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Dear FeynCalc users,

we agree that in many computations it is quite useful to work with SU(N)

matrices in the fundamental representation that carry explicit

fundamental indices. Although one can define such objects and do the

necessary contractions by hand, we decided to implement this as a new

feature in the upcoming FeynCalc 9. It is already available in the

development version.

<https://github.com/FeynCalc/feyncalc/commit/062cd8c8c6d3d62aed11945b496db5795545a319>

First of all, we changed the typesetting of all the SU(N) objects, such

that now adjoint indices are always upstairs, while the fundamental

indices are downstairs. An SU(N) matrix with explicit fundamental

indices is denoted by SUNTF. The full FeynCalcInternal representation is

SUNTF[{SUNIndex[a]},SUNFIndex[i],SUNFIndex[j]]

which corresponds to T^a_ij, where a is the adjoint, while i and j are

fundamental indices. Note the new head SUNFIndex that stands for

fundamental indices. Furthermore, unlike SUNT, SUNTF commutes with

everything, since it is not a matrix, but a particular element of a

matrix, i.e. c-number. The first argument is a list of SUNIndex objects,

hence to write say (T^a T^b T^c)_ij you can use

SUNTF[{SUNIndex[a], SUNIndex[b], SUNIndex[c] },SUNFIndex[i],SUNFIndex[j]]

As usual, for convenience you can use the FeynCalcExternal input style,

i.e. SUNT[{a},i,j] or SUNT[{a,b,c},i,j] are fine as well. A special case

is SUNT[a,i,j] which is automatically converted to SUNT[{a},i,j]

In addition to that we added the Kronecker delta in the fundamental

representation (SUNFDelta), described as

SUNFDelta[SUNFIndex[b],SUNFIndex[c]]. Its FeynCalcExternal name is SDF

and can be used as SDF[a,b].

A new function SUNFSimplify is responsible for working with fundamental

indices. It can contract SUNTFs that have common indices (i.e.

SUNTF[{a},i,j] SUNTF[{b},j,k] SUNTF[{c},k,l] becomes SUNTF[{a,b,c},i,l])

and perform contractions with Kronecker deltas like

SUNTF[{a},i,j] SUNFDelta[j,k] SUNTF[{b},k,l] -> SUNTF[{a,b},i,l]

SUNFDelta[i,j]SUNFDelta[j,i] -> SUNN

However, it is not needed to execute SUNFSimplify separately.

SUNSimplify automatically identifies expressions contatining SUNFIndex

and runs SUNFSimplify on them.

ComplexConjugate can handle SUNTF objects without problems,e.g.

ComplexConjugate[SUNTF[a, i, j]] -> SUNTF[a,j,i]

ComplexConjugate[SUNTF[{a,b,c}, i, j]] -> SUNTF[{c,b,a},j,i]

The same goes for FCRenameDummyIndices:

FCRenameDummyIndices[SUNTF[{a}, i, j] SUNTF[{b}, j, k]] ->

SUNTF[{a}, i, $AL[57]] SUNTF[{b}, $AL[57], k]

Now it is also very simple to convert FeynArts output that contains

color matrices to FeynCalc input:

SUNT[Index[Gluon, 2], Index[Colour, 3], Index[Colour, 5]]

/.{Index[Gluon,x_]:>SUNIndex[ToExpression["Glu"<>ToString[x]]],

Index[Colour,x_]:>SUNFIndex[ToExpression["Col"<>ToString[x]]],SUNT->SUNTF}

becomes

SUNTF[{SUNIndex[Glu2]}, SUNFIndex[Col3], SUNFIndex[Col5]].

Last but not least, we added new examples for tree level parton

processes in QCD that extensively use new SUNTF objects.

<https://github.com/FeynCalc/feyncalc/tree/master/FeynCalc/fcexamples/QCD>

If you are interested to help, please give the new SUNTF and SUNFDelta a

try and report possible issues.

Cheers,

Vladyslav

**Next message:**Vladyslav Shtabovenko: "Re: Polarization sums with dummy indices"**Previous message:**Nodoka Yamanaka: "Reduction of SUND"**Maybe in reply to:**Marco: "Issues with SUNSimplify"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Mail actions:**[ respond to this message ] [ mail a new topic ]

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