Put

c = a + b

V_{μ} = A_{μ} + \partial_{μ} π

and integrating cross terms by parts gives

L ≃ a A_{μ} □ A^{μ} + m^{2} A_{μ} A^{μ} - c (□ π)^{2} + m^{2} \partial_{μ} π \partial^{μ} π .

(a □ + m^{2}) A_{μ} = 0, □ (c □ + m^{2}) π = 0,

together with

\partial_{μ} A^{μ} = 0

ii

Solution

The scalar kinetic operator is proportional in momentum space to

k^{2} (c k^{2} - m^{2})

\frac{i}{k ^{2} ( c k ^{2} - m ^{2} )} = - \frac{i}{m ^{2}} (\frac{1}{k ^{2}} - \frac{c}{c k ^{2} - m ^{2}}),

up to the overall convention-dependent factor allowed in the question. Thus

c = a + b

. Under the physical relation

b = - a

, one has

c = 0

, so the extra massive pole and its ghost residue disappear.

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