Baryogenesis through neutrino oscillations: A unified perspective

Baryogenesis through neutrino oscillations is an elegant mechanism that has found several realizations in the literature corresponding to different parts of the model parameter space. Its appeal stems from its minimality and dependence only on physics below the weak scale. In this paper we show that by focusing on the physical time scales of leptogenesis instead of the model parameters, a more comprehensive picture emerges. The different regimes previously identified can be understood as different relative orderings of these time scales. This approach also shows that all regimes require a coincidence of time scales and this in turn translates to a certain tuning of the parameters, whether in mass terms or Yukawa couplings. Indeed, we show that the amount of tuning involved in the minimal model is never less than one part in $10^5$ according to a metric constructed from a combination of the sterile neutrino mass degeneracy and the Barbieri-Giudice tuning of the Yukawa coupling. Finally, we explore an extended model, where the tuning can be removed in exchange for the introduction of a new degree of freedom in the form of a leptophilic Higgs with a vacuum expectation value of the order of GeV.

Figure 1

The basic stages leading to the creation of a total lepton asymmetry from left to right: out-of-equilibrium scattering of LH leptons begin to populate the sterile neutrino abundance at order $\mathcal{O}(|F{|}^2)$; after some time of coherent oscillation, a small fraction of the sterile neutrinos scatter back into LH leptons to create an asymmetry in individual lepton flavors at order $\mathcal{O}(|F{|}^4)$; finally, at order $\mathcal{O}(|F{|}^6)$, a total lepton asymmetry is generated due to a difference in scattering rate into sterile neutrinos among the different active flavors.

Figure 2

Time evolution of the individual lepton flavor asymmetries and the total lepton asymmetry in different parameter regimes. The dashed vertical line indicates the electroweak phase transition. (Top) Regime I: The Yukawa couplings are small enough that washout effects are always irrelevant, (Center) Regime II: The Yukawa couplings are large enough that equilibration of the sterile neutrinos occurs at the electroweak phase transition, (Bottom) Regime III: Large lepton flavor dependence in the $L\to N^{†}$ rates, such that $\mathrm{\Gamma }(L_e\to N^{†})\ll \mathrm{\Gamma }(L_{\mu }\to N^{†})$, $\mathrm{\Gamma }(L_{\tau }\to N^{†})$. The Yukawa couplings are even larger than in regime II, such that $L_{\tau }$ and $L_{\mu }$ equilibrate completely with the sterile neutrinos at $T\gg T_{\mathrm{W}}$, and $L_e$ comes into equilibrium at $T\sim T_{\mathrm{W}}$.

Figure 3

Representative Feynman diagrams for sterile neutrino creation through (Top) $1\to 2$ processes, (Center) gauge boson $2\to 2$ scattering, (Bottom) quark $2\to 2$ scattering.

Figure 4

Illustration of the importance of flavor-dependent effects in generating a nonzero baryon asymmetry in regime I. The baryon asymmetry is plotted as a function of the ratio between the muon-sterile neutrino scattering rate and the corresponding tau-sterile rate. The asymmetry is maximized when the washout rates for $\mu$ and $\tau$ are substantially different, as predicted by Eq. (6). The mass splittings are $\mathrm{\Delta }{M}_N=3\times 10^{-8}$ (blue, short dash), $10^{-7}$ (purple, solid), $3\times 10^{-7}\mathrm{GeV}$ (black, long dash). The ratio of muon to tau rates is varied by changing the relative values of $\delta -\eta$, while the overall MNS $CP$ phase and other parameters are held fixed ($M_N=1\mathrm{GeV}$, $\omega =\pi /4-i/2$, $\delta +\eta =\pi /2$). The horizontal dashed line indicates the observed baryon asymmetry.

