67, 054001 (2003). [21] E. Groom et al. [Particle Data Group Collaboration], Eur. Phys. J. C 15, 1 (2000) and 2001 partial update for edition 2002 (URL: http://pdg.lbl.gov). [22] A. H. Fariborz and J. Schechter, Phys. Rev. D 60, 034002 (1999).

23

————————– SUNY Institute of Technology Conference on Theoretical High Energy Physics June 6th, 2002

————————– Gaussian Sum-Rules, Scalar Gluonium, and Instantons T.G. Steele a, 1 , D. Harnett a , G. Orlandini b a

b

Department of Physics & Engineering Physics, University of Saskatchewan, Saskatoon, Saskatchewan, S7N 5E2, Canada Dipartimento di Fisica and INFN Gruppo Collegato di Trento, Universit` a di Trento, I-38050 Povo, Italy

————————————————– Abstract Gaussian sum-rules relate a QCD prediction to a two-parameter Gaussianweighted integral of a hadronic spectral function, providing a clear conceptual connection to quark-hadron duality. In contrast to Laplace sum-rules, the Gaussian sum-rules exhibit enhanced sensitivity to excited states of the hadronic spectral function. The formulation of Gaussian sum-rules and associated analysis techniques for extracting hadronic properties from the sumrules are reviewed and applied to scalar gluonium. With the inclusion of instanton effects, the Gaussian sum-rule analysis results in a consistent scenario where the gluonic resonance strength is spread over a broad energy range below 1.6 GeV, and indicates the presence of gluonium content in more than one hadronic state.

1

Electronic address: [email protected]

25

Theoretical High Energy Physics

1

Introduction

The hadronic spectrum has too many scalar states above 1 GeV for a q q¯ nonet, as would be anticipated if gluonium states exist in the 1–2 GeV region [1]. Determining how this gluonium content is distributed among these scalar-isoscalar resonances is thus an important issue. In particular, the possibility that the observed hadronic states are mixtures of gluonium and quark mesons must be explored. Gaussian sum-rules are sensitive to the hadronic spectral function over a wide energy range, and are thus well-suited to studying the distribution of gluonium states. The simplest Gaussian sum-rule (GSR) [2] 1 G (ˆ s, τ ) = π


∞

t0

  1 (t − sˆ)2 √ ρ(t) dt , exp − 4τ 4πτ

τ >0

(1)

relates a QCD prediction on the left-hand side of (1) to the hadronic spectral function ρ(t) (with physical threshold t0 ) smeared over the energy range √ √ sˆ − 2 τ
t
sˆ + 2 τ , representing an energy interval for quark-hadron duality. An interesting aspect of the GSR is that the duality interval is constrained by QCD. A lower bound on this duality scale τ necessarily exists because the QCD prediction has renormalization-group properties that √ reference running quantities the the energy scale ν 2 = τ [2, 3]. Thus it is not possible to achieve the formal τ → 0 limit where complete knowledge of the spectral function could be obtained via lim G (ˆ s, τ ) =

τ →0

1 ρ (ˆ s) π

,

sˆ > t0

.

(2)

However, there is no theoretical constraint on the quantity sˆ representing the peak of the Gaussian kernel appearing in (1). Thus the sˆ dependence of the QCD prediction G (ˆ s, τ ) probes the behaviour of the smeared spectral function, reproducing the essential features of the spectral function. In particular, as sˆ passes through t values corresponding to resonance peaks, the Gaussian kernel in (1) reaches its maximum value, implying that Gaussian sum-rules weight excited and ground states equally. This behaviour

26

Steele, Harnett, and Orlandini

should be contrasted with Laplace sum-rules  1 R ∆2 = π 


∞

t0

 t exp − 2 ρ(t) dt , ∆ 

(3)

where excited states are damped by the exponential decay of the Laplace kernel. Thus, in comparison with Laplace sum-rules the Gaussian sum-rule (GSR) has an enhanced sensitivity to excited states of the spectral function. In this paper, the original formulation of Gaussian sum-rules [2] and analysis techniques for extracting spectral function hadronic features [3, 4] will be reviewed. These techniques will then be applied to scalar gluonium, where instanton contributions are known to be crucial for a consistent Laplace sum-rule analysis [5, 6]. Results of the GSR analysis indicate that the gluonium spectral strength is distributed across a broad energy range below 1.6 GeV [4].

2

Foundations of Gaussian Sum-Rules

The general formulation of GSRs will be reviewed in the context of scalar gluonium probed by the following correlation function.
2 Π(Q ) = i d4x eiq·x O|T {J(x), J(0)}|O , Q2 = −q 2 (4) π2 β(α)Gaµν (x)Gaµν (x) αβ0 Gaµν = ∂µ Aaν − ∂ν Aaµ + gf abc Abµ Acν    α 2  α 3 α(ν) 2 d = −β β (α) = ν − β + ... 0 1 dν 2 π π π 11 1 51 19 − nf , β1 = − nf , . . . β0 = 4 6 8 24 J(x) = −

(5) (6) (7) (8)

The current J(x) is renormalization-group invariant in the chiral limit of nf massless quarks as needed to probe physical (renormalization-group invariant) hadronic states.

27

Theoretical High Energy Physics

From the asymptotic form and assumed analytic properties of (4) follows a dispersion relation with three subtraction constants
Q6 ∞ ρ(t) 1 4 ′′ 2 2 ′ dt , Q2 > 0 Π(Q ) − Π(0) = Q Π (0) + Q Π (0) − 3 2 π t0 t (t + Q2 ) (9) where ρ(t) is the hadronic spectral function with physical threshold t0 , and the subtraction constant Π(0) has been included on the side of the equation containing the QCD prediction because it is determined by the low-energy theorem [7] 8π 2 Π(0) ≡ lim Π(Q ) = J . (10) Q2 →0 β0 The undetermined subtraction constants Π′ (0) and Π′′ (0) and field theoretical divergences in Π(Q2 ) proportional to Q4 are eliminated in an integerweighted family of Gaussian sum-rules1

 τ s − i∆) − (ˆ s − i∆)k Π(−ˆ s + i∆) (ˆ s + i∆)k Π(−ˆ Gk (ˆ s, τ ) ≡ B π i∆ (11) where k = −1, 0, 1, . . . and with the Borel transform B defined by  N d (−∆2 )N B ≡ lim . (12) Γ(N ) d∆2 N,∆2 →∞ ∆2 /N ≡4τ

Applying definition (11) to both sides of (9) annihilates the undetermined low-energy constants and the field theoretical divergence contained in Π(Q2 ). Using the identity 

  1 (∆2 )n −a n = for n ≥ 0 , (13) (−a) exp B 2 ∆ +a 4τ 4τ then leads to the following GSR family:   2


∞ −ˆ s −(ˆ s − t)2 1 1 k Π(0) = ρ(t)dt exp t exp Gk (ˆ s, τ )+δk −1 √ 4τ 4τ π 4πτ t0 (14) 1

This definition is a natural generalization of that given in [2]. To recover the original Gaussian sum-rule, we simply let k = 0 in (11).

28

Steele, Harnett, and Orlandini

Calculation of the Borel transform is achieved through an identity relating (12) to the inverse Laplace transform [2] B[f (∆2 )] =

1 −1 L [f (∆2 )] 4τ

(15)

where, in our notation, 1 L [f (∆ )] = 2πi −1

2




a+i∞ 2

f (∆ ) exp

a−i∞



∆2 4τ



d∆2

(16)

with a chosen such that all singularities of f lie to the left of a in the complex ∆2 -plane. With a change of variables, the calculation of the GSR reduces to [4] 


1 1 −(ˆ s + w)2 k s, τ ) = √ Π(w)dw (17) Gk (ˆ (−w) exp 4τ 4πτ 2πi Γ1 +Γ2 where Γ1 and Γ2 are the parabolae depicted in Figure 1.

Figure 1: Contour of integration Γ1 + Γ2 defining the Gaussian sum-rule. The wavy line on the negative real axis denotes the branch cut of Π(z).

3

Normalized Gaussian Sum-Rules

Studies of Gaussian sum-rules have traditionally focussed on their connection with finite-energy sum-rules as established through the diffusion 29

Theoretical High Energy Physics

equation

∂ 2 Gk (ˆ s, τ ) ∂Gk (ˆ s, τ ) = . 2 ∂ˆ s ∂τ In particular, the resonance(s) plus continuum model

(18)

ρ(t) = ρhad (t) + θ (t − s0 ) ImΠQCD (t) ,

(19)

when ρhad (t) is evolved through the diffusion equation (18), only reproduces the QCD prediction at large energies (τ large) if the resonance and continuum contributions are balanced through the finite-energy sum-rules [2]
s0 1 tn ρhad (t) dt , n = integer . (20) Fn (s0 ) = π t0

Within the resonance(s) plus continuum model (19), the continuum contribution to the GSRs is determined by QCD 


∞ −(ˆ s − t)2 1 1 cont k t exp Gk (ˆ s, τ, s0 ) = √ ImΠQCD (t)dt , (21) 4τ π 4πτ s0 and is thus combined with Gk (ˆ s, τ ) to give the total QCD contribution GQCD (ˆ s, τ, s0 ) ≡ Gk (ˆ s, τ ) − Gcont (ˆ s, τ, s0 ) k k

,

(22)

resulting in the final relation between the QCD and hadronic sides of the GSRs.  2 1 −ˆ s QCD Π(0) s, τ, s0 ) +δk −1 √ Gk (ˆ exp 4τ 4πτ

 (23)
∞ −(ˆ s − t)2 1 had k = ρ (t)dt t exp 4τ π t0 Integrating both sides of (23) reveals that the normalization of the GSRs is related to the finite-energy sum-rules
∞

GQCD (ˆ s, τ, s0 )dˆ s + δk −1 Π(0) k

1 = π


∞

t0

−∞

30

tk ρhad (t)dt .

(24)

Steele, Harnett, and Orlandini

Thus the diffusion equation analysis [2] relates the normalization of the GSR to the finite-energy sum-rules. Information independent of this relation is extracted from the normalized GSRs  2 1 −ˆ s √ Π(0) GQCD (ˆ s , τ, s ) + δ exp 0 k −1 4πτ k 4τ QCD s, τ, s0 ) ≡ (25) Nk (ˆ QCD Mk,0 (τ, s0 ) + δk −1 Π(0)
∞ sˆn Gk (ˆ s, τ, s0 )dˆ s , n = 0, 1, 2, . . . , (26) Mk,n (τ, s0 ) = −∞

which are related to the hadronic spectral function via   ∞ k −(ˆ s−t)2 1 had √1 ρ (t)dt t exp 4τ π 4πτ t0 ∞ 1 s, τ, s0 ) = dt . NkQCD (ˆ tk π ρhad (t) t0

4

(27)

Gaussian Sum-Rule Analysis Techniques

In the single narrow resonance model, ρhad (t) takes the form ρhad (t) = πf 2 δ(t − m2 )

(28)

where m and f are respectively the resonance mass and coupling. With such an ansatz, the normalized Gaussian sum-rule (27) becomes 

1 (ˆ s − m2 )2 QCD . (29) Nk (ˆ s, τ, s0 ) = √ exp − 4τ 4πτ Deviations from the narrow-width limit are proportional to m2 Γ2 /τ , so this narrow-width model may actually be a good numerical approximation. Phenomenological analysis of the single narrow resonance model proceeds from the observation that (29) has a maximum value (peak) at sˆ = m2 independent of the value of τ . The value of s0 is then optimized by minimizing the τ dependence of the sˆ peak position of the QCD prediction, and the resulting τ -averaged sˆ peak position leads to a prediction of the resonance mass [3]. The ρ meson illustrates this single-resonance analysis technique and demonstrates that GSRs can be used to predict resonance properties. The 31

Theoretical High Energy Physics

correlation function of the vector-isovector correlation function results in the following (k = 0) GSR     √   sˆ α ( τ) s0 − sˆ 1 QCD √ √ G0 (ˆ erf s, τ, s0 ) = 1+ + erf 16π 2 π 2 τ 2 τ  2 sˆ sˆ √ exp − C4v O4v  − 4τ 32π 2 τ πτ    2 sˆ2 sˆ 1 √ −1 + exp − C6v O6v  + 2 2τ 4τ 64π τ πτ    2 2 sˆ sˆ sˆ √ exp − C8v O8v  , −1 + − 2 2 6τ 4τ 128π τ πτ where π  2 αG − 8π 2 m ¯ q q (30) 3 896 3 C6v O6v  = − q q)2 (31) π α (¯ 81 and SU (2) symmetry along with the vacuum saturation hypothesis have been employed. For brevity, we refer to the literature [8] for the expressions for the dimension eight condensates, and simply use (30) to establish a convention consistent with [9]. Note that the running coupling is referenced √ to the energy scale τ , a point which will be discussed in the next Section. The non-perturbative QCD condensate contributions in (30) are exponentially suppressed for large sˆ. Since sˆ represents the location of the Gaussian peak on the phenomenological side of the sum-rule, the nonperturbative corrections are most important in the low-energy region, as anticipated by the role of QCD condensates in relation to the vacuum properties of QCD. This explicit low-energy role of the QCD condensates clearly exhibited for the Gaussian sum-rules is obscured in the Laplace sum-rules. The QCD inputs used for the ρ meson analysis are C4v O4v  =

Λ(3) = 300 MeV  q q = −fπ2 m2π αG2 = (0.045 ± 0.014) GeV4 , 2m ¯  896 3  π 1.8 × 10−4 GeV6 , fvs = 1.5 ± 0.5 C6v O6v  = −fvs 81 C8v O8v  = (0.40 ± 0.16) GeV8 , 

32

(32) (33) (34) (35)

Steele, Harnett, and Orlandini

consistent with the condensate parameters in [9]. The criteria of τ stability of sˆ peak predicts the following values for mρ and s0 [3] mρ = (0.75 ± 0.07) GeV ,

s0 = (1.2 ± 0.2) GeV2

(36)

in excellent agreement with the known value of the ρ mass. Furthermore, the phenomenological and QCD sides of the (normalized) Gaussian sumrules shown in Figure 2 are in superb agreement even for very small values of τ [3]. This agreement is particularly impressive since there are no free parameters corresponding to the normalization of the curves in this analysis.

Figure 2:

Comparison of the vector-current theoretical prediction for with the single resonance phenomenological model. The τ values used for the four pairs of curves, from top to bottom in the figure, are respectively τ = 0.5 GeV4 , τ = 1.0 GeV4 , τ = 2.5 GeV4 , and τ = 4.0 GeV4 . Note the almost complete overlap between the QCD prediction and phenomenological model.

N0QCD (ˆ s, τ, s0 )

33

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In more complicated resonance models, the sˆ peak position of the phenomenological model begins to develop τ dependence which is welldescribed by [3, 4] sˆpeak (τ, s0 ) = A +

C B + 2 τ τ

.

(37)

Analysis of how the sˆ peak “drifts” with τ in comparison with the behaviour (37) then allows optimization of s0 . After optimization of s0 , the resonance model parameters are extracted from various moments of the GSRs. For example, a square pulse2 centred at t = m2 with total width 2mΓ leads to the following (k = 0) normalized GSR [4]   

 sˆ − m2 − mΓ sˆ − m2 + mΓ 1 QCD √ √ − erf . erf s, τ, s0 ) = N0 (ˆ 4mΓ 2 τ 2 τ (38) Expansion of (38) for small Γ demonstrates that deviations from the narrow width limit (29) scale as m2 Γ2 /τ . The resonance parameters can then be determined by the following moment combinations [see (26)] of the righthand side of (38) [4] M0,1 = m2 M0,0 2  1 M0,1 M0,2 2 − = m2 Γ2 σ0 − 2τ ≡ M0,0 M0,0 3

(39) ,

(40)

where it is understood that the QCD expressions at the optimized value of s0 are used on the left-hand side.3 Moments also provide a method for testing the accuracy of agreement between the QCD and phenomenological sides of the normalized GSR beyond a simple χ2 measure which could be extracted from plots such as Figure 2. For example, a combination of third-order moments representing 2

This model would describe a broad, structureless feature in the hadronic spectral function. 3 The residual moment combinations leading to the resonance parameters must be τ independent, providing a consistency check on the analysis. Weak residual τ dependence is averaged over the τ range used to extract the optimized s0 from the peak-drift analysis.

34

Steele, Harnett, and Orlandini

the asymmetry of the sˆ dependence about its average value results in [4] (3) A0

M0,3 −3 = M0,0



M0,2 M0,0



M0,1 M0,0



+2



M0,1 M0,0

3

=0 ,

(41)

and hence a deviation of the QCD value of this moment from its value of (3) A0 = 0 in the square pulse model indicates a shortcoming of the model spectral function in comparison with QCD. The procedure for studying an N -parameter resonance model is easily generalized. The peak-drift analysis is used to optimize s0 , the lowest N moments are used to determine the resonance model parameters, and the next-highest moment combination is employed as a test of the accuracy of the model’s agreement with the QCD prediction.

