Investigating the sterile neutrino parameters with QLC in 3 + 1 scenario

In the scenario with four generation quarks and leptons and using a 3 + 1 neutrino model having one sterile and the three standard active neutrinos with a $4 \times 4$ unitary transformation matrix, $U_{PMNS_{4}}$, we perform a model-based analysis using the latest global data and determine bounds on the sterile neutrino parameters i.e. the neutrino mixing angles. Motivated by our previous results, where, in a quark-lepton complementarity (QLC) model we predicted the values of $\theta_{13}^{PMNS}=(9_{-2}^{+1})^{\circ}$ and $\theta_{23}^{PMNS}=(40.60_{-0.3}^{+0.1})^{\circ}$. In the QLC model the non-trivial correlation between $CKM_4$ and $PMNS_4$ mixing matrix is given by the correlation matrix $V_{c_{4}}$. Monte Carlo simulations are performed to estimate the texture of $V_{c4}$ followed by the calculation of $PMNS_4$ using the equation, $U_{PMNS_{4}}= (U_{CKM_{4}} . \psi_{4})^{-1}.V_{c_{4}}$, where $\psi_{4}$ is a diagonal phase matrix. The sterile neutrino mixing angles, $\theta_{14}^{PMNS}$, $\theta_{24}^{PMNS}$ and $\theta_{34}^{PMNS}$ are assumed to be freely varying between $(0-\pi/4)$ and obtained results which are consistent with the data available from various experiments, like No$\nu$A, MINOS, SuperK, Ice Cube-DeepCore. In further investigation, we analytically obtain approximately similar ranges for various neutrino mixing parameters $\mid{ U_{\mu 4}}\mid ^2$ and $\mid{ U_{\tau 4}}\mid ^2$.


