•3ϕS

The evaluated  next to leading order Lagrangian:

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

1/(f _ ϕ^(ó  ))^2 (c _ 2^( ) (i N _ 24^( ) (< Overscript[∇,^] _ μ Δ '6 u _ μ > '6 < χ _ - >) + N _ 22^( ) < Overscript[∇,^] _ μ Δ '6 Overscript[∇,^] _ μ 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 > + 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:

llu = CheckF[(WriteString["stdout", "."] ; UNMSplit[#, x, DropOrder -> 3]) & /@ Expand[lag], "SU3Weak3mesonSllu"] ;

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

lluu = NMExpand[llu] ;

Expand[lluu] // Length

239

Expand[lluu][[1]]

(i c _ 2^( ) N _ 24^( ) ℵ^3 (< ∂ _ μ(s^( )) '6 σ^6 '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] > '6 < χ '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->] >))/(8 (f _ ϕ^(ó  ))^5)

Redundant terms are discarded:

lld = (WriteString["stdout", "."] ; DiscardTerms[#, Retain -> {Particle[PhiMeson ] -> 3}, CommutatorReduce -> True, Method -> Coefficient]) & /@ Expand[lluu] ;

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

lld // Length

237

Remaining 'raw' quantites are given arguments:

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

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

Expand[ll] // Length

44

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

ll[[1]] // ExpandAll // NMExpand // Simplify

-1/(6 (f _ ϕ^(ó  ))^5) (i c _ 2^( ) N _ 24^( ) (2 3^(1/2) < Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 σ^8 > (m _ π^(ó  ))^2 + (m _ K^+^(ó  ))^2 (3 < Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 σ^3 > - 3^(1/2) < Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 σ^8 >) - (m _ K^0^(ó  ))^2 (3 < Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 σ^3 > + 3^(1/2) < Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 σ^8 >)) < Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 σ^6 '6 ∂ _ μ(Overscript[ϕ^( ), ->]) · Overscript[σ, ->] > ∂ _ μ s^( ) _ ó ^ó )

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

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

Expand[lll] // Length

210

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

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

lal[[1]]

-(i c _ 2^( ) N _ 21^( ) (m _ π^(ó  ))^2 < σ^6 '6 σ^i _ 1 '6 σ^i _ 2 '6 σ^i _ 3 > ∂ _ μ s^( ) _ ó ^ó  ϕ^( )^i _ 2 ϕ^( )^i _ 3 ∂ _ μ ϕ^( ) _ ó ^i _ 1)/(3 (f _ ϕ^(ó  ))^5)

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:

fields = {QuantumField[Particle[PhiMeson, RenormalizationState[0]], 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]}

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

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

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

res // Length

206

res // LeafCount

37161

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

resul // LeafCount

14505

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} ;

massshellrules = {Pair[Momentum[p1], Momentum[p4]] -> (MandelstamS - Pair[Momentum[p1], Momentum[p1]] - Pair[Momentum[p4], Momentum[p4]])/2, Pair[Momentum[p3], Momentum[p2]] -> (MandelstamS - ParticleMass[Pion, RenormalizationState[1]]^2 - ParticleMass[Pion, RenormalizationState[1]]^2)/2, Pair[Momentum[p1], Momentum[p2]] -> (MandelstamT - Pair[Momentum[p1], Momentum[p1]] - ParticleMass[Pion, RenormalizationState[1]]^2)/2, Pair[Momentum[p4], Momentum[p3]] -> (MandelstamT - Pair[Momentum[p4], Momentum[p4]] - ParticleMass[Pion, RenormalizationState[1]]^2)/2, Pair[Momentum[p1], Momentum[p3]] -> (MandelstamU - Pair[Momentum[p1], Momentum[p1]] - ParticleMass[Pion, RenormalizationState[1]]^2)/2, Pair[Momentum[p2], Momentum[p4]] -> (MandelstamU - Pair[Momentum[p4], Momentum[p4]] - ParticleMass[Pion, RenormalizationState[1]]^2)/2, Pair[Momentum[p3], Momentum[p3]] -> ParticleMass[Pion, RenormalizationState[1]]^2, Pair[Momentum[p2], Momentum[p2]] -> ParticleMass[Pion, RenormalizationState[1]]^2, MandelstamS + MandelstamT + MandelstamU -> 2 ParticleMass[Pion, RenormalizationState[1]]^2 + Pair[Momentum[p1], Momentum[p1]] + Pair[Momentum[p4], Momentum[p4]], -MandelstamS - MandelstamT - MandelstamU -> -(2 ParticleMass[Pion, RenormalizationState[1]]^2 + Pair[Momentum[p1], Momentum[p1]] + Pair[Momentum[p4], Momentum[p4]])} ;

