S^{(2)} = \frac{1}{2} \int \frac{d ^{4} k}{( 2 π ) ^{4}} A_{μ}^{a} (- k) [k^{2} δ^{μν} + (\frac{1}{ξ} - 1) k^{μ} k^{ν}] A_{ν}^{a} (k) .

P_{T}^{μν} = δ^{μν} - \frac{k ^{μ} k ^{ν}}{k ^{2}}, P_{L}^{μν} = \frac{k ^{μ} k ^{ν}}{k ^{2}},

the operator is

k^{2} (P_{T} + ξ^{- 1} P_{L})

. Its inverse, the gauge-boson propagator, is

D_{μν}^{ab} (k) = \frac{δ ^{ab}}{k ^{2}} (δ_{μν} + (ξ - 1) \frac{k _{μ} k _{ν}}{k ^{2}}) .

Therefore $X = 1$ and $Y = ξ - 1$ .

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