Jarlskog Determinant in Four Flavor Neutrino Oscillation Framework

Vivek Kumar Nautiyal^1,*, Bipin Singh Koranga², Sanjay Kumar Padaliya³, Neelam Das⁴ and Ashish Shrivastava²

¹Department of Physics, Babasaheb Bhimrao Ambedkar University, Lucknow-226025, India
²Department of Physics, Kirori Mal college (University of Delhi), Delhi-110007, India
³Department of Mathematics, S.G.R.R. (P.G.) College-248001, India
⁴Department of Physics, University of Lucknow, Lucknow-226007, India
E-mail: viveknautiyal01@gmail.com
*Corresponding Author

Received 20 January 2022; Accepted 11 February 2022; Publication 07 May 2022

Abstract

The weak CP voilation in leptonic sector is potentially one of the key aspect to probe the physics beyond the Standard Model (SM). This article majorly focus on the Dirac CP phase effect on the Jarlskog Deteminant in four flavor neutrino framework. The calculations for the upper and lower bound of the Jarlskog Determinant in $4\times 4$ neutrino four flavor mixing matrix has also been performed.

Keywords: CP violaton, Jarlskog determinant, PMNS Matrix.

1 Introduction

The experimental detection of neutrino oscillation came from Ray Davis’s Homestake experiment and Y. Fukuda et al. [1]. Several other collaborations have neutrino oscillation data obtained from the teresstrial and non-teresstial experiments. In attempt to interpret the neutrino oscillation data obtained from the atmospheric, solar and LSND collabration experiments we have to incorporate the four neutrino of definite mass.

The three different scale of neutrino mass-squared differences $\mathrm{△}{m}_{sun}^2\ll \mathrm{△}{m}_{atm}^2\ll \mathrm{△}{m}_{\mathit{LSND}}^2$ obtained by different neutrino oscillation experiments and LSND collabration [2]. It indicates the possibility of four neutrinos with definite mass to explain these data. Super- Kamiokande Collaboration, Y. Fukuda et al. [1] experimental data was in good agreement with two flavor oscillation i.e.${\nu }_{\mu }\to {\nu }_{ϵ}$ and provided evidence for neutrino oscillation. All these neutrino oscillation data could be consolidated by four neutrinos with definite mass by S.M. Bilenky et al. [3]. Thus four-neutrino models have been studied by earlier authors [15, 20, 21, 23]. The simultaneity in the oscillations of four different flavors of neutrinos demands an inclusion of a light sterile neutrino in the interpretation. The four neutrino mixing model are expected to violate the CP and T symmetry, the measurement of the violating factors require new generation very long baseline neutrino experiments. Some efforts are made in this directions suggest the possibilities to observe four neutrino oscillation. The better understanding of the nontrivial complex phases observed in $4\times 4$ lepton flavor mixing matrix can be achieved by investigating the specific parameters involved between CP violating phases and flavor $4\times 4$ lepton flavor mixing angle with rephrasing invariants of mixing matrix. Some attempts were made, but analytically correct result could not be achieved. The presence of $1-{\gamma }_5$ factors in Lagrangian indicates, parity violation is maximal in charged-current interactions. Similarly charge-conjugation invariance is maximally violated in mass mixing matrices [4]. CP violating nontrivial complex phases can be allocated in many different ways in $4\times 4$ lepton flavor mixing matrix. Maximal CP violation mainly depends on CP violating phases and mixing angles. In our present article, we have calculated Jarlskog determinant for different sample of ${\theta }_x,{\theta }_y$ and ${\theta }_z$ which are related with CP and T violating relative phases (${\delta }_{ij}$).

2 Four Neutrino Mixing Angles and Their Mass Squared Differences

The defect in atmospheric neutrino which depends on zenith-angle, was first observed [5] via transition of ${\nu }_{\mu }\to {\nu }_{\mu }$ with ${\mathrm{\Delta }}_{31}=(1-2)\times {10}^{-3}e{V}^2,si{n}^{2}2{\theta }_{23}=1.0$ mass difference and the mixing respectively. For normal and inverted mass hierarchy, neutrino mass square difference from the three neutrino data analysis of the Super-Kamiokande [6] with 90% CL were observed $1.9\times {10}^{-3}e{V}^2\le {\mathrm{\Delta }}_{31}\le 2.6\times {10}^{-3}e{V}^2$ and $1.7\times {10}^{-3}e{V}^2\le {\mathrm{\Delta }}_{31}\le 2.7\times {10}^{-3}e{V}^2$, respectively. The another evidence was obtained from the solar neutrino deficit [7], which is consistent with ${\nu }_{\mu }\to {\nu }_{\tau }/{\nu }_e$ transition. The out came from the Sudbury Neutrino Observatory SNO experiments [8] are compatible with the standard solar model [1] and strongly suggest the LMA solution, which is given as ${\mathrm{\Delta }}_{21}=7\times {10}^{-5}e{V}^2,si{n}^{2}2{\theta }_{12}=0.8.$ The others Solar neutrino experiments (Super-K, GALLEX, SAGE, SNO and GNO) show the neutrino oscillations, neutrino oscillation provide the most elegant explanation of all the data [9].

$${\mathrm{\Delta }}_{\mathit{solar}}=7_{-1.3}^{+5}\times {10}^{-5}e{V}^2,$$	(1)
$$ta{n}^2{\theta }_{\mathit{solar}}=0.4{}_{-0.1}{}^{+0.14}.$$	(2)

