Int J Theor PhysDOI 10.1007/s10773-014-2045-5

Invariance of Jarlskog Determinant Above the GUT Scale

Bipin Singh Koranga

Received: 25 November 2013 / Accepted: 5 February 2014© Springer Science+Business Media New York 2014

Abstract An asymmetry between the probabilities P (νμ → νe) and P (ν̄μ → ν̄e) would bedirect indication of CP violation at the fundamental level. Planck scale effects on neutrinomixing, we have derived the mixing angles of neutrino flavour due to Planck scale effects.It has been shown that Jarlskog determinant remains nearly invariant above the GUT scale.

Keywords Jarlskog determinant · Neutrino mixing · GUT scale

1 Introduction

The evidence of a deficit of detected solar neutrinos [1] indicates that electron neutrinosmust also participate in lepton mixing. The participation of all three neutrinos in leptonmixing raises the possibility of CP and T violation in neutrino oscillations. The emergenceof large mixing parameter in lepton sector indicate the potentially large CP and T violationare maximal for neutrinos in vacuum. A number of author [2–6] have explored the phe-nomenology of CP and T violation in neutrino oscillations for several different scenarios oflepton masses and mixing parameters. In both solar and atmospheric neutrino experimentscan traverse a significant of the earth. Long baseline accelerator and reactor experiment stilllonger baseline. CP violation arise as three or more generation [7, 8]. CP violation in neu-trino oscillation is interesting because it relates directly to CP phase parameter in the mixingfor n > 3 degenerate neutrino. We can write down the compact formula for the differenceof transition probability between conjugate channel.

�P(α,β) = P (νμ → νe)− P (ν̄μ → ν̄e), (1)

where(α, β) = (e, μ), (μ, τ), (τ, e).

The main physical goal in future experiment are the determination of the unknownparameter θ13 and upper bound sin22θ13 < 0.01 is obtained for the Ref. [9]. In particular,

B. S. Koranga (�)Kirori Mal College, University of Delhi, Delhi 110007, Indiae-mail: bipiniitb@rediffmail.com

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the observation of δ is quites interesting for the point of view that δ related to the origin ofthe matter in the universe. The determination of δ is the final goal of the future experiments.We get the analytical expression for �P(α,β) using the usual form of the MNS matrixparametrization [10].

U =⎛⎝

c12c13 s12c13 s13e−iδ

−s12c23 − c12s23s13eiδ c12c23 − s12s23s13e

iδ s23c13

s12s23 − c12c23s13eiδ −c12s23−s12s13s23e

iδ c23s13

⎞⎠ , (2)

where c and s denoted the cosine and sine of the respective notation, thus �P(α, β) invacuum can be written as

�P(α, β) = 16J (sin�21 sin�32 sin�31) . (3)

Here α and β denote different neutrino or anti-neutrino flavour

�ij = 1.27

(�ij

eV 2

) (L

Km

) (1GeV

E

), (4)

where �ij =(m2

i −m2j

)is the difference of ith and j th vacuum mass square eigenvalue,

E is the neutrino energy and L is the travel distance and the well known Jarlskog determinant[11], J is the standard mixing parametrization is given by

J = Im(Ue1U

∗e2U

∗μ1Uμ2

)

= 1

8sin2θ12sin2θ23sin2θ13 cos θ13sinδ, (5)

and the asymmetry parameter suggested by Cabibbo [12], as an alternative to measure CPviolation in the lepton sector

Acp = �P

P(νμ → νe)− P (ν̄μ → ν̄e). (6)

The purpose of this paper is to study the Planck scale effects on Jarlskog determinant. InSection 2, we discuss the neutrino mixing angle due to Planck scale effects. In Section 3,we give the conclusions.

