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

We study the parametrization and gauge dependences in Higgs field coupled to gravity context of asymptotic safety. use exponential derive fixed points for cosmological constant, Planck mass, mass its coupling, keeping arbitrary parameters $\ensuremath{\alpha}$ $\ensuremath{\beta}$, compare results with linear split. find that beta functions potential are expressed terms redefined such apparent dependence is absent. Only trace mode fluctuations couples it tends decouple large $\ensuremath{\beta}$ limit, but anomalous dimension becomes large, invalidating local approximation. This gives limitation parametrization. There also singularities some values well away from these, we rather stable critical exponents. thus there regions give exponents against change parameters. The coupling confirmed be irrelevant reasonable choice