Low-scale leptogenesis with minimal lepton flavor violation

We analyze the feasibility of low-scale leptogenesis where the inverse seesaw (ISS) and linear seesaw (LSS) terms are not simultaneously present. In order to generate the necessary mass splittings, we adopt a minimal lepton flavor violation (MLFV) hypothesis where a sterile neutrino mass degeneracy is broken by flavor effects. We find that resonant leptogenesis is feasible in both scenarios. However, because of a flavor alignment issue, MLFV-ISS leptogenesis succeeds only with a highly tuned choice of Majorana masses. For MLFV-LSS, on the other hand, a large portion of parameter space is able to generate sufficient asymmetry. In both scenarios we find that the lightest neutrino mass must be of order ${10}^{-2}\mathrm{eV}$ or below for successful leptogenesis. We briefly explore implications for low-energy flavor violation experiments, in particular $\mu \to e\gamma$. We find that the future MEG-II experiment, while sensitive to MLFV in our setup, will not be sensitive to the specific regions required for resonant leptogenesis.

Figure 1

Plot of the asymmetry generated in the ISS for the individual cases where no radiative corrections are considered (orange), where only ${\mathcal{N}}_1$ is considered (blue) and when both ${\mathcal{N}}_1$ and ${\mathcal{N}}_2$ are included (cyan). This was varied with ${\mu }_N$ (left) and ${\mu }_S$ (right). A resonant enhancement in the asymmetry occurs only if the next-to-leading order contributions are included. The red horizontal line indicates the asymmetry required to fit observations.

Figure 2

Plot of the asymmetry generated in the ISS when both ${\mathcal{N}}_1$ and ${\mathcal{N}}_2$ are included, as a function of ${\mu }_N$ (top left) and ${\mu }_S$ (top right). Points in purple correspond to the regime where ${\mu }_N>{\mu }_S$, while points in green to ${\mu }_S>{\mu }_N$. A resonant enhancement of sufficient size is generated for large values of ${\mu }_N$ as long as the hierarchy ${\mu }_N>{\mu }_S$ is satisfied. In order to match active neutrino data, ${\mu }_S$ which generates a tree-level contribution must be less than $\sim 100\mathrm{GeV}$, whereas ${\mu }_N$, which generates it at loop level, is allowed to be larger. The bottom figure compares the asymmetry generated from standard decay to oscillations (in orange) in the relevant regime ${\mu }_N>{\mu }_S$. While for some region of parameter space the two processes are of similar order, in the resonant regime the asymmetry due to oscillations is roughly 3 orders of magnitude smaller and is therefore a subdominant effect.

Figure 3

Plot of the decay width ${\mathrm{\Gamma }}_i\simeq \frac{m_i}{8\pi }\sum _l{h}_{li}^{*}{h}_{li}$ (blue) along with the mass splitting $\mathrm{\Delta }{m}_{i,j}$ (red) as a function of ${\mu }_N$ for ${\mu }_S={10}^{-4}\mathrm{GeV}$ and all radiative effects included. Here the mass splitting among opposite splitting SNs $\mathrm{\Delta }{m}_{i,j}^{\mathrm{OS}}$ (left) is compared to those with same sign splitting $\mathrm{\Delta }{m}_{i,j}^{\mathrm{SS}}$ (right) as defined in Eqs. (41), (40). While in both plots a region of resonance occurs, in the case on the left it occurs in a region of extremely strong washout; cf. Fig. 5. The inclusion of radiative effects allows for a resonance to occur between SNs with same-sign mass splittings in a region of minimized washout such that asymmetry generation can occur. In this region heavy SNs of opposite sign mass splitting grow in mass difference and their contribution becomes irrelevant.

Figure 4

Variation in the baryon asymmetry as a function of the lightest active neutrino mass $m_{{\nu }_1}$ (left) and varying the Wilson coefficient $n_1$ (right) for the case where all radiative spurion effects are included. In this scan we fixed ${\theta }_2^{\mathrm{c}}=0.7$, ${\mu }_S\simeq {10}^{-1}\mathrm{GeV}$ and ${\mu }_N\simeq 900\mathrm{GeV}$ and in the left plot set $n_1=1/16{\pi }^2$ and $m_{{\nu }_1}=0.01\mathrm{eV}$ in the right plot. As the lightest neutrino mass is reduced (becomes hierarchical) the asymmetry freezes out. There is a slight preference for light neutrino masses of $\mathcal{O}({10}^{-3}–{10}^{-2})\mathrm{eV}$. Larger values for $m_{{\nu }_1}$ lead to a degeneracy amongst the light neutrino masses and decreases the mass splitting generated by the inclusion of ${\mathcal{N}}_i$ decreasing the resonant enhancement. Increasing the size of the Wilson coefficient allows the mass splitting to be closer to resonance allowing for 1 or 2 orders of magnitude increased asymmetry.

