Transverse Momentum Moments

The transverse momentum moments (TMMs) are integrals of Transverse Momentum Dependent distributions (TMDs) over transverse momentum, weighted by powers of $k_T$. In this work, we establish robust relations between TMMs and collinear distributions. Specifically, we prove that the zeroth TMM corresponds to collinear twist-two distributions and derive a conversion factor to express it in the conventional $\overline{\text{MS}}$-scheme. The first and second moments are related to twist-three and twist-four collinear distributions, respectively. We discuss the applications of the zeroth, first, and second TMMs and provide phenomenological results for them based on current TMD extractions. These results open new avenues for the theoretical and phenomenological investigation of three-dimensional and collinear hadron structures.