Tetramaximal Neutrino Mixing and Its Implications on Neutrino Oscillations and Collider Signatures
Abstract
We propose a novel neutrino mixing pattern in terms of only two small integers and together with their square roots and the imaginary number . This ansatz is referred to as the “tetramaximal” mixing because it can be expressed as a product of four rotation matrices, whose mixing angles are all in the complex plane. It predicts , , and in the standard parametrization, and the Jarlskog invariant of leptonic CP violation is found to be . These results are compatible with current data and can soon be tested in a variety of neutrino oscillation experiments. Implications of the tetramaximal neutrino mixing on the decays of doublycharged Higgs bosons (for ) are also discussed in the triplet seesaw mechanism at the TeV scale, which will be explored at the upcoming LHC.
pacs:
PACS number(s): 14.60.Pq, 13.10.+q, 25.30.Pt1 Recent solar (1), atmospheric (2), reactor (3) and accelerator (4) neutrino experiments have convincingly verified the hypothesis of neutrino oscillation, a pure quantum phenomenon which can naturally occur if neutrinos are massive and lepton flavors are mixed. The mixing of lepton flavors is described by a unitary matrix , whose nine elements are commonly parametrized in terms of three rotation angles and three CPviolating phases. Defining three unitary rotation matrices in the complex (1,2), (1,3) and (2,3) planes as
(1) 
where and (for ), we can write out the standard parametrization of advocated by the Particle Data Group (5) and in Ref. (6):
(2)  
in which is a diagonal phase matrix which contains two nontrivial Majorana phases of CP violation. A global analysis of current neutrino oscillation data yields , and at the confidence level (7), but three phases of remain entirely unconstrained. The ongoing and forthcoming neutrino oscillation experiments will measure and . On the other hand, the neutrinoless doublebeta decay experiments will help to probe or constrain and .
The observed pattern of neutrino flavor mixing is certainly far beyond the imagination of many people. For instance, the trimaximal neutrino mixing proposed by Cabibbo (8),
(3) 
with being a complex cuberoot of unity (i.e., ), used to be a vivid ansatz in illustration of both large flavor mixing and maximal CP violation in the lepton sector; but it has been ruled out by current experimental data on neutrino oscillations. A simple modification of ,
(4)  
with and , which has been proposed by Harrison,
Perkins and Scott (9) and referred to as the tribimaximal
neutrino mixing matrix
The salient feature of is that its entries are all formed from small integers (, , and ) and their square roots, which are often suggestive of discrete flavor symmetries in the language of group theories. Then a natural question is whether one can construct a different but viable neutrino mixing pattern with fewer small integers. We find that the answer to this phenomenologically interesting question is affirmative: we may just use two small integers and together with their square roots and the imaginary number to build a neutrino mixing matrix which is compatible with current neutrino oscillation data. This new pattern, which will be referred to as the “tetramaximal” neutrino mixing, predicts
(5) 
and together with . Since is large and is maximal, the Jarlskog invariant of leptonic CP violation (13) turns out to be , which can give rise to appreciable effects of CP or T violation in longbaseline neutrino oscillations. Thus the tetramaximal neutrino mixing scenario is easily testable in a variety of neutrino oscillation experiments in the near future.
2 Now let us describe how to construct the new neutrino mixing matrix in terms of , and . We notice that the trimaximal mixing pattern can be decomposed as
(6) 
where and . Therefore, the tribimaximal neutrino mixing matrix arises from a product of four rotation matrices in the complex plane: three of them involve the rotation angle , and the fourth involves the rotation angle . The unique value of given above is crucial to assure that Eq. (6) can successfully reproduce the form of in Eq. (3) and then the form of in Eq. (4). Indeed, one happens to obtain from . This mysterious angle has a simple geometric explanation (14): it corresponds to the angle formed by two unequal diagonals from the same vertex of a cube.
