Light sterile neutrino and leptogenesis

Abstract

We studied models of leptogenesis where three right-handed Majorana neutrinos are involved and the minimal-extended seesaw mechanism including an additional singlet field produces four light neutrinos. This study shows that the type of mass ordering and heavy Majorana scales can be determined by inputting the simplest orthogonal matrix into the Casas-Ibarra (CI) representation of seesaw. The CP asymmetry produced from the decays of heavy neutrinos and the dilution mass are predicted in terms of the mass and mixing elements of the fourth neutrino. Upon the choice of CI matrix, the existence of a light sterile neutrino is required to explain the high-energy lepton asymmetry in light of phenomenological measurements. Although there are several free parameters attributable to an additional neutrino, the model can be in part constrained by low-energy experiments such as sterile neutrino searches and neutrinoless double-beta decays, as well as the observed baryon asymmetry in the universe.

A \(4\times 4\) unitary transformation

The standard parametrization of $3\times 3$ transformation matrix for three neutrinos can be expressed such as

$$\begin{aligned}{} & {} V~=~R\left( \theta _{23}\right) R\left( \theta _{13},\delta _{13}\right) R\left( \theta _{12}\right) \nonumber \\{} & {} \quad =\left( \begin{array}{ccc} c_{12}c_{13}&{}s_{12}c_{13}&{}s_{13}e^{-i\delta }\\ -s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta } &{} c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta } &{} s_{23}c_{13} \\ s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta } &{} -c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta } &{} c_{23}c_{13} \end{array}\right) \end{aligned}$$

(A.1)

where $s_{ij}$ and $c_{ij}$ denotes $\sin {\theta _{ij}}$ and $\cos {\theta _{ij}}$. For Majorana neutrinos, the transformation U is given by the product of V and $P_2$, where

$$\begin{aligned} P_2=\left( \begin{array}{ccc} e^{i\eta _1} &{} 0 &{} 0 \\ 0 &{} e^{i\eta _2} &{} 0 \\ 0 &{} 0 &{} 1 \end{array}\right) . \end{aligned}$$

(A.2)

The matrix V is called $U_\textrm{PMNS}$ and its elements are denoted as

$$\begin{aligned} U_\textrm{PMNS}=\left( \begin{array}{ccc} U_{e1} &{} U_{e2} &{} U_{e3} \\ U_{\mu 1} &{} U_{\mu 2} &{} U_{\mu 3} \\ U_{\tau 1} &{} U_{\tau 2} &{} U_{\tau 3} \end{array}\right) . \end{aligned}$$

(A.3)

When a sterile neutrino with $T_{3L}=0$ is added to the neutrino contents, the $4\times 4$ unitary transformation can be denoted by

$$\begin{aligned} U_F'=\left( \begin{array}{cccc} U'_{e1} &{} U'_{e2} &{} U'_{e3} &{} U'_{e4}\\ U'_{\mu 1} &{} U'_{\mu 2} &{} U'_{\mu 3} &{} U'_{\mu 4} \\ U'_{\tau 1} &{} U'_{\tau 2} &{} U'_{\tau 3} &{} U'_{\tau 4}\\ U'_{s1} &{} U'_{s2} &{} U'_{s3} &{} U'_{s4} \end{array}\right) , \end{aligned}$$

(A.4)

which consists of six rotations such as

$$\begin{aligned} U'_F= & {} R\left( \theta _{34},\delta _{34}\right) R\left( \theta _{24},\delta _{24}\right) R\left( \theta _{14}\right) \left( \begin{array}{cc} V &{} 0 \\ 0 &{} 1 \end{array}\right) \end{aligned}$$

(A.5)

where each $R(\theta _{ij})$ is a $4\times 4$ rotation matrix and the V is defined in Eq. (A.1). The elements of $U'_F$ are specified in terms of mixing angles and phases of the sterile neutrino:

