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Main Heavy quark physics
Heavy quark physicsAneesh V. Manohar, Mark B. Wise
Understanding the physics of heavy quarks gives physicists the unique opportunity to test the predictions of Quantum Chromodynamics and the Standard Model. Heavy Quark Physics provides an exciting introduction to this new area of high energy physics. Two leading experts in the field review the standard model, the basics of heavy quark spin-flavor symmetry, and detail how it can be applied to the classification of states, decays and fragmentation. The heavy quark effective theory is developed, including the radiative and 1/mQ corrections, and applied to the study of hadron masses, form factors, and inclusive decay rates. The authors also discuss the application of chiral perturbation theory to heavy hadrons. To aid the reader, many of the key calculations are performed step by step, and problems are provided at the end of each chapter. This lucid volume provides graduate students with an ideal introduction to the physics of heavy quarks, and more experienced researchers with an authoritative reference to the subject.
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Heavy Quark Physics Understanding the physics of heavy quarks gives physicists the unique opportunity to test the predictions of Quantum Chromodynamics and the Standard Model. This is the first introductory text to this exciting new area of high-energy physics. The book begins with a review of the standard model, followed by the basics of heavy quark spin-flavor symmetry and how it can be applied to the classification of states, decays, and fragmentation. Heavy quark effective theory is then developed, including the radiative and 1/m Q corrections, and applied to the study of hadron masses, form factors, and inclusive decay rates. The authors also discuss the application of chiral perturbation theory to heavy hadrons. Written by two world leaders, the presentation is clear, original, and thoroughly modern. To aid the reader, many of the key calculations are performed step by step, and problems and a concise review of the literature are provided at the end of each chapter. This lucid volume provides graduate students with an ideal introduction to the physics of heavy quarks, and more experienced researchers with an authoritative reference to the subject. Aneesh Manohar is Professor of Physics at the University of California, San Diego. After receiving his Ph.D. from Harvard University, Professor Manohar held positions at Harvard and the Massachusetts Institute of Technology before moving to the University of California, San Diego, where he has been since 1989. He has been a Scientific Associate at CERN, Geneva, a Visiting Fellow at Princeton University, and Iberdrola Visiting Professor at the University of Valencia. He was also awarded the A. P. Sloan Fellowship from 1987 to 1990. Mark Wise is the John A. McCone Professor of High Energy Physics at the California Institute of Technology. After receiving his Ph.D. from Stanford, he was a Junior Fellow at Harvard University before moving to the California Institute of Technology in 1983. Professor Wise also held the A. P. Sloan Fellowship from 1984 to 1987. C A M B R I D G E M O N O G R A P H S O N P A R T I C L E P H Y S I C S, NUCLEAR PHYSICS AND COSMOLOGY 10 General Editors: T. Ericson, P. V. Landshoff 1. K. Winter (ed.): Neutrino Physics 2. J. F. Donoghue, E. Golowich and B. R. Holstein: Dynamics of the Standard Model 3. E. Leader and E. Predazzi: An Introduction to Gauge Theories and Modern Particle Physics, Volume 1: Electroweak Interactions, the 'New Particles' and the Parton Model 4. E. Leader and E. Predazzi: An Introduction to Gauge Theories and Modern Particle Physics, Volume 2: CP-Violation, QCD and Hard Processes 5. C. Grupen: Particle Detectors 6. H. Grosse and A. Martin: Particle Physics and the Schrödinger Equation 7. B. Andersson: The Lund Model 8. R. K. Ellis, W. J. Stirling and B. R. Webber: QCD and Collider Physics 9. I. I. Bigi and A. I. Sanda: CP Violation 10. A. V. Manohar and M. B. Wise: Heavy Quark Physics Heavy Quark Physics A N E E S H V. M A N O H A R University of California, San Diego M A R K B. W I S E California Institute of Technology CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521642415 © Cambridge University Press 2000 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2000 This digitally printed version 2007 A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Manohar, Aneesh Vasant. Heavy quark physics / Aneesh V. Manohar, Mark B. Wise. p. cm. – (Cambridge monographs on particle physics, nuclear physics, and cosmology ; 10) ISBN 0-521-64241-8 (hb) 1. Quarks. 2. Heavy particles (Nuclear physics) I. Wise, Mark B., 1953– . II. Title. III. Series. QC793.5.Q252M36 2000 99-38167 539.7´2167 – dc21 CIP ISBN 978-0-521-64241-5 hardback ISBN 978-0-521-03757-0 paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. To our wives, Elizabeth and Jacqueline Contents Preface page ix 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 Review The standard model Loops Composite operators Quantum chromodynamics and chiral symmetry Integrating out heavy quarks Effective Hamiltonians for weak decays The pion decay constant The operator product expansion Problems References 1 1 8 15 17 24 25 31 32 42 43 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 Heavy quarks Introduction Quantum numbers Strong decays of excited heavy hadrons Fragmentation to heavy hadrons Covariant representation of fields The effective Lagrangian Normalization of states Heavy meson decay constants B̄ → D (∗) form factors c → form factors b → c form factors Problems References 44 44 45 48 52 54 58 60 61 63 70 72 72 75 vii viii Contents 3 3.1 3.2 3.3 3.4 3.5 3.6 Radiative corrections Renormalization in HQET Matching between QCD and HQET Heavy-light currents Heavy-heavy currents Problems References 77 77 84 87 95 99 100 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 Nonperturbative corrections The 1/m Q expansion Reparameterization invariance Masses b → c eν̄e decay B̄ → D (∗) eν̄e decay and Luke's theorem Renormalons v · A = 0 gauge NRQCD Problems References 102 102 104 105 107 112 115 123 124 127 129 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Chiral perturbation theory Heavy mesons gπ in the nonrelativistic constituent quark model B̄ → π eν̄e and D → π ēνe decay Radiative D ∗ decay Chiral corrections to B̄ → D (∗) eν̄e form factors Problems References 131 131 136 138 141 146 149 150 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 Inclusive weak decay Inclusive semileptonic decay kinematics The operator product expansion Differential decay rates Physical interpretation of 1/m 2b corrections The electron endpoint region |Vcb | from inclusive decays Sum rules Inclusive nonleptonic decays Bs − B̄ s mixing Problems References 151 151 157 164 166 168 173 175 178 181 186 187 Index 189 Preface We are entering an exciting era of B meson physics, with several new high luminosity facilities that are about to start taking data. The measurements will provide information on quark couplings and CP violation. To make full use of the experimental results, it is important to have reliable theoretical calculations of the hadronic decay amplitudes in terms of the fundamental parameters in the standard model Lagrangian. In recent years, many such calculations have been performed using heavy quark effective theory (HQET), which has emerged as an indispensible tool for analyzing the interactions of heavy hadrons. This formalism makes manifest heavy quark spin-flavor symmetry, which is exact in the infinite quark mass limit, and allows one to systematically compute the correction terms for finite quark mass. This text is designed to introduce the reader to the concepts and methods of HQET, developing them to the stage where explicit calculations are performed. It is not intended to be a review of the field, but rather to serve as an introduction accessible to both theorists and experimentalists. We hope it will be useful not just to those working in the area of heavy quark physics but also to physicists who work in other areas of high energy physics but want a deeper appreciation of HQET methods. We felt that if the book is to serve this role, then it is important that it not be too long. An effort was made to keep the book at the 200-page level and this necessitated some difficult decisions on which subjects were to be covered. The material presented here is not uniform in its difficulty. Section 1.8 on the operator product expansion, Section 4.6 on renormalons, and Chapter 6 on inclusive B decays are considerably more difficult than the other parts of the book. Although this material is very important, depending on the background of the reader, it may be useful to skip it on first reading. Chapter 3 involves some familiarity with radiative corrections in field theory as studied, for example, in a graduate course that discusses renormalization in quantum electrodynamics. Readers less comfortable with loop corrections can read through the chapter, accepting the results for the one-loop diagrams, without necessarily going through ix x Preface the detailed computations. A section on problems at the end of each chapter is intended to give the reader more experience with the concepts introduced in that chapter. The problems are of varying difficulty and most can be completed in a fairly short period of time. Three exceptions to this are Problem 2 of Chapter 3 and Problems 3 and 7 of Chapter 6, which are considerably more time-consuming. This book could serve as a text for a one-semester graduate course on heavy quark physics. The background necessary for the book is quantum field theory and some familiarity with the standard model. The latter may be quite modest, since Chapter 1 is devoted to a review of the standard model. The only references that are given in the text are to lattice QCD results or to experimental data that cannot be readily found by consulting the Particle Data Book (http://pdg.lbl.gov). However, at the end of each chapter a guide to some of the literature is given. The emphasis here is on the earlier papers, and even this list is far from complete. We have benefited from the comments given by a large number of our colleagues who have read draft versions of this book. Particularly noteworthy among them are Martin Gremm, Elizabeth Jenkins, Adam Leibovich, and Zoltan Ligeti, who provided a substantial number of valuable suggestions. Updates to the book can be found at the URL: http://einstein.ucsd.edu/hqbook. 1 Review The standard model of strong, weak, and electromagnetic interactions is a relativistic quantum field theory that describes all known interactions of quarks and leptons. This chapter provides a quick review of features of the standard model that are relevant for heavy quark systems, and of basic field theory techniques such as the operator product expansion. It will also serve the purpose of defining some of the normalization conventions and notation to be used in the rest of the book. 1.1 The standard model The standard model is a gauge theory based on the gauge group SU(3) × SU(2) × U(1). The SU(3) gauge group describes the strong color interactions among quarks, and the SU(2) × U(1) gauge group describes the electroweak interactions. At the present time three generations of quarks and leptons have been observed. The measured width of the Z boson does not permit a fourth generation with a massless (or light) neutrino. Many extensions of the minimal standard model have been proposed, and there is evidence in the present data for neutrino masses, which requires new physics beyond that in the minimal standard model. Low-energy supersymmetry, dynamical weak symmetry breaking, or something totally unexpected may be discovered at the next generation of high-energy particle accelerators. The focus of this book is on understanding the physics of hadrons containing a bottom or charm quark. The technically difficult problem is understanding the role strong interactions play in determining the properties of these hadrons. For example, weak decays can be computed by using a low-energy effective weak Hamiltonian. Any new physics beyond the standard model can also be treated by using a local low-energy effective interaction, and the theoretical difficulties associated with evaluating hadronic matrix elements of this interaction are virtually identical to those for the weak interactions. For this reason, most of the 1 2 Review discussion in this book will focus on the properties of heavy quark hadrons as computed in the standard model. The matter fields in the minimal standard model are three families of spin1/2 quarks and leptons, and a spin-zero Higgs boson, shown in Table 1.1. The index i on the Fermion fields is a family or generation index i = 1, 2, 3, and the subscripts L and R denote left- and right-handed fields, respectively, ψ L = PL ψ, ψ R = PR ψ, (1.1) where PL and PR are the projection operators PL = 1 (1 − γ5 ) , 2 PR = 1 (1 + γ5 ) . 