We study a model-independent parameterization of the vector pion form factor
that arises from the constraints of analyticity and unitarity. Our description
should be suitable up to s^(1/2) ~ 1.2 GeV and allows a model-independent
determination of the mass of the rho(770) resonance. We analyse the
experimental data on tau^- -> pion^- pion^0 nu_tau and e^+ e^- -> pion^+ pion^-
in this framework, and its consequences on the low-energy observables worked
out by chiral perturbation theory. An evaluation of the two pion contribution
to the anomalous magnetic moment of the muon, a_{mu}, and to the fine structure
constant, alpha(M_Z^2), is also performed.Comment: 5 pages, 2 figures. To appear in the proceedings of the High-Energy
  Physics International Conference on Quantum Chromodynamics QCD02, Montpellier
  (France), 2-9 July (2002

A. Pich

Abreu

Ackerstaff

Akhmetshin

Anderson

Asner

Barate

Bijnens

Cirigliano

De Troconiz

Ecker

Gasser

Gomez Dumm

Guerrero

Hagiwara

J. Portolés

Ochs

Pich

Portolès

Urech

Weinberg

English

arXiv

4 páginas, 2 figuras, 3 tablas.-- Comunicación presentada a la 9ª High-Energy Physics International Conference on Quantum ChromoDynamics (QCD 02) celebrada del 2 al 9 de Julio de 2002 en Montpellier (Francia).-- arXiv:hep-ph/0209224v1We study a model-independent parameterization of the vector pion form factor that arises from the constraints of analyticity and unitarity. Our description should be suitable up to √s  1.2 GeV and allows a model-independent determination of the mass of the (770) resonance. We analyse the experimental data on τ− → π−π0ντ and e+e− → π+π− in this framework, and its consequences on the low-energy observables worked out by chiral perturbation theory. An evaluation of the two pion contribution to the anomalous magnetic moment of the muon, aμ, and to the fine structure constant, α(MZZ2), is also performed.This work has
been supported in part by TMR, EC Contract
No. ERB FMRX-CT98-0169, by MCYT (Spain)
under grant FPA-2001-3031 and by ERDF funds
from the EU Commission.Peer reviewe

