The basic ideas and methods of chiral perturbation theory are briefly
reviewed. I discuss the recent attempts to build an effective Lagrangian in the
resonance region and summarize the known large-N_C constraints on the
low-energy chiral couplings.Comment: 6 pages. Invited talk at the 10th International Symposium on
  Meson-Nucleon Physics and the Structure of the Nucleon (MENU 2004), Beijing,
  China, August 29 - September 4, 200

A. PICH

Bijnens J.

Rosell I.

English

arXiv

The basic ideas and methods of chiral perturbation theory are briefly reviewed. I discuss the recent attempts to build an effective Lagrangian in the resonance region and summarize the known large-N-C constraints on the low-energy chiral couplings

Pich Zardoya, Antonio

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

arXiv:hep-ph/0410322v1  24 Oct 2004International Journal of Modern Physics Ac© World Scientific Publishing CompanyPRESENT STATUS OF CHIRAL PERTURBATION THEORY∗A. PICHDepartament de F´ısica Teo`rica, IFIC, Universitat de Vale`ncia–CSIC,Apt. Correus 22085, E-46071 Vale`ncia, Spain. E-mail: Antonio.Pich@ific.uv.esThe basic ideas and methods of chiral perturbation theory are briefly reviewed. I dis-cuss the recent attempts to build an effective Lagrangian in the resonance region andsummarize the known large–NC constraints on the low-energy chiral couplings.Keywords: Chiral Symmetry; Effective Field Theory; 1/NC Expansion1. Chiral SymmetryWith nf massless quark flavours, the QCD Lagrangian is invariant under globalSU(nf )L⊗SU(nf )R transformations of the left- and right-handed quarks in flavourspace. The symmetry group spontaneously breaks down to the diagonal subgroupSU(nf )L+R and n2f − 1 pseudoscalar massless Goldstone bosons appear in thetheory, which for nf = 3 can be identified with the eight lightest hadronic statesφa = {pi, K, η}. These pseudoscalar fields are usually parameterized through the3× 3 unitary matrix U(φ) = u(φ)2 = exp {iλaφa/f}.The Goldstone nature of the pseudoscalar mesons implies strong constraints ontheir interactions, which can be most easily analyzed on the basis of an effectiveLagrangian.1 Since there is a mass gap separating the pseudoscalar octet from therest of the hadronic spectrum, we can build an effective field theory containing onlythe Goldstone modes.2, 3 The low-energy effective Lagrangian can be organized interms of increasing powers of momenta (derivatives): L =?∑n L2n.It is convenient to consider an extended QCD Lagrangian, with quark currentscoupled to external Hermitian matrix-valued sources lµ, rµ, s, p. In addition togenerate the QCD Green functions, the external fields can be used to incorporatethe electromagnetic and semileptonic weak interactions, and the explicit breaking ofchiral symmetry through the quark masses: s =M+ . . . , M = diag(mu,md,ms).At lowest order in derivatives and quark masses, the most general effective La-grangian consistent with chiral symmetry has the form:4L2 = f24〈DµU †DµU + U †χ + χ†U〉 , χ ≡ 2B0 (s+ ip) , (1)∗Invited talk at the 10th International Symposium on Meson–Nucleon Physics and the Structureof the Nucleon (MENU 2004), Beijing, China, August 29 – September 4, 200412 A. PichTable 1. Phenomenological values of the renormalized couplingsLri (Mρ) in units of 10−3. The large–NC predictions obtained withinthe single-resonance approximation are