Whereas, the analysis of three neutrino global data came from the solar and KamLAND reactor was obtained [10] ${\mathrm{\Delta }}_{21}={7.5}_{-.20}^{+.19}\times {10}^{-3}e{V}^2$ and $ta{n}^2{\theta }_{12}=0.452{}_{-0.032}^{+0.035}$, respectively. Neutrino oscillation is also shown by the atmospheric neutrino experiments ( Kamiokande, Super-K ) and the best fit to the all data [1] is ${\mathrm{\Delta }}_{atmo}={2.0}_{-0.92}^{+1.0}\times {10}^{-3}e{V}^2$, $si{n}^{2}2{\theta }_{atmo}=0.4{}_{-0.10}^{+0.14}$. The upper bound on the third mixing angle ${\theta }_{13}$ was given by the CHOOZ reactor experiment [11] with the 90 % CL which was found as

$si{n}^2{\theta }_{13}$	$<0.20\mathit{\hspace{1em}}\text{for}|{\mathrm{\Delta }}_{31}|=2.0\times {10}^{-3}e{V}^2,$	(3)
$si{n}^2{\theta }_{13}$	$<0.16\mathit{\hspace{1em}}\text{for}|{\mathrm{\Delta }}_{31}|=2.5\times {10}^{-3}e{V}^2,$	(4)
$si{n}^2{\theta }_{13}$	$<0.14\mathit{\hspace{1em}}\text{for}|{\mathrm{\Delta }}_{31}|=3.0\times {10}^{-3}e{V}^2,$	(5)

The CP phase $\delta$ has not been constrained. The third mixing angle ${\theta }_{13}$ and associated mass difference from the two neutrino analysis of the MINOS data was obtained [12] $si{n}^2{\theta }_{13}<0.90$ and $|{\mathrm{\Delta }}_{31}|=(2.43\pm 0.13)\times {10}^{-3}e{V}^2$, respectively. The future neutrino experiments plan to measure the oscillation parameters precisely. The combined analysis of Day Bay, MINOS and Bugey-3 data [13] has excluded most of the parameter space of the mass squared difference ${\mathrm{\Delta }}_{41}=m_4^2-m_1^2$ and the mixing angle $si{n}^2{\theta }_{14}$ for the sterile neutrino, the latest global analysis of neutrino oscillation data indicates that a small region arround the best-fit value of [6]

$${\mathrm{\Delta }}_{41}=1.7e{V}^2.\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}si{n}^2{\theta }_{14}=0.019$$

(6)

3 Jarlskog in Four Flavor Neutrino

The sterile neutrino do not interact via weak interaction. For the flavor mixing of one sterile neutrino (${\nu }_{\mathrm{s}}$) and three active neutrinos (${\nu }_e,{\nu }_{\mu },{\nu }_{\tau }$), the matrix form of U [14, 15] can be written as

$$U=\left(\begin{array}{cccc}\hfill U_{s1}\mathit{\hspace{1em}}\hfill & \hfill U_{s2}\mathit{\hspace{1em}}\hfill & \hfill U_{s3}\mathit{\hspace{1em}}\hfill & \hfill U_{s4}\hfill \\ \hfill U_{e1}\mathit{\hspace{1em}}\hfill & \hfill U_{e2}\mathit{\hspace{1em}}\hfill & \hfill U_{e3}\mathit{\hspace{1em}}\hfill & \hfill U_{e4}\hfill \\ \hfill U_{\mu 1}\mathit{\hspace{1em}}\hfill & \hfill U_{\mu 2}\mathit{\hspace{1em}}\hfill & \hfill U_{\mu 3}\mathit{\hspace{1em}}\hfill & \hfill U_{\mu 4}\hfill \\ \hfill U_{\tau 1}\mathit{\hspace{1em}}\hfill & \hfill U_{\tau 2}\mathit{\hspace{1em}}\hfill & \hfill U_{\tau 3}\mathit{\hspace{1em}}\hfill & \hfill U_{\tau 4}\hfill \end{array}\right),$$

(7)

The Four dimension matrix U contains 6 mixing angles and 3 Dirac phase angles and 3 Majorana phase angle. The Jarlskog invariants of CP and T violation is given by:

$$J_{\alpha \beta }^{ij}\approx Im(U_{\alpha i}{U}_{\beta j}{U}_{\alpha j}^{*}{U}_{\beta i}^{*})$$

(8)

where, $\alpha ,\beta =s,e,\mu ,\tau$ and $i=1,2,3,4$.

Since $U$ is unitary matrix. So we get,

$$J_{\alpha \beta }^{ii}=J_{\alpha \alpha }^{ij}=J_{\beta \beta }^{ij}=J_{\alpha \beta }^{jj}=0$$

(9)

and

$$J_{\alpha \beta }^{ij}=-J_{\alpha \beta }^{ji}=J_{\beta \alpha }^{ij}=-J_{\beta \alpha }^{ji}$$

(10)

