Deviations to $\mu-\tau$ symmetry in a cobimaximal scenario

Neutrino mixing phenomena are resumed in the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [1, 2]. Global fits have reached the percent level in determining the mixing angles, confirming the massive nature of neutrinos. Also, there seem to be some hints leading to a non-zero $CP$ phase [3, 4, 5, 6, 7, 8], but no major information can be obtained for the Majorana $CP$ phases in oscillation experiments. Nevertheless, neutrinoless double beta decay experiments could be sensitive to the Majorana phases [9], which, in addition, could help elucidating the neutrino mass spectrum.

In the standard form, the PMNS matrix can be written in the following way

	$\displaystyle U_{\mathrm{PMNS}}$	$\displaystyle=$	$\displaystyle\left(\begin{array}[]{c@{\quad}c@{\quad}c}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta_{CP}}\\ -s_{12}c_{23}+c_{12}s_{23}s_{13}e^{i\delta_{CP}}&c_{12}c_{23}+s_{12}s_{23}s_{13}e^{i\delta_{CP}}&-s_{23}c_{13}\\ -s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta_{CP}}&c_{12}s_{23}-c_{23}s_{12}s_{13}e^{i\delta_{CP}}&c_{23}c_{13}\end{array}\right)$		(5)
			$\displaystyle\times{\mathrm{diag}}\left[1,e^{-i\frac{\beta_{1}}{2}},e^{-i\frac{\beta_{2}}{2}}\right]~{},$

in the case of Majorana neutrinos. Here, $s_{ij}$ and $c_{ij}$ stand for $\sin\theta_{ij}$ and $\cos\theta_{ij}$, respectively, with $\theta_{ij}$ denoting the mixing angles $\theta_{12}$, $\theta_{13}$, and $\theta_{23}$. The $CP$ phases are then represented by $\delta_{CP}$, $\beta_{1}$ and $\beta_{2}$, for Dirac and Majorana phases, respectively.

Within the theoretical efforts, a co-Bimaximal ($CBM$) mixing matrix has been motivated in the search of a symmetric structure in the masses and mixings of neutrinos [10, 11, 12, 13]. This form of the mixing matrix is obtained by considering $\theta_{23}=\pi/4$ and $\delta_{CP}=-\pi/2$ in Eq. (5), in addition with $\beta_{1,2}=0$, i.e.

$$U_{CBM}=\left(\begin{array}[]{c@{\quad}c@{\quad}c}c_{12}~{}c_{13}&s_{12}~{}c_{13}&-i~{}s_{13}\\ \frac{-1}{\sqrt{2}}(s_{12}-i~{}c_{12}~{}s_{13})&\frac{1}{\sqrt{2}}(c_{12}+i~{}s_{12}~{}s_{13})&\frac{-c_{13}}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}}(s_{12}+i~{}c_{12}~{}s_{13})&\frac{1}{\sqrt{2}}(c_{12}-i~{}s_{12}~{}s_{13})&\frac{c_{13}}{\sqrt{2}}\end{array}\right).$$

(6)

Such a simple form is consistent with current determinations of $\theta_{23}$ and $\delta_{CP}$ within $3\sigma$ error intervals, leaving $\theta_{13}$ and $\theta_{12}$ free of taking the experimental values. We can observe that the second and third rows in Eq. (6) display the following relation $|U_{\mu i}|=|U_{\tau i}|$, also known as a $\mu-\tau$ symmetry. The neutrino mass matrix can be obtained from $M_{\nu}^{0}=U_{CBM}{\mathrm{diag}}(m_{1},m_{2},m_{3})U_{CBM}^{\mathrm{T}}$, and can be resumed as [14]

$$M_{\nu}^{0}=\left(\begin{array}[]{c@{\quad}c@{\quad}c}m_{ee}&m_{e\mu}&m_{e\mu}^{*}\\ m_{e\mu}&m_{\mu\mu}&m_{\mu\tau}\\ m_{e\mu}^{*}&m_{\mu\tau}&m_{\mu\mu}^{*}\end{array}\right),$$

(7)

which presents a reflection symmetry between the $\mu$ and $\tau$ labels. Such a symmetric form is then an imprint of considering a $CBM$ mixing matrix in computing $M_{\nu}^{0}$, and could be considered as a starting point of any improved model of neutrinos masses and mixings, where the effective Majorana mass term remains invariant under the partially flavor changing $CP$ transformations

$$\nu_{eL}\rightarrow(\nu_{eL})^{c},~{}~{}~{}~{}~{}\nu_{\mu L}\rightarrow(\nu_{\tau L})^{c},~{}~{}~{}~{}~{}\nu_{\tau L}\rightarrow(\nu_{\mu L})^{c}~{},$$

