Next-to-Next-to-Leading-Order Logarithmic Corrections at Small Transverse Momentum in Hadronic Collisions

We study the region of small transverse momenta in $q\overline{q}$- and $\mathrm{gg}$-initiated processes with no colored particle detected in the final state. We present the universal expression of the $O\left({\alpha }_s^2\right)$ logarithmically enhanced contributions up to next-to-next-to-leading-order logarithmic accuracy. From there we extract the coefficients that allow the resummation of the large logarithmic contributions. We find that the coefficient known in the literature as $B^{\left(2\right)}$ is process dependent, since it receives a hard contamination from the one-loop correction to the leading-order subprocess. We present the general result of $B^{\left(2\right)}$ for both quark and gluon channels.