Leptogenesis and dark matter detection in a TeV scale neutrino mass model with inverted mass hierarchy

Abstract

Realization of the inverted hierarchy is studied in the radiative neutrino mass model with an additional doublet, in which neutrino masses and dark matter could be induced from a common particle. We show that the sufficient baryon number asymmetry is generated through resonant leptogenesis even for the case with rather mild degeneracy among TeV scale right-handed neutrinos. We also discuss the relation between this neutrino mass generation mechanism and low energy experiments for the DM direct search, the neutrinoless double β decay, and so on.

Appendix

We summarize the formulas of the reaction density used in the Boltzmann equations [68, 69] for the number density of N ₃ and the lepton number asymmetry.¹⁴ In order to give the expression for the reaction density of the relevant processes, we introduce dimensionless variables

$$ x=\frac{s}{M_3^2}, \qquad a_j=\frac{M_j^2}{M_3^2}, \qquad a_\eta=\frac{M_\eta^2}{M_3^2}, $$

(A.1)

where s is the squared center of mass energy. The reaction density for the decay of N _j can be expressed as

$$ \gamma_D^{N_j}=\frac{(hh^\dagger)_{jj}}{8\pi^3} M_3^4a_j \sqrt{a_j} \biggl(1-\frac{a_\eta}{a_j} \biggr)^2 \frac{K_1(\sqrt{a_j}z)}{z}, $$

(A.2)

where K ₁(z) is the modified Bessel function of the second kind.

The reaction density for the scattering process is expressed as

$$ \gamma(ab\rightarrow ij)=\frac{T}{64\pi^4}\int^\infty_{s_{\mathrm{min}}}ds\, \hat{\sigma}(s)\sqrt{s}K_1 \biggl(\frac{\sqrt{s}}{T} \biggr), $$

(A.3)

where s _min=max[(m _a +m _b )²,(m _i +m _j )²] and $\hat{\sigma}(s)$ is the reduced cross section. In order to give the concrete expression for the reaction density relevant to Eq. (24), we define the following quantities for convenience:

$$ \begin{aligned} \phantom{}& \frac{1}{D_i(x)}=\frac{x-a_i}{(x-a_i)^2+a_i^2c_i}, \\ &c_i= \frac{1}{64\pi^2} \biggl(\sum_{\alpha=e,\mu,\tau} |h_{\alpha i}|^2 \biggr)^2 \biggl(1- \frac{a_\eta}{a_i} \biggr)^4, \\ &\lambda_{ij}= \bigl[x-(\sqrt{a_i}+ \sqrt{a_j})^2 \bigr] \bigl[x-(\sqrt{a_i}- \sqrt{a_j})^2 \bigr], \\ &L_{ij}=\ln \biggl[\frac{x-a_i-a_j+ 2a_\eta +\sqrt{\lambda_{ij}}}{ x-a_i-a_j +2 a_\eta -\sqrt{\lambda_{ij}}} \biggr], \\ &L_{ij}^\prime=\ln \biggl[\frac{\sqrt{x}(x-a_i-a_j-2a_\eta) +\sqrt{\lambda_{ij}(x-4a_\eta)}}{ \sqrt{x}(x-a_i-a_j-2a_\eta) -\sqrt{\lambda_{ij}(x-4a_\eta)}} \biggr]. \end{aligned} $$

(A.4)

As the lepton number violating scattering processes induced through the N _i exchange, we have

$$\begin{aligned} \hat{\sigma}^{(2)}_N(x) =&\frac{1}{2\pi} \frac{(x-a_\eta)^2}{x^2} \Biggl[\sum_{i=1}^3 \bigl(hh^\dagger\bigr)_{ii}^2\frac{a_i}{x} \biggl\{\frac{x^2}{xa_i -a_\eta^2}+\frac{x}{D_i(x)} \\ &{}+\frac{(x-a_\eta)^2}{2D_i(x)^2} -\frac{x^2}{(x-a_\eta)^2} \biggl(1+\frac{x+a_i-2a_\eta}{D_i(x)} \biggr) \\ &{}\times \ln \biggl( \frac{x(x+a_i-2a_\eta)}{xa_i-a_\eta^2} \biggr) \biggr\} + \sum_{i>j}\mathrm{Re}\bigl[ \bigl(hh^\dagger\bigr)_{ij}^2\bigr] \\ &{}\times \frac{\sqrt{a_ia_j}}{x} \biggl\{ \frac{x}{D_i(x)}+\frac{x}{D_j(x)}+ \frac{(x-a_\eta)^2}{D_i(x)D_j(x)} \\ &{}+\frac{x^2}{(x-a_\eta)^2} \biggl(\frac{2(x+a_i-2a_\eta)}{a_j-a_i}- \frac{x+a_i-2a_\eta}{D_j(x)} \biggr) \\ &{}\times \ln\frac{x(x+a_i-2a_\eta)}{xa_i-a_\eta^2} \\ &{}+\frac{x^2}{(x-a_\eta)^2} \biggl(\frac{2(x+a_j-2a_\eta)}{a_i-a_j}- \frac{x+a_j-2a_\eta}{D_i(x)} \biggr) \\ &{}\times \ln\frac{x(x+a_j-2a_\eta)}{xa_j-a_\eta^2} \biggr\} \Biggr] \end{aligned}$$

