Nonzero from Charged Lepton Corrections
and the Atmospheric Neutrino Mixing Angle
(updated using the results of the global fits of 2013 data111The Addendum on pages Addendum: Analysis with the 2013 Data–Addendum: Analysis with the 2013 Data is not present in the published version of this paper. )
After the successful determination of the reactor neutrino mixing angle , a new feature suggested by the current neutrino oscillation data is a sizeable deviation of the atmospheric neutrino mixing angle from . Using the fact that the neutrino mixing matrix , where and result from the diagonalisation of the charged lepton and neutrino mass matrices, and assuming that has a i) bimaximal (BM), ii) tri-bimaximal (TBM) form, or else iii) corresponds to the conservation of the lepton charge (LC), we investigate quantitatively what are the minimal forms of , in terms of angles and phases it contains, that can provide the requisite corrections to so that , and the solar neutrino mixing angle have values compatible with the current data. Two possible orderings of the 12 and the 23 rotations in , “standard” and “inverse”, are considered. The results we obtain depend strongly on the type of ordering. In the case of “standard” ordering, in particular, the Dirac CP violation phase , present in , is predicted to have a value in a narrow interval around i) in the BM (or LC) case, ii) or in the TBM case, the CP conserving values being excluded in the TBM case at more than .
In the addendum we discuss the implications of the latest 2013 data.
Understanding the origin of the patterns of neutrino masses and mixing, emerging from the neutrino oscillation, decay, etc. data is one of the most challenging problems in neutrino physics. It is part of the more general fundamental problem in particle physics of understanding the origins of flavour, i.e., of the patterns of the quark, charged lepton and neutrino masses and of the quark and lepton mixing.
At present we have compelling evidence for the existence of mixing of three light massive neutrinos , , in the weak charged lepton current (see, e.g., ). The masses of the three light neutrinos do not exceed approximately 1 eV, eV, i.e., they are much smaller than the masses of the charged leptons and quarks. The three light neutrino mixing we will concentrate on in the present article, is described (to a good approximation) by the Pontecorvo, Maki, Nakagawa, Sakata (PMNS) unitary mixing matrix, . In the widely used standard parametrisation , is expressed in terms of the solar, atmospheric and reactor neutrino mixing angles , and , respectively, and one Dirac - , and two Majorana  - and , CP violation (CPV) phases:
The neutrino oscillation data, accumulated over many years, allowed to determine the parameters which drive the solar and atmospheric neutrino oscillations, , and , , with a high precision (see, e.g., ).
Furthermore, there were spectacular developments in the last 1.5 years in what concerns the angle (see, e.g., ). They culminated in a high precision determination of in the Daya Bay experiment with reactor :
Similarly the RENO, Double Chooz, and T2K experiments reported, respectively, , and evidences for a non-zero value of , compatible with the Daya Bay result. The high precision measurement on described above and the fact that turned out to have a relatively large value, have far reaching implications for the program of research in neutrino physics (see, e.g., ). After the successful measurement of , the determination of the absolute neutrino mass scale, of the type of the neutrino mass spectrum, of the nature - Dirac or Majorana, of massive neutrinos, as well as getting information about the status of CP violation in the lepton sector, are the most pressing and challenging problems and the highest priority goals of the research in the field of neutrino physics.
A global analysis of the latest neutrino oscillation data presented at the Neutrino 2012 International Conference , was performed in . The results on , and obtained in , which play important role in our further discussion, are given in Table 1.
|0.155||0.147 - 0.163||0.139 - 0.170||0.130 - 0.177|
|0.307||0.291 - 0.325||0.275 - 0.342||0.259 - 0.359|
|(NH)||0.386||0.365 - 0.410||0.348 - 0.448||0.331 - 0.637|
|(IH)||0.392||0.370 - 0.431||0.353 - 0.484 0.543 - 0.641||0.335 - 0.663|
|(NH)||3.39||2.42 - 4.27||—||—|
|(IH)||3.42||2.61 - 4.62||—||—|
An inspection of Table 1 shows that, in addition to the nonzero value of , the new feature which seems to be suggested by the current global neutrino oscillation data is a sizeable deviation of the angle from the value . This trend is confirmed by the results of the subsequent analysis of the global neutrino oscillation data performed in .
