Two Zero Mass Matrices and Sterile Neutrinos
Recent experimental data is indicative of the existence of sterile neutrinos. The minimal scheme that can account for the data and is consistent with cosmological observations is the 3+1 picture which consists of three predominantly active and one predominantly sterile neutrino with the fourth neutrino being heavier than the other three. Within this scheme there are two possibilities depending on whether the three light states obey normal or inverted hierarchy. In this paper we consider the two zero textures of the low energy neutrino mass matrix in presence of one additional sterile neutrino. We find that among 45 possible two zero textures for this case, 15 are consistent with all current observations. Remarkably, these correspond to the two-zero textures of a three active neutrino mass matrix. We discuss the mass spectrum and the parameter correlations that we find in the various textures. We also present the effective mass governing neutrinoless double beta decay as a function of the lowest mass.
Neutrino oscillation in standard three flavour picture is now well established from solar, atmospheric, reactor and accelerator neutrinos. The mass squared differences governing these oscillations are eV and eV. However, the reported observations of oscillations in LSND experiment  and recent confirmation of this by the MiniBooNE experiment  with oscillation frequency governed by a mass squared difference eV cannot be accounted for in the above framework. These results motivate the introduction of atleast one extra neutrino of mass eV to account for the three independent mass scales governing solar, atmospheric and LSND oscillations. LEP data on measurement of Z-line shape dictates that there can be only three neutrinos with standard weak interactions and so the fourth light neutrino, if it exists must be a Standard Model singlet or sterile. Recently this hypothesis garnered additional support from (i) disappearance of electron antineutrinos in reactor experiments with recalculated fluxes  and (ii) deficit of electron neutrinos measured in the solar neutrino detectors GALLEX and SAGE using radioactive sources . The recent ICARUS results  however, did not find any evidence for the LSND oscillations. But this does not completely rule out the LSND parameter space and small active-sterile mixing still remains allowed . Thus, the situation with sterile neutrinos remains quite intriguing and many future experiments are proposed/planned to test these results and reach a definitive conclusion .
Addition of one extra sterile neutrino to the standard three generation picture gives rise to two possible mass patterns – the 2+2 and 3+1 scenarios . Of these, the 2+2 schemes are strongly disfavored by the solar and atmospheric neutrino oscillation data . The 3+1 picture also suffers from some tension between observation of oscillations in antineutrino channel by LSND and MiniBooNE and non-observation of oscillations in the neutrino channels as well as in disappearance measurements. However, it was shown recently in  that a reasonable goodness-of-fit can still be obtained. Although introduction of more than one sterile neutrinos may provide a better fit to the neutrino oscillation data , the 3+1 scheme is considered to be minimal and to be more consistent with the cosmological data . Very recently combined analysis of cosmological and short baseline (SBL) data in the context of additional sterile neutrinos have been performed in [13, 14]. The analysis in  found a preference of the 3+1 scenario over 3+2 while the analysis in  shows that the status of the 3+2 scenario depends on the cosmological data set used and the fitting procedure and no conclusive statement can be made regarding whether it is favoured or disallowed. In fact the current cosmological observations of an weakly interacting relativistic "dark radiation" may actually prefer an additional sterile neutrino . If this radiation is attributed to extra neutrino species then the data gives the bound on the number of neutrinos as at 95% C.L. .
In this paper we consider the structure of the low energy neutrino mass matrices in presence of one extra sterile neutrino. The low energy mass matrix in the flavour basis is now complex symmetric with 10 independent entries and can be expressed in general as,
For three flavours the last row and the column would be absent and the mass matrix would contain 6 elements. In the context of three generations, a very remarkable result was obtained in  that there can be at the most two zeros in the low energy neutrino mass matrix in the flavour basis. For the three neutrino mass matrix there can be 15 possible two zero texture structures. These are the same as those shown in Table 1 for the four neutrino case after omitting the fourth row and column. Among these, only 7 textures corresponding to the A, B and C class were found to be compatible with the experimental data on neutrino oscillation . Normal Hierarchy (NH) was found to be allowed in all the textures whereas Inverted Hierarchy (IH) and Quasi-Degenerate (QD) solutions were allowed for the B and C classes. Various aspects of the two zero textures in the low energy neutrino mass matrix have been examined in [17, 18]. Recently, this analysis has been redone in [19, 20, 21, 22, 23] to take into account the recent results including the measured values of mixing angle by the reactor experiments . The analysis including the latest data and allowing the parameters to vary randomly in their 3 range shows that all 7 textures of the original analysis in  remain allowed . However, with the 1 range of parameters the scenarios become more constrained. With the oscillation parameters taken from  only A class with NH remain allowed while for oscillation parameters in  the textures belonging to and classes for IH and C for NH get excluded . This demonstrates that with precise determination of oscillation parameters the allowed scenarios would become more constrained. However, the situation may change altogether in presence of sterile neutrinos.
