Dark matter, baryogenesis and neutrino oscillations from right-handed neutrinos

We show that, leaving aside accelerated cosmic expansion, all experimental data in high energy physics that are commonly agreed to require physics beyond the Standard Model can be explained when completing the model by three right-handed neutrinos that can be searched for using present-day experimental techniques. The model that realizes this scenario is known as the Neutrino Minimal Standard Model ($\nu \mathrm{MSM}$). In this article we give a comprehensive summary of all known constraints in the $\nu \mathrm{MSM}$, along with a pedagogical introduction to the model. We present the first complete quantitative study of the parameter space of the model where no physics beyond the $\nu \mathrm{MSM}$ is needed to simultaneously explain neutrino oscillations, dark matter, and the baryon asymmetry of the Universe. The key new point of our analysis is leptogenesis after sphaleron freeze-out, which leads to resonant dark matter production, thus evading the constraints on sterile neutrino dark matter from structure formation and x-ray searches. This requires one to track the time evolution of left- and right-handed neutrino abundances from hot big bang initial conditions down to temperatures below the QCD scale. We find that the interplay of resonant amplifications, $CP$-violating flavor oscillations, scatterings, and decays leads to a number of previously unknown constraints on the sterile neutrino properties. We furthermore reanalyze bounds from past collider experiments and big bang nucleosynthesis in the face of recent evidence for a nonzero neutrino mixing angle ${\theta }_{13}$. We combine all our results with existing constraints on dark matter properties from astrophysics and cosmology. Our results provide a guideline for future experimental searches for sterile neutrinos. A summary of the constraints on sterile neutrino masses and mixings has appeared in Canetti et al. [Phys. Rev. Lett. 110, 061801 (2013)]. In this article we provide all details of our calculations and give constraints on other model parameters.

Figure 1

The thermal history of the Universe in the $\nu \mathrm{MSM}$.

Figure 2

Different constraints on $N_1$ mass and mixing. The blue region is excluded by x-ray observations, and the dark gray region $M_1<1\mathrm{keV}$ by the Tremaine-Gunn bound [114, 115, 137]. The points on the upper solid black line correspond to observed ${\Omega }_{\mathrm{DM}}$ produced in scenario I in the absence of lepton asymmetries (for ${\mu }_{\alpha }=0$) [19]; points on the lower solid black line give the correct ${\Omega }_{\mathrm{DM}}$ for $|{\mu }_{\alpha }|=1.24\times {10}^{-4}$, the maximal asymmetry we found. The region between these lines is accessible for $0\le |{\mu }_{\alpha }|\le 1.24\times {10}^{-4}$. We do not display bounds derived from ${\mathrm{Ly}}_{\alpha }$ forest observations because they depends on ${\mu }_{\alpha }$ in a complicated way, and the calculation currently includes considerable uncertainties [90].

Figure 3

Number of effective relativistic degrees of freedom $g_{*}$ as a function of temperature [166].

Figure 4

Contributions to the $N_I$ self-energies. Diagram (a) dominates for $T<v$, and diagram (b) for $T>v$. ${\Gamma }_N$ is obtained from the discontinuity of the diagrams [205], which can be computed by cutting it in various ways [167]. The gray self-energy blobs indicate that dressed lepton and Higgs propagators have to be used. Cuts through them reveal a large number of processes, which are summarized in Refs. 8, 24 and Appendix A of Ref. 6. Recently, it has been pointed out that current estimates suffer from an error $\mathcal{O}(1)$ due to infrared and collinear enhancements at high temperatures. Systematic approaches to include these effects can be found in Refs. 176, 206 for $T>M$ (relevant for baryogenesis) and [177, 207, 208] for $M>T$ (relevant for late-time asymmetries). We ignore this effect in our current study, as it is comparable to other uncertainties in the kinetic equations and would only slightly change the results for the relevant regions in the $\nu \mathrm{MSM}$ parameter space.

Figure 5

The functions $R^{(S)}(T,M)$ and $R_M^{(S)}(T,M)$ for $M=1.2\mathrm{GeV}$ (darkest curve), $M=1.6\mathrm{GeV}$, $M=2\mathrm{GeV}$, $M=2.5\mathrm{GeV}$, $M=3\mathrm{GeV}$, $M=3.5\mathrm{GeV}$, $M=4\mathrm{GeV}$ (lightest curve).

