I discuss application of the BLM method to obtain commensurate scale
relations connecting QCD exclusive amplitudes to other observables, in
particular the heavy quark potential.Comment: 7 pages, Latex, uses l-school.sty. Talk given at "New Nonperturbative
  Methods and Quantization on the Light Cone," Les Houches, France, 24 Feb.-7
  March 1997. To appear in the proceeding

Robertson, David G.

English

arXiv

I discuss application of the BLM method to obtain commensurate scale relations connecting QCD exclusive amplitudes to other observables, in particular the heavy quark potential

Otterbein University

Otterbein University Digital Commons @ Otterbein Physics Faculty Scholarship Physics 2-24-1997 Physical Coupling Schemes and QCD Exclusive Processes David G. Robertson Otterbein University Follow this and additional works at: https://digitalcommons.otterbein.edu/phys_fac  Part of the Physics Commons Repository Citation Robertson, David G., "Physical Coupling Schemes and QCD Exclusive Processes" (1997). Physics Faculty Scholarship. 4. https://digitalcommons.otterbein.edu/phys_fac/4 This Conference Proceeding is brought to you for free and open access by the Physics at Digital Commons @ Otterbein. It has been accepted for inclusion in Physics Faculty Scholarship by an authorized administrator of Digital Commons @ Otterbein. For more information, please contact digitalcommons07@otterbein.edu. LECTURE 10PHYSICAL COUPLING SCHEMES ANDQCD EXCLUSIVE PROCESSESDavid G. RobertsonDepartment of PhysicsThe Ohio State UniversityColumbus, OH 43210, USAOne of the most important problems in making reliable predictions in pertur-bative QCD is dealing with the dependence of the truncated perturbative serieson the choice of renormalization scale µ and scheme s for the QCD couplingαs(µ). Consider a physical quantity O, computed in perturbation theory andtruncated at next-to-leading order (NLO) in αs:O = αs(µ)[1 + (A1 +B1nf )αs(µ)pi+ · · ·], (1)where nf is the effective number of quark flavors. The finite-order expressiondepends on both µ and the choice of scheme used to define the coupling. In fact,Eq. (1) can be made to take on essentially any value by varying µ and the renor-malization scheme, which are a priori completely arbitrary. The scale/schemeproblem is that of choosing µ and the scheme s in an “optimal” way, so that anunambiguous theoretical prediction, ideally including some plausible estimateof theoretical uncertainties, can be made.(1)(1) The precise meaning of “optimal” in this context is connected to the minimiza-tion of remainders for the truncated series. As is well known, perturbation series inQCD are asymptotic, and thus there is an optimum number of terms that should becomputed for a given observable. In general, very little is known about the remain-ders in pQCD; however, if we assume that pQCD series are sign-alternating, then theremainder can be estimated by the first neglected (or last included) term. This term2 David G. RobertsonFor any given observable there is no rigorously correct way to make this choicein general. However, a particular prescription may be supported to a greater orlesser degree by general theoretical arguments and, a posteriori, by its successin practical applications. From these perspectives, a particularly successfulmethod for choosing the renormalization scale is that proposed by Brodsky,Lepage and MacKenzie [1]. In the BLM procedure, the renormalization scalesare chosen such that all vacuum polarization effects from fermion loops areabsorbed into the running couplings. A principal motivation for this choice isthat it reduces to the correct prescription in the case of Abelian gauge theory.Furthermore, the BLM scales are physical in the sense that they typically reflectthe mean virtuality of the gluon propagators. Another important advantage ofthe method is that it “pre-sums” the large and strongly divergent terms in thepQCD series which grow as n!