A new model for the QCD analytic running coupling, which incorporates the
effects due to the $\pi$ meson mass, is proposed. The properties of this
invariant charge in spacelike and timelike regions are examined. Its main
distinctive features are a finite infrared limiting value, which depends on the
pion mass, and the "plateau-like" behavior in the deep infrared domain of the
timelike region.Comment: Talk given at Eighth Workshop on Nonperturbative QCD, 7 - 11 June
  2004, Paris, France; 6 pages, 2 EPS figure

Nesterenko, A. V.

Papavassiliou, J.

English

arXiv

A. V. NESTERENKO

J. PAPAVASSILIOU

Crossref

THE QCD ANALYTIC EFFECTIVE CHARGE AND ITS DEPENDENCE ON THE PION MASS

A new model for the QCD analytic running coupling, which incorporates the effects due to the π meson mass, is proposed. The properties of this invariant charge in spacelike and timelike regions are examined. Its main distinctive features are a finite infrared limiting value, which depends on the pion mass, and the 'plateau-like' behavior in the deep infrared domain of the timelike region

Nesterenko, A.V.

Papavassiliou, Joannis

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

arXiv:hep-ph/0409220v1  18 Sep 2004THE QCD ANALYTIC EFFECTIVE CHARGEAND ITS DEPENDENCE ON THE PION MASSA.V. NESTERENKO1 and J. PAPAVASSILIOU21Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear ResearchDubna, 141980, Russian Federation2Departamento de F´ısica Teo´rica and IFIC, Centro Mixto,Universidad de Valencia-CSIC, E-46100, Burjassot, Valencia, SpainReceived 18 September 2004A new model for the QCD analytic running coupling, which incorporates the effects dueto the pi meson mass, is proposed. The properties of this invariant charge in spacelike andtimelike regions are examined. Its main distinctive features are a finite infrared limitingvalue, which depends on the pion mass, and the “plateau–like” behavior in the deepinfrared domain of the timelike region.Keywords: Nonperturbative QCD; dispersion relations; strong running coupling.1. IntroductionThe description of the infrared hadron dynamics remains a crucial challenge forelementary particle physics. Whereas perturbative calculations allow for a detailedstudy of the strong interaction processes at high energies, to date there is no reliabletheoretical method which would enable one to handle the hadron dynamics at lowenergies. However, many physical phenomena are tightly bound to the intrinsicallynonperturbative aspects of the strong interaction.The renormalization group (RG) method plays a fundamental role in the frame-work of Quantum Field Theory (QFT) and its applications. Usually, in order todescribe Quantum Chromodynamics (QCD) in the ultraviolet region, one appliesthe RG method together with perturbative calculations. Eventually, this leads toapproximate solutions of the RG equations, which are commonly used in the quanti-tative analysis of high energy processes. However, such solutions possess unphysicalsingularities in the infrared domain, a fact that contradicts the general principlesof local QFT and complicates the theoretical description of the intermediate andlow energy experimental data. Nevertheless, these difficulties can be avoided if onecomplements the perturbative results with a proper nonperturbative insight.For this purpose the relevant dispersion relations can be used. The idea of em-ploying the information “stored” in such integral representations together with per-12turbation theory forms the basis of the analytic approach to QFT. First it was pro-posed in the framework of Quantum Electrodynamics1 and lately it was extended2to QCD. Another important application of the dispersion relations is the analysisof the timelike QCD experimental data. Here, the dispersion relation for the AdlerD function3 enables one to handle the timelike strong interaction processes in aconsistent way.The effects due to the masses