Abstract
We revisit the perturbative expansion at high temperature and investigate its convergence by inspecting the renormalisation scale dependence of the effective potential. Although at zero temperature the renormalisation group improved effective potential is scale independent at one-loop, we show how this breaks down at high temperature, due to the misalignment of loop and coupling expansions. Following this, we show how one can recover renormalisation scale independence at high temperature, and that it requires computations at two-loop order. We demonstrate how this resolves some of the huge theoretical uncertainties in the gravitational wave signal of first-order phase transitions, though uncertainties remain stemming from the computation of the bubble nucleation rate.
Highlights
JHEP06(2021)069 been mooted that this may provide an experimental probe of particle physics beyond the Standard Model (BSM) which may be complementary and competitive with collider searches, see for e.g. refs. [17, 18] and references therein
We show an example of the renormalisation scale dependence of the gravitational wave spectrum from a first-order phase transition, in the real singlet-extended Standard Model
In the one-loop calculation, both the peak amplitude of the gravitational wave spectrum and the LISA signal-to-noise ratio vary by four orders of magnitude as the renormalisation scale varies over just a factor of four, a huge theoretical uncertainty
Summary
To demonstrate issues of scale dependence without unnecessary complications, we initially work with the simplest possible theory, that of a single real scalar field, φ, with a Z2 symmetry, φ → −φ. The cancellation only holds at leading order, receiving corrections from the oneloop running of parameters within the one-loop potential, and from two-loop corrections to the running of parameters in the tree-level potential. These are suppressed relatively by g2/(4π), and can be neglected in a one-loop analysis. At zero temperature the effective potential is independent of the renormalisation scale at one-loop order. This holds order-by-order in ,5 or equivalently in g2 for this theory. Our analysis has been carried out in the MS renormalisation scheme, but the conclusions hold independently of this because the one-loop (and two-loop) beta functions, as well as the logarithm of the Coleman-Weinberg potential, are independent of the renormalisation scheme [103]
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