5

Scalar Gluonium Gaussian Sum-Rules

The lowest two Gaussian sum-rules for scalar gluonium contain perturbative, condensate and instanton corrections, and are given by the following expressions [4]. 


s0 1 −(ˆ s − t)2 QCD s, τ, s0 ) = − √ G−1 (ˆ t dt exp 4τ 4πτ 0  

  t t 2 2 + 3a log × (a0 − π a2 ) + 2a1 log 2 ν2 ν2  2   2  −ˆ s c0 sˆ d0 sˆ 1 −b0 J + O6  − − 1 O8  exp +√ 4τ 2τ 4τ 2τ 4πτ  


s0
√  √ 16π 3 −(ˆ s − t)2 4 J2 ρ t Y2 ρ t dt −√ t exp dn(ρ)ρ 4τ 4πτ 0  2
2 −ˆ s 128π dn(ρ) −√ exp 4τ 4πτ (42)

35

Theoretical High Energy Physics




s0 1 −(ˆ s − t)2 2 = −√ t dt exp 4τ 4πτ 0  

  t t 2 2 × (a0 − π a2 ) + 2a1 log + 3a2 log 2 ν ν2 


s0 −(ˆ s − t)2 1 dt exp b1 J −√ 4τ 4πτ 0   2 1 −ˆ s d0 sˆ O8  c0 O6  − +√ exp 4τ 2τ 4πτ  


s0
√  √ 16π 3 −(ˆ s − t)2 2 4 −√ J2 ρ t Y2 ρ t dt dn(ρ)ρ t exp 4τ 4πτ

GQCD (ˆ s, τ, s0 ) 0

0

(43)

The perturbative coefficients in these expressions are  α 2  α 2  659 α a0 = −2 1+ + 247.480 π 36 π π   α 4  α 3 9 α , a2 = −10.1250 + 65.781 a1 = 2 π 4 π π

(44) (45)

as obtained from the three-loop MS calculation of the correlation function [10]. As a result of renormalization group scaling of the GSRs [3], the coupling in the perturbative coefficients is implicitly the running coupling √ at the scale ν 2 = τ in the MS scheme   β¯1 log L 1 1  ¯2  2 α(ν 2 ) ¯ (46) = − 2 + 3 β1 log L − log L − 1 + β2 π β0 L (β0 L) (β0 L)  2 βi ν 9 3863 (47) L = log , β¯i = , β0 = , β1 = 4 , β2 = 2 Λ β0 4 384 with ΛM S ≈ 300 MeV for three active flavours, consistent with current estimates of α(Mτ ) [1]. The condensate contributions in (42), (43) involve next-to-leading order [12] contributions4 from the dimension four gluon condensate J and 4

The calculation of next-to-leading contributions in [12] have been extended non-trivially to nf = 3 from nf = 0, and the operator basis has been changed from  2 αG to J.

36

Steele, Harnett, and Orlandini

leading order [13] contributions from gluonic condensates of dimension six and eight   O6  = gfabc Gaµν Gbνρ Gcρµ (48)       2 2 O8  = 14 αfabc Gaµρ Gbνρ − αfabc Gaµν Gbρλ (49)

  α 2  2 175 α α α 2 α 1+ , b1 = −9π , d0 = 8π 2 . , c0 = 8π b0 = 4π π 36 π π π π (50) The remaining terms in the GSRs (42), (43) represent instanton contributions obtained from single instanton and anti-instanton [11] (i.e., assuming that multi-instanton effects are negligible [14]) contributions to the scalar gluonic correlator [5, 6, 13, 15]. The quantity ρ is the instanton radius, n(ρ) is the instanton density function, and J2 and Y2 are Bessel functions in the notation of [16]. The instanton contributions to the Gaussian sum-rules can be interpreted [6] as naturally partitioning into an instanton continuum portion devolving from
 √  √ 1 inst 3 4 2 ImΠ (t) = −16π dn(ρ) ρ t J2 ρ t Y2 ρ t (51) π and a contribution which, like the Π(0) low-energy theorem (LET) term, appears only in the k = −1 sum-rule  2
−ˆ s 128π 2 dn(ρ) . (52) exp −√ 4τ 4πτ This asymmetric role played by the instanton is crucial in obtaining a consistent analysis from these two sum-rules. In the absence of instanton contributions the LET tends to dominate the left-hand side of (23), corresponding to a massless state in the single-narrow resonance model (29). However, the higher-weighted sum-rules are independent of the LET, and in the absence of instantons lead to a much larger mass scale in sum-rule analyses. This discrepancy between the LET-sensitive and LET-insensitive sum-rules has been shown to be resolved when instanton contributions are included in the Laplace sum-rules [5, 6]. A similar qualitative behaviour 37

Theoretical High Energy Physics

emerges from the Gaussian sum-rules, since the LET term in (23) and the LET-like instanton contribution (52) have the same functional form [i.e., each is proportional to exp (−ˆ s2 /(4τ ))] and occur with opposite sign in the left-hand side of (23). Thus there exists a cancellation between these effects, which is easily verified as being significant in the instanton liquid model [17] n(ρ) = nc δ(ρ − ρc ) nc = 8.0 × 10−4 GeV4

,

ρc =

(53) 1 GeV−1 0.6

,

(54)

along with a standard value for the gluon condensate [18] αG2  = (0.07 ± 0.01) GeV4

.

(55)

This qualitative argument is upheld by the detailed GSR analysis presented in [4]. However, the k = −1 GSR analysis is more sensitive to QCD uncertainties, justifying a focus on the k = 0 GSR for the remainder of this paper.

6

Gaussian Sum-Rule Analysis of Scalar Gluonium

A single narrow resonance model analysis of the k = 0 Gaussian sumrule results in a mass scale of approximately 1.3 GeV, but leads to poor agreement between the phenomenological model and QCD prediction as shown in Figure 3, indicating that the gluonium spectral function is poorly described by a single narrow resonance. Furthermore, in the narrow width model the second-order moment combination (40) should satisfy σ02 − 2τ = 0 ,

(56)

but Figure 4 shows a substantial deviation from this behaviour [4]. Since this moment combination is related to the width of the GSR, we conclude that the gluonium resonance strength must be distributed over a significant energy range. 38

Steele, Harnett, and Orlandini

Figure 3: Comparison of the theoretical prediction for N0QCD (ˆs, τ, s0 ) with the single narrow resonance phenomenological mode The τ values used for the three pairs of curves, from top to bottom in the figure, are respectively τ = 2.0 GeV4 , τ = 3.0 GeV4 , and τ = 4.0 GeV4 . Note the prominent disagreement between the QCD prediction and phenomenological model in the vicinity of the peaks.

The clear failure of the single narrow resonance model, indicative of distributed resonance strength significant enough to be resolved by the GSRs, is a significant conclusion in its own right, but various distributed resonance strength models have also been analyzed [4]. In order of increasing number of parameters (and increasing complexity) they are 1. Single non-zero width models (2 parameters: mass, width) 2. Two narrow resonance model (3 parameters: two masses, relative resonance strength) 3. Narrow resonance plus a non-zero width resonance models (4 parameters: two masses, one width, relative resonance strength)

39

Theoretical High Energy Physics

Figure 4: Plot of σ02 for the theoretical prediction (dotted curve) compared

with σ02 = 2τ for the single-resonance model (solid curve) for the k = 0 sumrule.

In the single non-zero width models, elaborations on the square pulse include a Gaussian resonance and a skewed Gaussian resonance models   2 (t − m2 ) ρ(t) ∼ exp − (57) 2Γ2   2 2 (t − m ) (58) ρ(t) ∼ t2 exp − 2Γ2 which are analytically and numerically simpler to analyze than a BreitWigner shape. In the Gaussian resonance models, the √ quantity Γ can be related to an equivalent Breit-Wigner width by ΓBW = 2 log 2Γ/m, and the t2 factor in the skewed Gaussian is chosen for consistency with (lowenergy) two-pion decay rates [7, 19]. The relevant moment combinations

40

Steele, Harnett, and Orlandini

for the Gaussian model (57) are [4]

(3) A0

where

M0,1 = m2 + Γ∆ M0,0 σ02 − 2τ = Γ2 − m2 Γ∆ − Γ2 ∆2   = m4 Γ − Γ3 ∆ + 3m2 Γ2 ∆2 + 2Γ3 ∆3  

 m4 exp − 2Γ2 2   ∆= π 1 + erf √m2

.

(59) (60) (61)

(62)

2Γ

The quantity ∆ is small, so the Gaussian model cannot accommodate large (3) values of the asymmetry A0 . However, the skewed Gaussian naturally leads to a larger asymmetry as reflected in the following results for the relevant moment combinations [4] M0,1 m2 (m4 + 3Γ2 ) + O(∆) = M0,0 m4 + Γ2 Γ2 (m8 + 3Γ4 ) σ02 − 2τ = + O(∆) (m4 + Γ2 )2 4m2 Γ6 (m4 − 3Γ2 ) (3) + O(∆) . A0 = (m4 + Γ2 )3

(63) (64) (65)

The results for the single non-zero resonance models are shown in Table 1, and indicate that the non-zero width models underestimate the QCD (3) value of the asymmetry A0 by at least an order of magnitude [4]. This failure suggests that further phenomenological models which can generate a larger asymmetry are required. A model containing two narrow resonances   (66) ρhad (t) = π f12 δ(t − m21 ) + f22 δ(t − m22 )

results in the following normalized Gaussian sum-rule (27)  

(ˆ s − m21 )2 1 QCD s, τ, s0 ) = √ r1 exp − N0 (ˆ 4τ 4πτ 

(ˆ s − m22 )2 . + r2 exp − 4τ 41

(67)

Theoretical High Energy Physics

mass (GeV) square pulse 1.30 ± 0.17 unskewed gaussian 1.30 ± 0.17 skewed gaussian 1.17 ± 0.15 QCD —

width (GeV) 0.59 ± 0.07 0.40 ± 0.05 0.49 ± 0.06 —

(3)

A0 (GeV6 ) 0 0.000342 0.00943 -0.0825

Table 1: The results of a k = 0 Gaussian sum-rules analysis of a variety of non-zero resonance width models. The quoted resonance parameters include (3) uncertainties introduced by the QCD input parameters, except for A0 which is obtained from the central values. For the Gaussian resonance models, the given width is the equivalent Breit-Wigner width where

f22 f12 , r = r1 + r2 = 1 (68) 2 f12 + f22 f12 + f22 parameterize the relative strength of the two resonances. This model can accommodate a large asymmetry parameter through an asymmetric distribution of resonance strength, as indicated by the following moment combinations [4] r1 =

1 M0,1 = (z + ry) M0,0 2 1 σ02 − 2τ = y 2 (1 − r2 ) 4 1 (3) A0 = − ry 3 (1 − r2 ) 4     1  S0 − 12τ 2 − 12τ σ02 − 2τ = y 4 1 − r2 1 + 3r2 16

(69) (70) (71) ,

(72)

where

z = m21 + m22 S0 ≡

,

M0,4 M0,3 M0,1 −4 M0,0 M0,0 M0,0

y = m21 − m22 , r = r1 − r2 2 4   M0,2 M0,1 M0,1 +6 −3 M0,0 M0,0 M0,0

(73) .

(74)

The lowest three moment combinations (69)–(71) are used to determine the three resonance parameters, and the final fourth-order residual combination (72) is used as a test of the agreement between QCD and the 42

Steele, Harnett, and Orlandini

phenomenological model. The resulting resonance parameters are [4] m1 = (0.98 ± 0.2) GeV, m2 = (1.4 ± 0.2) GeV, r1 = 0.28 ∓ 0.06, r2 = 1 − r1 .

(75)

Results from the fourth-order moment measure of the accuracy between QCD and the two-narrow resonance model is given in Table 2, and indicate an approximately 50% discrepancy between QCD and the phenomenological model [4]. S0 − 12τ 2 − 12τ (σ02 − 2τ ) QCD 0.170 GeV8 double resonance model 0.074 GeV8 Table 2: The results of a k = 0 Gaussian sum-rules analysis of the double narrow resonance model using central values of the QCD parameters.

As a final attempt to improve the agreement between the next-highest moment test of the agreement between QCD and the phenomenological model, consider a narrow resonance of mass m and relative strength rm , combined with a second resonance of mass M , width Γ and relative strength rM . Of course for a normalized GSR we must have rm + rM = 1, so this defines a four-parameter model. For example, when a square pulse is used to describe the second resonance the resulting normalized GSR is 

1 (ˆ s − m2 )2 QCD s, τ, s0 ) = rm √ N0 (ˆ exp − 4τ 4πτ

    2 sˆ − M + M Γ sˆ − M 2 − M Γ rM √ √ erf − erf , + 4M Γ 2 τ 2 τ (76) where rm + rM = 1. Four moment combinations are then needed to define the resonance parameters, and a fifth-order moment combination serves as a measure of the agreement between QCD and the phenomenological

43

Theoretical High Energy Physics

model [4] 1 M0,1 = (z + ry) + O (∆) M0,0 2  1  1 σ02 − 2τ = y 2 1 − r2 + Γ2 (z − y) (1 − r) + O (∆) 4 12     1 1 (3) A0 = − y 3 r 1 − r2 − Γ2 y 1 − r2 (z − y) + O (∆) 4 8     1  S0 − 12τ 2 − 12τ σ02 − 2τ = y 4 1 − r2 1 + 3r2 16   1 + Γ2 y 2 (1 + r) 1 − r2 (z − y) 8 1 + Γ4 (1 − r) (z − y)2 + O (∆) 40    1 (5) (3) A0 − 20τ A0 = − y 5 r 1 − r2 1 + r2 8   5 − Γ2 y 3 1 − r2 (1 + r)2 (z − y) 48   1 − Γ4 y 1 − r2 (z − y)2 + O (∆) 16

(77) (78) (79)

(80)

(81)

where the fifth-order asymmetry moment is (5) Ak

 2 Mk,5 Mk,4 Mk,1 Mk,3 Mk,1 = −5 + 10 Mk,0 Mk,0 Mk,0 Mk,0 Mk,0  3 5  Mk,1 Mk,2 Mk,1 +4 . − 10 Mk,0 Mk,0 Mk,0

(82)

The resulting resonance parameters and the QCD prediction and phenomenological values of the fifth-order asymmetry parameter are shown in Tables 3 and 4 [4]. An interesting feature of the results are that the wide resonance is consistently found to be lighter than the narrow resonance, and that the narrow plus skewed Gaussian model shows only a 20% deviation from the QCD value of the fifth order asymmetry parameter.

44

Steele, Harnett, and Orlandini

square gauss skewed

m (GeV) M (GeV) Width (GeV) rm 1.33 ± 0.18 1.23 ± 0.18 0.95 ± 0.12 0.6 ± 0.13 1.41 ± 0.19 1.23 ± 0.15 0.52 ± 0.06 0.49 ± 0.13 1.38 ± 0.13 1.06 ± 0.21 0.69 ± 0.07 0.44 ± 0.04

Table 3: Resonance parameters obtained in the various two-resonance scenarios. The label “square” denotes the narrow plus square pulse model, “gauss” refers to narrow plus Gaussian resonance model, and “skewed” indicates the narrow plus skewed Gaussian model. The mass M denotes the state associated with the quoted width, which corresponds to the equivalent Breit-Wigner width for the Gaussian models. The quoted resonance parameters include uncertainties introduced by the QCD input parameters.

(5)

square gauss skewed QCD

(3)

A0 − 20τ A0 −0.11 GeV10 −0.13 GeV10 −0.19 GeV10 −0.24 GeV10

Table 4:

Fifth-order residual moments as determined by QCD and the resonance models for central values of the QCD input parameters.

45

Theoretical High Energy Physics

7

Comparison of Distributed Resonance Strength Models

All the distributed resonance strength models considered in the previous section lead to the excellent agreement between the QCD and phenomenological sides of the normalized GSR shown in Figure 5, and clearly resolve the discrepancy evident in Figure 3 corresponding to the single resonance model [4]. However, a χ2 measure of the agreement between the theoretical and phenomenological curves in Figure 5 result in a χ2 which is an order of magnitude smaller in the two-resonance models [4]. The combination of this result with the Table 1 observation of at least an order of magnitude disagreement with the QCD value of the third-order asymmetry is compelling evidence in favour of a two-resonance scenario for distributed resonance strength, with an upper bound on the heavier mass of about 1.6 GeV obtained from Eq. (75) and Table 3. The various two-resonance models all have similar values of χ2 , which does not provide a useful criteria to distinguish between these models. However, the narrow plus skewed Gaussian resonance model leads to the best (≈ 20%) agreement with the QCD value of the next-order moment. In this scenario, the mass predictions and the pattern of a lighter broad resonance and a heavier narrow resonance is consistent with the identification of gluonium content in the f0 (1370) and f0 (1500).

8

Conclusions

In this paper, the formulation of Gaussian sum-rules has been reviewed. The key qualitative feature of Gaussian sum-rules is their enhanced sensitivity to excited states in comparison with Laplace sum-rules. Thus any resonance strong enough to stand out from the QCD continuum should reveal itself in a GSR analysis. The significance of normalized GSRs has been emphasized since they provide information independent of the the finite-energy sumrule constraint that arises from the evolution of GSRs through the diffusion equation [3]. Methods for obtaining phenomenological predictions from GSRs have been reviewed and applied to the ρ meson to demonstrate the predictive 46

Steele, Harnett, and Orlandini

Figure 5: Comparison of the theoretical prediction for N0QCD (ˆs, τ, s0 ) with the distributed resonance phenomenological models. The τ values used for the three pairs of curves, from top to bottom in the figure, are respectively τ = 2.0 GeV4 , τ = 3.0 GeV4 , and τ = 4.0 GeV4 . The almost perfect overlap between QCD and phenomenology is particularly impressive because there are no free parameters corresponding to the normalization of the QCD prediction and phenomenological models.

power of GSRs [3]. These techniques are easily adapted to a variety of resonance models, and through combinations of GSR moments, provide a natural criterion for testing the accuracy of the phenomenological model in comparison with QCD [4]. The important role of instanton effects in relation to the low-energy theorem has been illustrated for the GSRs of scalar gluonic currents. This analysis provides additional support for the interpretation of instanton effects first developed for Laplace sum-rules [6]. An identical, and natural, partitioning of instanton effects into a continuum and LET-like contribution resolves discrepancies between the LET-sensitive and LET-insensitive sum-rules in both the Laplace and Gaussian sum-rules. A detailed phenomenological analysis of the k = 0 GSR is presented 47

Theoretical High Energy Physics

because it is is less sensitive to QCD parameter uncertainties than the k = −1 (LET-dependent) GSR. The analysis clearly rules out a single narrow resonance scenario, and further indicates that gluonium resonance strength is spread over a broad energy region. In particular, evidence exists for two resonances below ≈ 1.6 GeV with a significant gluonium content, with the lighter resonance having a substantial width [4]. Such a scenario is consistent gluonium content of f0 (1370) and f0 (1500).

