•Pϕϕϕ

The evaluated leading order Lagrangian:

ll = ArgumentsSupply[Lagrangian[ChPTW3[2]], x, RenormalizationState[0], ExpansionOrder -> 3, DropOrder -> 3, DiagonalToU -> True] ;

Redundant terms are discarded:

lll = DiscardTerms[ll, Retain -> {Particle[PhiMeson , RenormalizationState[0]] -> 3, Particle[PseudoScalar[0] , RenormalizationState[0]] -> 1}, CommutatorReduce -> True, Method -> Expand] // Simplify

-1/(3 (f _ ϕ^(ó  ))^3) (c _ 5^( ) !, _ 0^( ) (< σ^6 '6 Overscript[p^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] > + < σ^6 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[p^( ), ->] · Overscript[σ, ->] > + 2 < σ^6 '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] > p^( )^0))

Generator matrices are traced:

llle = lll // ExpandU // CommutatorReduce[#, FullReduce -> True] & // Simplify ;

Indices are supplied:

$IsoIndicesCounter = 0 ;

llll = llle // IsoIndicesSupply // SUNReduce[#, FullReduce -> True] & // IndicesCleanup // CommutatorReduce[#, FullReduce -> True] & // Simplify

-1/(9 (f _ ϕ^(ó  ))^3) (2 c _ 5^( ) !, _ 0^( ) (2 p^( )^0 ϕ^( )^k1 (2 ϕ^( )^6 ϕ^( )^k1 + 3 d _ (k2 k3 k4)^(3) (d _ (6 k1 k4)^(3) + i f _ (6 k1 k4)^(3)) ϕ^( )^k2 ϕ^( )^k3) + ϕ^( )^k2 (2 d _ (k1 k2 k3)^(3) ϕ^( )^k1 (p^( )^k3 ϕ^( )^6 + p^( )^6 ϕ^( )^k3) + ϕ^( )^k3 (2 d _ (6 k1 k2)^(3) (p^( )^k3 ϕ^( )^k1 + p^( )^k1 ϕ^( )^k3) + 3 d _ (6 k1 k5)^(3) (d _ (k2 k5 k6)^(3) - i f _ (k2 k5 k6)^(3)) (i f _ (k3 k4 k6)^(3) p^( )^k4 ϕ^( )^k1 + d _ (k3 k4 k6)^(3) (p^( )^k4 ϕ^( )^k1 + p^( )^k1 ϕ^( )^k4)) + i (3 f _ (6 k1 k5)^(3) f _ (k2 k5 k6)^(3) f _ (k3 k4 k6)^(3) p^( )^k4 ϕ^( )^k1 + 2 f _ (6 k1 k2)^(3) p^( )^k1 ϕ^( )^k3 + 3 i d _ (k3 k4 k5)^(3) f _ (6 k1 k6)^(3) f _ (k2 k5 k6)^(3) (p^( )^k4 ϕ^( )^k1 + p^( )^k1 ϕ^( )^k4) + 3 d _ (k2 k5 k6)^(3) f _ (6 k1 k6)^(3) (i f _ (k3 k4 k5)^(3) p^( )^k4 ϕ^( )^k1 + d _ (k3 k4 k5)^(3) (p^( )^k4 ϕ^( )^k1 + p^( )^k1 ϕ^( )^k4)))))))

Calculation of the Feynman rule:

fields = {QuantumField[Particle[PseudoScalar[0], RenormalizationState[0]], SUNIndex[I1]][p1], QuantumField[Particle[PhiMeson, RenormalizationState[0]], SUNIndex[I2]][p2], QuantumField[Particle[PhiMeson, RenormalizationState[0]], SUNIndex[I3]][p3], QuantumField[Particle[PhiMeson, RenormalizationState[0]], SUNIndex[I4]][p4]}

{p^( )^I _ 1, ϕ^( )^I _ 2, ϕ^( )^I _ 3, ϕ^( )^I _ 4}

lal = Expand[llll] ;

melsimplified = If[Head[lal] === Plus, Plus @@ (IndicesCleanup[SUNReduce[FeynRule[#, fields]]] & /@ (List @@ lal)), Print["Error"]] ;

A check that different evaluations with specific components give the same result:

Table[(SUNReduce[#, FullReduce -> True] & /@ (melsimplified /. {I1 -> 3, I3 -> I2, I4 -> 6} /. I2 -> ii // Expand)), {ii, 8}]

{(4 i c _ 5^( ) !, _ 0^( ))/(3 (f _ ϕ^(ó  ))^3), (4 i c _ 5^( ) !, _ 0^( ))/(3 (f _ ϕ^(ó  ))^3), (4 i c _ 5^( ) !, _ 0^( ))/(3 (f _ ϕ^(ó  ))^3), (4 i c _ 5^( ) !, _ 0^( ))/(3 (f _ ϕ^(ó  ))^3), (4 i c _ 5^( ) !, _ 0^( ))/(3 (f _ ϕ^(ó  ))^3), (4 i c _ 5^( ) !, _ 0^( ))/(f _ ϕ^(ó  ))^3, (4 i c _ 5^( ) !, _ 0^( ))/(3 (f _ ϕ^(ó  ))^3), (4 i c _ 5^( ) !, _ 0^( ))/(3 (f _ ϕ^(ó  ))^3)}

Table[(SUNReduce[#, Explicit -> True, HoldSums -> False] & /@ Evaluate[(melsimplified /. {I1 -> 3, I2 -> ii, I3 -> ii, I4 -> 6} // Expand)]) // Simplify, {ii, 8}]

{(4 i c _ 5^( ) !, _ 0^( ))/(3 (f _ ϕ^(ó  ))^3), (4 i c _ 5^( ) !, _ 0^( ))/(3 (f _ ϕ^(ó  ))^3), (4 i c _ 5^( ) !, _ 0^( ))/(3 (f _ ϕ^(ó  ))^3), (4 i c _ 5^( ) !, _ 0^( ))/(3 (f _ ϕ^(ó  ))^3), (4 i c _ 5^( ) !, _ 0^( ))/(3 (f _ ϕ^(ó  ))^3), (4 i c _ 5^( ) !, _ 0^( ))/(f _ ϕ^(ó  ))^3, (4 i c _ 5^( ) !, _ 0^( ))/(3 (f _ ϕ^(ó  ))^3), (4 i c _ 5^( ) !, _ 0^( ))/(3 (f _ ϕ^(ó  ))^3)}

Converted by Mathematica  (July 10, 2003)