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arXiv · 2609.36880

Higher-curvature gravity in Weyl geometry: No-ghost and no-tachyon conditions

Tomoya Tachinami
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Abstract

Weyl geometry, in which the length scale can be chosen independently at each point of spacetime, provides a natural framework for extending general relativity through Weyl invariance, that is, local scale symmetry. We construct a gravitational theory whose Lagrangian is an arbitrary function of the scalar curvature in Weyl geometry and establish the conditions under which the theory is free of ghosts and tachyons around maximally symmetric vacua. For an arbitrary function, Weyl invariance requires a compensating scalar field in addition to the metric and the Weyl vector. By linearizing the action with an auxiliary field, we show that the Weyl vector couples to the scalar sector only through a single combination of the scalar fields. Consequently, the Weyl vector, shifted by the logarithmic gradient of this combination, becomes a Weyl-invariant massive vector field without gauge fixing. In the Einstein frame, given that the graviton is ghost-free, the no-tachyon condition for the Weyl vector, together with the no-ghost condition for the scalaron, the scalar mode associated with the higher-curvature terms, reduces to the positivity of the kinetic coefficient of the compensating scalar. When this coefficient vanishes, the scalaron does not propagate, leaving only the graviton and the massive vector. As an example, we show that for a Lagrangian containing constant, linear, and quadratic curvature terms, both a massive vector and a massive scalaron are necessarily present throughout the stability region, whereas without the linear term the scalaron becomes massless.

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Tomoya Tachinami. 2026-09-29. Higher-curvature gravity in Weyl geometry: No-ghost and no-tachyon conditions. https://arxiv.org/abs/2609.36880

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