of the lightest hadrons can be safely neglectedonly when one studies the QCD processes at high energies. But for the descriptionof the low energy dynamics the mass effects become essential. For example, this iscrucial for the description of the inclusive τ lepton decay. Both, the perturbativeresults and the dispersion relation for the Adler D function with the nonvanishingpion mass, are essential here for the correct interpretation of these data. However,no such mass effects have been taken into account in the analytic approach to QCDso far.In this talk we report recent progress4 on the incorporation of the effects due tothe π meson mass into the analytic approach to QCD, and the study of the behaviorof the developed running coupling in spacelike and timelike infrared domains.2. Strong Running Coupling in Spacelike and Timelike RegionsThe description of hadron dynamics in spacelike and timelike regions has been thesubject of many studies over a long period of time. The strong interaction processeswith the large spacelike momentum transfer q2 > 0 can be examined perturbativelyin the framework of the RG method (the metric with the signature (−1, 1, 1, 1) isused, so that positive q2 corresponds to a spacelike momentum transfer). But inorder to handle the processes in the timelike region (s = −q2 > 0), one has first torelate the perturbative results with the measured quantities.A substantial step towards the consistent description of the timelike data wasmade in Refs. 3, 5, and 6. Namely, it has been argued that the dispersion relationfor the Adler D functionD(q2) = q2∫∞4m2piR(s)(s+ q2)2ds (1)supplies a firm ground for comparing the perturbative results for D(q2) with theexperimental measurements of the R(s)–ratio of the e+e− annihilation into hadrons.Thus, the perturbative results for D(q2) can be continued into the timelike domainby making use of the relation inverse to (1) (see also Refs. 5 and 7)R(s) =12πilimε→0+∫ s−iεs+iεD(−ζ) dζζ. (2)The asymptotic ultraviolet behavior of the Adler D function can be computedby making use of the perturbation theory7,8D(q2) = Nc∑fQ2f[1 + d(q2)], d(q2) ≃ d1[αs(q2)π]+ d2[αs(q2)π]2+ · · · . (3)3Here d1 = 1, d2 ≃ 1.9857 − 0.1153nf, nf is the number of active quarks, Nc = 3is the number of colors, and Qf stands for the charge of the f -th quark. Since theintegral transformation (2) of Eq. (3) has to be performed every time when onedeals with the timelike experimental data, it is convenient to define9 the timelikeeffective charge α̂(s) in the same way, as R(s) relates to D(q2):α̂(s) =12πilimε→0+∫ s−iεs+iεα(−ζ) dζζ. (4)Here and further the strong running coupling in the spacelike and timelike domainsis denoted by α(q2) and α̂(s), respectively. The inverse relationα(q2) = q2∫∞4m2piα̂(s)(s+ q2)2ds (5)holds as well. Evidently, for the thorough description of the QCD processes at lowenergies the pion mass cannot be neglected in Eqs. (1) and (5).For the self–consistency of the method described above, one has first to bringthe perturbative approximation for the Adler D function (3) to conformity with thedispersion relation (1). Indeed, Eq. (1) implies that in the massless case (mπ = 0)D(q2) is the analytic function in the complex q2-plane with the only cut −∞ < q2 ≤0 along the negative semiaxis of real q2. However, the perturbative approximation ofd(q2) in Eq. (3) violates this condition. Fortunately, this discrepancy can be avoidedin the framework of the analytic approach to QCD.3. Massless Analytic Effective ChargeAs was mentioned in the Introduction, the dispersion relations provide one witha certain nonperturbative information about a quantity in hand, namely, with thedefinite analytic properties in the kinematic variable. Recently it was argued2 thatin the massless case the Ka¨lle´n–Lehmann spectral representation for the QCD in-variant chargeα(q2) =∫∞0̺(σ)σ + q2dσ (6)holds. It is precisely this condition that is