Journal of Physics: Conference Series OPEN ACCESS Non Unitary Neutrino Miing angles Matri in a Non Associative Gauge Model To cite this article: M Boussahel N Mebarki 5 J. Phys.: Conf. Ser. 59 View the article online for updates enhancements. This content was downloaded from IP address 46..4.5 on 4//8 at :
st Franco-Algerian Workshop on Neutrino Physics Journal of Physics: Conference Series 59 (5) doi:.88/74-6596/59// Non Unitary Neutrino Miing angles Matri in a Non Associative Gauge Model M Boussahel ()() N Mebarki () () Laboratoire de Physique Mathématique et Subatomique Mentouri University Constantine Algeria () Physics Department Faculty of Sciences Mohamed Boudiaf University M'sila Algeria E-mail: mboussahel@yahoo.com Abstract. A possible non-unitary neutrino-miing matri is discussed in the contet of a low energy limit of a recently proposed gauge model within the non associative geometry of Wulkenhaar's approach. Some bounds on certain physical parameters of the model are also obtained.introduction One of the greatest achievements of the non-commutative geometry ( NCG ) is the geometrization of the stard model [ ] many interesting topics have been treated outsting problem have bee solved understood within this approach so far. However Connes' prescription in NCG is compatible only with linear representations of the matri group which imposes very stringent constraints on gauge models. Indeed it was shown that [4 5] the only models which can be constructed in this approach are the stard model the Pati-Salam [6] the Pati-Mohapatra models [78]. Now if one takes into account the reality condition of the K -cycle [9] the last two models will be ruled out leaving at the end the stard model as the unique model compatible with Connes' prescription.. Recently Wulkenhaar has proposed a modification to the non-commutative geometry the differential geometry is formulated in terms of graded differential Lie algebras instead of a unital associative algebras as it is the case in Connes' approach [56]. Its application to a list of physical models has been successful. Among this list figure out the stard model [7] the flipped SU ( 5) U () [8] SO () models [ 9] left-right gauge model [] etc. The interesting feature of Wulkenhaar's approach or non-associative geometry ( NAG ) is the use of graded Lie algebra. On the other h the number of fermion families in nature the pattern of fermion masses miing angles are one of the most intriguing puzzles in modern particle physics. Over the last decade the -- etension of the stard model ( SM ) for the strong electroweak interactions based on the local gauge group SU c ( ) SU L () L U X () have been studied etensively [ 45678]. It provides an interesting attempt to answer the question on family replication. In fact this etension has among its best features that several models can be constructed so that anomaly cancellation is achieved by interplay between the families [ 4 5 6 7 8]. Moreover some models based on the -- local gauge structure are Content from this work may be used under the terms of the Creative Commons Attribution. licence. Any further distribution of this work must maintain attribution to the author(s) the title of the work ournal citation DOI. Published under licence by Ltd
