Success of collinear expansion in the calculation of induced gluon emission

TL;DR

This paper clarifies the validity and application of the collinear expansion in calculating induced gluon emission in DIS, reconciling different theoretical approaches and correcting misconceptions in previous work.

Contribution

It demonstrates the conditions under which collinear expansion is valid and connects various theoretical frameworks, correcting prior misconceptions about variable changes.

Findings

Collinear expansion is valid for <k_ ext{perp}^2> \\ll \\ell_ ext{perp}^2 \\ll Q^2.

The double-hard term in twist-four can be independently calculated.

Corrections beyond collinear approximation introduce a logarithmic factor difference.

Abstract

We clarify the confusion in a recent paper by Aurenche, Zakharov and Zaraket (AZZ) \cite{azz} over the procedure and region of validity of the collinear approximation in the twist expansion approach to induced gluon emission in deeply inelastic scattering (DIS) off a nucleus target. We point out that, in this approach to the semi-inclusive spectrum, the transverse momentum $\vec\ell_\perp$ of the induced gluon must be fixed in the collinear expansion in the transverse momentum $\vec k_\perp$ of the initial partons, therefore the result is valid for $<k_\perp^2> \ll \ell_\perp^2 \ll Q^2$. In the twist-four contribution, one can single out the double-hard term corresponding to collinear quark-gluon Compton scattering which can be calculated independently of the collinear expansion approach. We will discuss the connection between the collinear approximation in the twist expansion approach and the small $k_T$ approximation of the results in the Light-Cone Path Integral (LCPI) and Gyulassy-Levai-Vitev (GLV) opacity expansion approach. We point out the misconstrued variable change by AZZ before the $k_T$ expansion in LCPI and opacity expansion approach, without which one obtains the same result for the induced gluon spectrum under collinear approximation as in the twist expansion approach. We also show that corrections beyond the collinear approximation to the transverse momentum integrated gluon spectrum within the static potential model in the GLV approach give rise to a logarithmic factor difference from the result of collinear approximation.