Abstract
We study the parametrization and gauge dependences in the Higgs field coupled to gravity in the context of asymptotic safety. We use the exponential parametrization to derive the fixed points for the cosmological constant, Planck mass, Higgs mass and its coupling, keeping arbitrary gauge parameters $\ensuremath{\alpha}$ and $\ensuremath{\beta}$, and compare the results with the linear split. We find that the beta functions for the Higgs potential are expressed in terms of redefined Planck mass such that the apparent gauge dependence is absent. Only the trace mode of the gravity fluctuations couples to the Higgs potential and it tends to decouple in the large $\ensuremath{\beta}$ limit, but the anomalous dimension becomes large, invalidating the local potential approximation. This gives the limitation of the exponential parametrization. There are also singularities for some values of the gauge parameters but well away from these, we find rather stable fixed points and critical exponents. We thus find that there are regions for the gauge parameters to give stable fixed points and critical exponents against the change of gauge parameters. The Higgs coupling is confirmed to be irrelevant for the reasonable choice of gauge parameters.
Highlights
The functional renormalization group [1,2,3,4,5,6] is a powerful method to tackle nonperturbative phenomena in quantum field theory
The functional renormalization group has contributed towards asymptotically safe quantum gravity [27,28,29] which is formulated as a nonperturbative quantum field theory
A central object in the functional renormalization group is the effective average action Γk and its second-order functional derivative Γðk2Þ whose inverse form corresponds to the full propagator, so that in general the gauge fixing is required for the gauge field propagator in addition to the ultraviolet (UV) regularization
Summary
The functional renormalization group [1,2,3,4,5,6] is a powerful method to tackle nonperturbative phenomena in quantum field theory. We would like to study this problem by keeping the gauge parameters, and check how much the results depend on them Another possible problem is that it is found that there appears some unphysical poles in the cosmological constant in the beta functions for the gravity couplings if one uses the linear parametrization of the metric gμν 1⁄4 gμν þ hμν: ð1:1Þ. It turns out that if we use a redefined Planck mass, the beta functions for the scalar potential in the exponential parametrization become independent of the gauge parameters, and we can discuss the fixed points without apparent gauge dependence This is an advantage of the exponential parametrization, but the price is that the anomalous dimension of the Higgs scalar becomes large in general. The flow equations and anomalous dimension of the Higgs field are calculated in Appendix D
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