st Franco-Algerian Workshop on Neutrino Physics Journal of Physics: Conference Series 59 (5) doi:.88/74-6596/59// suitable to describe some neutrino properties because they include in a natural way most of the ingredients needed to eplain the masses miing in the neutrino sector [8 9 ]. Recently a very interesting -- model within the Wulkenhaar's approach of graded Lie algebra has been constructed. The goal of this paper is to study the non-unitarity of the neutrino-miing matri within the non-associative proposed model (NUNMAM ) in the low energy limit as a deviation of the stard. Some new bounds on the various physical parameters are also obtained. In setion we present the proposed non-associative model. In section we discuss the NUNMAM. Finally in section we draw our conclusions..brief description of the model The -- model is based on the gauge group SU () c SU () L U () N. The price to pay is the introduction of eotic quarks with electric charges 5 / 4 /. The main motivations to study this kind of model are: The natural prediction of three generations based on anomaly cancellation. Thus the family number must be three. Denoting by: Q u d J L e ee c ; () The triplet (resp. singlet) representations for the left ( L ) (resp. right ( R )) hed fields (leptons L quarks Q ) are: LL (); QL ( ) () LR (); L R ( ); L R ( ) () 5 QR ( ); Q R ( ); QR ( ) The numbers / in eq. (7) / / 5 / in eq. (7) are U() N charges. The normalization factor is introduced for practical reasons as it will be clear later. The electric charge operator Qe is defined in terms of the N charges as: Qe 8 N. X e (4) 8 are the usual Gell-Mann matrices. The other two leptons quarks generations L c ; Q sc J ; L c Q bt J (5) (6) belong to the representations: LL (); LL () LR (); L R ( ); L R ( ) LR ( ); L R ( ); LR ( ) QL ( ); QL ( ) 4 QR ( ); QR ( ); QR ( ) 4 QR ( ); QR ( ); QR ( ) (7) (8) (9) () () ()
st Franco-Algerian Workshop on Neutrino Physics Journal of Physics: Conference Series 59 (5) doi:.88/74-6596/59// Here u d s c b t J J J st for the wave function of the up down strange charm bottom top eotic quarks respectively. Now following ref. [] for the three generations the total internal Hilbert space is C 7 labelled by the elements: (QiL QiL QiL QiR QiR QiR LiL LiL LiL LiR LiR LiR )T () i QiL QiL QiL QiR QiR QiR C C LiL LiL LiL LiR LiR LiR C. are hermitian operators for which the eigenvalues on the states Qi are denoted by (4) The eigenvalues correspond eactly to the N charges of the commutative model. Similarly for leptons we assign operators coefficients (eigenvalues) arbitrary for which their action such that: (5) The action of the hermitian operators on the scalar matter vector fields i A respectively is denoted by: (6) (7) Regarding the mass matri M of the L -cycle it has the form: M M Q 8 8 M L (8) MQ M Q M Q M Q M Q M Q M L M L M L M L M Q (9) M L () ML M L
st Franco-Algerian Workshop on Neutrino Physics Journal of Physics: Conference Series 59 (5) doi:.88/74-6596/59// here M Q i M Li M (C) are the mass matrices of the fermions (quarks leptons) such that: M Q mu md M Q mb ms mc m J M Q mt mj mj () M L m e me M L M L m m m m () Now if we denote by: () the bosonic action S B containing the scalar fields bosons-scalar interactions: SB 6 g dtre( ) 6g d X X (4) here g is the U () N gauge couplings constants. After straightforward but tedious simplifications we obtain: (5) ˆ 8 g ˆ [tr d i A A 8 (6) ia 67 ia 4 tr ( d i A8 A ia 67 ia 45 4 ) tr ( d 4 ia 45 ia ia8 A 4 ]Tr M QQ M LL 4
