We analyse the odd-intrinsic-parity effective Lagrangian of QCD valid for
processes involving one pseudoscalar with two vector mesons described in terms
of antisymmetric tensor fields. Substantial information on the
odd-intrinsic-parity couplings is obtained by constructing the
vector-vector-pseudoscalar Green's three-point function, at leading order in
1/N_C, and demanding that its short-distance behaviour matches the
corresponding OPE result. The QCD constraints thus enforced allow us to predict
the decay amplitude omega -> pi gamma, the O(p^6) corrections to pi -> gamma
gamma and the slope parameter in pi -> gamma gamma^*.Comment: 4 pages, 1 figure. Talk given at QCD 03: High-Energy Physics
  International Conference in Quantum Chromodynamics, Montpellier, France, 2-8
  Jul 200

A. Pich

Bijnens

Ecker

Gasser

Hagiwara

J. Portolés

Knecht

Lepage

Moussallam

P.D. Ruiz-Femenía

Ruiz-Femenia

Weinberg

Wess

Witten

Nuclear Physics B - Proceedings Supplements

English

arXiv

4 páginas, 1 figura.-- Comunicación presentada en la 10ª High-Energy International Conference on Quantum ChromoDynamics celebrada del 2 al 8 de Julio de 2003 en Montpellier (Francia).-- arXiv:hep-ph/0309345v1We analyse the odd-intrinsic-parity effective Lagrangian of QCD valid for processes involving one pseudoscalar with two vector mesons described in terms of antisymmetric tensor fields. Substantial information on the odd-intrinsic-parity couplings is obtained by constructing the vector-vector-pseudoscalar Green's three-point function, at leading order in 1/NC, and demanding that its short-distance behaviour matches the corresponding OPE result. The QCD constraints thus enforced allow us to predict the decay amplitude ω → πγ, the O(p6) corrections to π → γγ and the slope parameter in π → γγ*.This work has been supported in part by TMR
EURIDICE, EC Contract No. HPRN-CT-2002-
00311, by MCYT (Spain) under grant FPA2001-
3031, and by ERDF funds from the EU.Peer reviewe