Introduction
After the completion of a few decades since the birth of Neutrino Physics and its experimental world, we are at a stage where we have unraveled various mysteries, including very strong evidence of neutrinos being massive and the existence of neutrino oscillation, but there are many issues that still need to be resolved. The recent results from Daya Bay, CHOOZ and other experiments [1][2][3][4][5] on the relatively large value of θ P M N S 13 , a clear 1 st -order picture of the three flavor lepton mixing matrix have emerged [6][7][8]. So, according to the current experimental situation we have measured all the quark and charged lepton masses, and the value of the difference between the squares of the neutrino masses ∆m 2 12 = m 2 2 − m 2 1 and |∆m 2 23 | = |m 2 3 − m 2 2 |. We also know the value of the quark mixing angles and the mixing 12 ≈ 13 • , θ CKM 23 ≈ 2.4 • and θ CKM 13 ≈ 0.2 • , a disparity-cum-complementarity between quark and lepton mixing angles is noticed. Since the quarks and leptons are fundamental constituents of matter and also that of the Standard Model(SM), the complementarity between the two of them is seen as a consequence of some symmetry at high energy scale. This complementarity popularly named 'Quark-Lepton Complementarity'(QLC) have been explored by several authors [9][10][11][12][13][14]. The relation is quite appealing to do the theory and phenomenology; however, it is still an open question, what kind of symmetry could be there between these fundamental particles of two sectors. The possible consequences of the QLC have been widely investigated in the literature. In particular, a simple correspondence between the U P M N S and U CKM matrices have been proposed and used by several authors [15][16][17][18] and analyzed in terms of a correlation matrix V c [19][20][21][22].
The fact that lepton flavors mix as well as oscillate leads to a new window of physics beyond the SM. Neutrino mixing may fill many voids of SM but still, there are few anomalies that could not be explained within the three flavor framework of neutrinos and points towards the existence of another flavor of neutrinos i.e.(sterile neutrino) with a mass ∼ eV scale. Sterile neutrinos are the singlets of the Standard Model gauge symmetries that can couple to the active neutrinos via mixing only. Till now there are bounds on the activesterile mixing, but there is no bound on the number of sterile neutrinos and on their mass scales. The existence of sterile neutrinos is investigated at different mass scales by various experiments; LSND, MiniBooNE, MINOS, Daya Bay, IceCube etc..
The main motivation behind this work is basically testing our model in the 4 th generation scenario. After the successful results obtained in our previous papers, we have tried to extend our model and complete analysis in 3 + 1 scenario. Along with that the major motivating factor that pushed us towards the extension of our model in 3+1 scenario is that the results obtained in our previous works [19][20][21][22](and references therein) are quite consistent with the recent results from NoνA [23] and IceCube [24] which give us new ray of hope in favour of our model and its stability. Taking into account the precise results obtained from various experiments on neutrino mixing angles for three generations our model and its predictions fit quite well. As we all are aware of the fact that with the pace of time we are entering into the better precision era, so considering that we have tried to extend our analysis to 3 + 1 scenario with more accurate Wolfenstein parameters for U CKM 4 and neutrino mixing parameters for U P M N S 4 preserving unitarity up to 4 th order. Using all the parameters that are available from the global data analysis we tried to investigate the structure of the 4 × 4 correlation matrix V c 4 , numerically. According to our investigations, there is a possibility of the existence and role of sterile neutrinos in the QLC, that helped us to give some constrained results for two sterile neutrino mixing angles i.e. θ P M N S 24 and θ 34 . On the stability of the framework that we have used in order to carry out our entire analysis which is the extension of SM in 4 th generation, one might argue that the four generation scenarios are strongly disfavoured. This is true, unless a substantial modification is realized for the scalar sector. However, such an extension of the Standard Model (with massive neutrinos) is excluded by several authors eg. in references [25,26]. As such, during the starting period of the discovery of Higgs particle, data was not so precise that the possibility of the 4 th generation coupling to the Standard Model Higgs doublet was not introduced in a different Beyond Standard Model scenarios. But, such options were ruled out as the LHC data develops gradually, which let many scenarios to go beyond SM(BSM). It has been noted that such 4 th generation is hidden during the single production of the Higgs Boson, while it shows up when one considers the double Higgs production i.e. gg → hh which can be considered in a different framework of a two Higgs doublet model (2HDM) [27][28][29][30]. This is the framework that we have taken to carry out our analysis which is well favored by the work done and published in 2018 [31]. In that work, they show that the current Higgs data does not eliminate the possibility of a sequential 4 th generation that gets their masses through the same Higgs mechanism as the first three generations.
In this paper, we have compared our data with the recent results provided by some ongoing experiments. Starting from the beginning of the sterile neutrino search by LSND and MiniBooNE anomalies we have covered a long distance till these recent experiments like NoνA, SuperK, MINOS, and Ice Cube-DeepCore and many more for the search of sterile neutrinos. In this paper we do not comment upon or explain the generation of sterile neutrinos or the 4 th generation quarks instead we have used the QLC model and done some numerical analysis to obtain bounds on the values of sterile neutrino parameters using previously formulated U P M N S 4 and U CKM 4 [32][33][34][35].
The paper is organized in five sections as follows. In the next section 2, we describe in brief the theory of the QLC model and show how this model fits in the 3+1 scenario. For the generation of 4×4 V c4 matrix different parametrizations were taken for formulation of U CKM 4 and U P M N S4 and all that is discussed in section 2. The investigation of correlation matrix (V c4 ), using Monte Carlo simulation is done in section 3. The PMNS matrix followed by the constrained values of sterile neutrino angles is obtained using the model equation in section 4 along with the results obtained using the QLC model are compared with bounds given by the global data analysis and various experiments. Finally, the conclusions are summarized in section 5. Various plots(contour and scattered plots) have been made in order to show the correlation between the θ 24 & θ 34 and | U µ4 | 2 & | U τ 4 | 2 as well as for sin 2 θ 24 and sin 2 θ 34 against sin 2 θ 12 and sin 2 θ 23 . We have also obtained the normal distribution function histograms for both θ 24 and θ 34 for different values of m t = 400/600GeV .

Theoretical Framework of QLC Model in 3+1 scenario
The mixing of quarks and leptons have always been of great interest and remains a mystery in particle physics. The search for symmetry or unification of quarks and leptons is one of the goals of particle physics, and many efforts are devoted toward this work. The bottom-up approach i.e., finding some phenomenological relations as well as their explanations, gives some clues on this issue. In the SM the mixing of quark and lepton sectors is described by the matrices U CKM and U P M N S . When we observe a pattern of mixing angles of quarks and leptons, and combine them with the pursuit for unification i.e. symmetry at some high energy leads the concept of quark-lepton complementarity i.e. QLC. Possible consequences of QLC have been widely investigated in the literature and in particular, a simple correspondence between the PMNS and CKM matrices have been proposed and analysed in terms of a correlation matrix V c . As long as quarks and leptons are inserted in the same representation of the underlying gauge group at some higher energy scale, we need to include in our definition of V c arbitrary but non-trivial phases between the quark and lepton matrices.
Hence, we will generalize the relation where V c is the correlation matrix defined as a product of U P M N S and U CKM . When sterile neutrinos are introduced in the 3 + N s schemes, where the N s (N s = 1 for one sterile mixing) is the number of new mass eigenstates. For this case, we define ψ 4 in place of ψ in equation 1. So, along with ψ 4 in QLC the above equation takes the form where the quantity ψ 4 is a diagonal matrix ψ 4 = diag(e ιψ i ) and the four phases of ψ i are set as free parameters because they are not restricted by present experimental evidences. The 3+1 active-sterile mixing scheme is a perturbation of the standard three-neutrino mixing in which the 3 × 3 unitary mixing matrix U is extended to a 4 × 4 unitary mixing matrix with | U e4 |, | U µ4 | and | U τ 4 | which leads to the generation of U P M N S 4 lepton mixing matrix. However, the addition of a 4 th generation to the standard model leads to a 4 × 4 quark mixing matrix U CKM 4 , which is an extension of the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix in the standard model.

