•2ϕAS

The evaluated  next to leading order Lagrangian:

lag = Lagrangian[ChPTW3[4]] /. CouplingConstant[ChPTW3[4], _ ? ((# < 14 || # === 36) &), ___] :> 0 ;

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

llu = CheckF[(WriteString["stdout", "."] ; UNMSplit[#, x, DropOrder -> 2]) & /@ Expand[lag], "SU3Weak2mesonSAllu"] ;

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

lux = Expand[NMExpand[llu]] ;

'raw' quantites are given arguments:

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

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

lla // Length

832

Redundant terms are discarded:

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

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

ll // LeafCount

23291

Expand[ll] // Length

267

DeclareUScalar[QuantumField[aa___, Particle[bb__], cc___][x_]] ; DeclareUScalar[UTrace1] ;

lll = (WriteString["stdout", "."] ; Simplify[NMExpand[ExpandAll[#]]]) & /@ ll ;

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

lal = (WriteString["stdout", "."] ; $IsoIndicesCounter = 0 ; IsoIndicesSupply[#]) & /@ Expand[lll] ;

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

Cases[lal, UTrace1[__ ? ((! FreeQ[{#}, QuantumField, Infinity, Heads -> True]) &)], Infinity, Heads -> True] // Length

0

Cases[lal /. _UTrace1 -> 1, _Umatrix, Infinity, Heads -> True] // Length

0

DeclareNonCommutative[UMatrix[a__]] ;

Calculation of the Feynman rule with FeynCalc:

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], QuantumField[Particle[Scalar[1], RenormalizationState[0]]][p4]}

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

res = ((WriteString["stdout", "."] ; Simplify[I * SUNReduce[FunctionalD[PhiToFC[#], fields], FullReduce -> True]]) & /@ lal) ;

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

res // LeafCount

21113

resu = Collect[res // Expand, {_DecayConstant, _UTrace1}] ;

resu // LeafCount

17573

resul = Collect[resu, HoldPattern[Plus[__ ? ((! FreeQ[{##}, Momentum | ParticleMass, Infinity, Heads -> True]) &)]]] ;

resul // LeafCount

11454

resul // Length

32

resul[[1]]

1/(f _ ϕ^(ó  ))^4 ((1/2 i c _ 2^( ) N _ 24^( ) p _ 4^μ _ 1 (m _ K^0^(ó  ))^2 - 1/2 i c _ 2^( ) N _ 24^( ) p _ 4^μ _ 1 (m _ K^+^(ó  ))^2) (< σ^3 . σ^I _ 3 > < σ^6 . σ^I _ 2 . σ^I _ 1 > + < σ^I _ 3 . σ^3 > < σ^6 . σ^I _ 2 . σ^I _ 1 > + < σ^3 . σ^I _ 2 > < σ^6 . σ^I _ 3 . σ^I _ 1 > + < σ^I _ 2 . σ^3 > < σ^6 . σ^I _ 3 . σ^I _ 1 >))

A few tests:

test = resul /. {I1 -> 7, I2 -> 3, I3 -> 3} // WriteOutUMatrices ;

subpar = Table[(ParticleMass[PseudoScalar[1], SUNIndex[i], RenormalizationState[0]] -> ParticleMass[Select[$IsoSpinProjectionRules, (! FreeQ[#, {i}] &)][[1]][[1]], RenormalizationState[0]]), {i, 8}] ;

udrules = {PionPlus -> Pion, PionZero -> Pion, KaonPlus -> Kaon, KaonZero -> Kaon} ;

