Can Leptonic Mixing Matrix have a Wolfenstein Form?

The Standard Model(SM) of particle physics has been an incredibly successful theory. Discovery of the 125-GeV scalar, if it is confirmed to be the SM Higgs boson will complete the SM [1, 2]. However, in spite of its astounding success we now know that SM cannot be the complete theory of nature. The discovery of neutrino oscillations was one of the conclusive proofs for the shortcoming of the SM [3, 4]. Ever since the discovery of neutrino oscillations our understanding of the neutrino oscillation parameters and hence in turn that of the leptonic mixing matrix is improving. The precision in measurement of certain mixing parameters has dramatically improved over the last decade [5, 6, 7]. This implies that neutrino physics is now entering the era of precision physics where the experimental data, in particular from neutrino oscillation experiments, can be used to rule out new physics models in a much more powerful way. In this work we confront one of the popular theoretical proposals, namely the possibility of leptonic mixing matrix having a Wolfenstein form at some high energy scale.

In its original form the ansatz hypothesized a “High Scale Mixing Unification” (HSMU) between the lepton and quark mixing matrices [8]. Here the unification of the two mixing matrices was hypothesized to happen at some high scale typically chosen to be the scale of Grand Unified Theories (GUTs). The Renormalization - Group (RG) evolution of the leptonic mixing angles and masses can then lead to the values of neutrino oscillation parameters with their experimental 3-$\sigma$ range. One of the key prediction of the HSMU hypothesis was prediction of a small yet non-zero value of $\theta_{13}$ leptonic mixing angle [8, 9, 10, 11, 12, 13, 14, 15]. This was due to the fact that $\theta_{13}=0$ is not a fixed point of the RG flow. Rather the RG evolution of the mixing angles from the high scale naturally leads to a small $\theta_{13}$, a fact later observed in experimental measurements [5, 6, 7]. After the experimental measurement of $\theta_{13}$ angle, the HSMU hypothesis was revisited for both Dirac and Majorana neutrinos and it was shown that indeed HSMU can be a good candidate proposal for understanding of the neutrino mixing and oscillation phenomenon consistent with the experimental data of that time [12, 13]. The scale of high energy unification as well as dependence on other parameters was also analyzed in these later works showing that the scale of unification does not need to be necessarily the GUT scale. Still later works expanded the idea further looking at the possibility whether the leptonic mixing matrix has a “Wolfenstein form” with hierarchical values of mixing angles at high scale, irrespective of whether or not they are exactly same as quark mixing angles [16, 17, 18, 19].

Overall, HSMU and its Wolfenstein form generalization have been shown to be consistent with successive sets of experimental data for increasingly precise determination of neutrino oscillation parameters over the past decade. However, as neutrino physics in entering the era of precision measurements, in this work we revisit it again to see if it still remains a viable possibility. The work-flow is presented in the rest of the paper in the following manner. Section 2 discusses the general framework of HSMU and RG evolution of neutrino oscillation parameters. Then in sections 3 and 4 we show our results of HSMU for Dirac and Majorana neutrinos. We find that the current oscillation data combined with other experimental constraints imply that HSMU ansatz is in severe tension with experiments. Further in section 5 we test whether or not the threshold corrections improve the negative results for HSMU ansatz. The next section 6 expands this unification hypothesis into Wolfenstein ansatz by introducing new free parameters. And finally we see conclusions from all the results’ interpretations, section 7.

We start with a general discussion of the HSMU hypothesis and the essential ingredients needed to have a large change in values of leptonic mixing angles over the course of RG evolution from high to low energy scale.

We begin our detailed analysis starting with the case of Dirac neutrinos. If neutrinos are Dirac fermions then by definition, one must add three right handed neutrinos, one for each generation, to the SM particle content. The simplest model for mass generation for Dirac neutrinos is through Higgs mechanism where the smallness of neutrino mass is due to small Yukawa couplings⁴⁴4In literature there exist various other mass generation mechanisms for Dirac neutrinos, interested readers can see Refs. [29, 30, 31, 32, 33, 34, 35, 36, 37, 38] for some of the recent works and Ref. [39] for a review.. As mentioned before, for HSMU one needs SUSY at the TeV scale which its simplest form can be implemented by embedding SM in MSSM with three additional superfields embedding the right handed neutrinos. The Lagrangians before and after SUSY breaking scale are given by

where $i,j,k$ are flavor indices, $\mathrm{Y_{\nu}}$ is the Yukawa matrix for the neutrinos, $\mathrm{\nu^{i}_{R}}$ is a right handed neutrino of flavor $i$ and $\mathbb{L},\mathbb{H}_{u},\mathbb{N}$ are the corresponding superfields. The RG equations for the evolution of neutrino masses and mixing parameters for this case can be found in [27].