Although , and , the deviations from these values are small, in fact we have , and , where we have used the relevant best fit values in Table 1. The value of and the magnitude of deviations of and from suggest that the observed values of , and might originate from certain “symmetry” values which undergo relatively small (perturbative) corrections as a result of the corresponding symmetry breaking. This idea was and continues to be widely explored in attempts to understand the pattern of mixing in the lepton sector (see, e.g., [10, 11, 12, 13, 14, 15, 16, 17]). Given the fact that the PMNS matrix is a product of two unitary matrices,
where and result respectively from the diagonalisation
of the charged lepton and neutrino mass matrices, it is usually assumed that
has a specific form dictated by a symmetry which
fixes the values of the three mixing angles in
that would differ, in general, by perturbative
corrections from those measured in the
PMNS matrix, while (and symmetry breaking effects
that we assume to be subleading) provide the requisite corrections.
A variety symmetry forms of have been explored
in the literature on the subject (see, e.g., ).
In the present study we will consider three widely used forms.
i) Tribimaximal Mixing (TBM) :
ii) Bimaximal Mixing (BM) :
iii) the form of resulting from the conservation of the lepton charge of the neutrino Majorana mass matrix  (LC):
where and .
We will define the assumptions we make on
and in full generality in Section 2.
Those assumptions allow us to cover, in particular, the case
of corrections from to the three widely used forms of
indicated above. We would like to notice here that
being the unity matrix,
i) in all three cases of interest of ;
ii) , if coincides with or , while can have an arbitrary value if is given by ;
iii) , for or , while if .
Thus, the matrix has to generate corrections
i) leading to compatible with the observations in all three cases of considered;
ii) leading to the observed deviation of from in the cases of or .
iii) leading to the sizeable deviation of from for or , if it is confirmed by further data that .
In the present article we investigate quantitatively what are the “minimal” forms of the matrix in terms of the number of angles and phases it contains, that can provide the requisite corrections to , and so that the angles in the resulting PMNS matrix have values which are compatible with those derived from the current global neutrino oscillation data, Table 1. Our work is a natural continuation of the study some of us have done in  and earlier in [11, 12, 13, 14].
The paper is organised as follows. In Section 2 we describe the general setup and we introduce the two types of “minimal” charged lepton “rotation” matrix we will consider: with “standard” and “inverse” ordering. The two differ by the order in which the 12 and 23 rotations appear in . In the same Section we derive analytic expressions for the mixing angles and the Dirac phase of the PMNS matrix in terms of the parameters of the charged lepton matrix both for the tri-bimaximal and bimaximal (or LC) forms of the neutrino “rotation” matrix . In Sections 3 and 4 we perform a numerical analysis and derive, in particular, the intervals of allowed values at a given C.L. of the neutrino mixing angle parameters , and , the Dirac phase and the rephasing invariant associated with , in the cases of the standard and inverse ordering of the charged lepton corrections. A summary and conclusions are presented in Section 5. Further details are reported in two appendices. In Appendix A we illustrate in detail the parametrisation we use for the standard ordering setup. Finally, in Appendix B we describe the statistical analysis used to obtain the numerical results.
2 General Setup
While neutrino masses and mixings may or may not look anarchical, the hierarchy of charged lepton masses suggests an ordered origin of lepton flavour. Given the wide spectrum of specific theoretical models, which essentially allows to account for any pattern of lepton masses and mixings, we would like to consider here the consequence for lepton mixing of simple, general assumptions on its origin. As we have indicated in the Introduction, we are interested in the possibility that the mixing angle originates because of the contribution of the charged lepton sector to lepton mixing [10, 11, 12, 13, 14, 15, 16]. The latter assumption needs a precise definition. In order to give it, let us recall that the PMNS mixing matrix is given by
where and are respectively the charged lepton and neutrino Majorana mass matrices (in a basis assumed to be defined by the unknown physics accounting for their structure) and and are diagonal with positive eigenvalues.
We will assume that the neutrino contribution to the PMNS matrix has , so that the PMNS angle vanishes in the limit in which the charged lepton contribution can be neglected, . This is a prediction of a number of theoretical models. As a consequence, can be parameterized as
where is a rotation by an angle in the block and , are diagonal matrices of phases. We will in particular consider specific values of and, in certain cases, of , representing the predictions of well known models.
The above assumption on the structure of is not enough to draw conclusions on lepton mixing: any form of can still be obtained by combining with an appropriate charged lepton contribution . However, the hierarchical structure of the charged lepton mass matrix allows to motivate a form of similar to that of , with , so that we can write: 333The use of the inverse in eqs. (11) and (12) is only a matter of convention. This choice allows us to lighten the notation in the subsequent expressions.
In fact, the diagonalisation of the charged lepton mass matrix gives rise to a value of that is small enough to be negligible for our purposes, unless the hierarchy of masses is a consequence of correlations among the entries of the charged lepton mass matrix or the value of the element , contrary to the common lore, happens to be sizable. In such a scheme, with no 13 rotation neither in the neutrino nor in the charged lepton sector, the PMNS angle is generated purely by the interplay of the 23 and 12 rotations in eqs. (10) and (11).