2 (1.2) Q iL , u iR , d Ri are the quark fields and L iL , eiR are the lepton fields. All the particles associated with the fields in Table 1.1 have been observed experimentally, except for the Higgs boson. The SU(2) × U(1) symmetry of the electroweak sector is not manifest at low energies. In the standard model, the SU(2) × U(1) symmetry is spontaneously broken by the vacuum expectation value of the Higgs doublet Table 1.1. Matter fields in the standard modela Field Q iL = u iL d Li u iR d Ri L iL = H= a SU(2) U(1) Lorentz 3 2 1/6 (1/2, 0) 3 1 2/3 (0, 1/2) 3 1 −1/3 (0, 1/2) 1 2 −1/2 (1/2, 0) 1 1 −1 (0, 1/2) 1 2 1/2 (0, 0) i νL eiL eiR SU(3) H+ H0 The index i labels the quark and lepton family. The dimensions of the SU(3) and SU(2) representations and their U(1) charge are listed in the second, third, and fourth columns, respectively. The transformation properties of the fermion fields under the Lorentz group SO(3, 1) are listed in the last column. 1.1 The standard model 3 H . The spontaneous breakdown of SU(2) × U(1) gives mass to the W ± and Z 0 gauge bosons. A single Higgs doublet is the simplest way to achieve the observed pattern of spontaneous symmetry breaking, but a more complicated scalar sector, such as two doublets, is possible. The terms in the standard model Lagrangian density that involve only the Higgs doublet + H (1.3) H= H0 are LHiggs = (Dµ H )† (D µ H ) − V (H ), (1.4) where Dµ is the covariant derivative and V (H ) is the Higgs potential V (H ) = λ † (H H − v 2 /2)2 . 4 (1.5) The Higgs potential is minimized when H † H = v 2 /2. The SU(2) × U(1) symmetry can be used to rotate a general vacuum expectation value into the standard form 0√ H = , (1.6) v/ 2 where v is real and positive. The generators of the SU(2) gauge symmetry acting on the Higgs (i.e., fundamental) representation are T a = σ a /2, a = 1, 2, 3, where the Pauli spin matrices are 0 1 0 −i 1 0 1 2 3 σ = , σ = , σ = , 1 0 i 0 0 −1 (1.7) (1.8) and the generators are normalized to Tr T a T b = δ ab /2. The U(1) generator Y is called hypercharge and is equal to 1/2 acting on the Higgs doublet (see Table 1.1). One linear combination of SU(2) × U(1) generators is left unbroken by the vacuum expectation value of the Higgs field H given in Eq. (1.6). This linear combination is the electric charge generator Q = T 3 + Y , where 1 0 , (1.9) Q = T3 + Y = 0 0 when acting on the Higgs representation. It is obvious from Eqs. (1.6) and (1.9) that QH = 0, (1.10) 4 Review so that electric charge is left unbroken. The SU(3) × SU(2) × U(1) symmetry of the standard model is broken to SU(3) × U(1) Q by the vacuum expectation value of H , where the unbroken electromagnetic U(1) Q is the linear combination of the original U(1) hypercharge generator, Y , and the SU(2) generator, T 3 , given in Eq. (1.9). Expanding H about its expectation value h + (x) √ (1.11) H (x) = v/ 2 + h 0 (x) and substituting in Eq. (1.5) gives the Higgs potential √ λ 2 2 V (H ) = (|h + | + |h 0 | + 2v Re h 0 )2 . 4 (1.12) The fields h + and Im h 0 are massless. This is an example of Goldstone's theorem. The potential has a continuous three-parameter family of degenerate vacua that are obtained from the reference vacuum in Eq. (1.6) by global SU(2) × U(1) transformations. [Of the four SU(2) × U(1) generators, one linear combination Q leaves the vacuum expectation value invariant, and so does not give a massless mode.] Field excitations along these degenerate directions cost no potential energy and so the fields h + and Im h 0 are massless. There √ is one massive scalar that is destroyed by the (normalized) real scalar field 2 Re h 0 . At tree level, its mass is λ v. (1.13) m Re h 0 = 2 Global SU(2) × U(1) transformations allow the space–time independent vacuum expectation value of H to be put into the form given in Eq. (1.6). Local SU(2) × U(1) transformations can be used to eliminate h + (x) and Im h 0 (x) completely from the theory, and to write 0 √ H (x) = . (1.14) v/ 2 + Re h 0 (x) This is the standard model in unitary gauge, in which the W ± and Z bosons have explicit mass terms in the Lagrangian, as is shown below. In this gauge, the massless fields h + and Im h 0 are eliminated, and so do not correspond to states in the spectrum of the theory. The gauge covariant derivative acting on any field ψ is Dµ = ∂µ + ig AµA T A + ig2 Wµa T a + ig1 Bµ Y, (1.15) where T A , A = 1, . . . , 8, are the eight color SU(3) generators T a , a = 1, 2, 3 are the weak SU(2) generators, and Y is the U(1) hypercharge generator. The generators are chosen to be in the representation of the field ψ on which the covariant derivative acts. The gauge bosons and coupling constants associated with 1.1 The standard model 5 these gauge groups are denoted AµA , Wµa , and Bµ and g, g2 , and g1 , respectively. The kinetic term for the Higgs field contains a piece quadratic in the gauge fields when expanded about the Higgs vacuum expectation value using Eq. (1.11). The quadratic terms that produce a gauge-boson mass are g22 v 2 1 1 v2 (1.16) (W W + W 2 W 2 ) + (g2 W 3 − g1 B)2 , mass 8 8 where for simplicity of notation Lorentz indices are suppressed. The charged W -boson fields L gauge-boson = W± = W1 ∓ iW2 √ 2 (1.17) have mass g2 v . 2 It is convenient to introduce the weak mixing angle θW defined by g1 g2 , cos θW = . sin θW = 2 2 2 2 g1 + g2 g1 + g2 MW = (1.18) (1.19) The Z -boson field and photon field A are defined as linear combinations of the neutral gauge-boson fields W 3 and B, Z = cos θW W 3 − sin θW B, A = sin θW W 3 + cos θW B. The Z boson has a mass at tree level g12 + g22 MW , v= MZ = 2 cos θW (1.20) (1.21) and the photon is massless. The covariant derivative in Eq. (1.15) can be reexpressed in terms of the mass-eigenstate fields as g2 Dµ = ∂µ + ig AµA T A + i √ Wµ+ T + + Wµ− T − 2 + i g12 + g22 (T3 − sin2 θW Q)Z µ + ig2 sin θW QAµ , (1.22) where T ± = T 1 ± i T 2 . The photon coupling constant in Eq. (1.22) leads to the relation between the electric charge e and the couplings g1,2 , g2 g1 e = g2 sin θW = , (1.23) g12 + g22 6 Review so the Z coupling constant g12 + g22 in Eq. (1.22) is conventionally written as e/(sin θW cos θW ). Outside of unitary gauge the H kinetic term also has a piece quadratic in the fields where the Goldstone bosons h + , Im h 0 mix with the longitudinal parts of the massive gauge bosons. This mixing piece can be removed by adding to the Lagrange density the 't Hooft gauge fixing term 1 ∂ µ Wµa + ig2 ξ (H † T a H − H † T a H ) 2ξ a 1 µ 2 ∂ Bµ + ig1 ξ (H † Y H − H † Y H ) , − 2ξ L gauge = − fix 2 (1.24) which gives the Lagrangian in Rξ gauge, where ξ is an arbitrary parameter. The fields h ± and Im h 0 have mass terms proportional to the gauge fixing constant ξ . In Feynman gauge ξ = 1 (the easiest for doing calculations), these masses are the same as those of the W ± and Z . Im h 0 and h ± are not physical degrees of freedom since in unitary gauge ξ → ∞ their masses are infinite and they decouple from the theory. SU(3) × SU(2) × U(1) gauge invariance prevents bare mass terms for the quarks and leptons from appearing in the Lagrange density. The quarks and leptons get mass because of their Yukawa couplings to the Higgs doublet, LYukawa = gui j ū iR H T Q L −gd d̄ iR H † Q L −gei j ēiR H † L L +h.c. j ij j j (1.25) where h.c. denotes Hermitian conjugate. Here repeated indices i, j are summed and the antisymmetric matrix is given by 0 1 = . (1.26) −1 0 Color indices and spinor indices are suppressed in Eq. (1.25). Since H has a vacuum expectation value, the Yukawa couplings in Eq. (1.25) give rise to the 3 × 3 quark and lepton mass matrices √ √ √ (1.27) Mu = vgu / 2, Md = vgd / 2, and Me = vge / 2. Neutrinos do not get mass from the Yukawa interactions in Eq. (1.25), since there is no right-handed neutrino field. Any matrix M can be brought into diagonal form by separate unitary transformations on the left and right, M → L D R † , where L and R are unitary, and D is real, diagonal and nonnegative. One can make separate unitary transformations on the left- and right-handed quark and lepton fields, while leaving the kinetic energy terms for the quarks, Q̄ iL i∂/ Q iL , ū iR i∂/ u iR , and d̄ iR i∂/ d Ri , and also those for 1.1 The standard model 7 the leptons, invariant. The unitary transformations are u L = U (u, L) u L , u R = U (u, R) u R , d L = U (d, L) d L , d R = U (d, R) d R , e L = U (e, L) eL , e R = U (e, R) eR . (1.28) Here u, d, and e are three-component column vectors (in flavor space) for the quarks and leptons, and the primed fields represent the corresponding mass eigenstates. The transformation matrices U are 3 × 3 unitary matrices, which are chosen to diagonalize the mass matrices ⎛ ⎞ mu 0 0 (1.29) U (u, R)† Mu U (u, L) = ⎝ 0 m c 0 ⎠ , 0 0 mt ⎛ ⎞ md 0 0 U (d, R)† Md U (d, L) = ⎝ 0 m s 0 ⎠ , (1.30) 0 0 mb and ⎛ ⎞ 0 me 0 U (e, R)† Me U (e, L) = ⎝ 0 m µ 0 ⎠ . 0 0 mτ (1.31) Diagonalizing the quark mass matrices in Eqs. (1.29) and (1.30) requires different transformations of the u L and d L fields, which are part of the same SU(2) doublet Q L . The original quark doublet can be rewritten as ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ u U (u, L) u L u ⎝ L⎠=⎝ ⎠ = U (u, L) ⎝ L ⎠ , (1.32) dL U (d, L) d L V d L where the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix V is defined by V = U (u, L)† U (d, L) . (1.33) It is convenient to reexpress the standard model Lagrangian in terms of the primed mass-eigenstate fields. The unitary matrices in Eq. (1.32) leave the quark kinetic terms unchanged. The Z and A couplings are also unaffected, so there are no flavor-changing neutral currents in the Lagrangian at tree level. The W couplings are left unchanged by U (u, L), but not by V , so that g2 g2 (1.34) √ W + ū L γ µ d L = √ W + ū L γ µ V d L . 2 2 As a result there are flavor-changing charged currents at tree level. The CKM matrix V is a 3 × 3 unitary matrix, and so is completely specified by nine real parameters. Some of these can be eliminated by making phase redefinitions of the quark fields. The u and d quark mass matrices are unchanged if one makes independent phase rotations on the six quarks, provided the same 8 Review phase is used for the left- and right-handed quarks of a given flavor. An overall equal phase rotation on all the quarks leaves the CKM matrix unchanged, but the remaining five rotations can be used to eliminate five parameters, so that V is written in terms of four parameters. The original Kobayashi-Maskawa parameterization of V is ⎞ ⎛ s1 c 3 s1 s3 c1 ⎟ ⎜ ⎟ ⎜ (1.35) V = ⎜ −s1 c2 c1 c2 c3 − s2 s3 eiδ c1 c2 s3 + s2 c3 eiδ ⎟ , ⎠ ⎝ −s1 s2 c1 s2 c3 + c2 s3 eiδ c1 s2 s3 − c2 c3 eiδ where ci ≡ cos θi , and si ≡ sin θi for i = 1, 2, 3. The angles θ1 , θ2 , and θ3 can be chosen to lie in the first quadrant, where their sines and cosines are positive. Experimentally it is known that these angles are quite small. The CKM matrix is real if δ = 0, so that δ = 0 is a signal of CP violation in the weak interactions. It describes the unitary transformation between the mass-eigenstate basis d i , and the weak interaction eigenstate basis d i . The standard notation for the masseigenstate fields is u 1 = u, u 2 = c, u 3 = t, d 1 = d, d 2 = s, d 3 = b. So far we have only considered the left-handed quark couplings to the gauge bosons. For the right-handed quarks there are no W -boson interactions in the standard model, and in the primed mass-eigenstate basis the couplings of the Z , photon, and color gauge bosons are flavor diagonal. The analysis for leptons is similar to that for quarks, with one notable difference – because the neutrinos are massless, one can choose to make the same unitary transformation on the left-handed charged leptons and neutrinos. The analog of the CKM matrix in the lepton sector can be chosen to be the unit matrix, and the leptons can be chosen to be simultaneously mass and weak eigenstates. We adopt the notation ν 1 = νe , ν 2 = νµ , ν 3 = ντ , e1 = e, e2 = µ, e3 = τ . From now on, we will use the mass-eigenstate basis for labeling the quark and lepton fields. 