The Cahn effect and Transverse Momentum Factorization Beyond Leading Power

We investigate the intrinsic transverse momentum distribution of partons in the proton via the Cahn effect, which arises within the framework of transverse momentum dependent (TMD) factorization. This effect leads to a characteristic azimuthal cos ϕ modulation between the leptonic and hadronic planes in semi-inclusive deep inelastic scattering (SIDIS). Employing perturbative QCD and TMD factorization theorems, we compute the TMD evolution of the Cahn function and examine its behavior beyond leading order. In particular, the equation-of-motion (EOM) relations are tested at next-to-leading order (NLO) in the QCD coupling, requiring explicit evaluation of rapidity and ultraviolet divergences in both the TMD soft factor and the TMD parton distribution functions. The validity of these relations provides both necessary and sufficient conditions for the consistency of TMD factorization at NLO and at next-to-leading power in the hard scale. We further analyze the proposed matching between low- and high-transverse-momentum regimes for the Cahn effect, a subject of ongoing theoretical interest. Our findings contribute to the endeavor TMD phenomenology and are directly relevant to SIDIS measurements at Jefferson Lab and the COMPASS experiment at CERN. Substantial progress is reported in the explicit verification of the underlying factorization theorems beyond leading power.