Hidden-sector neutrinos and freeze-in leptogenesis

Sterile neutrinos at the GeV scale can resolve several outstanding problems of the Standard Model (SM), such as the source of neutrino masses and the origin of the baryon asymmetry through freeze-in leptogenesis. However, they can be challenging to detect experimentally due to their small couplings to SM particles. In extensions of the SM with new interactions of the sterile neutrinos, they can be produced copiously at accelerators and colliders. We systematically investigate the impact of such novel interactions on the asymmetry from freeze-in leptogenesis. We find that the interactions tend to bring the sterile neutrinos into equilibrium at early times, leading to a significant reduction in the generated asymmetry. We also show that observable rates of several hidden-sector neutrino signatures, such as SM Higgs decays to pairs of sterile neutrinos, can be inconsistent with the observed baryon asymmetry and provide an opportunity to falsify freeze-in leptogenesis.

Figure 1

Feynman diagram illustrating one of the physical processes underlying freeze-in leptogenesis. The decays of a SM Higgs create coherent superpositions of RHN mass eigenstates, $N_I$, a subset of which subsequently annihilate back into a SM Higgs. The rate of the net process $L_{\beta }{H}^{*}\to L_{\alpha }{H}^{*}$ can differ from the $CP$-conjugate process when propagation phases are taken into account, giving rise to asymmetries in individual lepton flavors. These asymmetries are subsequently processed into a total baryon asymmetry by flavor-dependent washout effects.

Figure 2

Dimensionless equilibration times for the RHNs, $z_{\mathrm{eq}}$, as a function of the dark Higgs coupling to RHNs, $y$ and $M_{\varphi }=0$ (left panel); $M_{\varphi }=100\mathrm{GeV}$ (right panel). We have assumed that $\varphi$ is always in equilibrium with the SM. The different blue contours represent different values of the dark Higgs-SM Higgs quartic coupling: $\lambda =1$ (solidline); $\lambda ={10}^{-1}$ (dashedline); $\lambda ={10}^{-2}$ (dottedline). To facilitate comparison with the oscillation timescale $z_{\mathrm{osc}}$, we have indicated in red the values of $z_{\mathrm{osc}}$ corresponding to RHN squared mass splittings of $\mathrm{\Delta }{M}_{21}^2=10{\mathrm{keV}}^2$ (top) and $\mathrm{\Delta }{M}_{21}^2=1{\mathrm{GeV}}^2$ (bottom). We see that couplings $y\gtrsim {10}^{-4}$ cause the RHNs to come into equilibrium prior to oscillation and asymmetry generation for the indicated values of $\mathrm{\Delta }{M}_{21}^2$.

Figure 3

Abundances and asymmetries as a function of dimensionless time, $z$, for various couplings $y$ between RHNs and the dark scalar, $\varphi$. We have fixed $\lambda =0.1$, $M_{\varphi }=1\mathrm{GeV}$, $M_1=1\mathrm{GeV}$, $\mathrm{\Delta }M=3\times {10}^{-8}\mathrm{GeV}$, and the Yukawa couplings $F^{(I)}$ indicated in Eq. (28). In the left panel, the dashed lines indicate $\max (Y_N)$, and the solid lines indicate the asymmetry in the anomaly-free electron number, $\mathrm{\Delta }{Y}_e\equiv \mathrm{\Delta }{Y}_{B/3-L_e}$. We take the following values of $y$: $y=0$ (red lines), corresponding to the standard ARS scenario; $y=6\times {10}^{-6}$ (black lines); and $y={10}^{-5}$ (blue lines). It is evident that asymmetry generation is suppressed once $Y_N$ comes into equilibrium. Right panel: electron flavor asymmetry $\mathrm{\Delta }{Y}_e$ (solid lines) and total asymmetry $\mathrm{\Delta }{Y}_{B-L}$ (dotted lines) for various values of $y$: $y=0$ (top, red); $y=6\times {10}^{-6}$ (middle, black); and $y={10}^{-5}$ (bottom, blue). While the flavor asymmetries stop growing when the RHNs equilibrate, the reprocessing of the flavor asymmetries into a total $B-L$ asymmetry persists at later times.

