⚛️ phenomenology

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

This paper introduces PDFxTMDLib, a modern, high-performance C++ library that overcomes the customization and uncertainty analysis limitations of existing tools like LHAPDF and TMDLib by providing a flexible framework for both collinear and transverse momentum dependent parton distribution functions, which is validated through integration with the PYTHIA event generator and comparative studies.

Original authors: R. Kord Valeshabadi, S. Rezaie

Published 2026-02-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: R. Kord Valeshabadi, S. Rezaie

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the outcome of a massive, high-speed collision between two protons, like those happening at the Large Hadron Collider (LHC). To do this, you need to know exactly what's inside those protons. Protons aren't solid balls; they are chaotic bags of smaller particles called quarks and gluons (collectively known as "partons").

The problem is, these partons are moving wildly. Some are zooming straight ahead, while others are jittering side-to-side.

The Old Tools: The "Two Separate Maps"

Physicists have long used two different "maps" (software libraries) to navigate this chaos:

LHAPDF: This is the standard map for particles moving straight ahead. It's like a highway map that only cares about how fast a car is going forward. It's great, but it ignores the side-to-side swerving.
TMDLib: This is a specialized map for particles that are also swerving side-to-side (transverse momentum). However, this map is a bit of a mess. It's like a collection of different hand-drawn maps from different cartographers. Each one uses a different grid system, making it hard to switch between them or add new features. It's also rigid; if you wanted to draw a new kind of map, you couldn't easily do it.

The Problem: Scientists wanted a single, modern tool that could handle both the "straight highway" and the "swerving traffic" scenarios, was easy to customize, and could handle future, more complex traffic patterns.

The New Solution: PDFxTMDLib

The authors of this paper, R. Kord Valeshabadi and S. Rezaie, built PDFxTMDLib. Think of this as a universal, high-tech GPS system for particle physics.

Here is what makes it special, using some everyday analogies:

1. The "Universal Adapter" (Unified Framework)

Before, if you wanted to check the "straight" map and the "swerving" map, you had to use two different apps that didn't talk to each other.
PDFxTMDLib is like a universal adapter. It has one interface that works for both. Whether you are looking at a particle moving straight or one wobbling side-to-side, the library speaks the same language. It's like having one app that can show you a subway map, a bus map, and a bike-share map all in the same clean interface.

2. The "Lego System" (Modular Architecture)

Old libraries were like a pre-built plastic model kit. Once it was glued together, you couldn't change the wheels or the engine.
PDFxTMDLib is built like Lego. It separates the parts:

The Reader: How to read the map data.
The Interpolator: How to guess the speed between two known points on the map.
The Extrapolator: How to guess what happens outside the map's borders.
Because these are separate "Lego bricks," a scientist can swap out one brick (e.g., use a smarter way to guess speeds) without breaking the whole system. This makes it incredibly flexible for future discoveries.

3. The "Speed Demon" (High Performance)

In physics simulations, you might need to calculate these particle speeds billions of times for a single experiment. If your software is slow, the simulation takes weeks instead of days.
The authors used advanced coding techniques (like "compile-time polymorphism") to make this library fast. It's like upgrading from a bicycle to a sports car. In their tests, they simulated a famous particle collision (the Drell-Yan process) and found that their new library was about 5.6% faster than the old standard, while giving the exact same accurate results.

4. The "Error Check" (Uncertainty Handling)

Every map has a margin of error. Sometimes the data is fuzzy.
Old tools for the "swerving" particles (TMDs) didn't have a good way to calculate these errors automatically. You had to do it manually or use a separate, clunky tool.
PDFxTMDLib built this error-checking feature directly into the core. It's like a GPS that not only tells you the route but also says, "There's a 10% chance of traffic here," and does the math for you instantly.

5. The "Future-Proof" Design

The library is designed to be "extensible." Right now, it handles 2D (straight) and 3D (swerving) maps. But the authors built the engine so that in the future, if scientists discover a need for 4D or 5D maps (like "Double Parton Distributions"), they can just add a new dimension without rebuilding the whole car.

The Bottom Line

PDFxTMDLib is a modern, fast, and flexible software library that unifies two previously separate worlds of particle physics data.

For the Scientist: It saves time, reduces coding errors, and allows for complex, custom experiments that were previously too difficult to set up.
For the Field: It standardizes how we handle "swerving" particles, making sure everyone is using the same high-quality tools to understand the universe.

The authors have even made the code free and open-source, allowing anyone to download it, use it, and help improve it, ensuring that the "GPS" for the subatomic world keeps getting better.

1. Problem Statement

High-energy physics phenomenology relies heavily on Parton Distribution Functions (PDFs) to calculate cross-sections for particle collisions. Two primary frameworks are used:

Collinear Factorization: Uses Collinear PDFs (cPDFs) depending on longitudinal momentum fraction ( $x$ ) and factorization scale ( $\mu$ ).
$k_t$ -Factorization: Uses Transverse Momentum Dependent distributions (TMDs) or Unintegrated PDFs (UPDFs), which additionally depend on transverse momentum ( $k_t$ ).