Acknowledgments: TGS and DH are grateful for research funding from the Natural Sciences & Engineering Research Council of Canada (NSERC). Many thanks to Amir Fariborz and SUNY IT for careful organization and generous hospitality during the workshop on Theoretical High-Energy Physics.

References [1] K. Hagiwara et al., Phys. Rev. D66 (2002) 010001. [2] R.A. Bertlmann, G. Launer, E. de Rafael, Nucl. Phys. B250 (1985) 61. [3] G. Orlandini, T.G. Steele and D. Harnett, Nucl. Phys. A686 (261) 2001. [4] D. Harnett, T.G. Steele, Nucl. Phys. A695 (2001) 205. [5] E.V. Shuryak, Nucl. Phys. B203 (1982) 116; H. Forkel, Phys. Rev. D64 (2001) 034015. [6] D. Harnett, T.G. Steele, V. Elias, Nucl. Phys. A686 (2001) 393. [7] V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B191 (1981) 301. [8] V. Gim´enez, J. Bordes, J. Pe˜narrocha, Phys. Lett. B223 (1989) 251; S.N. Nikolaev, H.R. Radyushkin, Nucl. Phys. B213 (1983) 285; D.J. Broadhurst, S.C. Generalis, Phys. Lett. B165 (1985) 175. 48

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[9] V. Gim´enez, J. Bordes, J. Pe˜narrocha, Nucl. Phys. B357 (1991) 3. [10] K.G. Chetyrkin, B.A. Kneihl and M. Steinhauser, Nucl. Phys. B510 (1998) 61. [11] A. Belavin, A. Polyakov, A. Schwartz and Y. Tyupkin, Phys. Lett. B59 (1975) 85; G. ’t Hooft, Phys. Rev. D14 (1976) 3432. [12] E. Bagan and T.G. Steele, Phys. Lett. B234 (1990) 135. [13] V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B165 (1980) 67. [14] T. Sch¨aefer and E.V. Shuryak, Phys. Rev. Lett. 75 (1995) 1707. [15] B.V. Geshkenbein and B.L. Ioffe, Nucl. Phys. B166 (1980) 340; B.L. Ioffe and A.V. Samsonov, Phys. of Atom. Nucl. 63 (2000) 1527. [16] M. Abramowitz and I.E. Stegun, Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards Applied Mathematics Series, Washington) 1972. [17] E.V. Shuryak, Nucl. Phys. B203 (1982) 93. [18] S. Narison, Nucl. Phys. B (Proc. Supp.) 54A (1997) 238. [19] V.A. Novikov, M.A. Shifman, Z. Phys. C8 (1981) 43; M.A. Shifman, Z. Phys. C9 (1981) 347.

49

————————– SUNY Institute of Technology Conference on Theoretical High Energy Physics June 6th, 2002

————————– Possible Applications of Chiral Lagrangians in QCD Sum-Rules Amir H. Fariborz

1

Department of Mathematics/Science SUNY Institute of Technology Utica, New York, 13504-3050.

————————————————– Abstract In this article possible applications of chiral Lagrangians in QCD sumrules are discussed. Particularly, phenomenological modeling of the κ(900) resonance using a non-linear chiral Lagrangian is studied in some detail.

1

Electronic address: [email protected]

51

Theoretical High Energy Physics

QCD Lagrangian is formulated in terms of quarks and gluons, but the physical states observed in experiments are the hadrons. The objective of the quark hardon duality (QHD) is to provide a bridge between QCD and hadrons, and make it possible to describe properties of the hadrons using QCD Lagrangian [1]. Based on QHD, the inclusive hadronic cross sections at high energies coincide with cross sections calculated from QCD. Examples include processes such as e+ e− annihilation, hadronic τ decays and inclusive decays of heavy quarks. There are, however, ambiguities in the concept and the implementation of the QHD. For example, there is no standard way of defining the duality, or estimating the theoretical uncertainties. The two point correlation function in quantum field theory is usually employed in discussions of the QHD. For example, in e+ e− annihilation cross section, the relevant two point correlation function is Πµν (q) = i




d4 x eiqx
0|T Jµ (x)Jν† (0)|0

= (qµ qν − q 2 gµν )Π(q 2 )

(1)

which is transverse. The isovector vector current is Jµ =

¯ µd u¯γµ u − dγ √ 2

(2)

This correlation function is related to the spectral density ρ(s), which is an observable quantity and is defined as: ρ(s) =

12π Im[Π(s)] Nc

(3)

It is known that the ratio of the e+ e− annihilation cross sections is proportional to the hadronic contribution to the correlation function: R=

σ(e+ e− → hadrons) α Im[ΠHad. (q 2 )] σ(e+ e− → µ+ µ− )

(4)

In the deep Euclidean region, the field theory computation of the correlation function shows that ΠF.T. (Q2 ) = −

Nc Nc ln(Q2 ) ⇒ Im[Π(Q2 → −q 2 )] = − 2 12π 12π 52

(5)

Fariborz

in agreement with the measured R. This shows a connection between perturbative QCD and hadronic contribution in the deep Euclidean region. It is of course a non-trivial goal to see whether similar connections can be established between the properties of the low-lying hadrons and the QCD Lagrangian. This is elegantly formulated in the framework of the QCD sum rules by Shifman, Vainshtein and Zakharov (the SVZ sum rules) [2], and applied by number of authors to different processes. In this approach the correlation function of interest is calculated in field theory and is compared with the hadronic contribution to this correlation function. There are three types of contributions from field theory ΠF.T. (s) = ΠP ert. (s) + ΠN.P ert. (s) + ΠInst. (s)

(6)

where the terms on the right hand side correspond to the perturbative QCD contribution, the condensate expansion and the instanton contribution, respectively. The condensate expansion has the structure: ΠN.P ert. (q) = CQ (q)
¯ q q + CM (q)
¯ q Gq + CG2 (q)
G2  +CG3 (q)
G3  + C4q (q)
q 4  + · · ·

(7)

where the first term is dimension 3 quark condensate, followed by dimension 5 mixed quark-gluon condensate, dimension 4 and 6 gluon condensates, and dimension 6 quark condensate. The coefficients are determined from OPE. There are two main approaches for determining the values of the condensates; either from first principles in lattice QCD, or within the QCD sum rules framework. The quark condensate can be determined from the Gell-Mann-Oakes-Renner relation: ¯ = −m2 f 2 (mu + md )
¯ uu + dd π π

(8)

where fπ is the pion decay constant and is equal to 137 MeV, and mπ =137 MeV. The gluon condensate is determined from sum rule study of J/ψ
G2  ≈ 0.012 GeV4

(9)

But this estimate is not very accurate and deviations from this value is possible (up to 30% deviations from this value is expected [3]). The dimension 6 quark condensate is not directly known. In the QCD sum-rules 53

Theoretical High Energy Physics

study of the ρ meson, it is known that the factorization hypothesis
q 4  =
q 2 
q 2  + O



1 Nc



(10)

provides a consistent picture. However, this hypothesis is not tested independently, and it is therefore interesting to examine it through a different approach. It is also interesting to independently probe other condensates such as the gluon condensate of Eq. (9) which is not accurately known. The linear sigma model provides a framework in which these condensates can be probed. In particular the linear sigma model of ref. [4] that is formulated in terms of two meson nonets is an excellent framework in which the dimension 6 quark condensate can be probed. This is the first example of application of chiral Lagrangians in QCD sum-rules. We do not discuss this any further here (complete details are given in ref. [4]). In SVZ sum-rules, the imaginary parts of the correlation function calculated in field theory are matched to the hadronic contributions to the correlator. The matching is not unique and certain energy smearing with some weight function are normally used. If the hadron of interest is narrow and isolated from other resonances, the hadronic contribution to the correlator is parametrized by the Breit-Wigner function: Im Π

Had.

(s) = Im



 r

gr 2 mr − s − imr Gr



(11)

where the sum is over all contributing resonances. In the narrow width limit, this approaches to a sum of delta functions lim Im

Gr →0



 r



 gr = π gr δ(s − m2r ) 2 mr − s − imr Gr r

(12)

If the resonace is broad and it interferes with other nearby states, the above parametrization is no longer valid and a more general approach is needed. It turns out that in this case, the regulator Gr in the denominator become momentum dependent, and the important question is how to determine its energy dependence (13) Gr → Gr (s) Chiral Lagrangians provide a possible way to determine the hadronic contributions to the correlation function. This gives the second example of a 54

Fariborz

link between chiral Lagrangians and the QCD sum-rules. Here we discuss the case of κ(900) scalar meson and give some insight into its contribution to the scalar correlator. There are two iso-doublet scalar states below 2 GeV; the K0∗ (1430) which is an experimentally established state [5], and the κ(900) which is observed in some theoretical models including the non-linear chiral Lagrangian of ref.[6]. It is shown in [6] that although this state weakly couples to the πK channel, it has an important contribution to the πK scattering amplitude, without which, it is not possible to describe the πK scattering data. In the scattering amplitude, the κ(900) provides an important background for other contributing intermediate states. It is a non-trivial issue in QCD sum-rules to model the shape of a broad and interfering resonance. Obviously, the Breit-Wigner parametrization is no longer accurate for a broad resonance such as the κ(900). Using the non-linear chiral Lagrangian of ref. [6], here we explore possible ways of modeling κ(900) contribution to the scalar correlator. The imaginary part of the scalar correlator can be found by inclusion of all possible intermediate states, as described by the optical theorem. The single particle contributions are proportional to the propagator of the intermediate scalar: ImΠHad. (s) = Im



2 κ mκ

gκ − s − imκ G′

(14)

where gκ is a constant, and G′ in the denominator is an a priori unknown regulator. This leads to a Breit-Wigner parametrization of the scalar correlator (shown below with a subscript of “BW”): [ImΠHad. (s)]BW =

gκ mκ G′ (m2κ − s)2 + m2κ G′2

(15)

However, if the resonance is not narrow and isolated, we know that the Breit-Wigner shape is not a realistic parametrization due to a number of reasons, including, for example, lack of a proper threshold behavior, and lack of the knowledge of interference with nearby states. These problems are shown in figures 1 and 2. We see that as the resonance gets lighter and broader, the problem with the threshold effect becomes more serious, and therefore, the Breit-Wigner shape becomes less reliable. 55

Theoretical High Energy Physics

]BW/gk

10

G’=200 MeV

had.

Im[Π

G’=100 MeV

5

G’=300 MeV

0 0.5

1 Energy (GeV)

1.5

Figure 1: Contributions of the κ(900) to the scalar correlator using a Breit-Wigner parametrization, with mκ = 900 MeV and different values of G′ .

56

Fariborz

5

mκ=800 MeV

4

mκ=900 MeV

mκ=1000 MeV

Im[Π

had.

]BW/gk

3

2

1

0 0.5

1 Energy(GeV)

1.5

Figure 2: Contributions of the κ(900) to the scalar correlator using a Breit-Wigner parametrization, with G′ = 300 MeV and different values of mκ .

57

Theoretical High Energy Physics

The κ(900) is probed in πK scattering, and therefore, it is logical to compute the imaginary part of the scalar correlator by including the twoparticle intermediate states (i.e. virtual π and K). The exact connection between the imaginary parts of the scalar correlator, and the πK intermediate states should be derived from the optical theorem. Here, we take a more qualitative approach and try to estimates these imaginary parts. It seems reasonable to say that due to the intermediate pion and kaon interactions, the imaginary parts of the scalar correlator should be (at least in the first approximation) proportional to the imaginary parts of the πK scattering amplitude ImΠHad. (s) α ImT (s)

(16)

To compute the πK scattering amplitude T , different contributions must be taken into account. In the context of the chiral Lagrangian of ref. [6], this scattering amplitude is studied in detail. For orientation, let us consider a “toy” model in which κ(900) is the only contributing intermediate state that dominates the low energy region. In this case, the I=1/2 scattering amplitude is simply [6]: Tκ (s) = with

Gκ (s) =

m2κ

mκ Gκ (s) − s − imκ G′

(17)

 3 2 [s − (mπ + mK )2 ][s − (mπ − mK )2 ](s−m2π −m2K )2 γκπK 128mκ πs

(18) in which γκπK is the coupling of κ to the πK channel. The partial decay with of κ(900) is: Γ[κ → πK] = Gκ (m2κ ) (19)

Therefore, Gκ (s) can be rewritten as: 

m2κ [s − (mπ + mK )2 ][s − (mπ − mK )2 ](s − m2π − m2K )2

Γ[κ → πK] Gκ (s) =  s [m2κ − (mπ + mK )2 ][m2κ − (mπ − mK )2 ](m2κ − m2π − m2K )2

(20) The unitarity of the scattering amplitude demands that the regulator G′ in Eq. (17) depends on energy as G′ = Gκ (s) 58

(21)

Fariborz

1 MeV Γ πΚ=300

0.8

200 Γ πΚ=

MeV

Im[Π

Had.

]unitary/Fκ

0.6

90 Γ πΚ=

0.4

60 Γ πΚ=

MeV

MeV

0.2 Γ πΚ=30 MeV

0 0.5

1 Energy (GeV)

1.5

Figure 3: Contributions of the κ(900) to the scalar correlator using the unitary parametrization of Eq. (22), with mκ = 900 MeV and different values of Γ[κ → πK]. Therefore, using relation (16), the energy dependence of the κ(900) contribution to the scalar correlator (shown with subscript “unitary”) is [ImΠHad. (s)]unitary =

Fκ m2κ Gκ (s)2 (m2κ − s)2 + m2κ Gκ (s)2

(22)

where Fκ is a constant. Clearly, in this approach, the kinematics of scattering naturally provides a built-in threshold behavior. This is shown in figures 3 and 4. We see that as Γ[κ → πK] is increased, the scattering amplitude approaches a pure imaginary limit: lim

ΓπK large

Tk = i

(23)

which shows that no information about κ can be extracted from the scattering amplitude. In reality, however, we know that there are other intermediate states that contribute to the scattering amplitude and have a substantial interfer59

Theoretical High Energy Physics

1

mκ=900 MeV

0.8

mκ=1000 MeV

mκ=800 MeV

Im[Π

Had.

]unitary/Fκ

0.6

0.4

0.2

0 0.5

1 Energy (GeV)

1.5

Figure 4: Contributions of the κ(900) to the scalar correlator using the unitary parametrization of Eq. (22), with Γ[κ → πK] = 30 MeV and different values of mκ .

60

Fariborz

ence with the κ(900). In ref. [6] a detailed study of the scattering amplitude is given in which other contributing states such as the I=0 scalars, the vectors, as well as the φ4 contact interactions are all taken into account. Therefore, the s-channel κ(900) contribution in Eq. (17) is only a part of the total scattering amplitude. The best fit of the total scattering amplitude (that includes the contribution of all intermediate states) to the πK scattering data gives the unknown properties of the κ(900) in Eq. (17): mκ = 897 ± 2 MeV G′κ = 322 ± 6 MeV γκπK = 5.00 ± 0.07 GeV−1

(24)

This fit shows that

Γ(κ → πK) = 0.13 (25) G′ which is very different from a pure Briet-Wigner shape, for which this ratio is exactly one. This is a typical behavior of the low-lying scalars σ(550) and κ(900) investigated in refs. [6, 7]. This deviation from one, shows that the κ(900) enters into the scattering amplitude more as a background, and therefore, modeling its contribution to the scalar correlator is a non trivial problem. In this case, we show the imaginary part of the scalar correlator with a subscript “πK fit:” [ImΠHad. (s)]πKf it =

Fκ m2κ G′ G(s) (m2κ − s)2 + m2κ G′ 2

(26)

The energy dependence of this correlator is shown in figure 5 and is compared with the unitarity parametrization of (22). Both curves correspond to the case of fit (24) in which Γ[κ → πK] = 42 MeV and mκ = 897 MeV. We see that the parametrization (26), that includes the effect of the interference between κ(900) and other intermediate states, is smaller in magnitude but more uniformly distributed up to and beyond 1 GeV. Therefore, the effect of its interference with K0∗ (1430) cannot be ignored. This in contrast to the unitary parametrization (22) in which the dominant effect comes from low-energy region below 1 GeV. To further improve the computation of Im [ΠHad. ], the interference with the K0∗ (1430) should be taken into account. An appropriate parametrization for such a double resonance fit is given by the Ramsauer-Townsend 61

Theoretical High Energy Physics

1

0.8

Unitary

Im [Π

Had.

]/Fκ

0.6

0.4

0.2

0 0.5

πΚ fit

1 Energy (GeV)

1.5

Figure 5: Contributions of the κ(900) to the scalar correlator using the unitary parametrization of Eq. (22), and the parametrization described by Eqs. (26) together with the free parameters given in (24).

