Researchers have developed a theoretical framework to analyze transverse-momentum-dependent (TMD) fragmentation in electron-positron ($e^+e^-$) collisions and in semi-inclusive deep-inelastic scattering (SIDIS). This new approach focuses on measuring hadrons with respect to the thrust axis in $e^+e^-$, a technique already explored by the Belle experiment. The main advantage is that it allows for a more direct extraction of TMD fragmentation functions, avoiding the complexity of disentangling two TMD fragmentation functions that appear in conventional back-to-back hadron-pair measurements.

The work builds upon established factorization theorems and completes the operator-level formulation of soft ingredients, performing one-loop checks. Furthermore, it extends existing results for 1-jettiness factorization in SIDIS, where analogous measurements provide access to the TMD parton distribution functions of the incoming hadron. For phenomenology, nonperturbative effects are discussed, and a model is proposed that captures both event-shape dependence and correlations between event-shape and transverse-momentum measurements.

The study includes the resummation of transverse-momentum and thrust logarithms, exploring several schemes for treating the latter and implementing them in the artemide software. As an initial validation, the results were compared to simulated $e^+e^-$ data generated by Pythia8.3. It was found that the proposed nonperturbative model is flexible enough to describe the simulated data, with fitted parameters of the expected size in powers of $\Lambda_{\rm QCD}/Q$. In this test, the resummation of the logarithms of $q_T/(\tau Q)$ had little impact on the quality of the fit, but it did change the fitted parameters.