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 Incoming links: Euclidean adjoint Dirac Feynman rules

Take Euclidean gamma matrices with

{γ_{μ}, γ_{ν}} = 2 δ_{μν}

, and independent Grassmann fields

ψ, \overset{ˉ}{ψ}

. The minimally coupled Dirac action is

S_{ψ} = \int d^{4} x \overset{ˉ}{ψ}^{a} (γ_{μ} D_{μ} + m)_{a c} ψ^{c} .

Its mass term is the Dirac mass term, and the interaction is

- i g \overset{ˉ}{ψ}^{a} γ_{μ} A_{μ}^{b} (T_{ad}^{b})_{a c} ψ^{c}

. With Fourier convention

e^{i p \cdot x}

, the free kinetic operator is

i \neq p + m

S^{a c} (p) = δ^{a c} \frac{- i \neq p + m}{p ^{2} + m ^{2}}, V_{μ}^{b; a c} = + i g γ_{μ} (T_{ad}^{b})_{a c} = g f^{ba c} γ_{μ} .

The vertex sign follows by expanding

e^{- S_{ψ}}

, not

e^{i S_{ψ}}

. Momentum conservation supplies the usual Dirac delta function at each vertex. There is only a one-gluon fermion vertex, with no two-gluon seagull because the fermionic covariant derivative is first order. A closed fermion loop has the usual fermionic sign

- 1

. The identity

(i \neq p + m) (- i \neq p + m) = (p^{2} + m^{2}) I