Soft synchronous gauge in gravity: Principal value prescription

The synchronous gauge in gravity ([Formula: see text]) is ill-defined due to the singularity at [Formula: see text] in the graviton propagator. Previously, we studied “softening” this gauge by considering instead the gauge [Formula: see text], [Formula: see text], in the limit [Formula: see text]. We now explore the possibility of using a principal value prescription (not in the standard Cauchy sense), which amounts, roughly speaking, to replacing the singularities [Formula: see text], which then behave like distributions. An advantage of this prescription is the individual finiteness of the diagrammatic contributions at [Formula: see text]. We show that such a propagator follows upon adding to the action a gauge-violating term of a general form, which reduces to [Formula: see text] with a constant operator [Formula: see text] depending on [Formula: see text] and a metric functional [Formula: see text]. The contribution of the ghost fields to the effective action is analyzed. 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 [Formula: see text].