Ruiz Femenía, Pedro

Pich, Antonio

Portolés, Jorge

Digital.CSIC

arXiv:hep-ph/0309345v1  30 Sep 2003IFIC/03-43, FTUV/03-0930Phenomenology of the 〈VVP〉 Green’s function within the ResonanceChiral Theory ∗P. D. Ruiz-Femen´ıa, A. Pich and J. Portole´s aa Instituto de F´ısica Corpuscular, Universitat de Vale`ncia,Apartat Correus 22085, E-46071 Vale`ncia, SpainWe analyse the odd-intrinsic-parity effective Lagrangian of QCD valid for processes involving one pseudoscalarwith two vector mesons described in terms of antisymmetric tensor fields. Substantial information on the odd-intrinsic-parity couplings is obtained by constructing the vector-vector-pseudoscalar Green’s three-point function,at leading order in 1/NC , and demanding that its short-distance behaviour matches the corresponding OPE result.The QCD constraints thus enforced allow us to predict the decay amplitude ω → piγ, the O(p6) corrections topi → γγ and the slope parameter in pi → γγ∗.1. IntroductionEffective field theories of QCD have providedefficient ways to explore hadron dynamics in thoseregimes where we are not able to solve the full the-ory. In the very low-energy domain, chiral pertur-bation theory (χPT) [1,2] has achieved a remark-able success in describing the strong interactionsamong pseudoscalar mesons. Moving up to the1 GeV region the effects of vector resonances be-come dominant and must be accommodated inthe theory. Several works [3,4] have provideda sound procedure to include resonance stateswithin the chiral framework, namely the Reso-nance Chiral Theory (RχT). As the couplings en-tering the effective Lagrangian are not fixed bythe symmetry alone, one should rely on the phe-nomenology or, alternatively, construct theoreti-cal tools that could provide a meaningful way tocompare the results of the effective theory withthose of QCD. The pioneering work of Ref. [4]indicated that the analysis of Green’s functionsand form factors of QCD currents yields valuableinformation on the resonance sector.Recently, several authors have pushed forwardthis direction, either by using a Lagrangian withexplicit resonance degrees of freedom or withinthe framework of the lowest meson dominance∗Talk given by P. D. Ruiz-Femen´ıa at the High-EnergyPhysics International Conference in Quantum Chromody-namics (QCD 03), Montpellier, France, 2–8 July 2003.(LMD) approximation to the large number ofcolours (NC) limit of QCD [5–9]. In particular,the authors of Ref. [5] undertook a systematicstudy of several QCD three-point functions whichare free of perturbative contributions from QCDat short distances. Therefore, their OPE expan-sion should be more reliable when descending toenergies close to the resonance region. Under thishypothesis, it was shown [5] that while the ansatzderived from the LMD approach automaticallyincorporates the right short-distance behaviour ofQCD by construction, the same Green’s functionsas calculated with a resonance Lagrangian, in thevector-field representation, are incompatible withthe OPE outcome. Moreover the authors put for-ward that these discrepancies cannot be repairedjust by introducing local counterterms from thechiral Lagrangian L(6)χ , as it was done at O(p4)[4]. This result severely questions the usefulnessof the resonance effective theory beyond the ini-tial work of Ref. [4], and deserves further investi-gation.With this aim, we have reanalysed the vector-vector-pseudoscalar three-point function, thistime with the vector mesons described in terms ofantisymmetric tensor fields. This requires the in-troduction of an odd-intrinsic-parity effective La-grangian in the formulation of Ref. [3] contain-ing all allowed interactions between two vectorobjects (currents or resonances) and one pseu-doscalar meson. The details of the calculationcan be found in Ref. [10].2. RχT and the odd-intrinsic-parity sectorThe low-energy behaviour of QCD for the lightquark sector is ruled by the spontaneous breakingof chiral symmetry. The corresponding effectiverealization of QCD describing the interaction be-tween the Goldstone fields is χPT given, atO(p2),byL(2)χ = (F 2/4) 〈uµuµ + χ+〉 . (1)The inclusion of resonances as explicit degreesof freedom in the chiral framework was carriedout