SUPERLUMINAL NEUTRINOS IN THE FRAMEWORK OF EXTENDED STANDARD MODEL

Transcription

1 Fundamental Journal of Modern Physis Vol. 5, Issue, 013, Pages Published online at SUPEUMINA NEUTINOS IN THE FAMEWOK OF EXTENDED STANDAD MODE INDANATH BHATTACHAYYA Department of Mathematis Aharya Prafulla Chandra oy Government College Himahal Vihar, Matigara, Siliguri West Bengal, India Abstrat The onept of superluminal neutrinos originated from the first OPEA experiment is explored in light of the existene of imaginary mass, although the result of the future experiments negating suh onept does not affet the model proposed in this artile. It may be developed when the mass matrix of the Dira as well as Majorana neutrino field is diagonalized. One eigenvalue of suh diagonalized mass matrix is found to be imaginary, but large in magnitude; another one is obtained as very small but real. It is found that both of the resulting fields are the mixture of left handed as well as right handed fields unlie the earlier onept of seesaw mehanism. Suh treatment is also arried out for all three flavors of neutrinos, whih result six mass eigenstates, three imaginary and three real. The mass generation of neutrino in the framewor of left-right symmetri model is examined and the suitability of SO(4) model is indiated as it is onsistent in the present ontext. The osillation in the OPEA experiment is onsidered as the two fold proess in whih mass eigenstates doublet having imaginary mass is dominated over that having real mass and thus the neutrinos are found to be superluminal. Keywords and phrases: superluminal neutrinos, OPEA, speial relativity, imaginary mass of neutrino, left-right symmetri model. eeived June 13, Fundamental esearh and Development International

2 60 INDANATH BHATTACHAYYA 1. Introdution The first result of the OPEA experiment [1] (OPEA+ 1 ) has explored a new insight to the history of experimental as well as theoretial physis. An astonishing result of OPEA+ put a question mar on the feasibility of Speial Theory of elativity (ST) [4], although the phenomenon lie Cherenov radiation [5] ould be an evidene of superluminiity. No doubt, it is a revolutionary observation that neutrino may run faster than light. Therefore, either one has to repair the basi tools of the ST or to give a new model whih an explain the existene of the neutrino mass that is onsistent with OPEA+ observation. It is worth noting that the result of OPEA experiment has negated the possibility of the existene of superluminal neutrinos. In this artile, a model is proposed whih an inorporate both of the possibilities - superluminal as well as subluminal neutrinos. A series of efforts have been oming out just after publishing the result OPEA+ to address this important issue. In the ν µ ντ hannel of neutrino osillations OPEA + has measured the 5 neutrino veloity as ( v ) = [.48 ± 0.8 ( stat) ± 0.30 ( sys) ] 10, ontraditing the stringent limit obtained by SN1987A data in the energy range of few MeV [6, 7, 8]. It indiates that the muon neutrinos propagate faster than light in vauum. It is worth noting that ingli and Ma [9] interpreted this as a signal of orentz violation based on theoretial attempt [10, 11]. Hannested and Sloth [1] onsidered the superluminal behavior as the existene of light sterile neutrinos whih an propagate in the higher dimensional bul. Kehagias [13] loos that one as a loal effet aused by the salar filed originated from the earth. He predited the oupling of the salar field to the neutrinos an hange the baground metri, resulting the superluminal effet. Suh superluminosity was also explained in the framewor of extra-dimension [14]. Oda and Taira [15] onsidered this effet due to the existene of a new gauge field soured by the earth. The superluminal behavior of neutrinos is also tried to be explained by tahyoni behavior [16]. The idea of tahyoni neutrino has already been ome into existene [17, 18, 19]. Tahyons have their superluminal veloity not due to aeleration, but beause they are born with v >. There are another two ind of partiles having their veloity luminal and subluminal, nown as luxons and bradyons [0], respetively. 1 Throughout the literature OPEA+ stands for the result interpreting superluminal, whereas OPEA [, 3] is that in whih neutrinos found to be subluminal as usual ase.