Here we deal with four flavor framework by assuming that the sterile neutrino of eV range and the mixing of this sterile neutrino with three active neutrinos is light. By assuming one sterile neutrinos [23], the PMNS matrix $U_{4\times 4}$ is given by

$$U=R_{34}{R}_{24}{R}_{14}{R}_{23}{R}_{13}{R}_{12},$$

(11)

where $R_{ij}$ are rotations matrix in ij space,

$R_{12}$

$=\left(\begin{array}{cccc}\hfill c_{12}\mathit{\hspace{1em}}\hfill & \hfill s_{12}{e}^{-i{\delta }_{12}}\mathit{\hspace{1em}}\hfill & \hfill 0\mathit{\hspace{1em}}\hfill & \hfill 0\hfill \\ \hfill -s_{12}{e}^{i{\delta }_{12}}\mathit{\hspace{1em}}\hfill & \hfill c_{12}\mathit{\hspace{1em}}\hfill & \hfill 0\mathit{\hspace{1em}}\hfill & \hfill 0\hfill \\ \hfill 0\mathit{\hspace{1em}}\hfill & \hfill 0\mathit{\hspace{1em}}\hfill & \hfill 1\mathit{\hspace{1em}}\hfill & \hfill 0\hfill \\ \hfill 0\mathit{\hspace{1em}}\hfill & \hfill 0\mathit{\hspace{1em}}\hfill & \hfill 0\mathit{\hspace{1em}}\hfill & \hfill 1\hfill \end{array}\right)\mathit{\hspace{1em}}\text{and}$

$R_{34}$

$=\left(\begin{array}{cccc}\hfill 1\mathit{\hspace{1em}}\hfill & \hfill 0\mathit{\hspace{1em}}\hfill & \hfill 0\mathit{\hspace{1em}}\hfill & \hfill 0\hfill \\ \hfill 0\mathit{\hspace{1em}}\hfill & \hfill 1\mathit{\hspace{1em}}\hfill & \hfill 0\mathit{\hspace{1em}}\hfill & \hfill 0\hfill \\ \hfill 0\mathit{\hspace{1em}}\hfill & \hfill 0\mathit{\hspace{1em}}\hfill & \hfill c_{34}\mathit{\hspace{1em}}\hfill & \hfill s_{34}\hfill \\ \hfill 0\mathit{\hspace{1em}}\hfill & \hfill 0\mathit{\hspace{1em}}\hfill & \hfill -s_{34}\mathit{\hspace{1em}}\hfill & \hfill c_{34}\hfill \end{array}\right),$

where $c_{ij}=cos{\theta }_{ij},s_{ij}=sin{\theta }_{ij}{e}^{i{\delta }_{ij}}$.

