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.
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