Mass Difference for Charged Quarks from Asymptotically Safe Quantum Gravity

We propose a scenario to retrodict the top and bottom mass and the Abelian gauge coupling from first principles in a microscopic model including quantum gravity. In our approximation, antiscreening quantum-gravity fluctuations induce an asymptotically safe fixed point for the Abelian hypercharge leading to a uniquely fixed infrared value that is observationally viable for a particular choice of microscopic gravitational parameters. The unequal quantum numbers of the top and bottom quark lead to different fixed-point values for the top and bottom Yukawa couplings under the impact of gauge and gravity fluctuations. This results in a dynamically generated mass difference between the two quarks. To work quantitatively, the preferred ratio of electric charges of bottom and top in our approximation lies in close vicinity to the standard-model value of $Q_b/Q_t=-1/2$.

Figure 1

RG trajectory of standard-model couplings for $f_g=9.7\times {10}^{-3}$ and $f_y=1.188\times {10}^{-4}$ reaching $g(k_{\mathrm{IR}})=0.358$, $y_t(k_{\mathrm{IR}})=0.965$, and $y_b(k_{\mathrm{IR}})=0.018$ at $k_{\mathrm{IR}}=173\mathrm{GeV}$. We also plot $y_t^2-g_Y^2/3$ (pink, wide dashed), which approaches $y_{b*}^2$ (dotted) in the far UV; cf. Eq. (5).

Figure 2

IR values of retrodicted couplings $g_Y(k_{\mathrm{IR}})$, $y_t(k_{\mathrm{IR}})$, and $y_b(k_{\mathrm{IR}})$ at $k_{\mathrm{IR}}=173\mathrm{GeV}$ as a function of the two independent quantum-gravity contributions $f_g$ and $f_y$.

Figure 3

Bottom Yukawa coupling $y_b(k_{\mathrm{IR}})$ at $k_{\mathrm{IR}}=173\mathrm{GeV}$ as a function of the IR values of $g_Y(k_{\mathrm{IR}})$ and $y_t(k_{\mathrm{IR}})$.

Figure 4

Top Yukawa coupling $y_t(k_{\mathrm{IR}})$ at $k_{\mathrm{IR}}=173\mathrm{GeV}$ as a function of a nonuniversal gravity contribution $f_{g,\mathrm{nA}}/f_g$ (upper panel) and of a modified Planck scale $M_{\text{Planck}}$ (lower panel) for fixed $g_Y(k_{\mathrm{IR}})=0.358$ and $M_b=4.9\mathrm{GeV}$.

Figure 5

Top Yukawa coupling $y_t(k_{\mathrm{IR}})$ at $k_{\mathrm{IR}}=173\mathrm{GeV}$ as a function of the charge ratio $Q_b/Q_t$ for fixed $g_Y(k_{\mathrm{IR}})=0.358$ and $M_b=4.9\mathrm{GeV}$.

Figure 6

IR values of the retrodicted couplings $g_Y(k_{\mathrm{IR}})$, $y_t(k_{\mathrm{IR}})$, and $y_b(k_{\mathrm{IR}})$ at $k_{\mathrm{IR}}=173\mathrm{GeV}$ as a function of the two quantum-gravity contributions $f_g$ and $f_y$ at modified charge ratio $Q_b/Q_t=-2/3$ (left-hand panel) and $Q_b/Q_t=-1/3$ (right-hand panel).