New mixing schemes for (3+1) neutrinos
S. Dev, Desh Raj, Radha Raman Gautam, Lal Singh
TL;DR
This paper introduces new (3+1) neutrino mixing schemes that incorporate active and sterile neutrino mixing, featuring a structured matrix with perturbations to explain observed neutrino phenomena and correlations among mixing angles.
Contribution
It proposes novel mixing schemes with a factorized matrix structure, linking flavor symmetry with small perturbations to account for experimental neutrino data.
Findings
New mixing schemes with structured matrices
Correlations among neutrino mixing angles identified
Parameter space for neutrino mixing parameters calculated
Abstract
We propose new mixing schemes for (3+1) neutrinos which describe mixing among active-active and active-sterile neutrinos. The mixing matrix in these mixing schemes can be factored into a zeroth order flavor symmetric part and another part representing small perturbations needed for generating non-zero , nonmaximal , CP violation and active-sterile mixing. We find interesting correlations amongst various neutrino mixing angles and, also, calculate the parameter space for various parameters.
| Parameter | bound at CL |
|---|---|
| 0.0098 0.031 | |
| 0.0060 0.026 | |
| Mixing | ||
|---|---|---|
| pattern | ||
| TBM | 2 1 | 1 1 |
| BM | 1 | 1 |
| DM | ||
| HM | 1 | 1 |
| GRM1 | 1 | 1 |
| GRM2 | 1 | 1 |
| TFH1 | 1 1 | |
| TFH2 | 1 1 | |
| Mixing | ||
| pattern | ||
| TBM | ||
| BM | 1 | 1 |
| DM | 2 1 | 1 1 |
| HM | 1 | 1 |
| GRM1 | ||
| GRM2 | ||
| TFH1 | 1 1 | |
| TFH2 | 1 1 |
| Mixing scheme | |||||
|---|---|---|---|---|---|
| TBM | |||||||
|---|---|---|---|---|---|---|---|
| GRM2 | |||||||
| TBM | |||||||
| TFH1 | |||||||
| TFH2 | |||||||
| HM | |||||||
| GRM1 | |||||||
| GRM2 | |||||||
| TBM | |||||||
| BM | |||||||
| GRM1 | |||||||
| GRM2 | |||||||
| Mixing scheme | |||||||
|---|---|---|---|---|---|---|---|
| - | - | - | |||||
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New mixing schemes for (3+1) neutrinos
S. Dev ,1, Desh Raj ,2, Radha Raman Gautam ,2, Lal Singh ,2 sdev@[email protected]@[email protected]
(1*Department of Physics, School of Sciences, HNBG Central University, Srinagar, Uttarakhand 246174, INDIA.
2Department of Physics, Himachal Pradesh University, Shimla 171005, INDIA.*)
Abstract
We propose new mixing schemes for (3+1) neutrinos which describe mixing among active-active and active-sterile neutrinos. The mixing matrix in these mixing schemes can be factored into a zeroth order flavor symmetric part and another part representing small perturbations needed for generating non-zero , nonmaximal , CP violation and active-sterile mixing. We find interesting correlations amongst various neutrino mixing angles and, also, calculate the parameter space for various parameters.
1 Introduction
The discovery of massive neutrinos by Super-Kamiokande experiment [1] in the year 1998 has paved the way for new physics beyond the Standard Model (BSM) of particle physics. Various neutrino oscillation experiments in the last two decades have measured the three neutrino mixing angles and two (atmospheric and solar ) mass splittings rather precisely. However, several anomalies at Short Base Line (SBL) neutrino experiments indicate scale mass splitting. These anomalies were first reported by LSND experiment [2] in their anti-neutrino flux measurements and, subsequently, confirmed by MiniBooNE experiment [3] in both the neutrino and antineutrino modes. The recent MiniBooNE data [4] also support these anomalies. In addition, reactor experiments [5] and gallium solar neutrino experiments [6] strongly support these anomalies. A possible explanation of these anomalies would require, at least, one scale mass eigenstate in the neutrino sector and the decay width of Z boson would require the fourth neutrino state to be sterile. The recent global analysis [7] of neutrino oscillations in the presence of eV-scale sterile neutrinos, supports the explanation of reactor anomaly in terms of sterile neutrino oscillations in 3+1 scenario but disfavour sterile neutrino explanation of LSND anomaly. Reactor neutrino data favour sterile neutrino oscillation with 1.3eV2 and at the confidence level (CL)[8, 7].