cancelU = MandelstamU -> (ParticleMass[Kaon, RenormalizationState[1]])^2 + 2 (ParticleMass[Pion, RenormalizationState[1]])^2 + Pair[Momentum[p2], Momentum[p2]] - MandelstamS - MandelstamT ;

softPionLimit = {MandelstamS -> ParticleMass[Pion]^2, MandelstamT -> ParticleMass[Kaon]^2, MandelstamU -> Pair[Momentum[p4], Momentum[p4]]} ;

softPionLimit1 = {MandelstamS -> ParticleMass[Pion]^2, MandelstamT -> Pair[Momentum[p4], Momentum[p4]], MandelstamU -> ParticleMass[Kaon]^2} ;

I * Collect[Cancel[(test/I /. subpar /. udrules // MomentumExpand // ExpandScalarProduct) /. D -> Sequence[] //. massshellrules /. Pair[Momentum[p1], Momentum[p1]] -> ParticleMass[Kaon, RenormalizationState[1]]^2 /. Pair[Momentum[p1], Momentum[p1]]^2 -> ParticleMass[Kaon, RenormalizationState[1]]^4], {_DecayConstant, _ParticleMass}] /. _RenormalizationState -> Sequence[] // Simplify

1/(6 (f _ ϕ^(ó  ))^5) (i c _ 2^( ) (-8 N _ 21^( ) (m _ π^(ó  ))^4 - 12 N _ 22^( ) (m _ π^(ó  ))^4 - 24 N _ 23^( ) (m _ π^(ó  ))^4 + 16 N _ 21^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 12 N _ 22^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 24 N _ 23^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 20 s N _ 21^( ) (m _ π^(ó  ))^2 + 4 t N _ 21^( ) (m _ π^(ó  ))^2 + 4 u N _ 21^( ) (m _ π^(ó  ))^2 - 6 s N _ 22^( ) (m _ π^(ó  ))^2 + 6 t N _ 22^( ) (m _ π^(ó  ))^2 + 6 u N _ 22^( ) (m _ π^(ó  ))^2 - 12 s N _ 23^( ) (m _ π^(ó  ))^2 + 12 t N _ 23^( ) (m _ π^(ó  ))^2 + 12 u N _ 23^( ) (m _ π^(ó  ))^2 + 12 N _ 21^( ) p _ 4^2 (m _ π^(ó  ))^2 - 6 N _ 22^( ) p _ 4^2 (m _ π^(ó  ))^2 - 12 N _ 23^( ) p _ 4^2 (m _ π^(ó  ))^2 - 8 N _ 21^( ) (m _ K^(ó  ))^4 + 8 s N _ 21^( ) (m _ K^(ó  ))^2 + 2 t N _ 21^( ) (m _ K^(ó  ))^2 + 2 u N _ 21^( ) (m _ K^(ó  ))^2 - 3 t N _ 22^( ) (m _ K^(ó  ))^2 - 3 u N _ 22^( ) (m _ K^(ó  ))^2 - 6 t N _ 23^( ) (m _ K^(ó  ))^2 - 6 u N _ 23^( ) (m _ K^(ó  ))^2 - 12 N _ 21^( ) p _ 4^2 (m _ K^(ó  ))^2 + 6 N _ 22^( ) p _ 4^2 (m _ K^(ó  ))^2 + 12 N _ 23^( ) p _ 4^2 (m _ K^(ó  ))^2 + 3 N _ 19^( ) (-2 (m _ π^(ó  ))^4 + 2 ((m _ K^(ó  ))^2 - 2 s + t + u) (m _ π^(ó  ))^2 + 2 s^2 - t^2 - u^2 - 2 s (m _ K^(ó  ))^2 + t (m _ K^(ó  ))^2 + u (m _ K^(ó  ))^2 + p _ 4^2 (2 (m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 - 2 s + t + u)) + 3 N _ 20^( ) (6 (m _ π^(ó  ))^4 - 2 (2 (m _ K^(ó  ))^2 + s + t + u) (m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^4 + 2 s (m _ K^(ó  ))^2 + 2 t (m _ K^(ó  ))^2 + 2 u (m _ K^(ó  ))^2 + s t + s u - 2 t u + 2 p _ 4^2 (2 (m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 - 2 s + t + u))))