Figure 4

Electron flavor asymmetry, $\mathrm{\Delta }{Y}_e$, as a function of dimensionless time, $z$, for various RHN mass splittings $\mathrm{\Delta }M$ in GeV as indicated in the figure. We take $y={10}^{-5}$, with other parameters the same as in Fig. 3. We see that increasing $\mathrm{\Delta }M$ enhances the asymmetry at earlier times, which leads to a larger final asymmetry when oscillations are interrupted by RHN equilibration. When $\mathrm{\Delta }M$ is sufficiently large that $z_{\mathrm{osc}}<z_{\mathrm{eq}}$, however, the RHNs oscillate too early, and the asymmetry saturates at smaller values for larger splittings.

Figure 5

Electron flavor asymmetry, $\mathrm{\Delta }{Y}_e$, shown as a function of $y$ (left) for the indicated values of $\lambda$, and $\lambda$ (right) for the indicated values of $y$. Other parameters are the same as in Fig. 3. The points are the lepton-flavor asymmetries obtained from numerically solving the quantum kinetic equations, while the dashed lines indicate a $y^{-10}$ (left) power-law dependence and ${\lambda }^{-5}$ (right) power-law dependence.

Figure 6

Left panel: electron flavor asymmetry, $\mathrm{\Delta }{Y}_e$, shown as a function of $\mathrm{\Delta }M$. We have fixed $M_{\varphi }=1\mathrm{GeV}$, $M_1=1\mathrm{GeV}$, $\lambda =0.1$, $y={10}^{-5}$, and the Yukawa couplings $F^{(I)}$ indicated in Eq. (28). The points are the flavor asymmetries obtained from numerically solving the quantum kinetic equations, while the dashed lines indicate $\mathrm{\Delta }M$ and $\mathrm{\Delta }{M}^{-0.6}$ power-law dependence to facilitate comparison with analytic results. Right panel: dependence of $\mathrm{\Delta }{Y}_e$ on tree-level dark scalar mass, $M_{\varphi }$. We set $\mathrm{\Delta }M=3\times {10}^{-8}\mathrm{GeV}$ and otherwise keep all parameters apart from $M_{\varphi }$ the same.

Figure 7

Flavor and total $B-L$ asymmetries with $F^{(II)}$ defined in Eq. (29), and other parameters set to $\mathrm{\Delta }M=1.5\times {10}^{-3}\mathrm{GeV}$, $M_{\varphi }=1\mathrm{GeV}$, $\lambda =0.1$, and $y=3\times {10}^{-5}$. The final muon and tau asymmetries are suppressed by washout, but the electron asymmetry is protected, leading to a large $B-L$ asymmetry of comparable size to $\mathrm{\Delta }{Y}_e$. This maximizes the asymmetry when the Yukawa couplings, $F$, are large enough to be in the strong washout regime.

Figure 8

Left panel: baryon asymmetry as a function of $y$ obtained using the Yukawa texture approximately equal to $F^{(II)}$ from Eq. (29) (although with $\mathrm{Im}\omega$ adjusted slightly for each point), which gives the largest asymmetry in the strong washout regime by protecting the asymmetry in a single lepton flavor. We set $M_{\varphi }=1\mathrm{GeV}$, and for each value of $y$ and $\lambda$, we optimize the parameters $\mathrm{\Delta }M$ and $\mathrm{Im}\omega$ to give the maximum baryon asymmetry. The dashed lines indicate a $y^{-4}$ power-law dependence, and we find good agreement with the analytic prediction of Eq. (19). The observed baryon asymmetry is indicated with the dot-dashed line. For comparison, the optimized ARS asymmetry for this benchmark is $\mathrm{\Delta }{Y}_B=2\times {10}^{-7}$, and the dip seen around $y=5\times {10}^{-6}$ is due to flavor effects. Right panel: optimal $\mathrm{\Delta }M$ as a function of $y$, demonstrating that larger mass splittings (and earlier oscillation times) are preferred for larger couplings. The dashed lines indicate a $y^6$ power-law dependence to compare numerical results with the analytic prediction of Eq. (18).