in Ref. [3] for the even–intrinsic–parity sector(LV). For the odd–intrinsic–parity sector, threedifferent sources might contribute to the 〈VVP〉Green’s function :(i) the Wess-Zumino action ZWZ[v, a] [11],which is of O(p4) and fulfills the chiralanomaly,(ii) chiral invariant ǫµνρσ terms involving vec-tor mesons. Within the antisymmetric for-malism, the basis of odd-intrinsic-parity op-erators which comprise all possible verticesinvolving two vector resonances and onepseudoscalar (VVP), and vertices with onevector resonance and one external vectorsource plus one pseudoscalar (VJP) reads:O1VJP= ǫµνρσ 〈 {V µν , fρα+ }∇αuσ 〉 ,O2VJP= ǫµνρσ 〈 {V µα, fρσ+ }∇αuν 〉 ,O3VJP= i ǫµνρσ 〈 {V µν , fρσ+ }χ− 〉 ,O4VJP= i ǫµνρσ 〈V µν [ fρσ− , χ+] 〉 ,O5VJP= ǫµνρσ 〈 {∇αV µν , fρα+ }uσ 〉 ,O6VJP= ǫµνρσ 〈 {∇αV µα, fρσ+ }uν 〉 ,O7VJP= ǫµνρσ 〈 {∇σV µν , fρα+ }uα 〉 , (2)O1VVP= ǫµνρσ 〈 {V µν , V ρα}∇αuσ 〉 ,O2VVP= i ǫµνρσ 〈 {V µν , V ρσ}χ− 〉 ,O3VVP= ǫµνρσ 〈 {∇αV µν , V ρα}uσ 〉 ,O4VVP= ǫµνρσ 〈 {∇σV µν , V ρα}uα 〉 . (3)The corresponding resonance Lagrangianwill thus be defined asLoddV =7∑a=1caMVOaVJP+4∑a=1daOaVVP . (4)(iii) the relevant operators in the O(p6) Gold-stone chiral Lagrangian [12]. The success-ful resonance saturation of the chiral La-grangian couplings atO(p4) [3] might trans-late naturally to O(p6) couplings too, im-plying that they are generated completelythrough integration of vector resonances.Accordingly we do not include L(6)odd in ourevaluation.In summary we will proceed in the following byconsidering the relevant effective resonance the-ory (ERT) given by :ZERT[v, a, s, p] = ZWZ[v, a] + ZoddVχ [v, a, s, p] , (5)where ZoddVχ is generated by L(2)χ , LV and LoddV .3. Short-distance information on the odd-intrinsic-parity couplingsThe vector-vector-pseudoscalar QCD three-point function 〈VVP〉 is built from the octet vec-tor current and the octet pseudoscalar density,(ΠVVP)(abc)µν (p, q) =∫d4x∫d4y ei(p·x+q·y)×〈0|T [V aµ (x)V bν (y)P c(0) ] |0〉= ǫµναβ pαqβdabcΠVVP(p2, q2, r2) , (6)with the four-vector r = −(p+ q).When both momenta p, q in ΠVVP become si-multaneously large, the QCD calculation withinthe OPE framework gives, in the chiral limit andup to corrections of O(αs), [7]:limλ→∞ΠVVP ((λp)2, (λq)2, (λr)2)=−〈ψ¯ψ〉02λ4p2 + q2 + r2p2q2r2+O(1λ6), (7)VaqVbpPr=+ ++ +VaP+VbP+VaVbFigure 1. Diagrams entering the calculation ofthe VVP 3-point function with the ERT action.Double lines represent vector resonances, singlelines are short for pseudoscalar mesons.where 〈ψ¯ψ〉0 is the single flavour bilinear quarkcondensate. At leading order in the 1/NC ex-pansion of QCD, the three-point correlator in theeffective resonance theory given by ZERT is evalu-ated from the tree-level diagrams shown in Fig. 1.The LMD approximation, which assumes that asingle resonance in each channel saturates the re-quirements of QCD, is sufficient to satisfy theshort–distance constraint (7) up to order 1/λ4,provided the following conditions among the LoddVcouplings hold:4 c3 + c1 = 0 ,c1 − c2 + c5 = 0 ,c5 − c6 = NC64π2MV√2FV,d1 + 8 d2 = − NC64π2M2VF 2V+F 24F 2V,d3 = − NC64π2M2VF 2V+F 28F 2V. (8)As the couplings of the Effective Lagrangian donot depend on the masses of the Goldstone fieldsthe constraints above apply for non–zero pseu-doscalar masses too.Actually our 〈VVP〉 three-point function fullyreproduces the LMD ansatz suggested in Ref. [7] :ΠresVVP = −〈ψ¯ψ〉02(p2 + q2 + r2)− NC4pi2M4VF 2(p2 −M2V )(q2 −M2V )r2, (9)which has been successfully tested in previousworks [7,9]. The authors of Ref. [5] found thatthe same agreement with the short-distance QCDbehaviour could not be reached working with theresonance Lagrangian in the vector representa-tion, not even at the expense of introducing localcontributions from the O(p6) chiral Lagrangian.They then suggested that the problem may be in-herent to the effective Lagrangian approach andunlikely to be fixed just by using other represen-tations for the resonance fields; our result, de-rived in the antisymmetric tensor-field formula-tion with an odd-intrinsic-parity sector, contra-dicts this assertion, at least in what concerns the〈VVP〉 Green’s function.4. Phenomenology of intrinsic-parity vio-lating processes4.1. ω → piγAt tree-level, the intrinsic-parity