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)]

TL;DR

This paper critiques previous perturbative calculations of quark distributions in electrons for jet production, emphasizing the importance of nonperturbative QCD effects and advocating for phenomenological determination of these distributions.

Contribution

It highlights the significance of nonperturbative QCD contributions to quark distributions in electrons, challenging the perturbative approximation used in prior work.

Findings

Nonperturbative effects significantly impact quark distributions in electrons.

Resolved-electron contributions are essential for accurate cross section calculations.

Phenomenological methods are needed to determine quark distributions in electrons.

Abstract

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.

Equations

$$f_{q/l}(\xi ,{\mu }^2)$$

$${\mu }^2\frac{\partial }{\partial {\mu }^2}{f}_{q/l}=P_{q\gamma }\otimes f_{\gamma /l}+P_{ql}\otimes f_{l/l}.$$

$$[P\otimes f](\xi )={\int }_{\xi }^1\frac{dz}{z}P(\xi )f(\xi /z).$$

$${\mu }^2\frac{\partial }{\partial {\mu }^2}\left(\begin{array}{c}{f}_{q_i/l}\\ f_g\end{array}\right)=\left(\begin{array}{c}{P}_{q_i\gamma }\otimes f_{\gamma /l}\\ 0\end{array}\right)+\left(\begin{array}{c}{P}_{q_{i}l}\otimes f_{l/l}\\ 0\end{array}\right)+\sum _{q_j}\left(\begin{array}{cc}{P}_{q_i{q}_j}& 2P_{q_{i}g}\\ P_{g{q}_j}& P_{gg}\end{array}\right)\otimes \left(\begin{array}{c}{f}_{q_j/l}\\ f_{g/l}\end{array}\right),$$

$$f_{q/\gamma }(\xi ,{\mu }^2)=e_q^2\frac{\alpha }{2\pi }[{\xi }^2+(1-\xi {)}^2]\log \frac{{\mu }^2}{m_{\gamma }^2}.$$

Full text

In Ref. Abelof:2016pby , 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, $f_{q/l}(\xi,\mu^{2})$, where $\xi$ is the light-cone momentum fraction of the quark and $\mu$ is the renormalization scale. Such a contribution could be termed a “resolved-lepton” contribution. The distribution that was used in Ref. Abelof:2016pby is

[TABLE]

where $m_{l}$ is the lepton mass, $e_{q}$ is the electric charge of the quark, and $\alpha$ is the quantum-electrodynamics (QED) coupling constant. The single and double logarithms of $\mu$ cancel the $\mu$-dependence of other factors in the cross section at order $\alpha^{2}\alpha_{s}^{2}$.

In Ref. Abelof:2016pby , $f_{q/l}(\xi,\mu^{2})$ is derived by making use of the Dokshitzer, Gribov, Lipatov, Altarelli, Parisi (DGLAP) evolution equation Gribov:1972ri ; Lipatov:1974qm ; Dokshitzer:1977sg ; Altarelli:1977zs in the form

[TABLE]

Here, $f_{\gamma/l}(\xi,\mu^{2})$ is the distribution of a photon in a lepton, $f_{l/l}(\xi,\mu^{2})$ is the distribution of a lepton in a lepton, $P_{q\gamma}(z)$ and $P_{ql}(z)$ are the DGLAP splitting functions, and $\otimes$ denotes the convolution

[TABLE]

(In Eq. (2), we have absorbed factors of $\alpha$ into the definitions of the splitting functions.) In Ref. Abelof:2016pby , the splitting functions are evaluated to order $\alpha$ and order $\alpha^{2}$, respectively, and the QED distributions on the right side of Eq. (2) are evaluated at leading order in $\alpha$: $f_{\gamma/l}(\xi,\mu^{2})$ is the Weizsäcker-Williams distribution, and $f_{l/l}(\xi)=\delta(1-\xi)$. The distribution in Eq. (1) is obtained by integrating Eq. (2) with the boundary condition $f_{q/l}(\xi,m_{l}^{2})=0$.

In this comment, we point out that $f_{q/l}(\xi,\mu^{2})$ 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 $\tau$ lepton, then $f_{q/l}(\xi,m_{l}^{2})$ can be computed in QCD perturbation theory, and it can be evolved perturbatively from the scale $m_{l}^{2}$ to the scale $\mu^{2}$ in order to absorb logarithms of $\mu^{2}/m_{l}^{2}$ into $f_{q/l}(\xi,\mu^{2})$. In this case, the expression in Eq. (1) is a valid approximation for $f_{q/l}(\xi,\mu^{2})$ in that it captures the logarithmic contributions at leading-order in $\alpha$.111We note that the expression in Eq. (1) omits constant terms that arise in standard renormalization schemes, such as modified minimal subtraction. However, when the lepton is an electron or a muon, $f_{q/l}(\xi,\mu^{2})$ cannot be computed in QCD perturbation theory.

The nonperturbative nature of $f_{q/l}(\xi,\mu^{2})$ can be seen by considering its DGLAP evolution. When one considers QCD corrections, the evolution equation for $f_{q/l}(\xi,\mu^{2})$ contains additional contributions that arise from the emission of real and virtual gluons by the quark:

[TABLE]

where the sum over $q_{j}$ includes both quarks and antiquarks. Suppose that one were to follow the procedure in Ref. Abelof:2016pby , evolving $f_{q/l}$ from the scale $m_{l}$ to a hard-scattering scale. The splitting functions in Eq. (4) depend on $\alpha_{s}$ at scales $\mu$ that range from $m_{l}$ to the hard-scattering scale. If $\mu$ is sufficiently large, then the splitting functions can be computed in perturbation theory. However, if $\mu$ is less than a scale of order $\Lambda_{\mathrm{QCD}}$, then the perturbation expansion for the splitting functions fails, and the evolution of $f_{q/l}$ receives nonperturbative contributions. In the case of the electron or the muon, the range of $\mu$ 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. Abelof:2016pby requires only that collinear poles through order $\alpha^{2}$ be absorbed into $f_{q/l}(\xi,\mu^{2})$, a reliable calculation of $f_{q/l}(\xi,\mu^{2})$ 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 $\alpha_{s}$, 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 $f_{q/e}(\xi,\mu^{2})$ at a scale $\mu^{2}$ 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 $f_{q/e}(\xi,\mu^{2})$ 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 information on $f_{q/e}(\xi,\mu^{2})$. 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 $f_{q/e}(\xi,\mu^{2})$ 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 $m_{e}^{2}$ in the Weizsäcker-Williams distribution $f_{\gamma/e}(\xi,\mu^{2})$. A second logarithm arises when one integrates Eq. (2) from $m_{e}^{2}$ to $\mu^{2}$ 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 $m_{e}^{2}$. 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 $\mu$ that is considered in Ref. Abelof:2016pby , much of the large coefficient $\log^{2}(\mu^{2}/m_{e}^{2})$ 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 $f_{q/\gamma}$. In this case, the leading-order QED expression for the logarithmic contribution to the distribution that is analogous to Eq. (1) is

[TABLE]

The inadequacy of this leading-order logarithmic approximation is manifest in the logarithm of the photon mass $m_{\gamma}$. 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. Cornet:2004nb ; Aurenche:2005da ; Berger:2014rva .)