I * Collect[Cancel[(test/I /. subpar /. udrules /. p1 -> -p2 - p3 // MomentumExpand // ExpandScalarProduct) /. Pair[Momentum[p2, ___], Momentum[p3, ___]] -> (ParticleMass[Kaon, RenormalizationState[0]]^2 - Pair[Momentum[p2], Momentum[p2]] - Pair[Momentum[p3], Momentum[p3]])/2 /. {Pair[Momentum[p2, ___], Momentum[p2, ___], ___] -> ParticleMass[Pion, RenormalizationState[0]]^2, Pair[Momentum[p3, ___], Momentum[p3, ___], ___] -> ParticleMass[Pion, RenormalizationState[0]]^2}], {_DecayConstant, _ParticleMass}] // FullSimplify

1/(f _ ϕ^(ó  ))^4 (c _ 2^( ) (-4 N _ 19^( ) p _ 4^μ _ 1 (m _ π^(ó  ))^2 - 8 N _ 21^( ) p _ 4^μ _ 1 (m _ π^(ó  ))^2 - 2 N _ 22^( ) p _ 4^μ _ 1 (m _ π^(ó  ))^2 - 4 N _ 23^( ) p _ 4^μ _ 1 (m _ π^(ó  ))^2 + (2 N _ 19^( ) + 4 N _ 21^( ) + N _ 26^( )) p _ 4^μ _ 1 (m _ K^(ó  ))^2 - N _ 26^( ) p _ 2^μ _ 1 p _ 2 · p _ 4 - 2 N _ 19^( ) p _ 3^μ _ 1 p _ 2 · p _ 4 - N _ 26^( ) p _ 3^μ _ 1 p _ 2 · p _ 4 - 2 N _ 19^( ) p _ 2^μ _ 1 p _ 3 · p _ 4 - N _ 26^( ) p _ 2^μ _ 1 p _ 3 · p _ 4 - N _ 26^( ) p _ 3^μ _ 1 p _ 3 · p _ 4 + N _ 25^( ) ((p _ 2^μ _ 1 + p _ 3^μ _ 1) (p _ 2 · p _ 4 + p _ 3 · p _ 4) - p _ 4^μ _ 1 (m _ K^(ó  ))^2) + N _ 20^( ) (p _ 4^μ _ 1 (m _ K^(ó  ))^2 - (p _ 2^μ _ 1 - p _ 3^μ _ 1) (p _ 2 · p _ 4 - p _ 3 · p _ 4))))

Simplify /@ Collect[%, _CouplingConstant]

1/(f _ ϕ^(ó  ))^4 (c _ 2^( ) (-2 N _ 22^( ) p _ 4^μ _ 1 (m _ π^(ó  ))^2 - 4 N _ 23^( ) p _ 4^μ _ 1 (m _ π^(ó  ))^2 + 4 N _ 21^( ) p _ 4^μ _ 1 ((m _ K^(ó  ))^2 - 2 (m _ π^(ó  ))^2) - N _ 26^( ) (-p _ 4^μ _ 1 (m _ K^(ó  ))^2 + p _ 2^μ _ 1 (p _ 2 · p _ 4 + p _ 3 · p _ 4) + p _ 3^μ _ 1 (p _ 2 · p _ 4 + p _ 3 · p _ 4)) + N _ 25^( ) ((p _ 2^μ _ 1 + p _ 3^μ _ 1) (p _ 2 · p _ 4 + p _ 3 · p _ 4) - p _ 4^μ _ 1 (m _ K^(ó  ))^2) + N _ 20^( ) (p _ 4^μ _ 1 (m _ K^(ó  ))^2 - (p _ 2^μ _ 1 - p _ 3^μ _ 1) (p _ 2 · p _ 4 - p _ 3 · p _ 4)) - 2 N _ 19^( ) (p _ 3^μ _ 1 p _ 2 · p _ 4 + p _ 2^μ _ 1 p _ 3 · p _ 4 + p _ 4^μ _ 1 (2 (m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))))

UndeclareUScalar[QuantumField[aa___, Particle[bb__], cc___][x_]] ; UndeclareUScalar[UTrace1] ; UnDeclareNonCommutative[UMatrix[a__]] ;

Converted by Mathematica  (July 10, 2003)