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Bringing the Arguments of D Functions to Canonical Form

Introduction

In one-loop integrals corresponding to 22 scattering or 13 decay processes, the Passarino-Veltman functions depend on ten variables: 4 external masses, 4 internal masses, and 2 (out of 3) Mandelstam/Dalitz variables. It is conventional to present functions in publications with their arguments organized in the following way:

Due to numerous invariances of the function under permutations of its arguments, LoopIntegrate may not generate functions with this ordering. This tutorial illustrates the strategy to obtain functions with their arguments in canonical form for two examples:

    

(a)(b)

There are two issues to contend with in order to put the functions in canonical form.

1. Fixing the order of arguments

The first issue is getting the correct order, so that the external masses () occupy arguments 14 and the Mandelstam/Dalitz variables () occupy arguments 5 and 6. You can achieve this by entering the denominator specifiers in LoopIntegrate in the same order the corresponding internal lines are connected in a given graph. It does not matter which specifier comes first:

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You can replace the Lorentz dot products with masses and invariant energies using kinematic conditions. The function MandelstamRelations is a handy function to quickly generate a list of kinematic conditions for 22 scattering or 13 decay processes.

MandelstamRelations[{p1,p2,p3,p4,m1,m2,m3,m4}{s,t,u}]generate a list of replacement rules expressing dot products of 4-vectors , , , as Mandelstam invariants , , and masses , , , .

Rapidly generate 22 or 13 kinematic relations.

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2. Eliminating the dependent invariant

The second issue is to select the correct Mandelstam/Dalitz variable to eliminate using 4-momentum conservation:

This depends on the how the internal lines are connected in a given diagram: eliminate the invariant built out of the pair of momenta that enter/leave the diagram from vertices that are not joined by an internal propagator.

In the first diagram, eliminate since there is no internal line joining vertices and .
In the second diagram, eliminate since there is no line joining vertices and .

The option Eliminate to MandelstamRelations can be used to eliminate a given invariant:
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