Chiral low-energy constants (LECs) carry the information of short-distance
dynamics involving heavier degrees of freedom not present in the chiral
Lagrangian. Our knowledge of the LECs is all-important at phenomenological
level because their relevance in the prediction of hadronic observables at
low-energies and on the other side because they provide hints on the
construction of a dual theory of QCD in the low-energy regime. I review briefly
the status of these important couplings.Comment: 7 pages. Talk given at the 31st International Conference of
  Theoretical Physics, 5-11 Sep 2007, Ustron (Katowice, Poland

Portoles, J.

Acta Physica Polonica B

English

arXiv

7 pages.-- PACS nrs.: 11.15.Pg, 12.38.-t, 12.39.Fe.-- ISI Article Identifier: 000252187200017.-- ArXiv pre-print available at: http://arxiv.org/abs/0710.4405Presented at the XXXI International Conference of Theoretical Physics, "Matter to
the Deepest", Ustroń, Poland, September 5–11, 2007.Chiral low-energy constants (LECs) carry the information of short-distance dynamics involving heavier degrees of freedom not present in the chiral Lagrangian. Our knowledge of the LECs is all-important at phenomenological level because their relevance in the prediction of hadronic observables at low-energies and, on the other side, because they provide hints on the construction of a dual theory of QCD in the low-energy regime. I review briefly the status of these important couplings.This work has been supported in part by the EU MRTN-CT-2006-035482 (FLAVIAnet), by MEC (Spain) under grant FPA2004-00996 and by Generalitat Valenciana under grant ACOMP/2007/156.Peer reviewe

