•Two-vertex of fourth order in the chiral expansion

IsoVector[QuantumField[Particle[AxialVector[0], ___], ___], ___][_] := 0 ; IsoVector[QuantumField[Particle[Vector[0], ___], ___], ___][_] := 0 ; IsoVector[QuantumField[Particle[Vector[1], ___], ___], ___][_] := 0 ; QuantumField[Particle[Vector[1], ___], ___][_] := 0 ; IsoVector[QuantumField[Particle[Scalar[1], ___], ___], ___][_] := 0 ; QuantumField[Particle[Scalar[1], ___], ___][_] := 0 ;

LoadLagrangian[ChPT2[4]] ;

lag = Lagrangian[ChPT2[4]] + Lagrangian[ChPTEM2[4]] (* /. {CouplingConstant[ChPT2[4], _ ? (FreeQ[#, 1 | 2 | 3 | 4 | 5 | 6 | 8] &), ___] -> 0} *)

k _ 13^( ) (< Q '6 Q > '6 < Q '6 Q >) (f _ π^(ó  ))^4 + k _ 9^( ) (< Q '6 Q > '6 < Q '6 ÷„ '6 Q '6 ÷„^† >) (f _ π^(ó  ))^4 + k _ 1^( ) (< Q '6 ÷„ '6 Q '6 ÷„^† > '6 < Q '6 ÷„ '6 Q '6 ÷„^† >) (f _ π^(ó  ))^4 + k _ 14^( ) ((< (÷„ '6 χ^† + χ '6 ÷„^†) '6 Q > + < (÷„^† '6 χ + χ^† '6 ÷„) '6 Q >) '6 < Q >) (f _ π^(ó  ))^2 + k _ 11^( ) (< Q '6 Q > '6 (< ÷„ '6 χ^† > + < χ '6 ÷„^† >)) (f _ π^(ó  ))^2 + k _ 10^( ) (< Q '6 Q > '6 < ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† >) (f _ π^(ó  ))^2 + k _ 4^( ) (< ÷„^† '6 ÷s _ μ(÷„) '6 Q > '6 < ÷s _ μ(÷„) '6 ÷„^† '6 Q >) (f _ π^(ó  ))^2 + k _ 3^( ) (< ÷„^† '6 ÷s _ μ(÷„) '6 Q > '6 < ÷„^† '6 ÷s _ μ(÷„) '6 Q > + < ÷s _ μ(÷„) '6 ÷„^† '6 Q > '6 < ÷s _ μ(÷„) '6 ÷„^† '6 Q >) (f _ π^(ó  ))^2 + k _ 7^( ) (< Q '6 ÷„ '6 Q '6 ÷„^† > '6 (< χ '6 ÷„^† > + < χ^† '6 ÷„ >)) (f _ π^(ó  ))^2 + k _ 2^( ) (< Q '6 ÷„ '6 Q '6 ÷„^† > '6 < ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† >) (f _ π^(ó  ))^2 + k _ 12^( ) (< ÷s _ μ(Q _ L) '6 ÷s _ μ(Q _ L) > + < ÷s _ μ(Q _ R) '6 ÷s _ μ(Q _ R) >) (f _ π^(ó  ))^2 + k _ 8^( ) < (÷„^† '6 χ - χ^† '6 ÷„) '6 (÷„^† '6 Q '6 ÷„ '6 Q - Q '6 ÷„^† '6 Q '6 ÷„) > (f _ π^(ó  ))^2 + k _ 5^( ) (< (Q '6 ÷s _ μ(Q) - ÷s _ μ(Q) '6 Q) '6 ÷„^† '6 ÷s _ μ(÷„) > - < (Q '6 ÷s _ μ(Q) - ÷s _ μ(Q) '6 Q) '6 ÷s _ μ(÷„) '6 ÷„^† >) (f _ π^(ó  ))^2 + k _ 6^( ) < ÷s _ μ(Q) '6 ÷„ '6 ÷s _ μ(Q) '6 ÷„^† > (f _ π^(ó  ))^2 + L _ 7^( ) ((< χ '6 ÷„^† > - < ÷„ '6 χ^† >) '6 (< χ '6 ÷„^† > - < ÷„ '6 χ^† >)) + L _ 6^( ) ((< ÷„ '6 χ^† > + < χ '6 ÷„^† >) '6 (< ÷„ '6 χ^† > + < χ '6 ÷„^† >)) + L _ 4^( ) (< ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† > '6 (< ÷„ '6 χ^† > + < χ '6 ÷„^† >)) + L _ 1^( ) (< ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† > '6 < ÷s _ ν(÷„) '6 ÷s _ ν(÷„)^† >) + L _ 2^( ) (< ÷s _ μ(÷„) '6 ÷s _ ν(÷„)^† > '6 < ÷s _ μ(÷„) '6 ÷s _ ν(÷„)^† >) + H _ 2^( ) < χ^† '6 χ > + H _ 1^( ) (< L _ (μ ν) '6 L _ (μ ν) > + < R _ (μ ν) '6 R _ (μ ν) >) + k _ 15^( ) (γ^( ) _ (μ ν) '6 γ^( ) _ (μ ν)) < Q '6 Q > + L _ 5^( ) < ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† '6 (÷„ '6 χ^† + χ '6 ÷„^†) > + i L _ 9^( ) (< L _ (μ ν) '6 ÷s _ μ(÷„) '6 ÷s _ ν(÷„)^† > + < R _ (μ ν) '6 ÷s _ μ(÷„)^† '6 ÷s _ ν(÷„) >) + L _ 8^( ) (< ÷„ '6 χ^† '6 ÷„ '6 χ^† > + < χ '6 ÷„^† '6 χ '6 ÷„^† >) + L _ 3^( ) < ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† '6 ÷s _ ν(÷„) '6 ÷s _ ν(÷„)^† > + L _ 10^( ) < L _ (μ ν) '6 ÷„ '6 R _ (μ ν) '6 ÷„^† >