Figure 5

Plot of the washout as a function of ${\mu }_N$ (left) and ${\mu }_S$ (right) in the scenario where ${\mu }_N>{\mu }_S$ relevant for resonant leptogenesis. In blue the naïve washout is plotted while in red the effective washout which is defined in Eq. (68) relevant for situations with approximate lepton number conservation. As can be seen, naïvely the washout is overestimated by several orders of magnitude. In regions of large ${\mu }_N$ the effective washout is low enough (but still within the strong regime) for resonant leptogenesis to be feasible.

Figure 6

Plot of the $CP$ asymmetry into a specific flavor ${\epsilon }_{\alpha }=\sum _i{\epsilon }_{\alpha }^i$ generated in the ISS when both ${\mathcal{N}}_1$ and ${\mathcal{N}}_2$ are included, as a function of ${\mu }_N$ (left) and ${\mu }_S$ (right). Points in purple correspond to the regime where ${\mu }_N>{\mu }_S$, while points in green to ${\mu }_S>{\mu }_N$. At around ${10}^{-4}\mathrm{GeV}$ a natural resonance occurs generated by $\mathrm{\Delta }{m}_{i,j}^{\mathrm{OS}}$ in agreement with [43]. Once radiative effects are included an additional resonance occurs for large values of ${\mu }_N$ and in the regime ${\mu }_N>{\mu }_S$.

Figure 7

Plot of the $CP$ asymmetry into a specific flavor ${\epsilon }_{\alpha }=\sum _i{\epsilon }_{\alpha }^i$ generated in the ISS with no radiative effects included (green) and with ${\mathcal{N}}_1$ and ${\mathcal{N}}_2$ included (purple). This is varied with ${\mu }_N$ (left) and ${\mu }_S$ (right). A resonant enhancement in the asymmetry occurs only if the next-to-leading order contributions are included.

Figure 8

Plot of the baryon asymmetry as a function of the complex angle ${\theta }_2^{\mathrm{c}}$ for ${\delta }_{CP}=3\pi /2$ (left) and ${\delta }_{CP}=0$ (right). Here we have fixed ${\mu }_N\simeq 950\mathrm{GeV}$ and ${\mu }_S\simeq 10\mathrm{GeV}$ to be on resonance. In blue no radiative effects are included, in red only to leading order and in green next-to-leading order effects are included. The behavior of the asymmetry as a function of the complex angle is clear. At small values it freezes out to that provided by the Dirac phase ${\delta }_{CP}$; however, if only the leading radiative effects are included (red), the asymmetry tends to zero with decreasing ${\theta }_2^{\mathrm{c}}$ even for nonzero ${\delta }_{CP}$. This effect is similar to what occurs in the type-I MLFV scenario [51, 53, 89]. Asymmetry generation is possible both with low-energy phases as well as complex entries of the $R$ matrix where we loosely find the criterion $0.1\lesssim {\theta }_2^{\mathrm{c}}\lesssim 1$ for sufficient asymmetry generation when ${\delta }_{CP}=0$.

Figure 9

Plot of the ratios $R_{(i,j)[k,l]}$ for various combinations of flavor initial and final states as a function of the lightest neutrino mass $m_{{\nu }_1}$. Here the complex angle ${\theta }_2^{\mathrm{c}}$ has been switched off but the low-energy phase ${\delta }_{CP}$ is varied. Lines in red and orange correspond to the $CP$-conserving cases ${\delta }_{CP}=0$ and ${\delta }_{CP}=\pi$ respectively. In green is the maximally violating case of ${\delta }_{CP}=\pi /2$ or ${\delta }_{CP}=3\pi /2$. The shaded blue region corresponds to ${\delta }_{CP}=(0,2\pi )\backslash \{\pi /2,\pi ,3\pi /2\}$. All results are in full agreement with [46] for the $CP$-conserving cases.

Figure 10

Plot of the ratios $R_{(i,j)[k,l]}$ for various combinations of flavor initial and final states as a function of the lightest neutrino mass $m_{{\nu }_1}$. Here the low-energy phase ${\delta }_{CP}$ has been switched off but the complex angle ${\theta }_2^{\mathrm{c}}$ is varied. We find the same generic predictions as in the $CP$-conserving case of $R_{(\mu ,e)[\tau ,e]}>1$ (top left) $R_{(\mu ,e)[\tau ,\mu ]}<1$ (top right) and $R_{(\tau ,e)[\tau ,\mu ]}<1$ (bottom). For the ratios $R_{(\mu ,e)[\tau ,e]}$ and $R_{(\tau ,e)[\tau ,\mu ]}$ the introduction of $CP$ violation brings the ratios closer to one and this difference is most apparent with a hierarchical spectrum of light neutrinos. In orange ${\theta }_2^{\mathrm{c}}<0.1$, in red $0.1<{\theta }_2^{\mathrm{c}}<0.3$ and in burgundy ${\theta }_2^{\mathrm{c}}>0.3$.

