Next-to-next-to-leading power corrections to unpolarized Semi-Inclusive Deep Inelastic Scattering

TL;DR

This work extends rapidity-factorization methods to SIDIS to derive NNLP ($1/Q^2$) corrections to the unpolarized hadronic tensor and associated structure functions. It delivers analytic expressions for $F_{UU,T}$, $F_{UU,L}$, $F_{UU}^{ m cos\phi_h}$, and $F_{UU}^{ m cos2\phi_h}$ that include convolutions of $f_1$ with $D_1$ and $h_1^ot$ with $H_1^ot$, and rewrites results in both momentum and $b_T$ space. Numerical studies using Gaussian TMDs and standard collinear inputs show NNLP effects can be sizable, particularly at low to moderate $Q^2$, affecting the interpretation of azimuthal modulations and the longitudinal-to-transverse cross-section ratio $R_{ m SIDIS}$. Comparisons with HERMES and COMPASS illustrate partial agreement and highlight the need for refined phenomenology and future data from JLab and the EIC. The authors outline plans to extend the framework to polarized observables and to implement full TMD evolution, aiming for a more complete understanding of the 3D nucleon structure at subleading power.

Abstract

Semi-Inclusive Deep Inelastic Scattering (SIDIS) is a key tool for exploring the three-dimensional structure of the nucleon through Transverse Momentum Dependent parton distributions and fragmentation functions. While leading-power contributions to the SIDIS cross-section are well established, next-to-leading (NLP) of order $1/Q$ and next-to-next-to-leading power (NNLP) corrections of order $1/Q^2$ to the hadronic tensor have only recently begun to be systematically investigated. These corrections are essential for the reliable phenomenology and interpretation of modern high-precision data. In recent papers by one of the authors, NNLP corrections to Drell-Yan process were derived using rapidity factorization formalism. In the present work we extend this approach to SIDIS and obtain analytic expressions for the unpolarized structure functions. We derive NNLP corrections that include convolutions of unpolarized distributions, $f_1$, with unpolarized fragmentation functions, $D_1$, and Boer-Mulders functions, $h_1^\perp$, with Collins fragmentation functions, $H_1^\perp$. We compare our results with previous formulations, provide numerical studies, confront our predictions with HERMES and COMPASS measurements, and present predictions for future experiments at Jefferson Lab and the Electron-Ion Collider.

Figures (15)

• Figure 1: Kinematics of the SIDIS process $lp\to l^\prime h X$ in the one photon exchange approximation in the Trento frame Bacchetta:2004jz.
• Figure 2: Illustration of the the two frames used to describe the kinematics of the SIDIS process, as discussed in the text. The figure is from Ref. Boussarie:2023izj.
• Figure 3: $F_{UU,T}$ structure function for pion production off the proton at $Q^2=3$ GeV$^2$, $x=0.2$, $z_{h} = 0.3$. Left panel shows $\pi^+$ leading power contribution (blue line), next-to-next-to-leading power contribution from $f_1 D_1$ (orange dashed line), and from $h_1^\perp H_1^\perp$ (green dot-dashed line). Right panel shows next-to-next-to-leading power contribution from $h_1^\perp H_1^\perp$ for $\pi^+$ (blue line) and $\pi^-$ (orange dashed line) production.
• Figure 4: The ratio of $F_{UU,T}$ structure function over the leading power $F_{UU,T}^{\rm LP}$ for $\pi^+$ production off the proton at $Q^2=3$ GeV$^2$, $x=0.2$, $z_{h} = 0.3$.
• Figure 5: $F_{UU,L}$ for $\pi^+$ (left panel) and $\pi^-$ (right panel) production off the proton at $Q^2=5$ GeV$^2$ and $x=0.2$, $z_{h} = 0.3$. The contribution from the first term in Eq. \ref{['eq:FUUL']}, convolution of $f_1 D_1$, is shown as blue line, the second term convolution of $h_1^\perp D_1^\perp$ in Eq. \ref{['eq:FUUL']} is shown as orage dashed line.