Portolés, Jorge

Digital.CSIC

Vol. 38 (2007) ACTA PHYSICA POLONICA B No 11CHIRAL LOW-ENERGY CONSTANTS:STATUS AND PROSPECTS∗J. PortolésInstituto de Física Corpuscular, IFIC, CSIC-Universitat de ValènciaApt. Correus 22085, E–46071 València, Spain(Received October 22, 2007)Chiral low-energy constants (LECs) carry the information of short-distance dynamics involving heavier degrees of freedom not present in thechiral Lagrangian. Our knowledge of the LECs is all-important at phe-nomenological level because their relevance in the prediction of hadronicobservables at low-energies and, on the other side, because they providehints on the construction of a dual theory of QCD in the low-energy regime.I review briefly the status of these important couplings.PACS numbers: 11.15.Pg, 12.38.–t, 12.39.Fe1. IntroductionChiral symmetry of massless Quantum Chromodynamics has turned tobe a key tool in order to deal with the low-energy domain of strong inter-actions (typically E ∼ Mπ) where hadron dynamics is not appropriatelydescribed by partonic QCD. It is indeed the guiding principle in the con-struction of Chiral Perturbation Theory (χPT) [1–3] that intends to be thedual eﬀective theory of QCD, i.e. it describes a perturbative quantum ﬁeldtheory of strong interactions at low-energies.The χPT Lagrangian involves a perturbative expansion guided by powersof external momenta (p) and light quark masses (m), with p ∼ m, over ahadronic scale Λχ, driven by the loop expansion required by unitarity, p2/Λ2χor the one that rules the lightest heavier degrees of freedom omitted in theLagrangian, as the ρ(770) mass, p2/M2ρ . It includes the hadronic ﬁelds thatlive in this energy region, the multiplet of Goldstone bosons (π, K, η), andclassical auxiliary ﬁelds that help to determine Green functions satisfying theappropriate Ward identities. The theory can be systematically constructed∗ Presented at the XXXI International Conference of Theoretical Physics, “Matter tothe Deepest”, Ustroń, Poland, September 5–11, 2007.(3459)3460 J. Portolésby looking for all the operators that, with the ingredients above and up to aﬁxed order in theO(pn) expansion, satisfy the chiral symmetry constraint [4]:LχPT = LχPT2 + LχPT4 + LχPT6 + . . . . (1)LχPT2 embodies the spontaneous breaking of chiral symmetry and dependsonly on two parameters: F ∼ 92.4 MeV, related with the decay constant ofthe pion, and B0F2 = −〈0|ψψ|0〉, the vacuum expectation value of the lightquarks1.Higher orders in the expansion bring in the information of the dynamicsof short-distance contributions arising from heavier degrees of freedom thathave been integrated out, for instance, resonance states. As in any eﬀectiveﬁeld theory (EFT) this information is incorporated into the couplings of theoperators:LχPT4 =10∑i=1LiO(4)i , LχPT6 =90∑i=1CiO(6)i , (2)for SU(3). Explicit expressions for the operators can be read from Refs. [3,5].In Eq. (2) Li and Ci are the chiral LECs at O(p4) and O(p6) respectively;they are not provided by chiral symmetry. At present we do not know howto construct this Lagrangian directly from partonic QCD and, accordingly,we do not know how to determine the LECs from that framework. Hencewe have to use our knowledge on the foundations of EFTs. LECs in χPTshould receive contributions from the energy regime at or above the scalethat rules the chiral perturbative expansion [6]. It is reasonable to infer thatthe dynamics of the lightest meson resonances in the hadronic spectrum,that are not included explicitly in the Chiral Lagrangian, would provide thelarger contribution. Following this assumption the determination of the in-put of the resonance spectrum to the Li LECs in LχPT4 [7] showed that theyindeed saturate the values extracted from phenomenological analyses. Asa consequence, it is reasonable to think that the most important contribu-tion to the LECs is provided by the energy region immediately above theintegrated scale (E ∼ Λχ ∼Mρ).The determination of LECs is crucial for the predictability of low-energyhadronic observables determined using χPT. In the last ten years manyof these processes have been evaluated up to O(p6) ( [8] and referencestherein) but our ignorance of the values of the involved LECs lessens thepractical value of that enormous