Faddeev-Popov ghosts in quantum gravity beyond perturbation theory

We study the Faddeev-Popov ghost sector of asymptotically safe quantum gravity, which becomes nonperturbative in the ultraviolet. We point out that nonzero matter-ghost couplings and higher-order ghost self-interactions exist at a non-Gaussian fixed point for the gravitational couplings, i.e., in the ultraviolet. Thus the ghost sector in this nonperturbative ultraviolet completion does not keep the structure of a simple Faddeev-Popov determinant. We discuss implications of the new ghost couplings for the renormalization group flow in gravity, the form of the ultraviolet completion, and the relevant couplings, i.e., free parameters, of the theory.

Figure 1

These diagrams contribute to the $\beta$ function for a ghost-matter coupling. Dotted lines denote ghosts, spiralling lines metric propagators and full lines denote scalars.

Figure 2

These diagrams generate the matter-ghost coupling between ${\varphi }^2$ and a ghost and antighost and thereby remove the Gaussian fixed point in the corresponding $\beta$ function. Matter fields are denoted by thick lines, ghosts by dashed and metric fluctuations by spiralling lines. A regulator insertion exists on each of the internal propagators.

Figure 3

Here we plot the fixed-point value of $v$ as a function of $g$, for ${\eta }_N=-2$, ${\eta }_{\varphi }=-0.78$, cf. 30, for a regulator of the form Eq (17). The blue thick curve shows the value for $\lambda =0$, whereas the dashed purple curve shows the result for $\lambda =-0.5$.

Figure 4

These are the only two diagrams that induce four-ghost couplings, starting with a simple perturbative FP term in the action. Regulator insertions can be found on each of the internal lines.

Figure 5

Here we plot the fixed-point value at the shifted Gaussian fixed point for ${\eta }_c=-0.78$ and ${\eta }_N=-2$ as a function of $g$ for a regulator of the form Eq. (17). The blue thick curve shows the result for $\lambda =0$, whereas the purple dashed curve shows the result for $\lambda =-0.5$. Clearly, the value ${\chi }_{\mathrm{gh}}=0$ can only be reached by setting $g=0$.

Figure 6

We illustrate the RG flow in theory space in the case of asymptotic freedom (left panel) and asymptotic safety (right panel).

Figure 7

This six (anti) ghost diagram is constructed from the simple ghost-antighost-graviton vertex, only.