Figure 11

Plot of the branching ratio $\mathrm{BR}(\mu \to e\gamma )$ for ${\theta }_2^{\mathrm{c}}=0$ (left) and $0.1<{\theta }_2^{\mathrm{c}}<1$ (right) as a function of ${\mu }_N$. Points in blue correspond to the choice ${\delta }_{CP}=0$ whereas points in green correspond to ${\delta }_{CP}=3\pi /2$. In these plots the other LNV parameter ${\mu }_S$ has been fixed to the values ${10}^{-8}$, ${10}^{-5}$, ${10}^{-2}$ and 10 GeV. The ratio plateaus whenever ${\mu }_N<{\mu }_S$. The red dotted (solid) line corresponds to the current (future) sensitivity of MEG and MEG-II [91, 92], respectively, for this decay mode. Clearly, small values of the LNV parameters are experimentally accessible, while large values (necessary for MLFV-ISS resonant leptogenesis) correspond to a suppressed signal. The inclusion of low-scale $CPV$ through the Dirac phase has no effect on the prediction, while $CPV$ from ${\theta }_2^{\mathrm{c}}$ can alter the prediction by approximately an order of magnitude.

Figure 12

Plot of the asymmetry generated in the LSS for the three scenarios (left) and comparing the asymmetry generated from mixing to oscillations (right) as a function of ${\mathit{m}}_{\mathit{L}}$, the average of the nonzero entries of the LNV matrix $m_L$ in GeV. Points in blue correspond to no radiative corrections, points in orange are when ${\mathcal{N}}_1$ and ${\mathcal{S}}_1$ are included and points in cyan include all terms. Contrary to the case of MLFV-ISS, corrections at leading order are sufficient to generate the necessary asymmetry. The case without any radiative corrections is highly suppressed as the self-energy contributions to ${ϵ}_i^{\alpha }$ are identically zero. However a nonzero asymmetry is generated due to the differences in the decay widths generating slight deviations in the vertex contribution for each SN; cf. Eq. (47). Points in purple correspond to the estimated asymmetry generated due to oscillation effects between the heavy sterile neutrinos. The asymmetry due to oscillations is predicted to be of the same order as the asymmetry from standard thermal leptogenesis and modifies the predictions for the allowed range of couplings only slightly.

Figure 13

Plot of the effective washout to a specific lepton flavor $K_{\alpha }$ generated in the LSS as defined in Eq. (68). We find the washout is minimized for ${\mathit{m}}_{\mathit{L}}\simeq \mathcal{O}({10}^{-5}–{10}^{-3})\mathrm{GeV}$ in agreement with [43]. The washout which is related to the Yukawa couplings in the mass basis of the heavy SNs from Eq. (39), is dominantly controlled by the parameters of $m_L$ ($m_D$) when $\parallel m_L\parallel >\parallel m_D\parallel$ ($\parallel m_D\parallel >\parallel m_L\parallel$). For large values of ${\mathit{m}}_{\mathit{L}}$ the washout is not dependent on the size of the complex parameter $a_2^{\mathrm{c}}$ whereas for small values where the parameters of $m_D$ dominate, different values of $a_2^{\mathrm{c}}$ can produce different values of washout for the same choices in $m_L$.

Figure 14

Plot of the $CP$ asymmetry to a specific lepton flavor $\alpha$ as a function of ${\mathit{m}}_{\mathit{L}}$. Points in green correspond to when ${\mathcal{N}}_i$ and ${\mathcal{S}}_i$ are not included and points in purple correspond to when they are included (at lowest order and next-to-leading order the points are identical). Clearly a resonance due to their inclusion increases the overall $CP$ asymmetry by many orders of magnitude. The mass splittings induced in this scenario are directly proportional to the decay widths and therefore for any choice of the parameters within $m_L$ the mass splittings will always be in a resonant regime.

