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ISRN High Energy Physics

Volume 2012 (2012), Article ID 954272, 5 pages

http://dx.doi.org/10.5402/2012/954272

## CP Violating Tri-Bimaximal-Cabibbo Mixing

^{1}Institute of Mathematics, Statistics and Scientific Computation, IMECC-UNICAMP CP, 6065, 13083-859 Campinas, SP, Brazil^{2}Department of Physics and Astronomy, Rutherford Building, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand

Received 8 August 2012; Accepted 3 September 2012

Academic Editors: K. Cho, S. I. Eidelman, H. Hayashii, and W. Li

Copyright © 2012 D. V. Ahluwalia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In view of the new data from the Daya Bay and RENO collaborations, King has presented a very natural deformation of tri-bimaximal mixing. Here, we show that *L/E* flatness of the *e*-like event ratio in the atmospheric neutrino data, when coupled with King's observation that the smallest neutrino mixing angle, , seems to be related to the largest quark mixing angle (the Cabibbo angle ), leading to a CP violating tri-bimaximal-Cabibbo mixing. King's tri-bimaximal-Cabibbo mixing follows as a leading order approximation from our result.

The precise form of the neutrino mixing matrix, , that defines the relationship between the flavour and mass eigenstates, and , respectively [1, 2], reads and the knowledge of the masses for the underlying mass eigenstates arises from yet unknown physics. Nevertheless, once the parameters that determine the mixing matrix and the mass-squared differences are deciphered from the data, one can derive their phenomenological consequences on supernova explosions [3–6], on the synthesis of elements [7], on the cosmic microwave background, and, the distribution of large-scale structure [8]. In particular, if the neutrino mixing angle , then one can obtain CP violation in the neutrino sector with many interesting physical consequences [9–11].

The T2K, MINOS, and Double Chooz indications that the smallest neutrino mixing angle may be nonzero [12–14] has now been confirmed by the results of the Daya Bay and RENO collaborations [15, 16]. King has made the observation [17] that the smallest neutrino mixing angle seems to be related to the largest quark mixing angle, the Cabibbo angle [18], or equivalently to the Wolfenstein parameter, [2, 19]: (It is worth noting that Mohapatra and Smirnov had earlier conjectured King’s observation [20, Section 3.1].) To this observation, we now add that the , where is the neutrino source-detector distance and is the neutrino energy, flatness of the -like event ratio observed for atmospheric neutrinos [21] requires that This observation was first made in reference [22]. The value of obtained in [22] was also introduced recently as an Ansatz in [23].

Global analysis of neutrino oscillation data by two independent groups shows: (a) to be for the normal mass hierarchy while allowing for the full range for the inverted mass hierarchy [24], (b) with no significant difference between the normal and inverted mass hierarchies [25]. A detailed study of these two papers reveals that there is no statistically significant indication which disfavours . Regarding : (a) the first of the mentioned groups obtains for the normal mass hierarchy and for the inverted mass hierarchy (these values are consistent with ), while (b) the second group finds a slight preference for .

Both groups agree with the tri-bimaximal mixing value for the remaining angle [24, 25] With all the angles and phases thus fixed, the neutrino mixing matrix for the choice in (3) takes the form Its counterpart, , for is obtained by letting in . As a measure of CP violation, following [2], we define the asymmetries and find where all symbols have their usual meaning. The sign holds for . For , or equivalently , the matrix reduces to the standard tri-bimaximal mixing matrix [26]. (This may be compared with [27, equation ] that gives an interpolating matrix with as a variable. In one limit the interpolating matrix gives the bimaximal mixing [28–30] and in another it yields tri-bimaximal mixing [26].)

The result (7) is modified by matter effects [31, 32]. Its general features are studied in detail by various authors [11, 33–35]. In gravitational environment, the following argument suggests that one must expect a significant modification to the result (7). Neutrino oscillations provide us with a set of flavour oscillation clocks. These clocks must redshift according to the general expectations of the theory of general relativity. In gravitational environments of neutron stars, the dimensionless gravitational potential is (cf. for Earth, ). For a given source-detector distance and a given energy, the asymmetries for supernovae modeling must be accordingly modified [36–41] at the 20% level, or thereabouts.

