Soft-gluon resummation for high-$p_T$ inclusive-hadron production at COMPASS

We study the cross section for the photoproduction reaction $\gamma N\to hX$ in fixed-target scattering at COMPASS, where the hadron $h$ is produced at large transverse momentum. We investigate the role played by higher-order QCD corrections to the cross section. In particular we address large logarithmic “threshold” corrections to the rapidity-dependent partonic cross sections, which we resum to all orders at next-to-leading accuracy. In our comparison to the experimental data we find that the threshold contributions are large and improve the agreement between data and theoretical predictions significantly.

Figure 1

Photoproduction in lepton-nucleon scattering. The virtual photon is required experimentally to have low virtuality.

Figure 2

Schematic illustration of the factorization of the cross section close to threshold. The plane symbolizes the cut, which separates the amplitude and its complex conjugte. For each of the initial- and final-state partons there is a function (${\Delta }_{a,b,c},J_d$) describing soft-gluon emissions off these partons. The hard scattering part $H$ is depicted as amplitude and its complex conjugate. It can be completely separated from the soft function $S$, shown in the lower part of the figure. The double lines represent eikonal lines.

Figure 3

The two contours in Mellin-$N$ space for inverting the product of moments of the resummed partonic cross sections and the fragmentation functions. The crosses symbolize the poles of the fragmentation functions and LO cross sections on the real axis. $N_L$ is the position of the leftmost Landau pole.

Figure 4

Comparison of the LO, NLO and resummed (“resum”) calculations to COMPASS data. The error bars shown for the experimental data are the quadratic sums of statistical and systematic uncertainties. In addition, there is a 10% normalization uncertainty due to the luminosity determination. For scale $\mu =p_T/2$ we only show results for $p_T\ge 2\mathrm{GeV}$.

Figure 5

The direct contribution versus the resolved one in the $\overline{\mathrm{MS}}$ scheme for the photon’s parton distributions. The crosses denote the first-order expansions of the direct and resolved resummed cross sections.

Figure 6

Soft-gluon “$K$ factors” relative to NLO, as defined in Eq. (64), for COMPASS kinematics. The numbers labeling the curves refer to the superscript $n$ in $K^{(n)}$ in Eq. (64).

Figure 7

Ratio of production cross sections for $h^{-}$ over $h^{+}$: $d{\sigma }_{{\mu }^{+}d\to {\mu }^{{+}^{\prime }}+h^{-}}/d{\sigma }_{{\mu }^{+}d\to {\mu }^{{+}^{\prime }}+h^{+}}$. The error bars of the experimental data are statistical only.

Figure 8

Ratios of the $y$-dependent resummed and NLO cross sections, integrated over various $p_T$ bins. The cross sections are averaged over the ranges $[y-0.05:y+0.05]$.