Soft synchronous gauge: principal value prescription

The synchronous gauge in gravity ($g_{0 \lambda} = - \delta_{0 \lambda}$) is ill-defined due to the singularity at $p_0 = 0$ in the graviton propagator. Previously we studied "softening" this gauge by considering instead the gauge $n^\lambda g_{\lambda \mu} = 0$, $n^\lambda = (1, - \varepsilon (\partial^j \partial_j )^{- 1} \partial^k )$ in the limit $\varepsilon \to 0$. We now explore the possibility of using a principal value prescription (not in the standard Cauchy sense), which amounts, roughly speaking, to replacing singularities $p_0^{-j} \Rightarrow [ (p_0 + i \varepsilon )^{-j} + (p_0 - i \varepsilon )^{-j} ] / 2$, which then behave like distributions. We show that such a propagator follows upon adding to the action a gauge-violating term of a general form, which reduces to $\sim \int f_\lambda \Lambda^{\lambda \mu} f_\mu \d^4 x$ with a constant operator $\Lambda^{\lambda \mu}$ depending on $\partial$ and a metric functional $f_\lambda$. The contribution of the ghost fields to the effective action is analysed. For the required intermediate regularization, the discrete structure of the theory at small distances is implied. It is shown that the ghost contribution can be disregarded in the limit $\varepsilon \to 0$.