given in the last column.2, 3i Lri (Mρ) O(NC) Source LNC→∞i2L1 − L2 −0.6± 0.6 O(1) Ke4, pipi → pipi 0L2 1.4± 0.3 O(NC) Ke4, pipi → pipi 1.8L3 −3.5± 1.1 O(NC) Ke4, pipi → pipi −4.3L4 −0.3± 0.5 O(1) Zweig rule 0L5 1.4± 0.5 O(NC) FK : Fpi 2.1L6 −0.2± 0.3 O(1) Zweig rule 0L7 −0.4± 0.2 O(1) GMO, L5, L8 −0.3L8 0.9± 0.3 O(NC) Mφ, L5 0.8L9 6.9± 0.7 O(NC) 〈r2〉piV7.1L10 −5.5± 0.7 O(NC) pi → eνγ −5.4where DµU = ∂µU−irµU+iU lµ , 〈A〉 denotes the flavour trace of the matrix A andB0 is a constant, which, like f , is not fixed by symmetry requirements alone. Onefinds that f equals the pion decay constant (at lowest order) f = fpi = 92.4 MeV,while B0 is related to the quark condensate:B0 = −〈q¯q〉f2=M2pimu +md=M2K0ms +md=M2K±ms +mu. (2)With only two low-energy constants, the lowest-order chiral Lagrangian L2 encodesin a very compact way all the Current Algebra results obtained in the sixties.The symmetry constraints become less powerful at higher orders. At O(p4) weneed ten additional coupling constants Li to determine the low-energy behaviour ofthe Green functions:4L4 = L1 〈DµU †DµU〉2 + L2 〈DµU †DνU〉 〈DµU †DνU〉 + . . . (3)One-loop graphs with the lowest-order Lagrangian L2 contribute also at O(p4).Their divergent parts are renormalized by the L4 couplings, which introduces arenormalization scale dependence. The chiral loops generate non-polynomial contri-butions, with logarithms and threshold factors as required by unitarity, which arecompletely determined as functions of f and the Goldstone masses.The precision required in present phenomenological applications makes necessaryto include corrections of O(p6). This involves contributions from L4 at one-loop andL2 at two-loops, which can be fully predicted.5 However, the O(p6) chiral LagrangianL6 contains 90 (23) independent local terms of even (odd) intrinsic parity.5, 6 Thehuge number of unknown couplings limits the achievable accuracy. Clearly, furtherprogress will depend on our ability to estimate these chiral couplings, which encodethe underlying QCD dynamics.Several two-loop calculations have been already performed.7–12 Thus, the non-local O(p6) contributions (chiral logarithms) to many observables are known and,in some cases, the local ambiguities can be reduced to a few subtraction constantsusing dispersion relation techniques.Chiral Perturbation Theory 32. Resonance Chiral TheoryThe limit of an infinite number of quark colours is a very useful starting point tounderstand many features of QCD.13, 14 Assuming confinement, the strong dynamicsat NC →∞ is given by tree diagrams with infinite sums of hadron exchanges, whichcorrespond to the tree approximation to some local effective Lagrangian. Hadronicloops generate corrections suppressed by factors of 1/NC . At NC →∞, QCD has alarger symmetry U(3)L⊗U(3)R → U(3)L+R, and one needs to include in the matrixU(φ) a ninth Goldstone boson field, the η1. Resonance chiral theory provides thecorrect framework to incorporate the massive mesonic states.15Let us consider a chiral-invariant Lagrangian LR, describing the couplings ofresonance nonet multiplets V µνi (1−−), Aµνi (1++), Si(0++) and Pi(0−+) to the Gold-stone bosons:16LR =∑i{FVi2√2〈V µνi f+µν〉 +i GVi√2〈V µνi uµuν〉 +FAi2√2〈Aµνi f−µν〉+ cdi 〈Si uµuµ〉 + cmi 〈Si χ+〉 + i dmi 〈Pi χ−〉}, (4)where uµ ≡ i u†DµUu†, fµν± ≡ uFµνL u† ± u†FµνR u with FµνL,R the field-strengthtensors of the lµ and