Comment on"Single-inclusive jet production in electron-nucleon collisions through next-to-next-to-leading order in perturbative QCD"[Phys. Lett. B 763, 52--59 (2016)]

In the cross section for single-inclusive jet production in electron-nucleon collisions, the distribution of a quark in an electron appears at next-to-next-to-leading order. The numerical calculations in Ref. [1] were carried out using a perturbative approximation for the distribution of a quark in an electron. We point out that that distribution receives nonperturbative QCD contributions that invalidate the perturbative approximation. Those nonperturbative effects enter into cross sections for hard-scattering processes through resolved-electron contributions and can be taken into account by determining the distribution of a quark in an electron phenomenologically.

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 chromodynamics (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 q/l (ξ, µ 2 ), where ξ 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 where m l is the lepton mass, e q is the electric charge of the quark, and α is the quantumelectrodynamics (QED) coupling constant. The single and double logarithms of µ cancel the µ-dependence of other factors in the cross section at order α 2 α 2 s . In Ref. [1], f q/l (ξ, µ 2 ) is derived by making use of the Dokshitzer, Gribov, Lipatov, Altarelli, Parisi (DGLAP) evolution equation [2][3][4][5] in the form Here, f γ/l (ξ, µ 2 ) is the distribution of a photon in a lepton, f l/l (ξ, µ 2 ) is the distribution of a lepton in a lepton, P qγ (z) and P ql (z) are the DGLAP splitting functions, and ⊗ denotes the convolution (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 α 2 , respectively, and the QED distributions on the right side of Eq. (2) are evaluated at leading order in α: is the Weizsäcker-Williams distribution, and f l/l (ξ) = δ(1 − ξ). The distribution in Eq. (1) is obtained by integrating Eq. (2) with the boundary condition f q/l (ξ, m 2 l ) = 0. In this comment, we point out that f q/l (ξ, µ 2 ) receives nonperturbative QCD contributions that invalidate the expression for the distribution of a quark in an electron defined by Eq. (1).
If the lepton has a sufficiently large mass, as is the case for the τ lepton, then f q/l (ξ, m 2 l ) can be computed in QCD perturbation theory, and it can be evolved perturbatively from the scale m 2 l to the scale µ 2 in order to absorb logarithms of µ 2 /m 2 l into f q/l (ξ, µ 2 ). In this case, the expression in Eq. (1) is a valid approximation for f q/l (ξ, µ 2 ) in that it captures the logarithmic contributions at leading-order in α. 1 However, when the lepton is an electron or a muon, f q/l (ξ, µ 2 ) cannot be computed in QCD perturbation theory.
The nonperturbative nature of f q/l (ξ, µ 2 ) can be seen by considering its DGLAP evolution. When one considers QCD corrections, the evolution equation for f q/l (ξ, µ 2 ) contains additional contributions that arise from the emission of real and virtual gluons by the quark: where the sum over q j includes both quarks and antiquarks. Suppose that one were to follow the procedure in Ref. 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 α 2 be absorbed into f q/l (ξ, µ 2 ), a reliable calculation of f q/l (ξ, µ 2 ) requires that QCD corrections be taken into account. The concept that the short-distance part of the cross section can be computed at a fixed order in α s , while the parton distributions, when they are nonperturbative, cannot is, of course, familiar from other hard-scattering processes, such as deep-inelastic scattering.
The nonperturbative distribution for a quark in an electron f q/e (ξ, µ 2 ) at a scale µ 2 that is in the perturbative regime of QCD could, in principle, be determined phenomenologically by fitting cross-section predictions to data. A process that is particularly sensitive to f q/e (ξ, µ 2 ) is single-inclusive jet production in electron-electron scattering. Alternatively, 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-tion on f q/e (ξ, µ 2 ). Once the nonperturbative distribution for a quark in an electron has 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 q/e (ξ, µ 2 ) to nonperturbative QCD effects, the expression 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 m 2 e in the Weizsäcker-Williams distribution f γ/e (ξ, µ 2 ). A second logarithm arises when one integrates Eq. (2) from m 2 e to µ 2 using the perturbative expressions for the splitting functions. This procedure implies that quarks in the electron are generated by perturbative evolution all the way down to virtualities of order m 2 e . One would not expect a probe with a 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 log 2 (µ 2 /m 2 e ) in Eq. (1) comes from integration over virtualities that are smaller 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 q/γ . In this case, the leading-order QED expression for the logarithmic contribution to the distribution that is analogous to Eq. (1) is The inadequacy of this leading-order logarithmic approximation is manifest in the logarithm of the photon mass m γ . Of course, it is well established that the distribution of a quark in 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][7][8].)