Parton Fragmentation Functions Extracted with a Physics-Informed Neural Network

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.

💡 Research Summary

The paper presents a novel methodology for extracting parton fragmentation functions (FFs) that bypasses the need for predefined functional forms and incorporates the DGLAP evolution equations directly into the learning process of a Physics‑Informed Neural Network (PINN). Traditional FF extractions rely on parametrizing the FFs at an initial scale µ₀, then numerically evolving them with DGLAP equations; this introduces parametrization bias and incurs heavy computational costs because the evolution must be repeatedly evaluated during the fit.

In the proposed approach, a deep neural network is constructed to output the full set of FFs D_i^h(z, µ) for each parton flavor i (u, d, s, c, b, g) and for unidentified charged hadrons h± as functions of the momentum fraction z and factorization scale µ. The loss function consists of two components: (i) a data term that measures the χ² deviation between the network predictions and a comprehensive set of single‑inclusive e⁺e⁻ annihilation (SIA) measurements—including total cross sections, flavor‑tagged spectra (uds, c, b), and the longitudinal cross section dσ_L/dz which is only non‑zero at NLO and thus highly sensitive to the gluon FF—and (ii) an evolution term that penalizes violations of the time‑like DGLAP equations.

To evaluate the evolution term efficiently, the authors work in Mellin‑moment space. The Mellin transform turns the convolution in the DGLAP equations into a simple product, dramatically reducing computational effort. Automatic differentiation supplies the scale derivative ∂D/∂ln µ² required for the residual, while the strong coupling α_s(µ) and the NLO splitting kernels P_ij(z) are inserted directly. The inverse Mellin transform is performed using a deformed Bromwich contour to ensure numerical stability.

Training is performed on a large SIA data set from ALEPH, OPAL, DELPHI, SLD and other experiments, covering a wide range of centre‑of‑mass energies. The inclusion of flavor‑tagged and longitudinal data provides strong constraints on the gluon and heavy‑quark fragmentation components, which are typically poorly determined in SIA‑only fits. After training, the network yields FFs at the initial scale (µ₀≈5 GeV); these can be evolved to any higher scale simply by applying the learned DGLAP dynamics, without any additional numerical evolution step.

The authors validate the method through several tests. A closure test—generating pseudo‑data from a known FF set, training the PINN, and checking the recovery of the original FFs—demonstrates that the network can faithfully reproduce the underlying functions. They also compare the extracted FFs with established sets such as KRE, AKK08, and NNFF1.0, finding especially smooth and physically sensible behavior for the gluon FF at intermediate and large z, where traditional parametrizations often exhibit artifacts due to functional‑form bias.

To showcase phenomenological relevance, the extracted FFs are employed in next‑to‑leading‑order (NLO) calculations of inclusive charged‑hadron production in proton–proton and proton–antiproton collisions. Using the standard collinear factorization formula, the authors compute spectra at a broad array of collider energies (√s = 130 GeV up to 7 TeV) and compare with data from RHIC and the LHC. The predictions agree with measurements across the entire energy range, confirming that the PINN‑derived FFs are universal and evolve correctly under DGLAP.

Uncertainty quantification is addressed by generating Monte‑Carlo replicas of the experimental data and propagating them through the training pipeline, as well as by constructing a Hessian error matrix. This provides a reliable estimate of the FF uncertainties, which can be directly used in future global analyses that simultaneously fit PDFs and FFs or in the interpretation of upcoming Electron‑Ion Collider data.

In summary, the paper demonstrates that embedding fundamental QCD evolution equations into a neural‑network loss function yields a powerful, non‑parametric extraction of fragmentation functions. The method eliminates parametrization bias, reduces computational overhead by avoiding repeated DGLAP calls, and produces FFs that are immediately applicable to a wide variety of high‑energy processes. This PINN framework represents a significant step toward a unified, physics‑driven approach for extracting non‑perturbative quantities such as FFs and PDFs in modern particle‑physics phenomenology.