Threshold Resummation for Polarized High-$p_{T}$ Hadron Production at COMPASS

To obtain information about the nucleon’s gluon helicity distribution $\Delta g$ and to explore its contribution to the proton’s spin is the main focus of several current experiments. One of the probes employed for this purpose at CERN’s COMPASS experiment is $\mu N\rightarrow\mu^{\prime}hX$, where $h$ denotes a charged hadron produced at high transverse momentum. Kinematics for the process are chosen in such a way that the photons exchanged between the muon and the nucleon are almost real, so that the process effectively becomes $\gamma N\rightarrow hX$. Its double-longitudinal spin asymmetry $A_{LL}$ is directly sensitive to $\Delta g$, thanks to the presence of the photon-gluon fusion subprocess $\gamma g\to q\bar{q}$. COMPASS has recently presented data for the spin-averaged cross section Adolph:2012nm for the process, as well as for its spin asymmetry Adolph:2015hta ; Levillain:2015twa .

Thanks to the produced hadron’s large transverse momentum, the process $\gamma N\rightarrow hX$ may be treated with perturbative methods. As is well known Klasen:2002xb , hard photoproduction cross sections receive contributions from two sources, the “direct” ones, for which the photon interacts in the usual pointlike way in the hard scattering, and the “resolved” ones, for which the photon reveals its own partonic structure. Both contributions are of the same order in perturbation theory, starting at ${\cal O}(\alpha\alpha_{s})$, with the electromagnetic and strong coupling constants $\alpha$ and $\alpha_{s}$. Next-to-leading order (NLO, ${\cal O}(\alpha\alpha_{s}^{2})$) QCD corrections for the spin asymmetry for $\gamma N\rightarrow hX$ have been derived in Refs. deFlorian1998 and Jaeger2003 ; Jaeger2005 for the direct and resolved cases, respectively.

As discussed in MelaniesPaper , in the kinematic regime accessible at COMPASS perturbative corrections beyond NLO are important. This is because typical transverse momenta $p_{T}$ of the produced hadron are such that the variable $x_{T}=2p_{T}/\sqrt{S}$ (with $\sqrt{S}$ the muon-proton center-of-mass energy) is relatively large, $x_{T}\gtrsim 0.2$. This means that the partonic hard-scattering cross sections relevant for $\gamma N\rightarrow hX$ are largely probed in the “threshold”-regime, where the initial photon and parton have just enough energy to produce a pair of recoiling high-$p_{T}$ partons, one of which subsequently fragments into the observed hadron. The phase space for radiation of additional gluons then becomes small, allowing radiation of only soft and/or collinear gluons. As a result, the cancelation of infrared singularities between real and virtual diagrams leaves behind large double- and single-logarithmic corrections to the partonic cross sections. These logarithms appear for the first time at NLO and then recur with increasing power at every order of perturbation theory. Threshold resummation Sterman1987 ; Catani:1989ne allows to sum the logarithms to all orders to a certain logarithmic accuracy. It was applied to the spin-averaged cross section at COMPASS at next-to-leading logarithm (NLL) level in Ref. MelaniesPaper , where the resummation of both the direct and the resolved contribution was performed. The resummed result for the cross section was found to be significantly higher than the NLO one, by roughly a factor two. Comparison to the COMPASS data reported in Adolph:2012nm showed that this enhancement is crucial for achieving good agreement between data and the perturbative-QCD prediction.

In the light of this result, it is clear that threshold resummation should also be taken into account in the theoretical analysis of the spin asymmetry $A_{LL}$ measured at COMPASS Adolph:2015hta . $A_{LL}$ is the ratio of the spin-dependent cross section and the spin-averaged one. Since the latter has already been addressed in MelaniesPaper , we will in this paper examine threshold resummation for polarized scattering. As a first step, we will consider the direct contributions to the cross section, which are simpler to analyze and also formally dominate over the resolved ones near partonic threshold. We plan to complete our resummation study for $A_{LL}$ in a future publication by performing threshold resummation also for the resolved contribution in the spin-dependent case. We note that the resolved contribution to $\gamma N\rightarrow hX$ is structurally equivalent to hadronic scattering $pp\to hX$, for which threshold resummation was performed in the previous literature even for the polarized case deFlorian2007 . However, Ref. deFlorian2007 only addressed the simplified case when the cross section is integrated over all rapidities of the produced hadron, while in the present case we consider an arbitrary fixed rapidity. The techniques necessary for this were devloped in Almeida2009 ; MelaniesPaper and will be used here as well.