In this paper, we examine how many two zero textures are allowed by the current oscillation data in the low energy neutrino mass matrix when an extra sterile neutrino is present. We assume the known oscillation parameters are normally distributed with the peak at the best-fit value and 1 error as the width. First we check the status of the two zero texture solutions in the context of three generation mass matrices by this procedure. Then we check how much these conclusions change in the 3+1 scenario with one additional sterile neutrino. We also investigate if any new interesting correlations can be found specially for the sterile mixing angles. Finally, we discuss the changes expected in our result if the mass and mixing parameters are varied randomly in their 3 range instead of varying them in a Gaussian distribution peaked at the best-fit value.
Texture zero implies some of the elements are much smaller than the other elements of the mass matrix. Analysis of texture zeros puts restriction on the nature of the mass spectrum and can give rise to correlations between the mixing angles, masses and CP phases which may be confirmed or falsified by experimental observations. This can often help in understanding the underlying flavour symmetry . In case neutrino mass is generated by seesaw mechanism the texture zeros in the low energy mass matrix can be useful in identifying the possible high scale Yukawa matrices [28, 29, 30].
The plan of the paper goes as follows. In the next section we discuss our formalism. Section III discusses the results. We end in section IV with summary and conclusions.
We consider the 3+1 mass spectrum.
This can generate two possible mass orderings.
In one case the fourth neutrino is heavier than the
other three and in the other case the fourth neutrino is
lighter than the other three. LSND/MiniBooNE observations
dictate that the mass squared difference of the fourth state with
the three other states is eV.
However, the scheme in which the fourth state is lower
would be more disfavored from cosmology since there will be
three neutrino states with mass eV
which will contribute to the cosmological
energy density. Therefore, we consider the picture in which the
fourth state is heavier. Then, there are two possibilities shown in
which corresponds to a normal ordering among the active neutrinos (SNH). This gives
corresponding an inverted ordering among the active neutrinos (SIH) with the masses
|(NH or IH)||7.54||7.32 – 7.80||6.99 – 8.18|
|(NH or IH)||3.07||2.91 – 3.25||2.59 – 3.59|
|(NH)||2.43||2.33 – 2.49||2.19 – 2.62|
|(IH)||2.42||2.31 – 2.49||2.17 – 2.61|
|(NH)||2.41||2.16 – 2.66||1.69 – 3.13|
|(IH)||2.44||2.19 – 2.67||1.71 – 3.15|
|(NH)||3.86||3.65 – 4.10||3.31 – 6.37|
|(IH)||3.92||3.70 – 4.31||3.35 – 6.63|
|0.89||0.80 – 1.00||0.6 – 2|
|0.025||0.018 – 0.033||0.01 – 0.05|
|0.023||0.017 – 0.037||0.005 – 0.076|
We assume that the charged lepton mass matrix is diagonal and the mixing in the neutrino sector is solely responsible for the leptonic mixing. In the present case, the neutrino mixing matrix, can be parametrized in terms of six mixing angles (,,,,,), three Dirac type CP phases (,,) and three Majorana type CP phases (,,). The neutrino mass matrix in flavour basis is given by
where represent rotation in the ij generation space, for instance:
= , =
with and . The diagonal phase matrix has the form
The best-fit values and the 1 and 3 ranges of the oscillation parameters in the 3+1 scenario are given in Table 2. One can define three mass ratios
The two zero textures in the neutrino mass matrix give two complex equations viz.
where a, b, p and q can take the values , , and . The above eqn.(2.4) can be written as
The modulus of these quantities gives the magnitudes , while the argument determines the Majorana phases and .
Thus, the number of the free parameters is five, the lowest mass (NH) or (IH), three Dirac and one Majorana type CP phases. We can check for the two mass spectra in terms of the magnitude of the mass ratios , and as,
SNH which corresponds to , and
SIH which implies , and
Thus, it is which determines if the hierarchy among the
three light neutrinos is normal or inverted. Note that if
the three light neutrinos are quasi-degenerate then we will have .
