Threshold Resummation and Rapidity Dependence

TL;DR

This work extends threshold resummation to fully differential prompt-photon production by introducing a rapidity-aware formalism that preserves $\eta$-dependence. It combines Mellin moments in $x_T^2$ with a Fourier transform in rapidity, and develops NLL exponents alongside explicit ${\cal O}(\alpha_s)$-accurate hard-coefficient functions $C_{ab}^{(1)}(\hat{\eta})$, enabling a detailed comparison of the LOS and CMN resummation approaches. The authors implement a minimal-prescription inverse transform and perform numerical studies showing substantial reductions in scale dependence and notable enhancements at high $p_T$ and large $|\eta|$ within fixed-target kinematics. The results demonstrate that rapidity-resolved resummation yields results closely aligned with rapidity-integrated predictions while offering improved differential control, with implications for high-$x_T$ phenomenology and potential connections to jet data at large rapidities. Overall, the paper provides a comprehensive framework for rapidity-dependent threshold resummation and benchmarks two competing formalisms in a unified, phenomenologically relevant setting.

Abstract

We study the effects of threshold resummation on the rapidity dependence of single-particle-inclusive cross sections, using the prompt photon cross section as an example. We make use of the full resummation formula at next-to-leading logarithmic accuracy and develop a new technique for treating rapidity in resummation. We compare our phenomenological results with those of previous studies and discuss differences and similarities of the two existing resummation formalisms.

Figures (5)

• Figure 1: Contour in Mellin-$N$ space for inverting the resummed cross section. The asterisks denote the rightmost poles of the parton densities which have acquired an imaginary part through the Fourier transformation in rapidity. $N_L$ is the position of the Landau pole; see text.
• Figure 2: Threshold-resummed prompt photon cross sections, normalized to NLO, based on the formalisms of CMN (dashed) and LOS (solid). The parton densities were taken from grv. The E706 prompt photon data e706 are also shown in the same normalization.
• Figure 3: Rapidity dependence of the prompt photon cross section in pp and $\bar{\rm p}$p collisions at $\sqrt{S}=24.3$ GeV. Dashed lines are NLO, solid lines denote the cross section resummed to NLL accuracy. For each case, the results have been calculated for three choices of scales, $\mu_F = \mu_R = 2 p_T, p_T,p_T/2$, from lower to upper. The data are from UA6 ua6.
• Figure 4: Rapidity dependence of the prompt photon cross section in pN collisions at $\sqrt{S}=31.5$ GeV. Dashed and dotted lines are NLO, solid and dot-dashed lines denote the cross section resummed to NLL accuracy. For each case, the results have been calculated for five choices of scales, $\mu_F = \mu_R = 2 p_T, p_T,p_T/2$ (dashed and solid), and $\mu_F = 2 p_T, p_T/2$ at fixed $\mu_R=p_T$ (dotted and dot-dashed), from lower to upper.
• Figure 5: Threshold-resummed prompt photon cross sections, normalized to NLO, for p$\bar{\rm p}$ collisions at $\sqrt{S}=1800$ GeV.