A fast C++ implementation of thermal functions

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9 Citations (Scopus)


We provide a small C++ library with Mathematica and Python interfaces for computing thermal functions, defined JB/F(y2)≡ℜ∫0 x2log1∓e−x2+y2 dx,which appear in finite-temperature quantum field theory and play a role in phase-transitions in the early Universe, including baryogenesis, electroweak symmetry breaking and the Higgs mechanism. The code is available from https://github.com/andrewfowlie/thermal_funcs. Program summary: Program title: thermal_funcs Program Files doi: http://dx.doi.org/10.17632/5nzrr2jjpd.1 Licensing provisions: BSD 3-Clause Programming language: C++, C interface to Mathematica and SWIG interface to Python Nature of problem: Thermal functions appear in finite-temperature quantum field theory and influence phase transitions in the early Universe. They have no closed-form solution. Studying phase transitions requires repeated evaluations of, inter alia, thermal functions. We provide several fast and accurate methods, and first and second derivatives. Solution method: We implement numerical quadrature, a Bessel function representation, asymptotic solutions expressed in terms of zeta functions and polylogarithms, and approximations and limits.

Original languageEnglish
Pages (from-to)264-272
Number of pages9
JournalComputer Physics Communications
Publication statusPublished - Jul 2018
Externally publishedYes


  • Phase transitions
  • Special functions
  • Thermal functions


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