Unlike the three generation case discussed in [17, 18]
the lowest mass can not be determined in the four neutrino analysis
in terms of and since these ratios also depend on
through . Thus, we keep the lowest mass as a free parameter.
To find out the allowed two zero textures we adopt the following procedure.
We vary the lowest mass randomly from 0 to 0.5 eV. All the five mixing angles in Table 2 (apart from ) and the three mass squared differences are distributed normally about the best-fit values with the errors as given in Table 2. The three Dirac and one Majorana type CP phase as well as the remaining mixing angle are randomly generated. Then, we use the above conditions to find out which mass spectrum is consistent with the particular texture zero structure under consideration. We also calculate the three mass squared difference ratios
The ranges of these three ratios calculated from the experimental data are
The allowed textures are selected by checking that they give the ratios within the above range.
3 Results and Discussions
|Class||3 generation(Random)||3 generation(Gaussian)||3+1 generation (Gaussian)|
|B||NH, IH, QD||NH(, ), IH(, )||NH, IH, QD|
|C||NH, IH, QD||IH||NH, IH, QD|
In this section we present the results of our analysis. First we briefly discuss the results that we obtain for the two zero textures of the mass matrices. Next we present the results that we obtain for the 3+1 scenario i.e mass matrices.
3.1 Results for 3 neutrino mass matrix
For the 3 neutrino case, the lowest mass and the two Majorana phases can be determined from the mass ratios. Hence, the only unknown parameter is the Dirac type CP phase () which is generated randomly. All the other oscillation parameters are distributed normally, peaked at the best-fit and taking their one sigma error as width. We find that all 7 textures which were allowed previously remain so. However, the textures belonging to A class allow NH whereas for the B class, and admit NH and and allow IH solutions. Class C gets allowed only for IH. The D, E and F classes remain disallowed. In the 2nd column of Table 3 we summarize the results that we obtain for the two zero neutrino mass matrices with three active neutrinos using normal distribution of the oscillation parameters. The results obtained in this case are somewhat different from that obtained using random distribution of oscillation parameters. The reason for the difference stems from the different range of values of the atmospheric mixing angle used by these methods. If we assume a Gaussian distribution for around its best-fit then there is very less probability of getting the 3 range in the higher octant as these values lie near the tail of the Gaussian distribution. This disallow and for NH and and for IH . Similarly QD solutions for B class requires  and for a normal distribution of with the peak at present best-fit the 3 range extends upto and there is very little probability of getting values close to . Similarly for the C class NH and QD solutions are allowed only for values close to and hence is not admissible when Gaussian distribution of oscillation parameters about the best-fit value is assumed.
3.2 Results for 3+1 scenario
Adding one sterile neutrino, there exist in total forty five texture structures of the neutrino mass matrix which can have two zeros.
Among these the 9 cases with are disallowed as the mass matrix element contains the term which is large from the current data and suppresses the other terms. Hence, cannot vanish.
There are 21 cases where one has at least one zero involving the mass matrix element of the sterile part i.e where . This element is of the form,
The last term in this expression contains the product which is quite large as compared to first three terms and thus, there can be no cancellations. Thus, neutrino mass matrices with one of the zeros in fourth row or column are not viable.
The remaining cases are the 15 two zero cases for which none of the sterile components are zero. Thus, these also belong to the two zero textures of the three generation mass matrix. A general element in this category can be expressed as,
here, . We find all these 15 textures, presented in Table 1 get allowed with the inclusion of the sterile neutrino. This can be attributed to additional cancellations that the last term in eq. 3.2 induces. Table 3 displays the nature of the mass spectra that are admissible in the allowed textures.
In Fig.2 we present the values of vs the lowest mass for textures , , , , , , , and . This figure shows that for textures belonging to the A and E classes remains 1. Thus, these classes admit only NH solutions. The textures belonging to the D class allow NH and IH while the B,C,and F classes allow NH, IH and QD mass spectra.
The textures , , , , and are related by symmetry where for the four neutrino framework can be expressed as,
in such a way that
Note that for 3 generation case the angle in the partner textures linked by symmetry was related as . However, for the 3+1 case no such simple relations are obtained for the mixing angle . The angles and in the two textures related by symmetry are also different. For this case, the mixing angles for two textures linked by symmetry are related as
(16) (17) (18) (19)
The texture zero conditions together with the constraints imposed by the experimental data allow us to obtain correlations between various parameters specially the mixing angles of the 4 neutrino with the other three for the A and E classes. For the B, C, D and F classes one gets constraints on the effective mass governing .
