T Violation in Four Flavour Neutrino Oscillation in Planck Scale

Bipin Singh Koranga^* and Agam Kumar Jha

Department of Physics, Kirori Mal College (University of Delhi), Delhi-110007, India
E-mail: bipiniitb@rediffmail.com
*Corresponding Author

Received 27 August 2022; Accepted 03 September 2022; Publication 06 October 2022

Abstract

The Planck scale effects have been studied in the four flavour, we discuss the Planck scale effects in the four flavour neutrino sector on the asymmetry between T-conjugate oscillation probabilities. $\mathrm{\Delta }{P}_T=P({\nu }_{\alpha }\to {\nu }_{\beta })-P({\nu }_{\beta }\to {\nu }_{\alpha }),$ for four flavor framework. In this paper, we also discuss some aspect of T violation effects in four flavor neutrino oscillation above the GUT scale.

Keywords: T violation, four flavor mixing.

1 Introduction

The deficit of neutrinos flux suggested the tiny mass of neutrins. Neutrinos are not massless and mixing in the lepton sector, this indicates that there is T/CP violation. Three flavour neutrino oscillation probability in general depends on six parameter three mixing angles ${\theta }_{13},{\theta }_{12},{\theta }_{23}$, one CP violating phase $\delta$ and two independent mass square difference ${\mathrm{\Delta }}_{21}$ and ${\mathrm{\Delta }}_{31}$, The current best fit value of neutrino mixing angle and mass square difference from the neutrino experiments to be $si{n}^2{\theta }_{14}=0.019$, ${\mathrm{\Delta }}_{41}=1.70$ eV${}^2$ [1]. In search of neutrino oscillation, we have obtained the three mass square difference $\mathrm{△}{m}_{\mathit{sun}}^2\ll \mathrm{△}{m}_{\mathit{atm}}^2\ll \mathrm{△}{m}_{\mathit{LSND}}^2$ from atmospheric, solar neutrino and LSND collabration [2, 3, 4, 5]. Earlier study of T and CP violation for neutrinos have been given by [6, 7]. In this article, we will discuss the possible violation of Time reversal symmetry in four flavour neutrino oscillation. Four Flavour Neutrino Mixing beyond the GUT scale region in Section 2. In Section 3 give the Time reversal symmetry beyond the GUT scale. In Section 4, give the conclusions.

2 Planck Scale Effects in Four Flavour Neutrino Mixing

Neutrino mass squared differences and mixing angles beyond the GUT scale are studieded in earlier paper [8, 9, 10]. Mass matrix $M$ of neutrino is given by

$$\mathbf{M}=U^{*}\mathit{diag}(M_i)U^{†},$$

(1)

where, $M_i$, is the neutrino masses and $U_{\alpha i}$ is the usual mixing. Few of the parameters related to neutrino oscillation are known, the major expectation is given by the mixing elements $U$.

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

$U$	$=\mathit{diag}(e^{if1},e^{if2},e^{if3},e^{if4})R({\theta }_{34})R({\theta }_{24})\mathrm{\Delta }R({\theta }_{14})$
	$=R({\theta }_{13})R({\theta }_{12}){\mathrm{\Delta }}^{*}R({\theta }_{23})\mathit{diag}(e^{i\alpha },e^{i\beta },e^{i\gamma },1).$	(2)

The Dirac phase associates with the matrix $\mathrm{\Delta }=\mathit{diag}(e^{\frac{i\delta }{2}},1,1,e^{\frac{-i\delta }{2}}$). This leads to T/CP violation in neutrino oscillation $\alpha$, $\beta$ and $\gamma$ are the Majorana phases, which effects the neutrinoless double beta decay. $f_1,$ $f_2$, $f_3$and $f_4$ are the charged mixing angle in the charge lepton field. New mixing matrix beyond the GUT scale is given as [8, 9, 10]

$$U^{\prime }=U(1+i\delta \theta ),$$

(3)

where $\delta \theta$ is the form of hermition matrix, the first order neutrino mass square difference $\mathrm{\Delta }{M}_{ij}^{\prime }$, given by

$$\mathrm{\Delta }{M}_{ij}^{{}^{\prime 2}}=\mathrm{\Delta }{M}_{ij}^2+2.0(-M_{j}Re(m_{jj})+M_{i}Re(m_{ii})),$$	(4)
$$m=\mu U^t\lambda U,$$	(5)

and

$$\mu =\frac{v^2}{M_{pl}}.$$

The changed mixing matrix is

$$\delta {\theta }_{ij}=\frac{-Im(m_{ij})(M_i-M_j)+Re(m_{ij})(M_i+M_j)}{\mathrm{\Delta }{M}_{ij}^{{}^{\prime 2}}}.$$

(6)

Using Equation (3), we can compute four flavour neutrino mixing angles [11] as,

$si{n}^2{\theta }_{14}^{\prime }$	$={|{\mathrm{U}}_{\mathrm{s4}}^{\prime }|}^2,$	(7)
${\sin }^2{\theta }_{24}^{\prime }$	$={\displaystyle \frac{{|{\mathrm{U}}_{{\mathrm{e}}^{\prime }4}^{\prime }|}^2}{1-{|{\mathrm{U}}_{\mathrm{s4}}^{\prime }|}^2}},$	(8)
$si{n}^2{\theta }_{34}^{\prime }$	$={\displaystyle \frac{{|{\mathrm{U}}_{\mu 4}^{\prime }|}^2}{1-{|{\mathrm{U}}_{\mathrm{s4}}^{\prime }|}^2-{|{\mathrm{U}}_{\mathrm{s4}}^{\prime }|}^2}},$	(9)
${\sin }^2{\theta }_{13}^{\prime }$	$={\displaystyle \frac{{|{\mathrm{U}}_{\mathrm{s3}}^{\prime }|}^2}{1-{|{\mathrm{U}}_{\mathrm{s4}}^{\prime }|}^2}},$	(10)
$si{n}^2{\theta }_{12}^{\prime }$	$={\displaystyle \frac{{|{\mathrm{U}}_{\mathrm{s2}}^{\prime }|}^2}{1-{|{\mathrm{U}}_{\mathrm{s4}}^{\prime }|}^2-{|{\mathrm{U}}_{\mathrm{s3}}^{\prime }|}^2}},$	(11)
$si{n}^2{\theta }_{23}^{\prime }$	$={\displaystyle \frac{{|{\mathrm{U}}_{\mathrm{e3}}^{\prime }|}^2(1-{|{\mathrm{U}}_{\mathrm{s4}}^{\prime }|}^2)-({|{\mathrm{U}}_{\mathrm{s4}}^{\prime }|}^2{|{\mathrm{U}}_{\mathrm{e4}}^{\prime }|}^2)}{1-{|{\mathrm{U}}_{\mathrm{s4}}^{\prime }|}^2-{|{\mathrm{U}}_{\mathrm{e4}}^{\prime }|}^2}}$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{{|U_{s1}^{\prime }{U}_{e1}^{\prime }+U_{s2}^{\prime }{U}_{e2}^{\prime }|}^2(1-{|U_{s4}^{\prime }|}^2)}{(1-{|U_{s4}^{\prime }|}^2-{|U_{s3}^{\prime }|}^2)(1-{|U_{s4}^{\prime }|}^2-{|U_{e4}^{\prime }|}^2)}}.$	(12)

