Quantum-gravity-induced matter self-interactions in the asymptotic-safety scenario

We investigate the high-energy properties of matter theories coupled to quantum gravity. Specifically, we show that quantum gravity fluctuations generically induce matter self-interactions in a scalar theory. Our calculations apply within asymptotically safe quantum gravity, where our results indicate that the UV is dominated by an interacting fixed point, with nonvanishing gravitational as well as matter couplings. In particular, momentum-dependent scalar self-interactions are nonzero and induce a nonvanishing momentum-independent scalar potential. Furthermore, we point out that terms of this type can have observable consequences in the context of scalar-field driven inflation, where they can induce potentially observable non-Gaussianities in the cosmic microwave background.

Figure 1

The scalar-gravity-interaction vertices that are generated by the kinetic term for the scalar field allow to construct these two one-loop diagrams that generate scalar self-interactions.

Figure 2

Here we exemplify a matter $\beta$ function for the coupling $\rho$: Without coupling to gravity (blue line), this $\beta$ function will admit a Gaussian fixed point, as well as possibly a non-Gaussian one. Including gravity fluctuations, a term $\sim G$ will be added to the $\beta$ function, thus shifting the Gaussian to an interacting, non-Gaussian fixed point (red dashed line). A second non-Gaussian fixed point may or may not exist.

Figure 3

Here we show the diagrammatic expansion of the Wetterich equation, projected onto the ${\varphi }^2$ and ${\varphi }^4$ subspace of theory space, respectively. Diagrams 1a, 1b, and 1c contribute to ${\eta }_{\varphi }=-{\partial }_t\ln Z_{\varphi }$, whereas all diagrams 2 in principle contribute to ${\beta }_{{\rho }_i}$ and ${\beta }_{\lambda }$. Each loop contains a regulator insertion ${\partial }_t{R}_k$ that can be found on any of the internal lines.

Figure 4

Here we plot the fixed-point values of the matter self-couplings as a function of $g$ for a value of the cosmological constant $\lambda =1/10$. The blue circles show the first fixed point, the orange squares the second fixed point. The fixed point values approach each other for increasing values of $g$. Note that in fact, the values of ${\lambda }_{\varphi }$ at the second fixed point are small, but nonzero for $g>0$. Beyond $g=0.35$, we do not find any fixed points, i.e., the two fixed points annihilate.

Figure 5

Here we plot the four critical exponents at the two fixed points, for $\lambda =1/10$ and as a function of $g$. Clearly, at $g=0$, the critical exponents are given by the canonical dimensionality of the couplings $\overline{\rho }$ and $\lambda$. At $g>0$, they deviate significantly. Most importantly, there is one relevant critical exponent at the first fixed point (upper panel).

Figure 6

Metric loops will generically induce nonvanishing scalar self-interactions with an arbitrary even number of scalar fields.

Figure 7

Here we plot the value of ${\eta }_{\varphi }$ as a function of $g$ for $\lambda =1/10$ at the fixed points as in Fig. 4. Towards larger values of $g$, ${\eta }_{\varphi }$ becomes increasingly negative at both fixed points, implying that further matter couplings could be shifted into relevance.

Figure 8

Here we plot the RG flow towards the infrared, projected onto the Einstein-Hilbert plane for ${\rho }_a=47.094$, ${\rho }_b=24.771$, ${\rho }_c=66.120$, and ${\lambda }_{\varphi }=-127.502$, cf. Table . Clearly the flow resembles the flow in the pure Einstein-Hilbert truncation at vanishing matter couplings to a very high degree and admits trajectories passing very close to the Gaussian fixed point and yielding a constant value of the dimensionful Newton coupling and cosmological constant, in accordance with observations.