Figure 6

Values of $\Delta M$ and $\mathrm{Im}\omega$ that lead to the observed baryon asymmetry in scenarios I and II for different sterile neutrino masses $M=10$, 100 MeV, 1 and 10 GeV. The blue (shortest dashed) line corresponds to $M=10\mathrm{MeV}$, red (short dashed) to $M=100\mathrm{MeV}$, brown (long dashed) to $M=1\mathrm{GeV}$ and green (longest dashed) to $M=10\mathrm{GeV}$. The phases that maximize the asymmetry differ significantly for $\mathrm{Im}\omega \approx 0$ and away from that region. In the region $0.5<e^{\mathrm{Im}\omega }<1.5$ we choose ${\alpha }_2=\pi$, $\delta =0$, $\mathrm{Re}\omega =\frac{7}{10}\pi$ for normal hierarchy and ${\alpha }_2-{\alpha }_1=\pi$, $\delta =\pi$, $\mathrm{Re}\omega =\frac{3}{4}\pi$ for inverted hierarchy. Everywhere else we choose $\delta =\frac{3}{20}\pi$, $\mathrm{Re}\omega =\frac{1}{2}\pi$ for normal hierarchy and ${\alpha }_2-{\alpha }_1=\frac{11}{10}\pi$, $\delta =\frac{11}{20}\pi$, $\mathrm{Re}\omega =\frac{4}{5}\pi$ for inverted hierarchy. The upper panel shows the results for normal hierarchy, the lower panel for inverted hierarchy.

Figure 7

Constraints on the $N_{2,3}$ masses $M_{2,3}\simeq M$ and mixing $U^2=\mathrm{tr}({\theta }^{†}\theta )$ from baryogenesis in scenarios I and II: upper panel—normal hierarchy; lower panel—inverted hierarchy. In the region between the solid blue “BAU” lines, the observed BAU can be generated. The regions below the solid black “seesaw” line and dashed black “BBN” line are excluded by neutrino oscillation experiments and BBN, respectively. The areas above the green lines of different shades are excluded by direct search experiments, as indicated in the plot. The solid lines are exclusion plots for all choices of $\nu \mathrm{MSM}$ parameters; for the dashed lines the phases were chosen to maximize the BAU, consistent with the blue lines.

Figure 8

Constraints on the $N_{2,3}$ masses $M_{2,3}\simeq M$ and lifetime ${\tau }^{-1}\simeq \frac{1}{2}\mathrm{tr}{\Gamma }_N{|}_{T=1\mathrm{MeV}}$ from baryogenesis in scenarios I and II. In the region between the blue “BAU” lines, the observed BAU can be generated (solid line—normal hierarchy; dotted line—inverted hierarchy). The region above the solid black “BBN” line is excluded by BBN.

Figure 9

Values of $\delta M$ and $\mathrm{Im}\omega$ that lead to the lepton asymmetry required for dark matter production in scenario I for different singlet fermion masses, $M=2.5$, 4, 7, and 10 GeV and for normal hierarchy. The upper left panel corresponds to $M=2.5\mathrm{GeV}$, the upper right panel to $M=4\mathrm{GeV}$, the lower left panel to $M=7\mathrm{GeV}$ and the lower right panel to $M=10\mathrm{GeV}$. The phases that maximize the asymmetry differ significantly for $\mathrm{Im}\omega \approx 0$ and away from that region. We choose the phases ${\alpha }_2=\frac{\pi }{2}$, $\delta =\frac{3}{2}\pi$ in the region $0.5<e^{\mathrm{Im}\omega }<1.5$ and ${\alpha }_2=\frac{\pi }{5}$ and $\delta =0$ everywhere else.

Figure 10

Values of $\delta M$ and $\mathrm{Im}\omega$ that lead to the lepton asymmetry required for dark matter production in scenario I for different singlet fermion masses, $M=2.5$, 4, 7, and 10 GeV and inverted hierarchy. The upper left panel corresponds to $M=2.5\mathrm{GeV}$, the upper right panel to $M=4\mathrm{GeV}$, the lower left panel to $M=7\mathrm{GeV}$ and the lower right panel to $M=10\mathrm{GeV}$. The phases that maximize the asymmetry differ significantly for $\mathrm{Im}\omega \approx 0$ and away from that region. We choose ${\alpha }_2-{\alpha }_1=\frac{7}{5}$ and $\delta =\frac{3}{5}\pi$ in the region $0.5<e^{\mathrm{Im}\omega }<1.5$ and ${\alpha }_2-{\alpha }_1=0$, $\delta =\frac{9}{10}\pi$ everywhere else.