Figure 5

Baryon asymmetry (Bottom) and lepton washout rates at $T=140\mathrm{GeV}$ (Top) as a function of the Yukawa coupling magnitude, which is parametrized by $\mathrm{Im}\omega$. The plot shows the continuous evolution from regime I to regime II. The washout rates are shown for taus (top), muons, and electrons (bottom); the dashed line in the upper plot shows the Hubble scale at $T_{\mathrm{W}}=140\mathrm{GeV}$, while the dashed line in the lower plot shows the observed $Y_{\mathrm{\Delta }B}=8.6\times 10^{-11}$. The asymmetry is maximized around $\mathrm{\Gamma }(L_e\to N^{†})\sim \mathrm{\Gamma }(L_{\mu }\to N^{†})\approx H$. Other parameters held fixed: $M_N=1\mathrm{GeV}$, $\mathrm{\Delta }{M}_N=10^{-5}\mathrm{GeV}$, $\eta =-\pi /4$, $\delta =3\pi /4$, $\mathrm{Im}\omega >0$.

Figure 6

Baryon asymmetry (Bottom) and lepton washout rates at $T=140\mathrm{GeV}$ (Top) as a function of the Yukawa coupling magnitude, which is parametrized by $\mathrm{Im}\omega$. The plot shows the continuous evolution from regime I to regime III for parameters exhibiting strong destructive interference in $\mathrm{\Gamma }(L_e\to N^{†})$. The curves are the same as Fig. 5. The asymmetry is again maximized around $\mathrm{\Gamma }(L_e\to N^{†})\approx H$. Other parameters held fixed: $M_N=1\mathrm{GeV}$, $\mathrm{\Delta }{M}_N=10^{-3}\mathrm{GeV}$, $\eta =\delta =-\pi /4$, $\mathrm{Im}\omega >0$.

Figure 7

Fine-tuning required to obtain the observed baryon asymmetry, $Y_{\mathrm{\Delta }B}=8.6\times 10^{-11}$ according to the tuning measure Eq. (30). The other parameters used are (black, solid) $\eta =\delta =\mathrm{Re}\omega =\pi /4$; (purple, dotted) $\eta =\delta =-\mathrm{Re}\omega =-\pi /4$; (blue, dashed) $\eta =2.42$, $\delta =0.5$, $\mathrm{Re}\omega =\pi /2$. For all points, $\mathrm{\Delta }{M}_N$ is fixed by imposing the observed baryon asymmetry.

Figure 8

(Top) Ratio of the baryon asymmetry in the leptophilic Higgs model with $\mathrm{Im}\omega =1$ to the asymmetry in the minimal, single Higgs model with $\mathrm{Im}\omega$ set such that the $L_e$ washout rate is the same in both scenarios for each data point. The enhancement to the asymmetry from changing $⟨{\mathrm{\Phi }}_2⟩$ is quadratically larger than the tuned, minimal model. (Bottom) Baryon asymmetry in the leptophilic Higgs model (top, purple) and minimal model (bottom, black) with the $L_e$ washout rate the same in both scenarios for each data point. The dashed line indicates the observed baryon asymmetry of the universe. (Both) Other parameters are fixed at $M_N=1\mathrm{GeV}$, $\mathrm{\Delta }{M}_N=1\mathrm{GeV}$, $\eta =\delta =-\mathrm{Re}\omega =-\pi /4$.

Figure 9

Signal strength of SM Higgs decay to ${\tau }^{+}{\tau }^{-}$ as a function of the leptophilic Higgs mass $h_{\ell }$ with $\tan \beta =20$. The enhancement of the ${\tau }^{+}{\tau }^{-}$ signal strength comes from the modification of the SM Higgs coupling to taus (38). The horizontal solid line is the current CMS $7+8\mathrm{TeV}$ $2\sigma$ bound [30], and the horizontal dashed (dotted) lines show the $2\sigma$ reach for LHC14 at $300{\mathrm{fb}}^{-1}$ (ILC at 250 GeV, $250{\mathrm{fb}}^{-1}$). The reach estimates are from [31].

Figure 10

Feynman diagram for production of the leptophilic Higgs states at the LHC and their decays.

Figure 11

Temperature dependence of the coefficient ${\gamma }^{\text{av}}(T)$ in the sterile neutrino production rate as defined in Eq. (23). We show the contributions from $1\leftrightarrow 2$ scattering, $2\to 2$ scattering, and the total leading order rate. All values are taken from [24].