DFTT 61/98 hep-ph/ October 6, 1998 Is bi-maximal mixing compatible with the large angle MSW solution of the solar neutrino problem? C. Giunti I

Transcription

1 DFTT 6/98 hep-ph/9807 October 6, 998 Is bi-maximal mixing compatible with the large angle MSW solution of the solar neutrino problem? C. Giunti INFN, Sezione di Torino, and Dipartimento di Fisica Teorica, Universita di Torino, Via P. Giuria, I{05 Torino, Italy Abstract It is shown that the large angle MSW solution of the solar neutrino problem with a bi-maximal neutrino mixing matrix implies an energy-independent suppression of the solar e ux. The present solar neutrino data exclude this solution of the solar neutrino problem at 99.6% CL.

2 The possibility that the neutrino mixing matrix U has the bi-maximal mixing form U = 0 p B@ p 0 p CA () p has attracted a large attention [] after the presentation at the Neutrino '98 conference of the Super-Kamiokande evidence in favor of atmospheric neutrino oscillations with large mixing []. Neutrino bi-maximal mixing is capable of explaining in a elegant way the atmospheric neutrino anomaly [, 3, 4, 5], through! oscillations due to m ev and the solar neutrino problem (SNP) [6, 7, 8, 9, 0] through e! ; oscillations in vacuum due to m 0 0 ev []. As noted in [], the results of the recent analysis of solar neutrino data presented in [3] seem to imply that neutrino bi-maximal mixing may be also compatible at 99% CL with the large mixing angle (LMA) MSW [4] solution of the SNP [5] (see Fig. of [3]). Here I would like to notice that this conclusion seems to be in contradiction with the exclusion at 99.8% CL of an energy-independent suppression of the solar e ux presented in the same paper [3] (see section IV.D). The reason of this incompatibility is that bi-maximal mixing with the m ev corresponding to the LMA solution of the SNP implies an energy-independent suppression by a factor / of the solar e ux. This can be seen following the simple reasoning presented in [6]. The mixing of the neutrino states in vacuum is given X by (see, for example, [7]) j i = U k j k i ( = e; ; ) ; () k=;;3 where the states j i ( = e; ; ) describe neutrinos produced in weak interaction processes and the states j k i (k =;;3) describe neutrinos with denite masses m k. In the bi-maximal mixing scenario the numbering of the massive neutrinos is the usual one, i.e. such that m m m 3, and m ev for the solution of the atmospheric neutrino anomaly. If m ev for the LMA solution of the SNP, we have m 3 ' m ev. Solar neutrinos have energy E MeV and the ratio m 3 =E ' m 3 =E 0 9 ev is much larger than the matter induced potential V. 0 ev in the interior of the sun. Hence, the evolution equation of the heaviest massive neutrino 3 is decoupled from that of the two light neutrinos and (see, for example, [8]). Taking also in account that in the case of bi-maximal mixing U e3 = 0, one can see that an electron neutrino m m m is the dierence between the squared masses of the two massive neutrinos kj k j k and j. In the bi-maximal mixing scenario there are three massive neutrinos,, and 3. I want toemphasize from the beginning that I do not want to criticize at all the beautiful paper [3]. I am only concerned with the interpretation of its results.

