A = k^{2} + m^{2}

and

B = (p - k)^{2}

. Then

x A + (1 - x) B = [k - (1 - x) p]^{2} + Δ, Δ = x m^{2} + x (1 - x) p^{2} .

The identities for gamma matrices give

γ^{μ} (- i \neq k + m) γ_{μ} = i (d - 2) \neq k + d m .

After the shift

ℓ = k - (1 - x) p

, the term odd in

ℓ

integrates to zero. The rotationally symmetric loop integral is

\int \frac{d ^{d} ℓ}{( 2 π ) ^{d}} \frac{1}{( ℓ ^{2} + Δ ) ^{2}} = \frac{1}{( 4 π ) ^{d /2}} Γ (\frac{ϵ}{2}) Δ^{- ϵ /2} .

Consequently

Σ (\neq p) = - \frac{e ^{2}}{( 4 π ) ^{d /2}} Γ (\frac{ϵ}{2}) \int_{0}^{1} d x \frac{i ( 2 - ϵ ) ( 1 - x ) \neq p + ( 4 - ϵ ) m}{[ x m ^{2} + x ( 1 - x ) p ^{2} ] ^{ϵ /2}} .

Thus

C = i (2 - ϵ) (1 - x), F = 4 - ϵ, Δ = x m^{2} + x (1 - x) p^{2} .

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