In four flavour the standard form of U is

$U_{s1}$	$=(c_{14}{c}_{13}{c}_{12}),$
$U_{s2}$	$=(c_{14}{c}_{13}{s}_{12}{e}^{-i{\delta }_{12}}),$
$U_{s3}$	$=(c_{14}{s}_{13}{e}^{-i{\delta }_{13}}),$
$U_{s4}$	$=s_{14}{e}^{-i{\delta }_{14}},$
$U_{e1}$	$=(-c_{24}{c}_{23}{s}_{12}{e}^{i{\delta }_{12}}-c_{24}{s}_{23}{e}^{-i{\delta }_{23}}{s}_{13}{e}^{i{\delta }_{13}}{c}_{12}$
	$\mathrm{\hspace{1em}}-s_{24}{e}^{-i{\delta }_{24}}{s}_{14}{e}^{i{\delta }_{14}}{c}_{13}{c}_{12}),$
$U_{e2}$	$=(c_{24}{c}_{23}{c}_{12}-c_{24}{s}_{23}{e}^{-i{\delta }_{23}}{s}_{13}{e}^{i{\delta }_{13}}{s}_{12}{e}^{-i{\delta }_{12}}$
	$\mathrm{\hspace{1em}}-s_{24}{e}^{-i{\delta }_{24}}{s}_{14}{e}^{i{\delta }_{14}}{c}_{13}{s}_{12}{e}^{-i{\delta }_{12}}),$
$U_{e3}$	$=(c_{13}{c}_{24}{s}_{23}{e}^{-i{\delta }_{23}}-s_{24}{e}^{-i{\delta }_{24}}{s}_{14}{e}^{i{\delta }_{14}}{s}_{13}{e}^{-i{\delta }_{13}}),$
$U_{e4}$	$=(c_{14}{s}_{24}{e}^{-i{\delta }_{24}}),$
$U_{\mu 1}$	$=(c_{34}{s}_{23}{e}^{i{\delta }_{23}}{s}_{12}{e}^{i{\delta }_{12}}-c_{34}{c}_{23}{s}_{13}{e}^{i{\delta }_{13}}{c}_{12}$
	$\mathrm{\hspace{1em}}+s_{34}{e}^{-i{\delta }_{34}}{s}_{24}{e}^{i{\delta }_{24}}{c}_{23}{s}_{12}{e}^{i{\delta }_{12}}$
	$\mathrm{\hspace{1em}}+s_{34}{e}^{-i{\delta }_{34}}{s}_{24}{e}^{i{\delta }_{24}}{s}_{23}{e}^{-i{\delta }_{23}}{s}_{13}{e}^{i{\delta }_{13}}{c}_{12}$
	$\mathrm{\hspace{1em}}-s_{34}{e}^{-i{\delta }_{34}}{c}_{24}{s}_{14}{e}^{i{\delta }_{14}}{c}_{13}{c}_{12}),$
$U_{\mu 2}$	$=(-c_{34}{s}_{23}{e}^{i{\delta }_{23}}{c}_{12}-c_{34}{c}_{23}{s}_{13}{e}^{-i{\delta }_{13}}{s}_{12}{e}^{i{\delta }_{12}}$
	$\mathrm{\hspace{1em}}-s_{34}{e}^{-i{\delta }_{34}}{s}_{24}{e}^{i{\delta }_{24}}{c}_{23}{c}_{12}$
	$\mathrm{\hspace{1em}}+s_{34}{e}^{-i{\delta }_{34}}{s}_{24}{e}^{i{\delta }_{24}}{s}_{23}{e}^{-i{\delta }_{23}}{s}_{13}{e}^{i{\delta }_{13}}{s}_{12}{e}^{-i{\delta }_{12}}$
	$\mathrm{\hspace{1em}}-s_{34}{e}^{-i{\delta }_{34}}{c}_{24}{s}_{14}{e}^{i{\delta }_{14}}{c}_{13}{s}_{12}{e}^{-i{\delta }_{12}}),$
$U_{\mu 3}$	$=(c_{34}{c}_{23}{c}_{13}-s_{34}{e}^{-i{\delta }_{34}}{s}_{24}{e}^{i{\delta }_{24}}{s}_{23}{e}^{-i{\delta }_{23}}{c}_{13}$
	$\mathrm{\hspace{1em}}-s_{34}{e}^{-i{\delta }_{34}}{c}_{24}{s}_{14}{e}^{i{\delta }_{14}}{s}_{13}{e}^{-i{\delta }_{13}})$
$U_{\mu 4}$	$=(s_{34}{e}^{-i{\delta }_{34}}{c}_{24}{c}_{14}),$
$U_{\tau 1}$	$=(s_{34}{s}_{12}{s}_{23}+s_{34}{e}^{i{\delta }_{34}}{c}_{23}{s}_{13}{e}^{i{\delta }_{14}}{c}_{12}+c_{34}{s}_{24}{e}^{i{\delta }_{24}}{c}_{23}{s}_{12}{e}^{i{\delta }_{12}}$
	$\mathrm{\hspace{1em}}+c_{34}{s}_{24}{e}^{i{\delta }_{34}}{s}_{23}{e}^{-i{\delta }_{23}}{s}_{12}{e}^{i{\delta }_{12}}{c}_{12}$
	$\mathrm{\hspace{1em}}-c_{34}{c}_{24}{s}_{14}{e}^{i{\delta }_{14}}{c}_{13}{c}_{12}),$
$U_{\tau 2}$	$=(s_{34}{e}^{i{\delta }_{34}}{c}_{12}{s}_{23}{e}^{i{\delta }_{23}}+s_{34}{e}^{i{\delta }_{34}}{c}_{23}{s}_{13}{e}^{i{\delta }_{13}}{s}_{12}{e}^{-i{\delta }_{12}}$
	$\mathrm{\hspace{1em}}-c_{34}{s}_{24}{e}^{i{\delta }_{24}}{c}_{23}{c}_{12}$
	$\mathrm{\hspace{1em}}+c_{34}{s}_{24}{e}^{i{\delta }_{34}}{s}_{23}{e}^{-i{\delta }_{23}}{s}_{13}{e}^{i{\delta }_{13}}{s}_{12}{e}^{-i{\delta }_{12}}$
	$\mathrm{\hspace{1em}}-c_{34}{c}_{24}{s}_{14}{e}^{i{\delta }_{14}}{c}_{13}{s}_{12}{e}^{-i{\delta }_{12}}),$
$U_{\tau 3}$	$=(-s_{34}{e}^{i{\delta }_{34}}{c}_{23}{c}_{13}-c_{34}{s}_{24}{e}^{i{\delta }_{24}}{s}_{23}{e}^{-i{\delta }_{23}}{c}_{13}$
	$\mathrm{\hspace{1em}}-c_{34}{c}_{24}{s}_{14}{e}^{i{\delta }_{14}}{s}_{13}{e}^{-i{\delta }_{13}})$
$U_{\tau 4}$	$=(c_{34}{c}_{24}{c}_{14}),$

From Equations (9) and (10), we get nine independent $J_{\alpha \beta }^{ij}$, the magnitudes of $J_{\alpha \beta }^{ij}$ depends on mixing angles and six CP-violating phases. The explicit form of nine jarlskog $J_{\alpha \beta }^{ij}$ are