CKM 4 and P M N S 4 formulation
In order to calculate the texture of V c 4 we have used the U CKM 4 and U P M N S 4 taking reference from several works. Although the 4 th generation quarks are too heavy to produce in LHC, yet they may affect the low energy measurements, such as the quark t would contribute to b → s and b → d transitions, while the quark b would contribute similarly to c → u and t → c [32][33][34][35].
The CKM matrix in SM is a 3 × 3 unitary matrix while in the SM 4 (this is the simplest extension of the SM, and retains all of its essential features: it obeys all the SM symmetries and does not introduce any new ones), the U CKM 4 matrix is 4 × 4, matrix which can be shown as where all the elements of the matrix have their usual meanings except for b and t , which we have already defined above. In the presence of the sterile neutrino ν s , the flavor (ν α , α = e, µ, τ, s) and the mass eigenstates (ν i , i = 1, 2, 3, 4) are connected through a 4 × 4 unitary mixing matrix U, which depends on six complex parameters [36,37]. Such a matrix can be expressed as the product of six complex elementary rotations, which define six real mixing angles and six CP-violating phases. Of these phases three are of the Majorana type and are unobservable in oscillation processes, while the remaining three are of the Dirac type. A particularly convenient choice of the parametrization of the mixing matrix is where R ij andR ij represents a real and complex 4×4 rotation in the (i, j) plane, respectively containing the 2 × 2 sub matrices where

Numerical Simulation and Methodology
In the standard parametrization of U CKM and U P M N S in equation, we have inserted the observed/experimental values of the U CKM and U P M N S parameters, and obtained the probability density texture of the correlation matrix (V c ) using Monte Carlo method for 1 billion shots for each variable. We have fully used freedom of the unknown parameters like ψ and φ by varying them in the unconstrained spread [0 − 2π] with flat distribution. Now, we write this expression is the inverse of equation (2), which was used to estimate the texture of the correlation matrix V c 4 . Using equation 5 we have reverted back the results of the exercise in order to predict the unknown sector of U P M N S . In the inverse equation the generalised correlation matrix V c 4 thus obtained was basically used to be replaced by bimaximal(BM) and tribimaximal(TBM) matrices in the previous year by several authors [19][20][21][22](and references therein).
As such, this model procedure using numerical method of Monte Carlo and freedom of unknown parameters is having merit as its predictive power shown in our previous works published in quality journals [19][20][21][22].
The U CKM 4 matrix can be described, with appropriate choices for the quark phases, in terms of 6 real quantities and 3 phases. We have used the Dighe-Kim (DK) parameterization of the CKM 4 matrix [32][33][34][35]. This matrix can be calculated in the form of an expansion in powers of λ such that each element is accurate up to a multiplicative factor of [ where all the elements of U CKM are unitary up to O(λ 4 ). The above expansion corresponds to the Wolfenstein parametrization with C = √ ρ 2 + η 2 and δ ub = tan −1 ( η ρ ). Constraints on all the elements of CKM 4 matrix formulated using DK parametrization can be obtained by using the unitarity of the CKM 4 matrix. Through a variety of independent measurements, the SM 3 × 3 submatrix have been found to be approximately unitary. The values of CKM 4 parameters are taken from [32][33][34][35] where they perform the χ 2 -fit at two values of t mass i.e. m t = 400GeV & m t = 600GeV . The 4 thgeneration quark masses are constrained to a narrow band, which increases the predictability of the SM 4 . Along with that, they have also generated a fit for the 4 Wolfenstein parameters of the CKM matrix in the SM, in order to check for consistency with the standard fit. The results summarised are shown in  U P M N S for our analysis we have taken a basis where the charged lepton mass matrix is diagonal. Therefore, the lepton mixing matrix is simply U P M N S = U . So any complex symmetric 4 × 4 light neutrino mass matrix can be written as where M diag ν = diag(m 1 , m 2 , m 3 , m 4 ) is the diagonal form of the light neutrino mass matrix. The diagonalizing matrix U is the 4 × 4 version of the PMNS leptonic mixing matrix which is parametrized as where the rotation matrices R,R can be further parametrized as(for example R 24 andR 34 ) where C ij ≡ cos θ ij , S ij ≡ sin θ ij andS ij ≡ S ij e −ιφ ij and here, φ ij are the lepton Dirac CP phases. These phases are generalised as φ as these are unconstrained and we used the same range of spread for all [0 − 2π] with flat distribution.
The value of CP violation phases φ have been kept open varying freely between (0 − 2π) and the values used for θ 14 , θ 24 and θ 34 are assumed to vary freely between (0 − π/4). The reason behind this specific limit (0 − π/4) is that all the values obtained using our reference experiments i.e. NoνA, MINOS, SuperK, and IceCube-DeepCore [39][40][41][42][43] vary between this similar range so instead of taking a specific value we have take that whole range in our model. The  After performing the Monte Carlo simulations we estimated the texture of the correlation matrix (V c 4 ) for two different values of m t = 400GeV & 600GeV (where m t is the mass of t ) and implemented the same matrix in our inverse equation and obtained the constrained results for the sterile neutrino parameters. We obtained predictions for θ 24 and θ 34 and then compared our results with the current experimental bounds given by NoνA, MINOS, SuperK, and IceCube-DeepCore [39][40][41][42][43] experiments.