3 is created in the core of the sun as a superposition of the two light mass eigenstates and and, whatever happens during his propagation in the interior of the sun, its state when it emerges from the surface of the sun X is a linear combination of j i and j i: ji S = a k j k i ; (3) with k=; ja j + ja j =: (4) Since the massive neutrino states j k i propagate as plane waves, the state describing the neutrino detected on the Earth is X ji E = a k e ie kl j k i ; (5) k=; where L is the distance from the surface of the Sun to the detector on the Earth. The survival probability of solar electron neutrinos is then given by P sun e!e = jh eji E j : P e!e = X a k e ie kl h e j k i = X a k e iekl U k=; k=; ek : (6) Taking now into account the explicit values p U e == and U e = p = in the case of bi-maximal mixing and the fact that the neutrinos are extremely relativistic, we have P e!e = a a exp i m L E : (7) In the case of the LMA solution of the SNP m ev and the survival probability (7) oscillates with an oscillation length 4E=m 0 7 cm that is about one million times smaller than the Sun{Earth distance. Hence, the oscillations are not observable on the Earth because of averaging over the energy spectrum and only the average probability hp e!ei = (ja j + ja j )= (8) is observable. We have obtained the announced result: the LMA solution of the SNP in the bi-maximal mixing scenario implies an energy-independent suppression of the solar e ux of a factor =. Therefore, we have the apparent paradox that an energy-independent suppression of the solar e ux seems to be allowed at 99% CL by Fig. of Ref. [3] and is excluded at 99.8% CL in Section IV.D of the same paper. Notice that the two conclusions are based on the same set of data and the same theoretical calculation of the neutrino ux produced by thermonuclear reactions in the core of the sun [9]. The fact that the two cases refer to the same physical situation, i.e. an energyindependent suppression of the solar e ux, is also shown by the calculated in the 3

4 two cases. The of the right border of the LMA region 3 in Fig. of Ref. [3] is 4:3 +9: = 3:5, whereas the calculated in Section IV.D of the same paper for an energy-independent suppression of the solar e ux by a factor 0:48 is.0. Since this is the best t for an energy-independent suppression of the solar e ux, a value of =3:5 for a suppression factor 0:5 looks plausible. The solution of the apparent paradox explained above lies in a correct statistical interpretation of the allowed LMA region in Fig. of Ref. [3] and of the exclusion in Section IV.D of the same paper. The two cases have dierent statistical meanings. The allowed regions in Fig. of Ref. [3] are obtained under the assumption that the neutrino masses and mixing parameters are not known. In this case a general neutrino oscillation formula is used in the t, with the neutrino masses and mixing angles considered as free parameters. The best t in the LMA region happens to have a min = 4:3, which corresponds to a CL of 3.8% with DOF. Hence, a LMA solution is allowed at 3.8% CL and one can draw a 99% CL region corresponding to the parameters that have min +9:. The statistical analysis discussed in Section IV.D of Ref. [3] assumes that the solar e ux is suppressed by a constant factor that is the free parameter to be determined by the t. It happens that the best t has min =:0, which corresponds to a CL of 0.% with DOF. Hence, the hypothesis is excluded at 99.8% CL and no allowed region of the free parameter can be drawn. Since the two statistical analyses start from dierent assumptions, it is clear that they answer dierent questions and their conclusions cannot be compared. Moreover, it is important to notice that the test of the maximal mixing scenario with m ev does not correspond to either of the two statistical analyses. Indeed, if this scenario is assumed, we know that the solar e ux is suppressed by an energyindependent factor 0.5 and there is no parameter to t. Hence we test the hypothesis under consideration on the basis of its. The ' 3:5 indicated by Fig. of Ref. [3] implies a CL of 0.4% with 3 DOF. Therefore, the hypothesis is rejected at 99.6% CL. Notice that this exclusion is based only on the values of the elements U e, U e and U e3 of the neutrino mixing matrix. This means that also other types of neutrino mixing matrix, as those discussed in [0], are incompatible with the LMA solution of the SNP. In conclusion, I would like to emphasize that the allowed regions of the neutrino oscillation parameters calculated in the usual way (i.e. as Fig. of Ref. [3]) cannot be used to test a denite model (as the bi-maximal mixing model) because they have been obtained under dierent assumptions 4. In order to obtain allowed regions appropriate for model testing one must use the procedure described in [, ], i.e. one must consider each point of the parameter space as a model and perform a goodness of t testing with it. I think that it would be very useful if both types of allowed regions will be presented in future papers. 3 If U e3 =0,wehave sin =4jU ej ju ej (see []) and sin = corresponds to ju ej = ju ej = = p, as in the bi-maximal mixing matrix (). 4 They are useful if one wants to know the allowed range of the mixing parameters for other purposes. 4

