•The three-vertex Feynman rule

The leading order lagrangian in compact notation:

ulag = Lagrangian[HBChPT2[2]] // DotSimplify

i Overscript[Overscript[N^( ), _], →] . γ^μ . ÷s _ μ(Overscript[N^( ), →]) + 1/2 g _ A^( ) (-i Overscript[Overscript[N^( ), _], →] . γ^μ . (öÆ '6 ∂ _ μ(öÆ^†)) . Overscript[N^( ), →] + i Overscript[Overscript[N^( ), _], →] . γ^μ . (öÆ^† '6 ∂ _ μ(öÆ)) . Overscript[N^( ), →] - Overscript[Overscript[N^( ), _], →] . γ^μ . (öÆ '6 L^( ) _ μ '6 öÆ^†) . Overscript[N^( ), →] + Overscript[Overscript[N^( ), _], →] . γ^μ . (öÆ^† '6 R^( ) _ μ '6 öÆ) . Overscript[N^( ), →]) - Overscript[Overscript[N^( ), _], →] . Overscript[N^( ), →] m _ N^(ó  )

Expansion to full notation (the output is suppressed):

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

Actually it pays off rather well to expand and simplify already here (haven't checked if the expressions are actually correct):

la = (lag // NMExpand // DotSimplify // CommutatorReduce[#, FullSimplify -> True] & // Simplify) /. DropFactor[__] -> 1

1/(8 (f _ π^(ó  ))^2) (4 (2 i Overscript[Overscript[N^( ), _], →] . γ^μ . ∂ _ μ(Overscript[N^( ), →]) + g _ A^( ) Overscript[Overscript[N^( ), _], →] . γ^μ . a^( ) _ μ . Overscript[N^( ), →] + Overscript[Overscript[N^( ), _], →] . γ^μ . v^( ) _ μ . Overscript[N^( ), →] + 2 Overscript[Overscript[N^( ), _], →] . γ^μ . v^(s)^( ) _ μ . Overscript[N^( ), →] - 2 Overscript[Overscript[N^( ), _], →] . Overscript[N^( ), →] m _ N^(ó  )) (f _ π^(ó  ))^2 - 4 g _ A^( ) Overscript[Overscript[N^( ), _], →] . γ^μ . (∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->]) . Overscript[N^( ), →] f _ π^(ó  ) + i (Overscript[Overscript[N^( ), _], →] . γ^μ . (Overscript[π^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->]) . Overscript[N^( ), →] - Overscript[Overscript[N^( ), _], →] . γ^μ . (∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->]) . Overscript[N^( ), →]))

Isoindices are supplied.  NM is kept, but in some cases redundant:

$IsoIndicesCounter = 0 ;

lagi = la // IsoIndicesSupply // Simplify

1/(8 (f _ π^(ó  ))^2) (4 (2 i Overscript[Overscript[N^( ), _], →] . γ^μ . ∂ _ μ(Overscript[N^( ), →]) + g _ A^( ) Overscript[Overscript[N^( ), _], →] . γ^μ . a^( ) _ μ . Overscript[N^( ), →] + Overscript[Overscript[N^( ), _], →] . γ^μ . v^( ) _ μ . Overscript[N^( ), →] + 2 Overscript[Overscript[N^( ), _], →] . γ^μ . v^(s)^( ) _ μ . Overscript[N^( ), →] - 2 Overscript[Overscript[N^( ), _], →] . Overscript[N^( ), →] m _ N^(ó  )) (f _ π^(ó  ))^2 - 4 g _ A^( ) Overscript[Overscript[N^( ), _], →] . γ^μ . (∂ _ μ π^( ) _ ó ^i _ 1 '6 σ^i _ 1) . Overscript[N^( ), →] f _ π^(ó  ) + i (Overscript[Overscript[N^( ), _], →] . γ^μ . (π^( )^i _ 2 '6 σ^i _ 2 '6 ∂ _ μ π^( ) _ ó ^i _ 3 '6 σ^i _ 3) . Overscript[N^( ), →] - Overscript[Overscript[N^( ), _], →] . γ^μ . (∂ _ μ π^( ) _ ó ^i _ 4 '6 σ^i _ 4 '6 π^( )^i _ 5 '6 σ^i _ 5) . Overscript[N^( ), →]))

