Determine the preferable stronger convergence factor for gluon-mediated hard processes

Determine which stronger convergence-factor form provides the best numerical performance for gluon-mediated hard processes with Born singularities behaving as μ⁻⁴: the squared smooth factor \( [1+p_{\mathrm T}^2/\mu^2]^{-2} \) or the alternative factor involving \(Q^2_{(ai,b)}/\mu^4\), and establish whether either form is preferable to the theta-function implementation.

Background

For processes mediated by a t-channel gluon, such as $qq'\to qq'$, the underlying Born matrix element can diverge like $Q^4/\mu^4$ as the spurious-singularity scale $\mu$ tends to zero. The smooth convergence factor used for the quark-mediated example therefore may not provide sufficient damping, whereas a theta-function factor does suppress the relevant region.

The paper proposes two candidate stronger smooth factors: the square of the previously studied factor and a form proportional to $1/[1+p_{\mathrm T}^2Q^2_{(ai,b)}/\mu^4]$. Their integrated effects can be obtained, but their relative numerical performance is not investigated in the paper.