Kinematic power corrections for TMD factorization theorem of semi-inclusive deep-inelastic scattering

TL;DR

This work develops a gauge- and frame-invariant extension of the SIDIS TMD factorization theorem by summing kinematic power corrections (KPCs) that accompany the leading-twist, twist-two TMD distributions. The authors derive the complete twist-two hadron tensor within the TMD-with-KPCs framework, detailing the twist-two quark and antiquark correlators, their momentum-space representations, and the associated convolution structure that yields all SIDIS structure functions, including those activated by longitudinal photon contributions. They implement a Lorentz- and rapidity-factorization consistent convolution, with a finite integration domain in the parton momenta, and provide explicit expressions for the SIDIS structure functions in terms of twist-two TMD PDFs and FFs, connected through a common hard coefficient $C_{0, ext{DIS}}$ and rapidity scales $oldsymbol{ ext{ζ}}$, $ar{oldsymbol{ ext{ζ}}}$ (with $oldsymbol{ζ}ar{oldsymbol{ζ}}=Q^4$). Numerical studies using ART25 TMDs show KPCs contribute only a few percent at $Q\, ext{≈10 GeV}$ but can reach tens of percent at $Q\≈2$ GeV, indicating that including KPCs is essential for accurately describing current SIDIS data and potentially reconciling tensions among TMD extractions at low $Q$.

Abstract

We evaluate the complete set of kinematic power corrections (KPCs) to the leading power (LP) term of the transverse momentum dependent (TMD) factorization theorem for semi-inclusive deep-inelastic scattering (SIDIS) with a polarized target. This formulation restores the contributions of twist-two TMD distributions to all structure functions, including those that vanish at leading power, such as contributions of longitudinal photons. The resulting expressions are explicitly gauge- and frame-invariant, and inherit all key features of the standard TMD factorization framework, including the coefficient functions and the evolution equations. Numerical estimations show that KPCs contribute only a few percent at $Q\sim10$GeV, but can reach several tens of percents when $Q\sim 2$GeV. Consequently, accounting for kinematic power corrections can be vital for an accurate theoretical description of current SIDIS measurements.

Figures (6)

• Figure 1: Schematic representation of the "pyramid" of power corrections in the TMD factorization framework. The levels indicate specific orders in the $1/Q$-expansion, starting with LP at the top. The green bubbles represent power-suppressed terms of different types with three main categories. In the present work, we consider the TMD-with-KPCs approach, which consists in the summation of all KPCs, i.e. all terms along the facing edge.
• Figure 2: Description of the azimuthal angles according to the Trento conventions Bacchetta:2004jz.
• Figure 3: Integration domain for the convolution integral (\ref{['hadronTensor']}) in $\{\bm{k}_1^2,\bm{k}_2^2\}$(left) and $\{\xi,\zeta\}$(right) planes. The region $R_T$ is generated by the constraint $\bm{k}_1+\bm{q}_T=\bm{k}_2$, while the region $R_{\xi\zeta}$ is defined by the condition $0<\xi,\zeta<1$.
• Figure 4: Comparison of TMD-with-KPCs prediction to the pure LP TMD factorization expression for angle-integrated SIDIS cross-section as a function of $Q$. Solid lines indicate the comparison of $F_{UU,T}$ only, while dashed lines include the contribution $+\varepsilon F_{UU,L}$ term. The values of $(x,z,p_\perp)$ are indicated in the figure. The value of $\varepsilon$ is defined by (\ref{['def:varepsilon']}) and is $\varepsilon\sim 0.85-1.$ for the present kinematics.
• Figure 5: Comparison of TMD-with-KPCs prediction to the pure LP TMD factorization expression for angle-integrated SIDIS cross-section as a function of $p_\perp$. Solid lines indicate the comparison of $F_{UU,T}$ only, while dashed lines include the contribution $+\varepsilon F_{UU,L}$ term. The comparison is done for $z=0.5$ and values of $(x,Q)$ are indicated in the figure. The value of $\varepsilon$ is defined by (\ref{['def:varepsilon']}) and is $\varepsilon\sim 0.85-1.$ for the present kinematics.