Conformal dilaton gravity: Classical noninvariance gives rise to quantum invariance

When quantizing conformal dilaton gravity, there is a conformal anomaly which starts at two-loop order. This anomaly stems from evanescent operators on the divergent parts of the effective action. The general form of the finite counterterm, which is necessary in order to insure cancellation of the Weyl anomaly to every order in perturbation theory, has been determined using only conformal invariance. Those finite counterterms do not have any inverse power of any mass scale in front of them (precisely because of conformal invariance), and then they are not negligible in the low-energy deep infrared limit. The general form of the ensuing modifications to the scalar field equation of motion has been determined, and some physical consequences have been extracted.

Figure 1

Conformal case.

Figure 2

Non conformal case for $C<0$.

Figure 3

$f^{\prime }(0)>0$.

Figure 4

$f^{\prime }(0)<0$.