62

Fariborz

mechanism in which the K0∗ (1430) interferes with the κ(900) and other cross channel background contributions. This is given as [6]: T =

e2iδ m∗ Γ∗ + eiδ sinδ m2∗ − s − im∗ Γ∗

(27)

where δ is the background phase, and m∗ and Γ∗ are the mass and the decay width of the K0∗ (1430). Parametrization (27) guarantees that the scattering amplitude is unitary in the neighborhood of the K0∗ (1430). To compute δ, all cross channels should also be taken into account. We will continue this investigation in future works. The details of the scattering amplitude are given in ref. [6].

Acknowledgments: This work has been supported by grants from the State of New York/UUP Professional Development Committee, and the 2003 Faculty Summer Grant from the School of Arts and Sciences, SUNY Institute of Technology.

References [1] E.C. Poggio, H.R. Quinn and S. Weinberg, Phys. Rev. D13, 1958 (1976). [2] M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147, 385 (1979). [3] A. Di Giacomo, Nucl. Phys. Proc. Suppl. 47, 136 (1996). [4] D. Black, A.H. Fariborz, S. Moussa, S. Nasri and J. Schechter, Phys. Rev. D 64, 014031 (2001). [5] Review of Particle Physics, Phys. Rev. D66 (2002). [6] D. Black, A.H. Fariborz, F. Sannino and J. Schechter, Phys. Rev. D58, 054012 (1998). [7] F.Sannino and J. Schechter, Phys. Rev. D 52, 96 (1995); M. Harada, F. Sannino and J. Schechter, Phys. Rev. D 54, 1991 (1996); Phys . Rev. Lett. 78, 1603 (1997). 63

————————– SUNY Institute of Technology Conference on Theoretical High Energy Physics June 6th, 2002

————————– Magnetic Catalysis and Anisotropic Confinement in QCD V.A. Miransky

1

Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada

————————————————– Abstract The expressions for dynamical masses of quarks in the chiral limit in QCD in a strong magnetic field are obtained. A low energy effective action for the corresponding Nambu-Goldstone bosons is derived and the values of their decay constants as well as the velocities are calculated. The existence of a threshold value of the number of colors Ncthr , dividing the theories with essentially different dynamics, is established. For the number of colors Nc ≪ Ncthr , an anisotropic dynamics of confinement with the confinement scale much less than ΛQCD and a rich spectrum of light glueballs is realized. For Nc of order Ncthr or larger, a conventional confinement dynamics takes place. It is found that the threshold value Ncthr grows rapidly with the 2 > magnetic field [Ncthr > ∼100 for |eB| ∼(1 GeV) ]. In contrast to QCD with a nonzero baryon density, there are no principal obstacles for examining these results and predictions in lattice computer simulations.

1

Electronic address: [email protected]

65

Theoretical High Energy Physics

1

Introduction

Since the dynamics of QCD is extremely rich and complicated, it is important to study this theory under external conditions which provide a controllable dynamics. On the one hand, this allows one to understand better the vacuum structure and Green’s functions of QCD, and, on the other hand, there can exist interesting applications of such models in themselves. The well known examples are hot QCD (for a review see Ref. [1]) and QCD with a large baryon density (for a review see Ref. [2]). Studies of QCD in external electromagnetic fields had started long ago [3, 4]. A particularly interesting case is an external magnetic field. Using the Nambu-Jona-Lasinio (NJL) model as a low energy effective theory for QCD, it was shown that a magnetic field enhances the spontaneous chiral symmetry breakdown. The understanding of this phenomenon had remained obscure until a universal role of a magnetic field as a catalyst of chiral symmetry breaking was established in Refs. [5, 6]. The general result states that a constant magnetic field leads to the generation of a fermion dynamical mass (i.e., a gap in the one-particle energy spectrum) even at the weakest attractive interaction between fermions. For this reason, this phenomenon was called the magnetic catalysis. The essence of the effect is the dimensional reduction D → D −2 in the dynamics of fermion pairing in a magnetic field. In the particular case of weak coupling, this dynamics is dominated by the lowest Landau level (LLL) which is essentially D − 2 dimensional [5, 6]. The applications of this effect have been considered both in condensed matter physics [7, 8] and cosmology (for reviews see Ref. [9]). The phenomenon of the magnetic catalysis was studied in gauge theories, in particular, in QED [10, 11, 12, 13, 14, 15] and in QCD [16, 17, 18]. In the recent work [18], it has been suggested that the dynamics underlying the magnetic catalysis in QCD is weakly coupled at sufficiently large magnetic fields. In this paper, we study this dynamical problem rigorously, from first principles. In fact, we show that, at sufficiently strong magnetic fields, |eB| ≫ Λ2QCD , there exists a consistent truncation of the SchwingerDyson (gap) equation which leads to a reliable asymptotic expression for

66

Miransky

the quark mass mq . Its explicit form reads: 2/3

m2q ≃ 2C1 |eq B| (cq αs )






4Nc π exp − , 2 αs (Nc − 1) ln(C2 /cq αs )

(1)

where eq is the electric charge of the q-th quark and Nc is the number of colors. The numerical factors C1 and C2 equal 1 in the leading approximation that we use. Their value, however, can change beyond this approximation and we can only say that they are of order 1. The constant cq is defined as follows: 



e 1 (2Nu + Nd )   , cq =  eq  6π

(2)

where Nu and Nd are the numbers of up and down quark flavors, respectively. The total number of quark flavors is Nf = Nu  + Nd . The strong coupling αs in the last equation is related to the scale |eB|, i.e., 1 |eB| ≃ b ln 2 , αs ΛQCD

where b =

11Nc − 2Nf . 12π

(3) 

We should note that in the leading approximation the energy scale |eB| in Eq. (3) is fixed only up to a factor of order 1. As we discuss below, because of the running of αs , the value of the dynamical mass (1) grows very slowly with increasing the value of the background magnetic field. Moreover, there may exist an intermediate region of fields where the mass decreases with increasing the magnetic field. Another, rather unexpected, consequence is that a strong external magnetic field suppresses the chiral vacuum fluctuations leading to the (0) generation of the usual dynamical mass of quarks mdyn ≃ 300 MeV in QCD without a magnetic field. In fact, in a wide range of strong magnetic 2 < fields Λ2 < ∼ B ∼ (10TeV) (where Λ is the characteristic gap in QCD without the magnetic field; it can be estimated to be a few times larger (0) than ΛQCD ), the dynamical mass (1) remains smaller than mdyn . As it will be shown in Sec. 4, this point is intimately connected with another one: in a strong magnetic field, the confinement scale, λQCD , is much less than the confinement scale ΛQCD in QCD without a magnetic field. 67

Theoretical High Energy Physics

The central dynamical issue underlying this dynamics is the effect of screening of the gluon interactions in a magnetic field in the region of momenta relevant for the chiral symmetry breaking dynamics, m2q ≪ |k 2 | ≪ 

|eB|. In this region, gluons acquire a mass Mg of order Nf αs |eq B|. This allows to separate the dynamics of the magnetic catalysis from that of confinement. More rigorously, Mg is the mass of a quark-antiquark composite state coupled to the gluon field. The appearance of such mass resembles pseudo-Higgs effect in the 1 + 1 dimensional massive QED (massive Schwinger model) [19] (see below). Since the background magnetic field breaks explicitly the global chiral symmetry that interchanges the up and down quark flavors, the chiral symmetry in this problem is SU (Nu )L × SU (Nu )R × SU (Nd )L × SU (Nd )R × U (−) (1)A . The U (−) (1)A is connected with the current which is an anomaly free linear combination of the U (d) (1)A and U (u) (1)A currents. [The U (−) (1)A symmetry is of course absent if either Nd or Nu is equal to zero]. The generation of quark masses breaks this symmetry spontaneously down to SU (Nu )V × SU (Nd )V and, as a result, Nu2 + Nd2 − 1 gapless Nambu-Goldstone (NG) bosons occur. In Sec. 3, we derive the effective action for the NG bosons and calculate their decay constants and velocities. The present analysis is heavily based on the analysis of the magnetic catalysis in QED done by Gusynin, Miransky, and Shovkovy [11]. A crucial difference is of course the property of asymptotic freedom and confinement in QCD. In connection with that, our second major result is the derivation of the low energy effective action for gluons in QCD in a strong magnetic field [see Eq. (18) below]. The characteristic feature of this action is its anisotropic dynamics. In particular, the strength of static (Coulomb like) forces along the direction parallel to the magnetic field is much larger than that in the transverse directions. Also, the confinement scale in this theory is much less than that in QCD without a magnetic field. This features imply a rich and unusual spectrum of light glueballs in this theory. A special and interesting case is QCD with a large number of colors, in particular, with Nc → ∞ (the ’t Hooft limit). In this limit, the mass of gluons goes to zero and the expression for the quark mass becomes essentially different [see Eq. (25) in Sec. 5]. In fact, it will be shown that, for any value of an external magnetic field, there exists a threshold value 68

Miransky

(1 GeV)2 ]. Ncthr , rapidly growing with |eB| [e.g., Ncthr > ∼ 100 for |eB| > ∼ For Nc of the order Ncthr or larger, the gluon mass becomes small and irrelevant for the dynamics of the generation of a quark mass. As a result, expression (25) for mq takes place for such large Nc . The confinement scale in this case is close to ΛQCD . Still, as is shown in Sec. 5, the dynamics of chiral symmetry breaking is under control in this limit if the magnetic field is sufficiently strong. It is important that, unlike the case of QCD with a nonzero baryon density, there are no principal obstacles for checking all these results and predictions in lattice computer simulations of QCD in a magnetic field.

2

Magnetic catalysis in QCD

We begin by considering the Schwinger-Dyson (gap) equation for the quark propagator. It has the following form: 

G−1 (x, y) = S −1 (x, y) + 4πα s γ µ G(x, z)Γν (z, y, z ′ )Dνµ (z ′ , x)d4 zd4 z ′ , (4) where S(x, y) and G(x, y) are the bare and full fermion propagators in an external magnetic field, Dνµ (x, y) is the full gluon propagator and Γν (x, y, z) is the full amputated vertex function. Since the coupling αs related to the scale |eB| is small, one might think that the rainbow (ladder) approximation is reliable in this problem. However, this is not the case. Because of the (1+1)-dimensional form of the fermion propagator in the LLL approximation, there are relevant higher order contributions [10, 11]. Fortunately one can solve this problem. First of all, an important feature of the quark-antiquark pairing dynamics in QCD in a strong magnetic field is that this dynamics is essentially abelian. This feature is provided by the form of the polarization operator of gluons in this theory. The point is that the dynamics of the quark-antiquark  pairing is mainly induced in the region of momenta k much less than |eB|. This implies that the magnetic field yields a dynamical ultraviolet cutoff in this problem. On the other hand, while the contribution of (electrically neutral) gluons and ghosts in the polarization operator is proportional to k 2 , the fermion contribution is proportional to |eq B| [11]. As a result, the fermion contribution dominates in the relevant region with k 2 ≪ |eB|. 69

Theoretical High Energy Physics

This observation implies that there are three, dynamically very different, scale regions in this problem. The  first one is the region with the energy scale above the magnetic scale |eB|. In that region, the dynamics is essentially the same as in QCD without a magnetic field. In particular, the running coupling decreases logarithmically with increasing the energy scale there. The second region is that with the energy scale below the magnetic scale but much larger than the dynamical mass mq . In this region, the dynamics is abelian like and, therefore, the dynamics of the magnetic catalysis is similar to that in QED in a magnetic field. At last, the third region is the region with the energy scale less than the gap. In this region, quarks decouple and a confinement dynamics for gluons is realized. Let us first consider the intermediate region relevant for the magnetic catalysis. As was indicated above, the important ingredient of this dynamics is a large contribution of fermions to the polarization operator. It is large because of an (essentially) 1+1 dimensional form of the fermion propagator in a strong magnetic field. Its explicit form can be obtained by modifying appropriately the expression for the polarization operator in QED in a magnetic field [11]: N

P

AB,µν

f   |eq B| αs ≃ δ AB kµ kν − k2 gµν , 2 6π q=1 mq

for |k2 | ≪ m2q , (5)

N

P

AB,µν

f  |eq B| αs AB  µ ν 2 µν ≃− δ k k − k g , 2 π q=1 k

for m2q ≪ |k2 | ≪ |eB|, (6)

where gµν ≡ diag(1, 0, 0, −1) is the projector onto the longitudinal subspace, and kµ ≡ gµν kν (the magnetic field is in the x3 direction). Similarly, µν we introduce the orthogonal projector g⊥ ≡ g µν −gµν = diag(0, −1, −1, 0) µ µν ≡ g⊥ kν that we shall use below. Notice that quarks in a strong and k⊥ µν magnetic field do not couple to the transverse subspace spanned by g⊥ µ and k⊥ . This is because in a strong magnetic field only the quark from the LLL matter and they couple only to the longitudinal components of the gluon field. The latter property follows from the fact that spins of the LLL quarks are polarized along the magnetic field [10]. The expressions (5) and (6) coincide with those for the polarization operator in the massive Schwinger model if the parameter αs |eq B|/2 here is replaced by the dimensional coupling α1 of QED1+1 . As in the Schwinger 70

Miransky

model, Eq. (6) implies that there is a massive resonance in the kµ kν −k2 gµν component of the gluon propagator. Its mass is Mg2

=

Nf 

αs αs |eq B| = (2Nu + Nd ) |eB|. 3π q=1 π

(7)

This is reminiscent of the pseudo-Higgs effect in the (1+1)-dimensional massive QED. It is not the genuine Higgs effect because there is no complete screening of the color charge in the infrared region with |k2 | ≪ m2q . This can be seen clearly from Eq. (5). Nevertheless, the pseudo-Higgs effect is manifested in creating a massive resonance and this resonance provides the dominant forces leading to chiral symmetry breaking. Now, after the abelian like structure of the dynamics in this problem is established, we can use the results of the analysis in QED in a magnetic field [11] by introducing appropriate modifications. The main points of the analysis are: (i) the so called improved rainbow approximation is reliable in this problem provided a special non-local gauge is used in the analysis, and (ii) for a small coupling αs (α in QED), the relevant region of momenta in this problem is m2q ≪ |k 2 | ≪ |eB|. We recall that in the improved rainbow approximation the vertex Γν (x, y, z) is taken to be bare and the gluon propagator is taken in the one-loop approximation. Moreover, as we argued above, in this intermediate region of momenta, only the contribution of quarks to the gluon polarization tensor (6) matters. [It is appropriate to call this approximation the “strong-magnetic-field-loop improved rainbow approximation”. It is an analog of the hard-dense-loop improved rainbow approximation in QCD with a nonzero baryon density [20]]. As to the modifications, they are purely kinematic: the overall coupling constant in the gap equation α and the dimensionless combination Mγ2 /|eB| in QED have to be replaced by αs (Nc2 − 1)/2Nc and Mg2 /|eq B|, respectively. This leads us to the expression (1) for the dynamical gap. After expressing the magnetic field in terms of the running coupling, the result for the dynamical mass takes the following convenient form:
   eq  2 2/3 2C1   ΛQCD (cq αs ) exp



1 4Nc π ≃ . − e bαs αs (Nc2 − 1) ln(C2 /cq αs ) (8) As is easy to check, the dynamical mass of the u-quark is considerably larger than that of the d-quark. It is also noticeable that the values of m2q

71

Theoretical High Energy Physics

10

1

0.1

0.01 10

20

30

Figure 1: The dynamical masses of quarks as functions of ln(|eB|/Λ2QCD ) for Nc = 3 and two different values of Nf = Nu +Nd : (i) masses of u-quark (solid line) and d-quark (dash-dotted line) for Nu = 1 and Nd = 2; (ii) masses of u-quark (dashed line) and d-quark (dotted line) for Nu = 2 and Nd = 2. The result may not be reliable in the weak magnetic field region (shaded) where some of the approximations break. The values of masses are given in units of ΛQCD = 250 MeV.

the u-quark dynamical mass becomes comparable to the vacuum value (0) mdyn ≃ 300 MeV only when the coupling constant gets as small as 0.05. Now, by trading the coupling constant for the magnetic field scale |eB|, we get the dependence of the dynamical mass on the value of the external field. The numerical results are presented in Fig. 1 [we used C1 = C2 = 1 in Eq. (8)]. As one can see in Fig. 1, the value of the quark gap in a wide window of strong magnetic fields, Λ2QCD ≪ |eB| (Nc > ∼ 100 for |eB| ∼ 1 GeV ). In that case a conventional confinement dynamics, with the confinement scale λQCD ∼ ΛQCD , is realized. 82

Miransky

The dynamics of chiral symmetry breaking in QCD in a magnetic field has both similarities and important differencies with respect to the dynamics of color superconductivity in QCD with a large baryon density [2]. Both dynamics are essentially 1+1 dimensional. However, while the former is anisotropic [the rotational SO(3) symmetry is explicitly broken by a magnetic field], the rotational symmetry is preserved in the latter. This fact is in particular connected with that while in dense QCD quarks interact both with chromo-electric and chromo-magnetic gluons [20], in the present theory they interact only with the longitudinal components of chromo-electric gluons. This in turn leads to very different expressions for the dynamical masses of quarks in these two theories. Another important difference is that while the pseudo-Higgs effect takes place in QCD in a magnetic field, the genuine Higgs (Meissner-Higgs) effect is realized in color superconducting dense quark matter. Because of the Higgs effect, the color interactions connected with broken generators are completely screened in infrared in the case of color superconductivity. In particular, in the color-flavor locked phase of dense QCD with three quark flavors, the color symmetry is completely broken and, therefore, the infrared dynamics is under control in that case [26]. As for dense QCD with two quark flavors, the color symmetry is only partialy broken down to SU (2)c , and there exists an analog of the pseudo-Higgs effect for the electric modes of gluons connected with the unbroken SU (2)c . As a result, the confinement scale of the gluodynamics of the remaining SU (2)c group is much less than ΛQCD [27], like in the present case. The essential difference, however, is that, unlike QCD in a magnetic field, the infrared dynamics of a color superconductor is isotropic. Last but not least, unlike the case of QCD with a nonzero baryon density, there are no principal obstacles for examining all these results and predictions in lattice computer simulations of QCD in a magnetic field.