Pich, Antonio

Portolés, Jorge

Digital.CSIC

arXiv:hep-ph/0209224v1  19 Sep 20021Vector form factor of the pion : A model–independent approach ∗A. Pich a and J. Portole´s aaDepartament de F´ısica Teo`rica, IFIC, CSIC-Universitat de Vale`ncia,Edifici d’Instituts d’Investigacio´, Apt. Correus 22085, E-46071 Vale`ncia, SpainWe study a model–independent parameterization of the vector pion form factor that arises from the constraintsof analyticity and unitarity. Our description should be suitable up to√s ≃ 1.2GeV and allows a model–independent determination of the mass of the ρ(770) resonance. We analyse the experimental data on τ− →pi−pi0ντ and e+e− → pi+pi− in this framework, and its consequences on the low–energy observables worked outby chiral perturbation theory. An evaluation of the two pion contribution to the anomalous magnetic moment ofthe muon, aµ, and to the fine structure constant, α(M2Z), is also performed.1. IntroductionMatrix elements of QCD hadron currents inexclusive processes provide, from a phenomeno-logical point of view, a detailed knowledge onthe hadronization mechanisms. Their evaluation,however, is a long–standing problem due to thefact that it involves strong interactions in anenergy region driven by non–perturbative QCD.Within this framework semileptonic processes, asexclusive hadronic τ decays (τ− → H−ντ ) orhadronic cross sections out of electron–positronannihilation (e+e− → H0), furnish an excellentdynamical system to explore. In the StandardModel their amplitudes are generically given byM = C LµHµ , (1)where C is a factor containing the relevant cou-plings, Lµ is the leptonic matrix element, easilycalculable within the theory, andHµ = 〈H | Jµ eiLstrong | 0 〉 , (2)with Jµ the vector Vµ (e+e− → H0) or leftVµ − Aµ (τ− → H−ντ ) hadron current. Sym-metries help us to define a decomposition of Hµin terms of the allowed Lorentz structure of im-plied momenta and a set of functions of Lorentz∗IFIC/02−42 and FTUV/02-0919 reports. To appearin the proceedings of the High–Energy Physics Inter-national Conference on Quantum Chromodynamics, 2–9July (2002), Montpellier (France).invariants, the form factors FHi ,Hµ =∑i( . . . )iµ︸ ︷︷ ︸Lorentz struc.FHi (q2, . . .) . (3)Form factors are the goal of the hadronic matrixelements evaluation and, as can be noticed fromthe definition of Hµ in Eq. (2), are a strong in-teraction related problem in a non–perturbativeregime.In the last years experiments like ALEPH,CLEO-II, DELPHI, OPAL and CMD-2 [1,2,3,4]have provided and important amount and qual-ity of experimental data on exclusive channelswhich phenomenological analysis is now manda-tory. However most of these analyses are carriedout within modelizations (including simplifyingassumptions which may be are not well controlledfrom QCD itself [5]) that, while of importance toget an understanding of the involved dynamics,could give a delusive interpretation of data.The use of effective actions from QCD suppliesa powerful model–independent procedure to workwith. At very low energies [E ≪ Mρ, with Mρthe mass of the ρ(770) resonance] the most im-portant QCD feature is its chiral symmetry thatis realized in chiral perturbation theory (χPT) [6]with a long and successful set of predictions bothin strong and electroweak processes [7]. At higherenergies [E ∼ Mρ] resonance chiral theory is theanalogous effective theory [8], where the lightestresonance fields are kept as explicit degrees of2freedom. With the addition of dynamical con-straints coming from short–distance QCD, reso-nance chiral theory becomes a predictive model–independent approach. This framework can becombined with S–matrix theory properties. Ongeneral grounds local causality of the interactiontranslates into the analyticity properties of ampli-tudes and, correspondingly, of form factors. Be-ing analytic functions in complex variables the be-haviour of form factors at different energy scales isrelated and, moreover, they are completely deter-mined by their singularities. Dispersion relationsembody rigorously these properties and are theappropriate tool to enforce them.In this note we recall our work [9] on the vectorform factor of the pion in the model–independentapproach we have just sketched. We performa numerical analysis of the recent e+e− CMD-2 data [1] and we reanalyse the τ ALEPH data[2] when corrected by isospin breaking effects [10].The output of these analyses is a determinationof the ρ(770)±,0 masses, the low–energy param-eters of the vector pion form factor, data on theω(782) resonance, particularly the ρ− ω mixing,and a new evaluation of the two–pion contribu-tion to the anomalous magnetic moment of themuon aµ and to ∆α(M2Z).2. Vector form factor of the pionThe pion vector form factor, FV (s) is definedthrough〈pi+(p′)pi−(p) |V 3µ | 0 〉 = (p− p′)µ FV (s) , (4)where s = q2 = (p + p′)2 and V 3µ is the thirdcomponent of the vector current associated to theSU(3) flavour symmetry of the QCD Lagrangian.This form factor drives the isovector hadronicpart of e+e− → pi+pi− and, in the isospin limit, ofτ− → pi−pi0ντ . At very low energies, FV (s) hasbeen studied in the χPT framework up to O(p6)[11,12]. A successful study at the ρ(770) energyscale has been carried out in the resonance chiraltheory in Ref. [13].Analyticity and unitarity properties of FV (s)tightly constrain, on general grounds, the struc-ture of the form factor [9,13]. Elastic unitarityand Watson final–state theorem relate the imagi-nary part of FV (s) to the partial wave amplitudet11 for pipi elastic scattering, with angular momen-tum and isospin equal to one, asImFV (s+ iε) = eiδ11 sin(δ11)FV (s)∗ , (5)that