2 Neutrino Mixing Angle Due Above the GUT Scale

To calculate the effects of perturbation on neutrino observables. A natural assumption isthat unperturbed (0th order mass matrix M is given by

M = U∗diag|(Mi)U†, (7)

where, Uαi is the usual mixing matrix and Mi , the neutrino masses is generated by Grandunified theory. Most of the parameter related to neutrino oscillation are known, the majorexpectation is given by the mixing elements Ue3. We adopt the usual parametrization.

|Ue2||Ue1| = tanθ12 (8)

|Uμ3||Uτ3| = tanθ23 (9)

|Ue3| = sinθ13 (10)

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In term of the above mixing angles, the mixing matrix is

U = diag(eif 1, eif 2, eif 3)R(θ23)�R(θ13)�∗R(θ12)diag(eia1, eia2, 1). (11)

The matrix � = diag(e

1δ2 , 1, e

−iδ2

)contains the Dirac phase. This leads to CP violation

in neutrino oscillation a1 and a2 are the so called Majoring phase, which effects the neutrinoless double beta decay. f 1, f 2 and f 3 are usually absorbed as a part of the definition of thecharge lepton field. Planck scale effects will add other contribution to the mass matrix thatgives the new mixing matrix can be written as [13]

U′ = U(1 + iδθ),

=⎛⎝

Ue1 Ue2 Ue3Uμ1 Uμ2 Uμ3

Uτ1 Uτ2 Uτ3

⎞⎠

+ i

⎛⎝

Ue2δθ∗12 + Ue3δθ

∗23, Ue1δθ12 + Ue3δθ

∗23, Ue1δθ13 + Ue3δθ

∗23

Uμ2δθ∗12 + Uμ3δθ

∗23, Uμ1δθ12 + Uμ3δθ

∗23, Uμ1δθ13 + Uμ3δθ

∗23

Uτ2δθ∗12 + Uτ3δθ

∗23, Uτ1δθ12 + Uτ3δθ

∗23, Uτ1δθ13 + Uτ3δθ

∗23

⎞⎠ . (12)

Where δθ is a hermition matrix that is first order in μ [13–15]. The first order masssquare difference �M2

ij = M2i −M2

j ,get modified [13–15] as

�M′2ij = �M2

ij + 2(MiRe(mii)−MjRe(mjj )). (13)

The change in the elements of the mixing matrix, which we parametrized by δθ [13], isgiven by

δθij = iRe(mjj )(Mi +Mj)− Im(mjj)(Mi −Mj)

�M′2ij

. (14)

The above equation determine only the off diagonal elements of matrix δθij . The diag-onal element of δθij can be set to zero by phase invariance. Using (12), we can calculateneutrino mixing angle due to Planck scale effects,

|U ′e2|

|U ′e1|

= tan θ′12 (15)

|U ′μ3|

|U ′τ3|

= tan θ′23 (16)

|U ′e3| = sin θ

′13 (17)

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 dueto Planck scale effects [25–30] only θ13 and θ12 mixing angle have small deviation due toPlanck scale effects.

3 Jarlskog Determinant Due to Planck Scale Effects

Let us compute Jarlskog determinant due to new mixing due to Planck scale effects

J′ = Im

(U

′e1U

′∗e2U

′∗μ1U

′μ2

)

= Im((Ue1 + i(Ue2δθ∗12 + Ue3δθ13))((Ue2 − i(U∗

e1δθ∗12 + U∗

e3δθ13))

× ((U∗μ1 − i(Uμ2δθ12 + Uμ3δθ13))((Uμ2 + i(Uμ1δθ12 + Uμ3δθ

∗23) (18)

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We simplified Jarlskog determinant due to new mixing matrix

J′ = Im

(Ue1U

∗e2U

∗μ1Uμ2

) + Im(i(Uμ1Uμ2)(|Ue2|2δθ∗12

+Ue2Ue3δθ13 − |Ue1|2δθ∗12 − Ue1U∗e3δθ

∗23)

+ Im(i(U∗e1Ue2)(|Uμ1|2δθ12 + U∗

μ1Uμ3δθ∗23 − |Uμ2|2δθ12 − Uμ2U

∗μ3δθ13)

= J +�J

In term of mixing angle, we can write Jarlskog determinant in term of mixing parameterdue to Planck scale effects

J′ = 1

8sin2(θ12 + ε12)sin2(θ23 + ε23)sin2(θ13 + ε13) cos(θ13 + ε13)sinδ, (19)

Numerically % change of Jarlskog determinant is very small [16, 17], due to smallchange of mixing angle θ12 and θ13. From the expression of J