1.2 Loops Loop diagrams in the standard model have divergences from the high-momentum (ultraviolet) region of the momentum integrals. These divergences are interpreted by a renormalization procedure; the theory is regulated in some way and terms that diverge as the regulator is removed are absorbed into the definitions of the couplings and masses. Theories in which all divergences in physical quantities (e.g., S-matrix elements) can be removed in this way using a finite number of counterterms are called renormalizable. In the unitary gauge, ξ → ∞, the standard model is manifestly unitary (i.e., only physical degrees of freedom propagate because the "ghost" Higgs associated with h ± and Im h 0 have infinite 1.2 Loops 9 mass). The vector-boson propagator −i 2 gµν − kµ kν /MW,Z 2 k 2 − MW,Z (1.36) is finite as k → ∞, and naive power counting suggests that the standard model is not renormalizable. In the Feynman gauge, ξ = 1, the vector-boson propagator is gµν −i 2 , (1.37) 2 k − MW,Z which falls off as 1/k 2 , and naive power counting shows that the standard model is renormalizable. The potentially disastrous divergences that occur in the unitary gauge must cancel. However, unitarity is not manifest in the Feynman gauge because the unphysical degrees of freedom associated with h ± and Im h 0 are included as intermediate states in Feynman diagrams. The standard model is manifestly unitary in one gauge and manifestly renormalizable in another. Gauge invariance assures us that the theory is both unitary and renormalizable. In this book we will regularize Feynman diagrams by using dimensional regularization. Diagrams are calculated in n = 4 − dimensions, and the ultraviolet divergences that occur in four dimensions appear as factors of 1/ , as → 0. To review how dimensional regularization works, consider the quantum electrodynamics (QED) Lagrangian (0) 1 (0) (0)µν (0) (0) F + i ψ̄ (0) γ µ ∂µ − ie(0) A(0) − m (0) LQED = − Fµν µ ψ e ψ̄ ψ , (1.38) 4 which is part of the standard model Lagrangian. The superscript (0) is used to denote a bare quantity. Here (0) (0) = ∂µ A(0) Fµν ν − ∂ν A µ (1.39) is the bare electromagnetic field strength tensor. In n dimensions, the action SQED = d n x LQED (1.40) is dimensionless, since ei SQED is the measure in the Feynman path integral (we use units where h̄ = c = 1). It follows that the dimensions of the fields, the coupling constant e(0) , and the electron mass, m (0) e , are A(0) = (n − 2)/2 = 1 − /2, ψ (0) = (n − 1)/2 = 3/2 − /2, e(0) = (4 − n)/2 = /2, m (0) = 1. e (1.41) 10 Review The bare fields are related to the renormalized fields by 1 Aµ = √ A(0) µ , ZA 1 ψ = ψ (0) , Zψ 1 e = µ− /2 e(0) , Ze 1 (0) m . me = Zm e (1.42) The factor of µ− /2 is included in the relation between the bare and renormalized electric couplings so that the renormalized coupling is dimensionless. Here µ is a parameter with dimensions of mass and is called the subtraction point or renormalization scale of dimensional regularization. In terms of these renormalized quantities the Lagrange density is 1 LQED = − Z A Fµν F µν + i Z ψ ψ̄γ µ ∂µ − iµ /2 Z e Z A eAµ ψ 4 −Z m Z ψ m e ψ̄ψ, 1 = − Fµν F µν + i ψ̄γ µ ∂µ − iµ /2 eAµ ψ − m e ψ̄ψ + counterterms. 4 (1.43) It is straightforward to compute the renormalization constants Z A,ψ,e,m by using the formula for one-loop integrals in dimensional regularization, dnq (q 2 )α (2π )n (q 2 − M 2 )β i (α + n/2)(β − α − n/2) = n n/2 (−1)α+β (M 2 )α−β+n/2 , (1.44) 2 π (n/2)(β) and the Feynman trick for combining denominators, a1m 1 (M) 1 mn = (m 1 ) · · · (m n ) · · · an 1 1 × d x1 x1m 1 −1 · · · d xn xnm n −1 0 0 δ 1 − in= 1 xi , (1.45) (x1 a1 + · · · + xn an ) M where n mi . M= i =1 The Z 's are determined by the condition that time-ordered products of renormalized fields (i.e., Green's functions) be finite when expressed in terms of the 1.2 Loops 11 renormalized coupling and mass. This condition still leaves considerable freedom in how the Z 's are chosen. The precise way that the Z 's are chosen is called the subtraction scheme. The Z 's can be chosen to have the form Z =1+ ∞ Z p (e) p , (1.46) p=1 where the Z p (e) are independent of . This choice is called minimal subtraction (MS) because only the poles in are subtracted and no additional finite pieces are put into the Z 's. We will use the MS scheme, which is minimal subtraction followed by the rescaling µ2 → µ2 eγ /4π, where γ = 0.577 . . . is Euler's constant. The photon wavefunction renormalization Z A to order e2 can be determined by computing the photon–photon correlation function. There are two pieces to this order; the first is a tree-level contribution from the counterterm 1 − (Z A − 1)Fµν F µν . 4 (1.47) After truncating the external photon propagators, it gives i(Z A − 1)( pµ pν − p 2 gµν ), (1.48) where p is the photon four momentum. The second contribution is from the one-loop diagram Fig. 1.1, d n q Tr[γµ i(q/ + /p + m e )γν i (q/ + m e )] 2 . (1.49) (−1) (ie) µ (2π )n (q + p)2 − m 2e q 2 − m 2e The factor of (−1) arises from the closed fermion loop. The renormalization constant only depends on the 1/ pole, so the γ matrix algebra can be performed in four dimensions. Expanding µ =1+ ln µ + · · · , (1.50) one sees that µ can be set to unity for the infinite part of the diagram, and finite q p p p+q Fig. 1.1. One-loop vacuum polarization contribution to the photon propagator. 12 Review parts only depend logarithmically on µ. The denominators are combined using Eq. (1.45): 1 1 1 dx . (1.51) = 2 2 (q + p) − m 2e q 2 − m 2e 0 q 2 + 2xq · p + p 2 x − m 2e Making the change of variables k = q + px gives 1 dnk 1 −4e2 dx 2 n 2 2 (2π ) 0 k + p x(1 − x) − m 2e × 2kµ kν − k 2 − m 2e gµν − 2x(1 − x) pµ pν + p 2 x(1 − x)gµν . (1.52) Terms odd in k vanish upon integration and have been dropped. Evaluating the k integral using Eq. (1.44), keeping only the part proportional to 1/ (using ( /2) = 2/ + · · ·), and doing the x integral gives the divergent part of the one-loop contribution: 2 8e i (1.53) ( pµ pν − p 2 gµν ). 2 16π 3 For the photon two-point correlation function to be finite as → 0, the sum of Eqs. (1.53) and (1.48) must be finite. One therefore chooses 2 e 8 . (1.54) ZA = 1 − 3 16π 2 The wave-function renormalization constant Z ψ for the electron field ψ is obtained from the electron propagator. The counterterms (Z ψ − 1)ψ̄i∂/ ψ − (Z m Z ψ − 1)m e ψ̄ψ (1.55) i(Z ψ − 1)/p − i(Z m Z ψ − 1)m e (1.56) contribute to the propagator. In the Feynman gauge, the one-loop diagram Fig. 1.2 is dnq (−i)g µν /p + q/ + m e 2 µ (ie) γ i γ . (1.57) ν µ (2π )n q2 ( p + q)2 − m 2e q p p+q p Fig. 1.2. One-loop correction to the electron propagator. 1.2 Loops 13 Combining denominators and shifting the momentum integration as in the previous case gives 1 dnk −2m e + /p (1 − x) 2 dx . (1.58) 2e 2 n (2π ) 0 k 2 − m 2e x + p 2 x(1 − x) Performing the k integration by using Eq. (1.44) and then the x integration gives 1 i 2 (4e p (1.59) ) −2m + / e 16π 2 2 for the divergent contribution. The electron propagator is finite if 2 e Zψ = 1 − 2 16π 2 and e2 Zm = 1 − 6 16π 2 (1.60) (1.61) in the Feynman gauge. The remaining renormalization factor Z e can be determined by computing the ψ ψ̄ A three-point function to order e2 . The Feynman graph that has to be computed is the vertex renormalization graph of Fig. 1.3. The counterterm is 2 4 e . (1.62) Ze = 1 + 3 16π 2 √ Note that Z e = 1/ Z A to order e2 . The relation between the bare and renormalized couplings at order e2 is 2 e 4 , (1.63) e(0) = µ /2 eZ e = µ /2 e 1 + 3 16π 2 using Eqs. (1.62) and (1.42). The bare fields, coupling, and mass are independent of the subtraction point µ, which is an arbitrary quantity with dimensions of mass introduced so that the renormalized coupling is dimensionless. Since the bare Fig. 1.3. One-loop vertex correction. 14 Review coupling constant is independent of µ, d d 0 = µ e(0) = µ µ dµ dµ /2 eZ e = µ /2 eZ e 1 µ dZ e + β (e) + , 2 e Z e dµ (1.64) where the β function is defined by β(e) = µ de . dµ (1.65) This gives d ln Z e β(e) = − e − e . 2 d ln µ Using Eq. (1.62), 4 1 d µ e2 + · · · 3 16π 2 dµ e2 + ···, =− 12π 2 (1.66) d ln Z e = d ln µ (1.67) where the ellipses denote terms of higher order in e2 . The one-loop β function is e3 + ···, β(e) = − e + 2 12π 2 which is finite as (1.68) → 0, e3 + ··· . (1.69) 12π 2 The β function gives the µ dependence of the renormalized coupling e. Here µ is an arbitrary scale parameter, so physical quantities do not depend on µ. However, some choices for µ are more convenient than others for computations. Consider the cross section for σ (e+ e− → anything) at a center of mass energy squared, s = ( pe+ + pe− )2 m 2e . In QED this cross section is finite as m e → 0 and so for large s we neglect m e . The cross section has a power series expansion in the coupling e(µ), and it is independent of the subtraction point µ. The implicit µ dependence in the coupling is canceled by an explicit µ dependence in the Feynman diagrams. (One can see this by computing, e.g., the finite parts of Figs. 1.1–1.3.) Typically one finds that terms in the perturbation series have the form [α (µ) /4π ]n lnm s/µ2 , with m ≤ n, where β(e) = α (µ) = e2 (µ) 4π (1.70) is the (scale-dependent) fine structure constant. If s/µ2 is not of the order of unity, the logarithms can get large and cause a breakdown of perturbation theory. 1.3 Composite operators 15 One usually chooses µ2 ∼ s, which "minimizes" the higher-order terms in the perturbation expansion that have not been computed. With √ this choice of µ, one expects that perturbation theory is an expansion in α( s)/4π . When perturbation theory is valid we can use Eqs. (1.65) and (1.69) to solve explicitly for the dependence of the coupling on µ at one loop: 2 µ2 1 1 1 = 2 − . (1.71) ln 2 2 e (µ2 ) e (µ1 ) 12π µ21 The β function in Eq. (1.69) is positive, so e increases as µ increases, as can be seen explicitly from the solution in Eq. (1.71). 1.3 Composite operators Composite operators involve products of fields at the same space–time point. Consider, for example, the bare mass operator S (0) = ψ̄ (0) ψ (0) (x) . (1.72) Green's functions with an insertion of S (0) are usually divergent. An additional operator renormalization (beyond wave-function renormalization) is required to make the Green's functions finite. The renormalized operator S is S= Zψ 1 (0) (0) 1 (0) ψ̄ ψ = ψ̄ψ, S = ZS ZS ZS (1.73) where Z S is the additional operator renormalization. The operator S = ψ̄ψ + counterterms is conventionally denoted by just ψ̄ψ, with the counterterms implicit. Green's functions with insertions of S are finite in perturbation theory. The renormalization factor Z S can be computed from the three-point function of the time-ordered product of ψ, ψ̄, and S. It is simpler to use the one-particle irreducible Green's function rather than the full Green's function G to compute Z S . The counterterm contribution to the one-particle irreducible Green's function is Zψ − 1. ZS (1.74) The one-loop contribution to is shown in Fig. 1.4. The operator S contains no derivatives (and Z S is mass independent in the MS scheme), so Z S can be determined by evaluating Fig. 1.4 at zero external momentum (and neglecting the electron mass), giving d n q α iq/ iq/ β (−i)gαβ dnq 1 2 2 µ (ie) γ γ = −4ie + · · · , (1.75) (2π )n q 2 q 2 q2 (2π)n (q 2 )2 16 Review Fig. 1.4. One-loop graph with an insertion of a fermion-bilinear composite operator (denoted by ⊗) such as ψ̄ψ. where the ellipsis denote terms finite as → 0. Note that neglecting external momenta and the electron mass has produced an infrared (i.e., low momentum) divergence. Regulating this with a mass m by replacing q 2 in the denominator with (q 2 − m 2 ) gives 8e2 , 16π 2 (1.76) for the ultraviolet divergent part of Eq. (1.75). Adding Eqs. (1.74) and (1.76) together and using Eq. (1.60), we find that the 1/ divergence cancels, provided 2 e . (1.77) ZS = 1 + 6 16π 2 The anomalous dimension of the composite operator S is defined by γS = µ d ln Z S dµ (1.78) 6e2 . 