  The transverse momentum dependent (TMD) and collinear higher twist
theoretical factorization frameworks are the most frequently used approaches to
describing spin dependent hard cross sections weighted by and integrated over
transverse momentum. Of particular interest is the contribution from small
transverse momentum associated with the target bound state. In phenomenological
applications, this contribution is often investigated using transverse momentum
weighted integrals that sharply regulate the large transverse momentum
contribution, for example with Gaussian parametrizations. Since the result is a
kind of hybrid of TMD and collinear (inclusive) treatments, it is important to
establish if and how the formalisms are related in applications to weighted
integral observables. The suppression of a large transverse momentum tail, for
example, can potentially affect the type of evolution that is applicable. We
find that a naive version of a widely used identity relating the
$k_T^2$-weighted and integrated Sivers TMD function to a renormalized twist-3
function has strongly ambiguous ultraviolet contributions, and that corrections
to it are not necessarily perturbatively suppressed. We discuss the
implications for applications, arguing in particular that the relevant
evolution for transverse momentum weighted and integrated cross sections with
sharp effective large transverse momentum cutoffs is of the TMD form rather
than the standard renormalization group evolution of collinear correlation
functions.
