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APFEL: A PDF Evolution Library with QED corrections

Valerio Bertone, Stefano Carrazza, Juan Rojo

arXiv:1310.1394v3hep-phhep-exnucl-exnucl-th

TL;DR

Precision LHC predictions require PDFs that consistently include QCD and QED effects, while existing public QED-evolution tools had limited perturbative order or flexibility. APFEL addresses this gap with sequential, grid-based QCD⊗QED evolution and additional DIS and GUI modules. Its scope includes NNLO QCD and LO QED evolution, although photon-PDF theoretical uncertainties remain substantial.

  • Problem

    Precision LHC predictions require PDFs with consistent QCD and electroweak accuracy, but public QED-evolution support was limited in perturbative order and combination options.

  • Method

    APFEL solves coupled QCD⊗QED evolution sequentially using x-space interpolation and discretized Runge-Kutta evolution, with distinct QCD and QED PDF bases.

  • Results

    APFEL provides public NNLO QCD and LO QED PDF evolution in FFN and VFN schemes with pole or MS heavy-quark masses, plus DIS calculations and GUI tools.

  • Takeaways & Limitations

    APFEL offers a fast, flexible library for PDF analyses incorporating QED corrections and related DIS, plotting, and interface functionality.

  • Takeaways & Limitations

    Photon-PDF theoretical uncertainties are substantial and may require higher-order QED corrections for future more precise data.

Abstract

from arXiv · show

Quantum electrodynamics and electroweak corrections are important ingredients for many theoretical predictions at the LHC. This paper documents APFEL, a new PDF evolution package that allows for the first time to perform DGLAP evolution up to NNLO in QCD and to LO in QED, in the variable-flavor-number scheme and with either pole or MSbar heavy quark masses. APFEL consistently accounts for the QED corrections to the evolution of quark and gluon PDFs and for the contribution from the photon PDF in the proton. The coupled QCD+QED equations are solved in x-space by means of higher order interpolation, followed by Runge-Kutta solution of the resulting discretized evolution equations. APFEL is based on an innovative and flexible methodology for the sequential solution of the QCD and QED evolution equations and their combination. In addition to PDF evolution, APFEL provides a module that computes Deep-Inelastic Scattering structure functions in the FONLL general-mass variable-flavor-number scheme up to O($α_s^2$). All the functionalities of APFEL can be accessed via a Graphical User Interface, supplemented with a variety of plotting tools for PDFs, parton luminosities and structure functions. Written in Fortran 77, APFEL can also be used via the C/C++ and Python interfaces, and is publicly available from the HepForge repository.

Program Summary

APFEL is a publicly distributed PDF-evolution program for coupled QCD⊗QED DGLAP equations, with broad language-interface support and lightweight execution. It uses grid-based interpolation and Runge-Kutta methods, while requiring smooth initial PDF conditions.

  • Implementation: APFEL is written in Fortran 77 and exposes C/C++ and Python interfaces.
  • Distribution: The distributed program is approximately 2.4 MB, uses LHAPDF, requires no external files, and is released under the GNU Public License.
  • Capabilities: APFEL evolves unpolarized PDFs through NNLO in QCD and LO in QED within the variable-flavor-number scheme.The program supports both pole and MS heavy-quark masses.
  • Numerical method: The numerical method represents PDFs and splitting functions on an x-grid, discretizes the evolution equations, interpolates higher-order in x, and applies Runge-Kutta integration.
  • Performance: PDF tables with combined QCD⊗QED evolution typically require about 0.5 seconds after initialization on the specified Intel Core 2 Duo system.Initialization takes a few seconds.

1 Introduction

APFEL addresses the limited availability and flexibility of public QCD⊗QED PDF-evolution codes needed for precision LHC predictions. It combines sequential QCD and QED evolution with DIS and GUI modules, then validates the resulting tools against established benchmarks.

