Closed-form Laurent expansion of the additive plateau angularity integral

Derive a closed-form expression for the integrated rapidity-weighting function \(F_{\Pi,a}(\epsilon)=\int_{-\infty}^{\infty}\!\mathrm{d}\eta\,[f_{\Pi,a}(\eta;\omega,\eta_c,H)]^{2\epsilon}\) for general \(\epsilon\), where \(f_{\Pi,a}\) is the additive plateau angularity weighting, rather than leaving its \(\mathcal{O}(\epsilon^2)\) coefficient as a numerical integral.

Background

The integrated shape function $F_e(\epsilon)$ supplies the coefficients needed to construct the soft function. For the $L_p$-angularity and multiplicative plateau families, the paper obtains closed-form expressions, while the additive plateau angularity requires a remaining numerical integral at the order needed for NNLO.

The unresolved issue is specifically whether the general-$\epsilon$ integral for the additive plateau angularity can be evaluated in closed form. The authors provide the first relevant Laurent coefficient in closed form, but retain the second coefficient as an integral to be computed numerically.