Results
We have divided our results in two parts i.e. for m t = 400GeV and m t = 600GeV . The P M N S 4 matrix obtained in case of m t = 400GeV is  During the analysis of the above results, one can observe that values obtained by QLC(400/600 GeV) lies close to the results obtained from the NoνA and IceCube-DeepCore experiments. Numerically in all four equations 8, the effect of the sterile neutrino mixing parameters can be clearly seen on one another. We report interrelated behaviour between sin 2 θ P M N S    obtained using the QLC model. In order to analyse their impact more accurately we have made contour plots in the figure 7. If, the mixing angles are expressed in terms of the relevant matrix elements(eq. 8) then the limits on θ 34 and θ 24 becomes | U µ4 | 2 and | U τ 4 | 2 at the 99.7% C.L. i.e. 3σ range. This whole analysis is very less sensitive to θ 14 which is constrained to be small by reactor experiments [44] as well as the QLC model analysis via which the value obtained is smaller as compared to the other two angles. The contour plots shown in the corresponding figures 7 are depicting the correlation between θ 34 and θ 24 for m t = 400GeV & 600GeV and | U µ4 | 2 and | U τ 4 | 2 again for m t = 400GeV & 600GeV , respectively. These plots clearly depict constrained ranges of θ 34 and θ 24 as well as | U µ4 | 2 and | U τ 4 | 2 which are comparable to the experimental results obtained by NoνA and IceCube-DeepCore [23,24].

Conclusions
In the desire to understand the depth of the quark and lepton world and their varying scenario, the quark-lepton symmetry and unification field have drawn a lot of attention of many researchers in recent years. Out of the different aspects that imply the symmetry and unification in the quark and lepton sectors, the QLC relations between the mixing angles of the U CKM and U P M N S matrices have been considered very interesting and suggestive. Motivated by previous works towards its understanding, in this paper, we have made an attempt to take forward our previous work in a completely new direction to explain the QLC Model with sterile neutrinos. We have introduced a totally new approach of QLC involving the existence of sterile neutrino using 3+1 scenario.
The detailed analysis is done for the non-trivial relation between the U P M N S 4 and U CKM 4 mixing matrices along with the phase mismatch between the quarks and leptons via ψ 4 the diagonal matrix phase. Using all the parameters that are available from the global data analysis we have investigated the structure of the correlation matrix V c 4 , numerically for two different values of m t = 400GeV & 600GeV . We have obtained results for two sterile neutrino mixing angles i.e. θ P M N S 24 and θ P M N S 34 and two elements of the P M N S 4 mixing matrix i.e. | U µ4 | 2 and | U τ 4 | 2 then compared the upper limits with different experiments mentioned above. We have compared just the upper limits because only the upper limits of all the parameters are available collectively in the recent results from these experiments [39][40][41][42][43]. The whole analysis gives us the hint about the relevance of the sterile neutrinos with the quark-lepton unification and the model we have been using i.e. Quark-Lepton Complementarity. The complete understanding of such wide dissimilarity between the quark and lepton mixing patterns are considered to be one of the biggest challenges for the physics beyond the standard model.
In present world keeping in view all experiments these results are still favoured by the present experimental data within their measurement errors and if these QLC relations are not accidental, they strongly suggest the common connection between quarks and leptons at some high energy scale. Although it is very hard to understand these type of relations in ordinary bottom-up approaches, where the quarks and leptons are treated separately with no specific connection is seen between them. We do require some top-down approaches like the Grand Unified Theories(GUT) which sometimes also unify quarks and leptons and provide a framework to construct a model in which QLC relation can be embedded in a natural way. With this endeavor to perform numerical simulations in order to investigate the sterile neutrino mixing angles, one might have an eye to look forward towards the understanding of the QLC model in a better way. The connoisseur eye and deep knowledge with some theoretical ground can explain QLC relations precisely. The results obtained numerically and analytically in this paper can be noticed in a good agreement with the experimental data.