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arXiv · 2609.23704

Bi-scalar-tensor theory: a spatially covariant gravity approach

Jia-Jun Chen·Xian Gao·Tsutomu Kobayashi
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Abstract

We propose a novel method for constructing ghost-free multi-field higher-derivative scalar-tensor theories. The idea is to start from spatially covariant gravity and couple it to an additional scalar field. The resulting spatially covariant scalar field theory can be understood as a bi-scalar-tensor theory in the unitary gauge of one scalar. Within a broad class in which metric velocities enter through the extrinsic curvature and the additional scalar enters through temporal derivatives up to second order, we perform a Hamiltonian constraint analysis. We show that the theory generally propagates five physical degrees of freedom (DOFs), i.e., two tensor modes and three scalar modes. One of the scalar modes is the unwanted mode associated with the additional scalar. We then derive a degeneracy condition and an additional consistency condition. When both conditions hold, the unwanted mode is eliminated and the theory propagates four degrees of freedom. The resulting theories fall into two cases, depending on whether the preservation of the tertiary constraint closes without a new constraint or generates a quaternary constraint. We also study the two-field disformal transformation, derive its invertibility condition, and express it in the unitary gauge. Our construction provides a systematic approach to constructing bi- and multi-scalar-tensor theories with higher derivatives.

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Jia-Jun Chen, Xian Gao, Tsutomu Kobayashi. 2026-09-20. Bi-scalar-tensor theory: a spatially covariant gravity approach. https://arxiv.org/abs/2609.23704

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