Gravitationally Induced UV Completion of an $O(N)$ Scalar Theory

We investigate the ultraviolet completion of an $O(N)$ scalar field theory non-minimally coupled to gravity using the Wilsonian functional renormalization group in the proper-time formulation. Focusing on the spontaneously broken phase, we study the RG flow of the scalar potential and the non-minimal curvature coupling expanded around a running minimum. We identify two distinct classes of fixed-point solutions, one of which is ultraviolet attractive and characterized by a vanishing quartic coupling together with finite, interacting gravitational couplings. For a finite region of infrared initial conditions, the RG trajectories remain regular at all scales and approach this fixed point. This mechanism renders the theory asymptotically safe and leads to a flat scalar potential in the ultraviolet. We show that this mechanism is robust under changes of cutoff scheme and truncation, allowing the ultraviolet completion requirement to constrain the infrared values of the scalar couplings and the mass scale in the broken phase.

💡 Research Summary

The paper investigates whether gravity alone can render an O(N) scalar field theory ultraviolet (UV) complete when the scalars are non‑minimally coupled to curvature through a term F(ϕ)R. Using the Wilsonian functional renormalization group (FRG) in the proper‑time (PT) formulation, the authors derive flow equations for the dimensionless scalar potential u(x) and the non‑minimal coupling function f(x), together with the wave‑function renormalizations Z_L and Z_T for the longitudinal and transverse scalar modes. A spectrally adjusted type‑C cutoff is employed, which is known to give accurate critical exponents both for the Wilson‑Fisher fixed point and for pure gravity.

In the spontaneously broken phase the scalar potential is expanded around a running vacuum expectation value x₀(t) as
u_t(x)=u₀(t)+λ(t)(x−x₀(t))²,
where λ(t) is the effective quartic coupling. The flow of λ separates into a linear gravitational contribution β_g=−A(t)λ and a matter contribution β_m=η_L λ+b(t)λ². The coefficient A(t) depends on the dimensionless Newton coupling g(t)=1/𝑚̃_P(t) and on the first derivative f₁(t) of the non‑minimal function f(x)=f₀+f₁x. Crucially, for f₁>0 the sign of A can flip from negative (infrared) to positive (ultraviolet), providing an anti‑screening effect that drives λ towards smaller values at high scales. The matter coefficients b(t) and D(t) (entering η_L≈D λ²) are positive for the parameter range considered, so without the gravitational term λ would inevitably hit a Landau‑pole.

The authors identify two classes of fixed points. The first class contains the Gaussian fixed point (g=0, λ=0) and two interacting “Reuter‑type” fixed points with non‑zero Newton coupling but still λ=0. The second class includes a special fixed point with f₁=1/3, where x₀* and g* become non‑trivial functions of N, the wave‑function ratio w=Z_T/Z_L (denoted a), and the cutoff shape parameter m. At all fixed points η_L=η_T=0, leaving w as a free parameter. The crucial UV‑attractive fixed point is the one with vanishing quartic coupling, finite gravitational couplings, and a non‑zero f₁ that satisfies the condition
f₁(t) ≥ 1/