The implementation of HSMU ansatz in this case follows the general strategy discussed in previous section. We start with the know values of the quark masses, mixing parameters, gauge couplings etc at the low scale ($M_{Z}$). We use the SM RG equations for the evolution up to SUSY breaking scale (2 TeV) and then use the MSSM RG equations up to the high scale (GUT scale). At the GUT scale we fix the leptonic mixing angles and depending on the case (see discussion below) the CP phase also to be equal to the quark ones. The neutrino masses at GUT scale are taken as N.O. and are treated as free parameters. We then do RG running back to the low scale to obtain the RG evolved values of the neutrino oscillation parameters and compare with the global fit data [28].

In this case, we consider two sub-cases: one when there is no CP violation in the leptonic sector i.e the CP phase $\delta=0$) and the other when there is CP violation in the leptonic sector as well i.e. $\delta\neq 0$. In the second case following HSMU ansatz, at high scale we take the $\delta=\delta_{q}$. We analyze both cases one by one.

In this case we consider neutrinos to be Majorana particles whose mass is generated through Type-I seesaw mechanism. Current limits on neutrino mass [40, 41] imply that if all Yukawas are taken within their perturbative range then the Type-I seesaw scale has to be smaller than the GUT scale. Throughout this work we will take the seesaw scale to be $\sim 10^{12}$ GeV so that the neutrino Yukawa couplings remain throughout the entire range of RG running. In this case there are several scales and the Lagrangian used for RG running at each scale is shown in (4).

where $\mathbb{M}$ represents the Majorana mass matrix, $\mathbb{Y}$ is the Yukawa matrix and rest of the notation remains same as in (2). Furthermore, in (4), below seesaw scale, we have added the dimension five Weinberg operator ($\mathcal{L}_{5}$) obtained from integrating out the right handed neutrinos [42]. In its SUSY version it is given by

where $\Lambda$ is the cutoff scale for the effective operator which in our case is the seesaw scale and $\mathbb{\kappa}^{ij}$ is the effective coupling. After SUSY breaking the $\mathcal{L}_{5}$ becomes the canonical Weinberg operator added to SM Lagrangian [42].

At the HSMU scale, apart from the masses of neutrinos we now have additional three free parameters, namely the three phases, the usual $\delta$ CP as well as the two Majorana phases $\mathrm{\varphi}_{1}$ and $\mathrm{\varphi}_{2}$. Here again we have to consider two sub-cases- one when Majorana phases are zero and the other when they are non-zero. For simplicity, we are fixing $\delta$ to be equal to $\delta_{q}$ at high scale. We have check and verified that the case of no CP violation in Majorana case also leads to similar qualitative results as the case of $\delta\delta_{q}$ and $\mathrm{\varphi}_{1}=\mathrm{\varphi}_{2}=0$. Therefore, in order to avoid unnecessary repetition, we will not present it separately.

Before looking at the sub-cases we should point out that for Majorana neutrinos one also has an additional constraint coming from the experimental searches of neutrinoless double beta decay ($0\nu\beta\beta$). The $0\nu\beta\beta$ experiments provide constraints on a particular combination of neutrino masses and mixing parameters called the effective Majorana mass ($\mathrm{m_{\beta\beta}}$) which is given as

where $\mathrm{c_{ij}=\cos{\theta_{ij}}}$ and $\mathrm{s_{ij}=\sin{\theta_{ij}}}$ denote the sine and cosine of the mixing angles.

The decay rate of $0\nu\beta\beta$ process $\Gamma\propto|m_{\beta\beta}|^{2}$. The most stringent upper bound range on $\mathrm{m_{\beta\beta}}$ is set by KamLAND-Zen experiment and its value ranges from $0.065$ eV to $0.165$ eV [43] depending on the choice of the nuclear matrix elements. In this paper we will use the conservative upper bound, i.e., we will demand that the low scale value of $\mathrm{m_{\beta\beta}}<0.165$ eV.

