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

TL;DR

This paper demonstrates that an $O(N)$ scalar theory non-minimally coupled to gravity can achieve a gravity-induced UV completion via an interacting UV fixed point within a Wilsonian proper-time FRG framework. A consistent truncation around the running minimum reveals a fixed-point structure with $\lambda_* = 0$ and a line of fixed points controlled by gravity, yielding a UV-attractive critical surface and asymptotically safe trajectories. The study identifies a cutoff-insensitive critical line in the IR that separates UV-complete from singular flows and provides IR predictions for the scalar mass through a naturalness-inspired relation between IR initial data and the UV fixed point. The results imply that gravity alone can regulate scalar self-interactions, constrain IR parameters, and potentially pave the way for gravity-guided UV completions in more realistic matter sectors, with future work including gauge and Yukawa couplings to test robustness.

Abstract

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.

Figures (7)

• Figure 1: Fixed-point values for the family with $f_{1\ast}=1/3$, shown for $m=d/2+1$ and $\gamma=1$ as functions of $w_\ast=a$ for representative values of $N$. (a) $x_{0\ast}$ versus $w_\ast$. (b) $g_\ast$ versus $w_\ast$.
• Figure 4: Nontrivial critical exponents $\theta_5$ and $\theta_6$ associated with perturbations of the fixed point (\ref{['FPbello']}) (plus-sign branch), shown as functions of $w_\ast$ for representative values of $N$. (a) $\theta_5(w_\ast)$. (b) $\theta_6(w_\ast)$. While $\theta_5$ remains positive, $\theta_6$ changes sign at $w_\ast=\bar{w}_\ast(N)$.
• Figure 5: Projection of the IR coupling plane $(f_1(0),\lambda(0))$ at fixed $(x_0(0),g(0),w(0))$, illustrating the separation between UV-complete and singular trajectories. (a) Example for $g(0)=10^{-34}$, $x_0(0)=0.1$, $w(0)=1$ and $N=4$: the region corresponding to UV-complete trajectories (projection of $M_1$) is bounded by the critical line $\lambda(0)=h(f_1(0))$ (black). Outside this region, trajectories develop a singularity (projection of $M_2$). (b) The critical line $\lambda(0)=h(f_1(0))$ for several values of $g(0)$ at fixed $x_0(0)$ and $w(0)$. In the inset the full range up to $f_1(0)\to\infty$. The black dashed line is the asymptotic limit eq.(\ref{['critlineasint']}).
• Figure 8: Projection of representative RG trajectories onto the $(f_1,\lambda)$ plane for $t\gtrsim t_{\mathrm{tr}}$ (Planck and UV regime), obtained from initial conditions inside the UV-complete region. The example shown uses $N=4$, $g(0)=10^{-56}$, $w(0)=1$ and $x_0(0)=1$. Arrows indicate the direction of decreasing RG time (from UV to IR). In the Planck regime gravitational fluctuations induce a turnaround: instead of growing monotonically, both $\lambda$ and $f_1$ reach a maximum and then decrease towards the UV fixed point \ref{['FPbello']}. For $w(0)=1$ one finds $f_{1\ast}\approx 0.539$, marked by the black dot.
• Figure 10: Representative UV-complete runnings for $N=4$. (a) Running of $g(t)$, showing the crossover from the Gaussian regime to the non-Gaussian scaling regime around $t_{\mathrm{tr}}$. (b) Running of $x_0(t)$, which approaches a quasi-constant value in the fixed-point regime of Eq. (\ref{['FPbello']}).