Gauge coupling jump and small instantons from a large non-minimal coupling

TL;DR

This work shows that a large non-minimal coupling of a scalar to gravity can produce threshold effects that shift gauge couplings around an intermediate scale in both the Metric and Palatini formulations. These gauge-coupling jumps can enhance small instanton contributions and even induce a period of stronger QCD in the early universe, with significant implications for the QCD axion (quality problem, abundance, and isocurvature constraints) and for Higgs-inflation scenarios. The authors develop a Wilsonian, high-energy-decoupling argument and a dilute gas estimate to quantify the small-instanton effects, including the impact of quarks, unitarity bounds, and CP-violating versus CP-conserving threshold structures. The results suggest axion phenomenology can depart substantially from naïve expectations when a large non-minimal coupling is present, motivating further study of UV completions and broader scalar sectors.

Abstract

If a scalar field couples to the Ricci scalar with a large non-minimal coupling, the Standard Model coupling parameters can differ above and below an intermediate field range of the scalar due to the non-renormalizability. In this paper, we study, for the first time, the threshold effects on a gauge coupling in both Metric and Palatini formulations of gravity. We find that the gauge coupling naturally jumps around this intermediate scale since counter terms for the renormalization behave so. If the gauge coupling becomes strong with a large scalar field value due to this effect, there can be an enhanced small instanton contribution, the dilute gas approximation of which is justified because the gauge sector decouples when the scalar wave mode is very short. Using these findings, we discuss the QCD axion quality problem, the heavy QCD axion, the QCD axion abundance, and the suppression of isocurvature perturbations. We show that axion physics may differ substantially from naïve expectations when we introduce a large non-minimal coupling for any scalar field.

Figures (4)

• Figure 1: Two-loop diagram for the gluon wave function. With the large non-minimal coupling, this diagram contributes a non-trivial $\varphi$-dependent divergence.
• Figure 2: $(\partial_\varphi F)^2$, relevant to the correction to the gauge coupling$^{-2}$ by varying $h/M_{\rm pl}$ (top panel) and the canonically normalized $\varphi$ (bottom panel). We take $\xi=10^5$ in both figures. The black and red lines denote the case of metric and Palatini formulation respectively. The vertical lines in the top panel denote the $h/M_{\rm pl}=1/\xi, 1/\sqrt{\xi}$ from left to right.
• Figure 3: 2D slice of the configuration of the Higgs field with $\Lambda L/(2\pi)=256$ on $1024^3$ lattices. We generate the modes in the range $\Lambda/\epsilon<k<\Lambda$. The black dots denote the region with $\phi< \Lambda/(30\times 2\pi)$, which is highly subdominant. If $\Lambda/(30\times 2\pi) \sim M_{\rm pl}/\sqrt{\xi}$, in the yellow region the gauge coupling is constant and is $g$, while black dots have different gauge coupling. The perturbative unitarity is violated in black dots, while it is well preserved in the yellow region. Our instanton estimation is valid in the yellow regimes, while the contribution from black dots is neglected.
• Figure 4: Histogram of the $\varphi$ field values in the simulation (see also Fig. \ref{['fig:3']}). We use $\Lambda=256 (16) \times 2\pi/L$ for the blue (pink) one.