Currently, researchers rely on two separate libraries: LHAPDF for cPDFs and TMDLib for TMDs. The paper identifies several critical limitations in these existing tools:

Lack of Extensibility: Both libraries use fixed-dimensional interpolation (2D for cPDFs, 3D for TMDs), making it difficult to support higher-order distributions like Double Parton Distribution Functions (DPDFs).
Limited Customization: Users are confined to built-in algorithms and cannot easily implement custom interpolation, extrapolation, or file readers.
Inconsistency in TMDs: TMDLib lacks a standardized grid/file structure. Different TMD sets use different prescriptions, making it difficult to add new sets or perform consistent uncertainty propagation at the distribution level.
Missing Features in TMDs: While LHAPDF supports QCD coupling ( $\alpha_s$ ) and uncertainty quantification, TMDLib lacks standardized support for these features within the library itself.

2. Methodology and Architecture

The authors introduce PDFxTMDLib, a modern C++ library designed to unify cPDF and TMD handling within a single, extensible framework. The architecture is built on three core design principles:

Separation of Concerns: Clear interfaces for reading, interpolating, and extrapolating data.
Compile-Time Polymorphism: Utilizing the Curiously Recurring Template Pattern (CRTP) to minimize runtime overhead and maximize performance.
Type Erasure: Allowing flexible interface handling without traditional inheritance, enabling high performance while maintaining a clean API.

Key Architectural Components:

PDFSet Class: The primary user-facing template class (parameterized by CollinearPDFTag or TMDPDFTag). It unifies the interface for calculating cPDFs, TMDs, $\alpha_s$ , uncertainties, and correlations.
Factory Pattern: GenericCPDFFactory, GenericTMDFactory, and CouplingFactory instantiate specific objects based on YAML metadata, allowing users to bypass the PDFSet abstraction for direct, high-performance access.
GenericPDF Class: A template class encapsulating three pluggable strategies:
- IReader: Handles file I/O (supporting LHAPDF's lhagrid1 and a new TMD format).
- IInterpolator: Performs interpolation (e.g., bilinear, trilinear, bicubic).
- IExtrapolator: Handles values outside grid boundaries.
File Format Standardization: The library introduces lhagrid_tmd1, an extension of the LHAPDF lhagrid1 format. This standardizes the grid structure for TMDs (organizing data by $x$ , $k_t^2$ , $\mu^2$ , and flavor) to resolve the lack of standardization in the TMD community.

3. Key Contributions

Unified Framework: A single library capable of handling both collinear and transverse-momentum-dependent distributions, simplifying code maintenance for phenomenologists.
First-of-its-Kind TMD Features: PDFxTMDLib is the first library to provide standardized uncertainty quantification and QCD coupling ( $\alpha_s$ ) calculations directly at the TMD distribution level.
Extensibility to Higher Dimensions: The architecture is explicitly designed to scale to higher-dimensional distributions (e.g., DPDFs, DTMDs) via variadic templates and pluggable components, future-proofing the library.
Performance Optimization: By using CRTP and manual function dispatch, the library avoids the runtime polymorphism overhead typical of C++ virtual functions, resulting in faster execution.
Multi-Language Support: The library provides both a C++ API and a Python binding (pdfxtmd), facilitating easy integration into diverse research workflows.

4. Results and Validation

The authors validated PDFxTMDLib through numerical comparisons and performance benchmarks:

Numerical Accuracy:
- cPDFs: Comparisons against LHAPDF for sets like METAv10LHC, CT18LO, and EPPS16nlo showed perfect agreement across various kinematic points ( $x$ , $\mu^2$ ).
- TMDs: Comparisons against TMDLib for Parton Branching (PB) sets (e.g., PB-LO-HERAI+II-2020) demonstrated exact matching of distribution curves.
Phenomenological Validation (Drell-Yan Process):
- The library was integrated into the PYTHIA8 Monte Carlo event generator to simulate Drell-Yan processes ( $pp \to Z \to \mu^+\mu^-$ ) at 13 TeV.
- Results for total and differential cross-sections (invariant mass, rapidity, $p_T$ ) matched LHAPDF results perfectly.
Performance:
- In a simulation of 50 million Drell-Yan events, PDFxTMDLib reduced execution time by approximately 5.6% compared to LHAPDF (162,263s vs. 171,935s).
- This improvement is attributed to optimized interpolation algorithms and reduced computational overhead, despite the library's high level of abstraction.

5. Significance and Impact

PDFxTMDLib addresses a critical gap in high-energy physics software by providing a robust, standardized, and high-performance solution for PDF and TMD calculations.

Standardization: By introducing the lhagrid_tmd1 format and unified metadata handling, it lowers the barrier for adding new TMD sets and ensures consistency in uncertainty analysis.
Future-Proofing: Its modular design allows researchers to easily adapt to emerging theoretical models, such as double parton scattering, without rewriting core infrastructure.
Accessibility: The availability of Python bindings and a user-friendly API makes advanced PDF/TMD analysis accessible to a broader range of researchers, not just C++ experts.
Reliability: The library maintains the reliability of established physics calculations (by not "reinventing the wheel" for core physics logic) while significantly improving code readability, performance, and flexibility.

In conclusion, PDFxTMDLib represents a significant step forward in the computational infrastructure of particle physics, offering a modern alternative to legacy tools that better supports the evolving needs of precision phenomenology.