(A.5)

for $\ell_{\alpha}\eta^{\dagger}\rightarrow \bar{\ell}_{\beta}\eta$ and also

$$\begin{aligned} \hat{\sigma}^{(13)}_N(x) =&\frac{1}{2\pi} \Biggl[\sum _{i=1}^3\bigl(hh^\dagger \bigr)^2_{ii} \biggl\{ \frac{a_i(x^2-4xa_\eta)^{1/2}}{a_ix+(a_i-a_\eta)^2} \\ &{}+ \frac{a_i}{x+2a_i-2a_\eta} \\ &{}\times \ln \biggl(\frac{x+(x^2-4xa_\eta)^{1/2}+2a_i-2a_\eta}{ x-(x^2-4xa_\eta)^{1/2}+2a_i-2a_\eta} \biggr) \biggr\} \\ &{}+ \sum_{i>j}\mathrm{Re}\bigl[\bigl(hh^\dagger \bigr)_{ij}^2\bigr] \frac{\sqrt{a_ia_j}}{x+a_i+a_j-2a_\eta} \\ &{}\times \biggl\{\frac{2x+3a_i+a_j-4a_\eta}{a_j-a_i} \\ &{}\times \ln \biggl(\frac{x+(x^2-4xa_\eta)^{1/2}+2a_i-2a_\eta}{ x-(x^2-4xa_\eta)^{1/2}+2a_i-2a_\eta} \biggr) \\ &{}+ \frac{2x+a_i+3a_j-4a_\eta}{a_i-a_j} \\ &{}\times \ln \biggl(\frac{x+(x^2-4xa_\eta)^{1/2}+2a_j-2a_\eta}{ x-(x^2-4xa_\eta)^{1/2}+2a_j-2a_\eta} \biggr) \biggr\} \Biggr] \end{aligned}$$

(A.6)

for ℓ _α ℓ _β →ηη. The cross terms has no contribution if the CP phases are assumed to satisfy |sin2(φ _1,2−φ ₃)|=1. We adopt this possibility in the numerical analysis, for simplicity. Since another type of the lepton number violating process N _i N _j →ℓ _α ℓ _β induced by the η exchange could be suppressed for a small |λ ₅|, we can neglect them safely for the value of |λ ₅| used in this analysis.

As the lepton number conserving scattering processes which contribute to determine the number density of N ₃, we have

$$\begin{aligned} \hat{\sigma}^{(2)}_{N_iN_j}(x) =&\frac{1}{4\pi} \biggl[ \bigl(hh^\dagger\bigr)_{ii}\bigl(hh^\dagger \bigr)_{jj} \\ &{}\times \biggl\{\frac{\sqrt{\lambda_{ij}}}{x} \biggl(1+ \frac{(a_i-a_\eta)(a_j-a_\eta)}{(a_i-a_\eta)(a_j-a_\eta)+xa_\eta} \biggr) \\ &{}+\frac{a_i+a_j-2a_\eta}{x}L_{ij} \biggr\} \\ &{}- \mathrm{Re}\bigl[ \bigl(h h^\dagger\bigr)_{ij}^2\bigr] \frac{2\sqrt{a_ia_j}L_{ij}}{x-a_i-a_j+2a_\eta} \biggr] \end{aligned}$$

(A.7)

for $N_{i}N_{j} \rightarrow \ell_{\alpha}\bar{\ell}_{\beta}$ which are induced through the η exchange and also

$$\begin{aligned} \hat{\sigma}^{(3)}_{N_iN_j}(x) =& \frac{1}{4\pi} \frac{(x-4a_\eta)^{1/2}}{x^{1/2}} \biggl[\bigl|\bigl(hh^\dagger\bigr)_{ij}\bigr|^2 \biggl\{ \frac{\sqrt{\lambda_{ij}}}{x} \biggl(-2 \\ &{}+\frac{a_\eta(a_i- a_j)^2}{ (a_\eta-a_i)(a_\eta-a_j)x +(a_i-a_j)^2a_\eta} \biggr) \\ &{}+\frac{x^{1/2}}{(x-4a_\eta)^{1/2}} \biggl(1-2 \frac{a_\eta}{x} \biggr) L_{ij}^\prime \biggr\} \\ &{}- 2\mathrm{Re}\bigl[\bigl(h h^\dagger\bigr)_{ij}^2 \bigr] \frac{\sqrt{a_ia_j}(a_i+a_j-2a_\eta)L_{ij}^\prime}{ x(x-a_i-a_j-2a_\eta)} \biggr] \end{aligned}$$

(A.8)

for N _i N _j →ηη ^† which are induced through the ℓ _α exchange. The cross terms in these reduced cross sections can be neglected if the same assumption is made for the CP phases as Eqs. (A.5) and (A.6).