The recent Planck data [9] limit the effective number of relativistic degrees of freedom to (Planck TT+lowP+BAO) at CL and the sum of neutrino masses to be eV at the same confidence level. This is consistent with the bound given by the standard model of cosmology: . Although the cosmological bounds and latest Planck data disfavour the existence of scale sterile neutrinos, this apparent conflict can be resolved by considering the presence of new BSM physics . In this context it has been proposed [10] that suppressed effective mixing can lead to a decrease of if sterile neutrinos have self-interactions and the presence of additional gauge interactions can suppress the production of sterile neutrinos via flavor oscillations [11, 12]. Therefore, there is still possibility for the existence of eV scale sterile neutrinos. Also, from the theoretical standpoint, the sterile neutrinos could be the obvious candidates for right-handed neutrinos in the Standard Model of particle physics.
Super-Kamiokande has provided upper bounds on sterile neutrino parameters and at CL [13]. The recent data from reactor and other short and long baseline neutrino experiments such as MINOS[14], Daya Bay[15] etc. provide new bounds on active-sterile mixing and . Several ongoing and future long baseline experiments such as DUNE[16], T2HK[17], T2HKK[18] etc. may shed more light on neutrino oscillation physics and explore active-sterile mixing. The phenomenology and experimental constraints on (3+1) neutrinos have been reviewed in [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39].
The obvious next step is to build models that can give predictions on some neutrino parameters including the active-sterile non-trivial mixing and mass splitting of the order of scale as indicated by recent neutrino oscillation experiments. A four neutrino (3+1) scheme for explaining neutrino masses and mixing has large number of free parameters. Models with flavor symmetries can lead to specific structures of neutrino mass and mixing matrix with reduced number of free parameters. Neutrino models based on flavor symmetries have been extensively employed to explain the mixing matrix within the three neutrino framework. In lepton mass models, the residual flavor symmetries may remain intact even if the original flavor symmetry of the Lagrangian is broken. These different residual symmetries lead to different mixing matrices for the charged lepton and neutrino sectors. Neutrino mixing matrices based on residual flavor symmetries viz. tribimaximal mixing (TBM) [40], bimaximal mixing (BM) [41], hexagonal mixing (HM) [42], democratic mixing (DM) [43], golden ratio mixing I (GRM1) [44] and golden ratio mixing II (GRM2) [45] predict a vanishing reactor mixing angle () and a maximal atmospheric mixing angle () and, hence, need modifications to satisfy the data from current neutrino oscillation experiments. Two mixing schemes TFH1 (Toorop-Feruglio-Hagedorn 1) and TFH2, proposed in Ref. [46], predict nonzero and nonmaximal and need corrections to explain the three mixing angles, simultaneously. The TBM mixing matrix is given by
[TABLE]
Recent neutrino oscillation experiments have measured a non-zero which implies that (1,3) element of the neutrino mixing matrix () cannot be zero. Thus we need to modify the TBM mixing matrix to accommodate a non-zero . One simple possibility is to keep one of the columns of the TBM mixing matrix intact while modifying its remaining two columns within the unitarity constraints. This gives rise to three mixing patterns viz. , and which have their first, second and third columns identical to the TBM mixing matrix, respectively. mixing is given by
[TABLE]
reduces to the TBM mixing matrix in the special case and . mixing is given by
[TABLE]
mixing reduces to the TBM scheme in the special case and . This mixing scheme is generally known as the trimaximal mixing. mixing is given by
[TABLE]
reduces to the TBM mixing matrix in the special case and . The above partial mixing schemes have been successfully employed to explain the pattern of lepton mixing and have been extensively studied in the literature [47, 48]. Especially, the mixing gives a very good fit to the present neutrino oscillation data.
In the present work, we present new mixing schemes for (3+1) neutrinos which are essentially partial mixing schemes having either one row or one column of the mixing matrix to be the same as that of the popular mixing schemes like TBM, BM, DM, HM, GRM1, GRM2, TFH1 and TFH2. These mixing schemes accommodate active-active and active-sterile neutrino mixings. We, also, discuss general mixing schemes with one column or one row fixed with none of the mixing matrix elements equal to zero. A real mixing scheme with first or second column of the mixing matrix remaining the same as that of TBM has, already, been studied in Ref. [49] and it has been found that the mixing matrix with second column identical to TBM mixing matrix is the only viable case.
