Sudakov Resummation Effects in Prompt-Photon Hadroproduction

TL;DR

This paper investigates soft-gluon threshold effects in prompt-photon hadroproduction at fixed-target energies by performing next-to-leading-logarithmic (NLL) resummation in Mellin space and matching to fixed-order NLO. The authors derive the resummed expressions for the direct photon component, discuss the fragmentation contribution, and implement the Minimal Prescription for inverse Mellin transforms, achieving an NLO+NLL prediction. Numerically, the NLL corrections amplify cross sections near partonic threshold and, crucially, significantly reduce the renormalization and factorization scale dependence for realistic kinematics, while fragmentation contributions remain subdominant in the studied hadronic configurations. Comparisons to E706 and UA6 data show improved perturbative stability but do not fully resolve the observed excess at large x_T, suggesting non-perturbative effects or intrinsic k_T models may be needed for a complete description.

Abstract

We compute the effects of soft-gluon resummation, at the next-to-leading-logarithmic level, in the fixed-target hadroproduction cross section for prompt photons. We find in general that the corrections to the fixed next-to-leading-order results are large for large renormalization scales, and small for small scales. This leads to a significant reduction of the scale dependence of the results for most experimental configurations of interest. We compare our results to the recent measurements by the E706 and UA6 collaborations.

Figures (17)

• Figure 1: Behaviour of the LO and NLO terms ${\hat{\sigma}}_{ab\to d\gamma}^{(0)}$ and ${\hat{\sigma}}_{ab\to \gamma}^{(1)}$ (see Eq. (\ref{['pxsg']})) of the direct component of the prompt-photon cross-section. The contributions of the partonic channels $ab=q{\bar{q}}$ (left) and $ab=qg$ (right) are rescaled by the factor $(1-x^2)$ and plotted as a function of $x$. The renormalization, factorization and fragmentation scales are all set equal to $\mu^2=2\hbox{$E^2_T$}$, and $e_q=1$.
• Figure 2: Left (Right): the ${\cal O}(\alpha \alpha_s^2)$ contribution to the partonic prompt-photon cross section for the process $q\bar{q} \to \gamma+X$ ($qg \to \gamma+X$) plotted as a function of $\eta=(1-\hbox{$x_T$})/\hbox{$x_T$}$. The solid line represents the exact NLO result of Ref. Aurenche; the short-dashed line is the ${\cal O}(\alpha \alpha_s^2)$ piece of of the resummed result defined by Eqs. (\ref{['gammaresqq']}) and (\ref{['gammaresqg']}); the dot-dashed line is obtained from this last result by setting the constant $C^{(1)}_{q\bar{q} \to \gamma}$ ($C^{(1)}_{qg \to \gamma}$) to 0; the dashed line is obtained using Eq. (\ref{['eq:Cqashift']}) (Eq. (\ref{['eq:Cqgshift']})), with $A=2$. The renormalization, factorization and fragmentation scales were all set equal to $\mu^2=2\hbox{$E^2_T$}$, and $e_q=1$.
• Figure 3: Partonic cross-section for the processes $q\bar{q} \to \gamma X$ (left) and $q g \to \gamma X$ (right) (in pb/GeV, and for $\hbox{$E_T$}=10$ GeV). The dotted line is the LO result; the dashed line is the exact NLO result; the solid (dotdashed) lines correspond to the NLO+NLL result, with the coefficient $A$ defined in Eq. (\ref{['eq:Cqashift']}) (left) and in Eq. (\ref{['eq:Cqgshift']}) (right) equal to 0 (2). The number of flavours $N_f$ was set equal to 4 and we have taken $\Lambda_{ \rm QCD}^{(4)}=0.151$ GeV.
• Figure 4: Same as Fig. \ref{['fig:qares10']}, for $\hbox{$E_T$}=100$ GeV.
• Figure 5: Contribution of gluon resummation at order $\hbox{${\cal O}(\alpha\alpha_s^3)$}$ and higher, relative to the exact NLO result, for photon production via $q\bar{q}$ (left plot) and $qg$ (right plot) annihilation, in $pN$ collisions at $\sqrt{S}=31.5$ GeV ($E_{\rm beam}=530$ GeV). The solid (dashed) lines correspond to $A=0$ ($A=2$). The three sets of curves correspond to the choice of scale $\mu=\mu_F=2\hbox{$E_T$},\; \hbox{$E_T$}$ and $\hbox{$E_T$}/2$, in descending order, with PDF set CTEQ4M and $N_f=5$.