where,

$U_{\alpha 1}^{\prime }$	$=U_{\alpha 1}+{\displaystyle \sum _i}{U}_{\alpha i}({\displaystyle \frac{-Re(m_{i1})(M_i+M_1)-iIm(m_{i1})(M_i-M_1)}{M_i^2-M_1^2+2(M_{i}Re(m_{ii})-M_{1}Re(m_{11})}}.),$
$U_{\alpha 2}^{\prime }$	$=U_{\alpha 2}+{\displaystyle \sum _i}{U}_{\alpha i}({\displaystyle \frac{-Re(m_{i2})(M_i+M_2)-iIm(m_{i2})(M_i-M_2)}{M_i^2-M_2^2+2(M_{i}Re(m_{ii})-M_{4}Re(m_{22})}}.),$
$U_{\alpha 3}^{\prime }$	$=U_{\alpha 3}+{\displaystyle \sum _i}{U}_{\alpha i}({\displaystyle \frac{-Re(m_{i3})(M_i+M_3)-iIm(m_{i3})(M_i-M_3)}{M_i^2-M_3^2+2(M_{i}Re(m_{ii})-M_{4}Re(m_{33})}}.),$
${\mathrm{U}}_{\alpha 4}^{\prime }$	$={\mathrm{U}}_{\alpha 4}+{\displaystyle \sum _{\mathrm{i}}}{\mathrm{U}}_{\alpha \mathrm{i}}({\displaystyle \frac{-\mathrm{Re}({\mathrm{m}}_{\mathrm{i4}})({\mathrm{M}}_{\mathrm{i}}+{\mathrm{M}}_4)-\mathrm{iIm}({\mathrm{m}}_{\mathrm{i4}})({\mathrm{M}}_{\mathrm{i}}-{\mathrm{M}}_4)}{{\mathrm{M}}_{\mathrm{i}}^2-{\mathrm{M}}_4^2+2({\mathrm{M}}_{\mathrm{i}}\mathrm{Re}({\mathrm{m}}_{\mathrm{ii}})-{\mathrm{M}}_4\mathrm{Re}({\mathrm{m}}_{44})}}.),$
	$\mathrm{\hspace{1em}\hspace{1em}}\alpha =\mathrm{s},\mathrm{e},\mu ,\tau$

3 Time Reversal Symmetry due to Planck Scale Effects for Four Flavour Mixing

As far on T violation effects in four flavour framework, we find that a comparison of ${\nu }_{\alpha }\to {\nu }_{\beta }$ and ${\nu }_{\beta }\to {\nu }_{\alpha }$ oscillation probability. Time reversal symmetry is violated, if

$$\mathrm{\Delta }{P}_{\alpha \beta }^T=P({\nu }_{\alpha }\to {\nu }_{\beta })-P({\nu }_{\beta }\to {\nu }_{\alpha })\ne 0,$$

(13)

and

$$(\alpha ,\beta )=(e,\mu ),(\mu ,\tau ),(\tau ,e).$$

P(${\nu }_{\alpha }\to {\nu }_{\beta })$ and $P({\nu }_{\beta }\to {\nu }_{\alpha })$ is oscillation probabilities.

CP violation effects in neutrino oscillations, we find that a comparison of neutrino oscillation ${\nu }_{\alpha }\to {\nu }_{\beta }$ and ${\nu }_{\overline{\alpha }}\to {\nu }_{\overline{\beta }}$ anti-neutrino oscillation probability. CP symmetry is violated, if

$$\mathrm{\Delta }{P}_{\alpha \beta }^{CP}=P({\nu }_{\alpha }\to {\nu }_{\beta })-P(\overline{{\nu }_{\alpha }}\to \overline{{\nu }_{\beta }})\ne 0.$$

(14)

and

$$(\alpha ,\beta )=(e,\mu ),(\mu ,\tau ),(\tau ,e).$$

$\mathrm{\Delta }{P}_{\alpha \beta }^T$ and $\mathrm{\Delta }{P}_{\alpha \beta }^{CP}$ defined in Equations (13) and (14) are equal and given by

$$\mathrm{\Delta }{P}_{\alpha \beta }^T=\mathrm{\Delta }{P}_{\alpha \beta }^{CP}=16J(sin{\mathrm{\Delta }}_{21}sin{\mathrm{\Delta }}_{32}sin{\mathrm{\Delta }}_{31}).$$

(15)

here

$${\mathrm{\Delta }}_{ij}=1.27\left(\frac{{\mathrm{\Delta }}_{ij}}{e{V}^2}\right)\left(\frac{L}{Km}\right)\left(\frac{1GeV}{E}\right),$$

(16)

${\mathrm{\Delta }}_{ij}=(m_i^2-m_j^2)$ is neutrino mass square difference, L is baseline length, E is energy and J is the Jarlskog determinant [13] is given by