5 Acknowledgement I would like to thank Z.Z. Xing for bringing my attention to the problem under discussion. References [] V. Barger, S. Pakvasa, T.J. Weiler and K. Whisnant, hep-ph/ ; A.J. Baltz, A.S. Goldhaber and M. Goldhaber, hep-ph/ ; Y. Nomura and T. Yanagida, hep-ph/980735; G. Altarelli and F. Feruglio, hep-ph/ ; E. Ma, hepph/ ; N. Haba, hep-ph/980755; H. Fritzsch and Z.Z. Xing, hep-ph/98087; H. Georgi and S.L. Glashow, hep-ph/980893; S. Davidson and S.F. King, hepph/980896; R.N. Mohapatra and S. Nussinov, hep-ph/980830; hep-ph/980945; G. Altarelli and F. Feruglio, hep-ph/ [] T. Kajita, Talk presented at Neutrino '98 [3]; Y. Fukuda et al., Super-Kamiokande Coll., Phys. Rev. Lett. 8, 56 (998). [3] Y. Fukuda et al., Kamiokande Coll., Phys. Lett. B 335, 37 (994). [4] R. Becker-Szendy et al., IMB Coll., Nucl. Phys. B (Proc. Suppl.) 38, 33 (995). [5] W.W.M. Allison et al., Soudan Coll., Phys. Lett. B 39, 49 (997). [6] B.T. Cleveland et al., Ap. J. 496, 505 (998). [7] K.S. Hirata et al., Kamiokande Coll., Phys. Rev. Lett. 77, 683 (996). [8] W. Hampel et al., GALLEX Coll., Phys. Lett. B 388, 384 (996). [9] J.N. Abdurashitov et al., SAGE Coll., Phys. Rev. Lett. 77, 4708 (996). [0] Y. Suzuki, Talk presented at Neutrino '98 [3], 998. [] S.M. Bilenky and C. Giunti, preprint hep-ph/9800 (998). [] R.N. Mohapatra and S. Nussinov, hep-ph/ [3] J.N. Bahcall, P.I. Krastev and A.Yu. Smirnov, preprint hep-ph/98076 (998). [4] S.P. Mikheyev and A.Yu. Smirnov, Yad. Fiz. 4, 44 (985) [Sov. J. Nucl. Phys. 4, 93 (985)]; Il Nuovo Cimento C 9, 7 (986); L. Wolfenstein, Phys. Rev. D 7, 369 (978); ibid. 0, 634 (979). [5] See, for example: J.N. Bahcall, Talk presented at Neutrino '98 [3] (hepph/98086); A.Yu. Smirnov, Talk presented at Neutrino '98 [3] (hep-ph/980948). [6] S.M. Bilenky, C. Giunti and C.W. Kim, Phys. Lett. B 380, 33 (996). 5

6 [7] S.M. Bilenky and B. Pontecorvo, Phys. Rep. 4, 5 (978); S.M. Bilenky and S.T. Petcov Rev. Mod. Phys. 59, 67 (987); R.N. Mohapatra and P.B. Pal, Massive Neutrinos in Physics and Astrophysics, World Scientic Lecture Notes in Physics, Vol.4, World Scientic, Singapore, 99; C.W. Kim and A. Pevsner, Neutrinos in Physics and Astrophysics, Contemporary Concepts in Physics, Vol.8, Harwood Academic Press, Chur, Switzerland, 993. [8] T.K. Kuo and J. Pantaleone, Rev. Mod. Phys. 6, 937 (989). [9] J.N. Bahcall, S. Basu and M.H. Pinsonneault, Phys. Lett. B 433, (998). [0] H. Fritzsch and Z.Z. Xing, Phys. Lett. B 37, 65 (996); hep-ph/980734; K.S. Kang and S.K. Kang, Phys. Rev. D 56, 5 (997); M. Tanimoto, preprint hepph/ [] E. Gates, L.M. Krauss and M. White, Phys. Rev. D 5, 63 (995). [] S.M. Bilenky and C. Giunti, preprint hep-ph/ ; Nucl. Phys. B (Proc. Suppl.) 43, 7 (995). [3] Neutrino '98 WWW page: 6