(αsβ0)n, i.e., the infrared renormalons associatedwith coupling constant renormalization.Dependence on the renormalization scheme can be avoided by consideringrelations between physical observables only. By the general principles of renor-malization theory, such a relation must be independent of any theoretical con-ventions, in particular the choice of scheme in the definition of αs. A relationbetween physical quantities in which the BLM method has been used to fix therenormalization scales is known as a “commensurate scale relation” (CSR) [2].An important example is the generalized Crewther relation [2, 3], in which theradiative corrections to the Bjorken sum rule for deep inelastic lepton-protonscattering at a given momentum transfer Q are predicted from measurementsof the e+e− annihilation cross section at a commensurate energy scale√s ∝ Q.In this talk I summarize recent applications of the BLM procedure to obtainCSRs relating QCD exclusive amplitudes to other observables, in particular theheavy quark potential V (Q2) [4]. As we shall see, the heavy quark potential canbe used to define a physical coupling scheme which is quite natural for pertur-bative calculations. It may also be useful in the context of a nonperturbativeformulation of QCD based on light-cone quantization.BLM SCALE FIXINGThe term involving nf in Eq. (1) arises solely from quark loops in vacuumpolarization diagrams. In QED these are the only contributions responsiblefor the running of the coupling, and thus it is natural to absorb them into thedefinition of the coupling. The BLM procedure is the analog of this approachin QCD. Specifically, we rewrite Eq. (1) in the formO = αs(µ)[1 +(3β0B12)αs(µ)pi] [1 +(A1 +33B12)αs(µ)pi], (2)can take on essentially any value, however, by simply varying the scale and scheme,and thus its minimization is meaningless without invoking additional criteria.PHYSICAL COUPLING SCHEMES. . . 3correct to order α2s, where β0 = 11−2nf/3 is the lowest-order QCD beta func-tion. The first term in square brackets can then be absorbed by a redefinitionof the renormalization scale in the leading-order coupling, usingαs(µ∗) = αs(µ)[1− β0αs(µ)2piln (µ∗/µ) + · · ·]. (3)That is, the BLM procedure consists of defining the prediction for O at thisorder to beO = αs(µ∗)[1 +(A1 +33B12)αs(µ∗)pi+ · · ·], (4)whereµ∗ ≡ µe3B1 . (5)Note that knowledge of the NLO term in the expansion is necessary to fix thescale at LO. The scale occurring in the highest term in the expansion will ingeneral be unknown. A natural prescription is to set this scale to be the sameas that in the next-to-highest-order term.A very important feature of this prescription is that µ∗ is actually indepen-dent of µ. (This follows from considering the µ dependence of B1.) Thus pQCDpredictions using the BLM procedure are unambiguous.The same basic idea can be extended to higher orders, by systematicallyshifting nf dependence into the renormalization scales order by order. Theresult is that a generic perturbative expansionαs(µ)pi+ (A1 +B1nf )(αs(µ)pi)2+(A2 +B2nf + C2n2f)(αs(µ)pi)3+ · · · (6)is replaced by a series of the formαs(µ∗)pi+ A˜1(αs(µ∗∗)pi)2+ A˜2(αs(µ∗∗∗)pi)3+ · · · . (7)In general a different scale appears at each order in perturbation theory. Inaddition, the coefficients A˜n are constructed to be independent of nf , and sothe form of the expansion is unchanged as momenta vary across quark massthresholds. All effects due to quark loops in vacuum polarization diagrams areautomatically incorporated into the effective couplings.As discussed above, one motivation for this prescription is that it reducesto the correct result in the case of QED. In addition, when combined withthe idea of commensurate scale relations (see below), the BLM method can beshown to be consistent with the generalized renormalization group invarianceof Stu¨ckelberg and Peterman, in which one considers “flow equations” both inµ and in the parameters that define the scheme.To avoid scheme dependence, it is convenient to introduce physical effectivecharges, defined via some convenient observable, for use as an expansion pa-rameter. An expansion of a physical quantity in terms of such a charge is a4 David G. Robertsonrelation between observables and therefore must be independent of theoreticalconventions, such as the renormalization scheme, to any fixed order of pertur-bation theory. In practice, a CSR relating two observables A and B is obtainedby applying BLM scale-fixing to their respective perturbative predictions in,say, the MS scheme, and then algebraically eliminating αMS .A particularly useful scheme is furnished by the heavy quark potential V (Q2),which can be identified as the two-particle-irreducible amplitude for the scatter-ing of an infinitely heavy quark and antiquark at momentum transfer t = −Q2.The relationV (Q2) = −4piCFαV (Q)Q2, (8)with CF = (N2C − 1)/2NC = 4/3, then defines the effective charge αV (Q).This coupling provides a physically-based alternative to the usual MS scheme.Another useful charge is provided by the total e+e− → X cross section, via thedefinition R(s) ≡ 3Σe2q (1 + αR(√s)/pi) . The CSR relating αV and αR isαV (QV ) = αR(QR)(1− 2512αRpi+ · · ·), (9)where the ratio of commensurate scales is QR/QV = e23/12−2ζ3 ≃ 0.614 [4].Physical couplings like αV are of course renormalization-group-invariant, i.e.