Acknowledgments: I am grateful for support from the Natural Sciences and Engineering Research Council of Canada.

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References ∗

On leave of absence from Bogolyubov Institute for Theoretical Physics, 252143, Kiev, Ukraine.

[1] D. J. Gross, R. D. Pisarski, and L. G. Yaffe, Rev. Mod. Phys. 53, 43 (1981). [2] K. Rajagopal and F. Wilczek, in At the frontier of particle physics, Vol. 3, pp. 2061–2151, ed. M. Shifman (World Scientific, Singapore, 2002), arXiv:hep-ph/0011333. [3] S. Kawati, G. Konisi, and H. Miyata, Phys. Rev. D 28, 1537 (1983); S. P. Klevansky and R. H. Lemmer, Phys. Rev. D 39, 3478 (1989); H. Suganuma and T. Tatsumi, Annals Phys. 208, 470 (1991). [4] S. Schramm, B. Muller, and A. J. Schramm, Mod. Phys. Lett. A 7, 973 (1992). [5] V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, Phys. Rev. Lett. 73, 3499 (1994); Phys. Rev. D 52, 4718 (1995). [6] V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, Phys. Lett. B 349, 477 (1995). [7] K. Farakos, G. Koutsoumbas, and N. E. Mavromatos, Int. J. Mod. Phys. B 12, 2475 (1998); G. W. Semenoff, I. A. Shovkovy, and L. C. R. Wijewardhana, Mod. Phys. Lett. A 13, 1143 (1998); W. V. Liu, Nucl. Phys. B 556, 563 (1999); E. J. Ferrer, V. P. Gusynin, and V. de la Incera, Mod. Phys. Lett. B 16, 107 (2002); arXiv:cond-mat/0203217. [8] D. V. Khveshchenko, Phys. Rev. Lett. 87, 206401 (2001); E. V. Gorbar, V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, arXiv:condmat/0202422 (to appear in Phys. Rev. B). [9] V. A. Miransky, Prog. Theor. Phys. Suppl. 123, 49 (1996); Y. J. Ng, arXiv:hep-th/9803074; C. N. Leung, arXiv:hep-th/9806208; V. P. Gusynin, Ukr. J. Phys. 45, 603 (2000).

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[10] V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, Phys. Rev. D 52, 4747 (1995); Nucl. Phys. B 462, 249 (1996). [11] V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, Phys. Rev. Lett. 83, 1291 (1999); Nucl. Phys. B 563, 361 (1999). [12] C. N. Leung, Y. J. Ng, and A. W. Ackley, Phys. Rev. D 54, 4181 (1996); D. S. Lee, C. N. Leung, and Y. J. Ng, Phys. Rev. D 55, 6504 (1997); ibid. D 57, 5224 (1998). [13] D. K. Hong, Y. Kim, and S. J. Sin, Phys. Rev. D 54, 7879 (1996). [14] V. P. Gusynin and A. V. Smilga, Phys. Lett. B 450, 267 (1999). [15] J. Alexandre, K. Farakos and G. Koutsoumbas, Phys. Rev. D 64, 067702 (2001). [16] I. A. Shushpanov and A. V. Smilga, Phys. Lett. B 402, 351 (1997); N. O. Agasian and I. A. Shushpanov, Phys. Lett. B 472, 143 (2000). [17] V. C. Zhukovsky, V. V. Khudyakov, K. G. Klimenko, and D. Ebert, JETP Lett. 74, 523 (2001) [Pisma Zh. Eksp. Teor. Fiz. 74, 595 (2001)]; D. Ebert, V. V. Khudyakov, V. C. Zhukovsky, and K. G. Klimenko, Phys. Rev. D 65, 054024 (2002). [18] D. Kabat, K. Lee, and E. Weinberg, arXiv:hep-ph/0204120. [19] J. Schwinger, Phys. Rev. 125, 397 (1962); S. R. Coleman, R. Jackiw, and L. Susskind, Annals Phys. 93, 267 (1975); S. R. Coleman, Annals Phys. 101, 239 (1976). [20] D. T. Son, Phys. Rev. D 59, 094019 (1999); T. Schafer and F. Wilczek, Phys. Rev. D 60, 114033 (1999); D. K. Hong, V. A. Miransky, I. A. Shovkovy, and L. C. Wijewardhana, Phys. Rev. D 61, 056001 (2000); R. D. Pisarski and D. H. Rischke, Phys. Rev. D 61, 051501 (2000). [21] The fact that the pairing dynamics responsible for generating rather light quark masses (of order ΛQCD or less) decouples from the confinement dynamics may seem to be puzzling. As it will be shown in Sec. 85

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4, the confinement scale λQCD in QCD in a strong magnetic field is much less than ΛQCD , i.e., the quark masses are large if expressed in units of the new confinement scale. This point implies that the pairing dynamics is indeed weakly coupled. [22] V. A. Miransky, Dynamical Symmetry Breaking in Quantum Field Theories (World Scientific, Singapore, 1993). [23] H. Pagels and S. Stokar, Phys. Rev. D 20, 2947 (1979). [24] V. A. Miransky, I. A. Shovkovy, and L. C. R. Wijewardhana, Phys. Rev. D 62, 085025 (2000); ibid D 63, 056005 (2001). [25] D. T. Son and M. A. Stephanov, Phys. Rev. D 61, 074012 (2000); ibid D 62, 059902(E); S. R. Beane, P. F. Bedaque, and M. J. Savage, Phys. Lett. B 483, 131 (2000); K. Zarembo, Phys. Rev. D 62, 054003 (2000). [26] M. G. Alford, K. Rajagopal, and F. Wilczek, Nucl. Phys. B 537, 443 (1999). [27] D. H. Rischke, D. T. Son, and M. A. Stephanov, Phys. Rev. Lett. 87, 062001 (2001).

86

————————– SUNY Institute of Technology Conference on Theoretical High Energy Physics June 6th, 2002

————————– Supersymmetry in Anti de Sitter Space D.G.C. McKeon a, 1 , C. Schubert T. N. Sherry c, 3 a

b, 2

,

Department of Applied Mathematics, The University of Western Ontario, London, Ontario N6A 5B7 CANADA b Instituto de Fisica y Matematicas, Ciudad Universitaria de la Universidad Michoacana, C.P. 58040, Morelia, Michoacan, MEXICO c Department of Mathematical Physics, National University of Ireland Galway, Galway, IRELAND

————————————————– Abstract The supersymmetry (SUSY) algebras associated with AdSN (N = 2, 3, 4) are presented. Superspace models are given for AdS2 and AdS3 .

1

Electronic address: [email protected] Electronic address: [email protected] 3 Electronic address : [email protected]

2

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Theoretical High Energy Physics

N dimensional anti-de Sitter space is a surface of constant curvature embedded in N + 1 dimensions. It is described by the equation gµν xµ xν = a2

(1)

with diag gµν = (+, +, ..., +, −, −). Symmetry generators Jµν associated with invariances on this surface obey the algebra [Jµν , Jλσ ] = gµλ Jνσ + gνσ Jµλ − gµσ Jνλ − gνλ Jµσ .

(2)

The supersymmetric extension of this algebra involves the introduction of a Fermionic generator Q that is the “square root” of J µν . We begin by introducing the Dirac matrices γµ which satisfy {γ µ , γ ν } = 2g µν .

(3)

If Σµν = − 14 [γ µ , γ ν ], then Σµν satisfies the same algebra as in eq. (2). For AdS2 [1], we consider a 2D space defined by (x1 )2 − (x2 )2 + (x3 )2 = a2

(4)

with γ 1 = iτ1

γ 2 = τ2

γ 3 = iτ3 .

(5)

The charge conjugation matrix C is τ2 as it satisfies Cγ µ C −1 = −γ µT . A spinor Q transforming as Q → exp(Σµν ωµν )Q

(6)

under a Lorentz transformation has an associated spinor Q = Q† τ2 transforming as Q → Q exp(−Σµν ωµν ). (7) T

Furthermore, if Qc ≡ CQ , then Qc and Qc transform as Q and Q respectively. Since (Qc )c = Q, it is consistent to impose the Majorana condition Q = Qc ;

(8)

in this case Q = −Q∗ . We also define ˜ = −QT τ2 Q 88

(9)

McKeon, Schubert, and Sherry

˜ so that for a Majorana spinor, Qc = −Q. A suitable SUSY algebra for AdS2 is ˜ = 2Σµν Jµν {Q, Q}

(10a)

[Jµν , Q] = −Σµν Q

(10b)

in addition to eq. (2). Jacobi identities are satisfied due to the Fierz identity (τ a )ij (τ a )kl = 2δie δkj − δij δkl .

(11)

An N = 2 SUSY algebra for AdS2 is ˜ j } = −iǫij α ± 2δij Σµν Jµν {Qi , Q

(12a)

[Jµν , Qi ] = −Σµν Qi

(12b)

[α, Qi ] = ±iǫij Qj

(12c)

where α is an “internal symmetry” generator. For AdS3 [2], our space is defined by the surface x2 = (x0 )2 + (x1 )2 − (x2 )2 − (x3 )2 = a2 .

(13)

Dirac matrices are now 4 × 4; we take them to be µ

γ = with




0 λµ µ λ 0



µ

λµ = (1, iτ 2 , τ 1 , τ 3 ) and λ = (1, −iτ 2 , −τ 1 , −τ 3 ).

(14a)

(14b)

If now 1 ν µ σ µν = (λµ λ − λν λ ) 4 1 µ ν σ µν = − (λ λν − λ λµ ) 4

(15a) (15b)

then ν

µ

λµ λ + λν λ = 2g µν 89

(16)

Theoretical High Energy Physics

and [σ µν , σ λσ ] = g µλ σ νσ − ...

(17a)

[σ µν , σ λσ ] = g µλ σ νσ − ...

(17b)

We also find that surprisingly 1 σ µν = − ǫµνλσ σλσ 2

(18a)

1 σ ¯ µν = + ǫµνλσ σ λσ (ǫ0123 = +1). 2

(18b) T

Defining C by Cγ µ C −1 = −γ µT , so that if Qc ≡ CQ with Q = Q† (−iγ 2 γ 3 ), then it is feasible to impose the Majorana condition Q = Qc as well as the Weyl condition on Q. It is then possible to deal with a two component spinorial generator Q. Taking ˜ = QT τ2 Q

(19)

then two suitable superalgebras that are consistent with the Jacobi identities are ˜ = 2σ µν Kµν {Q, Q}

(20a)

[K µν , Q] = −σ µν Q

(20b)

˜ = 2σ µν K µν {Q, Q}

(21a)

and

[K µν , Q] = −σ µν Q.

(21b)

It has been necessary to use 1 J µν − 2  1 J µν + = 2

K µν = K

µν



1 1 µνλσ ǫ Jλσ = − ǫµνλσ Kλσ 2 2  1 1 µνλσ ǫ Jλσ = + ǫµνλσ Kλσ 2 2 

90

(22a) (22b)

McKeon, Schubert, and Sherry

in place of J µν in (20-21) on account of (18). Both K and K satisfy the algebra of eq. (2). For AdS4 , our surface is given by x2 = (x0 )2 − (x1 )2 − (x2 )2 − (x3 )2 + (x5 )2 = a2

(23)

and we take our Dirac matrices to be 0

Γ =




0 1 1 0



5

Γ =




−1 0 0 1



i

Γ =




0 τi −τ i 0



.

(24)

The charge conjugation matrix C satisfies CΓµ C −1 = +ΓµT . It is taken to be C = iΓ1 Γ3 . If now Q = Q† (−iΓ0 Γ5 ), then it is possible to impose the T Majorana condition Q = Qc ≡ CQ , but not a Weyl condition. Hence, if ˜ = QT C, then a suitable superalgebra is Q ˜ = −2ΣAB JAB {Q, Q} [J AB , Q] = −ΣAB Q.

(25a) (25b)

Again, all Jacobi identities are satisfied by this algebra. For superspace, we introduce a spinorial Grassmann coordinate θ. With AdS2 , θ is a two component Majorana spinor, for AdS3 it is a MajoranaWeyl spinor and for AdS4 it is a four component Majorana spinor. In AdS2 we can take ∂ ab Σ θ − (xa ∂ b − xb ∂ a ) ∂θ ∂ ; Q = γ a ∂a θ + γ a xa ∂ θ˜

J ab =

(26a) (26b)

two invariants are ˜ R2 = xa xa − θθ

(27a)

and

∂ . ∂θi The algebra of eq. (20) has the representation ∆ = xa ∂a + θi

J µν = −xµ ∂ ν + xν ∂ µ (no spin part!) 91

(27b)

(28a)

Theoretical High Energy Physics

and Q = λµ ∂ u θ + λ µ x µ

∂ ∂ θ˜

(28b)

which has the invariants ˜ (29a) R2 = xµ xµ − θθ ∂ ∆ = xµ ∂µ + θi i (29b) ∂θ Finally, for the AdS4 algebra of (25) we must introduce an additional Bosonic parameter β; we then have the representation




  ∂ ∂ Q = Γ ∂A + θ + ΓA xA − 3β ∂β ∂ θ˜ ∂ ΣAB θ − (xA ∂B − xB ∂A ) ; JAB = ∂θ invariants associated with this representation are A

(30a) (30b)

˜ R2 = x2 − 3β 2 − θθ

and

(31a)

∂ ∂ + θ˜ . (31b) ∂β ∂ θ˜ It is also possible to formulate models invariant under transformations associated with the SUSY algebras given above. For AdS2 , one example is the component field model ∆ = xA ∂A + β

 dA ˜  ab 1 S = Ψ Σ L + χ Ψ + Φ Lab Lab + χ(1 + χ) Φ ab 2 a 2

N  2 2 ˜ (32) − F + λN (1 + 2χ)Φ + 2ΦF + ΨΨ 







Here Ψ is a Majorana spinor, Φ and F are scalars, χ and λN are constants and Lab = −xa ∂b + xb ∂a . The SUSY transformations δΨ =







Σab Lab − (1 + χ) Φ − F ξ

˜ δΦ = ξΨ   δF = −ξ˜ Σab Lab + χ Ψ 92

(33)

McKeon, Schubert, and Sherry

are consistent with eq. (10). A superspace model associated with this algebra is S=



˜ + ρ)Φ(x, θ). d3 xd2 θδ(R2 − a2 )Φ(x, θ)(DD


∂ D ≡ −γ · ∂θ + γ · x ∂ θ˜

If

(34)



˜ ˜ Φ(x, θ) = φ(x) + λ(x)θ + F (x)θθ

(35)

∆Φ = ωΦ

(36)

with then (34) becomes S =







˜ Σab Lab + d3 xδ(x2 − a2 ) −λ

ρ−3 λ 2 

 1  ab φ L L + 2ω(ω − 1) φ − 2a2 F 2 ab 2a2 (12ω + ρ) 2 + 2(ρ − 1)φF + φ + (−1 − 2ω)φF . 2a2

+

(37)

A superfield interaction is provided by SI = gN



d3 kd2 θδ(R2 − a2 )ΦN (x, θ).

(38)

One cansimilarly generate models associated with AdS3 . For example, if  ∂ R = λ · ∂θ + x ∂ θ˜ , then S=



˜ d4 xd2 θδ(R2 − a2 )ΦRRΦ

(39)

becomes S =



˜ µν Lµν − 2)λ d4 xδ(x2 − a2 ) λ(σ

1 1 + 2 φ Lµν Lµν − ω 2 φ a 2 a2 2 F + (1 − ω)φF − 2 

93



(40)

Theoretical High Energy Physics

As in AdS4 superspace, θ has four (not two) components, it appears to be impossible to construct a superspace action associated with the algebra of eq. (25). Currently, AdS5 is under consideration.

References [1] D.G.C. McKeon and T. N. Sherry, unpublished report. [2] D.G.C. McKeon and C. Schubert, unpublished report.

94

————————– SUNY Institute of Technology Conference on Theoretical High Energy Physics June 6th, 2002

————————– D0 − D0 Mixing and CP-Violation in Charm Alexey A. Petrov

1

Department of Physics and Astronomy, Wayne State University Detroit, MI 48201

————————————————– Abstract 0

The Standard Model contribution to D0 − D mixing is dominated by the contributions of light s and d quarks. Neglecting the tiny effects due to b quark, both mass and lifetime differences vanish in the limit of SU (3)F symmetry. Thus, the main challenge in the Standard Model calculation of 0 the mass and width difference in the D0 − D system is to estimate the size of SU (3) breaking effects. We prove that D meson mixing occurs in the Standard Model only at second order in SU (3) violation. We consider the possibility that phase space effects may be the dominant source of SU (3) breaking. We find that y = (∆Γ)/(2Γ) of the order of one percent is natural in the Standard Model, potentially reducing the sensitivity to new physics of measurements of D meson mixing. We also discuss the possibility of observing lifetime differences and CP violation in charmed mesons both at the currently operating and proposed facilities.

