CP-Violation in Oscillations of Three Neutrino Generations: the Case of Degenerate Masses

Abstract

CP-symmetry violates in oscillations of neutrinos in vacuum in presence of three generations. In this case there is an inherent complex phase in PMNS-matrix. When any two neutrino masses degenerate the CP-violation vanishes. In the present article different mechanisms of CP-symmetry recovery are demonstrated in case of two degenerate neutrino mass-states.

Appendices

Appendix A

The derivation of the expression for the probability of neutrino oscillations in case of three neutrino generations is based on the work [2]. Let at the initial moment of time there is a neutrino with flavor $\alpha$ and momentum $p$. The propagation of the neutrino mass state in time and space in the context of the considering problem may be decribed by a plane wave: $|\nu_{k}(t,x)\rangle=e^{-iE_{k}t+ip_{k}x}|\nu_{k}\rangle$, where $k=1,2,3$, $E_{k}=\sqrt{p^{2}_{k}+m^{2}_{k}}$. Knowing the mixing matrix we can write for the evolution of the flavour state¹ :

$$|\nu_{\alpha}(t,x)\rangle=\sum_{k}U_{\alpha k}|\nu_{k}(t,x)\rangle$$

$${}=\sum_{k}U_{\alpha k}e^{-iE_{k}t+ip_{k}x}|\nu_{k}\rangle,\quad\alpha=e,\mu,\tau.$$

(A.1)

Since we consider the neutrino to have a definite momentum, $p_{i}=p$ for $i=1,2,3$. There is an opportunity to choose the other initial condition, namely the birth of neutrino with the definite energy $E$. However in the ultrarelativistic limit the both choices lead to the same result [7]. We get the amplitude of the oscillations:

$$A_{\nu_{\alpha}\rightarrow{\nu_{\beta}}}=\langle\nu_{\beta}|\nu_{\alpha}(t,x)\rangle$$

$${}=\sum_{k}U_{\alpha k}U^{*}_{\beta k}e^{-iE_{k}t+ipx}.$$

(A.2)

Then the probability of oscillations is equal:

$$P_{\nu_{\alpha}\rightarrow{\nu_{\beta}}}=|A_{\nu_{\alpha}\rightarrow{\nu_{\beta}}}|^{2}$$

$${}=\sum_{k,j}U_{\alpha k}U^{*}_{\beta k}U^{*}_{\alpha j}U_{\beta j}e^{-i(E_{k}-E_{j})t}.$$

(A.3)

Considering the neutrino ultrarelativistic, i.e. $m\ll p$, we obtain

$$(E_{k}-E_{j})t$$

$${}=\left(\sqrt{p^{2}+m^{2}_{k}}-\sqrt{p^{2}+m^{2}_{j}}\right)t$$

$${}\approx\left(p\left(1+\frac{m^{2}_{k}}{2p^{2}}\right)-p\left(1+\frac{m^{2}_{j}}{2p^{2}}\right)\right)t$$

$${}=\frac{\Delta m^{2}_{kj}}{2p}t\approx\frac{\Delta m^{2}_{kj}}{2E}L,$$

(A.4)

where $\Delta m^{2}_{kj}\equiv m^{2}_{k}-m^{2}_{j}$ and it is used, that in the ultrarelativistic case $E\simeq p$, $t\simeq L$.

Let us separate explicetly the constant term from from the oscillating term in the expression (A.3):

$$\sum_{k,j}U_{\alpha k}U^{*}_{\beta k}U^{*}_{\alpha j}U_{\beta j}e^{-i(E_{k}-E_{j})t}$$

$${}=\sum_{k}|U_{\alpha k}|^{2}|U_{\beta k}|^{2}$$

$${}+2\sum_{k>j}\textrm{Re}\{U_{\alpha k}U^{*}_{\beta k}U^{*}_{\alpha j}U_{\beta j}e^{-i\frac{\Delta m^{2}_{kj}}{2E}}L\},$$

