Chiral low-energy constants incorporate short-distance information from the
dynamics involving heavier degrees of freedom not present in the chiral
Lagrangian. We have studied the contribution of the lightest resonances to the
chiral low-energy constants, up to O(p^6), within a systematic procedure guided
by the large-Nc limit of QCD and also including short-distance asymptotic
constraints.Comment: 8 pages, 1 figure. Talk given at the Final Euridice Meeting
  "Effective Theories of colours and flavours : from EURODAPHNE to EURIDICE",
  24th-27th August 2006, Kazimierz (Poland

Portoles, J.

Acta Physica Polonica B

English

arXiv

8 pages, 1 figure.-- PACS nrs.: 11.15.Pg, 12.38.–t, 12.39.Fe.-- ISI Article Identifier: 000249819800009.-- ArXiv pre-print available at: http://arxiv.org/abs/hep-ph/0702179Talk given at the Final Euridice Meeting "Effective Theories of colours and flavours : from EURODAPHNE to EURIDICE", 24th-27th August 2006, Kazimierz (Poland).Chiral low-energy constants incorporate short-distance information from the dynamics involving heavier degrees of freedom not present in the chiral Lagrangian. We have studied the contribution of the lightest resonances to the chiral low-energy constants, up to O(p^6), within a systematic procedure guided by the large-Nc limit of QCD and also including short-distance asymptotic constraints.This work has been supported in part by HPRN-CT-2002-00311 (EURIDICE), by the EU MRTN-CT-2006-
035482 (FLAVIAnet), by MEC (Spain) under grant FPA2004-00996 and by Generalitat Valenciana under grants ACOMP06/098 and GV05/015.Peer reviewe

