Parity-violating scalar-tensor theory revisited: complete basis up to $d=4$ and Hamiltonian analysis

Abstract

We construct complete bases of parity-violating generally covariant scalar-tensor (GST) and spatially covariant gravity (SCG) monomials up to derivative grade $d=4$. Including all derivative partitions, each description contains one $d=3$ and thirteen $d=4$ linearly independent monomials. At the level of Lagrangians, we find six integration-by-parts relations that reduce polynomial-type Lagrangians to combinations of one $d=3$ and seven $d=4$ monomials with general coefficient functions. We derive the correspondence between the two bases, retaining general coefficients and total derivatives, and establish their independence under integrations by parts. Among the seven $d=4$ SCG monomials, five contain neither the lapse velocity nor the time derivative of the extrinsic curvature. For the two remaining monomials, we perform a Hamiltonian constraint analysis with a standard Einstein-Hilbert kinetic term and a lapse-dependent potential. On the regular branches considered, the lapse-velocity term propagates four degrees of freedom (DOFs), while the extrinsic-curvature-velocity term propagates six DOFs, whether present alone or together with the lapse-velocity term. These results provide a complete basis for constructing parity-violating scalar-tensor theories with three DOFs up to $d=4$ within this setting, with the parity-violating Lagrangian spanned by six GST polynomials: one $d=3$ and five $d=4$ polynomials.