Figure 9

Abundances (upper plot) and temperatures (lower plot) as functions of dimensionless time $z$, expressed as ratios to the values when in equilibrium with the SM, for $\varphi$ (dashed lines) and $N$ (solid lines). Here, we show benchmarks for which the hidden sector comes to local equilibrium before equilibrating with the SM. We take $\lambda =5\times {10}^{-6}$, $M_{\varphi }=M_N=1\mathrm{GeV}$, and consider two values of $y$: 0.01 (blue) and 0.5 (red). Note that when the hidden sector establishes local equilibrium, the abundances and temperatures rapidly approach the values predicted by the dark temperature, Eq. (43).

Figure 10

Abundances (upper plot) and temperatures (lower plot) as functions of dimensionless time $z$, expressed as ratios to the values when in equilibrium with the SM, for $\varphi$ (dashed lines) and $N$ (solid lines). Here, we show benchmarks for which $N$ equilibrates with $\varphi$ after $\varphi$ is already in equilibrium with the SM. We take $\lambda =3\times {10}^{-3}$, $M_{\varphi }=M_N=1\mathrm{GeV}$, and consider two values of $y$: ${10}^{-4}$ (blue) and $2\times {10}^{-2}$ (red).

Figure 11

Dimensionless time at which $N$ equilibrates with $\varphi$, $z_{\mathrm{h}.\mathrm{s}.\mathrm{eq}}$, defined as the time at which $T_N=0.9T_{\varphi }$. The solid curves show numerical results for the indicated values of $\lambda$, while the dashed lines show a $y^{-2}$ power-law dependence to facilitate comparison with analytic arguments provided in the text.

Figure 12

Dimensionless time dependence of the RHN abundance, $Y_N$, for $\lambda =0.5$ and the indicated values of $y$. The solid lines show the full solutions of the Boltzmann equations from Sec. 5a, while the dashed lines show the solutions to the Boltzmann equations where we have constrained $T_N=T_{\varphi }=T$ and $Y_{\varphi }=Y_{\varphi }^{\mathrm{eq}}$.

Figure 13

Dimensionless RHN equilibration time, $z_{\mathrm{eq}}$, defined such that $Y_N(z_{\mathrm{eq}})=0.9Y_N^{\mathrm{eq}}$ as a function of $y$ with $\lambda =0.5$. The solid lines show the full solutions of the Boltzmann equations from Sec. 5a, while the dashed lines show the solutions to the Boltzmann equations where we have constrained $T_N=T_{\varphi }=T$ and $Y_{\varphi }=Y_{\varphi }^{\mathrm{eq}}$. Both solutions exhibit a $z_{\mathrm{eq}}\propto y^{-2}$ dependence.

Figure 14

Dimensionless-time evolution of (top panel) hidden-sector abundances, (bottom panel) electron flavor and $B-L$ asymmetries, for $M_{\varphi }=1\mathrm{GeV}$, $\mathrm{\Delta }M=3\times {10}^{-8}\mathrm{GeV}$, $\lambda =5\times {10}^{-6}$, $y=5\times {10}^{-3}$, and SM Higgs coupling $F^{(I)}$ from Eq. (28).

Figure 15

Baryon asymmetry including the full treatment of hidden-sector equilibration for $M_{\varphi }=1\mathrm{GeV}$, $\mathrm{\Delta }M=3\times {10}^{-8}\mathrm{GeV}$, SM Higgs coupling $F^{(I)}$, and the indicated values of $\lambda$.

Figure 16

Left panel: baryon asymmetry as a function of $y$ obtained using the Yukawa texture $F^{(II)}$ from Eq. (29) including the full treatment of hidden-sector equilibration. We set $M_{\varphi }=1\mathrm{GeV}$, and for each value of $y$ and $\lambda$, we optimize the mass splitting to give the maximum baryon asymmetry. The dot-dashed line indicates the observed baryon asymmetry. The ARS asymmetry corresponding to $y=0$ is $\mathrm{\Delta }{Y}_B=2\times {10}^{-7}$.

