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We study the parametrization and gauge dependences in the Higgs field coupledto gravity in the context of asymptotic safety. We use the exponentialparametrization to derive the fixed points for the cosmological constant,Planck mass, Higgs mass and its coupling, keeping arbitrary gauge parameters$\alpha$ and $\beta$, and compare the results with the linear split. We findthat the beta functions for the Higgs potential are expressed in terms ofredefined Planck mass such that the apparent gauge dependence is absent. Onlythe trace mode of the gravity fluctuations couples to the Higgs potential andit tends to decouple in the large $\beta$ limit, but the anomalous dimensionbecomes large, invalidating the local potential approximation. This gives thelimitation of the exponential parametrization. There are also singularities forsome values of the gauge parameters but well away from these, we find ratherstable fixed points and critical exponents. We thus find that there are regionsfor the gauge parameters to give stable fixed points and critical exponentsagainst the change of gauge parameters. The Higgs coupling is confirmed to beirrelevant for the reasonable choice of gauge parameters.

Higgs scalar potential coupled to gravity in the exponential parametrization in arbitrary gauge