But here we consider the possibility of . In this case, we construct a new neutrino mixing pattern in terms of four rotation matrices whose mixing angles are all :
(7)  
where . It is clear that only contains two small integers and together with their square roots and the imaginary number . Because the mixing angle in each of the four rotation matrices of is , this neutrino mixing matrix can be referred to as the “tetramaximal” neutrino mixing pattern. Some discussions about the phenomenological consequence of are in order.

Comparing between Eqs. (2) and (7), we can easily obtain the values of three neutrino mixing angles as already listed in Eq. (5). It is also straightforward to calculate the Jarlskog invariant of CP violation from Eq. (7):
(8) On the other hand, we obtain from Eq. (2) with the help of Eq. (5). We are therefore left with or equivalently . Note that the maximal value of can only be achieved from the unrealistic trimaximal neutrino mixing pattern ; i.e., . We find that leptonic CP violation in the tetramaximal mixing case is about one third of (namely, ).

To figure out two Majorana CPviolating phases and , we may redefine the phases of three chargedlepton fields and three neutrino fields such that , , and in Eq. (7) become real and positive while properly appears in the other five elements of . This exercise will yield . A more straightforward way to determine and is to calculate the effective mass of the neutrinoless doublebeta decay by using Eqs. (2) and (5),
(9) and compare this result with the one which can be directly obtained from Eq. (7). We see no interference or cancellation in the latter procedure, and thus we simply arrive at from Eq. (9). Namely, .

The offdiagonal asymmetries of , which may serve as a simple description of the geometric structure of (15), are found to be
(10) which are about  and  axes of , respectively. More explicitly, and . Hence looks more symmetric about its  axis. The fact of in the tetramaximal mixing case implies that all the six unitarity triangles of in the complex plane are different from one another, although their areas are all equal to .

The tetramaximal neutrino mixing pattern shows an apparent  flavor symmetry, (for ), as one can directly see from Eq. (7). This result has an interesting implication on the flavor distribution of ultrahighenergy cosmic neutrinos at neutrino telescopes. Given the canonical source of cosmic neutrinos, where the neutrino flavor composition is
(11) due to the pionmuon decay chain arising from energetic or collisions (16), the condition of and will lead to an exact neutrino flavor democracy at a terrestrial neutrino telescope (17):
(12) Note that such a result can also be obtained from the tribimaximal neutrino mixing pattern , which provides the condition of and (18).
In short, the relatively large values of and predicted by this tetramaximal neutrino mixing scenario makes it easily testable in the forthcoming longbaseline (reactor and accelerator) neutrino oscillation experiments.
3 In the basis where the mass eigenstates of three charged leptons coincide with their flavor eigenstates, one may reconstruct the Majorana neutrino mass matrix by using the neutrino mixing matrix and three neutrino masses (for ):
(13) 
Taking account of the tetramaximal neutrino mixing pattern given in Eq. (7), we find that and hold. Namely,
(14) 
Such a specific texture of , which can give rise to the maximal CPviolating phase in neutrino oscillations (i.e., ), is possible to result from a certain flavor symmetry and its breaking mechanism (19). Here we focus our interest on the magnitudes of , because they can in principle be determined from a number of leptonnumberviolating processes in a given model. After a straightforward calculation, we obtain
(15)  
It is then easy to verify
(16) 
where and run over , and . Since the absolute mass scale of is unknown, we consider three special patterns of the neutrino mass spectrum allowed by current neutrino oscillation data: (1) normal hierarchy with ; (2) inverted hierarchy with ; and (3) near degeneracy with . Taking and (7) as typical inputs, we are then able to calculate for three different neutrino mass hierarchies by using Eq. (15). Our numerical results for are listed in TABLE I. Note that , the effective mass of the neutrinoless doublebeta decay, is found to be eV (normal hierarchy), eV (inverted hierarchy) or (near degeneracy) in this tetramaximal neutrino mixing ansatz.
