A study on the interplay between perturbative QCD and CSS/TMD formalism in SIDIS processes

TL;DR

The paper assesses the interplay between perturbative QCD and CSS/TMD resummation for SIDIS across the full transverse momentum range. It applies the CSS framework in $b_T$-space, decomposing the cross section into a resummed $W$ term and a regular $Y$ term, while incorporating a non-perturbative Sudakov factor and the $b_*$ prescription. The study finds that conventional Y-term matching fails in studied kinematics due to significant non-perturbative effects and that the perturbative expansion in the matching region is unreliable, with strong sensitivity to $b_{max}$ and $S_{NP}$. These results highlight the need for improved matching prescriptions and comprehensive experimental data to constrain non-perturbative inputs and validate resummation approaches in SIDIS.

Abstract

We study the Semi-Inclusive Deep Inelastic Scattering (SIDIS) cross section as a function of the transverse momentum, $q_T$. In order to describe it over a wide region of $q_T$, soft gluon resummation has to be performed. Here we will use the original Collins-Soper-Sterman (CSS) formalism; however, the same procedure would hold within the improved Transverse Momentum Dependent (TMD) framework. We study the matching between the region where fixed order perturbative QCD can successfully be applied and the region where soft gluon resummation is necessary. We find that the commonly used prescription of matching through the so-called Y-factor cannot be applied in the SIDIS kinematical configurations we examine. In particular, the non-perturbative component of the resummed cross section turns out to play a crucial role and should not be overlooked even at relatively high energies. Moreover, the perturbative expansion of the resummed cross section in the matching region is not as reliable as it is usually believed and its treatment requires special attention.

Figures (11)

• Figure 1: Perturbative contributions to the SIDIS cross sections, $d\sigma^{ASY}$, $d\sigma^{NLO}$ and $Y$ factor, corresponding to three different SIDIS kinematical configurations: on the left panel $\sqrt{s}=1$ TeV, $Q^2=5000$ GeV$^2$, $x=0.055$ and $z=0.325$; on the central panel a HERA-like experiment with $\sqrt{s}=300$ GeV, $Q^2=100$ GeV$^2$, $x=0.0049$ and $z=0.325$; on the right panel, a COMPASS-like experiment with $\sqrt{s}=17$ GeV, $Q^2=10$ GeV$^2$, $x=0.055$ and $z=0.325$.
• Figure 2: Resummed term of the SIDIS cross section including the non-perturbative contribution $S_{N\!P}$ in the Sudakov factor, calculated at three different values of $g_1$ and $g_{1f}$ and corresponding to the three different SIDIS kinematical configurations defined in Fig. \ref{['f1']}. Here $b_{max}= 1.0$ GeV$^{-1}$.
• Figure 3: The resummed cross section $W^{NLL}(q_T)$ corresponding to the three different SIDIS kinematical configurations defined in Fig. \ref{['f1']}. Here $b_{max}$ varies from $1.5$ GeV$^{-1}$ to $0.5$ GeV$^{-1}$, while $g_1$ and $g_{1f}$ are fixed at $g_1=0.3$ GeV$^2$, $g_{1f}=0.1$ GeV$^2$.
• Figure 4: The resummed term $W^{SIDIS}(b_*)\exp[S_{NP}(b_T)]$ as a function of $b_T$ corresponding to three different SIDIS kinematical configurations, $Q^2=5000$ GeV$^2$, $Q^2=100$ GeV$^2$, and $Q^2=10$ GeV$^2$. Here $b_{max}$ varies from $0.5$ GeV$^{-1}$ (left panel) to $1$ GeV$^{-1}$ (central panel), $1.5$ GeV$^{-1}$ (right panel). In order to compare different kinematical configurations, in this plot we fix $x$ and $z$ to values compatible with all of them: $x=0.055$ and $z=0.325$.
• Figure 5: The resummed term $b_T W^{SIDIS}(b_*)\exp[S_{NP}(b_T)]$ corresponding to the three different SIDIS kinematical configurations defined in Fig. \ref{['f1']}. Here $b_{max}$ varies from $1.5$ GeV$^{-1}$(solid line) to $1$ GeV$^{-1}$(dashed line), $0.5$ GeV$^{-1}$ (dotted line).