Our paper is organized as follows: In Section II we recall the general framework for the process $\gamma N\rightarrow hX$ in QCD perturbation theory. Section III collects all ingredients for the threshold resummed spin-dependent cross section. In Section IV we present phenomenological results for the spin-dependent and spin-averaged cross sections at COMPASS, as well as for the resulting longitudinal double-spin asymmetry. Finally, we conclude our paper in Sec. V.

We consider the process

where the lepton $\ell$ and the nucleon $N$ are both longitudinally polarized and where a charged hadron $h$ is observed at high transverse momentum $p_{T}$ (see Fig. 1). Demanding the scattered lepton $\ell^{\prime}$ to have a low scattering angle with respect to the incoming one, the main contributions come from almost on-shell photons exchanged between the lepton and the nucleon. The scattering may then be treated as a photoproduction process $\gamma N\rightarrow hX$, with the incoming lepton essentially serving as a source of quasi-real photons.

We introduce the spin-averaged and spin-dependent cross sections for the lepton-nucleon process as

where the superscripts $(++)$, $(+-)$ denote the helicities of the incoming particles. Using factorization, the differential spin-dependent cross section (as function of the hadron’s transverse momentum $p_{T}$ and pseudorapidity $\eta$) may be written as Jaeger2005 ; MelaniesPaper :

the sum running over all possible partonic channels $ab\rightarrow cX$. The $\Delta f_{b/N}\left(x_{n},\mu_{fi}\right)$ are the polarized parton distribution functions of the nucleon, which depend on the momentum fraction $x_{n}$ carried by parton $b$ and on an initial-state factorization scale $\mu_{fi}$. They can be written as differences of distributions for positive or negative helicity in a parent nucleon of positive helicity,

In Eq. (3) we have introduced also “effective” parton distributions in a lepton, $\Delta f_{a/\ell}\left(x_{\ell},\mu_{fi}\right)$, which we shall elaborate on further below. For now we just note that in terms of spin-dependence they are defined exactly as in (4). The $D_{h/c}(z,\mu_{ff})$ in (3) are the parton-to-hadron fragmentation functions that describe the hadronization of parton $c$ into hadron $h$, with $z$ being the fraction of the parton $c$’s momentum taken by the hadron and $\mu_{ff}$ a final-state factorization scale. Finally, the $d\Delta\hat{\sigma}_{ab\rightarrow cX}$ are the spin-dependent cross sections for the partonic hard-scattering processes $ab\rightarrow cX$. In analogy with (II) they are defined as

the indices now denoting the helicities of the incoming partons. The $d\Delta\hat{\sigma}_{ab\rightarrow cX}$ are perturbative and may hence be expanded in terms of the strong coupling constant $\alpha_{s}$,

Apart from the partonic kinematic variables that will be introduced shortly, they depend on the factorization scales and also on a renormalization scale $\mu_{r}$. We note that all formulas presented so far may be easily written for the spin averaged case by simply summing over helicities in (4),(5) instead of taking differences. This then gives the unpolarized cross section introduced in Eq. (II) in terms of the usual spin-averaged parton distributions $f_{a/\ell},f_{b/N}$ and partonic cross sections $d\hat{\sigma}_{ab\rightarrow cX}$.

In Eq. (3) we have introduced a number of kinematic variables. The partonic cross sections have been written differential in

with the Mandelstam variables

where $p_{a},p_{b},p_{c}$ are the four-momenta of the participating partons and where $S=(p_{\ell}+p_{n})^{2}$, with the lepton (nucleon) momentum $p_{\ell}$ ($p_{n}$). Furthermore,

where $x_{T}\equiv 2p_{T}/\sqrt{S}$, and the relationship between the hadron and the parton level center-of-mass system rapidities is

