•ϕϕA

lag = Lagrangian[ChPTW3[4]]

1/(f _ ϕ^(ó  ))^2 (c _ 2^( ) (N _ 13^( ) (< Δ '6 χ _ - > '6 < χ _ - >) + N _ 11^( ) (< Δ '6 χ _ + > '6 < χ _ + >) + N _ 7^( ) (< Δ '6 χ _ + > '6 < u _ μ '6 u _ μ >) + N _ 6^( ) (< Δ '6 u _ μ > '6 < χ _ + '6 u _ μ >) + i N _ 24^( ) (< Overscript[∇,^] _ μ Δ '6 u _ μ > '6 < χ _ - >) + N _ 8^( ) (< Δ '6 u _ μ '6 u _ μ > '6 < χ _ + >) + N _ 22^( ) < Overscript[∇,^] _ μ Δ '6 Overscript[∇,^] _ μ χ _ + > + N _ 12^( ) < Δ '6 χ _ - '6 χ _ - > + N _ 10^( ) < Δ '6 χ _ + '6 χ _ + > + i N _ 23^( ) (< Overscript[∇,^] _ μ Δ '6 χ _ - '6 u _ μ > + < Overscript[∇,^] _ μ Δ '6 u _ μ '6 χ _ - >) + i N _ 21^( ) (< Overscript[∇,^] _ μ Δ '6 χ _ + '6 u _ μ > - < Overscript[∇,^] _ μ Δ '6 u _ μ '6 χ _ + >) + N _ 20^( ) (< Overscript[∇,^] _ μ Δ '6 ω^(μ ν) '6 u _ ν > + < Overscript[∇,^] _ μ Δ '6 u _ ν '6 ω^(μ ν) >) + i N _ 17^( ) < Δ '6 u _ μ '6 f _ - _ (μ ν) '6 u _ ν > + i N _ 15^( ) < Δ '6 u _ μ '6 f _ + _ (μ ν) '6 u _ ν > + N _ 9^( ) (< Δ '6 χ _ - '6 u _ μ '6 u _ μ > - < Δ '6 u _ μ '6 u _ μ '6 χ _ - >) + N _ 5^( ) (< Δ '6 χ _ + '6 u _ μ '6 u _ μ > + < Δ '6 u _ μ '6 u _ μ '6 χ _ + >) + i N _ 16^( ) (< Δ '6 f _ - _ (μ ν) '6 u _ μ '6 u _ ν > + < Δ '6 u _ μ '6 u _ ν '6 f _ - _ (μ ν) >) + i N _ 14^( ) (< Δ '6 f _ + _ (μ ν) '6 u _ μ '6 u _ ν > + < Δ '6 u _ μ '6 u _ ν '6 f _ + _ (μ ν) >) + i N _ 19^( ) (< Overscript[∇,^] _ μ Δ '6 u _ μ '6 u _ ν '6 u _ ν > - < Overscript[∇,^] _ μ Δ '6 u _ ν '6 u _ ν '6 u _ μ >)))

First, UNMSplit is used to expand NM products of U matrices into meson fields:

Expand[NMExpand[lag]] // Length

27

llu = CheckF[(WriteString["stdout", "."] ; UNMSplit[#, x, DropOrder -> 2]) & /@ (Expand[NMExpand[lag]] /. UTrace[a_] :> 0 /; (Count[a, _LeftComponent | _RightComponent, Infinity] > 1)), "2MesonAllu"] ;

...........................

Expand[llu] // Length

1960

Redundant terms are discarded:

lld = (WriteString["stdout", "."] ; DiscardTerms[#, Retain -> {Particle[PhiMeson ] -> 2}, CommutatorReduce -> True, Method -> Coefficient]) & /@ Expand[llu] ;

........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

lld // Length

1647

lld // LeafCount

118552

Remaining 'raw' quantites are put on arguments:

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

ArgumentsSupply :: argxpr : Warning : The argument x is already in the expression.