shows that the phase of FV (s) must be δ11 .Thus analyticity and unitarity properties of theform factor are accomplished by demanding thatit should satisfy a n–subtracted dispersion rela-tion with the Omne`s solution [13]FV (s) = exp{n−1∑k=0skk!dkdsklnFV (s)|s=0+snpi∫ ∞4m2pidzznδ11(z)z − s− iε}. (6)This solution is strictly valid only below the in-elastic threshold (s < 16m2pi), however highermultiplicity intermediate states are suppressed byphase space and ordinary chiral counting. Theδ11(s) phase–shift, in Eq. (6), is rather well known,experimentally, up to E ∼ 2GeV [14].With an appropriate number of subtractions wecan parameterize FV (s) with the subtraction con-stants appearing in the first term of the exponen-tial in Eq. (6). In Ref. [9] we have used threesubtractions :FV (s) = exp{α1s +12α2s2+s3pi∫ Λ24m2pidzz3δ11(z)z − s− iε}, (7)where we have introduced an upper cut in theintegration, Λ. This cut–off has to be taken highenough not to spoil the, a priori, infinite intervalof integration, but low enough that the integrandis well known in the interval. The two subtractionconstants α1 and α2 (a third one is fixed by thenormalization FV (0) = 1) are related with thesquared charge radius of the pion 〈r2〉piV and thequadratic term cpiV in the low–energy expansionFV (s) = 1 +16〈r2〉piV s + cpiV s2 + O(s3). (8)The input of the δ11(s) phase–shift is includedas follows. Resonance chiral theory and vector3Source Mρ±(MeV) Mρ0(MeV)Our fit 775.9± 0.5 777.8± 0.7Ref. [17] 773.8± 0.6 772.6± 0.5Average [18] 775.9± 0.5Table 1Comparison of our results for Mρ± and Mρ0 withother recent figures. The average value [18] cor-responds to e+e− and τ data analyses only.0.751√ s  (GeV)0.511.5log |F V|2Our fitCMD−2 dataFigure 1. Comparison of experimental data frome+e− → pi+pi− by CMD-2 [1] with the results ofour fit.meson dominance provide a model–independentanalytic expression that describes properly theρ(770) contribution [13]δ11(s) = arctan{MρΓρ(s)M2ρ − s}, (9)with Γρ(s) the off–shell ρ(770) width as computedin Ref. [15]. This phase–shift is accurate up toE ∼ 1GeV. At higher energies heavier reso-nances with the same quantum numbers pop upand we use the available experimental data fromOchs [14].FV (s) endows the hadronic dynamics in thee+e− → pi+pi− process and, in the isospin limit inthe τ− → pi−pi0ντ decay. In Ref. [9] we analysedthe τ ALEPH [2] data where radiative correctionswere not taken into account. In this note wereanalyse these data when corrected for isospinSource 〈r2〉piV (GeV−2) cpiV (GeV−4)Our fit (τ) 11.0± 0.3 3.84± 0.03Our fit (e+e−) 11.5± 0.2 3.73± 0.02O(p6)χPT [12] 11.22± 0.41 3.85± 0.60Ref. [17] 11.17± 0.05 3.60± 0.03Table 2Comparison of our results for the low–energy pa-rameters of the pion form factor with other recentfigures.0.5 0.75 1 1.25√s  (GeV)01log |F V|2CLEO−II data ALEPH data Our fitGuerrero and Pich (Mρ = 0.776 GeV)Figure 2. Comparison of experimental data fromτ− → pi−pi0ντ decays by ALEPH [2] and CLEO-II [3] with the prediction of Ref. [13] and our fitto the ALEPH data.breaking effects recently computed [10]. Analo-gously we study the recent experimental data one+e− → pi+pi− en the ρ(770) energy region byCMD-2 [1]. To analyse the e+e− data we need toinclude the effect of the ω(782)→ pi+pi− processin our form factor. This we do through a ρ − ωmixing term defined as in Ref. [16].The results of our fit to CMD-2 data, withχ2/dof = 45.2/37, are shown in Fig. 1 whilethe Mρ0 mass and low–energy parameters aregiven in Tables 1 and 2. In addition we getMω = (781.8 ± 0.3)MeV, Γω = (9.3 ± 1.6)MeVand Θρω = (−3.3±0.5)×10−3GeV2. The analy-sis of the τ ALEPH data gives the results shownin Fig. 2. The fit has a χ2/dof = 30.2/21 andthe values ofMρ± and the low–energy parameterscan also be read in Tables 1 and 2. We also get4√smax(GeV) apipiµ |τ × 1010 apipiµ |e+e− × 10100.5 55.9± 0.5 56.7± 0.60.9 488± 7 475± 51.0 507± 7 490± 61.1 513± 8 494± 6Table 3Results for the two–pion contribution to aµ fromthe analyses of τ and e+e− data and for differentvalues of the√smax cut–off.√smax(GeV) 104∆α(M2Z)|τ 104∆α(M2Z)|e+e−0.9 31.9± 0.5 30.7± 0.41.0 34.0± 0.5 32.4± 0.41.1 34.8± 0.6 33.0± 0.5Table 4Results for the two–pion contribution to ∆α(M2Z)from the analyses of τ and e+e− data and fordifferent values of the√smax cut–off.∆Mρ±−ρ0 = (−1.9 ± 0.9)MeV and ∆Γρ±−ρ0 =(−0.2 ± 0.6)MeV, to be compared with the fig-ures in Ref. [18] : ∆Mρ±−ρ0 = (−0.4± 0.8)MeVand ∆Γρ±−ρ0 = (0.1± 1.9)MeV.Finally, in Tables 3 and 4 we show the re-sults for the two–pion vacuum polarization con-tribution to the anomalous magnetic moment ofthe muon, apipiµ and to the shift in the fine–structure constant ∆α(M2Z)pipi , for different val-ues of√smax (the upper limit of the hadronic in-variant mass in the dispersion integral that pro-vides the hadron vacuum polarization contribu-tion to both observables) and for the form factorscoming from the analyses of e+e− and τ data.Our results compare well with the recent compu-tation in Ref. [19].AcknowledgementsWe wish to thank S. Narison and his team forthe great organization of the QCD02 Conference.We also thank V. Cirigliano for his help withthe isospin breaking corrections. This work hasbeen supported in part by TMR, EC ContractNo. ERB FMRX-CT98-0169, by MCYT (Spain)under grant FPA-2001-3031 and by ERDF fundsfrom the EU Commission.REFERENCES1. R. R. Akhmetshin et al. 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Vector form factor of the pion : A model-independent approach

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Vector form factor of the pion : A model-independent approach

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