Research Journal of Recent Sciences ______ ______________________________ ______ ____ ___ ISSN 2277 - 2502 Vol. 1( 5 ), 7 2 - 7 4 , May (201 2 ) Res.J. Recent Sci. International Science Congress Association 72 Short Communication Deviation from Tri - bi - maximal mixings with Charged Lepton correction in case of Inverted and Normal Hierarchical Neutrino Mass Models Das Mrinal Kumar and Sarma Jayanta Kumar Department of Physics, Tezpur University, Napaam, Tezpur - 784028, Assam, INDIA Available online at: www.isca.in (Received 23 rd Feruary 2012 , revised 29 th Febru a ry 2012 , accepted 1 st March 2012 ) Abstract Deviation from tri - bi - maximal (TBM) mixing is an important topic in the recent neutrino physics. In this paper we try to get non - zero U e3 without disturbing the atmospheric and solar mixing angles. Present analysis is in case of inverted and normal hierarchical mass models using the charge lepton correction in the PMNS (Pontecorvo - Maki - Nakagawa - Sakata) matrix without CP violating phase. Her e we have shown that in this case one can perturb the TBM condition in order to generate a seizable value of U e3 in the range of 0.6 to 0.8, while at the same time keeping solar neutrino mixing near its measured value, which is close to and the atmosphere neutrino mixing close to . Keywords: Tri - bi - maximal mixing, Neutrino masses, charged lepton correction . I ntroduction The latest experimental data of the solar mixing angle points towards the specific form of lepton mixing matrix. This pattern is known as tri - bi - maximal mixing (TBM) or Harrison - Perkins - Scott ( HPS) 1 pattern, proposes flavour mixing matrix U PMNS as: . (1 ) Predictions of equation (1) viz. , , are consistent with atmospheric neutrino oscillation data. It also leads to . Recent values of and obtained from the global fits of neutrino oscillation are very close to those predicted by TBM are 2 - 7 : (2) and . (3) Even if nature has chosen TBM as lepton mixing scheme, one can expect deviation from it on very general grounds specially on vanishing , which is against the recent experimental data. To achieve experimental data deviation from TBM is required special on either by charge d lepton corrections, renormalization effects. If we give interest on the third mixing then we can see many author have reached none vanishing in model independent way in the 3σ range: (4) Similar values and ranges have been found in o ther independent analysis 6 - 9 . In the present paper, we try to investigate the possibility of breaking TBM with seizable value of U e3 as a result of deviation from this mixing in normal and inverted hierarchical models. Same type of work had been reported 5 in model independent way. In this paper, our aim is to implement the idea in different neutrino mass models which will be useful for any neutrino mass model using SO (10) GUT with seesaw mechanism. In our calculation, we play around the value suggested by equation (4) where we have not considered the CP phases. Main idea in the calculation is that, we try to keep the values of mixing fixed. For our analysis we have used the inverted and normal hierarchal mass models from ref 10 - 14 . These mass matrices are given by, , (5) Research Journal of Recen l Sciences ______ _ _ _______________________________ ______________ ______ __ ISSN 22 77 - 2502 Vol. 1(5), 7 2 - 7 4 , May (201 2 ) Res.J. Recent Sci. International Science Congress Association 73 Similarly, normal hierarchical mass matrix with TBM has the following form, (6) Methodology The PMNS matrix is, in general, product of two uni tary matrices (7) where diagonalizes the neutrino mass matrix and is associated with the diagonalization of the charged lepton mass matrix. Analysis on TBM has been carried out considering diagonal charged lepton mas s 15 . Also, analysis on charged lep ton correction unlike the present work have been carried out in different aspects 16 - 22 . For the present anal ysis neutrino mixing matrix without CP violation phase can be written as : (8) The unitary matrix which can be written as: (9) where we have used and . With this correction the small neutrino mass can be express considering non - diagonal charged lepton mass in seesaw mechanism. To incorporate this correction, first we calculate the elements of the f rom the relations (without CP phases) 23 - 25 . (10) and (11) where and (12) where . (13) Here , and . Different values of are taken from the given by e q uations (5) and (6). Elements of have been calculated from the relations given by 9 , and . (14) Result and Discussion U sing equation (10 - 13) elements of are calculated for both the cases normal as well as inverted. Then b y using the values of A, B and λ elements of are calculated 9 . After getting all the elements one can get the PMNS matrix with nonzero as shown in t able - 1, which is deviated from TBM condition as given by the equation (1). Table – 1 Values of A, B and λ Items Inverted hierarchical Normal hierarchical Range of A and B 0.2 - 5 0.2 - 5 Range of λ 0.104 - 0.247 0.104 - 0.247 λ value used 0.2 0.18 A and B value used 0.3 0.3 | U e3 | 2 0.064 0.049 The values of , and is taken from ref. as given in the Table - 2 below 14 : Table – 2 Values of , and Type Inverted hierarchical 0.0055 0.0165 0.05 Normal hierarchical 0.176 -- 0.03 The values of are calculated for both the cases viz. inverted hierarchical and normal hierarchical cases by taking consideration that the values of and still satisfy the conditions given in e q uations (2) and (3). Conclusion Tri - bi - maximal mixing provides a very close description of neutrino mixing angles. However the present hint of nonzero Research Journal of Recen l Sciences ______ _ _ _______________________________ ______________ ______ __ ISSN 22 77 - 2502 Vol. 1(5), 7 2 - 7 4 , May (201 2 ) Res.J. Recent Sci. International Science Congress Association 74 coming from analysis of global neutrino oscillation data may indicate that it is broken. Here, in this work, we try to use the charge lepton correction in PMNS matrix to get the deviation from TBM, which will be useful for model building ideas. There is a good scope for extension of this work wi th CP violating phases and also one can get the seizable values with running of renormalization group equation within this framework. Acknowledgement One of us (MKD) would like to thank the support of start - up grant of Tezpur University . References 1. Harrison P.F., Perkins D.H. and Scott W.G. , Tri - bimaximal mixing and the neutrino oscillation data, Phys. Lett . , B530 , 167 (2002) 2. Pakvasa S., Rodejohann W. and Weiler T.J. , Unitary Parametrization of Perturbations to Tribimaximal Neutrino Mixing , Phy. Rev . Lett . , 100 , 111801 (2008) 3. 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