Portolés, Jorge

Digital.CSIC

Vol. 38 (2007) ACTA PHYSICA POLONICA B No 9LARGE-NC ESTIMATE OF THE CHIRALLOW-ENERGY CONSTANTS∗J. PortolésInstituto de Física Corpuscular, IFIC, CSIC-Universitat de ValènciaApt. Correus 22085, 46071 València, Spain(Received July 2, 2007)Chiral low-energy constants incorporate short-distance informationfrom the dynamics involving heavier degrees of freedom not present inthe chiral Lagrangian. We have studied the contribution of the lightestresonances to the chiral low-energy constants, up to O(p6), within a sys-tematic procedure guided by the large-NC limit of QCD and also includingshort-distance asymptotic constraints.PACS numbers: 11.15.Pg, 12.38.–t, 12.39.Fe1. IntroductionChiral symmetry of massless Quantum Chromodynamics has become thekey tool in the study of the very low-energy domain of strong interactions(typically E ∼ Mπ), where QCD turns non-perturbative. It is indeed theguiding principle in the construction of Chiral Perturbation Theory (χPT),the eﬀective ﬁeld theory of QCD in this energy region [1–3]. The χPTLagrangian has a perturbative structure guided by powers of external mo-menta and light quark masses. It involves the multiplet of pseudoGoldstonebosons, i.e. pseudoscalar mesons (π, K, η), and classical auxiliary ﬁelds.The theory, up to a ﬁxed order in the expansion O(pn), can be obtained bya construction guided by chiral symmetry:LχPT = LχPT2 + LχPT4 + LχPT6 + . . . + LχPTn . (1)LχPT2 embodies the spontaneous breaking of the chiral symmetry and de-pends only on two parameters: F , the decay constant of the pion, andB0F2 = −〈0|ψψ|0〉, the vacuum expectation value of the light quarks; both∗ Presented at The Final EURIDICE Meeting “Effective Theories of Colours andFlavours: from EURODAPHNE to EURIDICE”, Kazimierz, Poland, 24–27 August,2006.(2789)2790 J. Portolésof them in the chiral limit. Higher orders in the expansion bring in the in-formation from short-distance contributions that have been integrated out,for instance resonance states. As in any eﬀective ﬁeld theory (EFT) this in-formation is incorporated into the low-energy constants (LECs) that weightthe operators of the theory:LχPT4 =10∑i=1LiO(4)i , LχPT6 =∑90i=1CiO(6)i , (2)for SU(3). Explicit expressions for the operators can be read fromRefs. [3,4].The scale that speciﬁes the chiral expansion, Λχ ∼MV (being MV the massof the ρ(770), the lightest hadron not included in the theory), indicates thatLECs in χPT should receive contributions from the energy regime at orabove that scale [5]. The determination of the contributions of the lightestmultiplets of resonances to the O(p4) LECs in LχPT4 [6] has shown that theyindeed saturate the values extracted from the phenomenological analyses.As a consequence it is reasonable to think that the most important contri-bution to the LECs is provided by the energy region immediately above theintegrated scale (E ∼ Λχ).2. The role of resonance chiral theoryAs illustrated in the O(p4) case [6, 7] a procedure to systematically de-termine the resonance contributions to the LECs in χPT is available. Essen-tially the idea is to construct a Lagrangian theory in terms of resonances,pseudoscalar mesons and auxiliary ﬁelds respecting the underlying chiralsymmetry. Then, upon integration of the heavier states, the χPT Lagrangianis recovered. The outcome of this ﬁrst step is that LECs are traded bythe equally unknown couplings of the resonance Lagrangian though, at thispoint, one may also notice relations between LECs. In a second stage infor-mation on the resonance couplings is obtained, whether from phenomenol-ogy or, more interestingly, by imposing theoretical constraints from the QCDasymptotic behaviour of form factors or Green functions.Contrarily to χPT, the lack of a mass gap between the spectrum of light-ﬂavoured resonances and the perturbative continuum prevents the construc-tion of an appropriate EFT to describe the interaction of resonances andpseudoscalar mesons. However there are several tools that allow us to grasprelevant features of QCD and to implement them in an EFT-like Lagrangianmodel. The two relevant basis are the following:(i) A theoremput forward byWeinberg [1] andworked out by Leutwyler [8]states that if one writes down the most general possible Lagrangian,including all terms consistent with assumed symmetry principles, andLarge-NC Estimate of the Chiral Low-Energy Constants 2791then calculates matrix elements with this Lagrangian to any given or-der of perturbation theory, the result will be the most general possibleS-matrix amplitude consistent with analyticity, perturbative unitarity,cluster decomposition and the principles of symmetry that have beenrequired.