Neutrino Oscillation and CPT Violation: A Four Flavour Matter Effects

Bipin Singh Koranga^1,* and Seema Rawat²

¹Department of Physics, Kirori Mal College (University of Delhi), Delhi-110007, India
²Department of Physics, Zakir Husain College (University of Delhi), Delhi-110007, India
E-mail: bipinkmcit@gmail.com
*Corresponding Author

Received 20 July 2025; Accepted 23 August 2025

Abstract

In a matter for the four flavour neutrino oscillation (3+1) scheme, the CPT violation is taken into consideration. Assuming that the mismatch between neutrino and anti-neutrino mass square discrepancies causes the CPT to be violated in matter. We suppose that the presence of matter causes the four flavour neutrino and anti-neutrino mass differences to square. For four flavour neutrino oscillation, we take into account the pertinent matter density profile. In this study, we calculate the upper bound constraint of the CPT violation in matter for a four flavour scheme. $|{\mathrm{\Delta }}_{21}^m-\overline{{\mathrm{\Delta }}_{21}^m}|=(1.21-3.44)\times {10}^{-3})e{V}^2,|{\mathrm{\Delta }}_{31}^m-\overline{{\mathrm{\Delta }}_{31}^m}|=(1.68-6.41)\times {10}^{-4})eV{}^2$ and $|{\mathrm{\Delta }}_{41}^m-\overline{{\mathrm{\Delta }}_{41}^m}|=(1.06-4.60)\times {10}^{-5})eV{}^2$.

In this paper, we examine the CPT violation in a framework with four flavours in matter.

Keywords: CPT violation, neutrino oscillation.

1 Introduction

Solar [1], atmospheric [2], reactor [3], and long-baseline neutrino experiments [4, 5] have all established the existence of neutrino oscillation. Results from LSND and MiniBoone indicate the existence of sterile neutrinos [7, 8]. The solar neutrino mass square difference is measured by the LSND experiment to be $0.2<{\mathrm{\Delta }}_{21}<100e{V}^2$ [6] and the MiniBooNe experiment to be $0.01<{\mathrm{\Delta }}_{21}<1.0e{V}^2$. These two investigations provide up new avenues for sterile neutrinos. By taking into account how neutrinos interact with matter, one can alter the likelihood of neutrinos oscillating in a vacuum [8]. One of the fundamental symmetries in particle physics is the CPT. The equality of the survival probabilities of neutrinos and anti-neutrinos in vacuum is implied by neutrino oscillation according to CPT conservation [9, 10]. Neutrino oscillation as a method of looking for CPT violations was initially put forth in Ref. [9]. Constraints on the solar and KamLAND data.

There is a signal for a CPT violation in the neutrino sector when $|{\mathrm{\Delta }}_{21}-\overline{{\mathrm{\Delta }}_{21}}|\le 1.1\times {10}^{-4}e{V}^2$ [11], which is a non-zero difference between ${\mathrm{\Delta }}_{21}$ and $\overline{{\mathrm{\Delta }}_{21}}|$. Prior to now, the majority of research on CPT violation [12–19]. For a four-flavor matter paradigm, we provide analytical expressions for three distinct neutrino mass square differences. In this essay, we would look at a potential CPT violation in a four-flavor framework. The following is a summary of the article. We talk about four flavour neutrino oscillation in vacuum with sterile neutrino in Section 2 of this article. We briefly touch on the CPT Violation four flavour neutrino oscillation in Section 3. Section 4 contains numerical results and Section 5 for the conclusion.

2 Four Flavor Neutrino Oscillation in Vacuum

In this part, we assume the sterile neutrino mix contains three distinct neutrinos and consider with four flavour neutrino oscillation. The PMNS matrix $U_{4\times 4}$ is provided by [20], and by adding one sterile neutrino [21],

$$U=R_{34}({\theta }_{34},{\delta }_{34})R_{24}({\theta }_{34})R_{14}({\theta }_{14},{\delta }_{14})R_{23}({\theta }_{23})R_{13}({\theta }_{13},{\delta }_{13})R_{12}({\theta }_{12}),$$

(1)

The rotations in ij space represented by the matrices $R_{ij}$

$$R_{ij}({\theta }_{ij},\delta )=\left(\begin{array}{cc}\hfill c_{ij}\hfill & \hfill s_{ij}{e}^{-i\delta }\hfill \\ \hfill -s_{ij}{e}^{i\delta }\hfill & \hfill c_{ij}\hfill \end{array}\right),$$

where $s_{ij}=sin{\theta }_{ij}$,$c_{ij}=cos{\theta }_{ij}$.