As we can see from Table 3, for many sets of $\mathrm{\varphi_{1}}$, $\mathrm{\varphi_{2}}$; along with $\mathrm{m_{\beta\beta}}$ we can bring in 4 out of 5 oscillation parameters inside their 3-$\sigma$ ranges. One of the mass squared difference could not be brought inside its allowed range for any choice of the free parameters. However, in past works it was shown that certain SUSY threshold corrections can have impact on the low scale values of the neutrino oscillation parameters, in particular the values of the mass square differences [9, 13]. The important threshold correction relevant to our analysis are given in Appendix B.

In order to see if inclusion of threshold corrections can lead to a viable parameter space consistent with current experimental data, we try to first see how that change the lower values of $\Delta\mathrm{m}^{2}_{\mathrm{atm}}$ and $\Delta\mathrm{m}^{2}_{\mathrm{sol}}$. For this we choose one of the most promising pair of values of $\mathrm{\varphi_{1}}$ $=100^{\circ}$ and $\mathrm{\varphi_{2}}$ $=50^{\circ}$ from Tab. 3, for which we were able to bring one mass squared difference inside its 3-$\sigma$ range. After we add the threshold corrections large enough to bring the other mass square difference inside its 3-$\sigma$ range, we find that the other mass squared difference which was already inside its allowed range, now moves out of the experimental 3-$\sigma$ range as shown in Figs. 8 and 9.

Fig.8 is plotted for the case when $\Delta\mathrm{m}^{2}_{\mathrm{atm}}$ is inside its 3-$\sigma$ range without threshold corrections and $\Delta\mathrm{m}^{2}_{\mathrm{sol}}$ is outside. For this case we choose the masses etc of the sparticles such that the threshold corrections are large enough to bring $\Delta\mathrm{m}^{2}_{\mathrm{sol}}$ inside its 3-$\sigma$ range, see eqs. (9) - (12) in Appendix B. After adding the corrections with appropriate parameters set, we find that the corrected $\Delta\mathrm{m}^{2}_{\mathrm{sol}}$ can indeed be brought inside its 3-$\sigma$ range, Fig.8(b). We plot this against one of the GUT scale mass- $\mathrm{m_{2}}$. The corresponding threshold corrections will naturally be added in $\Delta\mathrm{m}^{2}_{\mathrm{atm}}$ too. But for the same set of $\mathrm{m_{2}}$ values for which corrected $\Delta\mathrm{m}^{2}_{\mathrm{sol}}$ was brought in; the corrected $\Delta\mathrm{m}^{2}_{\mathrm{atm}}$ is significantly far from its 3-$\sigma$ range. Fig.8(a).

The same process is repeated in the Fig.9; except now $\Delta\mathrm{m}^{2}_{\mathrm{sol}}$ is inside and $\Delta\mathrm{m}^{2}_{\mathrm{atm}}$ outside its range, before threshold corrections are added. Even in this case, for the same range of $\mathrm{m_{2}}$ values the threshold corrected $\Delta\mathrm{m}^{2}_{\mathrm{sol}}$ shifts outside its 3-$\sigma$ range, Fig.9(a) when the corrected $\Delta\mathrm{m}^{2}_{\mathrm{atm}}$ is successfully brought in its respective 3-$\sigma$ range, Fig.9(b).

Thus in summary we have systematically analyzed all possible cases of HSMU ansatz, both for Dirac and Majorana neutrinos. Overall, we can conclude that HSMU ansatz is in conflict with the current 3-$\sigma$ allowed global fit ranges for neutrinos oscillation parameters and constraints from $0\nu\beta\beta$ decays in case of Majorana neutrinos. Today’s narrow 3-$\sigma$ experimental ranges of neutrino oscillation parameters do not allow HSMU hypothesis to be a plausible unification idea. In the next segment of the paper we discuss about an ansatz which generalizes the HSMU ansatz by imposing a looser set of demands on the high scale structure of the leptonic mixing matrix.

It was evident from previous sections that in no case the HSMU ansatz’s prediction for all low scale neutrino parameters is consistent with their current 3-$\sigma$ ranges. For this reason, we will like to consider another ansatz which tries to explain the hierarchical nature of neutrino mixing angles [16, 17, 18, 19]. This ansatz lifts the stringent restrictions on high scale leptonic mixing angles put by HSMU. In HSMU we consider that the leptonic mixing angles are exactly equal to those of quarks at the high scale (GUT scale). But instead, here we consider that the hierarchy in high scale quark mixing angle is duplicated in leptonic angles as well. This hierarchy is parameterized by “Wolfenstein-like” form. In this case the leptonic mixing angles are in the following pattern

where $\lambda$ is the leptonic Wolfenstein parameter which we define as the sine of the $\theta_{12}$ mixing angle. Wolfenstein parameter gives the neutrino mixing angles the following hierarchical structure-

We can vary $\theta_{23}$ and $\theta_{13}$ more finely by introducing $\mathrm{\alpha}$ and $\mathrm{\beta}$ such that $\alpha,\beta>0$, Eq. (7) gets modified into

Note that, choosing $\lambda$ to be equal to $0.22532$, $\mathrm{\alpha}$ to be equal to $0.698353$ and $\mathrm{\beta}$ to be equal to $0.264066$ we get back the HSMU ansatz.