(8)

where $(\nu_{\alpha L})^{c}$ stands for the charge conjugate of $\nu_{\alpha L}$. Theoretically, the origin of this pattern may be diverse. It could be related to a non-abelian discrete symmetry, type $A_{4}$, present at high energies, which should be broken at small energies to match as possible the experimental data (see for instance [13], and references therein). Furthermore, some corrections to this pattern have also been considered previously¹¹1Such corrections may arise, for example, from the renormalization group equations and/or from the charged lepton sector. to generate $\delta_{CP}$ and $\theta_{23}$ out of the maximal values [14, 15]. In most cases, any correction imposed to $U_{CMB}$ would have direct consequences on the symmetric structure of $M_{\nu}^{0}$.

In this work, we follow a different approach by considering a $CBM$ mixing matrix with Majorana $CP$ phases included in the standard form. We should expect that such phases may modify, or break, the symmetry of the mass matrix without modifying the mixing angles and the Dirac phase. Hence, in Sec. 2, we investigate the effects of the Majorana phases on the entries of the neutrino mass matrix that break the underlying symmetry. We modulate the deviations from the symmetric structure with two perturbation parameters which could impose some restrictions on the Majorana $CP$ phases. In Sec. 3, we follow a phenomenological approach to restrict the Majorana phases and study their effects in some physical observables. Finally, in Sec. 4 we present our main conclusions.

Motivated by the experimental values of the mixing angles, let us assume in the following a $PMNS$ mixing matrix of the form

$$U_{PMNS}=U_{CBM}\times{\mathrm{diag}}\left[1,e^{-i\frac{\beta_{1}}{2}},e^{-i\frac{\beta_{2}}{2}}\right]~{},$$

(9)

where $U_{CBM}$ is of the form in Eq. (6), and $\beta_{1}$ and $\beta_{2}$ the Majorana phases. The advantage of taking this parametrization is that it fulfills the current values of $\theta_{23}$ and $\delta_{CP}$ within $3\sigma$ or less, while $\theta_{12}$ and $\theta_{13}$ are free parameters, such that they can be easily accommodated within the $1\sigma$ experimental interval.

As can be expected, this form of the $PMNS$ matrix allows the mass matrix to deviate from the symmetric pattern observed in Eq. (7), which is directly induced by the Majorana phases. It should be clear that in the limit $\beta_{1,2}=0$, the symmetric pattern is restored.

Previous analysis [16, 17] has shown, in the case of a $\mu-\tau$ permutation symmetry, that departures from a symmetric mass matrix can be resumed by two parameters that measure the breaking of the equalities $m_{e\tau}=m_{e\mu}$ and $m_{\tau\tau}=m_{\mu\mu}$. In particular, some corrections to the Tri-Bi-Maximal²²2The corrections to the Tri-Bi-Maximal mixing usually include additional mixing angles and/or $CP$ phases. mixing matrix have been adopted [18, 19, 20], where the two breaking parameters of the $\mu-\tau$ permutation symmetry are related to correction angles. Then, we should anticipate that in the $CBM$ case, we could follow a similar approach since the inclusion of the Majorana phases would contribute to spoiling the reflection symmetry of the mass matrix, which, to the best of our knowledge, has not been considered previously. However, deepen on the origin of these phases is out of the scope of the present work.

Let us then modulate the deviations from a symmetric mass matrix in the form

$$M_{\nu}=M_{\nu}^{0}+\delta M\left(\delta,\epsilon\right)~{},$$

(10)

where $M_{\nu}^{0}$ follows the symmetric form in Eq. (7), displaying the imprints of the $CBM$ mixing, whereas

$$\delta M\left(\delta,\epsilon\right)=\left(\begin{array}[]{c@{\quad}c@{\quad}c}0&0&\delta\\ 0&0&0\\ \delta&0&\epsilon\end{array}\right)$$

(11)

modulates the deviations from the reflection symmetry. This matrix is written in terms of two breaking parameters, $\delta=m_{e\tau}-m_{e\mu}^{*}$ and $\epsilon=m_{\tau\tau}-m_{\mu\mu}^{*}$, where the mass matrix entries are now computed from $M_{\nu}=U_{PMNS}{\mathrm{diag}}(m_{1},m_{2},m_{3})U_{PMNS}^{\mathrm{T}}$ by using the Eq. (9). The novelty of our approach is then related to the fact of considering an underlaying reflection symmetry instead of the permutation symmetry in [18], which, in addition to our particular choice of the $PMNS$ matrix, lead to different expressions for the breaking parameters. It is worth noting that the corrected $CBM$ mixing matrix breaks explicitly the equalities $m_{\tau\tau}=m_{\mu\mu}^{*}$ and $m_{e\tau}=m_{e\mu}^{*}$, thus, the most advantageous texture to parametrize the correction matrix, with the minimum number of parameters, is as in Eq. (11). However, the texture of the correction mass matrix could be written in a different way, for instance, by locating the breaking parameters in the $\delta M_{\mu\mu}$, $\delta M_{e\mu}$ and $\delta M_{\mu e}$ entries, but their values would not change if the modulus of these parameters is taken. We can also define the two dimensionless parameters in the following way