While the assumption that is small, leading to eq. (11), is well motivated, textures leading to a sizeable are not excluded. In such cases, it is possible to obtain an “inverse ordering” of the and rotations in :
In the following, we will also consider such a possibility.
2.1 Standard Ordering
Consider first the standard ordering in eq. (11). We can then combine and in eqs. (10) and (11) to obtain the PMNS matrix. When doing that, the two 23 rotations, by the and angles, can be combined into a single 23 rotation by an angle . The latter angle is not necessarily simply given by the sum because of the possible effect of the phases in , (see further, eq. (68)). Nevertheless, the combination entering the PMNS matrix is surely a unitary matrix acting on the 23 block and, as such, it can be written as , where are diagonal matrices of phases and . Moreover, we can write , where and are diagonal matrices of phases that commute with the 12 transformations and either are unphysical or can be reabsorbed in other phases. The PMNS matrix can therefore be written as 
where the angle can have any value, is a diagonal matrix of unphysical phases, contains the two Majorana CPV phases, and contains the only Dirac CPV phase. The explicit relation between the physical parameters , and the original parameters of the model (, , and the two phases in ) can be useful to connect our results to the predictions of specific theoretical models. We provide it in Appendix A.
The observable angles in the standard PMNS parametrisation are given by
The rephasing invariant related to the Dirac CPV phase, which determines the magnitude of CP violation effects in neutrino oscillations , has the following well known form in the standard parametrisation:
At the same time, in the parametrisation given in eq. (13), we get:
The relation between the phases and present in the two parametrisations is obtained by equating eq. (15) and eq. (16) and taking also into account the corresponding formulae for the real part of . To leading order in , one finds the approximate relation (see further eqs. (27), (28) and eqs. (32) and (33) for the exact relations).
In this work we aim to go beyond the simplest cases considered already, e.g., in , where the charged lepton corrections to neutrino mixing are dominated only by the angle and is fixed at the maximal value , and consider the case in which is essentially free. A deviation of from can occur in models in which (BM, TBM) because of the charged lepton contribution to , or in models in which itself is not maximal (LC). This choice allows to account for a sizeable deviation of from the value , which appears to be suggested by the data . If we keep the assumption , the atmospheric mixing angle would be given by
This in turn would imply that the deviation from maximal atmospheric neutrino mixing corresponding to the observed value of is relatively small, as shown in Fig. 1.
As for the neutrino angle , we will consider two cases:
bimaximal mixing (BM): (as also predicted by models with approximate conservation of );
tri-bimaximal mixing (TBM): .
Since in the approach we are following the four parameters of the PMNS matrix (the three measured angles , , and the CPV Dirac phase ) will be expressed in terms of only three parameters (the two angles , and the phase ), the values of , , and will be correlated. More specifically, can be expressed as a function of the three angles, , and its value will be determined by the values of the angles. As a consequence, the factor also will be a function of , and , which will allow us to obtain predictions for the magnitude of the CP violation effects in neutrino oscillations using the current data on , and .
We note first that using eq. (14) we can express in terms of and :
As we have already indicated, we will use in the analysis which follows two specific values of (BM or LC); (TBM). Equation (19) will lead in each of the two cases to a new type of “sum rules”, i.e., to a correlation between the value of and the values of , and . In the case of bimaximal and tri-bimaximal , the sum rules have the form:
The expressions for in eqs. (20) and (22) are exact, while those given in eqs. (21) and (23) are obtained as expansions in the small parameter . The latter satisfies if and are varied in the intervals quoted in Table 1. To leading order in the sum rule in eq. (21) was derived in .
Taking for simplicity for the best fit values of the three angles in the PMNS matrix , and (see Table 1), we get:
Equating the imaginary and real parts of in the standard parametrisation and in the parametrisation under discussion one can obtain a relation between the CPV phases and . We find for the BM case ():
Since, as can be easily shown,
we indeed have to leading order in , and .
The expressions for and in eqs. (27) and (28) are exact. It is not difficult to check that we have . Using the result for , eq. (24), we can get expressions for and in terms of , and . We give below the result for :
Numerically we find for , and :
Therefore, we have . For fixed and , increases with the increasing of . However, cannot increase arbitrarily since eq. (20) and the measured values of and imply that the scheme with bimaximal mixing under discussion can be self-consistent only for values of , which do not exceed a certain maximal value. The latter is determined taking into account the uncertainties in the values of and in Section 3, where we perform a statistical analysis using the data on , , and as given in .