(ii) The inverse of the number of colours of the SU(NC) gauge group canbe taken as a perturbative expansion parameter [9]. Large-NC QCDshows features that resemble, both qualitatively and quantitatively,the NC = 3 case [10]. 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 theoryin terms of SU(3) (pseudoGoldstone mesons) and U(3) (heavier resonances)ﬂavour multiplets as active degrees of freedom. This has been established[6,7,11] systematically and devises what is known as Resonance Chiral The-ory (RχT) that shows the following main features:1. The construction of the operators in the Lagrangian is guided by chiralsymmetry for the pseudoGoldstone mesons and by unitary symmetryfor the resonances. The general structure of these couplings is:O = 〈R1R2 . . . Rm χ(pn) 〉 ∈ Lm︷ ︸︸ ︷RR...R(n) , (3)where Rj indicates a resonance ﬁeld and χ(pn) is a chiral structuredtensor, involving the pseudoGoldstone mesons and auxiliary ﬁelds only.Then, the usual chiral counting in χPT [1] represented by the powerof momenta can straightforwardly be applied to χ(pn). With thesesettings chiral symmetry is preserved upon integration of the reso-nance ﬁelds and, at the same time, the low-energy behaviour of theamplitudes is guaranteed.2. Symmetries do not provide information on the coupling constants asthese incorporate short-distance dynamics not included explicitly inthe Lagrangian. The latter is supposed to bridge between the energyregion below resonances (E≪MV ) and the parton regime (E≫MV ).This hypothesis indicates that it should match both regions and it sat-isﬁes, 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 asymptotic2792 J. Portoléstrend of form factors of QCD currents is estimated from the spectralstructure of two-point functions or the partonic make-up and it is en-forced on the couplings. This heuristic strategy is well supported bythe phenomenology [10–14].With this pattern the content of the theory is, schematically, given by:LRχT = LχPT2 +∑nLGBn>2 + LR , (4)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 pseudoGoldstone modes.RχT lacks an expansion parameter. There is of course the guide pro-vided by 1/NC that translates into the loop expansion, however there is nocounting that limits the number of operators with resonances that have to beincluded in the initial Lagrangian. However the number of resonance ﬁeldsto be kept relies fundamentally in the physical system that we are interestedin and the maximum order of the chiral tensor χ(pn) in Eq. (3) is very muchconstrained by the required 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 of 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 [15].To deal with this one can include more states that may delay the appearanceof that problem.3. Resonance contributions to the O(p6) LECsIn Ref. [11] we have constructed the RχT Lagrangian needed to evalu-ate the resonance contributions to the O(p6) LECs in Eq. (2). It has thefollowing structure:LRχT = LχPT2 + LGB4 + LGB6 + LRkin + LR(2) + LR(4) + LRR(2) + LRRR(0) + Lε . (5)The inclusion of spin-1 resonances has been performed within the antisym-metric tensor formalism. However, there are independent contributions tothe O(p6) LECs coming from odd-intrinsic-parity couplings involving spin-1resonances in the Proca formalism. These are included in Lε. In the aboverepresentation it can be shown [7] that, at O(p4), all local terms in LGB4Large-NC Estimate of the Chiral Low-Energy Constants 2793have to vanish in order not to upset the asymptotic behaviour of QCD cor-relators. A corresponding result at O(p6) is still lacking but we have alsoassumed that all the couplings in LGB6 are set to zero.LRχT in Eq. (5) involves 6 a priori unknown couplings in LR(2), 70 inLR(4), 38 in LRR(2) , 7 in LRRR(0) and 3 in Lε. Some additional work provides anenormous simpliﬁcation:(i) Upon integration of resonances not all couplings appear independentlyin the LECs. In general only several combinations of couplings inter-vene and to take into account this case one can perform suitable redef-initions of the ﬁelds. This procedure spoils in general the high-energybehaviour of the theory but it is correct for the evaluation of the LECs.Indeed the 70 couplings in LR(4) reduce to 23.