ll = (WriteString["stdout", "."] ; UNMSplit[#, x, DropOrder -> 2, DiagonalToU -> True]) & /@ lag ;

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

lll = ArgumentsSupply[ll /. {UMatrix[UChiralSpurionRight] -> UQuarkChargeMatrix[RenormalizationState[0], DiagonalToU -> True], UMatrix[UChiralSpurionLeft] -> UQuarkChargeMatrix[RenormalizationState[0], DiagonalToU -> True], UMatrix[UChiralSpurion] -> UQuarkChargeMatrix[RenormalizationState[0], DiagonalToU -> True]}, x, RenormalizationState[0], DiagonalToU -> True, ExpansionOrder -> 2, DropOrder -> 2] ;

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

llld = DiscardTerms[lll, Retain -> {Particle[Pion , RenormalizationState[0]] -> 2}, CommutatorReduce -> False, Method -> Expand] // Simplify ;

llle = ExpandU[llld, CommutatorReduce -> True] // Simplify

-1/(36 (f _ π^(ó  ))^2) (5 (e^( ))^4 (2 k _ 1^( ) + k _ 9^( )) (-(Overscript[öõ(3), ->] · Overscript[π^( ), ->])^2 + 3 Overscript[öõ(3), ->] × Overscript[π^( ), ->] · Overscript[öõ(3), ->] × Overscript[π^( ), ->] + Overscript[π^( ), ->] · Overscript[π^( ), ->]) (f _ π^(ó  ))^4 + 4 (e^( ))^2 (18 k _ 3^( ) (Overscript[öõ(3), ->] · ∂ _ μ(Overscript[π^( ), ->]))^2 + 9 k _ 4^( ) (Overscript[öõ(3), ->] · ∂ _ μ(Overscript[π^( ), ->]))^2 - 9 k _ 7^( ) (Overscript[öõ(3), ->] · Overscript[π^( ), ->])^2 (m _ π^(ó  ))^2 - 9 k _ 8^( ) (Overscript[öõ(3), ->] · Overscript[π^( ), ->])^2 (m _ π^(ó  ))^2 + 27 k _ 7^( ) Overscript[öõ(3), ->] × Overscript[π^( ), ->] · Overscript[öõ(3), ->] × Overscript[π^( ), ->] (m _ π^(ó  ))^2 + 27 k _ 8^( ) Overscript[öõ(3), ->] × Overscript[π^( ), ->] · Overscript[öõ(3), ->] × Overscript[π^( ), ->] (m _ π^(ó  ))^2 + 19 k _ 7^( ) Overscript[π^( ), ->] · Overscript[π^( ), ->] (m _ π^(ó  ))^2 + 9 k _ 8^( ) Overscript[π^( ), ->] · Overscript[π^( ), ->] (m _ π^(ó  ))^2 + 10 k _ 11^( ) Overscript[π^( ), ->] · Overscript[π^( ), ->] (m _ π^(ó  ))^2 + 2 k _ 14^( ) Overscript[π^( ), ->] · Overscript[π^( ), ->] (m _ π^(ó  ))^2 - 10 k _ 2^( ) ∂ _ μ(Overscript[π^( ), ->]) · ∂ _ μ(Overscript[π^( ), ->]) - 10 k _ 10^( ) ∂ _ μ(Overscript[π^( ), ->]) · ∂ _ μ(Overscript[π^( ), ->])) (f _ π^(ó  ))^2 + 144 (m _ π^(ó  ))^2 (2 (2 L _ 6^( ) + L _ 8^( )) Overscript[π^( ), ->] · Overscript[π^( ), ->] (m _ π^(ó  ))^2 - 2 L _ 4^( ) ∂ _ μ(Overscript[π^( ), ->]) · ∂ _ μ(Overscript[π^( ), ->]) - L _ 5^( ) ∂ _ μ(Overscript[π^( ), ->]) · ∂ _ μ(Overscript[π^( ), ->])))

