Parton Fragmentation Functions Extracted with a Physics-Informed Neural Network

TL;DR

This work introduces a Physics-Informed Neural Network (PINN) framework to extract unidentified charged-hadron fragmentation functions (FFs) at NLO by embedding the DGLAP evolution directly into neural-network training. Using comprehensive SIA data, including flavor-tagged and longitudinal observables, the PINN yields non-parametric FFs that satisfy $\partial D_i^{h^{\pm}}(z,Q)/\partial \ln Q^2 = (\alpha_s(Q)/(2\pi)) \sum_j P_{ji}(z,\alpha_s) \otimes D_j^{h^{\pm}}(z,Q)$ and can reproduce hadron spectra in $pp(\bar{p})$ collisions across RHIC and LHC energies. The results show reasonable agreement with existing parameterizations for singlet FFs in key $z$ ranges, while gluon and heavy-quark FFs are less constrained by SIA data alone, underscoring the value of including SIDIS and collider data for a more complete FF set. The approach demonstrates a promising path toward fast, accurate, and universally applicable non-perturbative quantities such as FFs and PDFs, with potential extensions to medium-modified evolution in heavy-ion environments.

Abstract

Reliable predictions of many high-energy strong interaction processes rely heavily on the non-perturbative parton fragmentation functions (FFs) extracted from existing experimental data. Conventional methods often require parameterized forms of FFs and additional scale evolution according to the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations. We introduce a novel approach to determining parton FFs using a Physics-Informed Neural Network (PINN). Unlike traditional methods, our approach does not require prior parameterized forms and directly integrates the DGLAP evolution equations into the neural network architecture, allowing the FFs to automatically satisfy these equations. We present new sets of parton FFs extracted from hadron spectra in electron-positron annihilation processes at next-to-leading order (NLO) in pQCD using this new technique. To validate our approach, we calculate charged hadron spectra in proton-(anti)proton collisions using the extracted FFs and demonstrate that the results align well with experimental data across a large range of colliding energies ($\sqrt{s}$ = 130, 200, 500, 630, 900, 1800, 2760, 5020, 5440, 7000 GeV). Our findings indicate that the PINN method not only simplifies the extraction process but also enhances the universal applicability of FFs across different energy scales. By eliminating the need for parameterized forms and additional DGLAP evolution, our approach represents a significant step forward toward fast and accurate extractions of non-perturbative quantities such as parton fragmentations functions and parton distribution functions.

Figures (15)

• Figure 1: The neural network framework divided into four modules for FFs extractions. The first module is the black dashed box representing the fully connected neural network. The second module is the black solid box representing the construction of the FFs and the Mellin transformation of the FFs. The third module is the blue solid box representing the auto-differentiation technique in deep learning, which can be used for differential computation in the DGLAP evolution equations. The fourth module is the red dashed box, representing the Physics-informed part of neural network. The primary purpose of this module is to allow the neural network to satisfy the DGLAP evolution equations and to compute the cross section of the SIA process in the Mellin space.
• Figure 2: Kinematic ranges in $(z, Q)$ in experimental SIA data used to determine the FFs.
• Figure 3: Comparison of the training results of the NN with the experimental data in FFs as function of $z$ and the Mellin moments, respectively. The uncertainty of the results is calculated from the 100 samples with a confidence interval of 2$\sigma$.
• Figure 4: The Mellin moments of the singlet quark ($D_{\Sigma}^{h^{\pm}}$) and gluon ($D_g^{h^{\pm}}$) fragmentation functions at the scale $Q = M_Z$, obtained from the PINN. The dashed straight lines indicate the momentum sum ($N=2$ Mellin moments).
• Figure 5: Upper panel: Comparison of the Mellin moments of the FFs at $Q = 14$ GeV. The green lines represent the moments evolved from the initial conditions in Fig. \ref{['baseline_91.2']} using the numerical DGLAP solver, while the magenta lines correspond to the direct PINN predictions at $Q = 14$ GeV. The dashed straight lines indicate the momentum sum ($N=2$). Lower panel: The ratios of the PINN predictions to the numerical solutions for moments of the singlet quark and gluon FFs.