$J_{se}^{13}$	$=c_{12}{c}_{13}{c}_{14}^2{c}_{23}{c}_{24}{s}_{12}{s}_{13}^2{s}_{14}{s}_{24}sin{\theta }_y$
	$\mathrm{\hspace{1em}}-c_{12}{c}_{13}^2{c}_{14}^2{c}_{23}{c}_{24}^2{s}_{12}{s}_{13}{s}_{14}{s}_{23}sin{\theta }_z$
	$\mathrm{\hspace{1em}}+c_{12}^2{c}_{13}{c}_{14}^2{c}_{24}{s}_{24}{s}_{13}{s}_{14}{s}_{23}sin\left({\theta }_y-{\theta }_z\right)$	(12)
$J_{se}^{24}$	$=c_{12}{c}_{13}{c}_{14}^2{c}_{23}{c}_{24}{s}_{12}{s}_{14}{s}_{24}sin{\theta }_y$
	$\mathrm{\hspace{1em}}-c_{13}{c}_{14}^2{c}_{24}{s}_{12}^2{s}_{13}{s}_{14}{s}_{23}{s}_{24}sin\left({\theta }_y-{\theta }_z\right)$	(13)
	$\mathrm{\hspace{1em}}{J}_{se}^{34}=c_{13}{c}_{14}^2{c}_{24}{s}_{13}{s}_{14}{s}_{23}{s}_{24}sin\left({\theta }_y-{\theta }_z\right)$	(14)
$J_{\tau s}^{13}$	$=c_{12}^2{c}_{13}{c}_{14}^2{c}_{23}{c}_{24}{c}_{34}{s}_{13}{s}_{14}{s}_{34}sin{\theta }_x$
	$\mathrm{\hspace{1em}}+c_{12}{c}_{13}{c}_{14}^2{c}_{23}{c}_{24}{c}_{34}^2{s}_{12}{s}_{13}^2{s}_{14}{s}_{24}sin{\theta }_y$
	$\mathrm{\hspace{1em}}+\left(c_{34}^2{s}_{24}^2-s_{34}^2\right)c_{12}{c}_{13}^2{c}_{14}^2{c}_{23}{s}_{12}{s}_{13}{s}_{23}sin{\theta }_z$
	$\mathrm{\hspace{1em}}-c_{12}{c}_{13}^2{c}_{14}^2{c}_{23}^2{c}_{34}{s}_{12}{s}_{13}{s}_{24}{s}_{34}sin\left({\theta }_x-{\theta }_y\right)$
	$\mathrm{\hspace{1em}}-c_{12}{c}_{13}{c}_{14}^2{c}_{24}{c}_{34}{s}_{12}{s}_{13}^2{s}_{14}{s}_{23}{s}_{34}sin\left({\theta }_x+{\theta }_z\right)$
	$\mathrm{\hspace{1em}}+c_{12}^2{c}_{13}{c}_{14}^2{c}_{34}^2{c}_{24}{s}_{13}{s}_{14}{s}_{23}{s}_{24}sin\left({\theta }_y-{\theta }_z\right)$
	$\mathrm{\hspace{1em}}-c_{12}{c}_{13}^2{c}_{14}^2{c}_{34}{s}_{12}{s}_{13}{s}_{23}^2{s}_{24}{s}_{34}sin\left({\theta }_x-{\theta }_y+{\theta }_z\right)$	(15)
$J_{\tau s}^{14}$	$=-c_{12}^2{c}_{13}{c}_{14}^2{c}_{23}{c}_{24}{c}_{34}{s}_{13}{s}_{14}{s}_{34}sin{\theta }_x$
	$\mathrm{\hspace{1em}}-c_{12}{c}_{13}{c}_{14}^2{c}_{23}{c}_{24}{c}_{34}^2{s}_{12}{s}_{13}{s}_{24}sin{\theta }_y$
	$\mathrm{\hspace{1em}}+c_{12}{c}_{13}{c}_{14}^2{c}_{24}{c}_{34}{s}_{12}{s}_{14}{s}_{23}{s}_{34}sin\left({\theta }_x+{\theta }_z\right)$
	$\mathrm{\hspace{1em}}-c_{12}^2{c}_{13}{c}_{14}^2{c}_{24}{c}_{34}^2{s}_{12}{s}_{13}{s}_{23}{s}_{24}sin\left({\theta }_y-{\theta }_z\right)$	(16)
$J_{\tau s}^{34}$	$=c_{13}{c}_{14}^2{c}_{23}{c}_{24}{c}_{34}{s}_{13}{s}_{14}{s}_{34}sin{\theta }_x$
	$\mathrm{\hspace{1em}}+c_{13}{c}_{14}^2{c}_{24}{c}_{34}^2{s}_{13}{s}_{14}{s}_{23}{s}_{24}sin\left({\theta }_y-{\theta }_z\right)$	(17)