The plan of this paper is as follows. Sec. II describes the general (3+1) neutrino framework. In Sec. III, we present partial neutrino mixing schemes and study their phenomenology. Sec. IV describes the general mixing schemes with one column/row fixed. Sec. V summarizes the main results of this work.
2 The (3+1) neutrino framework
The presence of sterile neutrino(s) affects the active neutrino mixing angles via the unitarity conditions of the mixing matrix i.e., , where and . Table 1 presents the bounds on active-sterile mixing matrix elements. The experimental mass splitting are eV2, eV2 for normal mass ordering and eV2 for inverted mass ordering at CL [50]. The fourth mass splitting is eV2 (best fit)[26]. The 3 ranges of the elements of the sub matrix of with the bounds presented in [27] without imposing the unitarity of are given by
[TABLE]
In the four (3+1) neutrino framework, there are three active and one sterile neutrinos. The corresponding neutrino mixing matrix is a unitary matrix. We use the following parametrization [32] for the neutrino mixing matrix:
[TABLE]
where matrix describes rotation in plane and diagonal phase matrix contains three Majorana type CP-violating phases. The advantage of such parametrization is that for vanishing active-sterile mixing the above parametrization reduces to the standard PMNS parametrization for three active neutrinos. The six neutrino mixing angles in terms of mixing matrix elements can be written as
[TABLE]
3 Partial Mixing Schemes
A partial mixing matrix () is defined as a unitary matrix with the th column (row) fixed to be , while keeping other three columns (rows) free within the unitarity constraints. The parameters and for different mixing schemes have been summarized in Table II . is the normalization constant. One can obtain a particular partial mixing matrix or by selecting the respective values of and listed in Table II. For example, choosing and for leads to a unitary matrix with its first column identical to TBM mixing matrix. Fig 1 shows the parameter space for and under the current neutrino oscillation data for and mixing schemes.
3.1 Mixing Scheme with first column fixed to
Here, we study the mixing scheme with first column fixed according to the well known mixing schemes:
[TABLE]
The most general mixing matrix with the first column fixed to be , can be written as
[TABLE]
where , and . The phase matrix diag contains three Majorana phases. The values of and for different popular mixing schemes are summarized in Table II. The above matrix has been derived in the Appendix. Fixing one row or column of the mixing matrix provides three independent constraints on the mixing angles and CP-violating phases. Comparing the magnitudes of the elements of mixing matrix with the unitary matrix in Eq. (6) imposes the following conditions:
[TABLE]
The first condition gives
[TABLE]
For mixing, it is clear from Eq. (10) that decreases with increase in and . Fig 1(a) shows that mixing is viable only for TBM and GRM2 partial mixings. Also, mixing predicts for TBM and for GRM2.
From the second condition , we have
[TABLE]
and from third condition , we have
[TABLE]
It is clear that the six mixing angles are not independent and are related as above. Using Eqs. (7) and (9), we obtain the following relations;
[TABLE]
The mixing matrix in Eq. (9) can be factorized as
[TABLE]
where
[TABLE]
The matrix represents one of the following mixing schemes: TBM, BM, DM, HM, GRM1, GRM2, TFH1, TFH2. reproduces different mixing schemes for different values of listed in Table II (except for TFH1 and TFH2). denote small rotations in the plane and represent perturbations to different mixing schemes. As an example, the partial mixing scheme with its first column fixed to the TBM values is obtained using Eqs. (9) and (14):
[TABLE]
where
[TABLE]
[TABLE]
For numerical analysis, we generate about points randomly. We vary the parameters and within the ranges and , respectively. Parameters and are chosen from Table II corresponding to the different partial mixing schemes. The experimental constraints on neutrino parameters from neutrino oscillation experiments are summarized in Eq. (5) and Table I which have been used to check the viability of above partial mixing schemes. Only TBM and GRM2 partial mixing schemes are allowed at 3 CL For . Fig. 2 shows the correlations among different neutrino mixing angles for mixing scheme with first column fixed to be the same as that of TBM. The correlation plot shown in Fig. 2(a) between and is in the form of band (in contrast to a line in the three neutrino case) due to the presence of extra parameters from sterile sector. varies inversely with (Fig. 2(b)) which is also clear from Eq. (10).