	$\displaystyle\hat{\delta}\equiv\frac{\delta}{m_{e\mu}^{*}}$	$\displaystyle=$	$\displaystyle\frac{-2}{1-iC}~{},$
	$\displaystyle\hat{\epsilon}\equiv\frac{\epsilon}{m_{\mu\mu}^{*}}$	$\displaystyle=$	$\displaystyle\frac{-2}{1-i\tilde{C}}~{}.$		(12)

It is direct to show that $C$ and $\tilde{C}$ are functions that can be expressed as

	$\displaystyle C$	$\displaystyle=$	$\displaystyle\frac{m_{3}c_{\beta_{2}}s_{13}-im_{1}c_{12}(s_{12}+ic_{12}s_{13})+im_{2}c_{\beta_{1}}s_{12}(c_{12}-is_{12}s_{13})}{m_{3}s_{\beta_{2}}s_{13}+im_{2}s_{\beta_{1}}s_{12}(c_{12}-is_{12}s_{13})}~{},$
	$\displaystyle\tilde{C}$	$\displaystyle=$	$\displaystyle\frac{m_{3}c_{\beta_{2}}c_{13}^{2}+m_{1}(s_{12}+ic_{12}s_{13})^{2}+m_{2}c_{\beta_{1}}(c_{12}-is_{12}s_{13})^{2}}{m_{3}s_{\beta_{2}}c_{13}^{2}+m_{2}s_{\beta_{1}}(c_{12}-is_{12}s_{13})^{2}}~{},$		(13)

where $s_{\beta_{i}}=\sin\beta_{i}$ and $c_{\beta_{i}}=\cos\beta_{i}$, for $i=1,2$, and $m_{1,2,3}$ the absolute values of neutrino masses. We can see from Eq. (2) that both parameters display the expected behavior as they depend on the Majorana $CP$ phases through the functions in Eq. (2). It is straightforward to show that in the limit case where $\beta_{1}=\beta_{2}=0$ such parameters vanish, restoring the symmetric pattern in Eq. (7). Hence, the size of the breaking of the reflection symmetry in the neutrino mass matrix is directly linked to the non-vanishing values of the Majorana phases. Small deviations from the symmetric pattern could be investigated by demanding $|\hat{\delta}|,|\hat{\epsilon}|\lesssim 1$ for the mass matrix, which is also called a slight or soft breaking of the $\mu-\tau$ symmetry [21, 22], and may help to resolve possible values of Majorana $CP$ phases. In addition, functions $C$ and $\tilde{C}$ also show a dependence on the three neutrino absolute masses. We can turn these expressions in terms of the two squared mass differences, according to the neutrino mass ordering, and the lightest neutrino mass, such that some particular cases should be analyzed.

Before presenting a full numerical analysis, let us consider, in the following, a semi-analytical approach and divide our discussion according to the neutrino mass ordering. For our numerical evaluations, we will follow the results in [6], but the same conclusions are obtained for other data sets.

In the case of the normal ordering ($NO$), we can adopt the approximate relations $|m_{1}|\ll|m_{2}|\approx\sqrt{\Delta m^{2}_{SOL}}\ll m_{3}\approx\sqrt{\Delta m^{2}_{ATM}}$, being $m_{1}=m_{0}$ the lightest neutrino mass, which we leave as a free parameter. The squared mass differences and the non-fixed angles are given at $1\sigma$ by $\Delta m^{2}_{SOL}=(7.42^{+0.21}_{0.20})\times 10^{-5}\mbox{ eV}^{2}$, $\Delta m^{2}_{ATM}=(2.510^{+0.027}_{-0.027})\times 10^{-3}\mbox{ eV}^{2}$, $\sin^{2}\theta_{12}=(3.04^{+0.12}_{-0.12})\times 10^{-1}$ and $\sin^{2}\theta_{13}=(2.246^{+0.062}_{-0.062})\times 10^{-2}$ [6].