3 SUPEUMINA NEUTINOS IN THE FAMEWOK 61 In the present artile, a theoretial model is proposed to address that superluminal issue. The motivation omes from the OPEA + result. Aording to the speial relativity any partile having its veloity greater than that of light (Tahyon) must have purely imaginary mass, and therefore, it is not possible to detet suh partile by any experiment. If the neutrinos are superluminal then they must be undetetable and should have imaginary mass ontraditing the smallness of the neutrino mass, evident from various experiments. Then one an thin there must be an underlying mehanism to have the neutrino veloity more than that of light in the ν µ ν τ hannel of OPEA + experiment. In this artile, a theoretial model is proposed to explain suh mehanism. First of all, it is onsidered that neutrino must have some imaginary mass, whih maes it tahyoni in nature to exeed its speed the value of. In the next setion, it is explained how one mass eigenstate of the neutrino may have the imaginary mass, whereas the other one beomes real in terms of the mass. In Setion 3, different mass generation mehanisms are outlined and it is mentioned that out of those whih one may be onsistent to the model proposed in the present artile. Finally, the underlying mehanism that maes the neutrino superluminal in the OPEA + experiment is desribed. The present model also inludes the possibility of the superluminiity being disproved by any other experiment.. Existene of Imaginary Mass of Neutrino Beyond the standard model, the seesaw mehanism in favor of neutrino mass generation is well aepted theory. It is believed that through seesaw the left handed neutrino aquires a very little mass, whereas, on other side, the right handed andidate has an extremely heavy mass. But the superluminal behavior of neutrino ompels us to reonsider that understanding. The neutrino mass inludes the Dira as well as Majorana mass terms. The agrangian of the neutrino field inorporating both of those mass terms is as follows: The ν and m m D ν ( ) ν ν h.. = + md m (1) ν ν are the spinors of the ative neutrinos, whereas ν and ν are those whih annot be observed in nature. The neutrino mass matrix, given in the equation (1) an be diagonalized to obtain the following expression.

4 6 INDANATH BHATTACHAYYA with introduing the angle θ so that φ The orresponding mass eigenvalues an be alulated as with ε 1, = 1. = m1φ1 φ1 + mφ () m tan θ = D. (3) m m ε1, ( ) ( ) = + ± + m 1, m m m m 4mD (4) The fields φ 1 and φ, present in the equation () have the form φ 1 = os θ( ν + ε1ν ) sin θ( ν + ε1ν ), (5) φ = sin θ( ν + εν ) + os θ( ν + εν ). (6) That is the basi building blo of the neutrino mass model. To emerge the neutrino as a purely Dira partile, m and m are taen to be zero, whereas purely Majorana field is evolved by onsidering m D = 0. The most interesting and well aepted ase is seesaw mehanism, in whih m >> md and m 0. In that ase mass eigenvalues beome m m D 1 m and m m. Here ase φ 1 is evolved as left handed neutrino and right handed antineutrino field, whereas φ beomes right handed neutrino and left handed antineutrino field that is hard to observe in the nature. Thus aording to the seesaw, the neutrino assoiated to φ 1 has a tiny mass and beomes left-handed in nature; but there suppose to exist another ind of right handed massive neutrino whih is yet to found in nature, may be due to its super massive struture. The well aepted seesaw mehanism fails to explain the superluminal nature of neutrinos. Therefore, a new theory is to be proposed whih will be able to interpret the OPEA + result. The basi assumption of the theoretial model proposed in this artile is that the field mass m = ν and ν are symmetri in every respet; whih follows the = m mm (say). In fat there is no good reason to assume that the

5 SUPEUMINA NEUTINOS IN THE FAMEWOK 63 neutrino mass assoiated with right handed neutrino is muh heavier than that of left handed neutrino, whih is a foundation of seesaw mehanism. This is quite omfortable to onsider both of those masses are same. In this irumstane equation (4) gives us m 1, = ε1, ( m M ± md ). (7) We are strongly motivated by the result of the OPEA + experiment and we must inorporate the possibility of the existene of imaginary mass of the neutrino in this model. The speial theory of relativity reveals that any ind of superluminal partile must have imaginary mass and vie versa. That is also true that the neutrino is not superluminal all the time, rather only one experiment shows the superluminal nature of neutrino and hene the imaginary mass; but there are several instants where the neutrinos are found to be subluminal having tiny mass. To fit suh riteria in this model it is assumed ε 1 = i, ε = 1 and m D < mm (but approximately equal), whih result m i( m + m ) m = m m 0. (8) 1 = M D, M D π This assumption readily maes θ =. The φ 1 and φ fields beome 4 φ 1 = φ Im = [( ν ν ) + i( ν + ν )], (9) 1 φ = φe = [ ν + ν ]. (10) φ e is the real neutrino field having tiny mass. This is not neessarily left handed, the right hand field is also inluded here. Suh field is, therefore, learly subluminal. But, the field φ Im bears imaginary mass and beomes superluminal nature. The neutrino mass evolved in that sense is also seesaw in nature, although the m 1 may not be high enough. Of ourse this is a new ind of seesaw mehanism, whih is essentially different from the earlier onept of seesaw mehanisms. Therefore, throughout the literature, we shall mention it as a new seesaw, whereas Type-I, Type- II and Type-III will be olletively mentioned as old seesaw. The above disussion was framed in the perspetive of neutrino with single flavor. In the nature three inds of neutrinos flavor have been observed. Therefore, in the general onsiderations the mass m D, m and m will be replaed by the 3 3