$UIndicesCounter = 0 ;

lagu = lagi // UIndicesSupply

1/(8 (f _ π^(ó  ))^2) (4 (2 i (Overscript[N^( ), _] _ n _ 9 '6 γ^μ '6 ∂ _ μ N^( ) _ n _ 9^ó ) + g _ A^( ) (Overscript[N^( ), _] _ n _ 11 '6 γ^μ '6 a^( ) _ μ '6 N^( ) _ n _ 11) + Overscript[N^( ), _] _ n13 '6 γ^μ '6 v^( ) _ μ '6 N^( ) _ n13 + 2 (Overscript[N^( ), _] _ n15 '6 γ^μ '6 v^(s)^( ) _ μ '6 N^( ) _ n15) - 2 (Overscript[N^( ), _] _ n16 '6 N^( ) _ n16) m _ N^(ó  )) (f _ π^(ó  ))^2 - 4 g _ A^( ) (Overscript[N^( ), _] _ n _ 1 '6 γ^μ '6 ∂ _ μ π^( ) _ ó ^i _ 1 '6 σ _ (n _ 1 n _ 2)^i _ 1 '6 N^( ) _ n _ 2) f _ π^(ó  ) + i (Overscript[N^( ), _] _ n _ 3 '6 γ^μ '6 π^( )^i _ 2 '6 σ _ (n _ 3 n _ 4)^i _ 2 '6 ∂ _ μ π^( ) _ ó ^i _ 3 '6 σ _ (n _ 4 n _ 5)^i _ 3 '6 N^( ) _ n _ 5 - Overscript[N^( ), _] _ n _ 6 '6 γ^μ '6 ∂ _ μ π^( ) _ ó ^i _ 4 '6 σ _ (n _ 6 n _ 7)^i _ 4 '6 π^( )^i _ 5 '6 σ _ (n _ 7 n _ 8)^i _ 5 '6 N^( ) _ n _ 8))

After using the commutation relations, some NMs are replaced with Times. Replacing UIndex with SUNIndex is necessary because FeynCalc doesn't (yet) support the index type UIndex.

lagc = lagu /. UIndex -> SUNIndex // CommutatorReduce // Simplify

1/(8 (f _ π^(ó  ))^2) (4 (g _ A^( ) (Overscript[N^( ), _]^n _ 11 '6 γ^μ) a^( ) _ μ N^( )^n _ 11 - 2 m _ N^(ó  ) Overscript[N^( ), _]^n16 N^( )^n16 + (Overscript[N^( ), _]^n13 '6 γ^μ) N^( )^n13 v^( ) _ μ + 2 (Overscript[N^( ), _]^n15 '6 γ^μ) N^( )^n15 v^(s)^( ) _ μ + 2 i (Overscript[N^( ), _]^n _ 9 '6 γ^μ) ∂ _ μ N^( ) _ ó ^n _ 9) (f _ π^(ó  ))^2 - 4 g _ A^( ) (Overscript[N^( ), _]^n _ 1 '6 γ^μ) N^( )^n _ 2 ∂ _ μ π^( ) _ ó ^i _ 1 σ _ (n _ 1 n _ 2)^i _ 1 f _ π^(ó  ) + i ((Overscript[N^( ), _]^n _ 3 '6 γ^μ) N^( )^n _ 5 π^( )^i _ 2 ∂ _ μ π^( ) _ ó ^i _ 3 σ _ (n _ 3 n _ 4)^i _ 2 σ _ (n _ 4 n _ 5)^i _ 3 - (Overscript[N^( ), _]^n _ 6 '6 γ^μ) N^( )^n _ 8 π^( )^i _ 5 ∂ _ μ π^( ) _ ó ^i _ 4 σ _ (n _ 6 n _ 7)^i _ 4 σ _ (n _ 7 n _ 8)^i _ 5))

fields = {QuantumField[DiracBar[Particle[Nucleon, RenormalizationState[0]]], SUNIndex[J1]][p1], QuantumField[Particle[Nucleon, RenormalizationState[0]], SUNIndex[J2]][p2], QuantumField[Particle[Pion, RenormalizationState[0]], SUNIndex[I3]][p3]}

{Overscript[N^( ), _]^J _ 1, N^( )^J _ 2, π^( )^I _ 3}

melsimplified = Simplify[FeynRule[lagc, fields]]

-(g _ A^( ) γ · p _ 3 σ _ (J _ 1 J _ 2)^I _ 3)/(2 f _ π^(ó  ))

Converted by Mathematica  (July 10, 2003)