(A.5)

where we used (A.4), and in the first term for $k=j$ we have $e^{-i(E_{k}-E_{j})t}=1$. By unitarity of the PMNS-matrix $\sum_{k}U_{\alpha k}U^{*}_{\beta k}=\sum_{j}U^{*}_{\alpha j}U_{\beta j}=\delta_{\alpha\beta}$, we obtain

$$\delta_{\alpha\beta}\cdot\delta_{\alpha\beta}=\left(\sum_{k}U_{\alpha k}U^{*}_{\beta k}\right)$$

$${}\times\left(\sum_{j}U^{*}_{\alpha j}U_{\beta j}\right)=\sum_{k,j}U_{\alpha k}U^{*}_{\beta k}U^{*}_{\alpha j}U_{\beta j}$$

$${}=\sum_{k=j}U_{\alpha k}U^{*}_{\alpha j}U^{*}_{\beta k}U_{\beta j}+\sum_{k\neq j}U_{\alpha k}U^{*}_{\beta k}U^{*}_{\alpha j}U_{\beta j}$$

$${}=\sum_{k}|U_{\alpha k}|^{2}|U_{\beta k}|^{2}+\sum_{k>j}U_{\alpha k}U^{*}_{\beta k}U^{*}_{\alpha j}U_{\beta j}$$

$${}+\sum_{k>j}U_{\alpha j}U^{*}_{\beta j}U^{*}_{\alpha k}U_{\beta k}=\sum_{k}|U_{\alpha k}|^{2}|U_{\beta k}|^{2}$$

$${}+\sum_{k>j}U_{\alpha k}U^{*}_{\beta k}U^{*}_{\alpha j}U_{\beta j}+\sum_{k>j}(U_{\alpha k}U^{*}_{\beta k}U^{*}_{\alpha j}U_{\beta j})^{*}$$

$${}=\sum_{k}|U_{\alpha k}|^{2}|U_{\beta k}|^{2}+2\sum_{k>j}\textrm{Re}\{U_{\alpha k}U^{*}_{\beta k}U^{*}_{\alpha j}U_{\beta j}\}$$

$${}=\delta_{\alpha\beta}.$$

(A.6)

Dividing the exponent (A.5) into the real and imaginary parts, eventually we get:

$$P_{\nu_{\alpha}\rightarrow{\nu_{\beta}}}=\delta_{\alpha\beta}$$

$${}-2\sum_{k>j}\textrm{Re}\{U_{\alpha k}U^{*}_{\beta k}U^{*}_{\alpha j}U_{\beta j}\}$$

$${}\times\left(1-\cos{\frac{\Delta m^{2}_{kj}}{2E}}L\right)$$

$${}-2\sum_{k>j}\textrm{Im}\{U_{\alpha k}U^{*}_{\beta k}U^{*}_{\alpha j}U_{\beta j}\}\sin{\frac{\Delta m^{2}_{kj}}{2E}L}$$

$${}=\delta_{\alpha\beta}-4\sum_{k>j}\textrm{Re}\{U_{\alpha k}U^{*}_{\beta k}U^{*}_{\alpha j}U_{\beta j}\}\sin^{2}{\frac{\Delta m^{2}_{kj}}{4E}L}$$

$${}-2\sum_{k>j}\textrm{Im}\{U_{\alpha k}U^{*}_{\beta k}U^{*}_{\alpha j}U_{\beta j}\}\sin{\frac{\Delta m^{2}_{kj}}{2E}L}.$$

(A.7)

Appendix B

Let us obtain the explicit form of the expression (5) for the oscillations $\nu_{e}\rightarrow\nu_{\mu}$:

$$\Delta P_{\nu_{e}\rightarrow\nu_{\mu}}$$

$${}=4\textrm{Im}\{U^{*}_{\mu 2}U_{e2}U_{\mu 1}U^{*}_{e1}\}\sin{\frac{\Delta m^{2}_{21}}{2E}L}$$

$${}+4\textrm{Im}\{U^{*}_{\mu 3}U_{e3}U_{\mu 1}U^{*}_{e1}\}\sin{\frac{\Delta m^{2}_{31}}{2E}L}$$

$${}+4\textrm{Im}\{U^{*}_{\mu 3}U_{e3}U_{\mu 2}U^{*}_{e2}\}\sin{\frac{\Delta m^{2}_{32}}{2E}L}.$$

(B.1)

