Drell-Yan as a probe of small x partons at the LHC

TL;DR

The paper addresses the problem of strong factorization-scale dependence in predictions for low-mass Drell–Yan production at high rapidity, which limits direct access to PDFs at very small $x$. The authors develop a method to fix the factorization scale by aligning the LO DGLAP evolution with the dominant NLO matrix element, identifying an optimal scale $\mu_F=\mu_0\approx 1.4\,M$ that absorbs most of the $\alpha_s\ln(1/x)$ enhancements into the PDFs. This yields predictions that are remarkably stable to variations in $\mu_F$ and shows NNLO corrections are small at this scale. Consequently, low-mass Drell–Yan measurements at the LHC, particularly at forward rapidities accessible to LHCb, can directly probe quark/antiquark PDFs in the ultra-small-$x$ region (down to $x\sim 10^{-5}$ for some kinematics) and, via DGLAP, constrain the low-$x$ gluon density. The work implies significant potential for improving global PDFs with LHC data, while highlighting the need to consider absorptive effects in evolution at the smallest $x$ values.

Abstract

The predictions of Drell-Yan production of low-mass, lepton-pairs, at high rapidity at the LHC, are known to depend sensitively on the choice of factorization and renormalization scales. We show how this sensitivity can be greatly reduced by fixing the factorization scale of the LO contribution based on the known NLO matrix element, so that observations of this process at the LHC can make direct measurements of parton distribution functions in the low x domain; x less than about 10^{-4}.

Figures (5)

• Figure 1: Diagrams (a,b) show NLO subprocesses for Drell-Yan production resulting from the splitting of the upper PDF (or 'right' PDF, in the notation of (\ref{['eq:5']})). Diagram (c) is the main NNLO subprocess $gg \to \bar{q}\gamma^* q$.
• Figure 2: (a) Sensitivity of $M=6$ GeV Drell-Yan $\mu^+\mu^-$ production, as a function of rapidity $Y$, to the choice of factorization scale: $\mu_F=M/2,\ M,\ 2M$, at LO, NLO, NNLO. (b) The bold lines correspond to the choice $\mu_F=\mu_0=1.4M$ which minimizes $C^{\rm NLO}_{\rm rem}$, and show the stability with respect to the variations $\mu=M/2,\ M,\ 2M$ in the scale of the PDFs convoluted with $C^{\rm NLO}_{\rm rem}$ -- the $\mu$ dependence is indicated by the symbolic equation at the top of the diagram. The dashed lines show that the stability disappears for other choices of $\mu_0$. The small crosses are the NNLO result. In this figure the renormalization scale is fixed at $\mu_R=M$.
• Figure 3: (a) Sensitivity of $M=6$ GeV Drell-Yan $\mu^+\mu^-$ production to the choice of renormalization scale: $\mu_R=M/2,\ M,\ 2M$, at LO, NLO, NNLO; and the fixed factorization scale $\mu_F=M$. (b) Stability of the NLO result is achieved for the optimal choice $\mu_F=\mu_0=1.4M$.
• Figure 4: The values of $x_1$ and $x_2$ of PDFs probed by the observation of Drell-Yan production, through the $qg$ channel, of a lepton-pair of mass $M=6$ GeV at a 7 TeV LHC. The gluon (quark) PDF is sampled in the $x$ region given by the dashed (continuous) curves. The distributions are shown for two different values of the rapidity, $Y$, of the lepton pair system. The two curves for each PDF at each rapidity correspond to two choices of the virtuality cutoff: namely $Q_0=1$ and $Q_0=2$ GeV.
• Figure 5: The 'LO+NLO' cross sections for the Drell-Yan production of a $M=6$ and $M=12$ GeV $\mu^+\mu^-$ pair, as a function of its rapidity, at 7 TeV, obtained using four different recent NLO sets of PDFs MSTWCT10NNPDF21ABM11. We show the 1$\sigma$ error corridor for the predictions obtained using the MSTW parton set.