$J$	$=Im(U_{e1}{U}_{e2}^{*}{U}_{\mu 1}^{*}{U}_{\mu 2})$
	$={\displaystyle \frac{1}{8}}sin2{\theta }_{12}sin2{\theta }_{23}sin2{\theta }_{13}cos{\theta }_{13}sin\delta ,$	(17)

Let us compute $\mathrm{\Delta }{P}_{\alpha \beta }^T$ and $\mathrm{\Delta }{P}_{\alpha \beta }^{CP}$ for mixing $U^{\prime }=U(1+i\delta \theta ).$

$${\mathrm{\Delta }}^{\prime }{P}_{\alpha \beta }^T={\mathrm{\Delta }}^{\prime }{P}_{\alpha \beta }^{CP}=16J^{\prime }(sin{\mathrm{\Delta }}_{21}^{\prime }sin{\mathrm{\Delta }}_{32}^{\prime }sin{\mathrm{\Delta }}_{31}^{\prime }),$$

(18)

where $J^{\prime }$ is the Jarlskog determiant for new mixing is given by [13]

$J^{\prime }$	$=Im(U_{e1}^{\prime }{U}_{e2}^{{}^{\prime }*}{U}_{\mu 1}^{{}^{\prime }*}{U}_{\mu 2}^{\prime })$
	$=Im(U_{e1}{U}_{e2}^{*}{U}_{\mu 1}^{*}{U}_{\mu 2})+Im(i(U_{\mu 1}{U}_{\mu 2})(|U_{e2}{|}^2\delta {\theta }_{12}^{*}+U_{e2}{U}_{e3}\delta {\theta }_{13}$
	$\mathrm{\hspace{1em}}-|U_{e1}{|}^2\delta {\theta }_{12}^{*}-U_{e1}{U}_{e3}^{*}\delta {\theta }_{23}^{*})+Im(i(U_{e1}^{*}{U}_{e2})(|U_{\mu 1}{|}^2\delta {\theta }_{12}$
	$\mathrm{\hspace{1em}}+U_{\mu 1}^{*}{U}_{\mu 3}\delta {\theta }_{23}^{*}-|U_{\mu 2}{|}^2\delta {\theta }_{12}-U_{\mu 2}{U}_{\mu 3}^{*}\delta {\theta }_{13})$
	$=J+\mathrm{\Delta }J$

The calculation of J Jarlskog determiant [12] for four flavour neutrino oscilation due to Planck scale region. which is given by replacing the neutrino matrix $U$ by new neutrino matrix $U^{\prime },$

$J_{se}^{{13}^{\prime }}$	$=Im((U_{s1}+i{\displaystyle \sum _i}{U}_{si}\delta {\theta }_{i1})(U_{e3}+i{\displaystyle \sum _i}{U}_{ei}\delta {\theta }_{i3})$
	$\mathrm{\hspace{1em}}\times (U_{s3}^{*}-i{\displaystyle \sum _i}{U}_{ei}^{*}\delta {\theta }_{i3}^{*})(U_{e1}^{*}-i{\displaystyle \sum _i}{U}_{ei}^{*}\delta {\theta }_{i1}^{*}))$
$J_{se}^{{24}^{\prime }}$	$=Im((U_{s2}+i{\displaystyle \sum _i}{U}_{si}\delta {\theta }_{i2})(U_{e4}+i{\displaystyle \sum _i}{U}_{ei}\delta {\theta }_{i4})$
	$\mathrm{\hspace{1em}}\times (U_{s4}^{*}-i{\displaystyle \sum _i}{U}_{ei}^{*}\delta {\theta }_{i4}^{*})(U_{e2}^{*}-i{\displaystyle \sum _i}{U}_{ei}^{*}\delta {\theta }_{i2}^{*}))$
$J_{se}^{{34}^{\prime }}$	$=Im((U_{s3}+i{\displaystyle \sum _i}{U}_{si}\delta {\theta }_{i3})(U_{e4}+i{\displaystyle \sum _i}{U}_{ei}\delta {\theta }_{i4})$
	$\mathrm{\hspace{1em}}\times (U_{s4}^{*}-i{\displaystyle \sum _i}{U}_{ei}^{*}\delta {\theta }_{i4}^{*})(U_{e3}^{*}-i{\displaystyle \sum _i}{U}_{ei}^{*}\delta {\theta }_{i3}^{*}))$
$J_{\tau s}^{{13}^{\prime }}$	$=Im((U_{\tau 1}+i{\displaystyle \sum _i}{U}_{\tau i}\delta {\theta }_{i1})(U_{s3}+i{\displaystyle \sum _i}{U}_{ei}\delta {\theta }_{i3})$
	$\mathrm{\hspace{1em}}\times (U_{\tau 3}^{*}-i{\displaystyle \sum _i}{U}_{\tau i}^{*}\delta {\theta }_{i1}^{*})(U_{s1}^{*}-i{\displaystyle \sum _i}{U}_{si}^{*}\delta {\theta }_{i1}^{*}))$
$J_{\tau s}^{{14}^{\prime }}$	$=Im((U_{\tau 1}+i{\displaystyle \sum _i}{U}_{\tau i}\delta {\theta }_{i1})(U_{s4}+i{\displaystyle \sum _i}{U}_{ei}\delta {\theta }_{i4})$
	$\mathrm{\hspace{1em}}(U_{\tau 4}^{*}-i{\displaystyle \sum _i}{U}_{\tau i}^{*}\delta {\theta }_{i4}^{*})(U_{s1}^{*}-i{\displaystyle \sum _i}{U}_{si}^{*}\delta {\theta }_{i1}^{*}))$
$J_{\tau s}^{{34}^{\prime }}$	$=Im((U_{\tau 3}+i{\displaystyle \sum _i}{U}_{\tau i}\delta {\theta }_{i3})(U_{s4}+i{\displaystyle \sum _i}{U}_{ei}\delta {\theta }_{i4})$
	$\mathrm{\hspace{1em}}(U_{\tau 4}^{*}-i{\displaystyle \sum _i}{U}_{\tau i}^{*}\delta {\theta }_{i4}^{*})(U_{s3}^{*}-i{\displaystyle \sum _i}{U}_{si}^{*}\delta {\theta }_{i3}^{*}))$
$J_{e\mu }^{{23}^{\prime }}$	$=Im((U_{e2}+i{\displaystyle \sum _i}{U}_{ei}\delta {\theta }_{i2})(U_{\mu 3}+i{\displaystyle \sum _i}{U}_{\mu i}\delta {\theta }_{i3})$
	$\mathrm{\hspace{1em}}\times (U_{e3}^{*}-i{\displaystyle \sum _i}{U}_{ei}^{*}\delta {\theta }_{i3}^{*})(U_{\mu 2}^{*}-i{\displaystyle \sum _i}{U}_{\mu i^{*}}\delta {\theta }_{i2}^{*}))$
$J_{e\mu }^{{24}^{\prime }}$	$=Im((U_{e2}+i{\displaystyle \sum _i}{U}_{ei}\delta {\theta }_{i2})(U_{\mu 4}+i{\displaystyle \sum _i}{U}_{\mu i}\delta {\theta }_{i4})$
	$\mathrm{\hspace{1em}}\times (U_{e4}^{*}-i{\displaystyle \sum _i}{U}_{ei}^{*}\delta {\theta }_{i4}^{*})(U_{\mu 2}^{*}-i{\displaystyle \sum _i}{U}_{\mu i}^{*}\delta {\theta }_{i2}^{*}))$
$J_{e\mu }^{{34}^{\prime }}$	$=Im((U_{e3}+i{\displaystyle \sum _i}{U}_{ei}\delta {\theta }_{i3})(U_{\mu 4}+i{\displaystyle \sum _i}{U}_{\mu i}\delta {\theta }_{i4})$
	$\mathrm{\hspace{1em}}\times (U_{e4}^{*}-i{\displaystyle \sum _i}{U}_{ei}^{*}\delta {\theta }_{i4}^{*})(U_{\mu 3}^{*}-i{\displaystyle \sum _i}{U}_{\mu i}^{*}\delta {\theta }_{i3}^{*}))$	(19)

