A simple matching procedure is proposed to extract constraints on effective
meson theories. In this way, a QCD prediction for the pion decay constant is
found, F(pi)=2 m(pi)/pi, i.e. approximately 90MeV. The same procedure also
determines other mesonic observables, like the decay width of the sigma meson
to two photons. Finally, some information which can be gained about the
hadronic light-by-light contributions to the muon anomalous moment are briefly
commented.Comment: 6 pages, 2 eps figure

Smith, Christopher

English

arXiv

A simple matching procedure is proposed to extract constraints on effective
meson theories. In this way, a QCD prediction for the pion decay constant is
found, F(pi)=2 m(pi)/pi, i.e. approximately 90MeV. The same procedure also
determines other mesonic observables, like the decay width of the sigma meson
to two photons. Finally, some information which can be gained about the
hadronic light-by-light contributions to the muon anomalous moment are briefly
commented.
 Comment: 6 pages, 2 eps figure

DIAL UCLouvain

arXiv:hep-ph/0207284v1  23 Jul 2002Very low energy matching of effectivemeson theories with QCDChristopher SmithInstitut de Physique The´orique, Universite´ Catholique de Louvain,Chemin du Cyclotron, 2, B-1348, Louvain-la-Neuve, BelgiumJuly 22, 2002AbstractA simple matching procedure is proposed to extract constraints oneffective meson theories. In this way, a QCD prediction for the piondecay constant is found, Fpi = 2mpi/pi ≈ 90 MeV. The same procedurealso determines other mesonic observables, like the decay width of thesigma meson to two photons.Finally, some information which can be gained about the hadroniclight-by-light contributions to the muon anomalous moment are brieflycommented.1When the electron field is integrated out from the QED action, onegets an effective theory where only photons can propagate. In addition,virtual fermion loops generate an infinite tower of self-interactions amongthe photons. Schwinger calculated the one-loop effective action, for constantelectromagnetic fields [1], to all ordersL(1)eff,fermion = −18pi2∫∞0dττ3e−m2τ[(eτ)2 abcosh (eaτ) cos (ebτ)sinh (eaτ) sin (ebτ)− 1](1)with a, b solutions of a2−b2 = E2−B2 and ab = E ·B. When expanded withrespect to the fermion mass (or α), one gets the four-photon interactions asdescribed by the well-known Euler-Heisenberg effective Lagrangian [2]L(1)EH,fermion =α290m4((FµνFµν)2 +74(Fµν F˜µν)2)with the definition F˜µν = 12εµνρσFρσ. Diagrammatically, the integration ofthe fermion fields can be represented asThe same treatment can be applied to get the photon effective couplingsgenerated by virtual quark loopsL(1)EH,quarks = e4Qα2Nc90m4Q((FµνFµν)2 +74(Fµν F˜µν)2)(2)where Nc is the number of colors, eQ is the quark charge (in unit of e) andmQ its mass.The starting point of the present letter is to assume that the relativestrength of the (FµνFµν)2 and (Fµν F˜µν)2 couplings is preserved through thecomplicate dressing by QCD of the quarks into hadrons. In other words, thehypothesis is that strong corrections will be absorbed into the quark mass,the only free parameter. The consequence is then that the same effectiveinteraction among the photons should be obtainable starting from an effec-tive meson theory. By matching the photon effective theories obtained by2integrating out meson fields to that obtained by integrating out the quarkfields (2), we will get a set of constraints on the parameters of the effectivemeson theory. Ultimately, the validity of the basic hypothesis will be testedby comparison with experiment.At very low energy, a few eV say, the contribution from pions will dom-inate. We will use the scalar QED Lagrangian for the charged