6 64 mass matries beome INDANATH BHATTACHAYYA M D, M and M. The general symmetri mass matries then M M D M = T (11) M D M with M M. Diagonalizing the M, we an obtain 6 mass eigenstates φ Im and φ e ( = 1,, 3) with assoiated 6 mass eigenvalues, alternatively imaginary and real. Thus eah flavor eigenstate of neutrinos orresponds an imaginary mass eigenvalue along with a real one. Therefore, it may be said that a flavor of neutrino forms a doublet of mass eigenstates φ e, φ Im. The φ e is bradyoni in nature and φ Im beomes tahyon; thus as a whole neutrino may be onsidered as elvisebrions [0]. 3. Generation of Neutrino Mass In the earlier days of the development of partile physis, the neutrino was supposed to be massless. Therefore, in the framewor of standard model there is no room for the generation of neutrino mass as SU( ) U ( 1) Y symmetry is broen down spontaneously. The onept of neutrino mass ame into existene when Ponteorvo proposed the neutrino osillation phenomena would lead to the existene of tiny neutrino mass [1]. Then there was need of a theoretial model whih ould explain the generation of neutrino mass as well as that of other elementary partiles. The effort was started to extend the standard model of eletro-wea interation theory. There are several models desribing the generation of neutrino mass by extending the standard model. The neutrino mass generation may be inorporated with a minimal extension of the standard model [, 3]. Neutrino mass ould also be generated in the extended standard model without inluding the right handed neutrino in the theory [4]. An elegant way to generate the neutrino mass is to inlude the right handed neutrinos in the model whih leads to the left right symmetri model [5, 6, 7]. When the right handed neutrino is inluded in the standard model the global B - symmetry beomes gaugable and the SU ( ) SU ( ) U ( 1 ) B- beomes the gauge group of the left right symmetri model [8, 9]. The old seesaw struture of neutrino mass [30] emerges in this left-right symmetri model.

7 SUPEUMINA NEUTINOS IN THE FAMEWOK 65 Eventually we would lie to disard the possibility whih leads to the old seesaw struture of neutrino mass. Therefore, B - symmetry breaing phenomena should not be taen into aount. Instead, we may thin about the neutrino mass generation in SO ( 4) model [31]. Aording to this model no Majorana mass is reated by the spontaneous breadown of SU( ) symmetry and most importantly, the Dira mass generated in this model is not essentially seesaw (old) in nature. Still the Majorana mass term, whih is approximately equal to the Dira mass term aording to this theory, are to be inorporated in the model. The Majorana field may be generated by introduing dimension five operator [3] in the unbroen agrangian. After the breaing of spontaneous symmetry, the neutrino gets Dira as well as Majorana mass terms defined in equation (1). The disussion in the previous setion reveals that the real mass of the neutrino is extremely small as the Dira and Majorana masses are very lose to eah other. On the other hand the magnitude of imaginary mass may not be high enough ompared to the other leptons, but muh higher than the real mass, whih results a new ind of seesaw. 4. Disussion Aording to the old seesaw model the right handed neutrino is hardly found in nature due to its extremely high mass. But as per the model proposed in this artile, one wing of a neutrino flavor annot be found not beause of its high mass, but as it has imaginary mass and thus beomes superluminal in nature. The other wing exists with its extremely small mass subjet to the experimental verifiation of its right handed nature. In other words, the present theoretial model annot disard the possibility of the existene of right handed neutrino even in the low energy range. In the old seesaw model, one mass eigenstate of neutrinos remains with a heavy mass, but may be sterile in nature sine ompletely right handed. The ative one having small mass onsists of purely left handed state. But aording to the model proposed in this artile, the mass eigenstate having high but imaginary mass is undeteted beause of its superluminal nature. In the framewor of this model, both of the mass eigenstates are mixture of left handed as well as right handed states. Therefore unlie the earlier onept the flavor eigenstates are the mixtures of all omplex mass eigenstates. et us explain the situation in more details in the view to explain the OPEA + result. In that experiment the neutrino osillation in the ν µ ντ has been taen into aount. Here the flavor eigenstates ν µ and ν τ are the mixtures of

