Comment on “Single-inclusive jet production in electron-nucleon collisions through next-to-next-to-leading order in perturbative QCD” 7 [Phys. Lett. B763, 52–59 (2016)] 1 0 2 Geoffrey T. Bodwin1,∗ and Eric Braaten2,† r a M 1High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA 2 2Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA 2 (Dated: March 24, 2017) ] h p Abstract - p In the cross section for single-inclusive jet production in electron-nucleon collisions, the distribu- e h [ tion of a quark in an electron appears at next-to-next-to-leading order. The numerical calculations 3 in Ref. [1] were carried out using a perturbative approximation for the distribution of a quark in v 4 an electron. We point out that that distribution receives nonperturbative QCD contributions that 1 2 8 invalidate the perturbative approximation. Those nonperturbative effects enter into cross sections 0 . forhard-scatteringprocessesthroughresolved-electron contributionsandcanbetakenintoaccount 1 0 by determining the distribution of a quark in an electron phenomenologically. 7 1 : v i X r a ∗[email protected] †[email protected] 1 In Ref. [1], the cross section for single-jet inclusive production in lepton-nucleon collisions is computed through next-to-next-to-leading order in perturbative quantum chromodynam- ics (QCD). That computation advances significantly the potential for precision comparisons between theory and experiment for this process. The cross section contains a contribution that is proportional to the distribution of a quark in a lepton, namely, f (ξ,µ2), where ξ q/l is the light-cone momentum fraction of the quark and µ is the renormalization scale. Such a contribution could be termed a “resolved-lepton” contribution. The distribution that was used in Ref. [1] is α 2 (1−ξ)(4+7ξ +4ξ2) µ2 f (ξ,µ2) = e2 +(1+ξ)logξ log2 q/l q 2π (cid:26)(cid:20) 6ξ (cid:21) m2 (cid:16) (cid:17) l (1−ξ)(2+5ξ −2ξ2) 8+15ξ −3ξ2 −8ξ3 µ2 + − − logξ −3(1+ξ)log2ξ log , (1) (cid:20) ξ 3ξ (cid:21) m2(cid:27) l where m is the lepton mass, e is the electric charge of the quark, and α is the quantum- l q electrodynamics (QED) coupling constant. The single and double logarithms of µ cancel the µ-dependence of other factors in the cross section at order α2α2. s In Ref. [1], f (ξ,µ2) is derived by making use of the Dokshitzer, Gribov, Lipatov, q/l Altarelli, Parisi (DGLAP) evolution equation [2–5] in the form ∂ µ2 f = P ⊗f +P ⊗f . (2) ∂µ2 q/l qγ γ/l ql l/l Here, f (ξ,µ2) is the distribution of a photon in a lepton, f (ξ,µ2) is the distribution of γ/l l/l a lepton in a lepton, P (z) and P (z) are the DGLAP splitting functions, and ⊗ denotes qγ ql the convolution 1 dz [P ⊗f](ξ) = P(ξ)f(ξ/z). (3) Z z ξ (In Eq. (2), we have absorbed factors of α into the definitions of the splitting functions.) In Ref. [1], the splitting functions are evaluated to order α and order α2, respectively, and the QED distributions on the right side of Eq. (2) are evaluated at leading order in α: f (ξ,µ2) γ/l is the Weizsa¨cker-Williams distribution, and f (ξ) = δ(1−ξ). The distribution in Eq. (1) l/l is obtained by integrating Eq. (2) with the boundary condition f (ξ,m2) = 0. q/l l Inthiscomment, wepointoutthatf (ξ,µ2)receivesnonperturbativeQCDcontributions q/l thatinvalidatetheexpression forthedistributionofaquarkinanelectrondefinedbyEq. (1). If the lepton has a sufficiently large mass, as is the case for the τ lepton, then f (ξ,m2) q/l l can be computed in QCD perturbation theory, and it can be evolved perturbatively from 2 the scale m2 to the scale µ2 in order to absorb logarithms of µ2/m2 into f (ξ,µ2). In this l l q/l case, the expression in Eq. (1) is a valid approximation for f (ξ,µ2) in that it captures the q/l logarithmic contributions at leading-order in α.1 However, when the lepton is an electron or a muon, f (ξ,µ2) cannot be computed in QCD perturbation theory. q/l The nonperturbative nature of f (ξ,µ2) can be seen by considering its DGLAP evolu- q/l tion. When one considers QCD corrections, the evolution equation for f (ξ,µ2) contains q/l additional contributions that arise from the emission of real and virtual gluons by the quark: ∂ f P ⊗f P ⊗f P 2P f µ2  qi/l =  qiγ γ/l+ qil l/l+  qiqj qig⊗ qj/l, (4) ∂µ2 fg 0 0 Xqj Pgqj Pgg fg/l           where the sum over q includes both quarks and antiquarks. Suppose that one were to follow j the procedure in Ref. [1], evolving f from the scale m to a hard-scattering scale. The q/l l splitting functions in Eq. (4) depend on α at scales µ that range from m to the hard- s l scattering scale. If µ is sufficiently large, then the splitting functions can be computed in perturbation