New Jarlskog Determinant from Physics above the GUT Scale

Transcription

1 EJTP 6, No. 2 (29) Electronic Journal of Theoretical Physics New Jarlskog Determinant from Physics above the GUT Scale Bipin Singh Koranga and S. Uma Sankar Department of Physics, Indian Institute of Technology Bombay, Mumbai 476, India Received 2 October 28, Accepted 16 October 28, Published 2 February 29 Abstract: We study the Planck scale effects on Jarlskog determiant. Quantum gravitational (Planck scale) effects lead to an effective SU(2) U(1) invariant dimension-5 Lagrangian involving neutrino and Higgs fields, which give rise to additional terms in neutrino mass matrix on electroweak symmetry breaking. We assume that gravitational interaction is flavor blind and compute the Jarlskog determiant due to Planck scale effects. In the case of neutrino sector, the strentgh of CP violation is measured by Jarlskog determiant. We applied our approach to study Jarlskog determinant due to the Planck scale effects. c Electronic Journal of Theoretical Physics. All rights reserved. Keywords: Quantum Gravity; Jarlskog Determinant; Planck scale; Higgs Fields PACS (28): 4.6.-m; y; 12.6.Cn; 12.6.Fr; g 1. Introduction The origin of CP violation is still mystery in particle physics. Recent advance to neutrino physics observation mainly of astrophysical observation suggested the existence of tiny neutrino mass. The experiments and observation have shown evidence for neutrino oscillation. The solar neutrino deficit has long been observed [1-4] the atmospheric neutrino anomaly has been found [5-7] and currently almost confirmed by IMB [8], and hence indicates that neutrinos are massive and there is mixing in lepton sector. Since there is a mixing in lepton sector, this indicates to imagine that there occurs CP violation in lepton sector. Several physicists have considered whether we can see CP violation effect in lepton sector through long baseline neutrino oscillation experiments. The neutrino oscillation probability, in general depends on six parameters two independent mass square difference Δ 21 and Δ 31, three mixing angles θ 12,θ 23,θ 13 and one CP violating phase δ. CP violation arise as three or more generation [9, 1]. CP violation in neutrino bipiniitb@rediffmail.com, bipiniitb@gmail.com

2 23 Electronic Journal of Theoretical Physics 6, No. 2 (29) oscillation is interesting because it relates directly to CP phase parameter in the mixing for n>3 degenerate neutrino. We can write down the compact formula for the difference of transition probability between conjugate channel. ΔP (α, β) =P (ν μ ν e ) P ( ν μ ν e ), (1) where (α, β) =(e, μ), (μ, τ), (τ,e). The main physical goal in future experiments are the determination of the unknown parameter θ 13 and upper bound sin 2 2θ 13 <.1 is obtained for the ref [11]. In particular, the observation of δ is quite interesting for the point of view that δ related to the origin of the matter in the universe. The determination of δ is the final goal of the future experiments. We get the analytical expression for ΔP (α, β) using the usual form of the MNS matrix parametrization [12]. c 12 c 13 s 12 c 13 s 13 e iδ U = s 12 c 23 c 12 s 23 s 13 e iδ c 12 c 23 s 12 s 23 s 13 e iδ s 23 c 13, (2) s 12 s 23 c 12 c 23 s 13 e iδ c 12 s 23 s 12 s 13 s 23 e iδ c 23 s 13 where c and s denoted the cosine and sine of the respective notation, thus ΔP (α, β) in vacuum can be written as ΔP (α, β) =16J (sinδ 21 sinδ 32 sinδ 31 ). (3) Here α and β denote different neutrino or anti-neutrino flavour where ( )( )( ) Δij L 1GeV Δ ij =1.27, (4) ev 2 Km E Δ ij =(m 2 i m 2 j)is the difference of i th and j th vacuum mass square eigenvalue, E is the neutrino energy and L is the travel distance and the well known Jarlskog determinant [13], J is the standard mixing parametrization is given by J = Im ( U e1 Ue2 U μ1 U ) μ2 = 1 8 sin2θ 12sin2θ 23 sin2θ 13 cosθ 13 sinδ, (5) and the asymmetry parameter suggested by Cabibbo [14], as an alternative to measure CP violation in the lepton sector ΔP A cp = (6) P (ν μ ν e ) P ( ν μ ν e ), The purpose of this paper is to study the Planck scale effects on Jarlskog Determinant. In Sec-2, we discuss the neutrino mixing angle due to Planck scale effects. In Sec-3, we give the conclusions.