st Franco-Algerian Workshop on Neutrino Physics Journal of Physics: Conference Series 59 (5) doi:.88/74-6596/59// [tr ( d i A A8 ia45 ia 5 ) tr ( d ia45 i A8 A ia67 5 ) tr ( d 5 ia ia 67 i A 8 A 5 )]Tr M QQ M L L [tr d 6 ia45 ia67 i A A8 6 (7) tr ( d ia i A 8 A ia67 6 ) tr ( d i A A8 ia ia 45 6 )]Tr M QQ M LL with N charges (8).. Fermionic action Concerning the fermionic action if the leptons wave functions L are represented by L ( LL L L L L LR L R L R ) than making the gauge fields redefinitions: ig W A A A ig (W iw ) ig W (9) A8 ig W 8 A 67 A6 A7 ig ( W 6 iw 7 ) ig U () A ig W A 45 A4 ia5 ig (W 4 iw 5 ) ig V () A we get the following fermionic interactions the vector gauge bosons scalar fields (Yukawa terms) are denoted by respectively: () () If we redefine the scalar fields: i i g g i 4 5 i i i i i 5 (4)
st Franco-Algerian Workshop on Neutrino Physics Journal of Physics: Conference Series 59 (5) doi:.88/74-6596/59// set: Tr M M M M Q Q Q Q L L L L 4 4 The lagrangian density Tr M Tr M Tr M M QQ MLL M LL QQ QQ M QQ ML L M LL (5) Tr M QQ M L L M LL 5 QQ 5 becomes: L 9 6 g 8 (6) 9 ( ) ( ) ( 4 4 ) ( 4 ) ( 5 ) ( ) ( ) ( ) ( 5 5 ) 4 4 ( 5 ) ( 6 ) ( ) (7) ( ) ( ) ( ) 5 5 6 6 6 6 ( 6 ) ( 4 ) ( ) ( A B ) A B A.B (8) (9) If we choose the vacuum epectation values of the dynamical scalar fields as: v 4 6 (4) the usual analysis shows that this set of VeV breaks the symmetry in one single step: SU () c SU () L U () N SU () c U () Q v or v the symmetry breaking chain SU () c SU () L U () N v SU () c SU () L U () N vor v SU () c U () Q For the particular value (4) becomes: (4) After the Higgs mechanism the scalar bosons masses take the following forms: (4) 6
st Franco-Algerian Workshop on Neutrino Physics Journal of Physics: Conference Series 59 (5) doi:.88/74-6596/59// Now for the charged gauge bosons in the basis W V U after the one step spontaneous symmetry breaking ( SSB ) Higgs mechanism the charged gauge bosons masses are: M W g v v M V g v v M U g v v (44). Neutral gauge bosons For the neutral gauge bosons one gets (after SSB ) in the basis W W W 8 the following symmetric non-diagonal mass matri M NGB : M M M M NGB M M M M M M (45) g v v v M g v v (46) gg M v v M g v v (47) gg v v v M g v v v (48) M If we set v v in order to have a one vanishing eigenvalue (representing the rest mass square of the photon M ) one has to have the constraint: M M M M M M M M (49) Consequently we deduce that: (5) The remaining non vanishing eigenvalues denoted by M M which can be identified with the mass square of the Z Z bosons respectively are given by: g M M Z 4 g v M M Z 4g g g v 4 g g (5) (5) Now if we introduce a miing angle such that: tan g (5) g if we want that the Z W gauge bosons masses are the same as the ones of the stard model then one can show easily that: cos w 4 tan tan 7 (54)
st Franco-Algerian Workshop on Neutrino Physics Journal of Physics: Conference Series 59 (5) doi:.88/74-6596/59// w is the Weinberg miing angle. Regarding the eigenstates related to M M Z M Z eigenvalues one can show that they are given by the following epressions: B B W W W 8 Z Z Z Z W W W 8 W W W 8 (55) (56) (57) tan (58) cot cot 5 B (59) / Z (6) / (6) Z / (6)..Neutral current If we denote by LNC the leptonic neutral currents lagrangian density coupled to both Z Z massive vector bosons such as: LNC g cos w L g i Li V Z g ALi Z 5 Li Z L i g VLi Z g ALi Z 5 Li Z (6) i g D t gvl Z D t t g VLZ D t t g AL Z g AL Z D t L gv Z D t t t L V Z L t g A Z D t t t (64) g ALZ D t t gvl Z gvl Z D t t g AL Z g ALZ D t t (65) with D D cos w tan tan tan (66) (67) 4 tan 5 tan (68) t B Z t B Z t B Z ( ) t B Z (69) (7) B Z Z ( ) (7) with 8