An examination of the matrix immediately shows that the expectation values of the and masses are identical. To the obtained above reproduces King’s result [17, equation (8)] for . The presented matrix not only accommodates the implications of the Daya Bay and RENO collaborations, but also the flatness of the -like event ratio seen in the atmospheric neutrino data while respecting all other known data on neutrino oscillations.

#### Acknowledgments

The result presented here was obtained on 10 May, 2012 and was presented the next day at a MatScience Seminar. The author would like to thank the Institute of Mathematical Sciences (MatScience, Chennai, India) for its hospitality and for its vibrant scholarly environment. The author is equally thankful to the Institute of Mathematics, Statistics and Scientific Computation (IMECC, Unicamp, Brazil) where the work was written during a sabbatical year.

#### References

- L. L. Chau and W. Y. Keung, “Comments on the parametrization of the Kobayashi-Maskawa matrix,”
*Physical Review Letters*, vol. 53, no. 19, pp. 1802–1805, 1984. View at Publisher · View at Google Scholar · View at Scopus - J. Beringer, J.-F. Arguin, R. M. Barnett et al., “Review of particle physics,”
*Physical Review D*, vol. 86, no. 1, Article ID 010001, 2012. View at Publisher · View at Google Scholar · View at Scopus - D. V. Ahluwalia-Khalilova, “Erratum: first prize essay for 1996: Neutrino oscillations and supernovae (General Relativity and Gravitation (1996) 28 (1161)),”
*General Relativity and Gravitation*, vol. 36, no. 9, pp. 2183–2187, 2004. View at Publisher · View at Google Scholar · View at Scopus - C. Lunardini, B. Müller, and H.-Th. Janka, “Neutrino oscillation signatures of oxygen-neon-magnesium supernovae,”
*Physical Review D*, vol. 78, no. 2, Article ID 023016, 2008. View at Publisher · View at Google Scholar · View at Scopus - H. Duan, G. M. Fuller, J. Carlson, and Y.-Z. Qian, “Simulation of coherent nonlinear neutrino flavor transformation in the supernova environment: correlated neutrino trajectories,”
*Physical Review D*, vol. 74, no. 10, Article ID 105014, 2006. View at Publisher · View at Google Scholar · View at Scopus - H. Duan, G. M. Fuller, J. Carlson, and Y. Z. Qian, “Flavor evolution of the neutronization neutrino burst from an O-Ne-Mg core-collapse supernova,”
*Physical Review Letters*, vol. 100, no. 2, Article ID 021101, 2008. View at Publisher · View at Google Scholar · View at Scopus - T. Yoshida, T. Kajino, H. Yokomakura, K. Kimura, A. Takamura, and D. H. Hartmann, “Neutrino oscillation effects on supernova light-element synthesis,”
*Astrophysical Journal*, vol. 649, no. 1 I, pp. 319–331, 2006. View at Publisher · View at Google Scholar · View at Scopus - J. Lesgourgues and S. Pastor, “Massive neutrinos and cosmology,”
*Physics Reports*, vol. 429, no. 6, pp. 307–379, 2006. View at Publisher · View at Google Scholar · View at Scopus - M. Y. Khlopov and S. T. Petcov, “Possible cosmological effect of CP-violation in neutrino oscillations,”
*Physics Letters B*, vol. 99, no. 2, pp. 117–121, 1981. View at Google Scholar · View at Scopus - P. H. Frampton, S. L. Glashow, and T. Yanagida, “Cosmological sign of neutrino CP violation,”
*Physics Letters*, vol. 548, no. 3-4, pp. 119–121, 2002. View at Publisher · View at Google Scholar · View at Scopus - A. B. Balantekin, J. Gava, and C. Volpe, “Possible CP-violation effects in core-collapse supernovae,”