A direct inspection in Eq. (2) shows that $m_{1}$ gives minor contributions to $C$ and $\tilde{C}$ as it is suppressed in the $NO$. Hence, it will not play a crucial role in determining $\hat{\delta}$ and $\hat{\epsilon}$, and can be safely neglected in a first approximation. We can also see that, for example, in the special case where $\beta_{1}=\beta_{2}=\beta$, we obtain $\hat{\delta}\approx\hat{\epsilon}\approx-2(1-i\cot\beta)^{-1}$. It is also easy to show that the combination where one of the Majorana phases is fixed to zero leaves a similar expression for the non-zero phase. Hence, we can observe that values of $\beta$ in the $CP$ conserving limit ($0,\pm\pi$) recover the symmetry in the neutrino mass matrix of Eq. (10). On the other hand, the maximal $CP$ violation case ($\pm\pi/2$) produces large deviations from the symmetric scenario since $|\hat{\delta}|\approx|\hat{\epsilon}|\approx 2$. We should then expect that, in the $NO$, if small deviations from the symmetric mass matrix are demanded, the Majorana phases should remain near the $CP$ conserving values. This can be observed in the left plot in Fig. 1, which are obtained for the full expressions.

For the inverted ordering ($IO$), we have the following relations: $|m_{1}|\approx\sqrt{\Delta m_{ATM}^{2}}$, $|m_{2}|\approx\sqrt{\Delta m_{SOL}^{2}+\Delta m_{ATM}^{2}}\gg m_{3}$. In this case, $m_{3}=m_{0}$ is the lightest neutrino mass, and $\Delta m^{2}_{SOL}=(7.42^{+0.21}_{0.20})\times 10^{-5}\mbox{ eV}^{2}$, $\Delta m^{2}_{ATM}=-(2.490^{+0.026}_{-0.028})\times 10^{-3}\mbox{ eV}^{2}$, $\sin^{2}\theta_{12}=(3.04^{+0.13}_{-0.12})\times 10^{-1}$ and $\sin^{2}\theta_{13}=(2.241^{+0.074}_{-0.062})\times 10^{-2}$ [6].

We observe in this case that, neglecting the lightest neutrino mass, the breaking parameters take the approximated form

	$\displaystyle\hat{\delta}$	$\displaystyle\approx$	$\displaystyle\frac{-2is_{\beta_{1}}e^{-i\beta_{1}}}{1-e^{-i\beta_{1}}\left(\frac{c_{12}s_{12}+ic_{12}^{2}s_{13}}{c_{12}s_{12}-is_{12}^{2}s_{13}}\right)}+{\mathcal{O}}\left(\sqrt{\frac{\Delta m_{SOL}^{2}}{\Delta m_{ATM}^{2}}}~{}\right)~{},$
	$\displaystyle\hat{\epsilon}$	$\displaystyle\approx$	$\displaystyle\frac{-2is_{\beta_{1}}e^{-i\beta_{1}}}{1+e^{-i\beta_{1}}\left(\frac{s_{12}+ic_{12}s_{13}}{c_{12}-is_{12}s_{13}}\right)^{2}}+{\mathcal{O}}\left(\sqrt{\frac{\Delta m_{SOL}^{2}}{\Delta m_{ATM}^{2}}}~{}\right)~{}.$		(14)

Within this approximation, we observe that the breaking parameters do not have a strong dependence on $\beta_{2}$, such that we should expect that this phase will not be restricted in our analysis of the values of $\hat{\delta}$ and $\hat{\epsilon}$. It is also important to note that the squared mass differences, and the mixing angles, will play a sub-leading role in the determination of these parameters, with the main contribution coming from $\beta_{1}$. Hence, we can verify that the exact symmetry limit is related to $CP$ conserving values ($0,\pm\pi$) of $\beta_{1}$, while deviations from the symmetric pattern are then governed in this limit by specific values of $\beta_{1}$. For instance, for $\beta_{1}\approx 0.1~{}(\sim 6^{\circ})$ we obtain $|\hat{\delta}|\approx|\hat{\epsilon}|\approx 0.2$. On the other hand, a maximal value of this phase $(\beta_{1}=\pm\pi/2)$ leads to a large deviation from the symmetric pattern, i.e., $|\hat{\delta}|\approx|\hat{\epsilon}|\approx 2$. The full numerical analysis is resumed in the right plot of Fig. 1.