16π 2 (1.79) so that γS = − Similar calculations can be performed for the vector and axial vector currents ψ̄γµ ψ and ψ̄γµ γ5 ψ, and one finds Z V = Z A = 1, so that the currents are not renormalized and their anomalous dimensions vanish at one loop. Note that Z = 1 means that the infinite part of Fig. 1.4 is canceled by wave-function renormalization, not that Fig. 1.4 is finite. The result Z = 1 arises because for m e = 0 both the axial and vector currents are conserved and the zero-component of these currents (integrated over all space) are charges Q A,V with commutation relations of the form [Q V , ψ] = −ψ, (1.80) for example. A conserved charge Q cannot be multiplicatively renormalized since that would spoil such commutation relations. In dimensional regularization with minimal subtraction, electron mass effects cannot induce a renormalization for the axial current because the renormalization factors are independent of particle masses. This is an example of a general result that "soft" symmetry 1.4 Quantum chromodynamics and chiral symmetry 17 breaking effects, i.e., symmetry breaking terms with operator dimensions less than four, do not affect renormalization in the MS scheme. The axial current is not conserved at one loop because of the axial anomaly. The divergence of the axial current is proportional to the dimension-four operator F F̃, so that symmetry breaking because of the anomaly is not soft. It produces an anomalous dimension for the axial current at two loops. We have considered a particularly simple example in which the operator S was multiplicatively renormalized, since there are no other gauge invariant local operators with the same quantum numbers. In general, one can have many different operators Oi with the same quantum numbers, and one needs a renormalization matrix, Oi(0) = Z i j O j . (1.81) This is referred to as operator mixing. In the MS scheme, Z i j is dimensionless, so operators can only mix with other operators of the same dimension. This greatly simplifies the analysis of operator mixing. In a general mass-dependent scheme, operators can also mix with operators of lower dimension. 1.4 Quantum chromodynamics and chiral symmetry The portion of the standard model that describes the strong interactions of quarks and gluons is called quantum chromodynamics (QCD). The QCD Lagrange density including for the moment only the "light" u, d, and s quark flavors is 1 A Aµν / − m q )q + counterterms, G + q̄(i D LQCD = − G µν 4 where q is the triplet of light quarks ⎛ ⎞ u q = ⎝d ⎠, s and m q is the quark mass matrix ⎞ 0 mu 0 mq = ⎝ 0 md 0 ⎠ . 0 0 ms (1.82) (1.83) ⎛ (1.84) A is the Here Dµ = ∂µ + ig AµA T A is the SU(3) color covariant derivative and G µν gluon field strength tensor, A G µν = ∂µ AνA − ∂ν AµA − g f ABC AµB ACν , (1.85) where the structure constants f ABC are defined by [T A , T B ] = i f ABC T C . The QCD renormalization factors can be calculated at order g 2 in a manner similar 18 Review Fig. 1.5. One-loop gluon contribution to the vacuum polarization. to that for QED. For example, quark wave-function and mass renormalization Z q and Z m are given by Fig. 1.2 with the photon replaced by a gluon. They can be obtained from the QED result by replacing e2 by g 2 T A T A , where T A T A = (4/3)11 for quarks in QCD. In the Feynman gauge, the order g 2 wave-function and mass renormalization factors are g2 g2 , Z . (1.86) Zq = 1 − = 1 − m 12π 2 2π 2 A major difference between QCD and QED occurs in the coupling constant renormalization. The β function for QCD is 2 g3 11 − Nq + O(g 5 ), (1.87) β(g) = − 16π 2 3 where Nq is the number of quark flavors. The quark contribution to the β function can be computed from Fig. 1.1 with the photon replaced by a gluon. It is obtained from the QED calculation by the replacement e2 → Nq g 2 /2, since Tr T A T B = δ AB /2 for each quark flavor in the loop. The other term in the β function is from gluon self-interactions, as in Fig. 1.5, and is not present in an Abelian gauge theory such as QED. The QCD β function is negative, as long as the number of quark flavors Nq is less than 16, so the QCD fine structure constant g 2 (µ) (1.88) 4π becomes smaller at larger µ, a phenomenon known as asymptotic freedom. At high energies, the coupling constant is small, and QCD perturbation theory should be reliable. We can explicitly solve for the µ-dependence of αs just as in QED: αs (µ) = αs (µ2 ) = 1 , 1/αs (µ1 ) + β0 ln µ22 /µ21 where β0 is proportional to the first term in the QCD β function, 33 − 2Nq β0 = . 12π (1.89) (1.90) Equation (1.89) is valid as long as µ1 and µ2 are large enough that the order g 5 terms in Eq. (1.87) can be neglected, i.e., as long as αs (µ1 ) and αs (µ2 ) are 1.4 Quantum chromodynamics and chiral symmetry 19 both small. It is convenient to introduce a subtraction-point independent constant QCD with dimensions of mass, defined by QCD = µe−1/[2β0 αs (µ)] . (1.91) Then our expression for the strong interaction fine structure constant becomes αs (µ) = 12π . (33 − 2Nq ) ln µ2 /2QCD (1.92) Equation (1.92) suggests that the QCD coupling constant diverges as µ → QCD . Of course, this expression for αs ceases to be valid when αs gets large. Nevertheless, one can still view QCD as the scale at which QCD becomes strongly coupled so that perturbation theory breaks down and nonperturbative effects become important. Experimentally, QCD is ∼200 MeV, and it sets the scale for nonperturbative strong interaction effects. One expects hadron masses such as the ρ meson mass to be dimensionless multiples of QCD . It is believed that QCD is a confining theory at long distances, i.e., the spectrum of physical states consists of color singlet states called hadrons; there are no colored hadrons. Bosonic hadrons are called mesons and fermionic hadrons are called baryons. The simplest ways to form color singlet combinations of the quark fields are q̄ α qα and ε αβγ qα qβ qγ . The u, d, and s quark masses are small compared with the scale QCD of nonperturbative strong interaction physics, and so it is useful to consider an approximation to QCD in which the masses of these light quarks are set to zero, and to do perturbation theory in m q about this limit. The limit m q → 0 is known as the chiral limit, because the light quark Lagrangian L light quarks / q = q̄ L i D / q L + q̄ R i D / qR = q̄ i D (1.93) has an SU(3) L × SU(3) R chiral symmetry qL → L qL qR → R qR, (1.94) [L ∈SU(3) L , R ∈SU(3) R ] under which the right- and left-handed quark fields transform differently. The Lagrange density in Eq. (1.93) also has a baryon number U(1) symmetry where the left- and right-handed quarks transform by a common phase, and an axial U(1) where all the left-handed quarks transform by a phase and all the right-handed quarks transform by the opposite phase. Although these axial U(1) transformations leave the Lagrange density invariant, they change the measure in the path integral, an effect known as the axial anomaly. Hence, the axial U(1) is not a symmetry of QCD. The chiral SU(3) L × SU(3) R symmetry of massless three-flavor QCD is spontaneously broken by the vacuum expectation value of quark bilinears j k q̄ R q L = v δ k j , (1.95) where v is of order 3QCD . [Here v should not be confused with the Higgs vacuum 20 Review expectation value.] The indices j and k are flavor indices, q 1 = u, q 2 = d, q 3 = s, and color indices are suppressed. If we make a SU(3) L × SU(3) R transformation q → q , j k q̄ R q L = v(L R † )k j . (1.96) Transformations with L = R leave the vacuum expectation value unchanged. Thus the nonperturbative strong interaction dynamics spontaneously breaks the SU(3) L × SU(3) R chiral symmetry to its diagonal subgroup SU(3)V . The eight j broken SU(3) L × SU(3) R generators transform the composite field q̄ R q Lk along symmetry directions, and so leave the potential energy unchanged. Fluctuations in field space along these eight directions are eight massless Goldstone bosons. We can describe the Goldstone boson fields by a 3 × 3 special unitary matrix (x), which represents the possible low-energy long-wavelength excitations j of q̄ R q L . Here vk j (x) ∼ q̄ R (x) q Lk (x) gives the local orientation of the quark condensate. has vacuum expectation value = 11. Under SU(3) L × SU(3) R transformations, → L R†. (1.97) The low-momentum strong interactions of the Goldstone bosons are described by an effective Lagrangian for (x) that is invariant under the chiral symmetry transformation in Eq. (1.97). The most general Lagrangian is f2 Tr ∂ µ ∂µ † + higher derivative terms, (1.98) 8 where f is a constant with dimensions of mass. There are no terms without any derivatives since Tr † = 3. At a low enough momentum the effects of the higher derivative terms can be neglected since they are suppressed by powers of 2 /2 , where p ptyp typ is a typical momentum and CSB is the scale associated CSB with chiral symmetry breaking, CSB ∼ 1 GeV. The field (x) is an SU(3) matrix and it can be written as the exponential 2i M , (1.99) = exp f Leff = of M, a traceless 3 × 3 Hermitian matrix. Under the unbroken SU(3)V subgroup (L = R = V ), → V V † , which implies that M → V M V † , i.e., M transforms as the adjoint representation. M can be written out explicitly in terms of eight Goldstone boson fields: ⎛ ⎞ √ √ 0 + + π K π / 2 + η/ 6 ⎜ ⎟ √ √ ⎜ ⎟ − 0 0 M =⎜ ⎟ . (1.100) π −π / 2 + η/ 6 K ⎝ ⎠ √ K− −2η/ 6 K̄ 0 1.4 Quantum chromodynamics and chiral symmetry 21 The factor of 2/ f is inserted in Eq. (1.99) so that the Lagrangian in Eq. (1.98) gives kinetic-energy terms for the Goldstone bosons with the standard normalization. In the QCD Lagrangian the light quark mass terms, Lmass = q̄ L m q q R + h.c., (1.101) transform under chiral SU(3) L × SU(3) R as (3̄ L , 3 R ) + (3 L , 3̄ R ). We can include the effects of quark masses (to first order) on the strong interactions of the pseudo-Goldstone bosons, π, K , and η, by adding terms linear in m q to Eq. (1.98) that transform in this way. Equivalently we can view the quark mass matrix itself as transforming like m q → Lm q R † under SU(3) L × SU(3) R . Then the Lagrange density in Eq. (1.101) is invariant under chiral SU(3) L × SU(3) R . With this transformation rule for m q , we include the effects of quark masses in the strong interactions of the π, K , and η by adding to Eq. (1.98) terms linear in † m q and m q that are invariant under SU(3) L × SU(3) R . This gives Leff = f2 Tr ∂ µ ∂µ † + v Tr(m q† + m q † ) + · · · . 8 (1.102) The ellipses in Eq. (1.102) represent terms with more derivatives or more insertions of the light quark mass matrix. The quark mass terms in the Lagrange density in Eq. (1.102) give masses to the Goldstone bosons 4v (m u + m d ), f2 4v = 2 (m u + m s ), f 4v = m 2K̄ 0 = 2 (m d + m s ), f m 2π ± = m 2K ± m 2K 0 (1.103) and hence the π, K , and η are referred to as pseudo-Goldstone bosons. The kaon masses are much larger than the pion masses, implying that m s m u,d . For the η − π 0 system there is a mass-squared matrix with elements 4v (m u + m d ), f2 4v = m 2π 0 η = √ (m u − m d ), 3f2 4v (4m s + m u + m d ) . = 3f2 m 2π 0 π 0 = m 2ηπ 0 m 2ηη Because m s (1.104) m u,d , the off-diagonal terms are small compared with m 2ηη . Hence, 22 Review up to corrections suppressed by (m u − m d )2 /m 2s , 4v (m u + m d ) f2 (1.105) 4v (4m s + m u + m d ) . 3f2 (1.106) m 2π 0 and m 2η It is interesting to note that the neutral pion mass is near the charged pion masses, not because m u /m d is near unity, but rather because m u − m d is small compared with m s . A more detailed study of mass relations, including electromagnetic corrections, leads to the expectation that m u /m d 1/2. The chiral Lagrangian in Eq. (1.102) contains two parameters, v with dimensions of (mass)3 and f with dimensions of mass. Since the quark masses always appear in conjunction with v it is not possible using the effective Lagrangian in Eq. (1.102) to determine the quark masses themselves. The effective theory describing the low-momentum interactions of the pseudo-Goldstone bosons only determines the ratios of quark masses, since v cancels out. Equation (1.102) is an effective Lagrangian that describes the low-momentum interactions of the pseudo-Goldstone bosons. One can use the effective theory to compute scattering processes, such as π − π scattering. Expanding out in terms of the meson fields, one finds that the Tr ∂µ ∂ µ † part of the Lagrangian has the four-meson interaction term, 1 Tr[M, ∂µ M][M, ∂ µ M]. 6f2 (1.107) Its tree-level matrix element (shown in Fig. 1.6) gives a contribution to the π–π scattering amplitude of the form M∼ 2 ptyp f2 , (1.108) 2 since where ptyp is a typical momentum. The amplitude is of the order of ptyp the vertex contains two derivatives. The mass terms also give a contribution of 2 ∼ m 2 . The contributions of higher derivative operators this form if we set ptyp π Fig. 1.6. Tree-level contribution to π–π scattering. 