In term of mixing angle and Dirac phases, we can write Jarlskog determinant $J_{\alpha \beta }^{i{j}^{\prime }}$ due to Planck scale are,

$J_{se}^{{13}^{\prime }}$	$={\displaystyle \frac{1}{16}}sin2{\theta }_{12}^{\prime }sin2{\theta }_{13}^{\prime }sin2{\theta }_{14}^{\prime }sin2{\theta }_{24}^{\prime }cos{\theta }_{14}^{\prime }cos{\theta }_{23}^{\prime }sin{\theta }_{13}^{\prime }sin{\theta }_y$
	$\mathrm{\hspace{1em}}-{\displaystyle \frac{1}{16}}sin2{\theta }_{12}^{\prime }sin2{\theta }_{13}^{\prime }sin2{\theta }_{14}^{\prime }sin2{\theta }_{23}^{\prime }cos{\theta }_{14}^{\prime }co{s}^2{\theta }_{24}^{\prime }sin{\theta }_z$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{8}}sin2{\theta }_{13}^{\prime }sin2{\theta }_{14}^{\prime }sin2{\theta }_{24}^{\prime }co{s}^2{\theta }_{12}^{\prime }cos{\theta }_{14}^{\prime }sin{\theta }_{23}^{\prime }sin({\theta }_y-{\theta }_z)$	(20)
$J_{se}^{{24}^{\prime }}$	$={\displaystyle \frac{1}{8}}sin2{\theta }_{12}^{\prime }sin2{\theta }_{14}^{\prime }sin2{\theta }_{24}^{\prime }cos{\theta }_{13}^{\prime }cos{\theta }_{14}^{\prime }cos{\theta }_{23}^{\prime }sin{\theta }_y$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{4}}sin2{\theta }_{13}^{\prime }sin2{\theta }_{14}^{\prime }si{n}^2{\theta }_{12}^{\prime }cos{\theta }_{14}^{\prime }sin{\theta }_{24}^{\prime }sin{\theta }_{23}^{\prime }sin({\theta }_y-{\theta }_z)$
$J_{se}^{{34}^{\prime }}$	$={\displaystyle \frac{1}{8}}sin2{\theta }_{13}^{\prime }sin2{\theta }_{14}^{\prime }sin2{\theta }_{24}^{\prime }cos{\theta }_{14}^{\prime }sin{\theta }_{23}^{\prime }sin({\theta }_y-{\theta }_z)$	(22)
$J_{\tau s}^{{13}^{\prime }}$	$={\displaystyle \frac{1}{8}}sin2{\theta }_{13}^{\prime }sin2{\theta }_{14}^{\prime }sin2{\theta }_{34}^{\prime }co{s}^2{\theta }_{12}^{\prime }cos{\theta }_{14}^{\prime }cos{\theta }_{23}^{\prime }cos{\theta }_{24}^{\prime }sin{\theta }_x$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{16}}sin2{\theta }_{12}^{\prime }sin2{\theta }_{13}^{\prime }sin2{\theta }_{14}^{\prime }sin2{\theta }_{24}^{\prime }cos{\theta }_{14}^{\prime }cos{\theta }_{23}^{\prime }co{s}^2{\theta }_{34}^{\prime }sin{\theta }_y$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{8}}(co{s}^2{\theta }_{34}^{\prime }si{n}^2{\theta }_{24}^{\prime }-si{n}^2{\theta }_{34}^{\prime })$
	$\mathrm{\hspace{1em}}\times sin2{\theta }_{12}^{\prime }sin2{\theta }_{13}^{\prime }sin2{\theta }_{23}^{\prime }cos{\theta }_{13}^{\prime }co{s}^2{\theta }_{14}^{\prime }sin{\theta }_z$
	$\mathrm{\hspace{1em}}-{\displaystyle \frac{1}{8}}sin2{\theta }_{12}^{\prime }sin2{\theta }_{13}^{\prime }sin2{\theta }_{34}^{\prime }cos{\theta }_{13}^{\prime }co{s}^2{\theta }_{14}^{\prime }co{s}^2{\theta }_{23}^{\prime }sin{\theta }_{24}^{\prime }$
	$\mathrm{\hspace{1em}}\times sin({\theta }_x-{\theta }_y)$
	$\mathrm{\hspace{1em}}-{\displaystyle \frac{1}{16}}sin2{\theta }_{12}^{\prime }sin2{\theta }_{13}^{\prime }sin2{\theta }_{14}^{\prime }sin2{\theta }_{34}^{\prime }cos{\theta }_{14}^{\prime }cos{\theta }_{24}^{\prime }$
	$\mathrm{\hspace{1em}}\times sin{\theta }_{13}^{\prime }sin{\theta }_{23}^{\prime }sin({\theta }_x+{\theta }_z)$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{8}}sin2{\theta }_{13}^{\prime }sin2{\theta }_{14}^{\prime }sin2{\theta }_{24}^{\prime }co{s}^2{\theta }_{12}^{\prime }cos{\theta }_{14}^{\prime }co{s}^2{\theta }_{34}^{\prime }$
	$\mathrm{\hspace{1em}}\times sin{\theta }_{23}^{\prime }sin({\theta }_y-{\theta }_z)$
	$\mathrm{\hspace{1em}}-{\displaystyle \frac{1}{8}}sin2{\theta }_{12}^{\prime }sin2{\theta }_{13}^{\prime }sin2{\theta }_{34}^{\prime }co{s}^2{\theta }_{14}^{\prime }cos{\theta }_{13}^{\prime }si{n}^2{\theta }_{23}^{\prime }$
	$\mathrm{\hspace{1em}}\times sin{\theta }_{24}^{\prime }sin({\theta }_x-{\theta }_y+{\theta }_z)$	(23)