Figure 15

Plot of the mass splitting $\mathrm{\Delta }{m}_{ij}$ as a function of ${\mathit{m}}_{\mathit{L}}$ in GeV when ${\mathcal{N}}_1$ and ${\mathcal{S}}_1$ are included. In red ${\mu }_N{n}_i={\mu }_S{s}_i=1/(4\pi {)}^2\mathrm{GeV}$, in green ${\mu }_N{n}_i={\mu }_S{s}_i=0.1\mathrm{GeV}$ and in blue ${\mu }_N{n}_i={\mu }_S{s}_i=1\mathrm{GeV}$. This is plotted against the decay width ${\mathrm{\Gamma }}_{i,j}$ in brown. As the combination ${\mathcal{N}}_1+{\mathcal{S}}_1\propto {\mathrm{\Gamma }}_{i,j}$ the mass splitting induced will always be on resonance independent of the value of $m_L$. Here there is no distinction between $\mathrm{\Delta }{m}_{i,j}^{\mathrm{SS}}$ and $\mathrm{\Delta }{m}_{i,j}^{\mathrm{OS}}$ and they both behave in a similar way.

Figure 16

Variation in the baryon asymmetry as a function of the lightest active neutrino mass $m_{{\nu }_1}$ (left) and varying the Wilson coefficients $n_i$ and $s_i$ (right) for the case where all radiative spurion effects are included. In this scan we fixed $a_2^{\mathrm{c}}=0.7$ and in the left plot set $n_i$, $s_i=1/16{\pi }^2$ whereas we fixed $m_{{\nu }_1}=0.01\mathrm{eV}$ in the right plot. In both plots similar behavior compared to MLFV-ISS is found.

Figure 17

Plot of the baryon asymmetry as a function of the complex parameter $a_2^{\mathrm{c}}$ for ${\delta }_{CP}=3\pi /2$ (let) and ${\delta }_{CP}=0$ (right). Here we have fixed $m_L\simeq {10}^{-4}\times \mathrm{diag}(1,2,5)\mathrm{GeV}$ to be in a region with sufficient washout suppression for necessary asymmetry generation. In blue no radiative effects are included and in green is the scenario where spurion effects are included. The behavior of the asymmetry as a function of the complex angle has consistent behavior as it is taken to zero. As before asymmetry generation is possible both with low-energy phases as well as complex entries of the $C$ matrix where smaller values of $a_2^{\mathrm{c}}$ are allowed compared to the ISS scenario.

Figure 18

Plot of the ratios $R_{(i,j)[k,l]}$ for various combinations of flavor initial and final states as a function of the lightest neutrino mass $m_{{\nu }_1}$. Here the complex parameter $a_2^{\mathrm{c}}$ has been switched off but the low-energy phase ${\delta }_{CP}$ is varied. Lines in red and orange correspond to the $CP$-conserving cases ${\delta }_{CP}=0$ and ${\delta }_{CP}=\pi$ respectively. In green is the maximally violating case of ${\delta }_{CP}=\pi /2$ or ${\delta }_{CP}=3\pi /2$. The shaded blue region corresponds to ${\delta }_{CP}=(0,2\pi )\backslash \{\pi /2,\pi ,3\pi /2\}$.

Figure 19

Plot of the ratios $R_{(i,j)[k,l]}$ for various combinations of flavor initial and final states as a function of the lightest neutrino mass $m_{{\nu }_1}$. Here the low-energy phase ${\delta }_{CP}$ has been switched off but the complex parameter $a_2^{\mathrm{c}}$ is varied. We find the same generic predictions as in the $CP$-conserving case of $R_{(\mu ,e)[\tau ,e]}>1$ (top left) $R_{(\mu ,e)[\tau ,\mu ]}\lesssim 1$ (top right) and $R_{(\tau ,e)[\tau ,\mu ]}\lesssim 1$ (bottom). Similar to the scenario above a cancellation occurs for the process $\tau \to e\gamma$ for specific values of the lightest neutrino mass $m_{{\nu }_1}$. In orange $a_2^{\mathrm{c}}<0.1$, in red $0.1<a_2^{\mathrm{c}}<0.3$ and in burgundy $a_2^{\mathrm{c}}>0.3$.

Figure 20

Plot of the branching ratio of $\mu \to e\gamma$ for ${\theta }_2^{\mathrm{c}}=0$ (left) and $0.1<a_2^{\mathrm{c}}<1$ (right) as a function of $m_L$. Points in blue correspond to the choice ${\delta }_{CP}=0$ whereas points in green correspond to ${\delta }_{CP}=3\pi /2$. The red dotted (solid) line corresponds to the current (future) sensitivity of MEG and MEG-II [91, 92] respectively for this decay mode. Currently very small values of the LNV parameter are probed for MLFV-LSS which correspond to a very large seperation of the LNV and LFV scales. In order to allow for necessary asymmetry generation we require approximately ${10}^{-5}\lesssim m_L/\mathrm{GeV}\lesssim {10}^{-3}$ which will not be probed by MEG-II in the near future. The inclusion of $CPV$ of any type does not significantly impact the predictions.