  • Motivation: Precision LHC predictions require PDFs incorporating QCD and electroweak corrections consistently with the hard-scattering calculations.
  • Research gap: Existing public QED-evolution support was limited: partonevolution provided only NLO QCD and lacked options for combining QCD and QED evolution equations.
  • Contribution: APFEL provides public PDF evolution through NNLO QCD and LO QED in FFN and VFN schemes, with pole or MS heavy-quark masses.
  • Method: APFEL sequentially solves QCD and QED evolution, allowing users to modify the ordering and heavy-quark-threshold choices.
  • Additional functionality: The library includes DIS structure-function calculations and a GUI with plotting tools for PDFs, parton luminosities, and structure functions.
  • Paper organization: The paper describes numerical techniques, functionalities, GUI access, benchmarking against public tools, and possible future developments.

2 DGLAP evolution with QED corrections

APFEL solves coupled QCD⊗QED DGLAP evolution by separating QCD and QED treatments, using specialized PDF bases and configurable evolution combinations. It includes photon contributions, heavy-quark threshold handling, and comparisons of sequential solutions whose differences are controlled by subleading terms.

  • Evolution strategy: The QED formulation includes the photon PDF, while QCD and QED splitting-function matrices act on a vector containing all parton distributions.The QED corrections add terms proportional to α and convoluted with PDFs.
  • Evolution strategy: APFEL solves QCD⊗QED DGLAP equations by determining separate QCD and QED evolution operators and combining them into a coupled operator.The combined operator evolves PDFs across both QCD and QED factorization scales.
  • QED evolution basis: The QED momentum sum rule includes the photon contribution and should remain valid at all scales when the input PDFs satisfy the momentum sum rule.Checking this equality provides an implementation test for QED-corrected DGLAP evolution.
  • QED evolution basis: A basis transformation reduces the QED system from thirteen coupled flavor equations to a three-by-three coupled system plus ten decoupled differential equations.Singlet distributions couple to the photon, whereas non-singlet distributions evolve multiplicatively without photon coupling.
  • Combining QCD and QED evolution: Because QCD and QED evolution operators do not commute, APFEL supports different orderings and combination prescriptions whose differences are beyond the implemented QED accuracy.The QavD solution is closest to the MRST04QED and partonevolution solutions, while QCED and QECD can receive numerically sizable subleading effects from large unresummed scale logarithms.
  • Deep-Inelastic Scattering: APFEL implements the FONLL general-mass variable-flavor-number scheme and also provides separate FFN and ZM-VFN predictions.This extends the package beyond PDF evolution to structure-function calculations across heavy-quark regimes.

3 Numerical techniques

APFEL solves the QCD and QED DGLAP equations by representing PDFs on interpolation grids, discretizing the equations, and integrating the resulting evolution equations numerically. Its grid structure improves efficiency while requiring special treatment of the large-x region.

  • Grid representation and evolution: APFEL represents PDFs and splitting functions on an x grid, uses higher-order interpolation, and solves the discretized evolution equations with Runge-Kutta methods.The same numerical strategy applies to both QCD and QED evolution.
  • Interpolation: The interpolation basis uses Lagrange functions of degree k with localized support around each grid node.The interpolation functions are nonzero only over a bounded interval around their associated node.
  • Evolution operators: On a logarithmic x grid, the evolution-operator matrix depends on node-index differences, so its first row determines the remaining entries.This reduces the number of integrals that must be computed by a factor Nx.
  • Evolution operators: The evolution operators satisfy coupled first-order linear differential equations in t and are solved with a fourth-order adaptive-step Runge-Kutta algorithm.No interpolation in t is used; arbitrary-x PDF values are obtained afterward through higher-order interpolation.
  • Grid choice and accuracy: A single logarithmic grid can lose accuracy at large x, so APFEL supports multiple interpolation grids tailored to different x regions.Using multiple grids slightly increases initialization and evolution time but improves large-x accuracy; the default uses three grids.

4 APFEL library documentation

The APFEL documentation describes its interfaces, installation, PDF-evolution workflow, customization options, and DIS module. It also presents a GUI that exposes most functionality and supports PDF and parton-luminosity plotting.