In order to gain some analytic insight into the results it is important to understand the mass scales involved in the problem. The solar mass scale is eV whereas the atmospheric mass scale is eV. Normal hierarchy among the active neutrinos implies corresponding to eV. It is also possible that implying 0.009 eV - 0.1 eV. We call this partial normal hierarchy. IH corresponds to . If on the other hand eV then which corresponds to quasi-degenerate neutrinos.
For these classes we find to be mainly in the range 0.0001 eV extending up to eV. Thus, these classes allow normal hierarchy (full or partial) among the 3 active neutrinos. These classes are characterized by the condition . for the four neutrino framework can be expressed as,
For smaller values of and NH the dominant contribution to the magnitude of the above term is expected to come from the last term . Therefore, very small values of is less likely to give for normal hierarchy. However, we get some allowed points in the small regime which implies smaller values of . can be approximated in the small limit as,
The maximum magnitude of the first two terms is . Then using typical values of ( eV) from the 3 range, we obtain ) in the small limit. This is true for all the textures in the A and E class. For the class we also simultaneously need . In the small limit approximate expression for is
and the first term i.e . While the other terms are of the order (0.006 - 0.007) which implies . This is reflected in the first and second panels of Fig.3 where the correlation of and with is depicted. As increases the contribution from the first three terms in increases and becomes larger for cancellation to occur. For , this increase in helps to achieve cancellation for higher values of and therefore stays almost the same. Similar argument also apply to the class which has .
For class, in addition we have . In the limit of small , can be approximated as,
As discussed earlier implies small ) in the limit of small . Thus, the contribution from the term is . The typical contribution from the last two terms is . This implies to be in the range (0.02 - 0.04) for smaller values of . This is reflected in the third panels of Fig.3 where we have plotted the correlation of with . Since with increasing , increases to make , does not increase further. Similar bounds on are also obtained for class.
As one approaches the QD regime then the terms containing the active neutrino masses starts contributing more. So for higher values of complete cancellation leading to even at the highest value of is not possible. This feature restricts to be 0.1 eV in A class.
Textures belonging to A and E class contains which is not possible for IH in the 3 generation case since the solar mixing angle is not maximal. In the 3+1 scenario a cancellation leading to is possible for IH but it requires values of in the higher side. It also contains a strong correlation in the Majorana phases. But in the other mass elements these constraints are not satisfied simultaneously and as a result the textures that contain do not admit inverted hierarchical mass spectrum.
Since , the effective mass ( = ) governing the neutrinoless double beta decay () is vanishing for these classes.
In the 3+1 scenario, B and C classes allow all three mass spectra – NH, IH and QD assuming the known oscillation parameters to be normally distributed (cf Table 3). In this case since in the textures related by symmetry is not correlated in a simple way, the value of this angle not being in the higher octant does not play a significant role as in the 3 generation case. Among these only allow few points for smaller values of for NH. For the IH solution, larger number of points are obtained corresponding to the lowest mass 0.01 eV as is seen from Fig.2. In these textures, for higher values of the lowest mass the active neutrino contribution to the matrix elements are larger and it is easier to obtain cancellations. Hence, textures belonging to these classes show a preference for QD solutions. For these textures the effective mass governing is non-zero. In the first row of Fig.4 we present the effective mass as a function of the lowest mass for the textures , and C for both NH and IH. These two merge at higher values of the lowest mass corresponding to the QD solution. The effective mass in these textures is eV for NH and 0.02 eV for IH. If no signal is seen in future experiments then large part of the parameter space belonging to these textures can be disfavoured.
These two textures are disallowed in the 3 generation case. However for the 3+1 scenario they get allowed. NH is admissible in all the textures belonging to these classes. The reason for this is the following.
In the three active neutrino scenario, the neutrino mass matrix in a block has the elements of the order of for normal hierarchy. Thus, in general these elements are quite large and cannot vanish . However, in the 3+1 case when there is one additional sterile neutrino, the neutrino mass matrix elements get contribution from the sterile part of the form where . This term is almost of the same order of magnitude and thus can cancel the active part, resulting into the possibility of vanishing elements in the block. Thus, the zero textures which were disallowed for NH are now allowed by the inclusion of sterile neutrino (3+1 case). In the case of IH, element for three active neutrinos is always of the order of and thus the textures , , , which requires were not allowed. However, for the 3+1 scenario the extra term coming due to the fourth state helps in additional cancellations and IH gets allowed in these (cf. Table 3). For class, IH for three active neutrino is disfavoured because of phase correlations. However, with the additional sterile neutrino this can be evaded making it allowed. In the bottom row of Fig.4 we present the effective mass governing for the textures , and as a function of the lowest mass. The texture allows lower values of for NH while for IH the lowest mass is largely eV. QD solution is not allowed in D class . For F class more points are obtained in the QD regime. Future experiments on would be able to probe these regions of parameter space.