and

$J_{e\mu }^{23}$	$=-\left(c_{12}^2{c}_{23}^2{s}_{24}^2-c_{12}^2{c}_{24}^2{s}_{23}^2-s_{12}^2{s}_{14}^2{s}_{24}^2+s_{12}^2{s}_{23}^2\right)$
	$\mathrm{\hspace{1em}}\times c_{13}{c}_{23}{c}_{24}{c}_{34}{s}_{13}{s}_{14}{s}_{34}sin{\theta }_x$
	$\mathrm{\hspace{1em}}+\left(c_{13}^2{c}_{34}^2{s}_{23}^2-c_{23}^2{c}_{34}^2{s}_{13}^2+s_{13}^2{s}_{14}^2{s}_{34}^2-s_{34}^2{s}_{23}^2\right)$
	$\mathrm{\hspace{1em}}\times c_{12}{c}_{13}{c}_{23}{c}_{24}{s}_{12}{s}_{14}{s}_{24}sin{\theta }_y$
	$\mathrm{\hspace{1em}}+\left(c_{24}^2{c}_{34}^2-s_{14}^2{c}_{24}^2{s}_{34}^2-c_{34}^2{s}_{14}^2{s}_{24}^2+s_{14}^2{s}_{24}^2{s}_{34}^2\right)$
	$\mathrm{\hspace{1em}}\times c_{12}{c}_{13}^2{c}_{23}{c}_{34}{s}_{12}{s}_{13}{s}_{23}sin{\theta }_z$
	$\mathrm{\hspace{1em}}+\left(c_{13}^2{s}_{24}^2-c_{24}^2{s}_{13}^2\right)$
	$\mathrm{\hspace{1em}}\times c_{12}{c}_{23}^2{c}_{34}{s}_{12}{s}_{13}{s}_{14}^2{s}_{24}{s}_{34}sin\left({\theta }_x-{\theta }_y\right)$
	$\mathrm{\hspace{1em}}+\left(c_{23}^2-c_{13}^2{s}_{24}^2-c_{24}^2{s}_{13}^2+s_{13}^2{s}_{14}^2{s}_{24}^2\right)$
	$\mathrm{\hspace{1em}}\times c_{12}{c}_{13}{c}_{24}{c}_{34}{s}_{12}{s}_{14}{s}_{23}{s}_{34}sin\left({\theta }_x+{\theta }_z\right)$
	$\mathrm{\hspace{1em}}-\left(c_{12}^2{c}_{23}^2{c}_{34}^2-c_{12}^2{c}_{23}^2{s}_{34}^2-c_{23}^2{c}_{34}^2{s}_{12}^2+s_{12}^2{s}_{14}^2{s}_{34}^2-s_{12}^2{s}_{23}^2{s}_{34}^2\right)$
	$\mathrm{\hspace{1em}}\times c_{13}{c}_{24}{s}_{13}{s}_{14}{s}_{23}{s}_{24}sin\left({\theta }_y-{\theta }_z\right)$
	$\mathrm{\hspace{1em}}+\left(c_{12}^2{c}_{13}^2{c}_{24}^2-c_{12}^2{c}_{24}^2{s}_{13}^2{s}_{14}^2-c_{13}^2{c}_{24}^2{s}_{12}^2{s}_{14}^2+c_{24}^2{s}_{12}^2{s}_{13}^2{s}_{14}^2\right)$
	$\mathrm{\hspace{1em}}\times c_{23}{c}_{34}{s}_{23}{s}_{24}{s}_{34}sin\left({\theta }_x-{\theta }_y+{\theta }_z\right)$
	$\mathrm{\hspace{1em}}-\left(c_{13}^2{c}_{24}^2{s}_{23}^2-c_{13}^2{s}_{23}^2{s}_{24}^2{s}_{14}^2-c_{24}^2{s}_{23}^2{s}_{13}^2{s}_{14}^2\right)$
	$\mathrm{\hspace{1em}}\times c_{12}{c}_{34}{s}_{12}{s}_{13}{s}_{24}{s}_{34}sin\left({\theta }_x-{\theta }_y+2{\theta }_z\right)$
	$\mathrm{\hspace{1em}}-c_{12}{c}_{13}^2{c}_{24}^2{c}_{34}{s}_{12}{s}_{13}{s}_{14}^2{s}_{24}{s}_{34}sin\left({\theta }_x+{\theta }_y\right)$
	$\mathrm{\hspace{1em}}+c_{12}{c}_{13}{c}_{23}^2{c}_{24}{c}_{34}{s}_{12}{s}_{13}^2{s}_{14}{s}_{23}{s}_{34}sin\left({\theta }_x-{\theta }_z\right)$
	$\mathrm{\hspace{1em}}+\left(c_{34}^2-s_{34}^2\right)c_{12}{c}_{13}{c}_{23}{c}_{24}{s}_{12}{s}_{13}^2{s}_{14}{s}_{23}^2{s}_{24}sin\left({\theta }_y-2{\theta }_z\right)$
	$\mathrm{\hspace{1em}}-\left(c_{12}^2-s_{12}^2\right)c_{13}{c}_{23}{c}_{24}{c}_{34}{s}_{13}{s}_{24}^2{s}_{14}{s}_{23}^2{s}_{34}sin\left({\theta }_x-2{\theta }_y+2{\theta }_z\right)$
	$\mathrm{\hspace{1em}}-c_{12}{c}_{13}{c}_{23}^2{c}_{24}{c}_{34}{s}_{12}{s}_{24}^2{s}_{14}{s}_{23}{s}_{34}sin\left({\theta }_x-2{\theta }_y+{\theta }_z\right)$
	$\mathrm{\hspace{1em}}+c_{12}{c}_{13}{c}_{24}{c}_{34}{s}_{12}{s}_{13}^2{s}_{14}{s}_{23}^2{s}_{24}^2{s}_{34}sin\left({\theta }_x-2{\theta }_y+3{\theta }_z\right)$	(18)
$J_{e\mu }^{24}$	$=c_{13}{c}_{14}^2{c}_{23}{c}_{24}{c}_{34}{s}_{12}^2{s}_{13}{s}_{14}{s}_{24}^2{s}_{34}sin{\theta }_x$
	$\mathrm{\hspace{1em}}+c_{12}{c}_{13}{c}_{14}^2{c}_{23}{c}_{24}{s}_{34}^2{s}_{12}{s}_{14}{s}_{24}sin{\theta }_y$
	$\mathrm{\hspace{1em}}-c_{12}{c}_{14}^2{c}_{23}^2{c}_{24}^2{c}_{34}{s}_{12}{s}_{13}{s}_{24}{s}_{34}sin\left({\theta }_x-{\theta }_y\right)$
	$\mathrm{\hspace{1em}}+c_{12}{c}_{13}{c}_{14}^2{c}_{24}{c}_{34}{s}_{12}{s}_{14}{s}_{23}{s}_{24}^2{s}_{34}sin\left({\theta }_x+{\theta }_y\right)$
	$\mathrm{\hspace{1em}}-c_{13}{c}_{14}^2{c}_{24}{s}_{12}^2{s}_{13}{s}_{14}{s}_{23}{s}_{24}{s}_{34}^{2}sin\left({\theta }_y-{\theta }_z\right)$
	$\mathrm{\hspace{1em}}+c_{12}{c}_{13}^2{c}_{14}^2{c}_{34}{s}_{12}{s}_{13}{s}_{23}^2{s}_{24}{s}_{34}sin\left({\theta }_x-{\theta }_y+2{\theta }_z\right)$
	$\mathrm{\hspace{1em}}-\left(c_{12}^2-s_{12}^2{s}_{13}^2\right)c_{14}^2{c}_{23}{c}_{24}^2{c}_{34}{s}_{23}{s}_{24}{s}_{34}sin\left({\theta }_x-{\theta }_y+{\theta }_z\right)$	(19)
$J_{e\mu }^{34}$	$=-c_{13}{c}_{14}^2{c}_{23}{c}_{24}{c}_{34}{s}_{13}{s}_{14}{s}_{24}^2{s}_{34}sin{\theta }_x$
	$\mathrm{\hspace{1em}}+c_{13}{c}_{14}^2{c}_{24}{s}_{13}{s}_{14}{s}_{23}{s}_{24}{s}_{34}^{2}sin\left({\theta }_y-{\theta }_z\right)$
	$\mathrm{\hspace{1em}}+c_{13}^2{c}_{14}^2{c}_{23}{c}_{24}^2{c}_{34}{s}_{23}{s}_{24}{s}_{34}sin\left({\theta }_x-{\theta }_y+{\theta }_z\right)$	(20)