Finally, the lower integration bounds in Eq. (3) are given by

An important aspect of photoproduction cross sections is that the quasi-real photon can interact in two ways. For the direct contributions (see Fig. 1), it participates directly in the hard-scattering, coupling in the usual pointlike way to quarks and antiquarks. However, as is well established, the photon may also itself behave like a hadron, revealing its own partonic structure in terms of quarks, antiquarks, and gluons, as shown in Fig. 2. The associated contributions are known as resolved photon contributions. The physical cross section is the sum of the direct and the resolved part:

Both contributions are captured by Eq. (3) by introducing an effective spin-dependent parton distribution for the lepton:

where $\Delta P_{\gamma\ell}(y)$ is the polarized Weizsäcker-Williams spectrum and $\Delta f_{a/\gamma}$ describes the distribution of parton $a$ inside the photon. Equation (13) also applies to the direct case, see Fig. 1, where parton $a$ is an elementary photon and hence

The Weizsäcker-Williams spectrum is given by deFlorian:1999ge :

where $\alpha$ is the fine structure constant. $\Delta P_{\gamma\ell}$ describes the (nearly) collinear emission of a polarized photon with momentum fraction $y$ by a polarized lepton with mass $m_{\ell}$. The low virtuality $Q^{2}$ of the photon is restricted by an upper limit $Q_{{\mathrm{max}}}^{2}$ that is determined by the experimental conditions.

In the direct case, there are three different LO subprocesses,

where the final-state particle in brackets is understood to remain unobserved, while the other parton fragments into the observed hadron. We note that the photon-gluon-fusion process is symmetric under exchange of $q$ and $\bar{q}$ in the final state. The spin-dependent cross sections for the LO subprocesses are deFlorian1998 :

with $C_{F}=4/3$, $T_{R}=1/2$ and the fractional electromagnetic charge $e_{q}$ of the quark.

In the resolved case, all $2\to 2$ QCD partonic processes contribute at LO:

where either of the final-state partons may fragment into the observed hadron. As the photon’s parton distributions $\Delta f_{a/\gamma}$ are of order $\alpha/\alpha_{s}$, both the direct and the resolved LO contributions are of order $\alpha^{2}\alpha_{s}$ for the $\ell N$ cross section (or of order $\alpha\alpha_{s}$ for the $\gamma N$ one) Klasen:2002xb . Note that the $\Delta f_{a/\gamma}$ contain a perturbative “pointlike” contribution that dominates at high $x_{\gamma}$, but also a nonperturbative “hadronic” piece that is associated with the photon converting into a vector meson and is important at low-to-mid $x_{\gamma}$.

As shown in Eq. (17), the LO partonic cross sections are proportional to $\delta(1-w)$. From (7) one finds that the invariant mass squared of the final state that recoils against the fragmenting parton is given by

The $\delta(1-w)$ at LO thus reflects the fact that the recoil consists of a single massless parton. At NLO, the partonic cross sections contain various types of distributions in $(1-w)$. Analytical expressions have been obtained in Refs. deFlorian1998 ; Jaeger2003 ; Jaeger2002 ; Aversa:1988vb ; Gordon:1994wu ; Hinderer:2015hra . For each process the result may be cast into the form

where the coefficients $A(v),B(v),C(v),F(v,w)$ depend on the process under consideration, and where the plus-distributions are defined as usual by

The function $F(v,w)$ in (20) contains all remaining terms without distributions in $(1-w)$. The terms with plus-distributions give rise to the large double-logarithmic corrections that are addressed by threshold resummation. Their origin lies in soft-gluon radiation, and they recur with higher power at every higher order of perturbation theory. For the $k$-th order QCD correction, the leading terms are proportional to $\alpha_{s}^{k}[\ln^{2k-1}(1-w)/(1-w)]_{+}$ (not counting the overall power of the partonic process in $\alpha_{s}$). Subleading terms are down by one or more powers of $\ln(1-w)$.

Both the direct and the resolved contributions have the structure shown in (20). In the following we discuss the all-order resummation of the threshold logarithms in the direct part of the cross section, which we separate from the resolved part adopting the $\overline{{\mathrm{MS}}}$ scheme. We perform the resummation to next-to-leading logarithm (NLL), which means that the three “towers” $\alpha_{s}^{k}[\ln^{2k-1}(1-w)/(1-w)]_{+}$, $\alpha_{s}^{k}[\ln^{2k-2}(1-w)/(1-w)]_{+}$, $\alpha_{s}^{k}[\ln^{2k-3}(1-w)/(1-w)]_{+}$ are taken into account to all orders in the strong coupling.