$$\begin{aligned}{} & {} \left( \begin{array}{c} U'_{e1} \\ U'_{\mu 1} \\ U'_{\tau 1} \\ U'_{s1} \end{array} \right) = \left( \begin{array}{c} V_{11} c_{14} \\ -V_{11} s_{14}s_{24}e^{-i \delta _{24}}+V_{21} c_{24} \\ -V_{11} s_{14}s_{34}c_{24}e^{-i \delta _{34}}-V_{21} s_{24}s_{34}e^{i (\delta _{24}-\delta _{34})} +V_{31} c_{34} \\ -V_{11} s_{14}c_{24}c_{34} - V_{21} s_{24}c_{34}e^{i \delta _{24}}-V_{31} s_{34}e^{i \delta _{34}} \end{array} \right) \nonumber \\{} & {} \left( \begin{array}{c} U'_{e2} \\ U'_{\mu 2} \\ U'_{\tau 2} \\ U'_{s2} \end{array} \right) = \left( \begin{array}{c} V_{12} c_{14} \\ - V_{12} s_{14}s_{24}e^{-i \delta _{24}}+V_{22} c_{24} \\ - V_{12} s_{14}s_{34}c_{24}e^{-i \delta _{34}}- V_{22} s_{24}s_{34}e^{i (\delta _{24}-\delta _{34})}+V_{32}c_{34} \\ - V_{12} s_{14}c_{24}c_{34}- V_{22} s_{24} c_{34}e^{-i \delta _{24}}-V_{32} s_{34}e^{i \delta _{34}} \end{array} \right) \nonumber \\{} & {} \left( \begin{array}{c} U'_{e3} \\ U'_{\mu 3} \\ U'_{\tau 3} \\ U'_{s3} \end{array} \right) = \left( \begin{array}{c} V_{13} c_{14} \\ -V_{13}s_{14}s_{24}+V_{23} c_{24}e^{-i \delta _{24}} \\ -V_{13} s_{14}s_{34}c_{24}e^{-i \delta _{34}} - V_{23}s_{24}s_{34}e^{i (\delta _{24}-\delta _{34})} +V_{33} c_{34} \\ -V_{13} s_{14}c_{24}c_{34} -V_{23} s_{24}c_{34}e^{i \delta _{24}} -V_{33}s_{34}e^{i \delta _{34}} \end{array} \right) \nonumber \\{} & {} \left( \begin{array}{c} U'_{e4} \\ U'_{\mu 4} \\ U'_{\tau 4} \\ U'_{s4} \end{array} \right) = \left( \begin{array}{c} s_{14} \\ s_{24}c_{14}e^{-i \delta _{24}} \\ s_{34}c_{14} c_{24}e^{-i \delta _{34}} \\ c_{14}c_{24}c_{34} \end{array}\right) \end{aligned}$$

(A.6)

For the transformation of Majorana neutrinos, the diagonal phase transformation with three Majorana phases also here is attached such that $U'=U'_FP_3$ with the following diagonal phase transformation:

$$\begin{aligned} P_3=\left( \begin{array}{cccc} e^{i\eta _1} &{} 0 &{} 0 &{} 0 \\ 0 &{} e^{i\eta _2} &{} &{} 0 \\ 0 &{} 0 &{} e^{i\eta _3} &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ \end{array}\right) . \end{aligned}$$

(A.7)

The individual rotation angle can be expressed in terms of the elements of $U'_F$,

$$\begin{aligned}{} & {} s_{14}^2=|U_{e4}'|^2 \nonumber \\{} & {} s_{24}^2=|U_{\mu 4}'|^2(1-|U_{e4}'|^2)^{-1} \nonumber \\{} & {} s_{34}^2=|U_{\tau 4}'|^2(1-|U_{e4}'|^2-|U_{\mu 4}'|^2)^{-1} \end{aligned}$$

(A.8)

where $|U_{s4}'|^2=1-|U_{e4}'|^2-|U_{\mu 4}'|^2-|U_{\tau 4}'|^2.$

Fig. 4

$Y_B$ vs. $m_{ee}$ for given choice of $(\delta _{24}, ~\delta _{34})$ in Choice $A'$. Majorana phases $\eta _2$ and $\eta _3$ run fully 0 to $\pi$ so as to cover the colored area. The blue dotted horizontal line at $(8.61\pm 0.05)\times 10^{-11}$ indicates the observed baryon asymmetry, while the red dotted vertical line at 21 meV indicates the upper bound of sensitivity of AMoRE II. The solid and dashed lines inside each shade mark the criteria, $\eta _2=0$ and $\eta _3=0$, respectively. Only positive results are presented

Fig. 5

Contours of $Y_B$ and $m_{ee}$ in $\eta _3-\eta _2$ space for given choice of $(\delta _{24}, ~\delta _{34})$ in Choice $A'$. The contour of $m_{ee}=28$ meV(24 meV) is represented by magenta(blue) solid line. The green curves $Y_B^\textrm{obs}$ correspond to the contours matched to the observation. The gray areas represent $|Y_B|<Y_B^\textrm{obs}$, while the orange and pink areas represent $Y_B>Y_B^\textrm{obs}$ and $Y_B<-Y_B^\textrm{obs}$, respectively

Fig. 6

$Y_B$ vs. $m_{ee}$ for given choice of $(\delta _{24}, ~\delta _{34})$ in Choice $B'$. Majorana phases $\eta _1$ and $\eta _2$ run fully 0 to $\pi$ so as to cover the colored area. The horizontal doted blue line at $(8.61\pm 0.05)\times 10^{-11}$ indicates the observed baryon asymmetry, while the vertical two lines at 21 meV and 49 meV indicate the upper bounds of sensitivities of AMoRE II and KamLAND2-Zen, respectively. The solid and dashed lines inside each shade mark the criteria $\eta _1=0$ and $\eta _2=0$, respectively. Only positive results are presented

Fig. 7

Contours of $Y_B$ and $m_{ee}$ in $\eta _2-\eta _1$ space for given choice of $(\delta _{24}, ~\delta _{34})$ in Choice $B'$, The contour of $m_{ee}=49$ meV(30 meV) is represented by magenta(blue) line. The green curves $Y_B^\textrm{obs}$ correspond to the contours matched to the observation. The gray areas represent $|Y_B|<Y_B^\textrm{obs}$, while the orange and pink areas represent $Y_B>Y_B^\textrm{obs}$ and $Y_B<-Y_B^\textrm{obs}$, respectively