We write down the elements attached to the each $\sin{\frac{\Delta m^{2}_{kj}}{2E}L}$:

$$\Delta m^{2}_{21}:U^{*}_{\mu 2}U_{e2}U_{\mu 1}U^{*}_{e1}$$

$${}=(c_{12}c_{23}-s_{12}s_{23}s_{13}e^{-i\delta})$$

$${}\times s_{12}c_{13}(-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta})c_{12}c_{13}$$

$${}=(-c_{12}c^{2}_{23}s_{12}-c^{2}_{12}c_{23}s_{23}s_{13}e^{i\delta}$$

$${}+s^{2}_{12}c_{23}s_{23}s_{13}e^{-i\delta}+s_{12}c_{12}s^{2}_{23}s^{2}_{13})$$

$${}=c_{12}c^{2}_{13}s_{12}(c_{23}s_{23}s_{13}(s^{2}_{12}e^{-i\delta}-c^{2}_{12}e^{i\delta})$$

$${}+s_{12}c_{12}((s_{23}s_{13})^{2}-c^{2}_{23}))$$

$${}\Longrightarrow\textrm{Im}\{U^{*}_{\mu 2}U_{e2}U_{\mu 1}U^{*}_{e1}\}$$

$${}=-c^{2}_{13}c_{12}c_{23}s_{13}s_{12}s_{23}\sin{\delta};$$

(B.2)

$$\Delta m^{2}_{31}:U^{*}_{\mu 3}U_{e3}U_{\mu 1}U^{*}_{e1}$$

$${}=s_{23}c_{13}s_{13}e^{-i\delta}(-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta})c_{12}c_{13}$$

$${}=c^{2}_{13}c_{12}s_{23}s_{13}(-s_{12}c_{23}e^{-i\delta}-c_{12}s_{23}s_{13})$$

$${}\Longrightarrow\textrm{Im}\{U^{*}_{\mu 3}U_{e3}U_{\mu 1}U^{*}_{e1}\}$$

$${}=c^{2}_{13}c_{12}c_{23}s_{13}s_{12}s_{23}\sin{\delta};$$

(B.3)

$$\Delta m^{2}_{32}:U^{*}_{\mu 3}U_{e3}U_{\mu 2}U^{*}_{e2}$$

$${}=s_{23}c_{13}s_{13}e^{-i\delta}(c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta})s_{12}c_{13}$$

$${}=s_{23}c^{2}_{13}s_{13}s_{12}(c_{12}c_{23}e^{-i\delta}-s_{12}s_{23}s_{13})$$

$${}\Longrightarrow\textrm{Im}\{U^{*}_{\mu 3}U_{e3}U_{\mu 2}U^{*}_{e2}\}$$

$${}=-c^{2}_{13}c_{12}c_{23}s_{13}s_{12}s_{23}\sin{\delta}.$$

(B.4)

Finally we obtain for the conversion $\nu_{e}$ in $\nu_{\mu}$:

$$\Delta P_{\nu_{e}\rightarrow\nu_{\mu}}$$

$${}=4c_{12}c^{2}_{13}c_{23}s_{13}s_{23}s_{12}\bigg{(}\sin{\frac{\Delta m^{2}_{31}}{2E}L}$$

$${}-\sin{\frac{\Delta m^{2}_{21}}{2E}L}-\sin{\frac{\Delta m^{2}_{32}}{2E}L}\bigg{)}\sin{\delta}.$$

(B.5)

Appendix C

We will show that an arbitrary unitary $3\times 3$-matrix of mixing can be made real when two neutrino mass states are degenerate. For this, we will consider its expansion to the matrices $W_{ij}$: $U=W_{12}W_{23}W_{13}$. The order of the product may be chosen arbitrary as in the section 3.1. The matrices $W_{ij}$ may be parametrized with one real angle and three complex phases. Let us take for example $W_{12}(\theta_{12},\phi_{1},\psi,\delta)$:

$$W_{12}$$

$${}=e^{i\phi_{1}/2}\begin{pmatrix}e^{i\psi}\cos{\theta_{12}}&e^{i\delta}\sin{\theta_{12}}&0\\ -e^{-i\delta}\sin{\theta_{12}}&e^{-i\psi}\cos{\theta_{12}}&0\\ 0&0&1\end{pmatrix}.$$

(C.1)

Defining $\psi=\psi_{1}+\delta_{1}$, $\delta=\psi_{1}-\delta_{1}$, let us write the matrix $W_{12}(\theta_{12},\phi_{1},\psi,\delta)=W_{12}(\theta_{12},\phi_{1},\psi_{1},\delta_{1})$ in the following way:

$$W_{12}=e^{i\phi_{1}/2}\begin{pmatrix}e^{i\psi_{1}}&0&0\\ 0&e^{-i\psi_{1}}&0\\ 0&0&1\end{pmatrix}$$

$${}\times\begin{pmatrix}\cos{\theta_{12}}&\sin{\theta_{12}}&0\\ -\sin{\theta_{12}}&\cos{\theta_{12}}&0\\ 0&0&1\end{pmatrix}\begin{pmatrix}e^{i\delta_{1}}&0&0\\ 0&e^{-i\delta_{1}}&0\\ 0&0&1\end{pmatrix}.$$