needed in order to bring the perturba-tive approximation for the Adler D function (3) in agreement with the dispersionrelation (1) and to satisfy the relations (4) and (5) in the limit of the masslesspion mπ = 0.This section is devoted to a brief overview of one of the massless models forthe QCD analytic running coupling.10 The distinguishing feature of this modelis that the analyticity requirement is imposed on the perturbative expansion ofthe β function. This way of incorporating the analyticity requirement into the RGformalism is consistent with the general definition of the QCD invariant charge.11,12The model10 has proved to be successful in the description of the hadron dynamics4of the both perturbative and intrinsically nonperturbative nature.13 The runningcoupling10 has also been rederived in Ref. 14 proceeding from different assumptions.Thus, the Ka¨lle´n–Lehmann representation (6) holds for the massless analyticinvariant charge (the relevant details can be found in Refs. 10 and 11)α(ℓ)an (q2) =4πβ0∫∞0ρ(ℓ)(σ)σ + zdσ, z =q2Λ2. (7)In this equation ρ(ℓ)(σ) denotes the ℓ-loop spectral densityρ(ℓ)(σ) = ρ(1)(σ) exp[−∫∞0P(ℓ)(ζ) ln∣∣∣∣1− ζσ∣∣∣∣ dζζ][cosψ(ℓ)(σ) +lnσπsinψ(ℓ)(σ)], (8)where ρ(1)(σ) = (1 + σ−1)/(ln2 σ + π2) stands for the one-loop spectral density,ψ(ℓ)(σ) = π∫∞σP(ℓ)(ζ) ζ−1dζ, P(ℓ)(σ) = R(ℓ)(σ)−R(1)(σ), andR(ℓ)(σ) = 12πilimε→0+ℓ−1∑j=0βj(4π)j+1{[α(ℓ)s (−σ − iε)]j+1−[α(ℓ)s (−σ + iε)]j+1}. (9)Here β0 = 11−2nf/3, β1 = 102−38nf/3, α(ℓ)s (q2) is the ℓ-loop perturbative runningcoupling. In the exponent in Eq. (8) the principal value of the integral is assumed.The model (7) shares all the advantages of the analytic approach: it contains nounphysical singularities, displays good higher-loop and scheme stability, and has noadjustable parameters. The massless analytic effective charge (7) incorporates theultraviolet asymptotic freedom with the infrared enhancement in a single expression,that plays a crucial part when applying this model to the study of quenched latticesimulation data.14 The detailed analysis of the properties of the invariant charge (7)and its applications can be found in Refs. 10–13.For the congruous description of the timelike hadron dynamics one has to employthe continuation (4) of the strong running coupling. This leads to the followingextension of the invariant charge (7) to the timelike domain10 (see also Ref. 9):α̂(ℓ)an (s) =4πβ0∫∞wρ(ℓ)(σ)dσσ, w =sΛ2, (10)where s = −q2 > 0, and the spectral density ρ(ℓ)(σ) is defined in Eq. (8). The plotsof the functions α(1)an (q2) and α̂(1)an (s) are shown in Figure 1. Note the asymmetrybetween these curves in the intermediate and low energy regions, which has tobe taken into account when one handles the experimental data (see review11 andreferences therein for the details).It is worth noting that the formulation of the analytic approach to QCD isstill essentially massless. The obtained results can be applied, for example, to thestudy of the experimental data at high energies, the pure gluodynamics, and thequenched lattice simulation data. However, for a consistent description of the in-frared hadron dynamics, the mass effects have to be incorporated into the analyticapproach to QCD.54. Massive Analytic Effective ChargeThe original dispersion relation for the Adler D function3 (1) with the nonvanish-ing mass of the π meson implies that D(q2) is the analytic function in the complexq2-plane with the only cut beginning at the two–pion threshold −∞ < q2 ≤ −4m2π.However, the approximation (3) violates this condition due to the spurious singular-ities of the perturbative running coupling. This disagreement can be avoided by im-posing the analyticity requirement of the form d(q2,m2π) =∫∞4m2piκ(σ)(σ + q2)−1dσon the function d(q2) in Eq. (3). Thus, the strong running coupling itself has tosatisfy the integral representationα(q2,m2π) =∫∞4m2pi̺(σ)σ + q2dσ. (11)Otherwise, one would encounter the