Extension of the non-pole technique to the twist-3 gluon distribution contribution in $pp$ collisions

Abstract

It is known that the Sivers effect is described as a twist-3 effect within the collinear factorization framework and has the characteristic feature that it arises from the pole contribution in a hard parton scattering. The calculation formalism that focuses only on the pole part was established in the 2000s and SSAs were calculated for many processes in $ep$ and $pp$ collisions. On the other hand, there also exists a contribution from twist-3 fragmentation functions regarded as the Collins effect. The Collins effect arises from the nonpole part of the hard scattering and its calculation is formulated in a somewhat different manner from that for the Sivers effect. In recent years, some attempts have been made to revisit the Sivers effect by applying the ``nonpole'' formalism developed for the Collins effect. For $ep$ collisions, this approach has successfully reproduced the known results. For $pp$ collisions, however, the presence of both initial-state-interaction and final-state-interaction makes the calculation more complicated and the known results have not yet been reproduced. In this paper, we provide a solution to this problem and develop the nonpole method so that it can be applied to the Sivers effect in any process.