task. Therefore it is mandatory to exploreprocedures that allow us to determine or, at least, provide reliable estimatesfor the values of chiral LECs up to O(p6).1 It happens that the loop expansion of the theory tells us that Λχ ∼ 4piF that is ofthe same order thanMρ. Hence there is only one scale in the perturbative expansionof the theory.Chiral Low-Energy Constants: Status and Prospects 34612. Tools: Large-NC and QCD asymptotic constraintsAs illustrated in the O(p4) case [7, 9], a procedure to systematicallydisclose the structure of the resonance contributions to the LECs in χPT isavailable. The key point is to construct a Lagrangian theory that includesresonances, Goldstone bosons and auxiliary ﬁelds respecting the underlyingchiral symmetry. There are several tools that allow us to grab importantproperties of QCD and to implement them in an EFT-like Lagrangian model.Two relevant features to consider are:(i) Weinberg [1] and Leutwyler [10] state that if one writes down the mostgeneral Lagrangian, including all terms consistent with assumed sym-metry principles, and then calculates matrix elements with this La-grangian to any given order of perturbation theory, the result will bethe most general possible S-matrix amplitude consistent with analyt-icity, perturbative unitarity, cluster decomposition and the principlesof symmetry that have been required.(ii) The 1/NC coupling (NC is the number of colours in QCD) can be takenas a perturbative expansion parameter [11]. Indeed large-NC QCDshows features that resemble, both qualitatively and quantitatively,the NC = 3 case [12]. In practice the consequences of this approachare that meson dynamics in the large-NC limit is described by treediagrams of an eﬀective local Lagrangian involving an inﬁnite spectrumof zero-width mesons.Both statements can be combined by constructing a Lagrangian in termsof SU(3) (Goldstone mesons) and U(3) (heavier resonances) ﬂavour multi-plets as explicit degrees of freedom respecting the underlying chiral symme-try. Then upon integration of the heavier states, the χPT Lagrangian is tobe recovered. This procedure has been systematically established [7, 9, 13]and devises what is known as Resonance Chiral Theory (RχT). Its contentis schematically given by:LRχT = LχPT2 +∑nLGBn>2 + LR , (3)where LGBn>2 has the same structure than LχPT4 , LχPT6 ,. . . in Eq. (2) thoughwith diﬀerent coupling constants, and LR involves terms with resonancesand their couplings to Goldstone modes:(1) The construction of the operators in the LR is guided by chiral sym-metry for the Goldstone mesons and by unitary symmetry for theresonances. The general structure of these couplings is:O = 〈R1R2 . . . Rmχ(pn)〉 ∈ Lm︷ ︸︸ ︷RR...R(n) , (4)3462 J. Portoléswhere Rj indicates a resonance ﬁeld and χ(pn) is a chiral structuredtensor, involving the pseudoscalar mesons and auxiliary ﬁelds only.With these settings chiral symmetry is preserved upon integration ofthe resonance ﬁelds and, at the same time, the low-energy behaviourof the amplitudes is guaranteed.(2) Symmetries do not provide information on the couplings in LR as theseincorporate short-distance dynamics not included explicitly in the La-grangian. The latter is supposed to bridge between the energy regionbelow resonances (E ≪MV ) and the parton regime (E ≫MV ). Thishypothesis indicates that it should match both regions and it satis-ﬁes, by construction, the chiral constraints. To suit the high-energybehaviour one can match, for instance, the OPE of Green functions(that are order parameters of chiral symmetry) with the correspondingexpressions evaluated within our theory. In addition, the asymptotictrend of form factors of QCD currents is estimated from the spectralstructure of two-point functions and it is enforced on the observables.This heuristic strategy is well supported by the phenomenology [12–16]and provides information on the resonance couplings.RχT does not have an expansion parameter, hence it does not admita conventional perturbative treatment. There is, of course, the guide pro-vided by 1/NC that translates into the loop expansion, however, there isno counting that limits the number of operators with resonances that haveto be included in the initial Lagrangian. The number of