where

$$\begin{array}{c}\hfill {\theta }_x={\delta }_{14}-{\delta }_{13}-{\delta }_{34}\hfill \\ \hfill {\theta }_y={\delta }_{14}-{\delta }_{12}-{\delta }_{24}\hfill \\ \hfill {\theta }_z={\delta }_{13}-{\delta }_{12}-{\delta }_{23}\hfill \end{array}$$

(21)

We assume the values of 6 CP-violating phases ${\delta }_{ij}$ are between 0 and 2$\pi$. Equations (12–20) are important to study the role of violation in T and CP within the framework of four-neutrino mixing. B. S. Koranga et al. [16, 18, 19, 22, 24] did some analysis on T and CP violation within three flavor framework for different parameterization and above GUT scale.

4 Results and Discussion

In numerical calculation, The active sterile neutrino mixing angle are ${\theta }_{14},{\theta }_{24}$ and ${\theta }_{34}$. In this calculation, we consider following value for sterile neutrino mixing angles [17], ${\theta }_{14}={3.6}^{\circ }$, ${\theta }_{24}=4^{\circ }$, ${\theta }_{34}={18.5}^{\circ }$. We consider mixing angles ${\theta }_{13}={10}^{\circ }$, ${\theta }_{23}={45}^{\circ }$, ${\theta }_{12}={34}^{\circ }$, ${\theta }_{34}={18.5}^{\circ }$, ${\theta }_{24}=4^{\circ }$, ${\theta }_{14}={3.6}^{\circ }$ [17]. The calculated values of Jarlskog determinant for different sample of ${\theta }_x,{\theta }_y$ and ${\theta }_z$ are presented in Tables 1 and 2. These three ${\theta }_x,{\theta }_y$ and ${\theta }_z$ angles are directly related with six Dirac CP phases which is associated with their mixing angles. On varying these three angles we obtained different values of nine Jarlskog determinant from which we took the maximum and the minimum value of Jarlskog determinant. The upper bound value of $J_{e\mu }^{23}=0.04510635487051475$ for ${\theta }_x={31}^{\circ },{\theta }_y=28{}^{\circ }$ and ${\theta }_z={90}^{\circ }$ and lower bound of $J_{se}^{13}=-0.039055825709320924$ for ${\theta }_x=0{}^{\circ },{\theta }_y=0{}^{\circ }$ and ${\theta }_z={90}^{\circ }$ are presented in Tables 1 and 2 respectively.

Table 1 Nine maximum Jarlskog Determinant values for various value of mixing angle. Current value of mixing angles ${\theta }_{23}={45}^{\circ }$, ${\theta }_{10}={10}^{\circ }$, ${\theta }_{12}={34}^{\circ }$, ${\theta }_{34}={18.5}^{\circ }$, ${\theta }_{14}={3.6}^{\circ }$, ${\theta }_{24}=4^{\circ }$

$J_{\alpha \beta }^{ij}$	Maximum	${\theta }_x$	${\theta }_y$	${\theta }_z$
$J_{\tau s}^{13}$	0.036685890619149106	${98}^{\circ }$	${170}^{\circ }$	${91}^{\circ }$
$J_{\tau s}^{14}$	0.014752208075303027	$0^{\circ }$	${90}^{\circ }$	${90}^{\circ }$
$J_{\tau s}^{34}$	0.002743458805405071	${90}^{\circ }$	${90}^{\circ }$	$0^{\circ }$
$J_{se}^{13}$	0.00040407451857360076	$0^{\circ }$	${90}^{\circ }$	$0^{\circ }$
$J_{se}^{24}$	0.0015695627093747612	$0^{\circ }$	${90}^{\circ }$	${180}^{\circ }$
$J_{se}^{34}$	0.0005262721896176136	$0^{\circ }$	${90}^{\circ }$	$0^{\circ }$
$J_{e\mu }^{23}$	0.04510635487051475	${31}^{\circ }$	${28}^{\circ }$	${90}^{\circ }$
$J_{e\mu }^{24}$	0.007314874300972153	$0^{\circ }$	${150}^{\circ }$	${74}^{\circ }$
$J_{e\mu }^{34}$	0.010142178442526652	${180}^{\circ }$	${90}^{\circ }$	$0^{\circ }$