*Physics Letters B*, vol. 662, no. 5, pp. 396–404, 2008. View at Publisher · View at Google Scholar · View at Scopus - K. Abe, Y. Hayato, T. Iida et al., “Search for differences in oscillation parameters for atmospheric neutrinos and antineutrinos at Super-Kamiokande,”
*Physical Review Letters*, vol. 107, no. 24, Article ID 241801, 2011. View at Publisher · View at Google Scholar · View at Scopus - P. Adamson, D. J. Auty, D. S. Ayres et al., “Improved search for muon-neutrino to electron-neutrino oscillations in MINOS,”
*Physical Review Letters*, vol. 107, no. 18, Article ID 181802, 2011. View at Publisher · View at Google Scholar · View at Scopus - Y. Abe, C. Aberle, T. Akiri et al., “Indication for the disappearance of reactor electron antineutrinos in the Double Chooz experiment,,”
*Physical Review Letters*, vol. 108, no. 13, Article ID 131801, 2012. View at Publisher · View at Google Scholar · View at Scopus - F. P. An, J. Z. Bai, A. B. Balantekin et al., “Observation of electron-antineutrino disappearance at Daya Bay,”
*Physical Review Letters*, vol. 108, no. 17, Article ID 171803, 2012. View at Publisher · View at Google Scholar · View at Scopus - J. K. Ahn, S. Chebotaryov, J. H. Choi et al., “Observation of reactor electron antineutrinos disappearance in the RENO experiment,”
*Physical Review Letters*, vol. 108, no. 19, Article ID 191802, 2012. View at Publisher · View at Google Scholar · View at Scopus - S. King, “Tri-bimaximal-cabibbo mixing,” http://arxiv.org/abs/1205.0506.
- N. Cabibbo, “Unitary symmetry and leptonic decays,”
*Physical Review Letters*, vol. 10, no. 12, pp. 531–533, 1963. View at Publisher · View at Google Scholar · View at Scopus - L. Wolfenstein, “Parametrization of the Kobayashi-Maskawa matrix,”
*Physical Review Letters*, vol. 51, no. 21, pp. 1945–1947, 1983. View at Publisher · View at Google Scholar · View at Scopus - R. N. Mohapatra and A. Y. Smirnov, “Neutrino mass and new physics,”
*Annual Review of Nuclear and Particle Science*, vol. 56, pp. 569–628, 2006. View at Publisher · View at Google Scholar · View at Scopus - Y. Fukuda, T. Hayakawa, E. Ichihara et al., “Evidence for oscillation of atmospheric neutrinos,”
*Physical Review Letters*, vol. 81, no. 8, pp. 1562–1567, 1998. View at Google Scholar · View at Scopus - D. V. Ahluwalia, Y. Liu, and I. Stancu, “CP-violation in neutrino oscillations and L/E flatness of the E-like event ratio at Super-Kamiokande,”
*Modern Physics Letters A*, vol. 17, no. 1, pp. 13–21, 2002. View at Publisher · View at Google Scholar · View at Scopus - X. Zhang and B.-Q. Ma, “A prediction of neutrino mixing matrix with CP violating phase,”
*Physics Letters B*, vol. 713, no. 3, pp. 202–205, 2012. View at Publisher · View at Google Scholar · View at Scopus - M. Tortola, J. Valle, and D. Vanegas, “Global status of neutrino oscillation parameters after recent reactor measurements,” http://arxiv.org/abs/1205.4018.
- G. L. Fogli, E. Lisi, A. Marrone, D. Montanino, A. Palazzo, and A. M. Rotunno, “Global analysis of neutrino masses, mixings, and phases: entering the era of leptonic CP violation searches,”
*Physical Review D*, vol. 86, no. 1, Article ID 013012, 2012. View at Publisher · View at Google Scholar · View at Scopus - P. F. Harrison, D. H. Perkins, and W. G. Scott, “Tri-bimaximal mixing and the neutrino oscillation data,”
