Quantum gravity contributions to the gauge and Yukawa couplings in proper time flow

Abstract

We derive quantum gravity contributions to the beta functions of the gauge and Yukawa couplings of a matter theory using the Schwinger proper-time flow equation. Working in the Einstein-Hilbert truncation, we investigate the gauge-fixing and regulator dependence of the corresponding renormalization group equations. We quantify the sensitivity of our results on unphysical parameters by evaluating the gravitational correction to the running matter couplings at the interactive fixed point of gravity and we compare our findings with existing determinations in alternative schemes. We finally confront the derived contributions with the typical size they should assume to generate observable low-scale predictions in the Standard Model and in several scenarios of new physics.

Figures (5)

• Figure 1: $f_y$ as a function of the regulator $m$ for $\tilde{G}=1$. Solid blue: $\alpha=1$, $\tilde{\Lambda}=0$; dashed blue: $\alpha=1$, $\tilde{\Lambda}=-4$; dotted red: $\alpha=0$, $\tilde{\Lambda}=0$; dot-dashed red: $\alpha=0$, $\tilde{\Lambda}=-4$.
• Figure 2: Dependence on the gauge-fixing parameter $\alpha$ of the gravity fixed point (a) $\tilde{G}^\ast$ and (b) $\tilde{\Lambda}^\ast$ in the minimal matter model. Dependence on the gauge-fixing parameter $\alpha$ of (c) $f_g$ and (d) $f_y$ in the minimal matter model. Blue solid line indicates the sharp regulator ($m\to \infty$), while dashed red line corresponds to a specific choice of the regulator parameter, $m=3$.
• Figure 3: Dependence on the gauge-fixing parameter $\alpha$ of the gravity fixed point (a) $\tilde{G}^\ast$ and (b) $\tilde{\Lambda}^\ast$ in the SM. Dependence on the gauge-fixing parameter $\alpha$ of (c) $f_g$ and (d) $f_y$ in the SM. Blue solid line indicates the sharp regulator ($m\to \infty$), while dashed red line corresponds to a specific choice of the regulator parameter, $m=3$.
• Figure 4: Isocontour lines of (a) $f_g$ and (b) $f_y$ in the $(\tilde{\Lambda}^*, \tilde{G}^*)$ plane, computed in the sharp-regulator limit of the proper time flow with gauge-fixing parameter $\alpha = 1$. The markers indicate the gravitational UV fixed points for the SM, $B-L$, SU(5), and SU(6) matter content. Isocontour lines of (a) $f_g$ and (b) $f_y$ in the $(\tilde{\Lambda}^*, \tilde{G}^*)$ plane, computed in the sharp-regulator limit of the proper time flow with gauge-fixing parameter $\alpha = 0$.
• Figure 5: Regulator dependence of the fixed-point values of the Einstein-Hilbert action, (a) $\tilde{G}^\ast$ and (b) $\tilde{\Lambda}^\ast$ in the minimal matter model for three values of the gauge-fixing parameter: $\alpha=0$ (solid blue), $\alpha=1$ (dashed red), and $\alpha=2$ (dotted green). (c) Same as (a), in the SM. (d) Same as (b), in the SM.