$k_T$ and threshold resummations

Abstract

We demonstrate that both the $k_T$ and threshold resummations can be performed in the Collins-Soper resummation formalism by evaluating soft gluon emissions with infrared cutoffs for the longitudinal and transverse loop momenta, respectively. The reason the $k_T$ resummation for a parton distribution function leads to suppression in the large $b$ region, $b$ being the conjugate variable of parton transverse momentum $k_T$, and the threshold resummation leads to enhancement in the large $N$ limit, $N$ being the moment of a distribution function, is a consequence of opposite directions of double-logarithm evolutions. The $k_T$ and threshold resummations for an energetic final-state jet give suppression. The switch of the threshold resummation from enhancement to suppression is attributed to a nonvanishing jet invariant mass. In the same framework we derive a unification of the $k_T$ and threshold resummations for a parton distribution function by requiring infrared cutoffs for both longitudinal and transverse loop momenta. This unified resummation exhibits suppression at large $b$, similar to the $k_T$ resummation, and exhibits enhancement at small $b$, similar to the threshold resummation.