packagex is hosted by Hepforge, IPPP Durham

Version 2.1, by Hiren H. Patel (
Accompanying paper: Comput.Phys.Commun. 197 (2015) 276. arXiv:1503.01469

Package-X is a Mathematica package for the analytic calculation and symbolic manipulation of one-loop Feynman integrals in relativistic quantum field theory.


Package-X follows the conventions summarized in the tables below.

Spacetime Conventions

Metric Signature
Spacetime dimension
Dirac matrix commutator
Fifth gamma matrix
Chiral projectors,
Levi-Civita symbol

Normalization of Loop Integrals

LoopIntegrate[num,k,{k+p0, m0},{k+p1, m1},]
    = ((i e^(-gamma_(E)epsilon))/((4pi)^(d/2)))^(-1)mu^(2epsilon)int(d^dk)/((2pi)^d)(num)/([(k+p_0)^2-m_0^2+i epsilon][(k+p_1)^2-m_1^2+i epsilon]...)

Arguments of the Passarino-Veltman Functions


General Tensor Operations and Symbols

LTensor (), LDot (.), LScalarQ Lorentz tensors and scalar products

Contract contract tensors with repeated indices

Transverse, Longitudinal project 2nd or higher rank Lorentz tensors

MandelstamRelations express scalar products in terms of Mandelstam invariants

MetricG(,)  ▪  LeviCivitaE(ε,,,) metric tensor and Levi-Civita symbol

Computing One-Loop Integrals

LoopIntegrate, LoopRefine routines for one loop integrals

LoopRefineSeries compute series expansions of loop integrals

Dim()  ▪  Eps(ϵ)  ▪  Mu(µ) dimensional regularization symbols

Dirac Algebra and Fermion Spinors

Spur compute traces over product of Dirac matrices

Projector project fermion self energy and vertex functions onto form factors

DiracMatrix, FermionLine (, ), FermionLineProduct, FermionLineExpand open fermion chains

Dirac1()  ▪  DiracG(γ)  ▪  DiracG5(γ5)  ▪  DiracPL(L)  ▪  DiracPR(R)  ▪  DiracS(σ,) objects in spinor space

Generated Symbols

PVA  ▪  PVB  ▪  PVC  ▪  PVD  ▪  PVX symbolic Passarino-Veltman functions

Special Functions and Abbreviations »

Kallenλ  ▪  DiscB  ▪  ScalarC0  ▪  ScalarD0  ▪