$J_{\tau s}^{{14}^{\prime }}$	$=-{\displaystyle \frac{1}{8}}sin2{\theta }_{13}^{\prime }sin2{\theta }_{14}^{\prime }sin2{\theta }_{34}^{\prime }co{s}^2{\theta }_{12}^{\prime }cos{\theta }_{14}^{\prime }cos{\theta }_{23}^{\prime }cos{\theta }_{24}^{\prime }sin{\theta }_x$
	$\mathrm{\hspace{1em}}-{\displaystyle \frac{1}{8}}sin2{\theta }_{12}^{\prime }sin2{\theta }_{13}^{\prime }sin2{\theta }_{24}^{\prime }co{s}^2{\theta }_{14}^{\prime }cos{\theta }_{23}^{\prime }co{s}^2{\theta }_{34}^{\prime }sin{\theta }_y$
	$\mathrm{\hspace{1em}}-{\displaystyle \frac{1}{8}}sin2{\theta }_{12}^{\prime }sin2{\theta }_{14}^{\prime }sin2{\theta }_{34}^{\prime }cos{\theta }_{13}^{\prime }cos{\theta }_{14}^{\prime }cos{\theta }_{24}^{\prime }sin{\theta }_{23}^{\prime }sin({\theta }_x+{\theta }_z)$
	$\mathrm{\hspace{1em}}-{\displaystyle \frac{1}{8}}sin2{\theta }_{12}^{\prime }sin2{\theta }_{13}^{\prime }sin2{\theta }_{24}^{\prime }co{s}^2{\theta }_{14}^{\prime }cos{\theta }_{12}^{\prime }co{s}^2{\theta }_{34}^{\prime }$
	$\mathrm{\hspace{1em}}\times sin{\theta }_{23}^{\prime }sin({\theta }_y-{\theta }_z)$	(24)
$J_{\tau s}^{{34}^{\prime }}$	$={\displaystyle \frac{1}{8}}sin2{\theta }_{13}^{\prime }sin2{\theta }_{14}^{\prime }sin2{\theta }_{34}^{\prime }cos{\theta }_{14}^{\prime }cos{\theta }_{23}^{\prime }cos{\theta }_{24}^{\prime }sin{\theta }_x$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{8}}sin2{\theta }_{13}^{\prime }sin2{\theta }_{14}^{\prime }sin2{\theta }_{24}^{\prime }cos{\theta }_{14}^{\prime }co{s}^2{\theta }_{34}^{\prime }sin{\theta }_{23}^{\prime }sin({\theta }_y-{\theta }_z)$	(25)
$J_{e\mu }^{{23}^{\prime }}$	$=-{\displaystyle \frac{1}{4}}(co{s}^2{\theta }_{12}^{\prime }co{s}^2{\theta }_{23}^{\prime }si{n}^2{\theta }_{24}^{\prime }-co{s}^2{\theta }_{12}^{\prime }co{s}^2{\theta }_{24}^{\prime }si{n}^2{\theta }_{23}^{\prime }$
	$\mathrm{\hspace{1em}}+si{n}^2{\theta }_{12}^{\prime }si{n}^2{\theta }_{14}^{\prime }si{n}^2{\theta }_{24}^{\prime }$
	$\mathrm{\hspace{1em}}+si{n}^2{\theta }_{12}^{\prime }si{n}^2{\theta }_{23}^{\prime })sin2\theta {}_{13}{}^{\prime }sin2\theta {}_{34}{}^{\prime }sin\theta {}_{14}{}^{\prime }cos\theta {}_{23}{}^{\prime }cos\theta {}_{24}{}^{\prime }sin\theta {}_x$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{4}}(co{s}^2{\theta }_{13}^{\prime }co{s}^2{\theta }_{34}^{\prime }si{n}^2{\theta }_{23}^{\prime }-co{s}^2{\theta }_{23}^{\prime }co{s}^2{\theta }_{34}^{\prime }si{n}^2{\theta }_{13}^{\prime }$
	$\mathrm{\hspace{1em}}+si{n}^2{\theta }_{13}^{\prime }si{n}^2{\theta }_{14}^{\prime }si{n}^2{\theta }_{34}^{\prime }$
	$\mathrm{\hspace{1em}}-si{n}^2{\theta }_{34}^{\prime }si{n}^2{\theta }_{23}^{\prime })sin2\theta {}_{12}{}^{\prime }sin2\theta {}_{24}{}^{\prime }sin\theta {}_{14}{}^{\prime }cos\theta {}_{13}{}^{\prime }cos\theta {}_{23}{}^{\prime }cos\theta {}_{24}{}^{\prime }sin\theta {}_y$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{8}}(co{s}^2{\theta }_{24}^{\prime }co{s}^2{\theta }_{34}^{\prime }-co{s}^2{\theta }_{24}^{\prime }si{n}^2{\theta }_{14}^{\prime }si{n}^2{\theta }_{34}^{\prime }$
	$\mathrm{\hspace{1em}}-co{s}^2{\theta }_{34}^{\prime }si{n}^2{\theta }_{14}^{\prime }si{n}^2{\theta }_{24}^{\prime }+si{n}^2{\theta }_{14}^{\prime }si{n}^2{\theta }_{24}^{\prime }si{n}^2{\theta }_{34}^{\prime })$
	$\mathrm{\hspace{1em}}\times sin2{\theta }_{12}^{\prime }sin2{\theta }_{13}^{\prime }sin2{\theta }_{23}^{\prime }cos{\theta }_{13}^{\prime }cos{\theta }_{34}^{\prime }sin{\theta }_z$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{4}}(si{n}^2{\theta }_{34}^{\prime }co{s}^2{\theta }_{13}^{\prime }-co{s}^2{\theta }_{24}^{\prime }si{n}^2{\theta }_{13}^{\prime })sin2{\theta }_{12}^{\prime }sin2{\theta }_{34}^{\prime }$
	$\mathrm{\hspace{1em}}\times co{s}^2{\theta }_{23}^{\prime }sin{\theta }_{13}^{\prime }si{n}^2{\theta }_{14}^{\prime }sin{\theta }_{24}^{\prime }sin({\theta }_x-{\theta }_y)$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{4}}(co{s}^2{\theta }_{23}^{\prime }-si{n}^2{\theta }_{24}^{\prime }co{s}^2{\theta }_{13}^{\prime }-co{s}^2{\theta }_{24}^{\prime }si{n}^2{\theta }_{13}^{\prime }$