The origin of is of course modeldependent. For simplicity, we assume that results from the triplet seesaw mechanism (20). By introducing an Higgs triplet into the standard model, we can write out the following renormalizable Yukawa interaction term:
(17) 
where
(18) 
Note that can also couple to the standardmodel Higgs doublet and thus violate lepton number by two units (21). When the neutral components of and acquire their vacuum expectation values and , respectively, the electroweak gauge symmetry is spontaneously broken and the resultant Majorana neutrino mass matrix reads . A clear signature of the triplet seesaw mechanism is the existence of doublycharged Higgs bosons . If the mass scale of is of TeV, then can be produced at the LHC via the DrellYan process or through the chargedcurrent process . Note that the masses of and are expected to be nearly degenerate in a class of triplet seesaw models (20); (21); (22), and thus only (for ) and decay modes are kinematically open. Note also that the leptonic channel becomes dominant when MeV is taken (22). Therefore, we concentrate on the samesign dilepton events of , which signify the lepton number violation and serve for the cleanest collider signatures of new physics (23). The branching ratio of turns out to be
(19) 
which is completely determined by the values of and . Taking account of Eq. (15), we can estimate the magnitude of for three special patterns of the neutrino mass spectrum chosen above. Our numerical results are listed in TABLE II. The measurement of these leptonnumberviolating decay modes at the LHC will help test the tetramaximal neutrino mixing scenario and distinguish it from other neutrino mixing patterns (24) in the TeVscale triplet seesaw mechanism.
4 Motivated by the principle of simplicity, we have proposed a novel neutrino mixing pattern in terms of only two small integers and together with their square roots and the imaginary number . Different from the tribimaximal mixing scenario, our tetramaximal mixing scenario can accommodate both nonvanishing and large CP violation. Its explicit predictions include , , , , and , which are compatible with current data and can soon be tested in a variety of neutrino oscillation experiments. We have also illustrated possible implications of the tetramaximal neutrino mixing on collider signatures by taking account of the TeVscale triplet seesaw mechanism. In particular, the branching ratios of leptonic decays of doublycharged Higgs bosons (for ) have been calculated for three special patterns of the neutrino mass matrix. The results are found to be encouraging and interesting.
The flavor symmetry behind the tetramaximal neutrino mixing pattern has to be seen. It is always possible to build a specific neutrino mass model from which such a flavor mixing pattern can be derived, although this kind of model building usually relies on some natural or contrived assumptions. All in all, the tetramaximal neutrino mixing can shortly be confronted with a number of precision neutrino experiments and even the LHC. A test of its many phenomenological consequences is therefore close at hand.
Acknowledgments: This work was supported in part by the National Natural Science Foundation of China.
Neutrino mass hierarchy  

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Neutrino mass hierarchy  

Footnotes
 Note that this pattern is quite similar to the democratic neutrino mixing pattern (10), although their consequences on and are quite different.
References
 SNO Collaboration, Q.R. Ahmad et al., Phys. Rev. Lett. 89, 011301 (2002).
 For a review, see: C.K. Jung et al., Ann. Rev. Nucl. Part. Sci. 51, 451 (2001).
 KamLAND Collaboration, K. Eguchi et al., Phys. Rev. Lett. 90, 021802 (2003); CHOOZ Collaboration, M. Apollonio et al., Phys. Lett. B 420, 397 (1998); Palo Verde Collaboration, F. Boehm et al., Phys. Rev. Lett. 84, 3764 (2000).
 K2K Collaboration, M.H. Ahn et al., Phys. Rev. Lett. 90, 041801 (2003).
 Particle Data Group, W.M. Yao et al., J. Phys. G 33, 1 (2006).
 H. Fritzsch and Z.Z. Xing, Phys. Lett. B 517, 363 (2001); Z.Z. Xing, Int. J. Mod. Phys. A 19, 1 (2004).
 A. Strumia and F. Vissani, hepph/0606054.
 N. Cabibbo, Phys. Lett. B 72, 333 (1978).
 P.F. Harrison, D.H. Perkins, and W.G. Scott, Phys. Lett. B 530, 167 (2002); P.F. Harrison and W.G. Scott, Phys. Lett. B 535.