Since from HSMU ansatz we already saw that only the case of Majorana neutrinos (with non-zero values of $\mathrm{\varphi_{1}}$, $\mathrm{\varphi_{2}}$) is close to being viable, therefore for Wolfenstein ansatz we will limit ourselves to this case only. We will follow the same strategy already discussed for HSMU but now with the relaxed condition of Wolfenstein ansatz on the high scale leptonic mixing angles. For the initial study of Wolfenstein ansatz, we analyze the effects of varying $\mathrm{\alpha}$ and $\mathrm{\beta}$ keeping Majorana phases to be zero and $\lambda$ equal to the value corresponding to the HSMU ansatz($\mathrm{\mathrm{\lambda_{HSMU}}}=0.22532$). Once we find a suitable pair of $\mathrm{\alpha}$, $\mathrm{\beta}$, we can study variations in $\lambda$. Only after thoroughly analyzing effects of $\mathrm{\alpha}$, $\mathrm{\beta}$ and $\lambda$ we will bring in non-zero Majorana phases into the picture. The CP phase $\delta$ is taken to be equal to that of quarks CP violation phase($\mathrm{\delta_{q}}$) in all the cases.

The High Scale Mixing Unification ansatz was one of the appealing possibilities to understand the deeper connection between lepton and quark sectors. The ansatz implies that at some high energy scale, usually taken as the GUT scale, the quark and leptonic mixing matrices are unified. The apparent differences between the two mixing matrices at the low scale of the experiments is then attributed to the large change in the leptonic mixing angles due to RG evolution from the high to the low scale. Such large RG evolution can be achieved in SuperSymmetric models like MSSM with the SUSY breaking scale in the few TeV range. Since the leptonic mixing angles and the neutrino mass square differences (observed in neutrino oscillations) RG evolve in a correlated fashion, this makes HSMU ansatz quite predictive. One of its early prediction was that $\theta_{13}$ angle should be non-zero but small [8], a fact later on confirmed by the experiments. In addition correlation between low scale values of $\theta_{13}$ and $\theta_{23}$ were also predicted [12, 13]. A generalization of the HSMU ansatz we call the Wolfenstein ansatz was also proposed in literature, where the strict requirement that at high scale the leptonic and quark mixing matrices are exactly same was relaxed. Instead the ansatz requires that the leptonic mixing angles at high scale should also have a hierarchical form, qualitatively similar to the one observed in the quark sector.

In this work we have thoroughly investigated both the HSMU and Wolfenstein ansatz, looking at the possibility of their compatibility with the current global fit results. We first started with the more stringent HSMU ansatz and investigated the various possibilities for both Dirac and Majorana neutrinos. For Dirac neutrinos we looked at the possibility of no CP violation as well as the possibility of CP violation in the leptonic sector. In both cases we found that for Dirac neutrinos, the current experimental data is completely incompatible with the HSMU hypothesis. We then looked at the Majorana cases, again looking at the possibility of zero and non-zero Majorana phases. For the case of zero Majorana phases, the results are similar to the Dirac case and is completely incompatible with current global fit results for the neutrino mixing angles. The case of non-zero Majorana phases was analysed in details and although we were able to have four of the neutrino oscillation parameters inside their current 3-$\sigma$ range but we found that all the five oscillation parameters and $m_{\beta\beta}$ can never be simultaneously brought inside their allowed values, even after taking into account the SUSY threshold corrections. Thus we concluded that although promising in past, the HSMU hypothesis is no longer compatible with the current global fit results. Finally, we analyzed the various possibilities for the Wolfenstein ansatz and again we found that all the six observables (the five neutrino oscillation ones + $m_{\beta\beta}$) cannot be brought simultaneously inside their currently allowed ranges, for any choice of the free parameters. Thus, we can finally conclude that the possibility of the leptonic mixing matrix having a hierarchical quark like form at a high scale is no longer viable.