Asymptotic safety of gravity-matter systems

We study the ultraviolet stability of gravity-matter systems for general numbers of minimally coupled scalars and fermions. This is done within the functional renormalization group setup put forward in [N. Christiansen, B. Knorr, J. Meibohm, J. M. Pawlowski, and M. Reichert, Phys. Rev. D 92, 121501 (2015).] for pure gravity. It includes full dynamical propagators and a genuine dynamical Newton’s coupling, which is extracted from the graviton three-point function. We find ultraviolet stability of general gravity-fermion systems. Gravity-scalar systems are also found to be ultraviolet stable within validity bounds for the chosen generic class of regulators, based on the size of the anomalous dimension. Remarkably, the ultraviolet fixed points for the dynamical couplings are found to be significantly different from those of their associated background counterparts, once matter fields are included. In summary, the asymptotic safety scenario does not put constraints on the matter content of the theory within the validity bounds for the chosen generic class of regulators.

Figure 1

Flow equation for the scale-dependent effective action ${\mathrm{\Gamma }}_k$ in diagrammatic representation. The double, dotted, solid and dashed lines correspond to the graviton, ghost, fermion and scalar propagators, respectively. The crossed circles denote the respective regulator insertions.

Figure 2

Vertex dressing of all three-point vertices used in this work. The vertex dressing consists of the respective wave function renormalizations, couplings and tensor structures. The first line in the figure depicts all pure gravity three-point vertices while the second line shows the ones with gravity-matter interactions.

Figure 3

Diagrammatic representation of the matter-induced flow of the graviton two-point function. Double, single and dashed lines represent graviton, fermion and scalar propagators, respectively; filled circles denote dressed vertices. Crossed circles are regulator insertions.

Figure 4

Diagrammatic representation of the matter-induced flow of the three-graviton vertex. Double, single and dashed lines represent graviton, fermion and scalar propagators, respectively; filled circles denote dressed vertices. Crossed circles are regulator insertions. All diagrams are symmetrized with respect to the interchange of external momenta $\mathbf{p}$.

Figure 5

Diagrammatic representation of the gravitationally induced flows of the matter two-point functions. Double, single and dashed lines represent graviton, fermion and scalar propagators, respectively; filled circles denote dressed vertices. Crossed circles are regulator insertions.

Figure 6

Fixed point values (left), real parts of the critical exponents (middle), and the anomalous dimensions evaluated at $p=k$ and $p=0$ (right) as functions of the number of scalars, $N_s$, respectively. The gray-shaded area in the left panel indicates where the regulator lies outside the reliability bounds defined in Sec. 3d due to a large graviton anomalous dimension (see right panel). The corresponding limiting number or scalars is given by $N_{s_{\text{trunc}}}$. The hatched regions in all three panels correspond to the $N_s$ regime where the UV fixed point does not exist. The green and red areas in the middle panel denote the region where the fixed point exhibits UV attractive direction and the region where it is fully repulsive, respectively. $N_{s_{\mathrm{stab}}}$ is the corresponding critical number. The colors of the curves in the middle panel indicate with which coupling of the left panel the corresponding eigenvector has the largest overlap. The anomalous dimension of the scalar ${\eta }_{\phi }$ (right panel) is zero due to the given graviton gauge.

Figure 7

Fixed point values (left), real parts of the critical exponents (middle), and the anomalous dimensions evaluated at $p=k$ and $p=0$ (right) as functions of the number of fermions, $N_f$, respectively. The colors of the curves in the middle panel indicate with which coupling of the left panel the corresponding eigenvector has the largest overlap. All quantities remain well behaved for any number of fermions. In particular, the fixed point stays attractive (middle panel) and the anomalous dimensions remain small (right panel).

Figure 8

Fixed point value $g^{*}$ as a function of $N_s$ for $N_f=0$, 5, 10, 15. The vertical lines denote the numbers of scalars for which the graviton anomalous dimension exceeds its critical value in the UV, for the respective number of fermions.

Figure 9

Fixed point values of the dynamical couplings $(g,{\lambda }_2)$ (solid lines) in comparison with their corresponding background counterparts $(\overline{g},\overline{\lambda })$ (dashed lines) as functions of the number of scalars $N_s$ (left panel) and the number of fermions $N_f$ (right panel). The fixed point values of the background couplings diverge for $N_s=60.8$ (left panel) and $N_f=68.6$ (right panel). The dotted regions denote the regimes beyond the latter divergences. $N_{s_{\mathrm{pole}}}$ denotes the number of scalars at which $\overline{\lambda }$ would run into the propagator pole if the identification ${\lambda }_2=\overline{\lambda }$, common in the background approximation is applied.

Figure 10

Logarithmic plot of the fixed point values of the background field couplings $(\overline{g},\overline{\lambda })$ as a function of the number of scalars $N_s$ (left) and as a function of the number of fermions $N_f$ (right) in comparison with results in background field approximation from [34] (DEP). Our background couplings behave very similarly to the couplings in [34]. The gray- and black-shaded areas denote the regimes where the fixed point for the couplings in [34] and for our couplings is lost, respectively.