Imaginary part of the active neutrino self-energy at T < 10 GeV

Takehiko Asaka, Mikko Laine and Mikhail Shaposhnikov

The files below contain data for the active neutrino interaction rate (or width, or damping rate), at T < 10 GeV, according to sec.3 of hep-ph/0612182. We have prepared a documentation file explaining the notation used and showing a number of plots for the interaction rate, both with and without the inclusion of hadronic effects.

It may be noted that at 5 GeV < T < 160 GeV the active neutrino interaction rate becomes simpler, because fermion masses can be largely omitted; the corresponding computation has been presented in 1605.07720 and is illustrated in fig.6 of that paper. Numerical results for the high-temperature regime (with M1 = 0) can be downloaded here. In the high-temperature regime it is not sufficient to use the Fermi model for the computation, and in fact other processes start to be important already at T ~ 3 GeV; we return to this further below.

It may also be noted that, as explained in hep-ph/0605209, the computation at T < 10 GeV contains substantial hadronic uncertainties. For illustration, results from two different computations are compared with each other here.

We now proceed to listing the date files. In accordance with the notation of the documentation file, the file names below indicate the direction of projection [Q or u, as "hatIQ" or "hatIu"], the value of M1 [in MeV], and flavour [alpha]. The normalization of the results is explained in the documentation file. The columns of the files are:

1: T / MeV
2: q / T
3: rate

Here are the files for small masses M1 << T:

For reference we also provide results for the case that all hadronic effects are omitted (i.e. the number of colours is set to zero):

Finally, consider larger masses, of the order of T or more. In this case there are two qualitative changes to be noted

* For T << M1, the functions are dominated by a vacuum decay width (note that active neutrinos are not on-shell; the results correspond to the on-shell point of sterile neutrinos). This means that in the units employed, when dimensionful quantities are normalized to the temperature, the results diverge (our current data is inaccurate when M1 > 100 T).

* For T >> GeV, the 2-loop self-energy diagrams described by the Fermi model are taken over by 1-loop diagrams, whose cuts correspond to 2 <-> 1 processes, particularly the coalescence of two neutrinos into a Z boson, or the coalescence of a neutrino and a charged lepton into a W boson. When this happens, however, it is also true that the active neutrino self-energy is no longer gauge independent. The results presented below correspond to the Feynman R_xi gauge; gauge dependence becomes substantial at the temperature above which a sudden increase of the imaginary part is seen. In gauge independent quantities, like the imaginary part of the right-handed neutrino self-energy, this gauge dependence is cancelled against other processes, such as (in Feynman gauge) the direct coalescence of a right-handed neutrino and an active neutrino into a Higgs or Z boson, or the coalescence of a right-handed neutrino and a charged lepton into a W boson.

With these remarks, here are the Feynman gauge results including 2 <-> 1 processes:

For the last set it is appropriate to repeat that our data becomes inaccurate when M1 > 100 T (T < 10 MeV).