contradiction between the dispersion relationfor the Adler D function (1) and its approximation (3).Fig. 1. The one-loop massless analytic run-ning coupling in spacelike (Eq. (7), q2 > 0)and timelike (Eq. (10), s = −q2 > 0) regions,z = −w = q2/Λ2.Fig. 2. The one–loop massive analytic run-ning coupling in spacelike (Eq. (12)) andtimelike (Eq. (13)) domains. The value Λ =623MeV and two active quarks are assumed.There are several ways to restore the spectral density ̺(σ) in the integral repre-sentations (6) and (11). Ultimately, this has led to different models for the strongrunning coupling within the analytic approach to QCD (see Refs. 10–12, 15). Themodel for the analytic invariant charge10 has proved to be successful in the de-scription of both perturbative and intrinsically nonperturbative strong interactionprocesses.11,13 We will therefore adopt in this work the spectral density (8).Thus, one arrives at the following integral representation for the massive analyticeffective chargeα(ℓ)an (q2,m2π) =4πβ0∫∞χρ(ℓ)(σ)σ + zdσ, z =q2Λ2, (12)where ρ(ℓ)(σ) denotes the ℓ-loop spectral density (8), and χ = 4m2π/Λ2. It turns outthat the nonvanishing mass of the π meson drastically affects the infrared behavior6of the running coupling. Indeed, instead of the infrared enhancement in the masslesscase (7), one has here the infrared finite limiting value for the effective charge (12)α(ℓ)0 (χ) = 4πβ−10∫∞χρ(ℓ)(σ)σ−1dσ. The continuation (4) of the running coupling (12)results in the following representation for the timelike massive effective charge:α̂(ℓ)an (s,m2π) =4πβ0∫∞wθ(σ − χ) ρ(ℓ)(σ) dσσ, w =sΛ2, (13)where s = −q2 ≥ 0, ρ(ℓ)(σ) is the ℓ-loop spectral density (8), and χ = 4m2π/Λ2.The effective charges (12) and (13) have a common finite value α(ℓ)0 (χ) in theinfrared limit. Then, the timelike coupling (13) has the “plateau” in the infrareddomain: α̂(ℓ)an (s,m2π) = α(ℓ)0 (χ), when 0 ≤√s ≤ 2mπ (see Fig. 2). The massless (10)and massive (13) timelike couplings coincide for√s > 2mπ, since the pion massaffects the coupling (13) only in the region 0 ≤ √s ≤ 2mπ, where it stops running.Finally, the developed model has been applied4 to the study of the experimentaldata on the inclusive τ lepton decay,16 that has given (at the one-loop level withtwo active quarks) the value Λ = (623± 81)MeV of the QCD scale parameter.AcknowledgmentsA.N. is grateful to I.L. Solovtsov for valuable discussions and comments. Support bygrants RFBR (02-01-00601, 04-02-81025), NS-2339.2003.2, and CICYT FPA20002-00612 is acknowledged.References1. P.J. Redmond, Phys. Rev. 112, 1404 (1958); N.N. Bogoliubov, A.A. Logunov, andD.V. Shirkov, Sov. Phys. JETP 37, 574 (1960).2. D.V. Shirkov and I.L. Solovtsov, Phys. Rev. Lett. 79, 1209 (1997).3. S.L. Adler, Phys. Rev. D10, 3714 (1974).4. A.V. Nesterenko and J. Papavassiliou, in preparation.5. A.V. Radyushkin, JINR Report No. 2–82–159, 1982; arXiv:hep-ph/9907228.6. N.V. Krasnikov and A.A. Pivovarov, Phys. Lett. B116, 168 (1982).7. S.G. Gorishny, A.L. Kataev, and S.A. Larin, Phys. Lett. B259, 144 (1991).8. L.R. Surguladze and M.A. Samuel, Phys. Rev. Lett. 66, 560 (1991).9. K.A. Milton and I.L. Solovtsov, Phys. Rev. D55, 5295 (1997); D59, 107701 (1999).10. A.V. Nesterenko, Phys. Rev. D62, 094028 (2000); D64, 116009 (2001).11. A.V. Nesterenko, Int. J. Mod. Phys. A18, 5475 (2003).12. A.V. Nesterenko and I.L. Solovtsov, Mod. Phys. Lett. A16, 2517 (2001).13. A.V. Nesterenko, Nucl. Phys. (Proc. Suppl.) B133, 59 (2004).14. F. Schrempp, J. Phys. G28, 915 (2002).15. D.V. Shirkov, Theor. Math. Phys. 132, 1309 (2002); arXiv:hep-ph/0408272.16. R. Barate et al. (ALEPH Collaboration), Eur. Phys. J. C4, 409 (1998);K. Ackerstaff et al. (OPAL Collaboration), ibid. C7, 571 (1999).

The QCD analytic effective charge and its dependence on the pion mass

The QCD analytic effective charge and its dependence on the pion mass

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