theory. However, if µ is less than a scale of order Λ , then the perturbation QCD expansion for the splitting functions fails, and the evolution of f receives nonperturbative q/l contributions. In the case of the electron or the muon, the range of µ includes a region in which perturbative QCD fails and nonperturbative effects dominate. Although the computation of the short-distance part of the cross section through the order of interest in Ref. [1] requires only that collinear poles through order α2 be absorbed into f (ξ,µ2), a reliable calculation of f (ξ,µ2) requires that QCD corrections be taken q/l q/l into account. The concept that the short-distance part of the cross section can be computed at a fixed order in α , while the parton distributions, when they are nonperturbative, cannot s is, of course, familiar from other hard-scattering processes, such as deep-inelastic scattering. The nonperturbative distribution for a quark in an electron f (ξ,µ2) at a scale µ2 that q/e is in the perturbative regime of QCD could, in principle, be determined phenomenologi- cally by fitting cross-section predictions to data. A process that is particularly sensitive to f (ξ,µ2) is single-inclusive jet production in electron-electron scattering. Alternatively, q/e with some sacrifice of sensitivity, one could make use of cross sections for single-jet inclusive production in electron-nucleon collisions. Lattice calculations might also provide informa- 1 We note that the expression in Eq. (1) omits constant terms that arise in standard renormalization schemes, such as modified minimal subtraction. 3 tion on f (ξ,µ2). Once the nonperturbative distribution for a quark in an electron has q/e been determined, it could be used to make reliable predictions for the resolved-electron contributions to hard-scattering processes. Because of the sensitivity of f (ξ,µ2) to nonperturbative QCD effects, the expression q/e in Eq. (1) can at best be regarded as a model for the distribution. One unphysical aspect of this model is its double-logarithmic dependence on the electron mass. There is a logarithm of m2 in the Weizsa¨cker-Williams distribution f (ξ,µ2). A second logarithm arises when e γ/e one integrates Eq. (2) from m2 to µ2 using the perturbative expressions for the splitting e functions. This procedure implies that quarks in the electron are generated by perturbative evolution all the way down to virtualities of order m2. One would not expect a probe with a e virtuality that is much less than a typical hadronic scale to be able to resolve the hadronic structure of the electron. For the range of µ that is considered in Ref. [1], much of the large coefficient log2(µ2/m2) in Eq. (1) comes from integration over virtualities that are smaller e than a typical hadronic scale of, say, 700 MeV. This feature of the model in Eq. (1) would tend to produce a significant overestimate of the contribution from quarks in the electron to the cross section for single-jet inclusive production in electron-nucleon collisions. Other nonperturbative effects that are not accounted for in the model could be substantial, as well. We note that a sensitivity to nonperturbative QCD effects arises in the same way in the case of the distribution of a quark in a real photon f . In this case, the leading-order QED q/γ expression for the logarithmic contribution to the distribution that is analogous to Eq. (1) is α µ2 f (ξ,µ2) = e2 [ξ2 +(1−ξ)2]log . (5) q/γ q2π m2 γ The inadequacy of this leading-order logarithmic approximation is manifest in the logarithm of the photon mass m . Of course, it is well established that the distribution of a quark in γ a real photon involves contributions that cannot be calculated in perturbation theory, but must, instead, be obtained from fits to experimental data. (See, for example, Refs. [6–8].) Acknowledgments We thank Frank Petriello and Yuri Kovchegov for helpful discussions. The work of E.B. was supported in part by the Department of Energy under grant de-sc0011726. The work of G.T.B. is supported by the U.S. Department of Energy, Division of High Energy Physics, 4 under Contract No. DE-AC02-06CH11357. The submitted manuscript has been created in part by UChicago Argonne, LLC, Operator of Argonne National Laboratory. Argonne, a U.S. Department of Energy Office of Science laboratory, is operated under Contract No. DE- AC02-06CH11357. The U.S. Government retains for itself, and others acting on its behalf, a paid-up nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Government. [1] G. Abelof, R. Boughezal, X. Liu and F. Petriello, Phys. Lett. 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