Recent Progress in Transverse Momentum-Dependent Parton Densities and Corresponding Parton Showers

The parton branching method is crucial for Monte Carlo generators, which are essential for high-energy physics predictions. We examine the impact of soft gluons on inclusive collinear and Transverse Momentum Dependent (TMD) parton densities. By applying the Parton-Branching (PB) method, we identify the non-perturbative Sudakov form factor with the integration range $z \to 1$, which is often neglected in collinear parton shower approaches. The significance of soft gluons is demonstrated through the transverse momentum spectrum of Drell-Yan lepton pairs, resulting in an intrinsic-$k_T$ distribution width that remains independent of $\sqrt{s}$, contrary to observations in collinear parton shower approaches. This behavior is attributed to the non-perturbative Sudakov form factor.