rµ flavour fields and χ± ≡ u†χu† ± uχ†u. The resonancecouplings FVi , GVi , FAi , cdi , cmi and dmi are of O(√NC).The lightest resonances have an important impact on the low-energy dynamics ofthe pseudoscalar bosons. Below the resonance mass scale, the singularity associatedwith the pole of a resonance propagator is replaced by the corresponding momen-tum expansion; therefore, the exchange of virtual resonances generates derivativeGoldstone couplings proportional to powers of 1/M2R. At lowest order in derivatives,this gives the large–NC predictions for the O(p4) couplings of chiral perturbationtheory:162L1 = L2 =∑iG2Vi4M2Vi, L3 =∑i{− 3G2Vi4M2Vi+c2di2M2Si},L5 =∑icdi cmiM2Si, L8 =∑i{c2mi2M2Si− d2mi2M2Pi}, (5)L9 =∑iFVi GVi2M2Vi, L10 =∑i{F 2Ai4M2Ai− F2Vi4M2Vi}.All these couplings are of O(NC), in agreement with the counting indicated inTable 1, while for the couplings of O(1) we get 2L1 − L2 = L4 = L6 = L7 = 0.Owing to the U(1)A anomaly, the η1 field is massive and it is often integratedout from the low-energy chiral theory. In that case, the SU(3)L ⊗ SU(3)R chiralcoupling L7 gets a contribution from η1 exchange:4, 16L7 = − d˜2m2M2η1, d˜m = − f√24. (6)4 A. Pich3. Short-Distance ConstraintsThe short-distance properties of the underlying QCD dynamics impose some con-straints on the low-energy parameters.15, 17 At leading order in 1/NC , the two-Goldstone matrix element of the vector current, is characterized byFV (t) = 1 +∑iFVi GVif2tM2Vi − t. (7)Since the vector form factor FV (t) should vanish at infinite momentum transfer t,the resonance couplings should satisfy∑iFVi GVi = f2 . (8)Similarly, the matrix element of the axial current between one Goldstone and onephoton is parameterized by the so-called axial form factor GA(t), which vanishes att→∞ provided that ∑i(2FVi GVi − F 2Vi)/M2Vi = 0 . (9)Requiring the scalar form factor FS(t), which governs the two-pseudoscalar matrixelement of the scalar quark current, to vanish at t→∞, one gets the constraints:184∑icdi cmi = f2 ,∑icmi (cmi − cdi) /M2Si = 0 . (10)Since gluonic interactions preserve chirality, the two-point function built froma left-handed and a right-handed vector quark currents ΠLR(t) satisfies an unsub-tracted dispersion relation. In the chiral limit, it vanishes faster than 1/t2 whent→∞; this implies the well-known Weinberg conditions:19∑i(F 2Vi − F 2Ai)= f2 ,∑i(M2ViF2Vi −M2AiF 2Ai)= 0 . (11)The two-point correlators of two scalar or two pseudoscalar currents would beequal if chirality was preserved. For massless quarks, ΠSS−PP (t) vanishes as 1/t2when t → ∞, with a coefficient proportional to αs 〈q¯Γq q¯Γq〉 ∼ αs 〈q¯q〉2 ∼ αsB20 .Imposing this behaviour, one gets:208∑i(c2mi − d2mi)= f2,∑i(c2miM2Si − d2miM2Pi)= 3 piαs f4/4 . (12)4. Single-Resonance ApproximationLet us approximate each infinite resonance sum with the first meson nonet con-tribution. This is meaningful at low energies where the contributions from higher-mass states are suppressed by their corresponding propagators. The resulting short-distance constraints are matching conditions between an effective theory below thescale of the second resonance multiplets and the underlying QCD dynamics.Chiral Perturbation Theory 5With this approximation, Eqs. (8), (9) and (11) determine the vector and axial-vector couplings in terms of MV and f :17FV = 2GV =√2FA =√2 f , MA =√2MV . (13)The scalar18 and