tParton: Implementation of next-to-leading order evolution of transversity parton distribution functions

TL;DR

This work addresses the public availability gap for transversity PDF evolution by introducing tParton, a Python package that implements LO and NLO DGLAP evolution using two complementary methods: Hirai-style real-space integration and Mellin-space moment inversion. It provides a rigorous theoretical framework for transversity evolution, including plus-distribution handling and Mellin-space solutions, and validates the implementations against established results and APFEL++. The authors report good numerical agreement (typically at the 1% level) between methods and languages, analyze sources of discrepancies, and offer guidance on accuracy and performance. The package, along with reproducible notebooks and Zenodo resources, enables practitioners to evolve transversity PDFs across scales with controlled precision, facilitating phenomenology and experimental analyses of nucleon transverse spin structure.

Abstract

We provide code to solve the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations for the nucleon transversity parton distribution functions (PDFs), which encode nucleon transverse spin structure. Though codes are widely available for the evolution of unpolarized and polarized PDFs, there are few codes publicly available for the transversity PDF. Here, we present Python code which implements two methods of solving the leading order (LO) and next-to-leading order (NLO) approximations of the DGLAP equations for the transversity PDF, and we highlight the theoretical differences between the two.

Figures (5)

• Figure 1: The GS-A distribution for $\Delta_T u_v+\Delta_T d_v$, evolved from 4 GeV$^2$ to 200 GeV$^2$ using both the Hirai method and the Vogelsang method at NLO. We also include the result of APFEL++, which was supplied by V. Bertone. The difference between our results and that of APFEL++ can be explained by different choices of $\alpha_S$, as discussed in the text. The bottom panel shows the absolute value of the difference between the other results and the benchmark, divided by the benchmark, where the benchmark is taken as "Python Hirai".
• Figure 2: The GS-A distribution for $x(\Delta_T \bar{u}-\Delta_T \bar{d})$, evolved from 4 GeV$^2$ to 200 GeV$^2$ using both the Hirai method and the Vogelsang method, as well as both choices of $\alpha_S$ at NLO. See Sec. \ref{['sec:dis']} for a discussion on different choices of $\alpha_S$. Cohen's method is used to degree 5 in both the Vogelsang curves, without much improvement in agreement at higher degrees (not shown).
• Figure 3: The GS-A distribution for $\Delta_T u_v+\Delta_T d_v$, evolved from 4 GeV$^2$ to 200 GeV$^2$ using both the Hirai method and the Vogelsang method at NLO, however with $\alpha_S$ being the numerical solution to the NLO $\alpha_S$ evolution equation, rather than the approximate analytical formula given in both Eq. (A.2) of Ref. hirai and Eq. (21) of Ref. Vogelsang97. The result of APFEL++ is the same as in Fig. \ref{['fig:1']}. The bottom panel shows the absolute value of the difference between the other results, including the APFEL++ result, and the benchmark, divided by the benchmark, where the benchmark is taken as "Python Hirai" (with numerical $\alpha_S$).
• Figure 4: The relative error in computing Fig. \ref{['fig:1']} as the degree of approximation and input PDF granularity $N_x'$ is varied in the Vogelsang method. The error is measured by the absolute value of the total area under the curve for the Vogelsang method minus the reference, divided by the area of the reference, where the reference is taken as "Python Hirai" in Fig. \ref{['fig:1']}.
• Figure 5: Relative error in computing Fig. \ref{['fig:1']} as $N_x$ and $N_t$ are varied in the Hirai method. The error is measured by the absolute value of the total area under the curve for the Hirai method with sub-optimal $N_x$ and $N_t$, minus the benchmark, divided by the benchmark, where the benchmark is taken as the total area under the curve of the Hirai method with the $N_x=3000$ and $N_t=500$. This is done with Python. Note that the colorscale is the same as in Fig. \ref{['fig:3_optimal_degree']}.