In the context of symmetry, the origin of the first eigenvector fixed as , can be seen as the invariance of the neutrino mass matrix under a symmetry: where the symmetry generator is defined as
[TABLE]
For the partial mixing corresponding to TBM, the generator and the corresponding mass matrix are given by
[TABLE]
where . Similarly, for the partial mixing corresponding to GRM2, we have
[TABLE]
where .
3.2 Mixing Scheme with second column fixed to
The general mixing scheme with the second column fixed to is given by
[TABLE]
where . From the condition , one finds
[TABLE]
mixing for TBM, TFH1 and TFH2 partial mixings predicts . For HM, GRM1 and GRM2 partial mixings , and , respectively. Eq. (23) implies, increases with increase in and which is opposite to mixing. The second condition implies
[TABLE]
and from third condition , we have
[TABLE]
Further from Eqs. (7) and (22), we have the following relations for mixing angles:
[TABLE]
It is clear that the neutrino mixing angle corresponding to mixing schemes and are related as
[TABLE]
The parametrization can be factorized as
[TABLE]
where denote complex rotations in the plane, diag is the phase matrix and given by
[TABLE]
represents one of the mixing schemes such as TBM etc.
For numerical analysis, the parameters and are varied randomly within the ranges and , respectively. and are chosen from Table II corresponding to a particular partial mixing scheme and experimental constraints on neutrino oscillation parameters are used to check the viability of these partial mixing schemes. For mixing schemes TBM, TFH1, TFH2, HM, GRM1 and GRM2 partial mixings are allowed at 3 CL. For scheme there are almost similar correlations among neutrino oscillation parameters for all viable partial mixings and in Fig. 3, we have plotted correlations among neutrino mixing angles for TBM partial mixing.
The generator corresponding to mass matrix which leads to a mixing scheme with the second column fixed to is given by
[TABLE]
For TBM, TFH1 and TFH2 partial mixing schemes, we have
[TABLE]
For HM partial mixing, we have
[TABLE]
For GRM1 partial mixing
[TABLE]
where .
For GRM2 partial mixing, the generator and the mass matrix are given by
[TABLE]
where .
3.3 Mixing Scheme with third row fixed to
Here, we discuss the mixing scheme where the third row of the mixing matrix is fixed to . The first condition with , implies
[TABLE]
The second condition gives
[TABLE]
which limits for TBM and GRM1, for BM and for GRM2 partial mixings. From third condition , we have
[TABLE]
The general mixing scheme with third row fixed to can be parametrized as
[TABLE]
where and . The mixing scheme can be factorized as
[TABLE]
where are complex rotations in the plane while and are phase matrices and given by
[TABLE]
reproduces different mixing schemes such as TBM for different values of and . For mixing scheme we find following relations:
[TABLE]
For the numerical analysis, we generate the parameters and randomly within the ranges ) and ), respectively. The parameters and are chosen from Table II and available experimental constraints on neutrino oscillation parameters are imposed to check the phenomenological viability of these mixing schemes. Only TBM, BM, GRM1 and GRM2 partial mixings are allowed for mixing scheme at 3 CL and the mixing angle is predicted to be zero for all these cases. There are similar correlations among neutrino oscillation parameters for all viable partial mixings under . Fig. 4, shows scatter plots amongst different neutrino mixing angles for TBM partial mixing under .
3.4 Mixing Scheme with second row fixed to
The general mixing scheme with second row fixed to is given by
[TABLE]
where . The three independent conditions and give
[TABLE]
respectively. This mixing scheme predicts which is not consistent with the recent global (3+1) neutrino oscillation data [26] and is, therefore, phenomenologically ruled out.
For the partial mixing schemes discussed above, and are fixed to the values listed in Table II while the other parameters , , , , and are free. We have not considered the parametrization in which the third column will be and in which the first row will be , since they predict vanishing (1,3) element of the neutrino mixing matrix which is experimentally ruled out.
The above parametrizations have six free parameters viz. , , , , and . Thus, the six neutrino mixing angles , , , , and the three -violating phases and can be expressed in terms of six free parameters. The mixing scheme is not viable, since, it leads to a vanishing contrary to the current neutrino oscillation data given in Table I. Therefore, we have three viable parametrizations viz., , and . The full allowed parameter space for the mixing schemes , and at 3 CL is given in Table III. Table IV gives the allowed ranges of various parameters at 3 CL for the viable partial mixing schemes.