(C.2)

Thus, we have separated two phase matrices and one rotation matrix around the third axis. Parametrization of matrices $W_{23}(\theta_{23},\phi_{2},\psi_{2},\delta_{2})$, $W_{13}(\theta_{13},\phi_{3},\psi_{3},\delta_{3})$ is carried out in a similar way. As it was mentioned, the unitary matrix $3\times 3$ has three independent phases and three independent angles. In our expansion of the matrix $U$ there are nine phases and three angles. Hence we conclude that three phases are arbitrary. We substitue (C.2) into the initial expansion of the matrix $U$, denoting the rotation matrices $\theta_{12},\theta_{23},\theta_{13}$ as $R_{12}$, $R_{23}$, $R_{13}$ respectively:

$$U=e^{i(\phi_{1}/2+\phi_{2}/2+\phi_{3}/2)}\begin{pmatrix}e^{i\psi_{1}}&0&0\\ 0&e^{-i\psi_{1}}&0\\ 0&0&1\end{pmatrix}$$

$${}\times R_{12}\begin{pmatrix}e^{i\delta_{1}}&0&0\\ 0&e^{-i\delta_{1}}&0\\ 0&0&1\end{pmatrix}\begin{pmatrix}1&0&0\\ 0&e^{i\psi_{2}}&0\\ 0&0&e^{-i\psi_{2}}\end{pmatrix}$$

$${}\times R_{23}\begin{pmatrix}1&0&0\\ 0&e^{i\delta_{2}}&0\\ 0&0&e^{-i\delta_{2}}\end{pmatrix}\begin{pmatrix}e^{i\psi_{3}}&0&0\\ 0&1&0\\ 0&0&e^{-i\psi_{3}}\end{pmatrix}$$

$${}\times R_{13}\begin{pmatrix}e^{i\delta_{3}}&0&0\\ 0&1&0\\ 0&0&e^{-i\delta_{3}}\end{pmatrix}.$$

(C.3)

Let us assume $\psi_{2}=\delta_{1}=\delta_{2}=0$ and define $\Delta_{1}=\psi_{3}+\delta_{3}$, $\Delta_{2}=\psi_{3}-\delta_{3}$. Then for the matrix $U$ we have:

$$U=e^{i(\phi_{1}/2+\phi_{2}/2)}\begin{pmatrix}e^{i\psi_{1}}&0&0\\ 0&e^{-i\psi_{1}}&0\\ 0&0&1\end{pmatrix}R_{12}$$

$${}\times R_{23}e^{i\phi_{3}/2}\begin{pmatrix}e^{i\Delta_{1}}\cos{\theta_{13}}&0&e^{i\Delta_{2}}\sin{\theta_{13}}\\ 0&1&0\\ -e^{-i\Delta_{2}}\sin{\theta_{13}}&0&e^{-i\Delta_{1}}\cos{\theta_{13}}\end{pmatrix}$$

$${}=e^{i(\phi_{1}/2+\phi_{2}/2)}\begin{pmatrix}e^{i\psi_{1}}&0&0\\ 0&e^{-i\psi_{1}}&0\\ 0&0&1\end{pmatrix}$$

$${}\times R_{12}R_{23}W_{13}(\theta_{13},\phi_{3},\Delta_{1},\Delta_{2}).$$

(C.4)

Let us assume that $m1=m3$; that is, the $|\nu_{1}\rangle$ and $|\nu_{3}\rangle$ states are degenerate. In just the same way as in Subsection 3.1, $U$(2) symmetry then appears. We now rotate the neutrino fields by means of a matrix $V_{13}$ such that $V^{\dagger}_{13}=W_{13}$. Considering that the rotations of charged-lepton fields eliminate the phases remaining on the left in the matrix $U$, we ultimately arrive at $U=R_{12}R_{23}$. Thus, the mixing matrix became real, so that there is no $CP$ violation. Obviously, this argument is valid in the case of degeneracy of any two neutrino states.