1

Electronic address: [email protected]

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1

Introduction

One of the most important motivations for studies of weak decays of charmed mesons is the possibility of observing a signal from new physics which can be separated from the one generated by the Standard Model (SM) interactions. The low energy effect of new physics particles can be naturally written in terms of a series of local operators of increasing dimension generating ∆C = 1 (decays) or ∆C = 2 (mixing) transitions. For D0 − D0 mixing these operators, as well as the one loop Standard Model effects, generate contributions to the effective operators that change D0 state into D0 state leading to the mass eigenstates ¯ 0
, |D12
= p|D0
± q|D

(1)

where the complex parameters p and q are obtained from diagonalizing the D0 − D0 mass matrix. The mass and width splittings between these eigenstates are parameterized by x≡

Γ2 − Γ1 m2 − m1 , y≡ , Γ 2Γ

(2)

where m1,2 and Γ1,2 are the masses and widths of D1,2 and the mean width and mass are Γ = (Γ1 + Γ2 )/2 and m = (m1 + m2 )/2. Since y is constructed from the decays of D into physical states, it should be dominated by the Standard Model contributions, unless new physics significantly modifies ∆C = 1 interactions. On the contrary, x can receive contributions from all energy scales, so it is usually conjectured that new physics can significantly modify x leading to the inequality x ≫ y. As we discuss later, this signal for new physics is lost if a relatively large y, of the order of a percent, is observed. It is known experimentally that D0 − D0 mixing proceeds extremely slowly, which in the Standard Model is usually attributed to the absence of superheavy quarks destroying GIM cancellations [1]. Another possible manifestation of new physics interactions in the charm system is associated with the observation of (large) CP-violation. This is due to the fact that all quarks that build up the hadronic states in weak decays of charm mesons belong to the first two generations. Since 2 × 2 Cabbibo quark mixing matrix is real, no CP-violation is possible in the dominant tree-level diagrams that describe the decay amplitudes. In the 96

Petrov

Standard Model CP-violating amplitudes can be introduced by including penguin or box operators induced by virtual b-quarks. However, their contributions are strongly suppressed by the small combination of CKM matrix elements Vcb Vub∗ . It is thus widely believed that the observation of (large) CP violation in charm decays or mixing would be an unambiguous sign for new physics. This fact makes charm decays a valuable tool in searching for new physics, since the statistics available in charm physics experiment is usually quite large. As in B-physics, CP-violating contributions in charm can be generally classified by three different categories: (I) CP violation in the decay amplitudes. This type of CP violation occurs when the absolute value of the decay amplitude for D to decay to a final state f (Af ) is different from the one of corresponding CP-conjugated amplitude (“direct CP-violation”); (II) CP violation in D0 − D0 mixing matrix. This type of CP violation is 2 ∗ manifest when Rm = |p/q|2 = (2M12 − iΓ12 )/(2M12 − iΓ∗12 ) = 1; and (III) CP violation in the interference of decays with and without mixing. This type of CP violation is possible for a subset of final states to which both D0 and D0 can decay. For a given final state f , CP violating contributions can be summarized in the parameter



q Af i(φ+δ)
Af

, (3)
= Rm e λf =
Af
p Af where Af and Af are the amplitudes for D0 → f and D0 → f transitions respectively and δ is the strong phase difference between Af and Af . Here φ represents the convention-independent weak phase difference between the ratio of decay amplitudes and the mixing matrix.

2

Experimental constraints

Presently, experimental information about the D0 − D0 mixing parameters x and y comes from the time-dependent analyses that can roughly be divided into two categories. First, more traditional studies look at the time dependence of D → f decays, where f is the final state that can be used to tag the flavor of the decayed meson. The most popular is the non-leptonic doubly Cabibbo suppressed decay (DCSD) D0 → K + π − . 97

Theoretical High Energy Physics

Time-dependent studies allow one to separate the DCSD from the mixing contribution D0 → D0 → K + π − , Γ[D0 (t) → K + π − ] = e−Γt |AK − π+ |2   2 √ Rm ′ ′ 2 2 2 (y + x )(Γt) , (4) × R + RRm (y cos φ − x sin φ)Γt + 4 where R is the ratio of DCS and Cabibbo favored (CF) decay rates. Since x and y are small, the best constraint comes from the linear terms in t that are also linear in x and y. A direct extraction of x and y from Eq. (4) is not possible due to unknown relative strong phase δ of DCS and CF amplitudes[2], as x′ = x cos δ + y sin δ, y ′ = y cos δ − x sin δ. This phase can be measured independently [3]. The corresponding formula can also −1 . be written [4] for D0 decay with x′ → −x′ and Rm → Rm 0 Second, D mixing can be measured by comparing the lifetimes extracted from the analysis of D decays into the CP-even and CP-odd final states. This study is also sensitive to a linear function of y via 2 Rm −1 τ (D → K − π + ) . − 1 = y cos φ − x sin φ + − τ (D → K K ) 2





(5)

Time-integrated studies of the semileptonic transitions are sensitive to the quadratic form x2 + y 2 and at the moment are not competitive with the analyses discussed above. The construction of a new tau-charm factory at Cornell (CLEO-c) will introduce new time-independent methods that are sensitive to a linear function of y. In particular, one can use the fact that heavy meson pairs produced in the decays of heavy quarkonium resonances have the useful property that the two mesons are in the CP-correlated states [5]. By tagging one of the mesons as a CP eigenstate, a lifetime difference may be determined by measuring the leptonic branching ratio of the other meson. The initial D0 D0 state is prepared as  1  |DD0
L = √ |D0 (k1 )D0 (k2 )
+ (−1)L |D0 (k2 )D0 (k1 )
, 2

(6)

where L is the relative angular momentum of two D mesons. There are several possible resonances at which CLEO-c will be running, for example 98

Petrov

ψ(3770) where L = 1 and the initial state is antisymmetric, or ψ(4114) where the initial D0 D0 state can be symmetric due to emission of additional pion or photon in the decay. In this scenario, the CP quantum numbers of the D(k2 ) can be determined. The semileptonic width of this meson should be independent of the CP quantum number since it is flavor specific. It follows that the semileptonic branching ratio of D(k2 ) will be inversely proportional to the total width of that meson. Since we know whether D(k2 ) is tagged as a (CP-eigenstate) D+ or and D− from the decay of D(k1 ) to Sσ , we can easily determine y in terms of the semileptonic branching ratios of D± . This can be expressed simply by introducing the ratio Γ[ψL → (H → Sσ )(H → Xl± ν)] , (7) RσL = Γ[ψL → (H → Sσ )(H → X)] Br(H 0 → Xlν) where X in H → X stands for an inclusive set of all final states. A deviation from RσL = 1 implies a lifetime difference. Keeping only the leading (linear) contributions due to mixing, y can be extracted from this experimentally obtained quantity, y cos φ = (−1)L σ

RσL − 1 . RσL

(8)

The current experimental upper bounds on x and y are on the order of a few times 10−2 , and are expected to improve significantly in the coming years. To regard a future discovery of nonzero x or y as a signal for new physics, we would need high confidence that the Standard Model predictions lie well below the present limits. As was recently shown[6], in the Standard Model x and y are generated only at second order in SU (3) breaking, (9) x , y ∼ sin2 θC × [SU (3) breaking]2 , where θC is the Cabibbo angle. Therefore, predicting the Standard Model values of x and y depends crucially on estimating the size of SU (3) breaking. Although y is expected to be determined by the Standard Model processes, its value nevertheless affects significantly the sensitivity to new physics of experimental analyses of D mixing[4]. Theoretical calculations of x and y, as will be discussed later, are quite uncertain, and the values near the current experimental bounds cannot be 99

Theoretical High Energy Physics

ruled out. Therefore, it will be difficult to find a clear indication of physics beyond the Standard Model in D0 − D0 mixing measurements alone. The only robust potential signal of new physics in charm system at this stage is CP violation. CP violation in D decays and mixing can be searched for by a variety of methods. For instance, time-dependent decay widths for D → Kπ are sensitive to CP violation in mixing (see Eq.(4)). Provided that the x and y are comparable to experimental sensitivities, a combined analysis of D → Kπ and D → KK can yield interesting constraints on CP-violating parameters[4]. Most of the techniques that are sensitive to CP violation make use of the decay asymmetry, ACP (f ) =





2

1 −

Af /Af



Γ(D → f ) − Γ(D → f ) =

2 .
Γ(D → f ) + Γ(D → f ) 1 +
Af /Af



(10)

Most of the properties of Eq.(10), such as dependence on the strong final state phases, are similar to the ones in B-physics[7]. Current experimental bounds from various experiments, all consistent with zero within experimental uncertainties, can be found in[8]. Other interesting signals of CP -violation that are being discussed in connection with tau-charm factory measurements are the ones that are using quantum coherence of the initial state. An example of this type of signal is a decay (D0 D0 ) → f1 f2 at ψ(3770) with f1 and f2 being the different final CP-eigenstates with CP |f1
= CP |f2
. This type of signals are very easy to detect experimentally. It is easy to compute this CP-violating decay rate for the final states f1 and f2 Γf1 f2

(2 + x2 − y 2 ) |λf1 − λf2 |2 + (x2 + y 2 ) |1 − λf1 λf2 |2 = Γf1 Γf2 (11) 2 (1 + x2 )(1 − y 2 ) 2Rm

The result of Eq. (11) represents a generalization of the formula given in Ref. [9]. It is clear that both terms in the numerator of Eq. (11) receive contributions from CP-violation of the type I and III, while the second term is also sensitive to CP-violation of the type II. Moreover, for a large set of the final states the first term would be additionally suppressed by SU (3) symmetry. For instance, λππ = λKK in the SU (3) symmetry limit. It 100

Petrov

is easy to see that only the second term survives if only CP violation in 2 2 the mixing matrix is retained, Γf1 f2 ∝ |1 − Rm | ∝ A2m . This expression is of the second order in CP-violating parameters. As it follows from the existing experimental constraints on rate asymmetries, CP-violating phases are quite small in charm system, regardless of whether they are produced by the Standard Model mechanisms or by some new physics contributions. In that respect, it looks unlikely that the SM signals of CP violation would be observed at CLEO-c with this observable. While the searches for direct CP violation via the asymmetry of Eq. (10) can be done with the charged D-mesons (which are self-tagging), investigations of the other two types of CP-violation require flavor tagging of the initial state. This severely cuts the available dataset. It is therefore interesting to look for signals of CP violation that do not require identification of the initial state. One possible CP-violating signal involves the observable  obtained by summing over the initial states, Γi = Γi + Γi for i = f, f .  A CP-odd observable that can be formed out of Γi is an asymmetry AUCP

Γf − Γf = .  Γf + Γf 



(12)

Note that this asymmetry does not require quantum coherence of the initial state and therefore is accessible in any D-physics experiment. The final states must be chosen such that AUCP is not trivially zero. As we shall see below, decays of D into the final states that are CP-eigenstates would result in zero asymmetry, while the final states like K + K ∗− or KS π + π − would not. A non-zero value of AUCP in Eq. (12) can be generated by both direct and indirect CP-violating contributions. These can be separated by appropriately choosing the final states. For example, indirect CP violating amplitudes are tightly constrained in the decays dominated by the Cabibbofavored tree level amplitudes, while singly Cabibbo suppressed amplitudes also receive contributions from direct CP violating amplitudes. Neglecting small CP-violation in the mixing matrix (Rm → 1) one obtains, AUCP = ×

Γf − Γf − Γf + Γf 2y + Γf + Γf + Γf + Γf Γf + Γf + Γf + Γf 



∗

cos φ ReAf Af − ReA∗f Af 

∗



+ sin φ ImAf Af + ImA∗f Af 101



.

(13)

Theoretical High Energy Physics

It is easy to see that, as promised, this asymmetry vanishes for the final states that are CP-eigenstates, as Γf = Γf and Γf − Γf = Γf − Γf .

3

Theoretical expectations

Theoretical predictions of x and y within and beyond the Standard Model span several orders of magnitude [10]. Roughly, there are two approaches, neither of which give very reliable results because mc is in some sense intermediate between heavy and light. The “inclusive” approach is based on the operator product expansion (OPE). In the mc ≫ Λ limit, where Λ is a scale characteristic of the strong interactions, ∆M and ∆Γ can be expanded in terms of matrix elements of local operators[11]. Such calculations yield x, y < 10−3 . The use of the OPE relies on local quarkhadron duality, and on Λ/mc being small enough to allow a truncation of the series after the first few terms. The charm mass may not be large enough for these to be good approximations, especially for nonleptonic D decays. An observation of y of order 10−2 could be ascribed to a breakdown of the OPE or of duality, but such a large value of y is certainly not a generic prediction of OPE analyses. The “exclusive” approach sums over intermediate hadronic states, which may be modeled or fit to experimental data[12]. Since there are cancellations between states within a given SU (3) multiplet, one needs to know the contribution of each state with high precision. However, the D is not light enough that its decays are dominated by a few final states. In the absence of sufficiently precise data on many decay rates and on strong phases, one is forced to use some assumptions. While most studies find x, y < 10−3 , Refs.[12] obtain x and y at the 10−2 level by arguing that SU (3) violation is of order unity, but the source of the large SU (3) breaking is not made explicit. In what follows we first prove that D0 −D0 mixing arises only at second order in SU (3) breaking effects. The proof is valid when SU (3) violation enters perturbatively. This would not be so, for example, if D transitions were dominated by a single narrow resonance close to threshold[6, 13]. Then we argue that reorganization of “exclusive” calculation by explicitly building SU (3) cancellations into the analysis naturally leads to values of y ∼ 1% if only one source of SU (3) breaking (phase space) is taken into 102

Petrov

account. The quantities M12 and Γ12 which determine x and y depend on matrix elements D0 | H w H w |D0
, where H w denote the ∆C = −1 part of the weak Hamiltonian. Let D be the field operator that creates a D0 meson and annihilates a D0 . Then the matrix element, whose SU (3) flavor group theory properties we will study, may be written as 0| D H w H w D |0
.

(14)

Since the operator D is of the form c¯u, it transforms in the fundamental representation of SU (3), which we will represent with a lower index, Di . We use a convention in which the correspondence between matrix indexes and quark flavors is (1, 2, 3) = (u, d, s). The only nonzero element of Di is D1 = 1. The ∆C = −1 part of the weak Hamiltonian has the flavor structure (¯ qi c)(¯ qj qk ), so its matrix representation is written with a fundamental index and two antifundamentals, Hkij . This operator is a sum of irreps contained in the product 3 × 3 × 3 = 15 + 6 + 3 + 3. In the limit in which the third generation is neglected, Hkij is traceless, so only the 15 and 6 representations appear. That is, the ∆C = −1 part of H w may be decomposed as 21 (O15 + O6 ), where ¯ ud) + s1 (¯ ¯ sc)(¯ ud) + (¯ uc)(¯ sd) + s1 (dc)(¯ uc)(dd) O15 = (¯ ¯ us) − s2 (¯ ¯ − s1 (¯ sc)(¯ us) − s1 (¯ uc)(¯ ss) − s21 (dc)(¯ 1 uc)(ds) , ¯ ud) − s1 (¯ ¯ O6 = (¯ sc)(¯ ud) − (¯ uc)(¯ sd) + s1 (dc)(¯ uc)(dd)

¯ us) + s2 (¯ ¯ − s1 (¯ sc)(¯ us) + s1 (¯ uc)(¯ ss) − s21 (dc)(¯ 1 uc)(ds) , (15)

ij and s1 = sin θC . The matrix representations H(15)ij k and H(6)k have nonzero elements

H(15)ij k : H(6)ij k :

H213 = H231 = 1 , H313 = H331 = −s1 ,

H213 = −H231 = 1 , H313 = −H331 = −s1 ,

H212 = H221 = s1 , H312 = H321 = −s21 ,

H212 = −H221 = s1 , H312 = −H321 = −s21 . (16) We introduce SU (3) breaking through the quark mass operator M , whose matrix representation is Mji = diag(mu , md , ms ) as being in the adjoint representation to induce SU (3) violating effects. We set mu = md = 0 103

Theoretical High Energy Physics

and let ms = 0 be the only SU (3) violating parameter. All nonzero matrix elements built out of Di , Hkij and Mji must be SU (3) singlets. We now prove that D0 −D0 mixing arises only at second order in SU (3) violation, by which we mean second order in ms . First, we note that the pair of D operators is symmetric, and so the product Di Dj transforms as a 6 under SU (3). Second, the pair of H w ’s is also symmetric, and the product Hkij Hnlm is in one of the reps which appears in the product 



(15 + 6) × (15 + 6)

S ′

= (15 × 15)S + (15 × 6) + (6 × 6)S

(17) ′

= (60 + 24 + 15 + 15 + 6) + (42 + 24 + 15 + 6 + 3) + (15 + 6) .

A direct computation shows that only three of these representations actually appear in the decomposition of H w H w . They are the 60, the 42, and the 15′ (actually twice, but with the same nonzero elements both times). So we have product operators of the form (the subscript denotes the representation of SU (3)) DD = D6 ,

H w H w = O60 + O42 + O15′ .

(18)

Since there is no 6 in the decomposition of H w H w , there is no SU (3) singlet which can be made with D6 , and no SU (3) invariant matrix element of the form (14) can be formed. This is the well known result that D0 −D0 mixing is prohibited by SU (3) symmetry. Now consider a single insertion of the SU (3) violating spurion M . The combination D6 M transforms as 6 × 8 = 24 + 15 + 6 + 3. There is still no invariant to be made with H w H w , thus D0 −D0 mixing is not induced at first order in SU (3) breaking. With two insertions of M , it becomes possible to make an SU (3) invariant. The decomposition of DM M is 6 × (8 × 8)S = 6 × (27 + 8 + 1) ′ = (60 + 42 + 24 + 15 + 15 + 6) +(24 + 15 + 6 + 3) + 6 .

