Data for thermally modified Standard Model masses and couplings

Mikko Laine, Philipp Schicho and York Schröder

The file below contains thermally modified masses and couplings, obtained from high-order RG equations as described in 1911.09123, with the Higgs expectation value evolved as can be deduced from 1508.07161.

General comments

The definition of thermally modified masses and couplings is not unique, but depends on the kinematics considered. The philosophy adopted here is to define the couplings as the effective parameters that appear in the dimensionally reduced effective theory, or its real-time generalization, the Hard Thermal Loop theory. Physically, this means that the couplings incorporate the influence of hard thermal fluctuations, but not of soft ones. To a fair approximation, such couplings can be obtained by running vacuum couplings to a thermal scale, ~ 2πT.

For the masses of vector bosons and fermions, we only list the "vacuum-like" part, involving the Higgs expectation value. In physical computations, the plasma contribution needs to be added, as explained in some detail in 2505.05206. In the file below, the plasma contribution is given for the SU(3) Debye mass, from the 2-loop computation in 1911.09123. In addition, the plasma contribution is included in the Higgs mass, incorporating lattice input from 1508.07161. Being a scalar particle, there is no tensor structure in the Higgs channel, so that in the last case the result is unique for any kinematics.

Apart from the renormalization scale, the Higgs expectation value needs to be evolved, in order to account for the electroweak phase transition. This introduces powerlike corrections. In order to avoid artificial singularities, we have not set the Higgs expectation value and consequently the vacuum-like masses to zero above the critical temperature, but rather to small finite values. However, these should have no effect in any practical computation (otherwise, IR divergences have not been properly cancelled).

Notation

As is also indicated in the preamble of the file, the columns are:

1: T / GeV
2: μ / T: renormalization scale
3: mu / MeV: up quark mass
4: md / MeV: down quark mass
5: ms / MeV: strange quark mass
6: mc / MeV: charm quark mass
7: mb / MeV: bottom quark mass
8: mt / MeV: top quark mass
9: mE / T: SU(3) Debye mass
10: g2s: SU(3) gauge coupling squared
11: g2w: SU(2) gauge coupling squared
12: g2p: U(1) gauge coupling squared
13: λ: Higgs quartic coupling
14: h2t: top Yukawa squared
15: h2b: bottom Yukawa squared
16: mW / MeV: W mass
17: mZ / MeV: Z mass
18: mh / MeV: Higgs mass
19: v / MeV: Higgs expectation value
20: me / MeV: electron mass
21: mμ / MeV: muon mass
22: mτ / MeV: tau mass

Here is the file: