•Two-vertex of fourth order in the chiral expansion

lag = Lagrangian[ChPTVirtualPhotons2[4]] /. QuantumField[Particle[(LeftComponent | RightComponent)[0], ___], ___, SUNIndex[0], ___] -> 0

(k _ 12^( ) (< Q '6 Q > '6 < Q '6 Q >) + k _ 13^( ) (< Q '6 ÷„ '6 Q '6 ÷„^† > '6 < Q '6 Q >) + k _ 14^( ) (< Q '6 ÷„ '6 Q '6 ÷„^† > '6 < Q '6 ÷„ '6 Q '6 ÷„^† >)) (f _ π^(ó  ))^4 + (k _ 5^( ) ((< ÷„^† '6 χ > + < χ^† '6 ÷„ >) '6 < Q '6 Q >) + k _ 6^( ) ((< ÷„^† '6 χ > + < χ^† '6 ÷„ >) '6 < Q '6 ÷„ '6 Q '6 ÷„^† >) + k _ 1^( ) (< ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) > '6 < Q '6 Q >) + k _ 2^( ) (< ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) > '6 < Q '6 ÷„ '6 Q '6 ÷„^† >) + k _ 7^( ) ((< (÷„ '6 χ^† + χ '6 ÷„^†) '6 Q > + < (÷„^† '6 χ + χ^† '6 ÷„) '6 Q >) '6 < Q >) + k _ 4^( ) (< ÷s _ μ(÷„)^† '6 Q '6 ÷„ > '6 < ÷s _ μ(÷„) '6 Q '6 ÷„^† >) + k _ 3^( ) (< ÷s _ μ(÷„)^† '6 Q '6 ÷„ > '6 < ÷s _ μ(÷„)^† '6 Q '6 ÷„ > + < ÷s _ μ(÷„) '6 Q '6 ÷„^† > '6 < ÷s _ μ(÷„) '6 Q '6 ÷„^† >) + k _ 11^( ) (< c _ μ^L Q _ L '6 c _ μ^L Q _ L > + < c _ μ^R Q _ R '6 c _ μ^R Q _ R >) + k _ 9^( ) (< ÷s _ μ(÷„)^† '6 (c _ μ^R Q _ R '6 Q - Q '6 c _ μ^R Q _ R) '6 ÷„ > + < ÷s _ μ(÷„) '6 (c _ μ^L Q _ L '6 Q - Q '6 c _ μ^L Q _ L) '6 ÷„^† >) + k _ 10^( ) < c _ μ^R Q _ R '6 ÷„ '6 c _ μ^L Q _ L '6 ÷„^† > + k _ 8^( ) (< (χ '6 ÷„^† - ÷„ '6 χ^†) '6 Q '6 ÷„ '6 Q '6 ÷„^† > + < (χ^† '6 ÷„ - ÷„^† '6 χ) '6 Q '6 ÷„^† '6 Q '6 ÷„ >)) (f _ π^(ó  ))^2 - 1/16 l _ 7^( ) ((< χ^† '6 ÷„ > - < ÷„^† '6 χ >) '6 (< χ^† '6 ÷„ > - < ÷„^† '6 χ >)) + 1/16 l _ 3^( ) ((< ÷„^† '6 χ > + < χ^† '6 ÷„ >) '6 (< ÷„^† '6 χ > + < χ^† '6 ÷„ >)) + 1/4 l _ 1^( ) (< ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) > '6 < ÷s _ ν(÷„)^† '6 ÷s _ ν(÷„) >) + 1/4 l _ 2^( ) (< ÷s _ μ(÷„)^† '6 ÷s _ ν(÷„) > '6 < ÷s _ μ(÷„)^† '6 ÷s _ ν(÷„) >) + 1/2 (h _ 1^( ) - h _ 3^( )) Re({χ}) + k _ 15^( ) (γ^( ) _ (μ ν) '6 γ^( ) _ (μ ν)) < Q '6 Q > + 1/4 (h _ 1^( ) + h _ 3^( )) < χ^† '6 χ > + 1/4 l _ 4^( ) (< ÷s _ μ(÷„)^† '6 ÷s _ μ(χ) > + < ÷s _ μ(χ)^† '6 ÷s _ μ(÷„) >) + l _ 5^( ) < G _ (μ ν)^R '6 G _ (μ ν)^L^† > - h _ 2^( ) (< G _ (μ ν)^L '6 G _ (μ ν)^L > + < G _ (μ ν)^R '6 G _ (μ ν)^R >) + 1/2 i l _ 6^( ) (< G _ (μ ν)^L '6 ÷s _ μ(÷„)^† '6 ÷s _ ν(÷„) > + < G _ (μ ν)^R '6 ÷s _ μ(÷„) '6 ÷s _ ν(÷„)^† >)