µ∂αV /∂µ = 0. However, the dependence of αV (Q) on Q is controlled by anequation which is formally identical to the usual RG equation. Since αV isdimensionless we must haveαV = αV(Qµ,αs(µ)). (10)Then µ∂αV /∂µ = 0 impliesQ∂∂QαV (Q) = βs(αs)∂αV∂αs≡ βV (αV ) , (11)whereβs = µ∂∂µαs(µ) . (12)This is formally a change of scheme, so that the first two coefficients β0 =11− 2nf/3 and β1 = 102− 38nf/3 in the perturbative expansion of βV are thestandard ones.EXCLUSIVE AMPLITUDES AND V (Q2)Exclusive processes are particularly challenging to compute in QCD becauseof their sensitivity to the unknown nonperturbative bound state dynamics ofthe hadrons. However, there is an extraordinary simplification which occursPHYSICAL COUPLING SCHEMES. . . 5when the hadrons are forced to absorb a large momentum transfer Q: onecan separate the nonperturbative long-distance physics associated with hadronstructure from the short-distance quark-gluon hard scattering amplitudes re-sponsible for the dynamical reaction. A meson form factor, for example, fac-torizes to leading order in 1/Q in the formFM (Q2) =∫ 10dx∫ 10dyφM (x, Q˜)TH(x, y,Q2)φM (y, Q˜) , (13)where φM (x, Q˜) is the meson distribution amplitude, which encodes the non-perturbative dynamics of the bound valence Fock state up to the resolutionscale Q˜, and TH is the leading-twist perturbatively-calculable amplitude forthe subprocess γ∗q(x)q(1− x)→ q(y)q(1 − y), in which the incident and finalmesons are replaced by valence quarks collinear up to the resolution scale Q˜.Contributions from nonvalence Fock states and the correction from neglectingthe transverse momenta in the subprocess amplitude from the nonperturbativeregime are higher twist, i.e., power-law suppressed. The transverse momentain the perturbative domain lead to the evolution of the distribution amplitudein Q˜ and to NLO corrections in αs. For further details and references see [5].It is straightforward to obtain CSRs relating exclusive amplitudes to, e.g.,the heavy quark potential. For the pion form factor, for example, we find [4]Fpi(Q2) =∫ 10dxφpi(x)∫ 10dyφpi(y)4piCFαV (Q∗V )(1 − x)(1 − y)Q2(1 + CVαV (Q∗V ))pi),(14)where CV = −1.91 andQ∗V 2 = (1−x)(1−y)Q2 is the virtuality of the exchangedgluon in the underlying hard scattering amplitude. Eq. (14) represents ageneral connection between the form factor of a bound-state system and theirreducible kernel that describes the scattering of its constituents.If we expand the QCD coupling about a fixed point [4], and assume that thepion distribution amplitude has the asymptotic form φpi(x) =√3fpix(1 − x),with fpi ≃ 93 MeV, then the integral over the effective charge in Eq. (14) canbe performed explicitly. We thus findQ2Fpi(Q2) = 16pif2piαV (e−3/2Q)(1− 1.91αVpi)(15)for the asymptotic distribution amplitude. In additionQ2Fγpi(Q2) = 2fpi(1− 53αV (e−3/2Q)pi)(16)for the γ → pi0 transition form factor. A further prediction resulting from thefactorized form of these results is that the normalization of the ratioRpi(Q2) ≡ Fpi(Q2)4piQ2|Fpiγ(Q2)|2 (17)= αV (e−3/2Q)(1 + 1.43αVpi)(18)6 David G. Robertsonis formally independent of the form of the pion distribution amplitude. TheNLO correction given here assumes the asymptotic distribution amplitude.A striking feature of these results is that the physical scale controlling theform factor in the αV scheme is very low: e−3/2Q ≃ 0.22Q, reflecting thecharacteristic momentum transfer experienced by the spectator valence quarkin lepton-meson elastic scattering. In order to compare these expressions todata, therefore, we require an Ansatz for αV at low scales. In Ref. [4] weconsider a parameterization of the formαV (Q) =4piβ0 ln(Q2+4m2gΛ2V) , (19)which effectively freezes αV to a constant value for Q2 ≤ 4m2g. A primary moti-vation for this is the observation that the data for exclusive amplitudes such asform factors, two-photon processes such as γγ → pi+pi−, and photoproductionat fixed θc.m. are consistent with the nominal scaling of the leading-twist QCDpredictions for momentum transfers down to a few GeV. This can be imme-diately understood if αV is slowly varying at low momentum. The