New C++ Library Unifies Parton Distribution Function Calculations

Researchers have introduced PDFxTMDLib, a high-performance C++ library designed to streamline the calculation of collinear and transverse momentum-dependent parton distribution functions in high-energy physics. Published in EPJ Research Infrastructures in February 2026, the study presents a unified framework that addresses specific limitations in existing tools such as LHAPDF and TMDLib. The library aims to provide greater customization and extensibility for physicists modeling the internal structure of hadrons, particularly in scenarios involving small-x physics or high-energy collisions where transverse momentum effects are significant.

Parton distribution functions, or PDFs, are fundamental quantities that characterize the momentum distribution of quarks and gluons within hadrons. These distributions are essential for predicting cross sections in high-energy collisions at facilities like the Large Hadron Collider. Two primary theoretical frameworks govern these calculations: collinear factorization, which assumes partons move parallel to the hadron’s momentum, and k_t-factorization, which incorporates transverse momentum effects. The new library supports both frameworks, allowing users to access collinear PDFs, which depend on the longitudinal momentum fraction and factorization scale, and TMDs, which additionally depend on the transverse momentum of the partons.

Existing libraries, including LHAPDF and TMDLib, have long served the physics community but face constraints in modern phenomenological studies. LHAPDF is widely used for collinear PDFs, while TMDLib handles transverse momentum-dependent distributions. However, both libraries rely on fixed-dimensional interpolation, which limits their ability to support higher-order distributions such as double parton distribution functions. Furthermore, users of these tools are often confined to built-in algorithms, with no straightforward method to implement custom interpolation or extrapolation strategies. For TMDs specifically, the lack of a standardized grid or file structure in TMDLib makes adding new sets a non-trivial task, and uncertainty propagation is not natively supported at the distribution level.

PDFxTMDLib addresses these shortcomings by employing modern C++ design principles, including separation of concerns, compile-time polymorphism, and type erasure. This architecture allows the library to achieve high computational performance while maintaining the flexibility needed for future theoretical developments. The core of the library is the PDFSet class, which provides a unified interface for computing various quantities, including PDFs, QCD coupling constants, and statistical measures such as uncertainties and correlations. By using template classes and factory patterns, the library can adapt its behavior to either collinear or TMD use cases without sacrificing efficiency.

A key innovation of the library is its modular architecture, which allows users to implement custom components for reading, interpolating, and extrapolating PDF data. The GenericPDF class encapsulates these strategies, enabling researchers to modify the evaluation process to suit specific research requirements. This design is explicitly intended to extend to higher-dimensional cases, such as double transverse momentum-dependent parton distribution functions, as theoretical models evolve. The library also introduces a new file format, lhagrid_tmd1, which is an extension of the standard lhagrid1 format to accommodate the additional dimensionality of TMDs, thereby standardizing storage and access where possible.

To validate the library’s accuracy and performance, the authors integrated PDFxTMDLib into the PYTHIA Monte Carlo event generator to simulate Drell–Yan cross sections. The Drell–Yan process, involving the production of a Z boson that decays into muon pairs, is a critical benchmark for investigating both collinear and transverse momentum-dependent distributions. The simulation was configured for proton–proton collisions at a center-of-mass energy of 13 TeV, generating 50 million events with specific phase-space constraints. The results were compared against those obtained using the LHAPDF library with the MSHT20nlo_as120 PDF set.

The numerical validation demonstrated that PDFxTMDLib provides results that are in perfect agreement with LHAPDF. The total Drell–Yan cross section was measured to be 1.695 × 10^-6 mb in both libraries, with negligible statistical uncertainties. Differential cross sections for the dilepton invariant mass, rapidity, and transverse momentum also showed excellent consistency, confirming the reliability of the new implementation. In terms of performance, PDFxTMDLib achieved a modest but notable improvement over LHAPDF, reducing the total execution time for 50 million events by approximately 5.6%. This efficiency gain is attributed to optimized algorithms that reduce computational overhead during event generation, a valuable feature for large-scale simulations where processing time is a limiting factor.

For transverse momentum-dependent distributions, the library was validated against TMDLib using specific parton-branching sets. The authors compared gluon and down quark distributions at fixed values of x and transverse momentum, observing full agreement between the two libraries. This confirms the correctness of the TMD implementation in PDFxTMDLib. However, the authors note that a direct performance comparison with TMDLib is not straightforward due to the latter’s structure as a collection of independent components, each with distinct computational characteristics. Therefore, the performance benchmarks are primarily focused on the collinear sector against LHAPDF.

One of the most significant contributions of PDFxTMDLib is the extension of uncertainty and QCD coupling calculations to the TMD domain. While these features have long been standard in collinear PDF libraries, they were previously unavailable in a standardized form for TMD analyses. PDFxTMDLib introduces an explicit ErrorType field in the metadata, following the LHAPDF convention, to enable consistent handling of uncertainty information. This allows for diagnostic and comparative studies of uncertainties directly at the distribution level, although the authors clarify that these estimates correspond to variations among set members and are not intended to represent uncertainties of physical cross sections.

The library is designed to be extensible, with a modular architecture that supports custom implementations of readers, interpolators, and extrapolators. This flexibility ensures that PDFxTMDLib can adapt to emerging theoretical developments and specialized applications in high-energy physics. The authors emphasize that the goal is not to replace existing tools like TMDLib, but to provide a unified and extensible framework that lowers the technical barrier for adding new TMD sets. The library currently supports a subset of TMD sets, specifically the parton-branching family, with plans to expand support progressively as updates are published on the project’s website.

PDFxTMDLib is available as open-source software, with the code hosted on GitHub and documentation provided online. The library requires a C++17 compatible compiler and CMake for installation, and it offers both C++ and Python interfaces to facilitate integration into various research workflows. The Python bindings allow users to utilize most of the library’s features easily, including uncertainty analysis and PDF calculation. By providing a robust and efficient framework for PDF computations, PDFxTMDLib contributes to the study of hadron structure and high-energy collision dynamics, supporting adaptable workflows for researchers in the field.

The introduction of PDFxTMDLib represents a significant step forward in the computational tools available for high-energy physics. By addressing the limitations of existing libraries and introducing new features for TMD analysis, the library enhances the precision and efficiency of phenomenological studies. As theoretical models continue to evolve, the extensible design of PDFxTMDLib positions it as a practical tool for future research, capable of accommodating more complex distribution functions and advanced computational requirements.

Subject of Research: PDFxTMDLib: A High-Performance C++ Library for Collinear and Transverse Momentum-Dependent Parton Distribution Functions

Article Title: PDFxTMDLib: A High-Performance C++ Library for Collinear and Transverse Momentum-Dependent Parton Distribution Functions

Article References: Valeshabadi, R. K., & Rezaie, S. (2026). PDFxTMDLib: A High-Performance C++ Library for Collinear and Transverse Momentum-Dependent Parton Distribution Functions. EPJ Research Infrastructures, 10(1), Article 3. https://doi.org/10.1007/s41781-025-00157-0

Image Credits: AI Generated

DOI: 10.1007/s41781-025-00157-0

Keywords: PDFxTMDLib, High-Performance, Library, Collinear, Transverse, Momentum-Dependent, Parton, Distribution, Functions, scientific research