Finally, we can also analyze the degenerate ordering, where absolute masses are the same order $|m_{1}|\approx|m_{2}|\approx|m_{3}|$. In this case, the breaking parameters show no strong dependence on the mass, but the limit is restricted to large values of the lightest neutrino mass, which is almost excluded from cosmological observations [23]. In this limit, the dependence of the breaking parameters on the mixing angles and $CP$ phases can be directly obtained from Eq. (2). As in the previous cases, the $CP$ conserving limit is related to the null values of the breaking parameters. A numerical inspection shows that, when we take $\beta_{1}=\beta_{2}=\beta$, these parameters will have a similar form to the first terms in Eq. (3), leading to $|\hat{\delta}|,|\hat{\epsilon}|\gtrsim 1$ for $(\beta=\pm\pi/2)$, which should be excluded in the search of small deviations. This pattern is also present in both $NO$ and $IO$, but leaves open questions about other different combinations in the values of $CP$ phases consistent with $|\hat{\delta}|,|\hat{\epsilon}|\lesssim 1$.

Our previous analysis may serve to confirm the expectations that the $CP$ conserving values of Majorana phases are linked to the symmetric structure of neutrino mass matrix, but also shows that we should investigate other possible combinations of $CP$ phases which may lead to small deviations from this symmetric pattern. To this aim, a full numerical analysis is also mandatory, considering different values of $m_{0}$, the mass orderings, and the deviations in the mixing angles from their central values. We show in Fig. 1 the different allowed regions of Majorana phases for different values of breaking parameters. In these plots, we run the lightest mass from zero to $0.4\mbox{ eV}$, in both mass orderings, and allowed the mixing angles to vary within the $3\sigma$ interval. We show, for comparison, three different cases: $|\hat{\delta}|,|\hat{\epsilon}|\lesssim 0.1$, $|\hat{\delta}|,|\hat{\epsilon}|\lesssim 0.3$, and $|\hat{\delta}|,|\hat{\epsilon}|\lesssim 0.5$. We can observe that other different combinations of Majorana phases are obtained, out of the $CP$ conserving limit, which leads to small (or slight) deviations from the symmetric mass matrix. As we could anticipate, the size of the allowed region depends on the selected values of the breaking parameters. It is worth noting that, while for the $NO$ both of the $CP$ phases are restricted to lie near the $CP$ conserving values, in the $IO$, the phase $\beta_{2}$ can reach a maximal value even for very small deviations. Such regions, and also the differences between different mass orderings, could be confronted with some physical observables as in the case of neutrinoless double beta decay.

In Fig. 2, we plot the matrix element of neutrinoless double beta decay ($|m_{ee}|$) depending on the lightest neutrino mass. We can observe that the combinations of Majorana phases obtained from the restrictions in $|\hat{\delta}|$ and $|\hat{\epsilon}|$ give specific regions in $|m_{ee}|$, which may be compared with forthcoming experimental observations. For comparison, we show in the left plot the allowed regions for the condition $|\hat{\delta}|,|\hat{\epsilon}|\lesssim 0.5$, which present a slight reduction compared to the case of totally free Majorana phases. In the right plot of Fig. 2, we present the region corresponding to the restriction $|\hat{\delta}|,|\hat{\epsilon}|\lesssim 0.1$. We observe, in this case, that some specific values of $|m_{ee}|$ would not be compatible with a slightly broken symmetry in the mass matrix.

We have centered our analysis on an indirect determination of the Majorana phases, which seem to be very restricted when a $\mu-\tau$ reflection symmetry is demanded in the mass matrix. These regions may be extended when this last requirement is relaxed, giving place to the possibility of having different combinations of values consistent with small deviations. In the $IO$, it is also possible to have one of the Majorana phases with a maximal value even for an exact symmetry in the mass matrix. Some of our results could be of theoretical interest in the search for models of neutrino masses and mixings.

A $CBM$ mixing matrix is of great theoretical interest as it is related to some discrete symmetries, which could help to understand the pattern of masses and mixings. We have shown, from a phenomenological approach, that $CP$ nonconserving values of the Majorana phases may break the symmetric structure of the neutrino mass matrix, regardless of the symmetry in the mixing matrix. By modulating the deviations with two breaking parameters, we found different combinations of Majorana phases consistent with small departures from the symmetric scenario, which could be of interest in the search for a perturbative treatment of the neutrino mass matrix, preserving the $CBM$ symmetry in the $PMNS$ matrix. In addition, we have also shown that simultaneous maximal values of Majorana phases are associated with large deviations from the symmetric limit, but, in the $IO$, it is possible to have one maximal Majorana phase consistent with the $\mu-\tau$ symmetry of the mass matrix. Restricted regions were obtained for the neutrinoless double beta decay amplitude, depending on the size of the deviations, which could be confronted with forthcoming results.

The authors acknowledge funding from Division General de Investigaciones (DGI) of the Santiago de Cali University under grant 935-621121-3068.