	$\mathrm{\hspace{1em}}+si{n}^2{\theta }_{13}^{\prime }si{n}^2{\theta }_{14}^{\prime }si{n}^2{\theta }_{24}^{\prime })$
	$\mathrm{\hspace{1em}}\times sin2{\theta }_{12}^{\prime }sin2{\theta }_{34}^{\prime }cos{\theta }_{13}^{\prime }cos{\theta }_{24}^{\prime }sin{\theta }_{23}^{\prime }sin({\theta }_x+{\theta }_z)$
	$\mathrm{\hspace{1em}}-{\displaystyle \frac{1}{4}}(co{s}^2{\theta }_{12}^{\prime }co{s}^2{\theta }_{23}^{\prime }co{s}^2{\theta }_{34}^{\prime }-co{s}^2{\theta }_{12}^{\prime }co{s}^2{\theta }_{23}^{\prime }si{n}^2{\theta }_{34}^{\prime }$
	$\mathrm{\hspace{1em}}-co{s}^2{\theta }_{23}^{\prime }co{s}^2{\theta }_{34}^{\prime }si{n}^2{\theta }_{12}^{\prime }+si{n}^2{\theta }_{12}^{\prime }si{n}^2{\theta }_{14}^{\prime }si{n}^2{\theta }_{34}^{\prime }$
	$\mathrm{\hspace{1em}}-si{n}^2{\theta }_{12}^{\prime }si{n}^2{\theta }_{23}^{\prime }si{n}^2{\theta }_{34}^{\prime })sin2\theta {}_{13}{}^{\prime }sin2\theta {}_{24}{}^{\prime }sin\theta {}_{14}{}^{\prime }sin\theta {}_{23}{}^{\prime }sin({\theta }_y-{\theta }_z)$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{4}}(co{s}^2{\theta }_{12}^{\prime }co{s}^2{\theta }_{13}^{\prime }co{s}^2{\theta }_{24}^{\prime }-co{s}^2{\theta }_{12}^{\prime }co{s}^2{\theta }_{24}^{\prime }si{n}^2{\theta }_{13}^{\prime }si{n}^2{\theta }_{14}^{\prime }$
	$\mathrm{\hspace{1em}}-co{s}^2{\theta }_{13}^{\prime }co{s}^2{\theta }_{24}^{\prime }si{n}^2{\theta }_{12}^{\prime }si{n}^2{\theta }_{14}^{\prime }+co{s}^2{\theta }_{24}^{\prime }si{n}^2{\theta }_{12}^{\prime }si{n}^2{\theta }_{13}^{\prime }si{n}^2{\theta }_{14}^{\prime })$
	$\mathrm{\hspace{1em}}\times sin2{\theta }_{23}^{\prime }sin2{\theta }_{34}^{\prime }sin{\theta }_{23}^{\prime }sin({\theta }_x-{\theta }_y+{\theta }_z)$
	$\mathrm{\hspace{1em}}-{\displaystyle \frac{1}{4}}(co{s}^2{\theta }_{13}^{\prime }co{s}^2{\theta }_{24}^{\prime }si{n}^2{\theta }_{23}^{\prime }-co{s}^2{\theta }_{13}^{\prime }si{n}^2{\theta }_{23}^{\prime }si{n}^2{\theta }_{24}^{\prime }si{n}^2{\theta }_{14}^{\prime }$
	$\mathrm{\hspace{1em}}-co{s}^2{\theta }_{24}^{\prime }si{n}^2{\theta }_{23}^{\prime }si{n}^2{\theta }_{13}^{\prime }si{n}^2{\theta }_{14}^{\prime })$
	$\mathrm{\hspace{1em}}\times sin2{\theta }_{12}^{\prime }sin2{\theta }_{34}^{\prime }sin{\theta }_{13}^{\prime }sin{\theta }_{24}^{\prime }sin({\theta }_x-{\theta }_y+2{\theta }_z)$
	$\mathrm{\hspace{1em}}-{\displaystyle \frac{1}{16}}sin2{\theta }_{12}^{\prime }sin2{\theta }_{13}^{\prime }sin2{\theta }_{24}^{\prime }sin2{\theta }_{34}^{\prime }cos{\theta }_{13}^{\prime }cos{\theta }_{24}^{\prime }si{n}^2{\theta }_{14}^{\prime }$
	$\mathrm{\hspace{1em}}\times sin({\theta }_x+{\theta }_y)$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{16}}sin2{\theta }_{12}^{\prime }sin2{\theta }_{13}^{\prime }sin2{\theta }_{23}^{\prime }sin2{\theta }_{34}^{\prime }sin{\theta }_{13}^{\prime }cos{\theta }_{24}^{\prime }$
	$\mathrm{\hspace{1em}}\times sin{\theta }_{14}^{\prime }sin({\theta }_x-{\theta }_z)$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{8}}(co{s}^2{\theta }_{34}^{\prime }-si{n}^2{\theta }_{34}^{\prime })sin2{\theta }_{24}^{\prime }sin2{\theta }_{23}^{\prime }sin2{\theta }_{34}^{\prime }sin{\theta }_{13}^{\prime }cos{\theta }_{12}^{\prime }$
	$\mathrm{\hspace{1em}}\times sin{\theta }_{24}^{\prime }sin{\theta }_{14}^{\prime }sin{\theta }_{23}^{\prime }sin({\theta }_y-2{\theta }_z)$
	$\mathrm{\hspace{1em}}-{\displaystyle \frac{1}{8}}(co{s}^2{\theta }_{12}^{\prime }-si{n}^2{\theta }_{12}^{\prime })sin2{\theta }_{13}^{\prime }sin2{\theta }_{23}^{\prime }sin2{\theta }_{34}^{\prime }si{n}^2{\theta }_{24}^{\prime }$
	$\mathrm{\hspace{1em}}\times cos{\theta }_{24}^{\prime }sin{\theta }_{14}^{\prime }sin{\theta }_{23}^{\prime }sin({\theta }_x-2{\theta }_y+2{\theta }_z)$