pionsLpi± = ∂µpi+∂µpi− −m2pipi+pi− − ieAµ(pi+∂µpi−− ∂µpi+pi−)+ e2AµAµpi+pi−and for the neutral pions, we introduce the coupling to two photonsLpi0 =12[∂µpi0∂µpi0 −m2pipi0pi0]+ gpiFµν F˜µνpi0We now integrate out the pion fields from Lpi±+Lpi0. This generates contactinteractions among photons:(it is understood that seagull interaction contributions are included).Schwinger computed the effective action obtained by integrating out thecharged pions, with the result [1]L(1)eff,pi±=116pi2∫∞0dττ3e−m2τ[(eτ)2 ab1sinh (eaτ) sin (ebτ)− 1](3)When expanded, the four-photon effective couplings are generatedL(1)EH,pi±=α290m4pi±(716(FµνFµν)2 +116(Fµν F˜µν)2)(4)3To this effective Lagrangian, we add the effective interaction generated bya neutral pion exchangeL(1)EH,pi0=g2pi2m2pi0(Fµν F˜µν)2(5)Combining (4) with (5), we getL(1)EH,pi =716α290m4pi±(FµνFµν)2 +(g2pi2m2pi0+116α290m4pi±)(Fµν F˜µν)2(6)The core of the method is to match this effective Lagrangian to thefermionic one (2), to extract a prediction for gpig2pi2m2pi0+ 116α290m4pi±716α290m4pi±=74→ gpi =α8mpi0m2pi±For the decay rate pi0 → γγ, this givesΓ(pi0 → γγ)=m3pi04pig2pi =α2256pim5pi0m4pi±= 7.7 eVTo be compared to the experimental value Γexp(pi0 → γγ)= (7.7± 0.6) eV[3]. The agreement is very good.If the description of pi0 → γγ in terms of the axial anomaly is used ([4],[5]), we get a determination of Fpigpi =αNc12piFpi=α8mpi0m2pi±→ Fpi =2Nc3pim2pi±mpi0= 91.9 MeV (7)again very close to the experimental value Fpi = 92.4 ± 0.3 MeV (obtainedfrom Γ (pi+ → µ+νµ), see [3]).A comment is in order. Usually, in chiral perturbation theory, the roleplayed bympi and Fpi is radically different: Fpi sets the scale of the Goldstoneboson interactions, while mpi is only a small explicit breaking of the sponta-neously broken symmetry; their physical content is therefore quite differentand a priori unrelated. On the other hand, if the fermionic character of theunderlying theory is assumed to be preserved through the passage from thequark picture to the hadron picture, the two turn out to be proportional.4As a by-product, we can also estimate the value of the constituent quarkmass for which the two descriptions match, i.e. for which the four-photoncouplings are the same in absolute magnitude. Setting mu = mde4uα2Nc90m4u+ e4dα2Nc90m4d=716α290m4pi±→ mu = mpi±4√(e4u + e4d) 487≈ 1.1×mpi±i.e. a constituent quark mass mu = md ≈ 153 MeV, quite close to mpi.Introducing Higher Mass ParticlesIn principle, one can introduce the η, η′,K±, ..., to get the effective La-grangianL(1)EH,PS =(g2pi2m2pi0+g2η2m2η+g2η′2m2η′+α21440(1m4pi±+1m4K±))(Fµν F˜µν)2+7α21440(1m4pi±+1m4K±)(FµνFµν)2 (8)with gη(′) the coupling constant for η(′) → γγ. We see that the pion con-tributions are by far the dominant ones: the corrections induced by heaviermesons are of a few percents, as can be seen by plugging in the experimen-tal values gη ≈ 6 × 10−6 MeV −1 and gη′ ≈ 8 × 10−6 MeV −1 [3]. Matching(8) to (2) and solving for Fpi, we find Fpi ≈ 96 MeV. This shows that thematching should not be expected to work to a better accuracy than roughly5%, despite the striking result (7).We now turn to the introduction of resonances, and in particular of thesigma meson. Our point of view is to consider the sigma as a resonanceoccurring in the two-pion channel. We take as effective LagrangianL(1)EH,pi0,σ=g2σ2m2σ(FµνFµν)2 +g2pi2m2pi0(Fµν F˜µν)2(9)with gσ the coupling constant for σ → γγ. To get this form, we assume thatin a first approximation, the scalar channel (i.e. (FµνFµν)2) is saturated bythe sigma, and we neglect the charged pion contribution to the pseudoscalarchannel (Fµν F˜µν)2 (which is roughly 10 times smaller than the neutral pion5contribution, see (6)). The effective photon Lagrangian (9) is now