Matches and mismatches in the descriptions of semi-inclusive processes at low and high transverse momentum

We investigate the transverse-momentum-dependence in semi-inclusive deep inelastic leptoproduction of hadrons. There are two different theoretical approaches to study this dependence, one for low and one for high transverse momentum of the observed hadron. We systematically investigate their connection, paying special attention to azimuthal distributions and to polarization dependence. In the region of intermediate transverse momentum, where both approaches are applicable, we find that their results match for certain observables but not for others. Interpolating expressions are discussed for the case where one has no matching. We then use power counting to determine which mechanism is dominant in various azimuthal and spin asymmetries that are integrated over the transverse momentum. Our findings have consequences for the extension of transverse-momentum-dependent factorization beyond leading twist. They also shed light on the problem of resumming logarithms of transverse momentum for azimuthal distributions. Our results can be carried over to the Drell-Yan process and to two-hadron production in e+e- annihilation