(19)

There are three elements of the 6 × 27 part which can give invariants with H w H w . Each invariant yields a contribution to D0 − D0 mixing proportional to s21 m2s . Thus, D0 − D0 mixing arises only at second order in the SU (3) violating parameter ms . 104

Petrov

We now turn to the contributions to y from on-shell final states, which result from every common decay product of D0 and D0 . In the SU (3) limit, these contributions cancel when one sums over complete SU (3) multiplets in the final state. The cancellations depend on SU (3) symmetry both in the decay matrix elements and in the final state phase space. While there are SU (3) violating corrections to both of these, it is difficult to compute the SU (3) violation in the matrix elements in a model independent manner. Yet, with some mild assumptions about the momentum dependence of the matrix elements, the SU (3) violation in the phase space depends only on the final particle masses and can be computed. We estimate the contributions to y solely from SU (3) violation in the phase space. We find that this source of SU (3) violation can generate y of the order of a few percent. The mixing parameter y may be written in terms of the matrix elements for common final states for D0 and D0 decays, y=

1  [P.S.]n D0 | H w |n
n| H w |D0
, Γ n

(20)

where the sum is over distinct final states n and the integral is over the phase space for state n. Let us now perform the phase space integrals and restrict the sum to final states F which transform within a single SU (3) multiplet R. The result is a contribution to y of the form  1 |n
ρn n| H w |D0
, D0 | H w ηCP (FR ) Γ n∈FR



(21)

where ρn is the phase space available to the state n, ηCP = ±1 [6]. In the SU (3) limit, all the ρn are the same for n ∈ FR , and the quantity in braces above is an SU (3) singlet. Since the ρn depend only on the known masses of the particles in the state n, incorporating the true values of ρn in the sum is a calculable source of SU (3) breaking. This method does not lead directly to a calculable contribution to y, because the matrix elements n|H w |D0
and D0 |H w |n
are not known. However, CP symmetry, which in the Standard Model and almost all scenarios of new physics is to an excellent approximation conserved in D decays, relates D0 |H w |n
to D0 |H w |n
. Since |n
and |n
are in a 105

Theoretical High Energy Physics

common SU (3) multiplet, they are determined by a single effective Hamiltonian. Hence the ratio D0 | H w |n
ρn n|H w |D0
0 0 n∈FR D | H w |n
ρn n|H w |D




yF,R = n∈FR =

| H w |n
ρn n|H w |D0
 0 n∈FR Γ(D → n)

n∈FR D



0

(22)

is calculable, and represents the value which y would take if elements of FR were the only channel open for D0 decay. To get a true contribution to y, one must scale yF,R to the total branching ratio to all the states in FR . This is not trivial, since a given physical final state typically decomposes into a sum over more than one multiplet FR . The numerator of yF,R is of order s21 while the denominator is of order 1, so with large SU (3) breaking in the phase space the natural size of yF,R is 5%. Indeed, there are other SU (3) violating effects, such as in matrix elements and final state interaction phases. Here we assume that there is no cancellation with other sources of SU (3) breaking, or between the various multiplets which occur in D decay, that would reduce our result for y by an order of magnitude. This is equivalent to assuming that the D meson is not heavy enough for duality to enforce such cancellations. Performing the computations of yF,R , we see[6] that effects at the level of a few percent are quite generic. Our results are summarized in Table 1. Then, y can be formally constructed from the individual yF,R by weighting them by their D0 branching ratios,    1 0 y= yF,R Γ(D → n) . Γ F,R n∈FR

(23)

However, the data on D decays are neither abundant nor precise enough to disentangle the decays to the various SU (3) multiplets, especially for the three- and four-body final states. Nor have we computed yF,R for all or even most of the available representations. Instead, we can only estimate individual contributions to y by assuming that the representations for which we know yF,R to be typical for final states with a given multiplicity, and then to scale to the total branching ratio to those final states. The total branching ratios of D0 to two-, three- and four-body final states can be extracted from the Review of Particle Physics[14]. Rounding to the 106

Petrov

yF,R /s21 −0.0038 −0.00071 0.032 0.031 0.020 0.016 0.04 −0.081 −0.061 −0.10 −0.14 0.51 0.57 −0.48 −0.11 −1.13 −0.07 −0.44 −0.13 3.3 2.2 1.9

Final state representation PP 8 27 PV 8A 8S 10 10 27 (V V )s-wave 8 27 (V V )p-wave 8 27 (V V )d-wave 8 27 (3P )s-wave 8 27 (3P )p-wave 8 27 (3P )form-factor 8 27 4P 8 27 27′

yF,R (%) −0.018 −0.0034 0.15 0.15 0.10 0.08 0.19 −0.39 −0.30 −0.48 −0.70 2.5 2.8 −2.3 −0.54 −5.5 −0.36 −2.1 −0.64 16 11 9.2

Table 1: Values of yF,R for some two-, three-, and four-body final states.

nearest 5% to emphasize the uncertainties in these numbers, we conclude that the branching fractions for P P , (V V )s-wave , (V V )d-wave and 3P approximately amount to 5%, while the branching ratios for P V and 4P are of the order of 10%[6]. We observe that there are terms in Eq. (23), like nonresonant 4P , which could make contributions to y at the level of a percent or larger. There, the rest masses of the final state particles take up most of the available energy, so phase space differences are very important. One can see that y on the order of a few percent is completely natural, and that anything an order

107

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of magnitude smaller would require significant cancellations which do not appear naturally in this framework. Cancellations would be expected only if they were enforced by the OPE, or if the charm quark were heavy enough that the “inclusive” approach were applicable. The hypothesis underlying the present analysis is that this is not the case.

4

Conclusions

We proved that if SU (3) violation may be treated perturbatively, then D0 − D0 mixing in the Standard Model is generated only at second order in SU (3) breaking effects. Within the exclusive approach, we identified an SU (3) breaking effect, SU (3) violation in final state phase space, which can be calculated with minimal model dependence. We found that phase space effects alone provide enough SU (3) violation to induce y ∼ 10−2 . Large effects in y appear for decays close to D threshold, where an analytic expansion in SU (3) violation is no longer possible. Indeed, some degree of cancellation is possible between different multiplets, as would be expected in the mc → ∞ limit, or between SU (3) breaking in phase space and in matrix elements. It is not known how effective these cancellations are, and the most reasonable assumption in light of our analysis is that they are not significant enough to result in an order of magnitude suppression of y, as they are not enforced by any symmetry arguments. Therefore, any future discovery of a D meson width difference should not by itself be interpreted as an indication of the breakdown of the Standard Model. At this stage the only robust potential signal of new physics in charm system is CP violation. We discussed several possible experimental observables that are sensitive to CP violation.

Acknowledgments It is my pleasure to thank D. Atwood, S. Bergmann, E. Golowich, Y. Grossman, A. Falk, Z. Ligeti, and Y. Nir for collaborations on the related projects.

108

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References [1] A. Datta, D. Kumbhakar, Z. Phys. C27, 515 (1985); A. A. Petrov, Phys. Rev. D56, 1685 (1997). [2] A. F. Falk, Y. Nir and A. A. Petrov, JHEP 9912, 019 (1999). [3] M. Gronau, Y. Grossman and J. L. Rosner, Phys. Lett. B 508, 37 (2001); J. P. Silva and A. Soffer, Phys. Rev. D 61, 112001 (2000). E. Golowich and S. Pakvasa, Phys. Lett. B 505, 94 (2001). [4] S. Bergmann, Y. Grossman, Z. Ligeti, Y. Nir, A. Petrov, Phys. Lett. B 486, 418 (2000). [5] D. Atwood and A. A. Petrov, arXiv:hep-ph/0207165. [6] A. F. Falk, Y. Grossman, Z. Ligeti and A. A. Petrov, Phys. Rev. D 65, 054034 (2002). [7] I. I. Bigi and A. I. Sanda, CP violation (Cambridge University Press, 2000). [8] D. Pedrini, J. Phys. G 27, 1259 (2001). [9] I. I. Bigi and A. I. Sanda, Phys. Lett. B 171, 320 (1986). [10] H. N. Nelson, in Proc. of the 19th Intl. Symp. on Photon and Lepton Interactions at High Energy LP99 ed. J.A. Jaros and M.E. Peskin, arXiv:hep-ex/9908021. [11] H. Georgi, Phys. Lett. B297, 353 (1992); T. Ohl, G. Ricciardi and E. Simmons, Nucl. Phys. B403, 605 (1993); I. Bigi and N. Uraltsev, Nucl. Phys. B 592, 92 (2001), for a recent review see A. A. Petrov, Proc. of 4th Workshop on Continuous Advances in QCD, Minneapolis, Minnesota, 12-14 May 2000, arXiv:hep-ph/0009160. [12] J. Donoghue, E. Golowich, B. Holstein and J. Trampetic, Phys. Rev. D33, 179 (1986); L. Wolfenstein, Phys. Lett. B164, 170 (1985); P. Colangelo, G. Nardulli and N. Paver, Phys. Lett. B242, 71 (1990); T.A. Kaeding, Phys. Lett. B357, 151 (1995). A. A. Anselm and Y. I. Azimov, Phys. Lett. B 85, 72 (1979); 109

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[13] E. Golowich and A. A. Petrov, Phys. Lett. B 427, 172 (1998). [14] D. E. Groom et al. [Particle Data Group Collaboration], Eur. Phys. J. C 15, 1 (2000).

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————————– SUNY Institute of Technology Conference on Theoretical High Energy Physics June 6th, 2002

————————– Extraction of |Vcb| via Inclusive Method F.A. Chishtie

1

F. R. Newman Lab for Elementary Particle-Physics, Cornell University, Ithaca, NY 14853.

————————————————– Abstract Renormalization Group (RG) and optimized Pad´ e-approximant methods are used to estimate the three-loop perturbative contribution to the inclusive semileptonic b → c decay rate. The perturbative decay rate, expressed in the pole mass scheme shows lesser renormalization scale dependence as compared to the M S scheme. Upon inclusion of the estimated three-loop contribution, we find the full decay rate to be 192π 3 Γ(b → cℓ− ν¯ℓ )/(G2F |Vcb |2 ) = 992 ± 198GeV5 . The errors are inclusive of theoretical uncertainties and nonperturbative effects. Ultimately, these perturbative contributions reduce the theoretical uncertainty in the extraction of the CKM matrix element |Vcb | from the measured inclusive semileptonic branching ratio.

1

Electronic Address: [email protected]

111

Theoretical High Energy Physics

The CKM matrix element |Vcb |, can be extracted from the inclusive semileptonic decay rate Γ (B → Xc ℓ− ν¯ℓ ). The advantage in considering the inclusive process is due to the fact that non-perturbative contributions are controllable. Therefore, an accurate perturbative determination of the b → cℓ− ν¯ℓ decay rate is of value in obtaining |Vcb | from data. The total perturbative inclusive rate for semileptonic B → Xc ℓ− ν¯ℓ and has been calculated to leading order in Heavy Quark Expansion (HQE) and to two-loop order in QCD [1]: 5

Γbc /κ = [mb ] F




m2c [1 − 1.67x(µ) −(8.9±0.3+3.48L(µ))x2 (µ)] (1) m2b 

where x(µ) ≡ αs (µ)/π, κ ≡ G2F |Vcb |2 /192π 3 ,

F (r) = 1 − 8r − 12r2 log(r) + 8r3 − r4 , L(µ) ≡ log(µ2 /mb mc ).

(2)

In the b → c transition, we find that the total inclusive rate expressed in terms of the b and c pole masses is better behaved [2] than in the M S scheme. This is in contrast to [3] where we find that the b → u rate has less renormalization scale dependence in the M S scheme than in the pole-mass scheme. In both cases the scale dependence, which is considerable, provides no optimal choice of renormalization scale µ. Consequently, we estimate the three-loop contributions to the above rate using Pad´ e approximants in order to reduce theoretical uncertainties like scale dependence and truncation error. The perturbative decay rate has the following general (scale-sensitive) form in powers of the strong coupling: S(x) = 1 + R1 x + R2 x2 + R3 x3 + . . . ,

(3)

where R1 and R2 are known as indicated in (1) and (2) . The (unknown) three-loop contribution term R3 is necessarily of the form: R3 = c0 + c1 L(µ) + c2 L2 (µ) + c3 L3 (µ).

112

(4)

Chishtie

Since the decay rate is renormalization group (RG) invariant, the following relation holds: dΓ (5) µ2 2 = 0. dµ This allows us to evaluate c1 , c2 and c3 exactly for both the decay rates. For the case of b → c we obtain c1 = −42.4 ± 1.3 , c2 = −7.25 , c3 = 0

(6)

The estimate for the RG-inaccessible coefficient c0 is obtained from the following estimate developed via asymptotic Pad´ e approximant methods [2]: (2 + k)R23 (7) R3P ade = (1 + k)R13 + R1 R2 The quantity k parametrizes a family of Pad´ e approximants as outlined in [2]. Since both the assumed form of the three-loop contribution and Pad´ e-estimated version are dependent on µ, we proceed by evaluating the following moments: Nj = (j + 2)

1

wj+1 R3 (w)dw

(8)

0

where log w = −L(µ). Hence, to estimate the value for c0 , we match the scale dependence of the known form (5) to the Pad´ e estimate (9), using the first four moments (N−1 , N0 , N1 , N2 ) in the perturbative (UV) region. This leads to four linear equations for the four three-loop coefficients {c0 , c1 , c2 , c3 }. For the b → u case the method works quite well with k = 0 [3]. However, in the case of b → c decay rate (pole mass scheme) the estimate obtained from k = 0 is ill-suited to the series in question [2]. A value of k for the b → c case can be obtained by finding an optimal k which minimizes the sum of the squares of the relative errors in predicting c1 and c2 , as given in (7). Applying the above procedure, we get the following estimated values of c0−3 for the b → c case (k = −0.94 for the central value of b0 = −8.9): c3 = 2.0 × 10−4 , c2 = −7.68, c1 = −39.7, c0 = −51.2. 113

(9)

Theoretical High Energy Physics

The above estimates of the RG-accessible coefficients c1 and c2 are within 7% error of their correct values (7). As a consistency check, we substitute the exact RG values into the moment equations to evaluate c0 . We then find that c0 = N−1 − c1 − 2c2 − 6c3 = −49.4 , 1 1 3 c0 = N0 − c1 − c2 − c3 = −50.1 , 2 2 4 1 2 2 c0 = N1 − c1 − c2 − c3 = −50.4 , 3 9 9 1 3 1 c0 = N2 − c1 − c2 − c3 = −50.6 . 4 8 32

(10)

which are all within 3% of the Pad´ e-estimated value in Eq.(11). Upon inclusion of the 3-loop contribution we see decreased renormalization scale dependence and the emergence of PMS (Principle of Minimal Sensitivity [4]) extrema for the decay rate [2, 3]. For the b → c transition, we find that the PMS value occurs at µ = 1 GeV and yields Γbc /κ = 1047 GeV5 . We find that this is remarkably close to the FAC (Fastest Apparent Convergence [5]) value which occurs at µ = 1.18 GeV and yields Γbc /κ = 1051 GeV5 . We take the PMS value of the decay rate as our central value. Further, we assume that the error in estimating c0 is the same as our largest error in estimating an RG-accessible coefficient for both cases. We then obtain the following value for the total decay rate for b → c case: Γbc /κ = 992 ± 198GeV5 .

(11)

This error estimate includes the uncertainties in the two-loop contribution (b0 ), the b-quark pole mass (mb = 4.9 ± 0.1 [6]), the strong coupling constant, the three loop estimate, and the uncertainty in estimating nonperturbative effects (for details see [2]). Thus, one can extract |Vcb | from (11) with a theoretical error of 10%.

Acknowledgments: I am grateful for support from the Natural Sciences and Engineering Research Council of Canada. 114

Chishtie

References [1] A. Czarnecki and K. Melnikov Phys. Rev. D 59 (1999) 014036. [2] M. R. Ahmady et al., Phys. Rev. D 65 (2002) 054021. [3] M. R. Ahmady, F. A. Chishtie, V. Elias and T. G. Steele, Phys. Lett. B 479 (2000) 201. [4] P. M. Stevenson, Phys. Rev. D 23 (1981) 2916. [5] G. Grunberg, Phys. Lett. B 95 (1980) 70 and Phys. Rev. D 29 (1984) 2315. [6] A.H. Hoang, Phys. Rev. D 61 (2000) 034005 and D 59 (1999) 014039.

115

————————– SUNY Institute of Technology Conference on Theoretical High Energy Physics June 6th, 2002

————————– Recent QCD Related Issues in B physics Mohammad R. Ahmady

1

Department of Physics, Mount Allison University, Sackville, NB E4L 1E6.

————————————————– Abstract In this talk, I discuss the gluon content of η ′ and its significance for the explanation of the unexpectedly large branching ratio for B decays to this meson. Specifically, I look at the nonspectator gluon fusion mechanism for the inclusive and exclusive η ′ production in two-body hadronic B decays.