′in (14), it is obvious that all

three mixing angle contributes to Jarlskog determent. A number of experiments constrained

it from above sin22 ¯θ13. < 0.17 limit was obtained by CHOOZ experiment [19] and similarlimit was given by Palo verde [20] (both reactor experiment). sin22θ13 < 0.26 limit wasgiven by K2K [21] and the following limit sin22θ13 < 0.15 was given by MINOS [18] bothbeing accelerator experiments. The T2K experiment [23] in a five year νμ run at the fullJ-PARC beam intensity, will be of the order of sin22θ13 < 0.006 (90 %CL). The Day Bayproject in china [22] could reach a sin22θ13 sensitivity below 0.01,while the RENO exper-iment in Korea [24] should reach a sensitivity around 0.02. Above GUT scale,the changein Jarlskog determent is very small less than 5 % only, this change due to only two mixingangle θ12 and θ13.

4 Conclusions

In reminder of this paper, we explore the neutrino mixing and Jarlskog determinant due toPlanck scale effects. In present paper,we study the Jarlsberg determinant above the GUTscale.In order to ensure that our discussion is relevant to theoretical, we will restrict ourconsiderations to neutrino mixing parameter due to Planck scale effects, which are notexperimental. In conclusion,we have ensure that above the GUT scale Jarlsberg determinantnearly invariant. Finally, we can wish one important comment, if future experiment find thenon zero value of CP phase so we can say there is possible CP asymmetry due to Planckscale effects.

References

1. Supper-Kamiokande Collaboration, Fukuda, Y., et al.: Phys. Lett. B 436, 33 (1998)2. Branco, G.C., Rebelo, M.N.: Acta Phys. Polon. B 38, 3819–3850 (2007)3. Nunokawa, H., Parke, S.J., Valle, J.W.F.: Progr. Part. Nucl. Phys. 60, 338–402 (2008)4. Klinkhamer, F.R.: Phys. Rev. D 73, 057301 (2006)5. Gonzalez, M.C., et al.: Phys. Rev. D 64, 096006 (2001)6. Xing, Z.-z.: Phys. Lett. B 487, 327–333 (2000)7. Kobayashi, M., Maskawa, T.: Prog. Theor. Phys. Rev. Lett. 45, 652 (1973)8. Varger, V., et al.: Phys. Rev. Lett. 45, 2084 (1980)9. CHOOZ Collaboration, Apollonio, M.: Phys. Lett. B 420, 397 (1998)

10. Review of Particle Physics: J. Phys. G 33, 156 (2006)11. Jarlskog, C.: Rev. Phys. Lett. 55, 1039 (1985)

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12. Cabibbo, N.: Phys. Lett. B 72, 333 (1978)13. Vissani, F., et al.: Phys. Lett. B 571, 209 (2003)14. Koranga, B.S., Narayan, M., Uma Sankar, S.: Phys. Lett. B 665, 63 (2008)15. Koranga, B.S., Narayan, M., Uma Sankar, S.: Fizika B 18, 219–226 (2009)16. Koranga, B.S.: Electron. J. Theor. Phys. 5, 133–140 (2008)17. Koranga, B.S., Uma Sankar, S.: Electron. J. Theor. Phys. 6, 229 (2009)18. MINOS Collaboration, Adsmson, P., et al.: Phys. Rev. Lett. 101, 131082 (2008)19. Apollonio, M., et al.: Phys. Lett. B 466, 415 (1999)20. Boehm, F., et al.: Phys. Rev. Lett. 84, 3764 (2000)21. Yamamoto, S., et al.: Phys. Rev. Lett. 84, 3764 (2006)22. Itow, Y., et al.: arXiv:hep-ex/0106.19 (2011)23. Guo, X., et al.: arXiv:hep-ex/0701029 (2007)24. Soo-Bong, K., et al.: arXiv:1204.0626 (2012)25. Koranga, B.S., Narayan, M.: Int. J. Theor. Phys. 52, 2209 (2013)26. Koranga, B.S.: Int. J. Theor. Phys. 51, 3688 (2012)27. Koranga, B.S., Kumar, V., Jha, A.K.: Int. J. Theor. Phys. 50, 2609 (2011)28. Koranga, B.S.: Int. J. Theor. Phys. 50, 35 (2012)29. Koranga, B.S.: Mod. Phys. Lett. A 25, 2183 (2010)30. Koranga, B.S.: E. J. Theor. Phys. 6 (2009)