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

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 na\"{i}ve expectations when we introduce a large non-minimal coupling for any scalar field.