Figure 17

SM Higgs branching fraction to RHN pairs, summed over two RHN flavors, with $M_{\varphi }=15\mathrm{GeV}$, $M_N=5\mathrm{GeV}$, and assuming $M_N=y{v}_{\varphi }$. The red shaded region corresponds to parameters incompatible with freeze-in leptogenesis, while the blue shaded region indicates where the mixing angle $\theta$ exceeds the level at which it is constrained by Higgs coupling measurements and direct searches.

Figure 18

SM Higgs branching fraction to RHN pairs, summed over two RHN flavors, with $M_{\varphi }=15\mathrm{GeV}$ and $v_{\varphi }$ treated as a free parameter (unrelated to RHN masses). The coupling $y$ is set to the largest value consistent with freeze-in leptogenesis, while the blue shaded region indicates where the mixing angle $\theta$ exceeds the level at which it is constrained by Higgs coupling measurements and direct searches.

Figure 19

SM Higgs product branching fraction $\mathrm{BF}(h\to \varphi \varphi )\mathrm{BF}(\varphi \to NN{)}^2$, with $M_{\varphi }=15\mathrm{GeV}$, $M_N=5\mathrm{GeV}$, and $v_{\varphi }=M_N/y$. The red and blue shaded regions are the same as in Fig. 17.

Figure 20

SM Higgs product branching fraction $\mathrm{BF}(h\to \varphi \varphi )\mathrm{BF}(\varphi \to NN{)}^2$, with $M_{\varphi }=15\mathrm{GeV}$, and $v_{\varphi }$ treated as a free parameter (unrelated to RHN masses). The coupling $y$ is set to the largest value consistent with freeze-in leptogenesis. The red region indicates parameters that are ruled out by measurements of the SM Higgs width, while the blue region indicates where the mixing angle $\theta$ exceeds the level at which it is constrained by Higgs coupling measurements and direct searches.

Figure 21

The $B$-meson product branching fraction $\mathrm{BF}(B\to K\varphi )\mathrm{BF}(\varphi \to NN)$, with $M_{\varphi }=2\mathrm{GeV}$, $M_N=0.5\mathrm{GeV}$, and $v_{\varphi }=M_N/y$. The red and blue shaded regions are the same as in Fig. 17.

Figure 22

The $B$-meson product branching fraction $\mathrm{BF}(B\to K\varphi )\mathrm{BF}(\varphi \to NN)$, with $M_{\varphi }=2\mathrm{GeV}$, $M_N=0.5\mathrm{GeV}$, and $v_{\varphi }$ treated as a free parameter (unrelated to RHN masses). The red and blue shaded regions are the same as in Fig. 20.

Figure 23

Ratio of the asymmetry factor with correct momentum-averaged asymmetry, ${\mathcal{A}}^{(\mathrm{full})}(z)$, compared to the corresponding factor $\mathcal{A}(z)$ from Eq. (12) with naive momentum averaging of the oscillation phase. We plot this ratio as a function of $⟨z_{\mathrm{osc}}⟩/z_{\mathrm{eq}}$ for $z_{\mathrm{eq}}=0.01$, although the curve looks identical for other values of $z_{\mathrm{eq}}$. We find that in the limit where oscillations occur after equilibration, the correct asymmetry is a factor of 7.5 larger than the naive averaging prediction, whereas the two methods agree within 15% for $⟨z_{\mathrm{osc}}⟩=z_{\mathrm{eq}}$.

Figure 24

Baryon asymmetry from freeze-out leptogenesis as a function of $y$ with initial condition $(R_N{)}_{IJ}(0)={\delta }_{IJ}$. We fix $\lambda =0.1$, $M_{\varphi }=10\mathrm{GeV}$, and $\mathrm{\Delta }{M}_N=2\times {10}^{-11}\mathrm{GeV}$, and the Yukawa coupling $F^{(I)}$ from Eq. (28).