1.4 Quantum chromodynamics and chiral symmetry 23 Fig. 1.7. One-loop contribution to π–π scattering. in the chiral Lagrangian are suppressed by more factors of the small momentum ptyp . What about loop diagrams? There are one-loop diagrams with two insertions of the π π π π vertex, such as Fig. 1.7. Each vertex gives a factor of p 2 / f 2 , the two meson propagators give a factor of 1/ p 4 , and the loop integration gives a factor of p 4 . The resulting amplitude in the MS scheme is M∼ 4 ptyp 16π 2 f 4 2 ln ptyp /µ2 . (1.109) 4 in the numerator is required by dimensional analysis, since The factor of ptyp there is a factor of f 4 in the denominator and the subtraction point µ, which also has dimensions of mass, only occurs in the argument of logarithms. The 16π 2 in the denominator typically occurs in the evaluation of one-loop diagrams. The one-loop diagram gives a contribution of the same order in the momentum expansion as operators in the chiral Lagrangian with four derivatives (or two insertions of the quark mass matrix). The total amplitude at order p 4 is the sum of one-loop diagrams containing order p 2 vertices and of tree graphs from the p 4 terms in the Lagrangian. The total p 4 amplitude is µ independent; the µ dependence in Eq. (1.109) is canceled by µ dependence in the coefficients of the p 4 terms in the Lagrangian. The pattern we have just observed holds in general. More loops give a contribution of the same order as a term in the Lagrangian with more derivatives. One can show that a graph with L loops, and n k insertions of vertices of order p k , produces an amplitude of order p D , where (see Problem 6) D = 2 + 2L + (k − 2) n k . (1.110) k Thus each loop increases D by two, and each insertion of a vertex of order p k increases D by k − 2. Note that k − 2 ≥ 0, since the Lagrangian starts at order p2 , so that each term in Eq. (1.110) is positive. Loop corrections and higher derivative operators are of comparable importance when the mass scale CSB that suppresses higher derivative operators is approximately equal to 4π f . The computation of pseudo-Goldstone scattering amplitudes in a momentum expansion using an effective Lagrangian is known as chiral perturbation theory. 24 Review Although the u, d, and s quark masses are small, the spectrum of QCD suggests that the theory contains quasi-particles that transform like u, d, and s under the unbroken SU(3)V group but have a larger mass of approximately 350 MeV. These quasi-particles are called constituent quarks, and the hadronic spectrum is consistent at least qualitatively with spectra calculated from nonrelativistic potential models for the interactions of constituent quarks. 1.5 Integrating out heavy quarks The top, bottom, and charm quark masses are m t 175 GeV, m b 4.8 GeV, and m c 1.4 GeV. For processes that occur at energies well below the masses of these quarks, it is appropriate to go over to an effective theory of the strong interactions where these heavy quarks are integrated out of the theory and no longer occur as explicit degrees of freedom in the Lagrangian. The effects of Feynman diagrams with a virtual heavy quark Q are taken into account by nonrenormalizable operators suppressed by factors of 1/m Q , and through shifts in the coupling constants of renormalizable terms in the effective Lagrangian. For definiteness, imagine integrating out the top quark and making a transition from the six-quark theory of the strong interactions to an effective five-quark theory. The strong coupling in the original theory with six flavors will be denoted by g (6) , and in the effective five-quark theory by g (5) . The relation between the two couplings is determined by ensuring that the scattering amplitudes computed in the five- and six-quark theories are the same. The general form of the relation is g (5) (µ) = g (5) [m t /µ, g (6) (µ)], since the g's are dimensionless. The power series expansion of g (5) in powers of g (6) has coefficients that, for µ very different from m t , contain large logarithms of m 2t /µ2 . If instead we pick µ = m t , g (5) has a power series expansion in g (6) (m t ) with coefficients that are not enhanced by any large logarithms. At tree level, g (5) (µ) = g (6) (µ), so one expects (1.111) g (5) (m t ) = g (6) (m t ) 1 + O αs(6) (m t ) . An explicit calculation shows that the one-loop term in this equation vanishes, so the first nontrivial contribution is at two loops. The strong coupling in the effective theory with n quarks is written as in Eq. (1.92), where the value of QCD now depends on which particular effective theory is being used (i.e., QCD → (n) QCD ). Equation (1.111) implies that, at leading order, the coupling constants are continuous at µ = m t . Combining this with Eq. (1.92), we find that (5) QCD = (6) QCD mt (6) QCD 2/23 . (1.112) 1.6 Effective Hamiltonians for weak decays 25 Integrating out the bottom and charm quarks to go over to effective four- and three-quark theories gives 2/25 m b (5) (4) , (1.113) QCD = QCD (5) QCD 2/27 m c (4) (3) . (1.114) QCD = QCD (4) QCD Equations (1.112)–(1.114) determine the most important influence of virtual heavy quarks on low-energy physics. For example, the proton mass m p is generated by nonperturbative dynamics in the effective three-quark theory so m p ∝ (3) QCD , where the constant of proportionality is independent of the heavy quark masses. Imagining that the value of the strong coupling is fixed at some veryhigh-energy scale (e.g., the unification scale), Eqs. (1.112)–(1.114) give the dependence of the proton mass on the heavy quark masses. For example, doubling the charm quark mass increases the proton mass by the factor 22/27 1.05. 1.6 Effective Hamiltonians for weak decays The strong and electromagnetic interactions conserve quark and lepton flavor, so many particles can only decay by means of the weak interactions. The simplest example of such a decay is the weak decay of a muon, µ → eνµ ν̄e . This decay is a purely leptonic process, since it does not involve any quark fields. The lowestorder graph for this decay has a single W boson exchanged, as shown in Fig. 1.8. The tree-level amplitude for the decay is g2 2 ū pνµ γα PL u( pµ ) ū( pe )γβ PL v pνe M(µ → eνµ ν̄e ) = √ 2 α β pµ − pνµ p µ − p νµ 1 αβ × , (1.115) g − 2 2 2 MW p µ − p νµ − M W νµ µ − e − νe Fig. 1.8. Lowest-order diagram for µ decay. 26 Review where g2 is the weak SU(2) coupling constant, and the W propagator has been written in the unitary gauge. The muon mass is much smaller than the W boson mass MW , so the momenta of all the leptons involved in µ decay are much smaller than MW . As a result, we can approximate the denominator of 2 2 , by −MW and neglect the factor of the W -boson propagator, ( pµ − pνµ )2 − MW 2 α β ( pµ − pνµ ) ( pµ − pνµ ) /MW in the numerator of the W -boson propagator. This approximation simplifies the decay amplitude to 4G F M(µ → eνµ ν̄e ) − √ ū pνµ γα PL u( pµ ) ū( pe )γ α PL v pνe , 2 (1.116) where the Fermi constant G F is defined by g22 GF . √ = 2 8MW 2 (1.117) The decay amplitude Eq. (1.116) is the same as that produced by the tree-level matrix element of the local effective Hamiltonian: 4G F HW = −LW = √ [ν̄µ γα PL µ][ēγ α PL νe ]. 2 (1.118) It is simpler to use an effective Hamiltonian description of the weak interactions in computing weak decay amplitudes at energies much smaller than MW and M Z , particularly if one wants to compute radiative corrections to decay amplitudes. Electromagnetic loop corrections to the µ → eν̄e νµ decay amplitude go partly into matrix elements of the Hamiltonian in Eq. (1.118) and partly into modifying the Hamiltonian itself. The corrections to the Hamiltonian are calculated by comparing amplitudes in the full theory with the W boson present as a dynamical field to amplitudes in the effective theory with the W boson removed. These corrections come from regions of loop momenta of order MW , since the effective Hamiltonian has been chosen to correctly reproduce the full Hamiltonian for momenta much smaller than MW . For this reason, the electron and muon masses occur as m e,µ /MW in the effective Hamiltonian, and they can be neglected at leading order. They are, of course, very important for the matrix elements of the effective Hamiltonian. Neglecting the electron and muon masses, we know that the Hamiltonian must be of the form in Eq. (1.118). In this limit electromagnetic corrections do not change chirality and so [ν̄µ γα PL µ][ēγ α PL νe ] and [ν̄µ γα PL νe ][ēγ α PL µ] are the only possible dimension-six operators that can occur. Terms with three gamma matrices between the fermion fields can be reduced to single gamma matrices by using the identity γα γβ γν = gαβ γν + gβν γα − gαν γβ − i where the sign convention is 0123 = 1. αβνη γ η γ5 , (1.119) Higher dimension operators are 1.6 Effective Hamiltonians for weak decays 27 negligible, being suppressed by powers of 1/MW . The Fierz operator identity, [ψ̄1 γα PL ψ2 ][ψ̄3 γ α PL ψ4 ] = [ψ̄1 γα PL ψ4 ][ψ̄3 γ α PL ψ2 ], (1.120) allows one to replace [ν̄µ γα PL νe ][ēγ α PL µ] by [ν̄µ γα PL µ][ēγ α PL νe ]. So beyond tree level the effective Hamiltonian is modified to 4G F MW , α(µ) [ν̄µ γ α PL µ][ēγα PL νe ], (1.121) HW = √ C µ 2 where µ is the subtraction point, and α is the electromagnetic fine structure constant. The only modification due to radiative corrections is the coefficient C, which is unity at tree level. Loop corrections at µ = MW with virtual loop momenta of order MW determine the deviation of the coefficient C from unity, so one expects C[1, α(MW )] = 1 + O [α (MW )]. (1.122) Any dependence of the matrix elements of the four-fermion operator [ν̄µ γα PL µ][ēγ α PL νe ] on the subtraction point is canceled by the µ dependence of C, so that physical quantities such as decay rates do not depend on µ. If the Hamiltonian above is used to calculate the muon decay rate, with µ = MW naively 2 ) in the perturbative one would think that there are large logarithms of (m 2µ /MW expansion of the matrix elements of the Hamiltonian. In fact we know that C is µ-independent and hence such logarithms do not occur. A simple explanation of this fact follows using the Fierz identity in Eq. (1.120), which allows us to rewrite the effective Hamiltonian in the form [ν̄µ γα PL νe ][ēγ α PL µ]. The neutrino fields do not interact electromagnetically, so the only renormalization is that of ēγ α PL µ. In the limit m e = m µ = 0, ēγ α PL µ is a conserved current and does not get renormalized. The electromagnetic coupling α is so small that even when it is multiplied by large logarithms, perturbation theory is usually adequate. However, this is not the case for the strong interactions. For an example in which such logarithms are important and must be summed, consider the effective Hamiltonian for nonleptonic b → c decays at tree level, 4G F ∗ [c̄α γµ PL bα ][d̄ β γ µ PL u β ]. (1.123) HW(c = 1) = √ Vcb Vud 2 In Eq. (1.123) α and β are color indices and repeated indices are summed. There is a contribution to the effective Hamiltonian for nonleptonic b → c decays where the d quark is replaced by a s quark. It has a coefficient that is suppressed by |Vus /Vud | ≈ 0.2 compared to Eq. (1.123). This "Cabibbo suppressed" contribution is neglected here. Also, we are focusing on c = 1 decays. There are nonleptonic decays where, at tree level, the final state has both a c and c̄ quark. For these decays the coefficient in the effective Hamiltonian HW(c = 0) is not smaller than Eq. (1.123). 28 Review Strong interaction loop corrections change the form of the Hamiltonian for b → c decays. An argument similar to that used for µ decay shows that there are two possible terms in the c = 1 effective Hamiltonian, MW MW 4G F ∗ , αs (µ) O1 (µ) + C2 , αs (µ) O2 (µ) , HW = √ Vcb Vud C1 µ µ 2 (1.124) where O1 (µ) = [c̄α γµ PL bα ][d̄ β γ µ PL u β ], O2 (µ) = [c̄β γµ PL bα ][d̄ α γ µ PL u β ]. (1.125) The coefficients C1,2 are determined by comparing Feynman diagrams in the effective theory with the W -boson integrated out with analogous diagrams in the full theory. At µ = MW we have from Eq. (1.123) that C1 [1, αs (MW )] = 1 + O[αs (MW )], C2 [1, αs (MW )] = 0 + O[αs (MW )]. (1.126) Subtraction-point dependence of the operators O1,2 cancels that in the coefficients C1,2 . Here O1,2 are local four-quark operators, and they must be renormalized to render their matrix elements finite. The relationship between bare and renormalized operators has the form Oi(0) = Z i j O j , (1.127) where i, j = {1, 2} and the repeated index j is summed over. Since the bare operator is µ independent, d (0) d d 0 = µ Oi (µ) = µ Z i j O j + Z i j µ O j , (1.128) dµ dµ dµ which implies that d O j = −γ ji Oi (µ), dµ (1.129) d µ γ ji = Z −1 Z ki . jk dµ (1.130) µ where Here γi j (g) is called the anomalous dimension matrix. It can be calculated order by order in the coupling constant from the Z 's. The subtraction-point independence of the weak Hamiltonian implies that 0=µ d d HW = µ (C j O j ), dµ dµ (1.131) 1.6 Effective Hamiltonians for weak decays yielding d µ Cj dµ 29 O j − C j γ ji Oi = 0. (1.132) Since the operators O1,2 are independent we conclude that µ d Ci = γ ji C j . dµ The solution to this differential equation is g(µ) T MW γ (g) , αs (µ) = P exp dg C j [1, αs (MW )]. Ci µ g(MW ) β(g) ij (1.133) (1.134) Here P denotes "coupling constant ordering" of the anomalous dimension matrices in the exponent, and γ T is the transpose of γ . It is straightforward to calculate the anomalous dimension matrix for O1,2 . At one loop, it is g 2 −1 3 . (1.135) γ (g) = 3 −1 8π 2 It is convenient to diagonalize this matrix by forming the linear combinations of operators O± = O1 ± O2 . (1.136) Using the Fierz identity in Eq. (1.120), it is evident that O+ is symmetric under interchange of the d and c quark fields, whereas O− is antisymmetric. Under an SU(2) flavor group under which the d and c quark fields form a doublet, O− is a singlet and O+ is a triplet. The c, d mass difference breaks this flavor symmetry. Quark masses do not affect the renormalization constants Z i j , so mixing between O+ and O− is forbidden by this symmetry. In terms of O± the effective weak Hamiltonian is M M 4G F W W ∗ HW = √ Vcb Vud C+ , αs (µ) O+ (µ) + C− , αs (µ) O− (µ) , µ µ 2 (1.137) where C± [1, αs (MW )] = 1 + O[αs (MW )]. 