resonance ﬁeldsto be kept relies essentially in the physical system that we are interestedin. Moreover, the maximum order of the chiral tensor χ(pn) in Eq. (4) isconstrained by the high-energy behaviour.As commented above, large-NC requires, already at NC → ∞, an inﬁ-nite spectrum in order to match the leading QCD logarithms, though we donot know how to implement this in a model-independent way. The usualapproach in RχT is to include the lightest resonances because their phe-nomenological relevance, though there is no conceptual problem that pre-vents the addition of a ﬁnite number of multiplets. This cut in the spectrummay produce inconsistencies in the matching procedure outlined above [17].To deal with this, one can include more multiplets that may delay the ap-pearance of that problem.3. Status and prospectsI comment brieﬂy the present status and several developments on ourknowledge of chiral LECs.Chiral Low-Energy Constants: Status and Prospects 34633.1. O(p4)Contributions from the lightest multiplets of vector resonances (JPC =1−−) to SU(2) [2] and vector, axial-vector (1++), scalar (0++) and pseu-doscalar (0−+) resonances to SU(3) [7] LECs are in general good agreementwith those values extracted from observables [18], showing that those reso-nances saturate the phenomenological values. This has been lately conﬁrmedby the inclusion of one multiplet of resonances with 2++ and 1+− quantumnumbers [19] that are shown to play, quantitatively, a minor role.A study of the strange quark mass dependence of the O(p4) SU(2) LECs(ℓri ) up to O(p6) has also been recently performed [20]. The authors essen-tially obtain:ℓri = δi7F 28B0ms+ ai + bix+O(x2) , i = 1, . . . 7 , (5)where x = M2K/(16π2F 2) and ms is the mass of the strange quark. Hereai = ai(Lj , lK) arise at O(p4) and bi = bi(Lj , Ck, lK) at O(p6), with lK =log(M2K/µ2). The dependence on the strange quark or kaon mass is explicitlystated above. In addition, and for later discussion, the bi parameters canbe written as bi = p0,i + p1,ilK + p2,il2K with an explicit dependence p0,i =p0,i(LjLk, Cl), p1,i = p1,i(Lj).3.2. O(p6)In Ref. [13] we have constructed the RχT Lagrangian needed to evaluatethe resonance contributions to theO(p6) LECs in Eq. (2). It can be shown [9]that, at O(p4), all local terms in LGB4 (see Eq. (3)) have to vanish in ordernot to spoil the asymptotic behaviour of QCD correlators. A correspondingresult at O(p6) is still lacking but we have also assumed that all the couplingsin LGB6 are set to zero.LRχT in Eq. (3) involves 124 a priori unknown couplings. Some addi-tional work provides an enormous simpliﬁcation:(i) Upon integration of resonances not all couplings appear independentlyin the LECs. In general only several combination of couplings inter-vene and, to take into account this case, one can perform suitableredeﬁnitions of the ﬁelds. This procedure may upset the high-energybehaviour of the theory but it is correct for the evaluation of the LECs.Indeed, the unknown couplings are reduced to 77.3464 J. Portolés(ii) The next step is to enforce short-distance information, i.e. the lead-ing behaviour at large momenta, for two and three-point functionsand form factors. This procedure, set in Ref. [9], relies on the well-known properties of partonic scattering or asymptotic QCD [21]. Two-current correlators and associated form-factors provide 19 new con-straints on couplings, while the three-point Green functions studiedtill now: 〈VAP〉 [15] and 〈SPP〉 [13, 16], give 6 and 5 independent re-strictions, respectively. We are left with 47 couplings. Further studiesalong this line can diminish even more number of unknown constants.Notwithstanding, we can already determine fully, at this point, the res-onance contribution to the O(p6) couplings C78 and C89 (that appear inπ → ℓνℓγ and π → ℓνℓγ∗, respectively), C87 (in 〈AµAν〉), C88 and C90 (inF πV (q2) and the q2 dependence of the form factors in Kℓ3), C38 (in 〈SS〉)and C12 and C34 (in Fπ,KV (q2) and fK0π−+ (0)).The evaluation of resonance contributions to Cj carried out in Ref. [13]can be termed as an NC → ∞ evaluation (with a cut spectrum). It is in-teresting to notice that when the values obtained for Cj (namely CRj ) aresubstituted in the expressions obtained in Ref. [20] for