pseudoscalar parameters are obtained from (10) and (12):15cm = cd =√2 dm = f/2 , MP =√2MS (1− δ)1/2 . (14)The last relation involves a small correction δ ≈ 3 piαsf2/M2S ∼ 0.08αs, which wecan neglect together with the tiny effects from light quark masses.Inserting these predictions into Eqs. (5), one finally gets all O(NC p4) chiralperturbation theory couplings, in terms of MV , MS and f :2L1 = L2 =14L9 = −13L10 =f28M2V, (15)L3 = − 3 f28M2V+f28M2S, L5 =f24M2S, L8 =3 f232M2S. (16)The last column in Table 1 shows the results obtained with MV = 0.77 GeV,MS = 1.0 GeV and f = 92 MeV. Also shown is the L7 prediction in (6), takingMη1 = 0.80 GeV. The agreement with the measured values is a clear success of thelarge–NC approximation. It demonstrates that the lightest resonance multiplets giveindeed the dominant effects at low energies.The study of other Green functions provides further matching conditions be-tween the hadronic and fundamental QCD descriptions. Clearly, it is not possibleto satisfy all of them within the single-resonance approximation. A useful general-ization is the so-calledMinimal Hadronic Ansatz, which keeps the minimum numberof resonances compatible with all known short-distance constraints for the problemat hand.21 Some O(p6) chiral couplings have been already analyzed in this way, bystudying an appropriate set of three-point functions.22–255. Subleading 1/NC CorrectionsThe large–NC limit provides a very successful description of the low-energy dynam-ics.15 However, we are still lacking a systematic procedure to incorporate next-to-leading contributions in the 1/NC counting. Up to now, the effort has concentratedin pinning down the most relevant subleading effects, such as the resonance widthswhich regulate the corresponding poles in the meson propagators,26 or the role offinal state interactions in the physical amplitudes.18, 26–28Quantum loops including virtual resonance propagators constitute a major tech-nical challenge.29, 30 Their ultraviolet divergences require higher dimensional coun-terterms, which could generate a problematic behaviour at large momenta. Thus, itis necessary to investigate the short-distance QCD constraints at the next-to-leadingorder in 1/NC . A first step in this direction is the recent one-loop calculation of thevector form factor in the resonance chiral theory.30 Further work towards a moreformal renormalization procedure is in progress.6 A. PichAcknowledgementsThis work has been supported by EU HPRN-CT2002-00311 (EURIDICE), MCYT(FPA-2001-3031) and Generalitat Valenciana (GRUPOS03/013, GV04B-594).References1. S. Weinberg, Physica A96, 327 (1979).2. G. Ecker, Prog. Part. Nucl. Phys. 35, 1 (1995).3. A. Pich, Rep. Prog. Phys. 58, 563 (1995); arXiv:hep-ph/9802419.4. J. Gasser and H. Leutwyler, Nucl. Phys. B250, 465, 517, 539 (1985).5. J. Bijnens, G. Colangelo and G. Ecker, Ann. Phys. 280, 100 (2000); JHEP 9902, 020(1999); Phys. Lett. B441, 437 (1998).6. J. Bijnens, L. Girlanda and P. Talavera, Eur. Phys. J. C23, 539 (2002).7. G. Colangelo et al., Nucl. Phys. B603, 125 (2001); Phys. Rev. Lett. 86, 5008 (2001);Phys. Lett. B488, 261 (2000); Phys. Rep. 353, 207 (2001).8. J. Bijnens et al., Nucl. Phys. B508, 263 (1997); B517, 639 (1998); B489, 387 (1997);B585, 293 (2000); B568, 319 (2000); B598, 665 (2001); B602, 87 (2001); B669, 341(2003); Phys. Lett. B374, 210 (1996); B480, 71 (2000); JHEP 9805, 014 (1998); 0203,046 (2002); 0310, 061 (2003); 0401, 050 (2004); 0405, 036 (2004).9. S. Bellucci et al., Nucl. Phys. B23, 80 (1994); B31, 413 (1994); U. Burgi, Phys. Lett.B377, 147 (1996); Nucl. Phys. B479, 392 (1996).10. M. Knecht et al., Nucl. Phys. B471, 445 (1996); B457, 513 (1995).11. P. Post and K. Schilcher, Eur. Phys. J. C25, 427 (2002); Nucl. Phys. B599, 30 (2001).12. E. Golowich and J. Kambor, Phys. Rev. D58, 036004 (1998); Nucl. Phys. B447, 373(1995); S. Durr and J. Kambor, Phys. Rev. D61, 114025 (2000).13. G. ’t Hooft, Nucl. Phys. B72, 461 (1974); B75, 461 (1974).14. E. Witten, Nucl. Phys. B160, 57 (1979).15. A. Pich, arXiv:hep-ph/0205030.16. G. Ecker, J. Gasser, A. Pich and E. de Rafael, Nucl. Phys. B321, 311 (1989).17. G. Ecker et al., Phys. Lett. B223, 425 (1989).18. M. Jamin, J.A. Oller and A. Pich, Nucl. Phys. B622, 279 (2002); B587, 331 (2000).19. S. Weinberg, Phys. Rev. Lett. 18, 507 (1967).20. M. Golterman and S. Peris, Phys. Rev. D61, 034018 (2000).21. M. Knecht and E. de Rafael, Phys. Lett. B424, 335 (1998); S. Peris et al., JHEP9805, 011 (1998); 0201, 024 (2002); Phys. Rev. Lett. 86, 14 (2001).22. B. Moussallam, Phys. Rev. D51, 4939 (1995); Nucl. Phys. B504, 381 (1997).23. M. Knecht and A. Nyffeler, Eur. Phys. J. C21, 659 (2001).24. P.D. Ruiz-Femen´ıa, A. Pich and J. Portole´s, JHEP 0307, 003 (2003); V. Cirigliano,G. Ecker, M. Eidemu¨ller, A. Pich and J. Portole´s, Phys. Lett. B596, 96 (2004).25. J. Bijnens, E. Ga´miz, E. Lipartia and J. Prades, JHEP 0304, 055 (2003).26. F. Guerrero and A. Pich, Phys. Lett. B412, 382 (1997); D. Go´mez Dumm, A. Pichand J. Portole´s, Phys. Rev. D62, 054014 (2000); A. Pich and J. Portole´s, Phys. Rev.D63, 093005 (2001); J.J. Sanz-Cillero and A. Pich, Eur. Phys. J. C27, 587 (2003).27. E. Pallante and A. Pich, Phys. Rev. Lett. 84, 2568 (2000); Nucl. Phys. B592, 294(2000); E. Pallante, A. Pich and I. Scimemi, Nucl. Phys. B617, 441 (2001).28. T.N. Truong, Phys. Rev. Lett. 61, 2526 (1988); A. Dobado et al., Phys. Lett. B235,134 (1990); T. Hannah, Phys. Rev. D55, 5613 (1997); J.A. Oller and E. Oset, Phys.Rev. D60, 074023 (1999); A. Go´mez and J.R. Pela´ez, Phys. Rev. D65, 054009 (2002).29. O. Cata` and S. Peris, Phys. Rev. D65, 056014 (2002).30. I. Rosell, J.J. Sanz-Cillero and A. Pich, JHEP 0408, 042 (2004).

Present status of chiral perturbation theory

6 páginas, 1 tabla.-- Comunicación presentada en el 10º International Symposium on Meson-Nucleon Physics and the Structure of the Nucleon (MENU 2004) celebrado del 29 de Agosto al 4 de Septiembre de 2004 en Beijing (China).-- arXiv:hep-ph/0410322v1The basic ideas and methods of chiral perturbation theory are briefly reviewed. I discuss the recent attempts to build an effective Lagrangian in the resonance region and summarize the known large–NC constraints on the low-energy chiral couplings.This work has been supported by EU HPRN-CT2002-00311 (EURIDICE), MCYT
(FPA-2001-3031) and Generalitat Valenciana (GRUPOS03/013, GV04B-594).Peer reviewe

Pich, Antonio

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Present Status of Chiral Perturbation Theory

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PRESENT STATUS OF CHIRAL PERTURBATION THEORY

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Present Status of Chiral Perturbation Theory

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