Baryogenesis via GeV scale hot sterile neutrino oscillations

Jacopo Ghiglieri and Mikko Laine

The files below contain data and results related to 1711.08469, in which a numerical solution of rate equations describing GeV scale sterile neutrino oscillations at temperatures above 100 GeV was presented. The CP violation present in this dynamics leads to the generation of lepton and baryon asymmetries. (More recently the results have been extended to lower temperatures and broader parameter ranges in 1811.01971, where also the rate coefficients were determined more precisely at T < 150 GeV.)

Starting from the more trivial ingredients, the background equation of state, determining the expansion rate of the Universe, was taken over from here. Thereby the heat capacity has a peak around T ~ 160 GeV, which leads to small features in the production process.

An essential role in the solution is played by so-called rate coefficients, denoted by Q, R, S in 1703.06087, where they were determined at leading order in Standard Model couplings, including a resummation of 1 + n ↔ 2 + n scatterings and the inclusion of all 2 ↔ 2 scatterings. Employing the notation in eqs. (5.5)-(5.7) of 1711.08469, in which the leading mass dependence has been factored out, a number of tables are provided. The temperature and comoving momentum grids are collected in the file

The columns in this file are:

1: x = log(Tmax/T) with Tmax = 107 GeV
2: T / MeV (300 values)
3: kT / T (249 values)

The actual coefficients are collected in six other files, with the line number always corresponding to the coordinates given in axes.dat. The columns of these files are:

1: Q11
2: Q12
3: Q21
4: Q22

Actually, there is no dependence on the indices within our approximation, so all columns have the same value. (This would not be the case if larger masses M ~ gT were considered, or if we went to lower temperatures, say T < 100 GeV for M = 1 GeV.) For M = 1 GeV, here are the files:

Note that at k < 0.3T the determination of the coefficients contains theoretical uncertainties and, in some cases, numerical ripples. This domain does give a noticeable contribution to the final lepton asymmetries, and would merit further thought (more elaboration can be found in the conclusions and appendix A of 1711.08469).

In addition, we list the phase factors from eq. (4.4) of 1711.08469, i.e. ∫0x d x' Ĥfast(x'). This is specific to the masses considered; here are the results for "case 1" from below (to be precise, the lower bound was set to 0 → −10-10):


With the coefficients from above, the evolution equations can be integrated. We tabulate results for the final baryon asymmetry for various benchmark parameter values. The notation follows that in eqs. (5.2)-(5.3) of 1711.08469. Numerical integration is non-trivial and, with given coefficients, errors on the 2% level cannot be excluded. However this numerical uncertainty is less than the theoretical uncertainty from NLO corrections to various coefficient functions, and from NLO and NNLO corrections to susceptibilities, known to be on a 20% level. Cases 22 and 23 have furthermore been flagged with "∼", given that temperatures T < 160 GeV play an important role for these extremely degenerate cases and that therefore the rate coefficients pertinent to the Higgs phase should be determined more accurately.

case M1 / GeV M2 / GeV hierarcy Re(z) Im(z) φ1 δ YB [10-10]
1 0.7688 0.7776 IH 2.444 −3.285 −1.857 −2.199 +1.38
2 0.7688 0.7776 IH 2.444 −3.285 −1.857 +2.199 +1.54
3 0.7688 0.7776 IH 2.444 −3.285 +1.857 +2.199 −0.88
4 0.7688 0.7776 IH 2.444 +3.285 −1.857 +2.199 +0.88
5 0.7688 0.7776 IH 2.444 +3.285 +1.857 +2.199 −1.38
6 0.7688 0.7776 NH 2.444 −3.285 −1.857 −2.199 +0.08
7 0.7688 0.7776 NH 2.444 −3.285 −1.857 +2.199 −0.01
8 0.7688 0.7776 NH 2.444 −3.285 +1.857 +2.199 −0.09
9 0.7688 0.7776 NH 2.444 +3.285 −1.857 +2.199 +0.18
10 0.7688 0.7776 NH 2.444 +3.285 +1.857 +2.199 −0.08
11 0.7727 0.7737 NH 2.444 −3.285 −1.857 −2.199 +0.30
12 0.77315 0.77325 NH 2.444 −3.285 −1.857 −2.199 +1.06
13 0.773195 0.773205 NH 2.444 −3.285 −1.857 −2.199 +3.25
14 0.7731995 0.7732005 NH 2.444 −3.285 −1.857 −2.199 +8.34
15 0.7688 0.7776 IH 2.444 −3.285 −1.857 −1.400 +1.53
16 0.7688 0.7776 IH 2.444 −3.285 −1.857 −0.700 +1.14
17 0.7688 0.7776 IH 2.444 −3.285 −1.857 +0.000 +0.83
18 0.7688 0.7776 IH 2.444 −3.285 −1.857 +0.700 +1.11
19 0.7688 0.7776 IH 2.444 −3.285 −1.857 +1.400 +1.55
20 0.7776 0.7688 IH 2.444 −3.285 −1.857 −2.199 −1.38
21 0.7776 0.7688 IH 2.444 +3.285 +1.857 +2.199 +1.38
22 0.7732−0.5∗10-8 0.7732+0.5∗10-8 IH 2.444 −5.3 −1.857 −2.199 ∼1.50
23 0.7732−0.5∗10-13 0.7732+0.5∗10-13 IH 2.444 −1.5 −1.857 −2.199 ∼0.98


For case 1, we also show the solution as a function of the temperature, in accordance with figs. 1-2 of 1711.08469. The columns are explained on the first rows of the file:

In addition, the evolution of various components of the density matrix is given, for five characteristic values of the comoving momentum (k/T ≈ 1,2,3,4,5):