No-scale Brans-Dicke Gravity -- ultralight scalar boson & heavy inflaton

It is very much intriguing if the Planck scale $M_{\rm{Pl}}$ is not a fundamental parameter. The Brans-Dicke gravity is nothing but the theory where the Planck scale $M_{\rm{Pl}}$ is indeed an illusional parameter. The theory predicts a massless scalar boson whose exchanges between matters induce unwanted long range forces. We solve this problem imposing there is no dimensionful parameter in the theory, even at the quantum level. We further extend the theory by including a $R^2$ term and a non-minimal coupling of the Standard Model Higgs to gravity, as their coefficients are dimensionless. This extension provides a heavy inflaton field that is consistent with all cosmological observations, with a potential very similar to that of the Starobinsky model. The inflaton necessarily decays into the massless scalar bosons, resulting in a non-negligible amount of dark radiation in the present universe. We demonstrate that the inflation model yields a sufficiently high reheating temperature for successful leptogenesis, and we also discuss a possible candidate for dark matter.

💡 Research Summary

The paper proposes a “no‑scale” version of Brans‑Dicke (BD) gravity in which the Planck mass is not a fundamental constant but is generated dynamically by the vacuum expectation value of a scalar field ϕ. In the conventional BD theory the scalar ϕ couples non‑minimally to curvature (ξ ϕ² R) and inevitably yields a massless scalar degree of freedom χ (the logarithm of ϕ) that mediates a long‑range fifth force, in conflict with precision tests of gravity. The authors eliminate this problem by enforcing strict scale‑invariance: every dimensionful parameter in the Standard Model (SM) – the Higgs mass term, right‑handed neutrino Majorana masses, etc. – is replaced by a coupling to ϕ². Consequently, after a Weyl transformation to the Einstein frame the χ field does not appear in any tree‑level interaction with SM fields.

Quantum corrections could re‑introduce χ‑matter couplings through running couplings. To prevent this, the authors adopt a scale‑invariant renormalisation prescription: the UV cutoff (or renormalisation scale) is taken to be proportional to √(ξ) ϕ, i.e. Λ = c √(ξ) ϕ with c < 1. All β‑functions are then evaluated at a fixed reference scale c Mₚₗ, which guarantees that the shift symmetry of χ survives at the quantum level. Thus χ remains a completely decoupled, ultralight (effectively massless) scalar that does not generate a fifth force.

The theory is then extended by adding a dimension‑less R² term and, later, a non‑minimal Higgs‑curvature coupling |H|²R. The R² term is linearised with an auxiliary field X, leading to a Lagrangian containing two scalar degrees of freedom: the massless χ and a massive combination Θ that depends on both ϕ and X. After the Weyl rescaling, the kinetic term for Θ becomes singular at Θ = 1, which flattens the scalar potential near this point. Defining a canonically normalised inflaton σ through Θ = cosh²(σ/√6 Mₚₗ), the potential takes the form

V(σ) = (9 λ/4 ξ² α) Mₚₗ⁴