	$\mathrm{\hspace{1em}}-{\displaystyle \frac{1}{8}}sin2{\theta }_{12}^{\prime }sin2{\theta }_{23}^{\prime }sin2{\theta }_{34}^{\prime }cos{\theta }_{12}^{\prime }cos{\theta }_{23}^{\prime }sin{\theta }_{24}^{\prime }cos{\theta }_{34}^{\prime }sin{\theta }_{14}^{\prime }$
	$\mathrm{\hspace{1em}}\times sin{\theta }_{34}^{\prime }sin({\theta }_x-2{\theta }_y+{\theta }_z)$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{16}}sin2{\theta }_{12}^{\prime }sin2{\theta }_{13}^{\prime }sin2{\theta }_{24}^{\prime }sin2{\theta }_{34}^{\prime }sin{\theta }_{13}^{\prime }sin{\theta }_{14}^{\prime }si{n}^2{\theta }_{23}^{\prime }$
	$\mathrm{\hspace{1em}}\times sin{\theta }_{24}^{\prime }sin({\theta }_x-2{\theta }_y+3{\theta }_z)$	(26)
$J_{e\mu }^{{24}^{\prime }}$	$={\displaystyle \frac{1}{16}}sin2{\theta }_{13}^{\prime }sin2{\theta }_{14}^{\prime }sin2{\theta }_{24}^{\prime }sin2{\theta }_{34}^{\prime }cos{\theta }_{14}^{\prime }cos{\theta }_{23}^{\prime }sin{\theta }_{14}^{\prime }sin{\theta }_{24}^{\prime }sin{\theta }_x$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{8}}sin2{\theta }_{12}^{\prime }sin2{\theta }_{14}^{\prime }sin2{\theta }_{24}^{\prime }cos{\theta }_{13}^{\prime }cos{\theta }_{14}^{\prime }cos{\theta }_{23}^{\prime }si{n}^2{\theta }_{34}^{\prime }sin{\theta }_x$
	$\mathrm{\hspace{1em}}-{\displaystyle \frac{1}{4}}sin2{\theta }_{12}^{\prime }sin2{\theta }_{24}^{\prime }co{s}^2{\theta }_{14}^{\prime }co{s}^2{\theta }_{23}^{\prime }cos{\theta }_{24}^{\prime }cos{\theta }_{34}^{\prime }sin{\theta }_{13}^{\prime }$
	$\mathrm{\hspace{1em}}\times sin{\theta }_{34}^{\prime }sin({\theta }_x-{\theta }_y)$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{16}}sin2{\theta }_{12}^{\prime }sin2{\theta }_{14}^{\prime }sin2{\theta }_{24}^{\prime }sin2{\theta }_{34}^{\prime }cos{\theta }_{13}^{\prime }sin{\theta }_{23}^{\prime }$
	$\mathrm{\hspace{1em}}\times sin{\theta }_{24}^{\prime }sin({\theta }_x+{\theta }_y)$
	$\mathrm{\hspace{1em}}-{\displaystyle \frac{1}{8}}sin2{\theta }_{13}^{\prime }sin2{\theta }_{14}^{\prime }sin2{\theta }_{24}^{\prime }cos{\theta }_{14}^{\prime }si{n}^2{\theta }_{12}^{\prime }sin{\theta }_{23}^{\prime }$
	$\mathrm{\hspace{1em}}\times si{n}^2{\theta }_{34}^{\prime }sin({\theta }_y-{\theta }_z)$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{8}}sin2{\theta }_{12}^{\prime }sin2{\theta }_{13}^{\prime }sin2{\theta }_{34}^{\prime }cos{\theta }_{13}^{\prime }co{s}^2{\theta }_{14}^{\prime }si{n}^2{\theta }_{23}^{\prime }$
	$\mathrm{\hspace{1em}}\times sin{\theta }_{24}^{\prime }sin({\theta }_x-{\theta }_y+2{\theta }_z)$
	$\mathrm{\hspace{1em}}-{\displaystyle \frac{1}{4}}(co{s}^2{\theta }_{12}^{\prime }-si{n}^2{\theta }_{12}^{\prime }si{n}^2{\theta }_{13}^{\prime })sin2{\theta }_{23}^{\prime }sin2{\theta }_{24}^{\prime }co{s}^2{\theta }_{14}^{\prime }cos{\theta }_{34}^{\prime }$
	$\mathrm{\hspace{1em}}\times cos{\theta }_{24}^{\prime }sin{\theta }_{34}^{\prime }sin({\theta }_x-{\theta }_y+{\theta }_z)$	(27)
$J_{e\mu }^{{34}^{\prime }}$	$=-{\displaystyle \frac{1}{16}}sin2{\theta }_{13}^{\prime }sin2{\theta }_{14}^{\prime }sin2{\theta }_{24}^{\prime }sin2{\theta }_{34}^{\prime }cos{\theta }_{14}^{\prime }cos{\theta }_{23}^{\prime }sin{\theta }_{24}^{\prime }sin{\theta }_x$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{8}}sin2{\theta }_{13}^{\prime }sin2{\theta }_{14}^{\prime }sin2{\theta }_{24}^{\prime }cos{\theta }_{14}^{\prime }sin{\theta }_{23}^{\prime }si{n}^2{\theta }_{34}^{\prime }sin({\theta }_y-{\theta }_z)$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{4}}sin2{\theta }_{23}^{\prime }sin2{\theta }_{24}^{\prime }co{s}^2{\theta }_{13}^{\prime }co{s}^2{\theta }_{14}^{\prime }cos{\theta }_{24}^{\prime }cos{\theta }_{34}^{\prime }$
	$\mathrm{\hspace{1em}}\times sin{\theta }_{34}^{\prime }sin({\theta }_z+{\theta }_x-{\theta }_y)$	(28)
${\theta }_x$	$={\delta }_{14}-({\delta }_{13}+{\delta }_{34})$
${\theta }_y$	$={\delta }_{14}-({\delta }_{12}+{\delta }_{24})$
${\theta }_z$	$={\delta }_{13}-({\delta }_{12}+{\delta }_{23})$	(29)