Limit[% /. softPionLimit1 //. massshellrules /. _RenormalizationState -> Sequence[], ParticleMass[Pion] -> 0] // Simplify

(i c _ 2^( ) (6 N _ 20^( ) p _ 4^2 (p _ 4^2 - (m _ K^(ó  ))^2) - (m _ K^(ó  ))^2 (2 N _ 21^( ) (3 (m _ K^(ó  ))^2 + 5 p _ 4^2) - 3 (N _ 22^( ) + 2 N _ 23^( )) (p _ 4^2 - (m _ K^(ó  ))^2))))/(6 (f _ ϕ^(ó  ))^5)

I * Collect[Cancel[(test/I /. subpar /. udrules // MomentumExpand // ExpandScalarProduct) /. massshellrules /. Pair[Momentum[p1], Momentum[p1]] -> ParticleMass[Kaon, RenormalizationState[1]]^2 /. Pair[Momentum[p1], Momentum[p1]]^2 -> ParticleMass[Kaon, RenormalizationState[1]]^4 /. cancelU /. RenormalizationState[1] -> RenormalizationState[0]], {_DecayConstant, _ParticleMass}] // Simplify

1/(6 (f _ ϕ^(ó  ))^5) (i c _ 2^( ) (-6 N _ 21^( ) (m _ K^(ó  ))^4 - 3 N _ 22^( ) (m _ K^(ó  ))^4 - 6 N _ 23^( ) (m _ K^(ó  ))^4 + 24 N _ 21^( ) (m _ π^(ó  ))^2 (m _ K^(ó  ))^2 + 12 N _ 22^( ) (m _ π^(ó  ))^2 (m _ K^(ó  ))^2 + 24 N _ 23^( ) (m _ π^(ó  ))^2 (m _ K^(ó  ))^2 + 6 s N _ 21^( ) (m _ K^(ó  ))^2 + 3 s N _ 22^( ) (m _ K^(ó  ))^2 + 6 s N _ 23^( ) (m _ K^(ó  ))^2 + 2 N _ 21^( ) p _ 2^2 (m _ K^(ó  ))^2 - 3 N _ 22^( ) p _ 2^2 (m _ K^(ó  ))^2 - 6 N _ 23^( ) p _ 2^2 (m _ K^(ó  ))^2 - 12 N _ 21^( ) p _ 4^2 (m _ K^(ó  ))^2 + 6 N _ 22^( ) p _ 4^2 (m _ K^(ó  ))^2 + 12 N _ 23^( ) p _ 4^2 (m _ K^(ó  ))^2 - 24 s N _ 21^( ) (m _ π^(ó  ))^2 - 12 s N _ 22^( ) (m _ π^(ó  ))^2 - 24 s N _ 23^( ) (m _ π^(ó  ))^2 + 4 N _ 21^( ) p _ 2^2 (m _ π^(ó  ))^2 + 6 N _ 22^( ) p _ 2^2 (m _ π^(ó  ))^2 + 12 N _ 23^( ) p _ 2^2 (m _ π^(ó  ))^2 + 12 N _ 21^( ) p _ 4^2 (m _ π^(ó  ))^2 - 6 N _ 22^( ) p _ 4^2 (m _ π^(ó  ))^2 - 12 N _ 23^( ) p _ 4^2 (m _ π^(ó  ))^2 + 3 N _ 19^( ) (-2 (m _ π^(ó  ))^4 + 2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 2 s (m _ π^(ó  ))^2 + 4 t (m _ π^(ó  ))^2 + s^2 - 2 t^2 - p _ 2^4 - s (m _ K^(ó  ))^2 + 2 t (m _ K^(ó  ))^2 - 2 s t + p _ 2^2 (-2 (m _ π^(ó  ))^2 - (m _ K^(ó  ))^2 + 2 s + 2 t + p _ 4^2) - p _ 4^2 (-4 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 + 3 s)) - 3 N _ 20^( ) (-2 (m _ π^(ó  ))^4 + 2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 2 s (m _ π^(ó  ))^2 + 4 t (m _ π^(ó  ))^2 + s^2 - 2 t^2 - s (m _ K^(ó  ))^2 + 2 t (m _ K^(ó  ))^2 - 2 s t + 2 p _ 4^2 (-4 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 + 3 s) - p _ 2^2 (-2 (m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2 + s - 2 t + 2 p _ 4^2))))

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

Converted by Mathematica  (July 10, 2003)