Transverse momentum resummation and analytic continuation into the deep infrared

Abstract Extracting information on the three-dimensional structure of hadrons from transverse-momentum data is a challenging task, requiring the separation of perturbative contributions under QCD control from genuinely non-perturbative effects, unavoidably affected by phenomenological bias. Standard approaches model transverse momentum dependent (TMD) observables in impact-parameter space and compare them with data after Fourier inversion, generating an interplay between perturbative terms, non-perturbative models, and prescriptions used to extend perturbation theory beyond its natural domain. In this work, a consistent methodology to perform the Fourier inversion analytically is proposed, achieving a solid resummation of TMD parton distributions and cross sections directly in transverse-momentum space. The resummation is carried out to next-to-leading logarithmic (NLL) accuracy, validated against alternative prescriptions, and tested on experimental data. This framework provides a systematic procedure to analytically continue observables into the deep infrared, identifying two primary sources of non-perturbative effects: the low energy behavior of the strong coupling and of the parton distribution functions. This points to a paradigm shift in TMD phenomenology, where the structure of hadrons emerges from the dynamics of QCD itself rather than from flexible parametrizations with only indirect physical interpretation.