  • Interfaces and installation: APFEL is written in Fortran 77 and can also be accessed through C/C++ and Python interfaces.The documentation focuses on the C/C++ interface while noting that Fortran 77 and Python usage is similar.
  • Interfaces and installation: After installation, users can call APFEL routines directly or start an interactive console through the apfel shell script.The apfel-config script supplies compiler flags for custom makefiles.
  • PDF evolution workflow: PDF evolution first precomputes evolution operators during initialization and then evolves PDFs between an initial scale Q0 and a final scale.The initialization needs to be performed only once after general evolution settings are defined.
  • Customization: Users can customize the combined evolution through QCD, QED, and two averaged QCD⊗QED solution options, which are equivalent up to subleading O(ααs) terms.The available choices are QCD, QED, QavDP, and QavDS.
  • DIS module: The DIS module computes neutral- and charged-current observables and can use APFEL-evolved PDFs or interface directly with LHAPDF.This module is separate from, but compatible with, APFEL PDF evolution.
  • Graphical User Interface: The GUI exposes almost all APFEL features without coding, supports Monte Carlo and Hessian PDF uncertainties, and provides customizable plots and saved data.PDF and parton-luminosity GUI results were verified against corresponding PDF benchmark results.

5 Validation and benchmarking

APFEL is benchmarked against established QCD and QCD⊗QED evolution codes, showing strong agreement for QCD evolution and identifying the most reliable coupled-solution strategy. Its DIS structure-function module also agrees closely with FONLLdis.

  • 5.1 QCD evolution: APFEL agrees with HOPPET for NNLO QCD PDF evolution using both pole and MS masses, with differences at most ∼0.02% at large x.The comparison covers the full x range and uses Les Houches benchmark settings.
  • 5.2.1 Comparison with partonevolution: NLO QCD-only evolution from APFEL and partonevolution shows good agreement under the shared benchmark settings.This establishes the baseline before QED corrections are included.
  • 5.2.1 Comparison with partonevolution: With combined QCD⊗QED evolution, quark and gluon differences are at most O(0.01%), while photon-PDF differences stay below a few percent except at largest x.The comparison uses APFEL’s QCED, QECD, and QavD solution options against partonevolution.
  • 5.2.1 Comparison with partonevolution: QCED and QECD can differ from partonevolution by up to 70% for the photon PDF because unresummed scale logarithms enhance formally subleading terms.The largest quark and gluon discrepancy is up to 0.5% at large x.
  • 5.3 Consistency of the coupled solution: Using 100 evolution steps reduces differences among QavD, QECD, and QCED to at most 2%, supporting QavD as the most reliable solution.The analysis attributes the improvement to avoiding potentially large unresummed logarithms through small-step evolution.
  • 5.4 DIS structure functions: APFEL’s DIS structure functions agree with FONLLdis below the percent level for F2^c and at most 1.3% for F2^l.The authors describe this as sufficient accuracy for phenomenology.

6 Conclusions and outlook

APFEL provides public NNLO QCD plus LO QED PDF evolution with flexible interfaces, heavy-quark schemes, and a FONLL DIS module. The paper reports accurate DIS benchmarking and outlines extensions for fragmentation functions, polarized PDFs, resummation, and higher-order QED effects.

  • APFEL combines NNLO QCD and LO QED corrections for DGLAP PDF evolution, supporting FFN and VFN schemes with pole or MSbar heavy-quark masses.
  • The library adds neutral- and charged-current DIS structure functions through O(α_s^2) in the FONLL scheme.
  • APFEL is accessible through Fortran 77, C/C++, Python, and a graphical interface with plotting tools, and is publicly distributed through HepForge and its repository.
  • The authors motivate APFEL as a useful tool for PDF fitting because precision LHC predictions require consistently including QED and electroweak effects.
  • Planned extensions include time-like evolution, polarized PDFs, alternative factorization schemes, small-x resummation, pure electroweak corrections, and higher-order QED effects.
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