2 At any other subtraction point g(µ) γ± (g) MW , αs (µ) = exp dg C± [1, αs (MW )], C± µ g(MW ) β(g) (1.138) (1.139) 30 Review where g2 + O(g 4 ), 4π 2 g2 γ− (g) = − 2 + O(g 4 ), 2π γ+ (g) = (1.140) and β (g) is given by Eq. (1.87). Provided µ QCD , the strong coupling αs (µ) is small over the range of integration in Eq. (1.139) and higher-order terms in g can be neglected in γ± and β. This gives MW 1 αs (MW ) a± C± , (1.141) , αs (µ) = µ 2 αs (µ) where a+ = 6 , 33 − 2Nq a− = − 12 . 33 − 2Nq (1.142) Expressing αs (MW ) in terms of αs (µ) using Eq. (1.89), the perturbative power series expansions of C± have the form 1 + a1 αs (µ) ln(MW /µ) + a2 αs2 (µ) ln2 (MW /µ) + · · · . 2 (1.143) The expression for C± in Eq. (1.143) sums all leading logarithms of the form αsn (µ) lnn (MW /µ), neglecting subleading logarithms of order αsn lnn−1 (MW /µ). The series of subleading logarithms can be summed by using two-loop renormalization group equations, and so on. The subtraction-point dependence in the coefficients C± cancels that in the matrix elements of the operators O± so any value of µ can be used. However, if ptyp is the typical momentum in a nonleptonic 2 ), for decay, the matrix elements of O± will contain large logarithms of (µ2 / ptyp µ very different from ptyp . Roughly these logarithms come from integrations over momenta in the region between ptyp and µ. They are summed by scaling the coefficients down from the subtraction point MW to one of order ptyp , which moves the logarithms from the matrix elements of O± to the coefficients C± . The exponents a± in Eq. (1.142) depend on the number of quark flavors Nq . It is convenient to integrate out the top quark at the same time as the W boson so that Nq = 5. For inclusive weak decay of a hadron containing a b quark, the typical momenta of the decay products are of the order of the b-quark mass, and the large logarithms of (MW /m b )2 are summed by evaluating the coefficients C± at µ = m b . In this case, C+ (m b ) = 0.42, C− (m b ) = 0.70, using αs (MW ) = 0.12 and αs (m b ) = 0.22. (1.144) 1.7 The pion decay constant 1.7 31 The pion decay constant Weak pion decay π − → µν̄µ determines the value for the parameter f that occurs in the chiral Lagrangian for pion strong interactions in Eq. (1.98). Neglecting electromagnetic corrections, the effective Hamiltonian for π − → µν̄µ decay is 4G F Heff = √ Vud [ūγα PL d][µ̄γ α PL νµ ]. 2 (1.145) Here color indices on the quark fields are suppressed. The current ūγα PL d is conserved in the limit m u,d → 0, and consequently its strong interaction matrix elements are subtraction-point independent. Taking the π − → µν̄µ matrix element of Eq. (1.145) gives the pion decay amplitude √ (1.146) M = −i 2G F Vud f π ū( pµ )/pπ PL v pνµ , where the pion decay constant, f π , is the value of the pion-to-vacuum matrix element of the axial current, 0|ūγ α γ5 d|π − ( pπ ) = −i f π pπα . (1.147) The measured pion decay rate gives f π 131 MeV. In Eq. (1.147) the pion field is normalized by using the standard covariant norm: π( pπ )|π( pπ ) = 2E π (2π)3 δ 3 (pπ − pπ ). Parity invariance of the strong interactions implies that only the axial current part of the left-handed current contributes in Eq. (1.146). In the limit m u,d,s = 0, global SU(3) L transformations are a symmetry of QCD. The conserved currents associated with this symmetry can be derived by considering the change in the QCD Lagrangian under infinitesimal local SU(3) L transformations, L = 1 + i LA T A , (1.148) with space–time dependent infinitesimal parameters LA (x). The change in the QCD Lagrange density, Eq. (1.93), under this transformation is A µ A δLQCD = −JLµ ∂ L , (1.149) A JLµ = q̄ L T A γµ q L (1.150) where are the conserved currents associated with SU(3) L transformations. We also know how left-handed transformations act on the meson fields in . The change in the chiral Lagrange density under an infinitesimal left-handed transformation on the in Eq. (1.98) is A µ A ∂ L , δLeff = −JLµ (1.151) 32 Review where if2 Tr T A ∂µ † . 4 Comparing Eqs. (1.150) and (1.152) gives A JLµ =− (1.152) f2 Tr T A ∂µ † + · · · , (1.153) 4 where the ellipses are contributions from higher derivative terms in the chiral Lagrangian. Matrix elements of the quark current involving the pseudo-Goldstone boson can be calculated by expanding in terms of M on the right-hand side of Eq. (1.153). In particular, the part linear in M yields the tree-level relation f = f π . Loops and higher derivatives operators in the chiral Lagrangian give corrections to the relation between f and f π . The kaon decay constant is defined by q̄ L T A γµ q L = −i 0|ūγ α γ5 s|K − ( p K ) = −i f K p αK . (1.154) The measured K − → µν̄µ decay rate determines f K to be ∼25% larger than the pion decay constant, f K 164 MeV. At leading order in chiral SU(3) L × SU(3) R , f = f K = f π , and the 25% difference between f π and f K is the typical size of SU(3)V breaking arising from the nonzero value of the strange quark mass. At higher orders in chiral perturbation theory, the Noether procedure for finding the representation of q̄ L T A γµ q L in terms of pseudo-Goldstone boson fields becomes ambiguous. Total derivative operators in the chiral Lagrangian give can A A A 3 ), a contribution to the current JLµ (although not to the charges Q L = d x JL0 even though they are usually omitted from the chiral Lagrangian because they do not contribute to pseudo-Goldstone boson S-matrix elements. Note that at leading order in chiral perturbation theory there are no possible total derivative operators since ∂ µ (Tr † ∂µ ) = 0. 1.8 The operator product expansion The operator product expansion (OPE) is an important tool in particle physics and condensed matter physics, and it will be applied later in this book to describe inclusive B decay and to discuss sum rules. The use of the operator product expansion is best illustrated by an explicit example. In this section, the OPE will be applied to the study of deep inelastic lepton–proton scattering. The main purpose of the discussion is to explain the use of the OPE, so the presentation of the phenomenology of deep inelastic scattering will be kept to a minimum. The basic deep inelastic scattering process is (k) + proton( p) → (k ) + X ( p + q), in which an incoming lepton with momentum k scatters off a target proton, to produce an outgoing lepton with momentum k , plus anything X . The Feynman graph in Fig. 1.9 is the leading term in an expansion in the 1.8 The operator product expansion 33 k′, E′ k, E q = k - k′ X p Fig. 1.9. The basic diagram for deep inelastic lepton–hadron scattering. The virtual photon momentum is q. The final hadronic state is not measured and is denoted by X . electromagnetic fine structure constant α. The traditional kinematical variables used to describe the inclusive scattering process are the momentum transfer Q 2 = −(k − k)2 , and the dimensionless variable x defined by x= Q2 , 2p · q (1.155) where q = k −k . Note that for deep inelastic scattering, Q 2 > 0. It is also useful to define ω = 1/x. The deep inelastic scattering cross section is the inclusive cross section in the limit that Q 2 is large with x fixed. The total cross section is obtained from squaring the amplitude represented by Fig. 1.9 and performing the appropriate phase space integrations. The lepton and photon parts of this amplitude as well as the phase space integrals can easily be computed. The nontrivial quantity is the square of the hadronic part of the diagram, which is µ ν (2π )4 δ 4 (q + p − p X ) p|Jem (0)|X X |Jem (0)| p, (1.156) X µ where the sum is over all possible final states X , and Jem is the electromagnetic current. For convenience momentum and spin labels on the state vectors are suppressed. A spin average over the proton states | p is also understood. It is conventional to define the hadronic tensor µ 1 µν ν (1.157) d 4 x eiq · x p| Jem (x), Jem (0) | p. W ( p, q) = 4π Inserting a complete set of states gives 1 µν µ ν W ( p, q) = d 4 x eiq · x p|Jem (x)|X X |Jem (0)| p 4π X ν µ − p|Jem (0)|X X |Jem (x)| p , (1.158) where the sum on X is a sum over all final states, as well as an integral over the 34 Review allowed final state phase space. Translation invariance implies that µ µ p|Jem (x)|X = p|Jem (0)|X ei( p− p X ) · x , µ µ X |Jem (x)| p = X |Jem (0)| pei( p X − p) · x . (1.159) Inserting Eq. (1.159) into Eq. (1.158) gives 1 µ ν (2π )4 δ 4 (q + p − p X ) p|Jem (0)|X X |Jem (0)| p W µν ( p, q) = 4π X ν µ (0)|X X |Jem (0)| p . (1.160) −(2π)4 δ 4 (q + p X − p) p|Jem The only allowed final states are those with p 0X ≥ p 0 , since the baryon number is conserved. For q 0 > 0, only the first delta function in Eq. (1.160) can be satisfied, and the sum in Wµν reduces to the expression in Eq. (1.156) involving the hadronic currents, and the energy-momentum conserving delta function (up to a factor of 1/4π). Since only the first term in Eq. (1.158) contributes, one could µ ν (0) have defined Wµν in Eq. (1.157) simply as the matrix element of Jem (x)Jem without the commutator. The reason for using the commutator is that then Wµν has a nicer analytic structure when continued away from the physical region. The most general form of Wµν consistent with current conservation, parity and time-reversal invariance is qµ qν p · q qµ p · q qν F2 + pν − , pµ − Wµν = F1 −gµν + 2 2 q p·q q q2 (1.161) where F1,2 can be written as functions of x and Q 2 . Here F1,2 are called structure functions. The Q 2 dependence of the structure functions can be calculated in quantum chromodynamics. The starting point in the derivation is the time-ordered product of two currents: µ µν ν (x) Jem (0) . (1.162) t ≡ i d 4 xeiq · x T Jem The proton matrix element of tµν , Tµν = p|tµν | p, (1.163) can also be written in terms of structure functions, qµ qν p · q qµ p · q qν T2 + pν − . pµ − Tµν = T1 −gµν + 2 2 q p·q q q2 (1.164) The analytic structure of T1,2 as a function of ω for fixed Q 2 is shown in Fig. 1.10. There are cuts in the physical region 1 ≤ |ω|. The discontinuity across the 1.8 The operator product expansion 35 Im ω −1 1 Re ω Fig. 1.10. The analytic structure of Tµν in the complex ω plane. The discontinuity across the cuts 1 ≤ |ω| ≤ ∞ is related to Wµν . right-hand cut for T1,2 is F1,2 , Im T1,2 (ω + i, Q 2 ) = 2π F1,2 (ω, Q 2 ). (1.165) [The discontinuity across the left-hand cut gives the structure functions for deep inelastic scattering off antiprotons.] The key idea that permits the computation of Tµν in certain limiting cases is the operator product expansion. Consider the time-ordered product of two local operators separated in position by z: T [Oa (z) Ob (0)] . (1.166) For small z, the operators are at practically the same point. In this limit, the operator product can be written as an expansion in local operators, Cabk (z)Ok (0). (1.167) T [Oa (z) Ob (0)] = k The coefficient functions depend on the separation z. Low-momentum (compared with 1/z) matrix elements of the left-hand side are completely equivalent to matrix elements of the right-hand side. Thus one can replace the product T [Oa (z)Ob (0)] in the computation of matrix elements by the expansion in Eq. (1.167), where the coefficients Cabk (z) are independent of the matrix elements, provided that the external states have momentum components that are small compared with the inverse separation 1/z. In QCD, the coupling constant is small at short distances because of asymptotic freedom. Thus the coefficient functions can be computed in perturbation theory, since all nonperturbative effects occur at scales that are much larger than z, and do not affect the computation of the coefficient functions. The momentum space version of the operator product expansion is for the product (1.168) d 4 z eiq · z T [Oa (z) Ob (0)]. 