MATAD-ng

MATAD is a form code for three-loop MAssive TADpole integral reduction originally written by M.Steinhauser and published in Comput.Phys.Commun. 134 (2001) 335-364.

MATAD-ng is based on the same code by M.Steinhauser but use FORM4 PolyRatFun for intermediate expressions and allow to produce reduction without expansion in ep in terms of master integrals and gamma functions.

For origianl MATAD usage and examples see M.Steinhauser lectures at CAPP 2009 https://www.ttp.kit.edu/~ms/capp09.pdf

Master integrals

To use reduction at arbitrary d only following 9 master integrals needed:

With expansion of Euler gamma functions near d for remaining trivial integrals

Gam(n,x)=Gamma(n+(2-d/2)*x)*Exp(ep*x*EulerGamma), iGam(n,x)=Exp(-ep*x*EulerGamma)/Gamma(n+(2-d/2)*x)

Each integral is divided by Exp(-ep*EulerGamma) and Zeta[2] is present in final results.

Extension to weight 6

To use new results for integrals expansion up to weight 6 disable expansion in form and instead use attached mathematica files with master integrals substitution rules mtdw6.m and tables with HPL's of up to weight 6 of argument Exp[I Pi/3]. Real parts stored in nhplRe.m and imaginary in nhplIm.m with 20000 numerical digits precision.

Usage

Reduction

Calculate one-loop integral using MATAD notation for massive propagators s1m and for massles propagator 1/p1.p1

#-
#include matad-ng.hh
* We use s1m for massive denominator and 1/p1.p1 for massless
L ex1loop = s1m^2*p1.p1;
* 1 - one-loop reduction
#call matad(1)
* expansion upto ep^3
#call exp4d(3)
hide;
.sort

Calculate three-loop integral using auxiliary topology definition, mass distribution determined by symbol for example [MM00MM]

L ex3loopB = tad3l([MM00MM],2,2,2,1,1,1);
* Convert to original MATAD notation
#call FromAuxTopo
* 3 - three-loop reduction
#call matad(3)

Expand result in ep up to O(ep) near d=4-2*ep

#call exp4d(1)

Dimension shifts

Here we express two-loop master integral miT1 in d=(4+2)-2*ep dimensions in terms of integrals in d=4-2*ep dimensions

L f1 = miT1;
#call shift4plus(2)

And result is

f1 =
   + Gam(1,1)^2*rat(-48,d^6 - 13*d^5 + 64*d^4 - 148*d^3 + 160*d^2 - 64*d)
   + miT1*rat(3,d^2 - d);

where miT1 is two-loop master integral in four dimensions and d=4-2*ep

Tests

To check correctness of results with package supplied test cases with results obtained with the help of original MATAD package. To run tests use command $ form tests.frm and file tests.err will be produced if results of calculation do not match.