In a similar way we obtain for the TBM case ():
The results for and we have derived are again exact and, as can be shown, satisfy . Using the above expressions and the expression for given in eq. (22) and neglecting the corrections due to , we obtain and . With the help of eq. (25) we can express and in terms of , and . The result for reads:
For the best fit values of , and , we find:
Thus, in this case or . For and the same values of and we get and .
The fact that the value of the Dirac CPV phase is determined (up to an ambiguity of the sign of ) by the values of the three mixing angles , and of the PMNS matrix, eqs. (30) and (34), are the most striking predictions of the scheme considered with standard ordering and bimaximal and tri-bimaximal mixing in the neutrino sector. For the best fit values of , and we get and or in the cases of bimaximal and tri-bimaximal mixing, respectively. These results imply also that in the scheme with standard ordering under discussion, the factor which determines the magnitude of CP violation in neutrino oscillations is also a function of the three angles , and of the PMNS matrix:
This allows to obtain predictions for the range of possible values of using the current data on , and . We present these predictions in Section 3. The predictions we derive for and will be tested in the experiments searching for CP violation in neutrino oscillations, which will provide information on the value of the Dirac phase .
We would like to note that the sum rules we obtain in the BM (LC) and TBM cases, eqs. (30) and (34), differ from the sum rules derived in  using van Dyck and Klein type discrete symmetries (, , , etc.), and in  on the basis of GUT and , and symmetries. More specifically, for the values of , and , compatible with current global neutrino oscillation data, for instance, the predictions for the value of the CPV phase obtained in the present study differ from those found in [23, 24]. The same comment is valid also for the possible ranges of values of and found by us and in . Our predictions for agree with the ones reviewed in  in the context of charged lepton corrections, once we take the particular case .
2.2 Inverse Ordering
As anticipated, we also study for completeness the case where the diagonalisation of the charged lepton mass matrix gives rise to the inverse ordering in eq. (12). The PMNS matrix, in this case, can be written as 
where unphysical phases have been eliminated, contains the two Majorana phases, and . Unlike in the case of standard ordering, it is not possible to combine the 23 rotation in the neutrino and charged lepton sector and describe them with a single parameter, . After fixing and , we therefore have, in addition to the Majorana phases, four independent physical parameters, two angles and two phases, one more with respect to the case of standard ordering. In particular, it is not possible anymore to write the mixing matrix in terms of one physical Dirac CPV phase only. Thus, in this case the four parameters of the PMNS matrix (the three angles , and and the Dirac CPV phase ) will be expressed in terms of the four parameters of the inverse ordering parametrisation of the PMNS matrix, eq. (37). We have for , and :
Given that the expressions for and do not depend on the value of , they will be the same for bimaximal and tri-bimaximal mixing (in both cases ):
where the phase . The expression (41) for is approximate, the corrections being of the order of or smaller.
For each value of the phase , any value of and in the experimentally allowed range at a given C.L., can be reproduced for an appropriate choice of , and . This is not always the case for the solar neutrino mixing angle , as we will see in Section 4. Using eqs. (39), can be expressed in terms of and as follows:
bimaximal mixing (BM), :
tri-bimaximal mixing (TBM), :
The expressions for in eqs. (42) and (44) are exact, while those given in (43) and (45) are obtained as expansions in in which the terms up to and , respectively, were kept. Note that the corrections to the approximate expressions for are negligibly small, being . This together with eq. (43) and the ranges of allowed values of and quoted in Table 1 suggests that the bimaximal mixing scheme considered by us can be compatible with the current () data on and only for a very limited interval of negative values of close to ().
It follows from eqs. (42) and (44) that the value of is determined by the values of the PMNS angles and . At the same time, depends on two parameters: and . This implies that the values of and are correlated, but cannot be fixed individually using the data on .
It is not difficult to derive also the expressions for the factor in terms of the inverse ordering parameters in the two cases of values of of interest:
We have not discussed here the LC case (conservation of the lepton charge ) as it involves five parameters (, , , and two CPV phases). At the same time, the “minimal” LC case with is equivalent to the standard ordering case with BM mixing (i.e., with ) analised in detail in the previous subsection.
As in the case of the standard ordering, to obtain the CPV phase of the standard parametrisation of the PMNS matrix from the variables of these models, that is the function , we equate the imaginary and real parts of in the two parametrisations.
3 Results with Standard Ordering
In the numerical analysis presented here, we use the data on the neutrino mixing parameters obtained in the global fit of  to constrain the mixing parameters of the setup described in Section 2. Our goal is first of all to derive the allowed ranges for the Dirac phase , the factor and the atmospheric neutrino mixing angle parameter . We will also obtain the allowed values of and . We start in this Section by considering the standard ordering setup, and in particular the two different choices for the angle : (BM and LC), (TBM).