Figure 11

Constraints on the $N_{2,3}$ masses $M_{2,3}\simeq M$ and mixing $U^2=\mathrm{tr}({\theta }^{†}\theta )$ in scenario I. The lepton asymmetry at $T=100\mathrm{MeV}$ can be large enough that the resonant enhancement of $N_1$ production is sufficient to explain the observed ${\Omega }_{\mathrm{DM}}$ inside the dashed blue and red lines for normal and inverted neutrino mass hierarchy, respectively. The regions below the “seesaw” lines are excluded by neutrino oscillation experiments for the indicated choice of hierarchy.

Figure 12

Estimate of the maximal lepton asymmetry that can be created in scenario I around $T=100\mathrm{MeV}$. The blue small dashed line corresponds to normal hierarchy, the red long dashed line to inverted hierarchy. This plot was generated using the maximal values found in the data files used to produce Figs. 9, 10.

Figure 13

Constraints on the $N_{2,3}$ masses $M_{2,3}\simeq M$ and mixing $U^2=\mathrm{tr}({\theta }^{†}\theta )$ in the constrained $\nu \mathrm{MSM}$ (scenario I): upper panel—normal hierarchy; lower panel—inverted hierarchy. In the region between the solid blue “BAU” lines, the observed BAU can be generated. The lepton asymmetry at $T=100\mathrm{MeV}$ can be large enough that the resonant enhancement of $N_1$ production is sufficient to explain the observed ${\Omega }_{\mathrm{DM}}$ inside the solid red “DM” line. The $CP$-violating phases were chosen to maximize the asymmetry at $T=100\mathrm{MeV}$. The regions below the solid black “seesaw” line and dashed black “BBN” line are excluded by neutrino oscillation experiments and BBN, respectively. The areas above the green lines of different shades are excluded by direct search experiments, as indicated in the plot. The solid lines are exclusion plots for all choices of $\nu \mathrm{MSM}$ parameters; for the dashed lines the phases were chosen to maximize the late-time asymmetry, consistent with the red line.

Figure 14

Same as Fig. 13, but with a different set of $CP$-violating phases. The plot illustrates how the “BAU” line moves when the phases are changed; it can cross through points deep inside the maximal “BAU” region, while the phase still allows for DM production. The sign of the BAU is opposite in the two disjoint “BAU” regions.

Figure 15

Constraints on the $N_{2,3}$ masses $M_{2,3}\simeq M$ and lifetime ${\tau }^{-1}\simeq \frac{1}{2}\mathrm{tr}{\Gamma }_N{|}_{T=1\mathrm{MeV}}$ in the constrained $\nu \mathrm{MSM}$ (scenario I). In the region between the blue “BAU” lines, the observed BAU can be generated. The lepton asymmetry at $T=100\mathrm{MeV}$ can be large enough that the resonant enhancement of $N_1$ production is sufficient to explain the observed ${\Omega }_{\mathrm{DM}}$ inside the red “DM” line. The $CP$-violating phases were chosen to maximize the asymmetry at $T=100\mathrm{MeV}$. Solid lines—normal hierarchy; dotted lines—inverted hierarchy.

Figure 16

Constraints on the $N_{2,3}$ masses $M_{2,3}\simeq M$ and parameter $\mathrm{Im}\omega$ in the constrained $\nu \mathrm{MSM}$ (scenario I): upper panel—normal hierarchy; lower panel—inverted hierarchy. In the region between the solid blue “BAU” lines, the observed BAU can be generated. The lepton asymmetry at $T=100\mathrm{MeV}$ can be large enough that the resonant enhancement of $N_1$ production is sufficient to explain the observed ${\Omega }_{\mathrm{DM}}$ inside the solid red “DM” line. The $CP$-violating phases were chosen to maximize the asymmetry at $T=100\mathrm{MeV}$.

Figure 17

The Higgs expectation value as a function of temperature for $m_H=126\mathrm{GeV}$.