The results presented above are obtained by varying the known oscillation parameters given in Table 2 as distributed normally around their best-fit and with a width given by the 1 range of the parameters. There is a finite probability of getting the points in the 3 range of this Gaussian distribution although more points are selected near the best-fit values. However, note that for some of the parameters the 3 range obtained in this procedure is different from that presented in Table 2. Thus, our results may change if we vary the parameters randomly in their 3 range as we have seen in the 3 generation case. In Fig. 5 we show the allowed values of as a function of the lowest mass for the case where all the parameters are varied randomly in their 3 range. We find that lower values of the smallest mass get disfavoured by this method. The main reason for this is that if we use the Gaussian method then the allowed 3 range of the mixing angle is from (0.002 - 0.048) while that of is from (0.001 - 0.06). Thus, smaller values of and are possible which helps in achieving cancellation conditions for smaller values of or . But if the parameters are varied randomly in the 3 range presented in Table 2 then such smaller values of the angles are not allowed and consequently no allowed points are obtained for smaller values of masses. In particular, we obtain (NH) or (IH) 0.01 eV in all the textures However, main conclusions presented in Table 3 regarding the nature of the allowed mass spectrum for the 3+1 scenario remain unchanged though the allowed parameter space gets reduced. Specially for the A, E and C classes very few points get allowed. Fully hierarchical neutrinos () are not possible in any of the textures. Textures belonging to the B and F classes give more points in the QD regime. D class allows partial NH or IH.
Recent experimental observations make a case for enlarging the scope of three flavour oscillations to include one or more sterile neutrinos with mass around 1 eV. Although induction of more than one sterile neutrino may provide a better fit to the oscillation data the cosmological observations may be more consistent with the three active and one sterile picture with the sterile neutrino being heavier. With the addition of one sterile neutrino the parameter space describing neutrino masses and mixing at low energy increases to include four independent masses, six mixing angles and six phases. The low energy mass matrix in the flavour basis now consists of 10 independent elements as opposed to six elements for the three generation case. It is well known that for the three generation case the neutrino mass matrix in flavour basis can have at the most two zeros. In this work, we have considered the two zero mass matrices in the framework of three active and one sterile neutrino. We find many distinctive features in this case as compared to the three neutrino scenario. For the 3+1 case there can be 45 possible two zero textures as opposed to 15 for the 3 generation case. Among these 45 possible two-zero textures only 15 survive the constraints from global oscillation data. Interestingly these 15 cases are the 15 two-zero textures that are possible for three active neutrino mass matrices. While for the three active neutrino case only 7 of these were allowed addition of one sterile neutrino make all 15 cases allowed as the sterile contribution can be instrumental for additional cancellations leading to zeros. All the allowed textures admit NH. The classes B, C, F also allow IH and QD solutions in addition. The results are summarized in Table 3.
If we vary the mass and mixing parameters normally peaked at the best-fit value and 1 error as the width then we find solutions for smaller values of (NH) and (IH). In this case for the textures with i.e A class and E class we obtain correlations between the mixing angles , , and the lowest mass scale . For these textures the effective mass responsible for neutrinoless double beta decay is zero. For the other allowed textures we present the effective mass measured in neutrinoless double beta decay as a function of the smallest mass scale. If however, the known oscillation parameters are varied randomly in their allowed 3 range then although the main conclusions deduced above regarding the allowed mass spectra in various textures remain the same the allowed parameter space reduces in size. In particular, we obtain a bound on the smallest mass as eV and completely hierarchical neutrinos are no longer allowed.
In this work we have concentrated on the two zero textures. However it is possible that the neutrino mass matrix in the 3+1 picture may allow the presence of more than two zeros . These results can be useful in probing underlying flavour symmetries and also for obtaining textures of Yukawa matrices in presence of light sterile neutrinos should their existence be confirmed in future experiments.
The authors thank Werner Rodejohann for his involvement in the initial stages of the work and many useful discussions. They would also like to thank Anjan Joshipura for helpful discussions.
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