Expand[ll] // Length

740

lle = (WriteString["stdout", "."] ; DiscardTerms[#, Retain -> {Particle[PhiMeson , RenormalizationState[0]] -> 2, Particle[AxialVector[0], RenormalizationState[0]] -> 1}, CommutatorReduce -> True, Method -> Expand]) & /@ Expand[ll] ;

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

lle // Length

346

Generator matrices are traced:

llle = ExpandU[lle, CommutatorReduce -> True] ;

Indices are supplied:

$IsoIndicesCounter = 0 ;

Expand[llle] // Length

986

llll = CheckF[(WriteString["stdout", "."] ; # // CommutatorReduce[#, FullReduce -> True] & // IsoIndicesSupply // SUNReduce // IndicesCleanup // NMExpand // CommutatorReduce[#, FullReduce -> True] & // Simplify) & /@ Expand[llle], "2MesonAllll"] ;

Calculation of the Feynman rule:

fields = {QuantumField[Particle[AxialVector[0], RenormalizationState[0]], LorentzIndex[μ1], SUNIndex[I1]][p1], QuantumField[Particle[PhiMeson, RenormalizationState[0]], SUNIndex[I2]][p2], QuantumField[Particle[PhiMeson, RenormalizationState[0]], SUNIndex[I3]][p3]}

{A^( ) _ μ _ 1^I _ 1, ϕ^( )^I _ 2, ϕ^( )^I _ 3}

llll // Length

846

melsimplified = (WriteString["stdout", "."] ; Simplify[IndicesCleanup[I * SUNReduce[FunctionalD[PhiToFC[#], fields], Fullreduce -> True]]]) & /@ llll ;

..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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

melsimplified /. {I1 -> 7, I2 -> 3, I3 -> 3} // SUNReduce[#, Explicit -> True, HoldSums -> False] & // Contract // Collect[#, {_DecayConstant, _ParticleMass}, If[FreeQ[#, _DecayConstant], Simplify[#], #] &] & // Simplify

1/(f _ ϕ^(ó  ))^4 (2 c _ 2^( ) (p _ 2^μ _ 1 + p _ 3^μ _ 1) ((N _ 8^( ) + 2 N _ 9^( ) - N _ 22^( ) - 2 N _ 23^( )) (m _ π^(ó  ))^2 - (2 N _ 6^( ) - N _ 8^( ) + 2 N _ 9^( ) - N _ 22^( ) - 2 N _ 23^( ) - 2 N _ 24^( )) (m _ K^+^(ó  ))^2 + (2 N _ 5^( ) + 2 N _ 6^( ) + N _ 8^( ) + 2 N _ 9^( ) - N _ 22^( ) - 2 N _ 23^( ) - 2 N _ 24^( )) (m _ K^0^(ó  ))^2 - 2 N _ 20^( ) p _ 2 · p _ 3))

melsimplified /. {I1 -> 7, I2 -> 3, I3 -> 3} // SUNReduce // SUNReduce // SUNReduce // SUNReduce // SUNReduce // SUNReduce // Collect[#, {_DecayConstant, _ParticleMass}, If[FreeQ[#, _DecayConstant], Simplify[#], #] &] & // Simplify

1/(f _ ϕ^(ó  ))^4 (2 c _ 2^( ) (p _ 2^μ _ 1 + p _ 3^μ _ 1) ((N _ 8^( ) + 2 N _ 9^( ) - N _ 22^( ) - 2 N _ 23^( )) (m _ π^(ó  ))^2 - (2 N _ 6^( ) - N _ 8^( ) + 2 N _ 9^( ) - N _ 22^( ) - 2 N _ 23^( ) - 2 N _ 24^( )) (m _ K^+^(ó  ))^2 + (2 N _ 5^( ) + 2 N _ 6^( ) + N _ 8^( ) + 2 N _ 9^( ) - N _ 22^( ) - 2 N _ 23^( ) - 2 N _ 24^( )) (m _ K^0^(ó  ))^2 - 2 N _ 20^( ) p _ 2 · p _ 3))

Converted by Mathematica  (July 10, 2003)