36 Review In the limit that q → ∞, the Fourier transform in Eq. (1.168) forces z → 0, and again the operator product can be expanded in terms of local operators with coefficient functions that depend on q. For large q, Cabk (q)Ok (0). (1.169) d 4 zeiq · z T [Oa (z) Ob (0)] = k This expansion is valid for all matrix elements, provided q is much larger than the characteristic momentum in any of the external states. We will use the Fourier transform version of the operator product expansion, Eq. (1.169). The product of two electromagnetic currents in Eq. (1.162) can be expanded in terms of a sum of local operators multiplied by coefficients that are functions of q. This expansion will be valid for proton matrix elements, Eq. (1.163), provided that q is much larger than the typical hadronic mass scale QCD . The local operators in the operator product expansion for QCD are quark and gluon operators with arbitrary dimension d and spin n. An operator with µ ···µ spin n and dimension d can be written as Od,n1 n , where Od,n is symmetric and traceless in µ1 · · · µn . The matrix element of Od,n in the spin-averaged proton S[ p µ1 · · · p µn ]. S acts on a tensor to project target is proportional to m d−n−2 p out the completely symmetric traceless component. The power of m p follows from dimensional analysis, since a proton state with the conventional relativistic normalization has dimension minus one. The coefficient functions in the operator product expansion are functions only of q. Thus the free indices on the operator O must be either µ, ν or be contracted with q α . Every index on O contracted with q α produces a factor of p · q, which is of the order of Q 2 in the deep inelastic limit. An index µ or ν is contracted with the lepton momentum, and produces a factor of p · k or p · k , both of which are also of the order of Q 2 in the deep inelastic limit. In addition, since tµν has dimension two, the coefficient of O must have dimension [mass]2−d in the operator product expansion. Thus, the contribution of any operator O to the differential cross section is of the order of µ ···µn Cµ1 ···µn Od,n1 q µ1 qµ µ ···µ · · · n Q 2−d Od,n1 n , Q Q qµ qµ p µ1 · · · p µn , → 1 · · · n Q 2−d m d−n−2 p Q Q ( p · q)n 2−d d−n−2 Q mp , → Qn Q 2+n−d Q 2−t n n →ω =ω , mp mp → (1.170) where the twist t is defined as t = d − n = dimension − spin. (1.171) 1.8 The operator product expansion 37 Table 1.2. Dimension, spin, and twist for the basic objects in the QCD Lagrangian Parameter q G µν Dµ Dimension Spin Twist 3/2 1/2 1 2 1 1 1 1 0 The most important operators in the operator product expansion are those with the lowest possible twist. Twist-two operators contribute a finite amount to the structure functions in the deep inelastic limit, twist three contributions are suppressed by m p /Q, and so on. The fundamental fields in QCD are quark and gluon fields, so the gauge invariant operators in the operator product expansion can be written in terms of quark fields q, the gluon field strength G µν , and the covariant derivative D µ . Table 1.2 lists the basic objects, with their dimension and twist. Any gauge invariant operator must contain at least two quark fields, or two gluon field strength tensors. Thus the lowest possible twist is two. A twist-two operator has either two q's or two G µν 's and an arbitrary number of covariant derivatives. The indices of the covariant derivatives are not contracted, because an operator such as D 2 has twist two, whereas the traceless symmetric part of D α D β has twist zero. The first step in doing an operator product expansion is to determine all the linearly independent operators that can occur. We have just seen that the leading operators are twist-two quark and gluon operators. We will simplify the analysis by considering not the electromagnetic current but rather Jµ = q̄γµ q for a single quark flavor q. Results for the realistic case can be obtained by summing over flavors weighted by the square of quark charges. The Lorentz structure of the quark operators must be either q̄γ µ q or q̄γ µ γ5 q in the limit that light quark masses can be neglected, because the operator product J µ J ν does not change chirality. The conventional basis for twist-two quark operators is: ↔ µ2 ↔ µn 1 i n−1 µ1 ···µn = S q̄ γ µ1 D · · · D q , (1.172) Oq,V 2 2 µ1 ···µn Oq,A 1 = 2 i 2 n−1 ↔µ2 S q̄ γ µ1 D ↔µn ··· D γ5 q , (1.173) where ↔µ → ← Ā D B = Ā D µ B − Ā D µ B. (1.174) 38 Review µ ···µ 1 n have matrix elements proportional to the proton spin, The operators Oq,A and so do not contribute to spin-averaged scattering. The tower of twist-two gluon operators needed for scattering from unpolarized protons is µ ↔µn−1 1 i n−2 µ1 ···µn µ α ↔ 2 =− S G A1 D . . . D G Aα µn . (1.175) Og,V 2 2 We will only compute the operator product expansion to lowest order in αs , so the gluon operators do not occur. The most general form for t µν consistent with current conservation and using only twist-two operators is ∞ qµ qν 2n qµ1 · · · qµn µ1 ···µn tµν = −gµν + 2 2 C (1) j,n O j,V 2 n q (−q ) j = q,g n = 2,4,... ∞ qµ qµ1 q ν q µ2 + − gµµ1 − g νµ 2 q2 q2 n = 2,4,... × 2n qµ3 · · · qµn µ1 ···µn 2C (2) , j,n O j,V (−q 2 )n−1 j = q,g (1.176) (2) where the unknown coefficients are C (1) j,n and C j,n , and the factors of two and signs have been chosen for later convenience. The second step in doing an operator product expansion is to determine the (2) coefficients of the operators, C (1) j,n and C j,n . The best way to do this is to evaluate enough on-shell matrix elements to determine all the coefficients. Since we have argued that the coefficients can be computed using any matrix elements, we will evaluate the coefficients by taking matrix elements in on-shell quark and gluon states. We will only illustrate the computation of the coefficients to lowest nontrivial order, i.e., (αs )0 , in this chapter. A generic term in the operator product expansion can be written as J J ∼ C q Oq + C g O g , (1.177) where q and g refer to quark and gluon operators. Taking the matrix element of both sides in a free quark state gives q|J J |q ∼ Cq q|Oq |q + C g q|Og |q. (1.178) The electromagnetic current is a quark operator. Thus the left-hand side is of the order of (αs )0 . The matrix element q|Oq |q is also of the order of (αs )0 , whereas the matrix element q|Og |q is of the order of (αs )1 since there are at least two gluons in Og , each of which contributes a factor of the QCD coupling constant g to the matrix element. Thus, one can determine Cq to leading order by taking the quark matrix element of both sides of the operator product expansion, neglecting the gluon operators. 1.8 The operator product expansion q,ν q,µ q,ν 39 q,µ p+q p-q p,s p,s p,s p,s Fig. 1.11. The lowest order diagrams contributing to the quark matrix element of the product of two electromagnetic currents. As mentioned previously, we work in a theory with a single quark flavor with charge one. The quark matrix element of the left-hand side of the operator product expansion, Eq. (1.169), is given by the Feynman graphs in Fig. 1.11, Mµν = i ū( p, s) γ µ i /p + q/ /p − q/ γ ν u( p, s) + i ū( p, s) γ ν i γ µ u( p, s). ( p + q)2 ( p − q)2 (1.179) Note that there is an overall factor of i because we are computing i times the time-ordered product in Eq. (1.162). The crossed diagram (second term) can be obtained by the replacement µ ↔ ν, q →−q from the direct diagram (first term), so we concentrate on simplifying the first term. Expanding the denominator gives 2p · q 2 2 2 = q 2 (1 − ω) , (1.180) ( p+q) = 2 p·q +q = q 1 + 2 q since p 2 = 0 for an on-shell massless quark. The numerator can be simplified using the γ matrix identity in Eq. (1.119): ū( p, s)γ µ ( /p + q/ )γ ν u( p, s) = ū( p, s)[( p + q)µ γ ν + ( p + q)ν γ µ −g µν ( /p + q/ ) + i µναλ ( p + q)α γλ γ5 ]u( p, s). (1.181) For an on-shell massless quark, /p u( p, s) = 0, ū( p, s) γλ u( p, s) = 2 pλ , ū( p, s) γλ γ5 u( p, s) = 2h pλ , (1.182) where h is the quark helicity. Thus the /p and µναλ pα γλ γ5 terms both give zero. For spin-averaged matrix elements the sum over helicities gives zero and so we neglect the part of Mµν proportional to h. Combining the various terms and using (1 − ω)−1 = ∞ n=0 ωn (1.183) 40 Review gives Mµν = − ∞ 2 ωn [( p + q)µ p ν + ( p + q)ν p µ − g µν p · q]. (1.184) q2 n = 0 To complete the operator product expansion, we need the free quark matrix element of the right-hand side of the operator product. The matrix element of the quark operators of Eq. (1.172) in a free quark state with momentum p is µ ···µn 1 q ( p)|Oq,V |q ( p) = S[ p µ1 · · · p µn ] = p µ1 · · · p µn , (1.185) since p 2 = 0. The factors of i and 2 in Eqs. (1.172) and (1.173) were chosen so that no such factors appear in the matrix elements. We determine the coefficient functions for the spin-independent terms in the operator product expansion. Including the crossed diagram, the spin-independent terms on the left-hand side of the operator product are Mµν = − ∞ 2 ωn [( p + q)µ p ν + ( p + q)ν p µ − g µν p · q] q 2 n=0 + (µ ↔ ν, q → − q, ω → − ω) , (1.186) since ω is odd in q. The crossed diagram causes half the terms to cancel, so that the matrix element is ∞ ∞ 4 4 Mµν = − 2 ωn 2 p µ p ν − 2 ωn (q µ p ν + q ν p µ − g µν p · q) q n = 0,2,4 q n = 1,3,5 ∞ 2n ( p · q)n p · qq µ p · qq ν 8 µ ν p p − − =− 2 q n = 0,2,4 (−q 2 )n q2 q2 ∞ 4 q µq ν 2n ( p · q)n+1 µν − 2 −g . (1.187) + q n = 1,3,5 (−q 2 )n q2 Equation (1.187) can be rewritten in the form µν M ∞ 8 2n q µ3 · · · q µn+2 =− 2 q n = 0,2,4 (−q 2 )n q µ q µ1 q ν q µ2 µµ1 νµ2 g pµ1 · · · pµn+2 × g − − q2 q2 ∞ 2n q µ1 · · · q µn+1 4 q µq ν µν −g pµ1 · · · pµn + 1 , (1.188) − 2 + q n = 1,3,5 (−q 2 )n q2 which separates the q and p dependence. The coefficient functions in the operator product depend only on q, and the matrix elements depend only on p. We have separated the operator product into 1.8 The operator product expansion 41 pieces which depend only on q and only on p. By comparing with Eq. (1.185), we can write Eq. (1.188) as ∞ 2n q µ3 · · · q µn+2 8 µν M =− 2 q n = 0,2,4 (−q 2 )n q µ q µ1 q ν q µ2 µµ1 νµ2 p|Oq,V µ1 ···µn+2 | p × g − − g q2 q2 ∞ 2n q µ1 · · · q µn+1 q µq ν 4 µν p|Oq,V µ1 ···µn+1 | p −g + − 2 q n = 1,3,5 (−q 2 )n q2 (1.189) so that t µν ∞ 2n q µ3 · · · q µn q µ q µ1 q ν q µ2 µµ1 νµ2 g g =2 − − (−q 2 )n−1 q2 q2 n = 2,4,6 ∞ q µq ν 2n q µ1 · · · q µn µν −g Oq,V µ1 ···µn . +2 + (−q 2 )n q2 n = 2,4,6 Oq,V µ1 ···µn (1.190) This is the operator product expansion for the spin-independent part of tµν , i.e., the part involving only vector operators. Only vector operators with n even occur in the operator product expansion, because t µν is even under charge conjugation. Comparing with the most general form for the operator product in Eq. (1.176), (1,2) = 1. Considering a gluon we see that at lowest order in αs the coefficients Cq,n matrix element gives C g,n = 0 at lowest order in αs . At higher orders in αs the coefficient functions and the operator matrix elements depend on a subtraction point µ. Since the physical quantity tµν is independent of the arbitrary choice of subtraction point, a renormalization group equation similar to that for coefficients in the weak nonleptonic decay Hamiltonian in Eq. (1.132) can be derived for the coefficients C (1,2) j,n . At µ = Q there are no large logarithms in the coefficients C (1,2) j,n . Therefore, we have that (1,2) Cq,n [1, αs (Q)] = 1 + O[αs (Q)], (1.191) (1,2) C g,n [1, αs (Q)] = 0 + O[αs (Q)]. However, at µ = Q there are large logarithms of Q/ QCD in the nucleon matrix element of the twist-two operators. It is convenient to use the renormalization group equations that the C (1,2) j,n satisfy and the initial conditions in Eqs. (1.191) to move the Q dependence from the matrix elements into the coefficients by scaling the subtraction point down to a value µ Q. It is this calculable Q dependence that results in the dependence of the structure functions T1,2 and hence F1,2 on Q, without which they would just be functions of x. So quantum chromodynamics predicts a calculable logarithmic dependence of the structure functions F1,2 on Q, 42 Review Fig. 1.12. The proton structure function F2 (x, Q 2 ), measured in deep inelastic muon scattering by the NMC Collaboration [M. Arneodo et al., Phys. Lett. 364B (1995) 107]. The data is shown as a function of Q 2 for different values of x. For clarity, the plots for different x values are offset by one unit vertically, so that what is plotted is F2 + N x , where N x is an integer equal to 1 for x = 0.5, 2 for x = 0.35, etc. which has been verified experimentally. The fact that this dependence is weak at large Q is a consequence of asymptotic freedom. In free field theory the structure functions F1,2 are independent of Q, which is called scaling. The logarithmic Q dependence is usually called a scaling violation. Some experimental data showing the approximate scaling of F2 are shown in Fig. 1.12. 