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

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

lll = ArgumentsSupply[ll, 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] /. $Substitutions // Simplify ;

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

-1/(36 (f _ π^(ó  ))^2) (5 (e^( ))^4 (k _ 13^( ) + 2 k _ 14^( )) (-(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 _ 6^( ) (Overscript[öõ(3), ->] · Overscript[π^( ), ->])^2 (m _ π^0^(ó  ))^2 + 27 k _ 6^( ) Overscript[öõ(3), ->] × Overscript[π^( ), ->] · Overscript[öõ(3), ->] × Overscript[π^( ), ->] (m _ π^0^(ó  ))^2 + 36 k _ 8^( ) Overscript[öõ(3), ->] × Overscript[π^( ), ->] · Overscript[öõ(3), ->] × Overscript[π^( ), ->] (m _ π^0^(ó  ))^2 + 10 k _ 5^( ) Overscript[π^( ), ->] · Overscript[π^( ), ->] (m _ π^0^(ó  ))^2 + 19 k _ 6^( ) Overscript[π^( ), ->] · Overscript[π^( ), ->] (m _ π^0^(ó  ))^2 + 2 k _ 7^( ) Overscript[π^( ), ->] · Overscript[π^( ), ->] (m _ π^0^(ó  ))^2 - 10 k _ 1^( ) ∂ _ μ(Overscript[π^( ), ->]) · ∂ _ μ(Overscript[π^( ), ->]) - 10 k _ 2^( ) ∂ _ μ(Overscript[π^( ), ->]) · ∂ _ μ(Overscript[π^( ), ->])) (f _ π^(ó  ))^2 + 36 l _ 3^( ) Overscript[π^( ), ->] · Overscript[π^( ), ->] (m _ π^0^(ó  ))^4)

IsoIndicesCounter = 0 ;

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

1/(36 (f _ π^(ó  ))^2) (5 (e^( ))^4 (k _ 13^( ) + 2 k _ 14^( )) ((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 _ 5^( ) (π^( )^k1)^2 (m _ π^0^(ó  ))^2 - 2 k _ 7^( ) (π^( )^k1)^2 (m _ π^0^(ó  ))^2 - 36 k _ 8^( ) (π^( )^k1)^2 (m _ π^0^(ó  ))^2 + 36 k _ 8^( ) δ _ (3 k1)^(2) δ _ (3 k2)^(2) π^( )^k1 π^( )^k2 (m _ π^0^(ó  ))^2 + k _ 6^( ) (9 (π^( )^3)^2 - 46 (π^( )^k1)^2 + 27 δ _ (3 k1)^(2) δ _ (3 k2)^(2) π^( )^k1 π^( )^k2) (m _ π^0^(ó  ))^2 - 18 k _ 3^( ) (∂ _ τ1 π^( ) _ ó ^3)^2 + 9 k _ 4^( ) (∂ _ τ1 π^( ) _ ó ^3)^2 + 10 k _ 1^( ) (∂ _ τ1 π^( ) _ ó ^k1)^2 + 10 k _ 2^( ) (∂ _ τ1 π^( ) _ ó ^k1)^2) (f _ π^(ó  ))^2 - 36 l _ 3^( ) (m _ π^0^(ó  ))^4 (π^( )^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] & // Simplify

1/(9 (f _ π^(ó  ))^2) (2 (5 (e^( ))^4 (k _ 13^( ) + 2 k _ 14^( )) (δ _ (3 i _ 1)^(2) - 1) (f _ π^(ó  ))^4 - (e^( ))^2 (10 k _ 5^( ) (m _ π^0^(ó  ))^2 + 46 k _ 6^( ) (m _ π^0^(ó  ))^2 + 2 k _ 7^( ) (m _ π^0^(ó  ))^2 + 36 k _ 8^( ) (m _ π^0^(ó  ))^2 - 36 k _ 6^( ) δ _ (3 i _ 1)^(2) (m _ π^0^(ó  ))^2 - 36 k _ 8^( ) δ _ (3 i _ 1)^(2) (m _ π^0^(ó  ))^2 + 10 k _ 1^( ) p _ 1 · p _ 2 + 10 k _ 2^( ) p _ 1 · p _ 2 - 18 k _ 3^( ) p _ 1 · p _ 2 δ _ (3 i _ 1)^(2) + 9 k _ 4^( ) p _ 1 · p _ 2 δ _ (3 i _ 1)^(2)) (f _ π^(ó  ))^2 - 9 l _ 3^( ) (m _ π^0^(ó  ))^4))

Converted by Mathematica  (July 10, 2003)