1

Electronic address: [email protected]

117

Theoretical High Energy Physics

1

Introduction

Two-body hadronic B decays have been at the focus of a lot of theoretical investigations. This is mostly due to the importance of these decay channels in search for CP violation outside the K meson systems and for the better undersatnding of the QCD effects in nonleptonic decays. Two prominent examples of these transition modes are η ′ production in B meson decays to a final state containing a strange quark and the color-suppressed neutral B decays to D◦ π ◦ . The inclusive B → Xs η ′ and its exclusive counterpart B → Kη ′ were measured by the CLEO collaboration sometime ago[1]. A naive estimate of the latter decay could be obtained via a comparison with the known two-body decay B → D¯◦ π + ,i.e., |Vub Vus |2 Γ(B + → K + η ′ ) ≈ ≈ 10−4 , Γ(B + → D¯◦ π + ) |Vcb |2

(1)

which combined with the experimental measurement BR(B + → D¯◦ π + ) = (5.3 ± 0.5) × 10−3 results in a partial decay width: Γ(B + → K + η ′ ) ≈ 10−7 .

(2)

The more recent experimental measurement of this channel can be summarized as follows [2, 3, 4]: −6 [CLEO], B(B ± → K ± η ′ ) = (80+10 −9 ± 7) × 10 +6.2+9.3 −6 = (77.9−5.9−8.7 ) × 10 [Belle], −6 = (67 ± 5 ± 5) × 10 [BaBar].

(3)

We observe that the naive Standard Model expectation is around two orders of magnitude smaller than the experimental values. There are various proposed mechanisms, both within and without the Standard Model, to explain this unexpectedly large branching ratio. One such mechanism utilizes the fact that, because of the U (1)A anomaly, η ′ coupling to two gluons is expected to be quite strong and therefore, its production in hadronic B decays could be due to the fusion of gluons[5]. It is for this same reason that the gluon content of η ′ is quite relevant to this class of the nonleptonic B decays. 118

Ahmady

2

The gluon fusion mechanism

Our proposed mechanism is a non-spectator process in which η ′ is produced via fusion of the gluon from QCD penguin b → sg ∗ and another one emitted by the light quark inside B meson. The effective Hamiltonian for this transition consists of three ingredients: 1) The effective neutral current flavor changing vertex b → sg which is as follows[6] GF gs [E0 s¯(q 2 gµν − qµ qν )γ ν (1 − γ5 )T a b A(b → sg) = − iλt √ 2 8π 2 (4) − E0′ s¯imb σµν q ν (1 + γ5 )T a b] , where λt = Vtb Vts∗ and q is the gluon four-momentum. Here, we take into account only the dominant chromo-electric operator. 2) The gluon-gluon-psuedoscalar meson (η ′ in this case) vertex which can be written as: Aµσ (gg → η ′ ) = H(q 2 , p2 , m2η′ )δ ab ǫµσαβ qα pβ .

(5)

q and p are four-momenta of the two gluons and H is the relevant form factor which contains a factor of αs implicitly. One can make an estimate of H(0, 0, m2η′ ) ≈ 1.8 GeV−1 using the decay mode ψ → η ′ γ which is expected to proceed mainly via on-shell gluons. However, contrary to H(q 2 , p2 , m2η′ ) ≈ H(0, 0, m2η′ ) assumption utilized in the literature, it is argued that the momentum dependence of H could be quite significant resulting in a suppression by an order of magnitude. We show that the proposed non-spectator mechanism can produce a large enough branching ratio which could match the observed value when such suppression factor is taken into account. 3) The emission of gluon by the light quark. Putting together the above three terms results in the nonspectator effective Hamiltonian for the b → sη ′ transition: Hef f = CH(¯ sγµ (1 − γ5 )T a b)(¯ q γσ T a q) where

GF αs E0 , C = λt √ 2 2π 119

1 µσαβ ǫ qα p β , p2

(6)

(7)

Theoretical High Energy Physics

In fact, using this effective Hamiltonian with a constant gluon-gluon-η ′ form factor one obtains: BR(B → Xs η ′ ) = 4.7 × 10−3 2.0 ≤ pη′ ≤ 2.7 GeV .

(8)

Inserting a momentum dependent form factor H(q 2 , 0, m2η′ ) =

H(0, 0, m2η′ ) , (q 2 /m2η′ − 1)

(9)

in (6) results in the reduction of the above estimate by more than an order of magnitude to BR(B → Xs η ′ ) ≈ 1.3 × 10−4 , which is in the same ballpark as the experimental results. We note that the magnitude and momentum behavior of H are crucial elements for the success of the gluon fusion mechanism. In fact, one can show that the nonperturbative QCD effects do not change the leading momentum behavior of the gluongluon-η ′ form factor reflected in (9)[7]. The magnitude of this form factor however, is very much sensitive to the gluon content of η ′ .

3

The gluon content of η ′

Exactly how much glue is in the make up of the η ′ meson has been at the focus of numerous research work. The SU (3)F singlet axial vector current is not conserved even in the chiral limit due to the non-zero anomalous contributions: 3αs ˜ αβ , Gαβ G qγ µq − (10) ∂µ J5µ = 2im¯ 4π where Gαβ is the gluonic field strength tensor. The presence of the second term for the dominant flavor-singlet component of the η ′ meson naturally leads to a significant glueball constituent for this pseudoscalar particle. In fact, this non-zero anomaly is the underlying reason for η ′ to be substantially heavier than the lowest-lying octet of the pseudoscalar mesons. The SU (3)F singlet and octet mesons have the following flavor wavefunctions: 1 1 η1 = √ (u¯ u + dd¯ + s¯ s) , η8 = √ (u¯ u + dd¯ − 2s¯ s) . 3 6 120

(11)

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Even though η ′ meson is dominantly made of the singlet combination η1 but includes a small component of η8 as well |η ′ >= cos θp |η1 > + sin θp |η8 > ,

(12)

where θp is the pseudoscalar mixing angle. In fact, one can extend this picture with including a gluonium component for η ′ : |η ′ >= A|η1 > +B|η8 > +C|gluonium > .

(13)

A global fit to all available experimental data excludes C = 0 and is consistent with up to 26% fractional value for the glue component[8]. A more systematic method for this investigation has been put forward by C. Rosenzweig, A. Salomone and J. Schechter[9] where an extension of the effective chiral lagrangian for pseudoscalar mesons which satisfies the axial vector anomaly is used to obtain the glue component of η ′ .

4

The exclusive B → Kη ′ decay mode

To calculate the exclusive decay mode, one can re-arrange (6) via Fierz transformation: 1 q γ ρ (1 − γ5 )b) (¯ sγµ (1 − γ5 )T a b)(¯ q γσ T a q) = [(¯ sγσ γρ γµ (1 − γ5 )q)(¯ 9 q (1 − γ5 )b) +(¯ sγσ γµ (1 + γ5 )q)(¯ 1 sγσ σρη γµ q)(¯ − (¯ q σ ρη (1 − γ5 )b) 2 +color octect] , (14) and by using the definition of the decay constants for the B and K mesons in the context of factorization, it is possible to express the matrix element for B → Kη ′ decay as the following: < η ′ K|Hef f |B >= −i

2CHfB fK (pB .qpK .p − pB .ppK .q) , 9p2

(15)

Consequently, the exclusive decay rate can be obtained as Γ(B → Kη ′ ) =


 C 2 H 2 fB2 fK2 2 2 3 2 2 2 2 3p | p | + (m + | p | )(p − p ) , | p | ′ K K 0 K η 0 486πp4 (16)

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where | pK | and p0 are the three momentum of the K meson 2



1

2 (m2B + m2K − m2η′ ) 2   − mK | pK | = , 4m2B

(17)

and the energy transfer by the gluon emitted from the light quark in the B meson rest frame, respectively. Inserting p2 ≈ −Λ2QCD ≈ −0.32 GeV2 , p0 = 0.3 GeV, fB = 0.2 GeV and αs = 0.2 in (16) results in the following exclusive branching ratio: B(B → Kη ′ ) = 7.1 × 10−5 ,

(18)

We observe that the gluon fusion mechanism can actually reproduce the experimental results in (3) with reasonable values for the model parameters.

5

B ± → π ±η ′ decays and direct CP violation

Based on the gluon fusion mechanism, one can predict that the charged decay modes B ± → π ± η ′ could exhibit a potentially significant direct CP violation[10]. This is due to the fact that for these channels, the tree and penguin amplitudes receive a CKm suppression factor of the same order of magnitude. The nonspectator contribution to B → πη ′ can be obtained from (15) by replacing fK with fπ and C with C ′ , where GF αs ∗ (Vtb Vtd∗ [E(xt ) − E(xc )] + Vub Vud [E(xu ) − E(xc )]) , (19) C′ = √ 2 2π and the coefficient function E is defined as 2 xi (18 − 11xi − x2i ) x2 (15 − 16xi + 4x2i ) E(xi ) = − ln xi + i ln x + , i 3 6(1 − xi )4 12(1 − xi )3 (20) 2 2 with xi = mi /mW and mi being the internal quark mass[6] Thus, including the expression for the tree amplitudes, which are derived by using the factorization assumption, the total matrix element for B − → π − η ′ can be 122

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written as: < η ′ π − |Hef f |B − > = −i

2C ′ HfB fπ (pB .qpπ .p − pB .ppπ .q) 9p2

′ GF ∗ [a1 fπ F0B→η (m2π )(m2B − m2η′ ) + i √ Vub Vud 2 + a2 fηu′ F0B→π (m2η′ )(m2B − m2π )] . (21)

a1 = 1.03 and a2 = 0.12 are the combinations of the Wilson coefficients of the current-current operators. fηu′ and F0B→P , P = η ′ , π, are defined as follows: uγ µ (1 − γ5 )u|0 >= ifηu′ pµη′ , (22) < η ′ (pη′ )|¯ u¯γ µ (1 − γ5 )u|B(pB ) >= m2 − m2 [ (pB + pP )µ − B 2 P q µ ]F1B→P (q 2 ) q 2 2 mB − mP µ B→P 2 + q F0 (q ) , (23) q2 where q = pB − pP . The Wolfenstein parametrization and the unitarity triangle convention for the phases of the quark mixing matrix elements, i.e., < P (pP )

|

Vtd = Aλ3 (1 − ρ − iη) = |Vtd |e−iβ , Vub = Aλ3 (ρ − iη) = |Vub |e−iγ , (24) are used in our calculations. Besides the different weak (CP odd) phases that enter nonspectator (penguin) and tree amplitudes, an overall strong (CP even) phase θs for the gluon fusion amplitude is assumed. In fact, one possible source of this strong phase could be the form factor which parametrizes the g −g −η ′ vertex. Inserting H = |H|eiθs (|H| ≈ 1.8GeV−1 in (21) and using the same input values for various parameters as in B → Kη ′ case, we obtain the following branching ratio for B − → π − η ′ −

− ′

−6

B(B → π η ) = 1.26 × 10



|Vub | 0.003

2

[1

|Vtd |2 |Vtd | Cos(β + γ)) 2 − 1.3 |Vub | |Vub | |Vtd | + 1.06(Cos(θs ) − 0.65 Cos(θs + β + γ))] (25) . |Vub | + 0.49(1 + 0.42

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Theoretical High Energy Physics

The first term in Eq. (25) is due to current-current operators (tree) while the second term is the contribution of the nonspectator gluon fusion process (penguin) and the last entry is the cross term. The branching ratio for the CP conjugate process B + → π + η ′ is obtained from the Eq. (25) by changing the sign of the weak angles β and γ. Consequently, the CP asymmetry which is defined as Acp =

Γ(B − → π − η ′ ) − Γ(B + → π + η ′ ) , Γ(B − → π − η ′ ) + Γ(B + → π + η ′ )

(26)

is proportional to Sin(β + γ) = Sin(α) and Sin(θs ). One can show that there is a correlation between the branching ratio B(B − → π − η ′ ) and the direct CP asymmetry Acp when θs takes on values in the range (−π , π). It is quite interesting that a large asymmetry is possible due to the comparable nonspectator and tree contributions to this process. For example, for ρ = 0.25 and η = 0.20 (i.e. |Vub | = 0.0028) , Acp could be as large as 75% and B(B − → π − η ′ ) = 3.4 × 10−6 if θs = 90◦ . A larger branching ratio B(B − → π − η ′ ) = 1.2 × 10−5 is possible for ρ ≈ 0 and η = 0.46 (i.e. |Vub | = 0.0040) however, the asymmetry is somewhat smaller around 20%. On the other hand, for the preferred values ρ = 0.12 and η = 0.34 (i.e. |Vub | = 0.0031), the resulting branching ratio and asymmetry are B(B − → π − η ′ ) = 2.9 × 10−6 , Acp = −0.41 ,

(27)

for θs ≈ −55◦ , which is the case if we assume that the absorptive part of the gluon-gluon-η ′ form factor H is responsible for the strong phase.

Acknowledgments: I am grateful to the Mount Allison University for financial support. Also, I would like to thank SUNYIT and in paricular Professor Amir Fariborz for the hospitality and the superb organization of the conference.

References [1] Jim Smith, talk given at the 7th International Symposium on Heavy Flavor Physics, Santa Barbara CA (July 1997); B. Behren, talk given 124

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at the Princeton BaBar meeting (March 1997); P. Kim, talk given at FCNC 97, Santa Monica CA (Feb,1997); F. W¨ urthwein, CALT-682121, HEP-EX/9706010. [2] S. J. Richichi et al.(CLEO Collaboration), Phys. Rev. Lett. 85, 520 (2000). [3] K. F. Chen (Belle Collaboration), talk at the 31th International Conference on High Energy Physics, Amsterdam, Netherlands, July, 2002. [4] A. Bevan (BaBar Collaboration), talk at the 31th International Conference on High Energy Physics, Amsterdam, Netherlands, July, 2002. [5] M. R. Ahmady, E. Kou, and A. Sugamoto, Phys. Rev. D 58, 014015 (1998). [6] T. Inami and C. S. Lim, Progr. Theor. Phys. 65 (1981) 297. [7] M. R. Ahmady, V. Elias and E. Kou, Phys. Rev. D57, 7034 (1998). [8] E. Kou, Phys. Rev. D63, 054027 (2001). [9] C. Rosenzweig, A. Salomone and J. Schechter, Phys. Rev. D24, 2545 (1981). [10] M. R. Ahmady and E. Kou, Phys. Rev. D59, 054014 (1999).

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SCHEDULE Thursday June 6, 2002 • 8:00 - 8:45 Registration/Continental Breakfast • 8:45 Welcome: – Dr. R. Sarner, Executive Vice President for Academic Affairs, SUNY Institute of Technology – Dr. D. Murphy, Dean, School of Arts and Sciences, SUNY Institute of Technology • Session I: QCD [D.G.C. McKeon, Chair] – 9:00 - 9:35 V. Elias (University of Western Ontario), “Resolving RG-Mass Dependence in Perturbative QCD via AllOrders Summation of RG-Accessible Logarithms” – 9:40 - 10:15 J. Schechter (Syracuse University), “Study of Scalar Mesons and Related Radiative Decays” • 10:20 - 10:45 Break • Session II: QCD (continue) [M.R. Ahmady, Chair] – 10:45 - 11:20 T.G. Steele (University of Saskatchewan), “Gaussian Sum Rules, Scalar Gluonium, and Instantons” – 11:25 - 12:00 A.H. Fariborz (SUNY Institute of Technology), “Possible Applications of Chiral Lagrangians in QCD SumRules” • 12:05 - 1:45 Lunch • Session III: Field Theory [T.N. Sherry, Chair] – 1:45 - 2:20 V.A. Miransky (University of Western Ontario), “Magnetic Catalysis and Anisotropic Confinement in QCD” 127

Theoretical High Energy Physics

– 2:25 - 3:05 D.G.C. Mckeon (University of Western Ontario), “Supersymmetry in AdS2 and AdS3 ” • 3:10 - 3:40 Break • Session IV: B and C Physics [A.H. Fariborz, Chair] – 3:40 - 4:15 M.S. Alam (SUNY at Albany), “Charmed Baryon Spectroscopy: A Test of the Quark Model of Hadrons” – 4:20 - 4:55 A. Petrov (Wayne State University), “D-Dbar Mixing and CP-Violation in Charm” – 5:00 - 5:35 F.A. Chishtie (Cornell University), “Towards full O(αs2 ) QCD Corrections to B → ππ and B → πK Decays Under the Factorization Hypothesis” – 5:40 - 6:15 M.R. Ahmady (Mount Allison University), “Recent QCD Related Issues in B-Physics” • Reception 6:45-9:30

128

LIST OF PARTICIPANTS M. R. Ahmady H. Benaoum F. Chishtie T. Converse V. Elias A.H. Fariborz A. Girard A. Hsie S. Karamov P. Lein D.G.C. McKeon V. A. Miransky J. Pecina A. Petrov T. Phi S. Qazi A. Reimann C. Rosenzweig E. Rusjan J. Schechter T. Sherry A. Squires T.G. Steele

Mount Allison University Syracuse University Cornell University SUNY Institute of Tech Univ. of Western Ontario SUNY Institute of Tech SUNY Morrisville SUNY Institute of Tech Syracuse University MVCC Univ. of Western Ontario Univ. of Western Ontario Rome Research Lab Wayne State University Mount Allison University SUNY Institute of Tech SUNY Institute of Tech Syracuse University SUNY Institute of Tech Syracuse University National Univ. of Ireland Mount Allison University Univ. of Saskatchewan

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[email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]

Theoretical High Energy Physics

130 Front row from the left: Amir Fariborz, Mohammad Ahmady, Joe Schechter, Volodya Miransky, Tan-Trao Phi, Farrukh Chishtie, Hachemi Benaoum, Saj Alam, Alexey Petrov. Back row from the left: Tom Steele, Aaron Reimann, Amgad Squires, Victor Elias, Tom Sherry, Paul Lein