As discussed in the Introduction, measurements of cross sections and spin asymmetries for the photoproduction process $\mu N\rightarrow hX$ are carried out in the COMPASS experiment Adolph:2012nm ; Adolph:2015hta at CERN. We therefore present our phenomenological results for COMPASS kinematics. COMPASS uses a longitudinally polarized muon beam with mean beam energy of $E_{\mu}=160$ GeV, resulting in $\sqrt{S}=17.4$ GeV. Both deuteron and proton targets are available. COMPASS imposes the cut $Q^{2}_{{\mathrm{max}}}=1$ GeV² on the virtuality of the exchanged photon which we use in the Weizsäcker-Williams spectrum (15). As in COMPASS, we also implement the cuts $0.2\leq y\leq 0.9$ on the fraction of the lepton’s momentum carried by the photon, and $0.2\leq z\leq 0.8$ for the fraction of the energy of the virtual photon carried by the hadron. Finally, charged hadrons are detected in COMPASS if their scattering angle is between $10\leq\theta\leq 120$ mrad, corresponding to $-0.1\leq\eta\leq 2.38$ in the hadron’s pseudorapidity. We integrate over this range.

Our default choice for the helicity parton distributions is the set of deFlorian:2014yva (referred to as DSSV2014). We adopt the fragmentation functions of Ref. fDSS (DSS) throughout this work. In the calculations of the NLL resummed unpolarized cross sections we follow Ref. MelaniesPaper and use the numerical code of that work. Unless stated otherwise, we employ the unpolarized parton distribution functions of Ref. Martin:2009iq (referred to as MSTW). For comparisons we will also present results for the NLO resolved contributions, for which we will adopt the unpolarized and polarized photonic parton distributions of Refs. Gluck:1991jc and Gluck:1992zq , respectively. We furthermore choose all factorization/renormalization scales to be equal, $\mu_{r}=\mu_{fi}=\mu_{ff}\equiv\mu$. We usually choose $\mu=p_{T}$, except when investigating the scale dependence of the theoretical predictions.

We have studied the impact of threshold resummation at next-to-leading logarithmic level on the spin-dependent cross section for $\gamma N\rightarrow hX$ at high transverse momentum $p_{T}$ of the hadron $h$, and on the resulting double-longitudinal spin asymmetry $A_{LL}$. For the present work, we have implemented the resummation only for the direct contribution to the cross section. For the kinematics relevant for the COMPASS experiment we find that the spin-dependent cross section receives much smaller enhancements by resummation than the spin-averaged one treated in Ref. MelaniesPaper . As a result, threshold effects do not cancel in the double-spin asymmetry, and the prediction for $A_{LL}$ decreases when resummation is taken into account. Definite conclusions about the impact of resummation on the spin asymmetry will become possible only when also the resummation for the resolved component has been carried out, which we plan to do in future work. Only then will an extraction of the proton’s gluon helicity distribution $\Delta g$ become meaningful. We also note that the scale dependence of the perturbative cross section remains uncomfortably large even when resummation for the direct piece is taken into account. In order to improve this it may, eventually, be necessary to extend resummation to next-to-next-to-leading logarithmic level, following the techniques developed in Ref. Hinderer:2014qta .

Comparison to the recent COMPASS data Adolph:2015hta shows that the theoretically predicted spin asymmetries fail to reproduce the data well. Especially for the proton target the data show a nearly vanishing asymmetry, while the theoretical result appears to be always clearly positive. In fact, it is worth stressing that each of the theoretical results shown in Fig. 6 predicts a larger spin asymmetry for the proton than for the deuteron, in contrast to the trend seen in the data. This feature of the theoretical predictions is likely no accident, as a simple study of the LO direct contributions shows Julius . Clearly, future work is needed in order to clarify in how far the leading-twist perturbative-QCD framework can accommodate a larger spin asymmetry for $\mu d\rightarrow\mu^{\prime}hX$ than for $\mu p\rightarrow\mu^{\prime}hX$.