Matter matters in asymptotically safe quantum gravity

We investigate the compatibility of minimally coupled scalar, fermion and gauge fields with asymptotically safe quantum gravity, using nonperturbative functional renormalization group methods. We study $d=4,5$ and 6 dimensions and within certain approximations find that for a given number of gauge fields there is a maximal number of scalar and fermion degrees of freedom compatible with an interacting fixed point at positive Newton coupling. The bounds impose severe constraints on grand unification with fundamental Higgs scalars. Supersymmetry and universal extra dimensions are also generally disfavored. The standard model and its extensions accommodating right-handed neutrinos, the axion and dark-matter models with a single scalar are compatible with a fixed point.

Figure 1

Diagrams contributing to ${\eta }_h$. Spirals denote gravitons, thick lines fermions, dashed lines scalars and curly lines gauge fields. A crossed circle denotes an insertion of ${\partial }_t{R}_k$.

Figure 2

Diagrams contributing to ${\eta }_D$. Analogous diagrams with external scalar and vector fields determine ${\eta }_S$ and ${\eta }_V$.

Figure 3

The shaded triangles are the areas in the $N_S$-$N_D$ plane compatible with a gravitational fixed point for $N_V=0$ (darkest triangle in bottom left corner), $N_V=6,12,24,45$ (from bottom to top).

Figure 4

Left and middle: Position of the fixed point as a function of the number of scalar fields. Right: Critical exponents. Note the logarithmic scales. All with type II cutoff and one-loop anomalous dimensions.

Figure 5

The graviton (left), ghost (middle) and scalar (right) anomalous dimensions as functions of the number of scalar fields.

Figure 6

The values of ${\tilde{G}}_{*}$ (dots) and ${\tilde{\mathrm{\Lambda }}}_{*}$ (squares) as functions of the number of Dirac fields, at one loop with type II cutoff.

Figure 7

The critical exponents ${\theta }_{1,2}$ as functions of the number of Dirac fields.

Figure 8

The graviton (left), ghost (middle) and fermion (right) anomalous dimensions as functions of the number of fermion fields, all at one loop and with type II cutoff.

Figure 9

The points in the $N_S$-$N_D$ plane compatible with a gravitational fixed point with two relevant directions for $N_V=0$. The line represents the perturbative bound (40). Lighter shades of gray mean smaller ${\eta }_h$; black means ${\eta }_h>10$.

Figure 10

Position of the fixed point (left and middle) and critical exponents (right) as a function of number of vector fields.

Figure 11

The graviton (left), ghost (middle) and vector (right) anomalous dimensions as functions of the number of vector fields.

Figure 12

The region compatible with the existence of a gravitational fixed point with ${\tilde{G}}_{*}>0$ and two attractive directions for $d=4$, $N_V=12$. The line represents the perturbative bound (40). Lighter shades of gray mean smaller ${\eta }_h$; black means ${\eta }_h>10$.

Figure 13

Left: Existence region of the fixed point connected to the one without matter, in the plane $N_D=0$. Right: Existence region of a second real fixed point with two relevant directions, in the same plane. The line $N_D=0$ in Fig. 12 corresponds to the line $N_V=12$ in these two figures.

Figure 14

The region compatible with the existence of a gravitational fixed point with ${\tilde{G}}_{*}>0$ and two attractive directions for $d=5$ (top) and $d=6$ (bottom) and $N_V=12$. Lighter shades of gray mean smaller ${\eta }_h$; black means ${\eta }_h>10$.