onlygenerated by virtual pi0 and σ exchanges, which are particles respectivelyassociated with the axial [4] and trace anomaly [6].By matching (9) with (2), we can relate gσ to gpig2σ/2m2σg2pi/2m2pi0=47→gσgpi=√47mσmpi0≈ 2.8 (10)for mσ ≈ 530 MeV [7]. From (10), the sigma width to two photons ispredicted to beΓ (σ → γγ) =m3σ4pig2σ =17pim5σm2pi0g2pi =α2448pim5σm4pi±≈ 4.1+3.5−2.1 keV (11)with mσ = (530± 70) MeV [7]. This is compatible withΓ (σ → γγ) = (3.8± 1.5) keV (12)found by a partial-wave analysis of γγ → pipi [8]. Note, by the way, thatbecause mσ appears to the fifth power in (11), this formula is in fact quiteefficient in constraining mσ once Γ (σ → γγ) is known. For instance, fromthe experimental value (12), one finds mσ = 520+36−49 MeV.In conclusion, the assumption that the sigma resonance nearly saturatesthe two-pion scalar channel gives a reasonable prediction for Γ (σ → γγ).Conclusions and PerspectivesIn this letter, we have shown how to get information on the coupling con-stant of effective meson theories from QCD. As said, the present frameworkrelies entirely on the assumption that the strong interactions do not renor-malize the relative strength of the (FµνFµν)2 and (Fµν F˜µν)2 couplings (orat least that this renormalization is small). The fact that we found reason-able predictions for Fpi and Γ (σ → γγ) seems to validate this assumption.Further work should clarify the range of validity of the present method.Encouraged by the success of the matching at O(α2/m4), one couldnow undertake an analysis at the next order. At O(α3/m8), the fermioniceffective Lagrangian is [1], [9]L(1)DKR,quarks = e6QNcpiα3315m8Q(−4 (FµνFµν)3 −132(FαβFαβ)(Fµν F˜µν)2)6At that order, the integration of the pseudoscalars is more involved, and itremains to be seen which kind of constraints can emerge. Also, the extensionto non-constant electromagnetic fields may lead to interesting results. Allthese questions are left for future studies.We close this letter with a comment on the computation of the hadroniclight-by-light corrections to the muon anomalous moment. Even if it is truethat the momentum configuration in that case and in our case is quite differ-ent, our approach can offer an interesting limiting case in which the varioustheoretical models can be tested. For instance, in the present approach, itappears that the pions and the constituent quarks do not contribute simul-taneously. The same is true for the sigma meson and the charged pions. Inthe context of the muon anomalous moment, the constituent quark, chargedpion and sigma meson contributions are all considered at the same time (seefor example [10]). Whether this leads to double-counting or not is, in ouropinion, not settled.Acknowledgments: I am grateful to J. Pestieau and S. Trine for usefuldiscussions. I acknowledge financial support from the I.I.S.N.References[1] J. Schwinger, Phys. Rev. 82, 664 (1951).[2] H. Euler, W. Heisenberg, Z. Phys. 98, 714 (1936).[3] D.E. Groom et al, Eur. Phys. J. C15, 1 (2000).[4] S. L. Adler, Phys. Rev. 177, 2426 (1969).[5] J.S. Bell, R. Jackiw, Nuovo Cim. 60A, 46 (1969).[6] M. S. Chanowitz, J. Ellis, Phys. Rev. D7, 2490 (1973).[7] M. Ishida, S. Ishida, T. Komada and S.-I. Matsumoto, proceedings of9th International Conference on Hadron Spectroscopy (Hadron 2001),Protvino, Russia (hep-ph/0110358 ), and references cited there.[8] M. Boglione, M.R. Pennington, Eur. Phys. J. C9, 11 (1999).[9] D. Dicus, C. Kao, W. Repko, Phys. Rev. D57, 2443 (1998)[10] E. de Rafael, Phys. Lett. B322, 239 (1994); M. Hayakawa, T. Kinoshita,A. I. Sanda, Phys. Rev. D54, 3137 (1996).7

Very low energy matching of effective meson theories with QCD

Abstract

Similar works

Full text

Available Versions

DIAL UCLouvain