Variable Planck mass from the gauge invariant flow equation

Using the gauge invariant flow equation for quantum gravity we compute how the strength of gravity depends on the length or energy scale. The fixed point value of the scale-dependent Planck mass in units of the momentum scale has an important impact on the question, which parameters of the Higgs potential can be predicted in the asymptotic safety scenario for quantum gravity? For the standard model and a large class of theories with additional particles the quartic Higgs coupling is an irrelevant parameter at the ultraviolet fixed point. This makes the ratio between the Higgs boson and the top-quark mass predictable.

Figure 1

Contour plot of the fixed point value of $v_{+}$ in the $({\tilde{\mathcal{N}}}_M,{\tilde{\mathcal{N}}}_U)$ plane. In the red region for negative ${\tilde{\mathcal{N}}}_U$ and ${\tilde{\mathcal{N}}}_M$ no constant scaling solution is found. In the yellow region the scaling solution is unstable due to $w_{+}<0$. The green region admits a stable constant scaling solution. As one moves toward the region of large positive ${\tilde{\mathcal{N}}}_U$ and negative ${\tilde{\mathcal{N}}}_M$, corresponding to a large number of scalar fields $N_S$, the fixed point value $v_{+}$ approaches one. For $v$ close to one our approximations are no longer reliable. We indicate the location of pure gravity, the standard model, as well as SU(5) and SO(10) GUT models with a number of scalar fields varying between $N_S=50$ and 100.

Figure 2

Contour plot of the fixed point value of $w_{+}$ in the $({\tilde{\mathcal{N}}}_M,{\tilde{\mathcal{N}}}_U)$ plane. The strength of gravity at the fixed point $w_{+}^{-1}$ increases as one approaches the excluded red and yellow regions for large negative ${\tilde{\mathcal{N}}}_U$ and ${\tilde{\mathcal{N}}}_M$. For sufficiently negative ${\tilde{\mathcal{N}}}_U$ the fixed point gravity becomes unstable as ${\tilde{\mathcal{N}}}_M$ turns negative (red region). For the small value of $w_{+}$ near the boundary of the excluded red and yellow regions one may have doubts on the validity of the truncation.

Figure 3

Contour plot of the fixed point value of $u_{+}$ in the $({\tilde{\mathcal{N}}}_M,{\tilde{\mathcal{N}}}_U)$ plane. Roughly the fixed point potential or cosmological constant $u_{+}$ is positive for positive ${\tilde{\mathcal{N}}}_U$ and negative for negative ${\tilde{\mathcal{N}}}_U$.

Figure 4

Contour plot of the allowed (green) and excluded (red) regions for the new fixed point. We indicate line plots of $v_{-}=1$ (red dotted line) and $w_{-}=0$ (blue solid line) in the $({\tilde{\mathcal{N}}}_M,{\tilde{\mathcal{N}}}_U)$ plane. For the yellow region the fixed point $u_{-}$, $w_{-}$ is unstable. Only for the green region with ${\tilde{\mathcal{N}}}_U$ and ${\tilde{\mathcal{N}}}_M$ somewhat smaller than zero is the new fixed point stable. As the unstable yellow region is approached, our truncation is not expected to remain valid.

Figure 5

Fixed point values of $v_{*}$, $u_{*}$, and $w_{*}$ as functions of $N_S$ for SU(5) (solid lines) and SO(10) (dashed lines) GUT models. We display both possible constant scaling solutions $(u_{+},w_{+},v_{+})$ (blue lines in upper parts of the figures) and $(u_{-},w_{-},v_{-})$ (red lines in lower parts of the figures). For the Reuter fixed point the truncation becomes insufficient as $v_{+}$ moves close to one for large $N_S$. For the new fixed values no stable solutions are found for large $N_S$ due to $w_{-}<0$. For small $w$ our approximations are doubtful. The gray regions are not allowed due to the conditions $v_{*}<1$ and $w_{*}>0$. The vertical dotted and dot-dashed lines are at $N_S=47$ and $N_S=139$ at which $w_{-*}=0$, respectively.

Figure 6

Graviton induced anomalous dimension $A$, given by Eq. (242), for SU(5) and SO(10) GUT models as a function of the number of scalars $N_S$.

Figure 7

Critical exponents (real parts of eigenvalues of $T^{(12)}$, $T^{(34)}$, and $T^{(56)}$) as functions of $N_S$ in SU(5) and SO(10) GUTs. For small values of $N_S$ the critical exponents are not far from the values given by the canonical mass dimension of the associated parameters. For the SO(10) model the curves for small $N_S$ from top to bottom correspond approximately to $u_0$, $w_0$, ${\tilde{m}}_H^2$, ${\xi }_H$, ${\lambda }_H$, and $w_2$. In particular, the critical exponent for the quartic coupling of the Higgs scalar is negative, making this irrelevant parameter predictable. As $N_S$ increases, the quartic scalar coupling becomes first more and more irrelevant, with decreasing ${\theta }_5$. For $N_S$ close to 40 the eigenvalues of the stability matrix develop an imaginary part, and the absolute value of the critical exponents starts to increase. For large $N_S$ our approximations no longer remain valid.

Figure 8

Contour plots for the real parts of ${\theta }_5$ (left) and ${\theta }_6$ (right). The red (no constant scaling solution) and yellow (unstable solution due to $w_{+}<0$) regions are excluded. The pink dashed line shows the change of $N_S$ for the SU(5)-GUT model. It corresponds to the plots for ${\theta }_5$ and ${\theta }_6$ in Fig. 7. For $N_S\ge 43$ in SU(5) GUT both ${\theta }_5$ and ${\theta }_6$ have an imaginary part, so that their real parts are degenerate.

Figure 9

Contour plot of the fixed point value of $v_{-}$ in the $({\tilde{\mathcal{N}}}_M,{\tilde{\mathcal{N}}}_U)$ plane. For the yellow region on the right of the figure there is no stable solution due to $v_{-}>1$. For the red region no constant scaling solution is found.

Figure 10

Contour plot of the fixed point value of $w_{-}$ in the $({\tilde{\mathcal{N}}}_M,{\tilde{\mathcal{N}}}_U)$ plane. For the yellow region the new fixed point has unstable gravity due to $w_{-}<0$. The combination of the yellow and red regions in Figs. 9 and 10 leads to the excluded red and yellow regions in Fig. 4.

Figure 11

Contour plot of the fixed point value of $u_{-}$ in the $({\tilde{\mathcal{N}}}_M,{\tilde{\mathcal{N}}}_U)$ plane.