4 General Mixing Schemes with one row or one column fixed
In this section, we discuss partial mixing schemes with one column or one row fixed and none of the mixing matrix element equal to zero. Here, we study the phenomenology of mixing scheme keeping the first column or first row fixed.
4.1 Mixing Scheme with one column fixed to
Here, we discuss the possibility of having any one of the columns of mixing matrix fixed to . Any column of the mixing matrix fixed to gives three independent conditions on the magnitudes of the elements of mixing matrix viz.,
[TABLE]
where is the normalization factor. Here, we consider the general mixing matrix of the form
[TABLE]
where is the rotation matrix in the - plane with as the phase angle. and are two diagonal phase matrices. In the above mixing matrix, the three phases () can be associated with the Majorana-type CP-violating phases and can be extracted out. Using three conditions from Eq. (41), the general mixing matrix with first column fixed to becomes
[TABLE]
where
[TABLE]
with
[TABLE]
4.2 Mixing Scheme with one row fixed to
A mixing scheme with first row fixed to leads to the following three independent condition on the magnitudes of the elements of neutrino mixing matrix:
[TABLE]
where is the normalization factor. Considering the mixing scheme of the form
[TABLE]
where the three phases () associated with the Majorana-type CP-violating phases can be extracted out. Using conditions from Eq. (46), a neutrino mixing scheme with first row fixed is given by
[TABLE]
where
[TABLE]
with
[TABLE]
For the numerical analysis, about points are generated randomly. Comparing Eq. (6) with Eq. (43) for mixing and Eq. (48) for mixing, the corresponding six neutrino mixing angles can be calculated by using Eq. (7). The parameters are generated randomly within the range ) and phases are generated within the range ). For mixing scheme the parameters , and are varied randomly and for mixing these parameters are varied within the ranges: . Using available experimental constraints on neutrino oscillation parameters, we obtain the correlations among various parameters as shown in Fig. 5 for mixing and Fig. 6 for mixing. The allowed ranges for various parameters at 3 CL are given in Table V. By following the same procedure, we can obtain the mixing matrix with any column (row) fixed to ().
5 Summary
In this work, we presented new mixing schemes for (3+1) neutrinos. These include partial mixing schemes having one column or one row of the mixing matrix fixed to that of the popular mixing schemes such as TBM, BM, DM, HM, GRM1, GRM2, TFH1 and TFH2. These mixing schemes are useful to describe active-active and active-sterile neutrino mixings. These partial mixing schemes are obtained by modifications to exact mixing schemes such as TBM, DM, BM, etc. and can be factored into two parts: and . The part represents one of the popular mixing scheme like TBM etc. having flavor symmetric origin and the part can be considered as a correction to . We calculated the experimentally allowed parameter space for the parameters and . For mixing with first column fixed, only TBM, GRM2 partial mixing schemes are allowed at 3 CL while for mixing with second column fixed, partial mixings TBM, HM, TFH1, TFH2, GRM1 and GRM2 are allowed at 3 CL. For mixing having third row fixed, the allowed partial mixings are: TBM, BM, GRM1, and GRM2. We, also, studied the phenomenology of general mixing schemes for neutrinos with one column or one row fixed such that the fixed column or row has no vanishing element.
**Acknowledgements
**The research work of S. D. is supported by the Council of Scientific and Industrial Research (CSIR), Government of India, New Delhi vide grant No. 03(1333)/15/EMR-II. S. D. gratefully acknowledges the kind hospitality provided by IUCAA, Pune. R. R. G. acknowledges the financial support provided by the Council of Scientific and Industrial Research (CSIR), Government of India, New Delhi vide grant No. 13(8949-A)/2017-Pool.
6 Appendix: The parametrization
Here, we derive the mixing matrix with first column fixed to :
[TABLE]
The unknown mixing matrix elements can be written as sum of real and imaginary terms and using the orthogonality of columns, we obtain
[TABLE]
The parameters , and can be related to Majorana phases which can be factored out as a Majorana phase matrix. Therefore, substituting in above equation and using the unitarity constraints , we get
[TABLE]
where
[TABLE]
In total, there are six free parameters in the mixing matrix , viz. and . These parameters can be further reparametrized in terms of six angles and as
[TABLE]
With these redefinitions, the most general mixing matrix of type becomes
[TABLE]
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