scaling ofthe exclusive amplitude then follows that of the subprocess amplitude TH witheffectively fixed coupling.The parameters ΛV and m2g are determined by fitting to a lattice determina-tion of V (Q2) [6] and to a value of αR advocated in [7] using Eq. (9). We findΛV ≃ 0.16 GeV and m2g ≃ 0.2 GeV2. With these values, the prediction for Fγpiis in excellent agreement with the data for Q2 in the range 2–10 GeV2. Wealso reproduce the scaling and normalization of the γγ → pi+pi− cross sectionat large momentum transfer. However, the normalization of the space-like pionform factor Fpi(Q2) obtained from electroproduction experiments is somewhathigher than that predicted by Eq. (15). This discrepancy may actually be dueto systematic errors introduced by the extrapolation of the γ∗p→ pi+n data tothe pion pole. What is at best measured in electroproduction is the transitionamplitude between a mesonic state with an effective space-like massm2 = t < 0and the physical pion. It is theoretically possible that the off-shell form factorFpi(Q2, t) is significantly larger than the physical form factor because of its biastowards more point-like qq valence configurations in its Fock state structure.These and related issues are discussed elsewhere [8].In any case, we find no compelling argument for significant higher-twist con-tributions in the few-GeV regime from the hard scattering amplitude or theendpoint regions, since such corrections would violate the observed scaling be-havior of the data.SUMMARYAs we have emphasized, the αV scheme is quite natural when analyzing QCDprocesses perturbatively. By definition, it automatically incorporates quarkPHYSICAL COUPLING SCHEMES. . . 7(as well as the corresponding gluon) vacuum polarization contributions intothe coupling; thus the coefficients in a perturbative expansion do not changeas momenta vary across quark mass thresholds. It is directly connected toone of the most useful observables in QCD, the heavy quark potential, which isaccessible on the lattice as well as phenomenologically via the spectrum of heavyquarkonium. Finally, the scale-setting problem in QCD appears much lessmysterious from this point of view: the scale appropriate for each appearance ofαV in a Feynman diagram is just the momentum transfer of the correspondingexchanged gluon.(2) This prescription is equivalent to the BLM procedure.It may also prove useful in the context of nonperturbative calculations basedon the light-cone formalism. A light-cone Hamiltonian expressed in terms ofαV , so that it reproduces covariant perturbation theory with αV appearinginside the momentum integrals, should be very well suited to studying, e.g.,heavy quark systems: the effects of the light quarks and higher Fock stategluons that renormalize the coupling are already contained in αV . At highmomentum scales αV should be computable via perturbation theory, while atlow scales a semi-phenomenological Ansatz would be necessary. As we havediscussed, exclusive processes can provide a valuable window for determiningαV in the low-energy domain.It is a pleasure to thank S. J. Brodsky, C.-R. Ji and A. Pang for the veryenjoyable collaboration which led to this work. I also thank the organizers,in particular P. Grange´, for putting together such a stimulating and enjoyablemeeting. This work was supported in part by a grant from the U.S. Departmentof Energy.References[1] S. J. Brodsky, G. P. Lepage, and P. B. Mackenzie, Phys. Rev. D 28, 228(1983).[2] H. J. Lu and S. J. Brodsky, Phys. Rev. D 48, 3310 (1993).[3] S. J. Brodsky, G. T. Gabadadze, A. L. Kataev and H. J. Lu, Phys. Lett.372B, 133 (1996).[4] S. J. Brodsky, C.-R. Ji, A. Pang and D. G. Robertson, “Optimal renor-malization scale and scheme for exclusive processes,” SLAC–PUB–7473,hep-ph/9705221.[5] S. J. Brodsky and G. P. Lepage, in Perturbative Quantum Chromodynam-ics, A. H. Mueller, Ed. (World Scientific, 1989).[6] C. T. H. Davies et. al., Phys. Rev. D 52, 6519 (1995).[7] A. C. Mattingly and P. M. Stevenson, Phys. Rev. D 49, 437 (1994).[8] S. J. Brodsky and D. G. Robertson, manuscript in preparation.[9] H. J. Lu, Ph. D. thesis, Stanford University (1992).(2) There are complications which arise when gluon self-couplings are present.These are discussed in [9].

Physical Coupling Schemes and QCD Exclusive Processes

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Physical Coupling Schemes and QCD Exclusive Processes

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