4 Conclusions

We discussed some importance of T violation in four flavour neutrino oscillation beyond the GUT scale. We have presented four flavour neutrino mixing and possible T violation term ${\mathrm{\Delta }}^{\prime }{P}_{\alpha \beta }^T$ above the GUT scale. In four flavour neutrino oscillation above the GUT scale region [10]. The mixing angle changes in ${\theta }_{14},{\theta }_{24}$ and ${\theta }_{34}$ above the GUT scale, are very small. But the change in ${\theta }_{23}$ is very large for large range of values of the majaorona phases $\alpha ,\beta$ and $\gamma$. In four flavour mixing gives the range of mixing angle ${\theta }_{12}^{\prime }={\theta }_{12}\pm {3.0}^{\circ }$, ${\theta }_{12}^{\prime }={\theta }_{12}\pm {45}^{\circ }$ [12] modified mass square difference ${\mathrm{\Delta }}_{21}^{\prime }={\mathrm{\Delta }}_{21}\pm (1.0+0.5)\times {10}^{-5}$ eV${}^2$ [10], for Planck scale $M_{pl}\approx 2.0\times {10}^{19}$ GeV. In this study, beyond the GUT scale region, we have obtained, solar mixing angle ${\theta }_{12}$, atmoshpheric angle ${\theta }_{23}$ and solar neutrino mass square difference ${\mathrm{\Delta }}_{21}^{\prime }$ are more effective for Time Reversal symmetry violation. We would like to conclude that in planck scale region, two mixing angle ${\theta }_{12}$, ${\theta }_{23}$ and solar mass square difference ${\mathrm{\Delta }}_{21}^{\prime }$ will more effective for Time Reversal symmetry violation.

References

[1] S. Gariazzo et al., JHEP 1706, 135 (2017).

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[3] SNO Collaboration, B. Aharmim et al., Phys. Rev. C 75, 045502 (2007).

[4] Super-Kamiokande Collaboration, J. Hosaka, et al., Phys. Rev. D 74, 32002 (2006).

[5] T2K Collaboration, Y. Itow F. Ardellier et al., arXiv:hep-ex/0106019.

[6] N. Cabibbo, Phys. Lett. B 72, 333 (1978).

[7] V. Barger, K. Whisnant, R.J.N. Phillips, Phys. Rev. Lett., 2084 (1980).

[8] B.S. Koranga, S. Uma Sankara, M. Narayan, Phys. Letts. B 665, 63–66 (2008).

[9] F. Vissani et al., Phys. Lett. B571, 209, (2003).

[10] Bipin Singh Koranga, Mod. Phys. Lett. A25, 1–6 (2010).

[11] B.S. Koranga, V.K. Nautiyal, A.K. Jha, M. Narayan, Int. J. Theor. Phys. (2021). https://doi.org/10.1007/s10773-021-04811-2.

[12] Bipin Singh Koranga and S. Uma Sankar, Electron. J. Theor. Phys. 5, 1–6 (2009).

[13] B.S. Koranga and V.K. Nautiyal, Int. J. Theor. Phys., 60, 3548–3565, 9 (2021).

Biographies

Bipin Singh Koranga is an Associate Professor in the Department of Physics, Kirori Mal College, University of Delhi. He has been with the Theoretical Physics Group, IIT Bombay since 2001 and received the Ph.D. degree in physics (neutrino masses and mixings) from the Indian Institute of Technology Bombay in 2007. He has been teaching basic courses in physics and mathematical Physics at the graduate level for the last 15 years. His research interests include the origin of universe, physics beyond the standard model, theoretical nuclear physics, quantum mechanical neutrino oscillation and few topics related to astrology. He has published over 50 scientific papers in various International Journals and three book in international publishers. His present research interest includes the neutrino mass models and related phenomenology. He is also a life member of Indian Physics Society.

Agam Kumar Jha is an Associate Professor in the Department of Physics, Kirori Mal College, University of Delhi. He earned his Ph.D. degree in physics (High Energy Particle Physics) from University of Delhi, Delhi. He has been teaching basic courses in Physics at graduate and postgraduate level for the last 18 years. He has published several scientific papers in various international journals of repute and also presented his works at national and international conferences. His research interests include the Quark Gluon Plasma (QGP) and Neutrino Physics.

Journal of Graphic Era University, Vol. 10_2, 217–228.
doi: 10.13052/jgeu0975-1416.1029
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