Determination of unpolarized TMD distributions from the fit of Drell-Yan and SIDIS data at N$^4$LL

TL;DR

This paper performs a global extraction of unpolarized TMD distributions by fitting Drell–Yan and SIDIS data within TMD factorization at N$^4$LL accuracy. Using the artemide framework and the $\zeta$-prescription, it implements a flavor-dependent non-perturbative model and large-$x$ resummation to simultaneously determine the unpolarized TMDPDFs, TMDFFs, and the Collins–Soper kernel, along with transverse momentum moments. The results show good agreement with the data, reveal a larger CS kernel than some prior extractions (yet compatible with lattice and earlier SIDIS-inclusive analyses), and identify specific anomalies tied to collinear inputs. Importantly, the work provides first-ever estimates of the second transverse momentum moment for TMD distributions, offering new insights into $\langle \bm{k}_T^2\rangle$ and its flavor dependence, and confirms backward compatibility with collinear PDFs/FFs through zeroth TMMs.

Abstract

We present a fit of the transverse momentum spectrum for Drell-Yan and semi-inclusive deep inelastic scattering data, based on transverse momentum dependent (TMD) factorization at N$^4$LL accuracy. Our analysis shows good agreement with the data and confirms the findings of previous studies. Based on this, we extract the unpolarized TMD parton distribution functions, the TMD fragmentation functions, and the Collins-Soper kernel. Compared to earlier works, our study incorporates several improvements, including large-$x$ resummation, flavor and fragmentation function dependence, among others. Additionally, we supplement our extraction with an analysis of the transverse momentum moments of the extracted distributions.

Figures (37)

• Figure 1: Contour plot for values of the $\chi^2/N_{\text{NP}}$ obtained in the fit at different values of the transition scales. The contours indicate the lines of the same $\chi^2/N_{\text{NP}}$, with the value indicated on the line. The red crosses indicate the current set of transition parameters with $\chi^2/N_{\text{NP}}=1.017$. The red circles indicate the values of the transition parameters with best value of $\chi^2/N_{\text{NP}}$, which is $0.987$ for the left plot, and $1.008$ for the right plot.
• Figure 2: The kinematic range in $Q$ and $x$ covered by the used data. A darker shade indicates a denser distribution of data.
• Figure 3: Correlation matrix for parameters of the fit.
• Figure 4: The values of $\chi^2/N_{\text{pt}}$ obtained for fits with sets of data selected with different values of cut parameters $\delta$, eq. (\ref{['def:data-selection']}), and $\delta_\perp$, eq. (\ref{['delta_perp']}). The black (orange, green) points indicate the values for total (only DY, only SIDIS) data set. The numbers indicate the number of data points for each case.
• Figure 5: Comparison of the CS kernel extracted in this work (labeled as ART25) with other determinations, namely: (upper left panel) extraction from fits of data made in refs. Moos:2023yfa (ART23), Scimemi:2019cmh (SV19), Bacchetta:2025ara (MAPNN), Bacchetta:2024qre (MAP24) and Bacchetta:2022awv (MAP22); (upper right panel) lattice computations performed in refs. Bollweg:2024zet (BGMZ24), Avkhadiev:2023pozAvkhadiev:2024mgd (ASWZ24), Shu:2023cot (SSSV23), LatticePartonLPC:2022eev (LPC22). For the SSSV23 analyses only the extraction made with pion are presented for clarity; (lower left panel) The dot-dashed curve is the CS kernel implementedin ref. Billis:2024dqq. The solid orange and gray dashed curves are the CS kernel determined from the fit of the energy-energy correlator in the back-to-back regime Kang:2024djaCuerpo:2025zde.(lower right panel) The curve labeled as "CASCADE" correspond to the CS kernel used in the parton branching approach within the CASCADE generator CASCADE:2021bxeBermudezMartinez:2020tys determined by the method of ref. BermudezMartinez:2022ctj. The orange line shows the result of the computation of the CS kernel in the instanton vacuum model Liu:2024sqj. The HSO curve corresponds to a fit made within the hadron-structure oriented approachAslan:2024nqg. The blue dotted line is the CS kernel determined via analyses of thrust distribution in single inclusive electron-positron annihilation data in ref. Boglione:2023duo.