violating tran-sition ω → πγ receives contributions from boththe VJP and VVP terms of LoddV (direct and ρ-mediated diagrams respectively). If we plug inthe QCD constraints, Eq. (8), we find a full pre-diction for this process :Γ(ω → πγ) = α192Mω(1− m2piM2ω)3×[NC4π2M2ωF 2− M2ωM2V(1 +m2piM2ω) ]2. (10)The direct and the ρ exchange diagrams almostcontribute to similar extent to this process. Thismeans that contrary to what we would expectfrom VMD, the ωρπ coupling does not saturatethe decay ω → πγ. This has immediate conse-quences to other channels where VMD alone wasthought to be the relevant mechanism of decay,as in ω → π+π−π0, where the direct amplitudecompetes in size with the intermediate meson ex-change term [10].Varying the parameter F from the bare valueF0 ≃ 87 MeV to the dressed one (i.e. thepion decay constant), Fpi ≃ 92.4 MeV [13], weget that Γ(ω → πγ) ranges from 0.703 MeV to0.524 MeV, with MV = Mρ = 771.1 MeV andMω = 782.6 MeV [13]. This 5–30% deviationfrom the experimental value, Γ(ω → πγ)|exp =(0.734±0.035)MeV, is in accordance with the ex-pected size of next-to-leading 1/NC corrections.Also the ρ → πγ decay widths, related withω → πγ by a SU(3)V-symmetry factor, are ex-tracted from the analysis [10].4.2. pi → γγIn the chiral limit, the amplitude for the π →γγ process is non-vanishing and exactly predictedby the ABJ anomaly. The odd-intrinsic-parity in-teractions among vector resonances introduced inSection 2 generate O(p6) chiral corrections to thisprocess. Only the two-resonance driven diagramsurvives after the short-distance conditions areapplied. The correction induced into the π → γγwidth gives :Γ(π → γγ) = α264 π3 F 2m3pi [ 1 − ∆]2 , (11)where∆ =4π23F 2M2Vm2piM2V≃ 0.006 . (12)This result provides a tiny 1% correction to thewidth, and it is perfectly compatible with the ex-perimental uncertainty, Γ(π → γγ)|exp = (7.7 ±0.6) eV.4.3. pi → γγ∗The π → γγ∗ amplitude is usually written asa slope parameter α which modifies the on-shellbehaviour:Mpi→γγ∗ = Mpi→γγ(1 + αk∗2), (13)where k∗ is the off-shell photon momentum. Theinteractions contained in LoddV yield a contribu-tion to the parameter α that amountsαodd =1M2V[1− 4π2F 2NCM2V]≃ 1.36 GeV−2 ,which is smaller than the VMD estimate, αVMD =1/M2V ≃ 1.68 GeV−2. The chiral loops contribu-tions to this slope, αχ ≃ 0.26 GeV−2 were calcu-lated in Ref. [14]. We can add both contributionsto get m2piα ≃ 0.029, to be compared with theaveraged value m2piα|exp = 0.032 ± 0.004 in thePDG [13]. αodd has been extended beyond theLMD approximation by the inclusion of a secondvector resonance into the 〈VVP〉 ansatz, Eq. (9),in Ref. [5]. The latter is in fact needed to havethe right 1/k∗2 behaviour for large k∗ [15,16] inthe form factor Fpiγγ∗(k∗).AcknowledgementsWe wish to thank S. Narison and his teamfor the organization of the QCD 03 conference.This work has been supported in part by TMREURIDICE, EC Contract No. HPRN-CT-2002-00311, by MCYT (Spain) under grant FPA2001-3031, and by ERDF funds from the EU.REFERENCES1. S. Weinberg, Physica 96A (1979) 327.2. J. Gasser and H. Leutwyler, Ann. of Phys.(NY) 158 (1984) 142.3. G. Ecker, J. Gasser, A. Pich and E. de Rafael,Nucl. Phys. B321 (1989) 311.4. G. Ecker, J. Gasser, H. Leutwyler, A. Pichand E. de Rafael, Phys. Lett. B223 (1989)425.5. M. Knecht and A. Nyffeler, Eur. Phys. J. C21 (2001) 659.6. J. Bijnens, E. Gamiz, E. Lipartia andJ. Prades, JHEP 0304 (2003) 055.7. B. Moussallam, Phys. Rev. D 51 (1995) 4939.8. B. Moussallam, Nucl. Phys. B 504 (1997)381.9. M. Knecht, S. Peris, M. Perrottet and E. deRafael, Phys. Rev. Lett. 83 (1999) 5230.10. P. D. Ruiz-Femenia, A. Pich and J. Portole´s,JHEP 0307 (2003) 003.11. J. Wess and B. Zumino, Phys. Lett. B37(1971) 95;E. Witten, Nucl. Phys. B 223 (1983) 22.12. J. Bijnens, L. Girlanda and P. Talavera, Eur.Phys. J. C 23 (2002) 539.13. K. Hagiwara et al. [Particle Data Group Col-laboration], Phys. Rev. D 66 (2002) 010001.14. J. Bijnens, A. Bramon and F. Cornet, Phys.Rev. Lett. 61 (1988) 1453.15. Joaquim Prades, private communication.16. G. P. Lepage and S. J. Brodsky, Phys. Lett.B87 (1979) 359.

Phenomenology of the <VVP> Green's function within the resonance chiral theory

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