Table 2 Nine minimum Jarlskog Determinant values for various value of mixing angle. Current value of mixing angles ${\theta }_{23}={45}^{\circ }$, ${\theta }_{10}={10}^{\circ }$, ${\theta }_{12}={34}^{\circ }$, ${\theta }_{34}={18.5}^{\circ }$, ${\theta }_{14}={3.6}^{\circ }$, ${\theta }_{24}=4^{\circ }$

$J_{\alpha \beta }^{ij}$	Minimun	${\theta }_x$	${\theta }_y$	${\theta }_z$
$J_{\tau s}^{13}$	$-$0.0019163137495424884	${180}^{\circ }$	${90}^{\circ }$	${180}^{\circ }$
$J_{\tau s}^{14}$	$-$0.018052733828834214	${128}^{\circ }$	${139}^{\circ }$	${38}^{\circ }$
$J_{\tau s}^{34}$	$-$0.00047328592409829115	$0^{\circ }$	$0^{\circ }$	${90}^{\circ }$
$J_{se}^{13}$	$-$0.039055825709320924	$0^{\circ }$	$0^{\circ }$	${90}^{\circ }$
$J_{se}^{24}$	$-$0.00016456357872771316	$0^{\circ }$	${180}^{\circ }$	${90}^{\circ }$
$J_{se}^{34}$	$-$0.0005262721896176136	$0^{\circ }$	$0^{\circ }$	${90}^{\circ }$
$J_{e\mu }^{23}$	$-$0.009549376384925656	${90}^{\circ }$	$0^{\circ }$	${180}^{\circ }$
$J_{e\mu }^{24}$	$-$0.007283419390778245	${180}^{\circ }$	${180}^{\circ }$	${102}^{\circ }$
$J_{e\mu }^{34}$	$-$0.010143317279290433	${168}^{\circ }$	$0^{\circ }$	${102}^{\circ }$

5 Conclusions

Jarlskog invariance plays a crucial role in developing a better understanding of flavors of neutrino. It has potency to tune the magnitude of CP and T violation in both quark and lepton sectors. In present article, we have calculated the upper and lower bound of Jarlskog Determinant in a four flavor framework. The upper bound and lower bound values of Jarlskog determinant is found out to be $J_{e\mu }^{23}=0.04510635487051475$ for ${\theta }_x={31}^{\circ },{\theta }_y=28{}^{\circ }$ and ${\theta }_z={90}^{\circ }$ and $J_{se}^{13}=-0.039055825709320924$ for ${\theta }_x=0{}^{\circ },{\theta }_y=0{}^{\circ }$ and ${\theta }_z={90}^{\circ }$ respectively. Since, these Jarlskog determinant appear in the imaginary part of the expression of oscillation probability amplitude [15, 25], hence it does not contribute to oscillation probability. Whereas it plays a crucial role when we deal with the leptonic sector CP and T violation. The contribution of Jarlskog determinant $J_{e\mu }^{23}$ and $J_{e\mu }^{13}$ in CP violation, found to be more significant among the others nine Jarlskog determinant.

Acknowledgement

One of the authors BSK thanks Ratindra Gautam for the preperation of manuscript.

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Biographies

Vivek Kumar Nautiyal is a graduate from Lucknow University, Lucknow, India. He received the M.Tech. degree in applied optics from Indian Institute of Technology Delhi, India in 2014. He has been in constant touch with the nuclear and particle physics for 8 years and earned the Ph.D. degree in physics. He has more than three years of teaching experience in graduation and postgraduation levels. His research interests include the nuclear and particle physics, neutrino physics.

Bipin Singh Koranga is a graduate from Kumaun University, Nainital, India. He has been with the Theoretical Physics Group, IIT Bombay since 2006 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 12 years. His research interests include the origin of universe, Physics beyond the standard model, theoretical nuclear physics, quantum mechanical neutrino oscillation.

Sanjay Kumar Padaliya is presently Head, Department of Mathematics, S.G.R.R. (P.G) College, Dehradun, India. He received the Ph.D. degree in mathematics (fixed point theory) from Kumaun University, Nainital, India. He has been teaching basic courses in mathematics at graduate and postgraduate levels for the last 20 years. His present research interest includes the fixed point theory and fuzzy analysis. He is also a life member of Indian Mathematical Society, Ramanujan Mathematical Society, and International Academy of Physical Sciences.

Neelam Das is a graduate and postgraduate from Rani Durgavati Vishwavidyalaya, Jabalpur M.P, India. He has been in constant touch with the nuclear physics since 2012 and pursuing her doctoral degree from Lucknow University, Lucknow. He has more than 10 years of teaching experience in graduation level. His research interests is in nuclear and neutrino physics.

Ashish Shrivastava is a graduate from Kirori Mal College, Delhi University, Delhi, India. Now he is pursuing his Master of Science from National Institute of Technology, Agartala, India. His research interests is in neutrino physics.

Journal of Graphic Era University, Vol. 10_2, 83–94.
doi: 10.13052/jgeu0975-1416.1022
© 2022 River Publishers

Abstract

1 Introduction

2 Four Neutrino Mixing Angles and Their Mass Squared Differences

3 Jarlskog in Four Flavor Neutrino

4 Results and Discussion

5 Conclusions

Acknowledgement

References

Biographies