3 Electronic Journal of Theoretical Physics 6, No. 2 (29) Neutrino Mixing Angle and Mass Square Differences Above the GUT Scale To calculate the effects of perturbation on neutrino observables. The calculation developed in an earlier paper [15]. A natural assumption is that unperturbed ( th order mass matrix) is given by M = U diag(m i )U, (7) where, U αi is the usual mixing matrix and M i, the neutrino masses is generated by Grand unified theory. Most of the parameters related to neutrino oscillation are known, the major expectation is given by the mixing elements U e3. We adopt the usual parameterizations. In term of the above mixing angles, the mixing matrix is U e2 U e1 = tanθ 12 (8) U μ3 U τ3 = tanθ 23 (9) U e3 = sinθ 13 (1) U = diag(e if1,e if2,e if3 )R(θ 23 )ΔR(θ 13 )Δ R(θ 12 )diag(e ia1,e ia2, 1). (11) The matrix Δ = diag(e 1δ 2, 1,e iδ 2 ) contains the Dirac phase. This leads to CP violation in neutrino oscillation a1 anda2 are the so called Majorana phase, which effects the neutrinoless double beta decay. f1, f2andf3 are usually absorbed as a part of the definition of the charge lepton field. Planck scale effects will add other contribution to the mass matrix that gives the new mixing matrix can be written as [15] U = U(1 + iδθ), where δθ is a hermitian matrix that is first order in μ[16,17]. The first order mass square difference ΔMij 2 = M i 2 Mj 2,get modified [16,17] as ΔM 2 ij =ΔM ij 2 +2(M ire(m ii ) M j Re(m jj )). (12) The change in the elements of the mixing matrix, which we parameterized by δθ[15], is given by δθ ij = ire(m jj)(m i + M j ) Im(m jj )(M i M j ) ΔM 2 ij. (13) The above equation determines only the off diagonal elements of matrix δθ ij. The diagonal element of δθ ij can be set to zero by phase invariance. Using Eq(12), we can calculate neutrino mixing angle due to Planck scale effects,

4 232 Electronic Journal of Theoretical Physics 6, No. 2 (29) U e2 U e1 = tanθ 12 (14) U μ3 U τ3 = tanθ 23 (15) U e3 = sinθ 13 (16) As one can see from the above expression of mixing angle due to Planck scale effects, depends on new contribution of mixing U = U(1 + iδθ). To see the mixing angle due to Planck scale effects [14,16] only θ 13 and θ 12 mixing angle have small deviation due to Planck scale effects. 3. Jarlskog Determinant Due to Planck Scale Effects Note from eq(14), that the correction term depends crucially on the type of neutrino mass spectrum. For a hierarchical or invert hierarchial spectrum the correction is negligible. Hence we consider a degenerate neutrino spectrum and take the common neutrino mass to 2 ev, which is the upper limit from the tritium decay experiment [18]. Let us compute Jarlskog determiant due to new mixing due to Planck scale effects ( ) J = Im U e1 U e2 U μ1 U μ2 = Im((U e1 + i(u e2 δθ 12 + U e3δθ 13 ))((U e2 i(u e1 δθ 12 + U e3 δθ 13)) ((U μ1 i(u μ2δθ 12 + U μ3 δθ 13 ))((U μ2 + i(u μ1 δθ 12 + U μ3 δθ 23 ) (17) We simplified Jarlskog determiant due to new mixing matrix J = Im ( U e1 U e2u μ1u μ2 ) +Im(i(Uμ1 U μ2 )( U e2 2 δθ 12 +U e2 U e3 δθ 13 U e1 2 δθ 12 U e1 U e3δθ 23) +Im(i(U e1u e2 )( U μ1 2 δθ 12 + U μ1u μ3 δθ 23 U μ2 2 δθ 12 U μ2 U μ3δθ 13 ) = J +ΔJ In terms of mixing angle, we can write Jarlskog determiant in terms of mixing parameter due to Planck scale effects J = 1 8 sin2(θ 12 + ɛ 12 )sin2(θ 23 + ɛ 23 )sin2(θ 13 + ɛ 13 )cos(θ 13 + ɛ 13 )sinδ, (18) We define the percentage change in Jarlskog determinant due to Planck scale effects