p0,i, the diﬀerent com-binations of Cj couplings vanish (except for i = 7) [22]. Something similarhappens with the combinations of LiLj products under additional condi-tions. The reason of these unexpected cancellations is still not understoodand it would imply that, in the large-NC limit, resonance saturation of O(p4)SU(2) chiral LECs is already achieved at leading order. This would be verymuch satisfactory both for the 1/NC expansion and for the determinationof resonance contributions to chiral LECs. It is nice to have something leftto understand for the future.I wish to thank Michał Czakon, Henryk Czyż and Janusz Gluza for theexcellent organization (but for the weather) of the XXXI International Con-ference of Theoretical Physics in Ustroń, Poland. This work has been sup-ported in part by the EU MRTN-CT-2006-035482 (FLAVIAnet), by MEC(Spain) under grant FPA2004-00996 and by Generalitat Valenciana undergrant ACOMP/2007/156.REFERENCES[1] S. Weinberg, Physica A 96, 327 (1979).[2] J. Gasser, H. Leutwyler, Ann. Phys. 158, 142 (1984).[3] J. Gasser, H. Leutwyler, Nucl. Phys. B250, 465 (1985).Chiral Low-Energy Constants: Status and Prospects 3465[4] G. Ecker, Prog. Part. Nucl. Phys. 35, 1 (1995) [hep-ph/9501357].[5] J. Bijnens, G. Colangelo, G. Ecker, J. High Energy Phys. 9902, 020 (1999)[hep-ph/9902437].[6] H. Georgi, Nucl. Phys. B361. 339 (1991).[7] G. Ecker, J. Gasser, A. Pich, E. de Rafael, Nucl. Phys. B321, 311 (1989).[8] J. Bijnens, Prog. Part. Nucl. Phys. 58, 521 (2007) [hep-ph/0604043].[9] G. Ecker, J. Gasser, H. Leutwyler, A. Pich, E. de Rafael, Phys. Lett. B223,425 (1989).[10] H. Leutwyler, Ann. Phys. 235, 165 (1994) [hep-ph/9311274].[11] G. ’t Hooft, Nucl. Phys. B75, 461 (1974); E. Witten, Nucl. Phys. B160, 57(1979).[12] S. Peris, M. Perrottet, E. de Rafael, J. High Energy Phys. 9805, 011 (1998)[hep-ph/9805442]; M. Knecht, S. Peris, M. Perrottet, E. de Rafael, Phys. Rev.Lett. 83, 5230 (1999) [hep-ph/9908283]; S. Peris, B. Phily, E. de Rafael, Phys.Rev. Lett. 86, 14 (2001) [hep-ph/0007338].[13] V. Cirigliano, G. Ecker, M. Eidemüller, R. Kaiser, A. Pich, J. Portolés, Nucl.Phys. B753, 139 (2006) [hep-ph/0603205].[14] M. Knecht, A. Nyffeler, Eur. Phys. J. C21, 659 (2001) [hep-ph/0106034];A. Pich, Proceedings of the Phenomenology of Large NC QCD, ed. R. Lebed,World Scientific, Singapore 2002, p. 239 [hep-ph/0205030]; G. Amorós,S. Noguera, J. Portolés, Eur. Phys. J. C27, 243 (2003) [hep-ph/0109169];P. D. Ruiz-Femenía, A. Pich, J. Portolés, J. High Energy Phys. 0307, 003 (2003)[hep-ph/0306157]; J. Portolés, P. D. Ruiz-Femenía, Nucl. Phys. Proc. Suppl.131, 170 (2004) [hep-ph/0311251]; J. Portolés, Nucl. Phys. Proc. Suppl. 164,292 (2007) [hep-ph/0509279].[15] V. Cirigliano, G. Ecker, M. Eidemüller, A. Pich, J. Portolés, Phys. Lett. B596,96 (2004) [hep-ph/0404004].[16] V. Cirigliano, G. Ecker, M. Eidemüller, R. Kaiser, A. Pich, J. Portolés,J. High Energy Phys. 0504, 006 (2005) [hep-ph/0503108].[17] J. Bijnens, E. Gamiz, E. Lipartia, J. Prades, J. High Energy Phys. 0304, 055(2003) [hep-ph/0304222].[18] G. Amorós, J. Bijnens, P. Talavera, Nucl. Phys. B602, 87 (2001)[hep-ph/0101127].[19] G. Ecker, C. Zauner, Eur. Phys. J. C52, 315 (2007) [hep-ph/0705.0624];G. Ecker, Acta Phys. Pol. B 38 (2007), these proceedings.[20] J. Gasser, C. Haefeli, M. A. Ivanov, M. Schmid, Phys. Lett. B652, 21 (2007)[hep-ph/0706.0955].[21] M. Froissart, Phys. Rev. 123, 1053 (1961); S. Weinberg, Phys. Rev. Lett. 18,507 (1967); T. Das, V.S. Mathur, S. Okubo, Phys. Rev. Lett. 19, 859 (1967);G.P. Lepage S.J. Brodsky, Phys. Rev. D22, 2157 (1980).[22] C. Haefeli, private communication.

Chiral low-energy constants: Status and prospects

Chiral low-energy constants (LECs) carry the information of shortdistance dynamics involving heavier degrees of freedom not present in the chiral Lagrangian. Our knowledge of the LECs is all-important at phenomenological level because their relevance in the prediction of hadronic observables at low-energies and, on the other side, because they provide hints on the construction of a dual theory of QCD in the low-energy regime. I review briefly the status of these important couplings

Chiral Low-Energy Constants: Status and Prospects

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