*Physics Letters B*, vol. 530, no. 1-4, pp. 167–173, 2002. View at Publisher · View at Google Scholar · View at Scopus - I. Stancu and D. V. Ahluwalia, “L/E-flatness of the electron-like event ratio, in Super-Kamiokande and a degeneracy, in neutrino masses,”
*Physics Letters B*, vol. 460, no. 3-4, pp. 431–436, 1999. View at Google Scholar · View at Scopus - F. Vissani, “A Study of the scenario with nearly degenerate Majorana neutrinos,” http://arxiv.org/abs/hep-ph/9708483.
- D. V. Ahluwalia, “On reconciling atmospheric, LSND, and solar neutrino-oscillation data,”
*Modern Physics Letters A*, vol. 13, no. 28, pp. 2249–2264, 1998. View at Google Scholar · View at Scopus - V. Barger, S. Pakvasa, T. J. Weiler, and K. Whisnant, “Bi-maximal mixing of three neutrinos,”
*Physics Letters B*, vol. 437, no. 1-2, pp. 107–116, 1998. View at Google Scholar · View at Scopus - L. Wolfenstein, “Neutrino oscillations in matter,”
*Physical Review D*, vol. 17, no. 9, pp. 2369–2374, 1978. View at Publisher · View at Google Scholar · View at Scopus - S. Mikheev and A. Y. Smirnov, “Resonance amplification of oscillations in matter and spectroscopy of solar neutrinos,”
*Soviet Journal of Nuclear Physics*, vol. 42, pp. 913–917, 1985. View at Google Scholar - J. Gava and C. Volpe, “Collective neutrino oscillations in matter and CP violation,”
*Physical Review D*, vol. 78, no. 8, Article ID 083007, 2008. View at Publisher · View at Google Scholar · View at Scopus - J. P. Kneller and G. C. McLaughlin, “Three flavor neutrino oscillations in matter: flavor diagonal potentials, the adiabatic basis, and the CP phase,”
*Physical Review D*, vol. 80, no. 5, Article ID 053002, 2009. View at Publisher · View at Google Scholar · View at Scopus - L. S. Kisslinger, E. M. Henley, and M. B. Johnson, “Neutrino oscillation in matter and parameters s
_{13},*δ*_{cp},”*International Journal of Modern Physics E*, vol. 21, no. 7, Article ID 1250065, 2012. View at Publisher · View at Google Scholar · View at Scopus - D. V. Ahluwalia and C. Burgard, “Gravitationally induced neutrino-oscillation phases,”
*General Relativity and Gravitation*, vol. 28, no. 10, pp. 1161–1170, 1996. View at Google Scholar · View at Scopus - D. V. Ahluwalia and C. Burgard, “Interplay of gravitation and linear superposition of different mass eigenstates,”
*Physical Review D*, vol. 57, no. 8, pp. 4724–4727, 1998. View at Google Scholar · View at Scopus - K. Konno and M. Kasai, “General relativistic effects of gravity in quantum mechanics: a case of ultra-relativistic, spin 1/2 particles,”
*Progress of Theoretical Physics*, vol. 100, no. 6, pp. 1145–1157, 1998. View at Google Scholar · View at Scopus - J. Wudka, “Mass dependence of the gravitationally induced wave-function phase,”
*Physical Review D*, vol. 64, no. 6, Article ID 065009, 2001. View at Publisher · View at Google Scholar · View at Scopus - B. Mukhopadhyay, “Neutrino asymmetry around black holes: neutrinos interact with gravity,”
*Modern Physics Letters A*, vol. 20, no. 28, pp. 2145–2155, 2005. View at Publisher · View at Google Scholar · View at Scopus - P. Singh and B. Mukhopadhyay, “Gravitationally induced neutrino asymmetry,”
*Modern Physics Letters A*, vol. 18, no. 11, pp. 779–785, 2003. View at Publisher · View at Google Scholar · View at Scopus