 H. Fritzsch and Z.Z. Xing, Phys. Lett. B 372, 265 (1996); Phys. Lett. B 440, 313 (1998); Phys. Rev. D 61, 073016 (2000).
 Z.Z. Xing, Phys. Lett. B 533, 85 (2002).
 For recent reviews with extensive references, see: H. Fritzsch and Z.Z. Xing, Prog. Part. Nucl. Phys. 45, 1 (2000); Altarelli and F. Feruglio, New J. Phys. 6, 106 (2004); R.N. Mohapatra and A.Yu. Smirnov, Ann. Rev. Nucl. Part. Sci. 56, 569 (2006); A. Strumia and F. Vissani, in Ref. (7).
 C. Jarlskog, Phys. Rev. Lett. 55, 1039 (1985); H. Fritzsch and Z.Z. Xing, Nucl. Phys. B 556, 49 (1999).
 T.D. Lee, “New Insights to Old Problems”, hepph/0605017.
 Z.Z. Xing, Phys. Rev. D 65, 113010 (2002).
 See, e.g., Z.Z. Xing, Phys. Rev. D 74, 013009 (2006); Z.Z. Xing and S. Zhou, Phys. Rev. D 74, 013010 (2006); Z.Z. Xing, Nucl. Phys. B (Proc. Suppl.) 168, 274 (2007); Nucl. Phys. B (Proc. Suppl.) 175176, 421 (2008); and references therein.
 Z.Z. Xing and S. Zhou, arXiv:0804.3512 [hepph].
 J.G. Learned and S. Pakvasa, Astropart. Phys. 3, 267 (1995).
 T. Kitabayashi and M. Yasue, Phys. Lett. B 621, 133 (2005); I. Aizawa, T. Kitabayashi, and M. Yasue, Phys. Rev. D 72, 055014 (2005); Nucl. Phys. B 728, 220 (2005); Z.Z. Xing, H. Zhang, and S. Zhou, Phys. Lett. B 641, 189 (2006); T. Baba and M. Yasue, Phys. Rev. D 77, 075008 (2008).
 M. Magg and C. Wetterich, Phys. Lett. B 94, 61 (1980); J. Schechter and J.W.F. Valle, Phys. Rev. D 22, 2227 (1980); T.P. Cheng and L.F. Li, Phys. Rev. D 22, 2860 (1980); R.N. Mohapatra and G. Senjanovic, Phys. Rev. D 23, 165 (1981).
 W. Chao, Z. Si, Z.Z. Xing, and S. Zhou, arXiv:0804.1265 [hepph].
 See, e.g., K. Huitu, J. Maalampi, A. Pietila, and M. Raidal, Nucl. Phys. B 487, 27 (1997); B. Dion et al., Phys. Rev. D 59, 075006 (1999); E.J. Chun, K.Y. Lee, and S.C. Park, Phys. Lett. B 566, 142 (2003); A.G. Akeroyd and M. Aoki, Phys. Rev. D 72, 035011 (2005); A. Hektor et al., Nucl. Phys. B 787, 198 (2007); T. Han, B. Mukhopadhyaya, Z. Si, and K. Wang, Phys. Rev. D 76, 075013 (2007); C.S. Chen, C.Q. Geng, and D.V. Zhuridov, arXiv:0801.2011.
 W.Y. Keung and G. Senjanovic, Phys. Rev. Lett. 50, 1427 (1983).
 See, e.g., J. Garayoa and T. Schwetz, arXiv:0712.1453; M. Kadastik, M. Raidal, and L. Rebane, arXiv:0712.3912; A.G. Akeroyd, M. Aoki, and H. Sugiyama, arXiv:0712.4019; P. Fileviez Pérez, T. Han, G.Y. Huang, T. Li, and K. Wang, arXiv:0803.3450; P. Ren and Z.Z. Xing, arXiv:0805.4292.