IsoIndicesCounter = 0 ;

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

1/(36 (f _ π^(ó  ))^2) (5 (e^( ))^4 (2 k _ 1^( ) + k _ 9^( )) ((π^( )^3)^2 + π^( )^k1 (3 δ _ (3 k1)^(2) δ _ (3 k2)^(2) π^( )^k2 - 4 π^( )^k1)) (f _ π^(ó  ))^4 + 4 (e^( ))^2 (-10 k _ 11^( ) (π^( )^k1)^2 (m _ π^(ó  ))^2 - 2 k _ 14^( ) (π^( )^k1)^2 (m _ π^(ó  ))^2 + 9 k _ 8^( ) ((π^( )^3)^2 - 4 (π^( )^k1)^2 + 3 δ _ (3 k1)^(2) δ _ (3 k2)^(2) π^( )^k1 π^( )^k2) (m _ π^(ó  ))^2 + k _ 7^( ) (9 (π^( )^3)^2 + π^( )^k1 (27 δ _ (3 k1)^(2) δ _ (3 k2)^(2) π^( )^k2 - 46 π^( )^k1)) (m _ π^(ó  ))^2 - 18 k _ 3^( ) (∂ _ τ1 π^( ) _ ó ^3)^2 - 9 k _ 4^( ) (∂ _ τ1 π^( ) _ ó ^3)^2 + 10 k _ 2^( ) (∂ _ τ1 π^( ) _ ó ^k1)^2 + 10 k _ 10^( ) (∂ _ τ1 π^( ) _ ó ^k1)^2) (f _ π^(ó  ))^2 + 144 (m _ π^(ó  ))^2 (-4 L _ 6^( ) (m _ π^(ó  ))^2 (π^( )^k1)^2 - 2 L _ 8^( ) (m _ π^(ó  ))^2 (π^( )^k1)^2 + (2 L _ 4^( ) + L _ 5^( )) (∂ _ τ1 π^( ) _ ó ^k1)^2))

fields = {QuantumField[Particle[PseudoScalar[2], RenormalizationState[0]], SUNIndex[i1]][p1], QuantumField[Particle[PseudoScalar[2], RenormalizationState[0]], SUNIndex[i2]][p2]}

{π^( )^i _ 1, π^( )^i _ 2}

$ConstantIsoIndices = Union[$ConstantIsoIndices, {i1, i2}] ;

amp4 = (-I FeynRule[llll, fields]) /. i2 -> i1 // SUNReduce[#, FullReduce -> True] & // FullSimplify

1/(9 (f _ π^(ó  ))^2) (2 (5 (e^( ))^4 (2 k _ 1^( ) + k _ 9^( )) (δ _ (3 i _ 1)^(2) - 1) (f _ π^(ó  ))^4 + (e^( ))^2 (-2 (23 k _ 7^( ) + 18 k _ 8^( ) + 5 k _ 11^( ) + k _ 14^( )) (m _ π^(ó  ))^2 - 10 (k _ 2^( ) + k _ 10^( )) p _ 1 · p _ 2 + 9 (4 (k _ 7^( ) + k _ 8^( )) (m _ π^(ó  ))^2 + (2 k _ 3^( ) + k _ 4^( )) p _ 1 · p _ 2) δ _ (3 i _ 1)^(2)) (f _ π^(ó  ))^2 - 72 (2 L _ 6^( ) + L _ 8^( )) (m _ π^(ó  ))^4 - 36 (2 L _ 4^( ) + L _ 5^( )) p _ 1 · p _ 2 (m _ π^(ó  ))^2))

Converted by Mathematica  (July 10, 2003)