Production of heavy sterile neutrinos from vector boson decay at electroweak temperatures

In the standard model extended with a seesaw mass matrix, we study the production of sterile neutrinos from the decay of vector bosons at temperatures near the masses of the electroweak bosons. We derive a general quantum kinetic equation for the production of sterile neutrinos and their effective mixing angles, which is applicable over a wide range of temperature, to all orders in interactions of the standard model and to leading order in a small mixing angle for the neutrinos. We emphasize the relation between the production rate and Landau damping at one-loop order and show that production rates and effective mixing angles depend sensitively upon the neutrino’s helicity. Sterile neutrinos with positive helicity interact more weakly with the medium than those with negative helicity, and their effective mixing angle is not modified significantly. Negative helicity states couple more strongly to the vector bosons, but their mixing angle is strongly suppressed by the medium. Consequently, if the mass of the sterile neutrino is $\lesssim 8.35\mathrm{MeV}$, there are fewer states with negative helicity produced than those with positive helicity. There is an Mikheyev-Smirnov-Wolfenstein-type resonance in the absence of lepton asymmetry, but due to screening by the damping rate, the production rate is not enhanced. Sterile neutrinos with negative helicity freeze out at $T_f^{-}\simeq 5\mathrm{GeV}$, whereas positive helicity neutrinos freeze out at $T_f^{+}\simeq 8\mathrm{GeV}$, with both distributions far from thermal. As the temperature decreases, due to competition between a decreasing production rate and an increasing mixing angle, the distribution function for states with negative helicity is broader in momentum and hotter than that for those with positive helicity. Sterile neutrinos produced via vector boson decay do not satisfy the abundance, lifetime, and cosmological constraints to be the sole dark matter component in the Universe. Massive sterile neutrinos produced via vector boson decay might solve the ${}^7\mathrm{Li}$ problem, albeit at the very edge of the possible parameter space. A heavy sterile neutrino with a mass of a few MeV could decay into light sterile neutrinos, of a few keV in mass, that contribute to warm dark matter. We argue that heavy sterile neutrinos with lifetime $\le 1/H_0$ reach local thermodynamic equilibrium.

Figure 1

One-loop contributions to the self-energy. The neutral current tadpole is proportional to the lepton (and quark) asymmetry.

Figure 2

${\gamma }^{-}(q)/M_W$ given by Eq. (4.30) vs $y=q/T$ for $\tau =M_W/T=1$, 2, 3 respectively.

Figure 3

${\gamma }^{+}(q)M_W/M_s^2$ given by Eq. (4.31) vs $y=q/T$ for $\tau =M_W/T=1$, 2, 3 respectively.

Figure 4

$I^{-}(\tau ,y)$ given by Eq. (4.32) vs $y=q/T$ for $\tau =M_W/T=1$, 2, 3 respectively.

Figure 5

$I^{+}(\tau ,y)$ given by Eq. (4.33) vs $y=q/T$ for $\tau =M_W/T=1$, 2, 3 respectively.

Figure 6

$J^{-}(\tau ,y)$ Eq. (5.4) vs $y=q/T$ for $\tau =M_W/T=1$, 2 respectively.

Figure 7

$J^{+}(\tau ,y)$ Eq. (5.9) vs $y=q/T$ for $\tau =M_W/T=1$, 2 respectively.

Figure 8

Rates $R^{-}(y,\tau )$ [see Eqs. (7.8) and (7.9)] vs $\tau$ for $y=0.5$, 1, 3 for $M_s/M_W={10}^{-4}$.

Figure 9

Rates $R^{-}(y,\tau )$ vs $y$ for $\tau =2$, 5, 10 for $M_s/M_W=10^{-}4$.

Figure 10

Asymptotic distribution function $F^{-}(y)$ [see Eqs. (7.10) and (7.11)] vs $y$.

Figure 11

$R^{+}(\tau ,y)$ given by Eq. (7.18) vs $y=q/T$ for $\tau =1$, 3, 5 respectively.

Figure 12

$R^{+}(y,\tau )$ vs $\tau$ for $y=1$, 3, 5.

Figure 13

Asymptotic distribution function $F^{+}(y)$ in Eq. (7.19) vs $y=q/T$.

Figure 14

Total distribution function $f(M_s,y)$ in Eq. (7.24) multiplied by $y^2$ vs $y=q/T$ for $M_s=1;10\mathrm{MeV}$.