Parity-violating scalar-tensor theory and the Qi-Xiu

We investigate the parity-violating scalar-tensor theory and pay special attention to terms that are free of the Ostrogradsky ghost in the unitary gauge, i.e., when the scalar field possesses a timelike gradient. We exhaustively identify the generally covariant scalar-tensor theory (GST) monomials with parity violation up to $d=4$, where $d$ is the total number of derivatives in the unitary gauge. According to the correspondence between GST terms and the spatially covariant gravity (SCG) terms in the unitary gauge, we also exhaustively identify the SCG monomials with parity violation up to $d=4$, where the Lie derivatives of the extrinsic curvature and the lapse function are necessarily introduced. We find a total of 9 independent parity-violating SCG monomials, of which 7 contain no higher-order Lie derivatives and are thus automatically free of ghosts, while 2 involve Lie derivatives of the extrinsic curvature and the lapse function and are thus potentially dangerous. By explicitly deriving their generally covariant correspondence, we obtain 7 independent scalar-tensor terms dubbed the ``Qi-Xiu'' Lagrangians, which are the most general parity-violating scalar-tensor theories that are ghost-free in the unitary gauge up to $d=4$. Our results include the existing theories in the literature, such as the Chern-Simons term and the chiral scalar-tensor theories, as special cases.