8 66 INDANATH BHATTACHAYYA φ = φe + iφ Im and φ = φe + iφ Im. In the low energy limit, only the real part of 1 φ and φ are expeted to tae part in the osillation phenomena. But in the intermediate state, we annot exlude the possibility of the partiipation of φ Im assuming ( E ) = ( p) m1 ( = 1, ), where p >> m1 but 1 φ Im and E remains in the same energy range as that of initial ν µ. Thus the neutrino osillation may 1, e e our in two different hannels, one is onventional ( φ φ ) and the other is 1 Im φ Im ( φ, ). In the OPEA + experiment of ν µ ντ osillation sine the masses 1, Im Im of the ( φ φ ) hannel are imaginary, it is obvious that the neutrinos ross the luminal limit. On the other hand the OPEA experiment [, 3] resulted subluminal 1, e e neutrinos beause neutrino osillations must tae plae in the ( φ φ ) hannel. Therefore, it an be onsidered that the neutrino osillation is a two fold proess, one ours within our light one and another taes plae outside of it, without hanging the overall energy range. If the osillation event in the superluminal region is dominated by that of subluminal region then the overall neutrino speed due to ν µ ν τ osillation rosses the luminal barrier; in the reverse ase the neutrino speed remains subluminal. The OPEA + experiment found the neutrino speed greater than that of light sine here superluminal event would dominate over the subluminal one. Measuring the neutrino speed less than that of light by OPEA experiment annot lead to negate the revolutionary result of OPEA+, rather one an say that the subluminal event should dominate over the superluminal event in that ase. eferenes [1] T. Adam et al., arxiv: (011) [first version]. [] T. Adam et al., arxiv: (01) [revised version]. [3] M. Antolleno et al., arxiv: (01). [4] A. Einstein, Annalen Phys. 17 (1905), 891. [5] P. A. Cherenov, Dol. Aad. Nau Ser. Fiz. (1934), 451. [6] J.. Ellis, N. Harries, A. Meregaglia, A. ubbia and A. Saharov, Phys. ev. D 78 (008), ; arxiv: [hep-ph].

9 SUPEUMINA NEUTINOS IN THE FAMEWOK 67 [7] M. J. ongo, Phys. ev. D 36 (1987), 376. [8] D. Fargion and D. D Armiento, arxiv: [9] Z. ingli and M. Ma, arxiv: [10] Z. ingli and M. Ma, arxiv: , [11]. Zhou and B. Ma, Chen. Phys. C 35 (011), 957. [1] S. Hannestad and M. S. Sloth, arxiv: [13] A. Kehagias, arxiv: [14] A. Niolaidis, arxiv: [15] I. Oda and M. Tair, arxiv: [16] G. Fienberg, Phys. ev. 159 (1967), 1089; O. M. Bilaniu and E. C. G. Sudarshan, Phys. Today (5) (1969), 43. [17] J. Ciborowsi and J. embielinsy, Eur. Phys. J. C 8 (1999), 159. [18] G. J. Stephenson Jr., T. Goldman and B. H. J. MKellar, Phys. ev. D 6 (000), ;. N. Mohapatra and S. Nussinov, Phys. ett. B 395 (1997), 63. [19] A. G. Cohen and S.. Glashow, Phys. ev. ett. 107 (011), [0] O. I. Chashhina and Z. K. Silagadze, arxiv: [1] B. Ponteorvo, JETP 6 (1957), 49 and JETP 7 (1958), 17. [] A. Zee, Phys. ett. B 93 (1980), 389. [3] K. S. Babu, Phys. ett. B 03 (1988), 13. [4]. N. Mohapatra, arxiv: hep-ph / [5] J. C. Pati and A. Salam, Phys. ev. D 10 (1974), 75. [6]. N. Mohapatra and J. C. Pati, Phys. ev. D 11 (1975), 566 and 558. [7] G. Senjanovi and. N. Mohapatra, Phys. ev. D 1 (1975), 150. [8] B. Brahmahari, E. Ma and U. Sarar, Phys. ev. ett. 91 (003), [9]. N. Mohapatra and. E. Marsha, Phys. ett. B 91 (1980),. [30]. N. Mohapatra and G. Senjanovi, Phys. ev. ett. 44 (1980), 91. [31] I. Bhattaharyya, Commun. Theor. Phys. 54 (010), 305. [3] S. Weinberg, Phys. ev. ett. 43 (1979), 1566.