5 Electronic Journal of Theoretical Physics 6, No. 2 (29) P = J 1 (19) J GUT The Majorana phases a 1 and a 2 have a non-trivial effect on the Planck scale corrections. We show the results as contour plots of J in the a 1 a 2 plane. In our calculation, we used best fit values of mixing angles, θ 12 =34,θ 23 =45 o, θ 13 =1 o.we considered non zero value of CP phase and we took δ =45 o, 9 o. In Fig. 1 and Fig. 3 for θ 13 =1 o. For the value, which is the upper limit coming for CHOOZ experiments, note that there is reasonable range of Majorama phases, where Jarlskog determinant change 5% only, this change due to two mixing angles. In this paper, we studied Planck scale effects the Jarlskog determinant. MNS matrix and Jarlskog detergent, which is signal for CP violation in neutrino oscillation. We have obtained, due to Planck scale effects, two mixing angle θ 12 and θ 13 extra contributes to Jarlskog determinant. References [1] J.Madore, S.Schraml,P.Schupp and J.Wess, Eur.Phys.Jour.C16(2)161 [2] M.R.Douglas and N.A.Nekrasov,Rev.Mod.Phys.73,(21)977. [3] GALLEX Collaboration, W. Hampal et al., Phys. Lett B 447, 127 (1999). [4] SAGE Collaboration, J. N Abdurashitor et al., astro-ph/ [5] Kamiokande Collaboration, Y. Fukuda et al.,phys. Rev. Lett 82, 181 (1999). [6] Homestake Collaboration, B.T. Cleveland, et al., Astrophys.J.496:55-526,(1998) [7] Kamiokande Collaboration, K. S Hirata et al., Phys.Lett. B 25, 416 (1998). [8] IMB Collaboration, D. Casper et al., Phys. Rev. Lett 66, 2561 (1991). [9] MACRO Collaboration, M. Ambrosio et al.,phys. Lett B 434, 451 (1998). [1] KamLAND Collaboration, K.E Guchi et al.,phys.rev.lett.9:2182(23). [11] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. Rev. Lett. 45, 652 (1973). [12] V. Varger et al., Phys.Rev.Lett 45, 284 (198). [13] CHOOZ Collaboration, M. Apollonio, Phys. Lett B 42, 397 (1998). [14] Review of Particle Physics, J. of Physics. G 33, 156 (26). [15] C. Jarlskog, Phys. Rev. Lett.55, 139 (1985). [16] N. Cabibbo, Phys.Lett B 72,333 (1978). [17] F. Vissani et al.,phys.lett. B571, 29, (23). [18] Bipin Singh Koranga, Mohan Narayan and S. Uma Sankar, Phys.Lett.B665,63 (28). [19] Bipin Singh Koranga, Mohan Narayan and S. Uma Sankar, arxiv:hep-ph/ [2] Ch. Weinneimer et al.,phys.lett.b 46, 219 (1999); V. M Labashev et al.,phys.lett. B 46,227 (1999).

6 234 Electronic Journal of Theoretical Physics 6, No. 2 (29) a a2 Fig. 1 J due to Planck scale effects as function of Majorana phase for δ =45 o a a2 Fig. 2 J due to Planck scale effects as function of Majorana phase for δ =9 o.