•Four-vertex

The leading order lagrangian in raw form:

Lagrangian[ChPT2[2]]

1/4 (f _ π^(ó  ))^2 (< ÷„ '6 χ^† > + < χ '6 ÷„^† > + < ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† >)

The expanded lagrangian:

ll = ArgumentsSupply[Lagrangian[ChPT2[2]], x, RenormalizationState[0], ExpansionOrder -> 4, DropOrder -> 4] ;

Redundant terms are discarded:

lll = DiscardTerms[ll, Retain -> {Particle[Pion , RenormalizationState[0]] -> 4}, Method -> Expand] // Simplify

1/(48 (f _ π^(ó  ))^2) (< Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] > (m _ π^(ó  ))^2 - 2 < Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] > + < Overscript[π^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] > + 3 < Overscript[π^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] > - < ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] > + < ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] > - 2 < ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] >)

Matrices are traced:

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

(4 (Overscript[π^( ), ->] · ∂ _ μ(Overscript[π^( ), ->]))^2 + (Overscript[π^( ), ->] · Overscript[π^( ), ->])^2 (m _ π^(ó  ))^2 - 4 Overscript[π^( ), ->] · Overscript[π^( ), ->] ∂ _ μ(Overscript[π^( ), ->]) · ∂ _ μ(Overscript[π^( ), ->]))/(24 (f _ π^(ó  ))^2)

Indices are supplied:

$IsoIndicesCounter = 0 ;

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

(π^( )^k1 ((m _ π^(ó  ))^2 π^( )^k1 (π^( )^k2)^2 + 4 ∂ _ τ1 π^( ) _ ó ^k2 (π^( )^k2 ∂ _ τ1 π^( ) _ ó ^k1 - π^( )^k1 ∂ _ τ1 π^( ) _ ó ^k2)))/(24 (f _ π^(ó  ))^2)

Calculation of the Feynman rule:

fields = FieldsSet[QuantumField[Particle[Pion, RenormalizationState[0]]]]

{π^( )^I _ 1, π^( )^I _ 2, π^( )^I _ 3, π^( )^I _ 4}

melsimplified = Simplify[FeynRule[llll, fields]]

-1/(3 (f _ π^(ó  ))^2) (i ((-(m _ π^(ó  ))^2 + p _ 1 · p _ 2 + p _ 1 · p _ 3 - 2 p _ 1 · p _ 4 - 2 p _ 2 · p _ 3 + p _ 2 · p _ 4 + p _ 3 · p _ 4) δ _ (I _ 1 I _ 4) δ _ (I _ 2 I _ 3) + (-(m _ π^(ó  ))^2 + p _ 1 · p _ 2 - 2 p _ 1 · p _ 3 + p _ 1 · p _ 4 + p _ 2 · p _ 3 - 2 p _ 2 · p _ 4 + p _ 3 · p _ 4) δ _ (I _ 1 I _ 3) δ _ (I _ 2 I _ 4) - ((m _ π^(ó  ))^2 + 2 p _ 1 · p _ 2 - p _ 1 · p _ 3 - p _ 1 · p _ 4 - p _ 2 · p _ 3 - p _ 2 · p _ 4 + 2 p _ 3 · p _ 4) δ _ (I _ 1 I _ 2) δ _ (I _ 3 I _ 4)))

Converted by Mathematica  (July 10, 2003)