st Franco-Algerian Workshop on Neutrino Physics Journal of Physics: Conference Series 59 (5) doi:.88/74-6596/59//.non-unitary neutrino miing mass matri Now we use the non-stard neutrino interactions (NSI ) approach which consists to describe the neutrino new interactions by parameterizing the effects of the new physics in neutrinos oscillation. The NSI effective lagrangian density is: effnsi G F [l Pl ][ P P LR l P (7) l P l P R ( 5 ) / L ( 5 ) /. Here l e encodes the deviation from the stard model interactions between the neutrino of flavor with component P-hed of leptons l P can resulting in a neutrinos of flavour. It is very important to mention that some of the parameters be etracted for eample from the epressions of the differential cross sections for the elastic scattering processes e e. In fact in the contet of the non associative model of ref. [] described in the previous section after direct but tedious calculations one gets the following e : epression of the differential cross section ddy d e me E i m (7) [ i ( y ) e i y dy 4 E me is the electron rest mass E is the incident neutrino energy y Te / E is the rapidity ( Te is the electron recoil energy at the final state) (Ω Ω + Ω Ω )/( ) (Ω Ω Ω Ω )/( (Ω Ω Ω Ω + Ω Ω )/( ) ) (74) (75) with Ω + Ω Ω + Ω (76) A Ω (77) g L g L g L g L g g V Z B g A Z A g B g A B V Z A Z c w c w c w c w g L gv Z cw g L g c w V Z g g L g AZ c w (78) g L g (79) c w AZ Now it is easy to show that: + + ( ) Ω Ω Ω Ω (8) ) (8) CVe / s w s w sin w 9 (8)
st Franco-Algerian Workshop on Neutrino Physics Journal of Physics: Conference Series 59 (5) doi:.88/74-6596/59// M V M Z M V M M V M w Z (8) / / (84) are the relative left right-hed non-unitary parameters ( g L g R are the total left right hed couplings in the stard model charged neutral currents). 4. Conclusions Through this paper we have derived some of the parameters encoding the deviation from the unitarity of the Pontecorvo-Maki-Nakagawa-Sakata modified neutrino miing matri (PMNS) in the contet of the low energy limit of the non associative geometry model developed in ref. [4] (more developments of the subect other phenomenological study are under investigation). Acknowledgement We are very grateful to the Algerian Ministry of education research DGRSDT for the financial support. References [] Connes A Lott J 99 Nucl. Phys. B 8 (Proc. Suppl.) 9 [] Connes A994 Noncommutative geometry Academic Press New York [] Schucker T Zylinski J M 995 J.Geom.Phys. 6 7 [4] Lizzi F Mangano G Miele G Sparano G 996 Mod. Phys. Lett. A 56 [5] Brout R 998 Nucl.Phys. 65 (Proc.Suppl.) [6] Pati J C Salam A 974 Phys.Rev. D 75 [7] Mohapatra R N Pati J C 975 Phys. Rev. D 566 [8] Mohapatra R N Senanovic G 98 Phys. Rev. D 65 [9] Connes A 995 J.Math.Phys. 6 694 []Mebarki N Harrat M Boussahel M 7 Int. J. Mod. Phys. A 679 []Wulkenhaar R 997 Non-associative geometry-unified models based on L-cycles PhD Thesis Leipzig University []Wulkenhaar R 997 J.Math.Phys. 8 58 []Wulkenhaar 997 Phys. Lett. B 9 9 [4]Wulkenhaar R 998 Int.J.Mod.Phys. A 67 [5]Wulkenhaar R 999 Int. J. Mod. Phys. A 4 559 [6]Benslama A Mebarki N JHEP 8 [7]Pisano F Pleitez V 99 Phys. Rev. D 46 4 [8]Frampton P H 99 Phys.Rev.Lett. 69 889 [9]Pleitez V Tonasse M D 99 Phys.Rev. D 48 5 []Ng D 994 Phys. Rev. D 49 485 []Monetro J C Pisano F Pleitez V 99 Phys. Rev. D 47 98 []Long H N 996 Phys. Rev. D 5 47 []Pleitez V 996 Phys. Rev. D 5 54 [4]Gutierrez D A Ponce W A Sanchez L A 6 Int.J.Mod.Phys. A 7 [5]Kitabayashi T Yasuè M Phys. Rev. D 6 956 [6]Okamoto Y Yasuè M 999 Phys. Lett. B 466 67 [7]Kitabayashi T Phys. Rev. D 64 57 [8]Kitabayashi T Yasuè M Phys. Lett. B 49 6 [9]Kitabayashi T Yasuè M Phys. Rev. D 67 56 []Boussahel M Mebarki N Int. J. Mod. Phys. A 6 87