(ii) The next step is to enforce additional short-distance information, i.e.the leading behaviour at large momenta, for two and three-point func-tions and form factors. This procedure, set in Ref. [7], relies in well-known features of partonic scattering or asymptotic QCD [16]. Two-current correlators and associated form-factors provide 19 new con-straints on couplings, while the three-point Green functions studiedtill now: 〈VAP〉 [13] and 〈SPP〉 [11, 14], give 6 and 5 independentrestrictions, respectively.Hence we can already determine fully the resonance contribution to theO(p6) couplings C78 and C89 (that appear in π → ℓνℓγ and π → ℓνℓγ∗, re-spectively), C87 (in 〈AµAν〉), C88 and C90 (in FπV (q2) and the q2 dependenceof the form factors in Kℓ3), C38 (in 〈SS〉) and C12 and C34 (in Fπ,KV (q2) andfK0π−+ (0)).It is signiﬁcant to observe that, as shown in the analysis of the 〈SPP〉Green function, the use of a Lagrangian theory like RχT though involved [11]brings more information (encoded in the symmetries of the Lagrangian) thanthe use of a parametric ansatz [14].4. Kℓ3 decays: determination of fK0π−+(0)Kℓ3 decays have the potential to provide one of the most accurate de-terminations of the Vus CKM element. The main uncertainty in extractingthis parameter comes from theoretical calculations of the vector form factorfK0π−+ (0) deﬁned by:〈π−(p) | sγµu |K0(q) 〉 = fK0π−+ (t)(q + p)µ + fK0π−−(t) (q − p)µ , (6)2794 J. Portoléswith t = (q − p)2. Deviations of fK0π−+ (0) from unity (the octet symmetrylimit) are of second order in the SU(3) breaking:fK0π−+ (0) = 1 + fp4 + fp6 + . . . (7)The ﬁrst correction is O(p4) in χPT, through one loop calculation and nolocal terms, and it gives fp4 = −0.0227, essentially without uncertainty [17].At O(p6) two-loop, one-loop and local terms contribute and the latter makethe determination more uncertain. Loops give f loopsp6(Mρ)=0.0093(5) [14,18].The explicit form for the tree-level contribution is:f treep6 (Mρ) = 8(M2K −M2π)2F 2π[(Lr5(M2ρ ))2F 2π− Cr12(M2ρ )− Cr34(M2ρ )], (8)that involves LECs of χPT both at O(p4) and O(p6). In Ref. [14], andusing the method outlined in these proceedings, we have determined thecontribution of scalar and pseudoscalar resonances to the LECs present inEq. (8), and we get:f treep6 (Mρ) = −(M2K −M2π)22M4S(1−M2SM2P)2. (9)As can be seen in Fig. 1, it produces a tiny result: f treep6(Mρ) = −0.002(12)1 1.1 1.2 1.3 1.4 1.5-0.03-0.02-0.0100.010.02MS (GeV)f treep6 (Mρ)L5 × L5/F2pi−(C12 + C34)Fig. 1. We display f treep6 (Mρ) as a function of MS for MP = 1.3GeV (solid line).We also plot the two components: the dashed line represents the term proportionalto L5×L5, while the dotted line represents the term proportional to −(C12 +C34).The cancellation between both contributions is very large.due to a strong cancellation between both terms. We end up with the ﬁnalresult:fK0π−+ (0) = 0.984(12) . (10)Large-NC Estimate of the Chiral Low-Energy Constants 2795In Table I we compare our result with other determinations coming fromdiﬀerent sources. This comparison shows a clear pattern: both quenchedand unquenched lattice results are in very good agreement with the quarkmodel prediction by Leutwyler and Roos [17], while our analytic determina-tion of fK0π−+ (0) shows a clear tension. It is also interesting to notice thetiny modiﬁcation that unquenching produces in the lattice results though,as shown in Ref. [18], chiral logarithms are very much important in the de-termination of fK0π−+ (0). A better understanding is still required on thisissue.TABLE IComparison of different predictions for fK0pi−+ (0). The value quoted for Ref. [18]has been modified as explained in Ref. [14].Reference fK0pi−+ (0)Quark model [17] 0.961(8)Lattice (quenched) [19] 0.960(9)Lattice (unquenched) [20] 0.968(11)Lattice (unquenched) [21] 0.961(5)(Chiral + [17]) [18] 0.971(10)Kπ scalar f.f. [22] 0.974(11)Ours [14] 0.984(12)5. PerspectiveThe study of the resonance energy region is essential for the understand-ing of hadron phenomenology driven by non-perturbative QCD: hadronictau decays, ﬁnal-state interactions, kaon decays, etc.RχT is a systematic setting that allows to implement known features ofQCD into a Lagrangian framework. We have shown that it provides sensibleresults but much more work is still needed to reach a better understandingboth on the phenomenology and on the underlying ideas that join togetherin this formulation.I wish to thank M. Krawczyk, G. Pancheri and H. Czyż for the excellentorganization of the Final Euridice meeting. This work has been supported inpart by HPRN-CT-2002-00311 (EURIDICE), by the EU MRTN-CT-2006-035482 (FLAVIAnet), by MEC (Spain) under grant FPA2004-00996 and byGeneralitat Valenciana under grants ACOMP06/098 and GV05/015.2796 J. PortolésREFERENCES[1] S. Weinberg, Physica A 96, 327 (1979).[2] J. Gasser, H. Leutwyler, Annals Phys. 158, 142 (1984).[3] J. Gasser, H. 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Large-Nc estimate of the chiral low-energy constants

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