Soft background fields at next-to-leading power in transverse momentum dependent SIDIS with jets

TL;DR

This paper develops a detailed factorization framework for SIDIS processes involving jets at small transverse momenta, incorporating soft modes at next-to-leading power to improve the understanding of twist-3 distributions and azimuthal asymmetries.

Contribution

It introduces a novel factorization formula at NLP using the background field method with soft modes, enabling better definition of twist-3 operators and form factors for SIDIS with jets.

Findings

Derived a factorization formula at NLP including soft modes.

Identified conditions for higher-order agreement with previous approaches.

Presented phenomenological results for azimuthal asymmetry involving twist-3 distributions.

Abstract

We present a factorization formula for the $e + h \rightarrow e + \text{jet} + X$ cross section at small transverse momenta up to next-to-leading power (NLP), derived using the background field method (BFM) with explicit inclusion of soft modes. We discuss the relation between soft modes and overlap subtractions at NLP, showing that the inclusion of soft modes enables a definition of twist-3 operators with properly subtracted rapidity and endpoint divergences. We also derive the effective current operator at NLP and identify additional structures compared to previous approaches based on soft-collinear effective theory and the BFM without soft modes. Nevertheless, for the hadronic tensor, we find agreement at the perturbative order that we are working at, and identify a condition that needs to be satisfied for this agreement to extend to higher orders. Furthermore, we construct the most general form factors for this process, taking into account the polarization of the initial states. These involve perturbatively-calculable jet functions, opening a path to precise determinations of twist-3 hadronic distributions. Finally, our formalism is illustrated with phenomenological results for a specific azimuthal asymmetry, whose leading-logarithmic contribution depends solely on a twist-3 hadronic distribution and a twist-2 jet function.