1.9 Problems 1. Consider an SU(5) gauge theory with a scalar field that transforms in the adjoint representation → U U † , U ∈ SU(5). Suppose gets the vacuum expectation value ⎡ 2 0 ⎢ ⎢ 0 2 ⎢ ⎢ = v ⎢ 0 0 ⎢ ⎢ ⎣ 0 0 0 0 0 0 2 0 0 −3 0 0 0 0 ⎤ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥. ⎥ 0 ⎥ ⎦ 0 −3 (a) What is the unbroken subgroup H of SU(5)? (b) What are the H quantum numbers of the massive SU(5) gauge bosons? 1.10 References 43 2. If there are N generations of quarks and leptons, show that the CKM matrix contains (N − 1)2 real parameters. 3. Calculate the vertex renormalization constant Z e given in Eq. (1.62). 4. Calculate to order g 2 the renormalization matrix Z i j defined in Eq. (1.127) for the operators O1 and O2 defined in Eq. (1.125). Use it to deduce the anomalous dimension matrix in Eq. (1.135). 5. Calculate the cross section σ (π + π − → π + π − ) at center of mass energy E to leading order in the chiral perturbation theory expansion. 6. In chiral perturbation theory, any Feynman diagram contributing to π–π scattering has L loops, n k insertions of vertices of order p k , and Nπ internal pion lines. The resulting amplitude is of order p D , where D = (powers of p in numerator) − (powers of p in denominator). Using the identity L = Nπ − k n k + 1, derive Eq. (1.110) for D. 7. Calculate the decay amplitude for K − → π 0 eν̄e at leading order in chiral perturbation theory. 8. (a) Calculate the semileptonic free quark decay rate (b → ceν̄e ). (b) Using the renormalization group improved effective Hamiltonian in Eq. (1.124), calculate the nonleptonic free quark decay rate (b → cd ū). Neglect all masses except those of the b and c quarks. 1.10 References The material in this chapter can be found in many textbooks. Some suggested references are: J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics, McGraw-Hill, 1964 J.D. Bjorken and S.D. Drell, Relativistic Quantum Fields, McGraw-Hill, 1964 C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, 1980 M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995 S. Weinberg, The Quantum Theory of Fields, Vol I: Foundations and Vol II: Modern Applications, Cambridge University Press, 1995 R. Balian and J. Zinn-Justin (editors), Methods in Field Theory, Les Houches Session XXVIII, North-Holland, 1976 H. Georgi, Weak Interactions and Modern Particle Theory, Benjamin/Cummings, 1984 J.F. Donoghue, E. Golowich, and B.R. Holstein, Dynamics of the Standard Model, Cambridge University Press, 1992 2 Heavy quarks The light u, d, and s quarks have masses m q that are small compared to the scale of nonperturbative strong dynamics. Consequently, it is a good approximation to take the m q → 0 limit of QCD. In this limit QCD has an SU(3) L × SU(3) R chiral symmetry, which can be used to predict some properties of hadrons containing these light quarks. For quarks with masses m Q that are large compared with the scale of nonperturbative strong dynamics, it is a good approximation to take the m Q → ∞ limit of QCD. In this limit QCD has spin-flavor heavy quark symmetry, which has important implications for the properties of hadrons containing a single heavy quark. 2.1 Introduction The QCD Lagrangian in Eq. (1.82) describes the strong interactions of light quarks and gluons. As discussed in Sec. 1.4, there is a nonperturbative scale QCD that is dynamically generated by QCD. A color singlet state, such as a meson made up of a quark–antiquark pair, is bound by the nonperturbative gluon dynamics. If the quarks are light, the typical size of such a system is of the order of −1 QCD . Consider a Q q̄ meson that contains a heavy quark with mass m Q QCD , and a light quark with mass m q QCD . Such a heavy-light meson also has a typical size of the order of −1 QCD , as for mesons containing only light quarks. The typical momentum transfer between the heavy and light quarks in the Q q̄ meson arising from nonperturbative QCD dynamics is of the order of QCD . An important consequence of this fact is that the velocity v of the heavy quark is almost unchanged by such strong interaction effects, even though the momentum p of the heavy quark changes by an amount of the order of QCD , since v = p/m Q . A similar argument holds for any hadron containing a single heavy quark Q. In the limit m Q → ∞, the heavy quark in the meson can be labeled by a velocity four-vector v that does not change with time. The heavy quark behaves like a static 44 2.2 Quantum numbers 45 external source that transforms as a color triplet, and the meson dynamics reduces to that of light degrees of freedom interacting with this color source. One sees immediately that the mass of the heavy quark is completely irrelevant in the limit m Q → ∞, so that all heavy quarks interact in the same way within heavy mesons. This leads to heavy quark flavor symmetry: the dynamics is unchanged under the exchange of heavy quark flavors. The 1/m Q corrections take into account finite mass effects and differ for quarks of different masses. As a result, heavy quark flavor symmetry breaking effects are proportional to (1/m Q i − 1/m Q j ), where Q i and Q j are any two heavy quark flavors. The only strong interaction of a heavy quark is with gluons, as there are no quark–quark interactions in the Lagrangian. In the m Q → ∞ limit, the static heavy quark can only interact with gluons via its chromoelectric charge. This interaction is spin independent. This leads to heavy quark spin symmetry: the dynamics is unchanged under arbitrary transformations on the spin of the heavy quark. The spin-dependent interactions are proportional to the chromomagnetic moment of the quark, and so are of the order of 1/m Q . Heavy quark spin symmetry breaking does not have to be proportional to the difference of 1/m Q 's, since the spin symmetry is broken even if there are two heavy quarks with the same mass. The heavy quark SU(2) spin symmetry and U(Nh ) flavor symmetries (for Nh heavy flavors) can be embedded into a larger U(2Nh ) spin-flavor symmetry in the m Q → ∞ limit. Under this symmetry the 2Nh states of the Nh heavy quarks with spin up and down transform as the fundamental representation. We will see in Sec. 2.6 that the effective Lagrangian can be written in a way that makes this symmetry manifest. 2.2 Quantum numbers Heavy hadrons contain a heavy quark as well as light quarks and/or antiquarks and gluons. All the degrees of freedom other than the heavy quark are referred to as the light degrees of freedom . For example, a heavy Q q̄ meson has an antiquark q̄, gluons, and an arbitrary number of q̄q pairs as the light degrees of freedom. Although the light degrees of freedom are some complicated mixture of the antiquark q̄, gluons, and q̄q pairs, they must have the quantum numbers of a single antiquark q̄. The total angular momentum of the hadron J is a conserved operator. We have also seen that the spin of the heavy quark S Q is conserved in the m Q → ∞ limit. Therefore, the spin of the light degrees of freedom S defined by S ≡ J − S Q (2.1) is also conserved in the heavy quark limit. The light degrees of freedom in a hadron are quite complicated and include superpositions of states with different particle numbers. Nevertheless, the total spin of the light degrees of freedom is a good quantum number in heavy hadrons. We will define the quantum numbers j, 46 Heavy quarks + *+ Ds , D s cs 0 D ,D cu *0 D+, D*+ cd Fig. 2.1. Flavor SU(3) weight diagram for the spin-0 pseudoscalar and spin-1 vector cq̄ mesons. The corresponding bq̄ mesons are the B̄ 0s , B̄ − , and B̄ 0 , and their spin-1 partners. The vertical direction is hypercharge, and the horizontal direction is I3 , the third component of isospin. s Q , and s as the eigenvalues J2 = j ( j + 1), S2Q = s Q (s Q +1), and S2 = s (s + 1) in the state H (Q) . Heavy hadrons come in doublets (unless s = 0) containing states with total spin j± = s ± 1/2 obtained by combining the spin of the light degrees of freedom with the spin of the heavy quark s Q = 1/2. These doublets are degenerate in the m Q → ∞ limit. If s = 0, there is only a single j = 1/2 hadron. Mesons containing a heavy quark Q are made up of a heavy quark and a light antiquark q̄ (plus gluons and q q̄ pairs). The ground state mesons are composed of a heavy quark with s Q = 1/2 and light degrees of freedom with s = 1/2, forming a multiplet of hadrons with spin j = 1/2 ⊗ 1/2 = 0 ⊕ 1 and negative parity, since quarks and antiquarks have opposite intrinsic parity. These states are the D and D ∗ mesons if Q is a charm quark, and the B̄ and B̄ ∗ mesons if Q is a b quark. The field operators which annihilate these heavy quark mesons ∗(Q) , respectively. The light antiquark with velocity v are denoted by Pv(Q) and Pvµ ¯ can be either a ū, d, or s̄ quark, so each of these heavy meson fields form a 3̄ representation of the light quark flavor group SU(3)V . The SU(3) weight diagram for the 3̄ mesons is shown in Fig. 2.1. In the nonrelativistic constituent quark model, the first excited heavy meson states have a unit of orbital angular momentum between the constituent antiquark and the heavy quark. These L = 1 mesons have s = 1/2 or 3/2, depending on how the orbital angular momentum is combined with the antiquark spin. The s = 1/2 mesons form multiplets of spin parity 0+ and 1+ states named (for Q = c) D0∗ and D1∗ , and the s = 3/2 mesons form multiplets of 1+ and 2+ states named (for Q = c) D1 and D2∗ . Properties of the s = 1/2 and s = 3/2 states are related in the nonrelativistic constituent quark model, but not by heavy quark symmetry. Baryons containing a heavy quark consist of a heavy quark and two light quarks, plus gluons and q q̄ pairs. The lowest-lying baryons have s = 0 and s = 1 and form 3̄ and 6 representations of SU(3)V , which are shown in Figs. 2.2 and 2.3, respectively. We can easily understand this pattern in the nonrelativistic 2.2 Quantum numbers 47 + Λc cud Ξ0c Ξ+c cds cus Fig. 2.2. Flavor SU(3) weight diagram for the 3̄ spin-1/2 c [qq] baryons. The corres0 ponding b [qq] baryons are the 0b , − b and b . The vertical direction is hypercharge, and the horizontal direction is I3 , the third component of isospin. Σc , Σ c 0 *0 Σ+c , Σ*+ c *++ Σ++ c , Σc cdd cud cuu Ξc′ 0, Ξ*0 c Ξc′ +, Ξ*+ c cds cus Ω0c , Ω*0 c css Fig. 2.3. Flavor SU(3) weight diagram for the 6 spin-1/2 and spin-3/2 c (qq) baryons. The corresponding b (qq) baryons are the spin-1/2 b−,0,+ , b−,0 , and − b , and their spin-3/2 partners. The vertical direction is hypercharge, and the horizontal direction is I3 , the third component of isospin. constituent quark model. In this model the ground-state baryons have no orbital angular momentum and the spatial wave function for the two light constituent quarks is symmetric under their interchange. The wave function is also completely antisymmetric in color. Fermi statistics then demands that for s = 0, where the spin wave function is antisymmetric, the SU(3)V flavor wave function is also antisymmetric, and hence transforms as (3 × 3)antisymmetric = 3̄. For s = 1, the SU(3)V flavor wave function is symmetric and hence transforms as (3 × 3)symmetric = 6. The s = 0 ground-state baryons have positive parity and total spin of 1/2, and the spinor fields that destroy these states are denoted by (Q) v . The s = 1 ground-state baryons have positive parity and come in a doublet of states with total spins of 1/2, and 3/2. We denote the fields that destroy these ∗(Q) , respectively. The spectrum of excited baryons is more states by v(Q) and vµ 48 Heavy quarks complicated than in the meson sector. In the nonrelativistic constituent quark model, the L = 1 baryons come in two types; states with the unit of orbital angular momentum between the two light quarks, and states with the unit of orbital angular momentum between the light quark pair and the heavy quark. The latter are expected to be lower in mass. The lowest-mass hadrons containing c and b quarks are summarized in Tables 2.1 and 2.2, respectively. 2.3 Strong decays of excited heavy hadrons In many cases the two members of a doublet with spin of the light degrees of freedom s can decay by means of a single pion emission to the two members of another lower-mass doublet with spin of the light degrees of freedom s . The orbital angular momentum of the emitted pion (L , L z ) is re