In four flavor there are three Dirac CP-violating phase ${\delta }_{ij}.$ The explicit form of U is

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

(2)

where

$U_{e1}$	$=(c_{14}{c}_{13}{c}_{12}),$
$U_{e2}$	$=(c_{14}{c}_{13}{s}_{12}),$
$U_{e3}$	$=(c_{14}{s}_{13})e^{-i{\delta }_{13}},$
$U_{e4}$	$=s_{14}{e}^{i{\delta }_{14}}{e}^{i{f}_1},$
$U_{\mu 1}$	$=(-s_{24}{c}_{14}{c}_{13}{e}^{i{\delta }_{14}}-c_{24}{s}_{23}{s}_{13}{c}_{12}{e}^{i{\delta }_{13}})c_{12}-c_{24}{c}_{23}{s}_{12}{e}^{-i\delta },$
$U_{\mu 2}$	$=(c_{24}{c}_{23}{c}_{12}-s_{12}(s_{24}{s}_{14}{c}_{13}{e}^{i{\delta }_{14}}+c_{24}{s}_{23}{s}_{13}{e}^{-i{\delta }_{13}}),$
$U_{\mu 3}$	$=c_{24}{s}_{23}{c}_{13}-s_{24}{s}_{14}{c}_{13}{e}^{-i\delta ({\delta }_{14}-{\delta }_{13})},$
$U_{\mu 4}$	$=(s_{14})e^{-i{\delta }_{14}},$
$U_{\tau 1}$	$=c_{12}[-s_{34}{c}_{24}{s}_{14}{c}_{13}{e}^{-i({\delta }_{34}-{\delta }_{14})}-s_{13}{e}^{i{\delta }_{13}}(c_{34}{c}_{23}-s_{34}{s}_{24}{s}_{23}{e}^{-i{\delta }_{34}})]$
	$\mathrm{\hspace{1em}}+s_{12}(s_{34}{s}_{24}{c}_{23}{e}^{-i{\delta }_{34}}+s_{12}(s_{34}{s}_{24}{c}_{23}{e}^{-i{\delta }_{34}}+c_{34}{s}_{23}),$
$U_{\tau 2}$	$=-c_{12}(s_{34}{s}_{24}{c}_{23}{e}^{-i{\delta }_{34}}+c_{34}{s}_{23})-s_{12}[s_{34}{c}_{24}{s}_{14}{s}_{13}{e}^{-i({\delta }_{34}-{\delta }_{14})}$
	$\mathrm{\hspace{1em}}+s_{13}{e}^{i{\delta }_{13}}(c_{34}{c}_{23}-s_{34}{s}_{24}{s}_{23}{e}^{-i{\delta }_{34}})],$
$U_{\tau 3}$	$=-c_{13}(c_{34}{c}_{23}-s_{34}{s}_{24}{s}_{23}{e}^{-i{\delta }_{34}})-s_{34}{c}_{24}{s}_{14}{s}_{13}{e}^{-i({\delta }_{34}-{\delta }_{13})},$
$U_{\tau 4}$	$=(s_{34}{c}_{24}{c}_{14})e^{i{\delta }_{34}},$
$U_{s1}$	$=c_{12}[-c_{34}{c}_{24}{s}_{14}{c}_{13}{e}^{i{\delta }_{14}}+s_{13}{e}^{i{\delta }_{13}}(s_{34}{c}_{23}{e}^{i{\delta }_{34}}+c_{34}{s}_{24}{s}_{23})]$
	$\mathrm{\hspace{1em}}-s_{12}(-c_{34}{s}_{24}{c}_{23}+s_{34}{s}_{23}{e}^{i{\delta }_{34}})$
$U_{s2}$	$=c_{12}(-c_{34}{s}_{24}{c}_{23}+s_{34}{s}_{23}{e}^{i{\delta }_{34}})$
$U_{s3}$	$=(-s_{34}{c}_{23}{c}_{13}-c_{34}{s}_{24}{s}_{23}{c}_{13}-c_{34}{c}_{24}{s}_{14}{s}_{13}{e}^{-i\delta })e^{i\gamma }{e}^{i{f}_4}$
$U_{s4}$	$=(c_{34}{c}_{24}{c}_{14})e^{i{f}_4},$

If we suppose that there is one sterile neutrino, then [21] gives the probability of four flavour neutrino oscillation in a vacuum

$P_{{\nu }_{\alpha }\to {\nu }_{\beta }}$	$={\displaystyle \sum _i}|U_{\mu 1}{|}^2|U_{e1}{|}^2+2{\displaystyle \sum _{i<j}}[Re(U_{ei}{U}_{\mu j}{U}_{ej}^{*}{U}_{\mu i}^{*})cos{\overline{\mathrm{\Delta }}}_{ij}$
	$\mathrm{\hspace{1em}}-Im(U_{ei}{U}_{\mu j}{U}_{ej}^{*}{U}_{\mu i}^{*})sin{\overline{\mathrm{\Delta }}}_{ij}],$	(3)

where ${\overline{\mathrm{\Delta }}}_{ij}={\mathrm{\Delta }}_{ij}L/2E$ , baseline length of particular experiment is L.

3 Neutrino Mass Square Difference in Matter for Four Flavor Framework

The Hamiltonian $H_{vacuum}$ in the propagation of neutrinos in vacuum is given by

$$H_{vacuum}=\left(\begin{array}{cccc}\hfill E_1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill E_2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill E_3\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill E_4\hfill \end{array}\right),$$

(4)

where $E_k(k=1,2,3,4)$ are the energies of the neutrino mass eigenstates k with mass $m_k;$

$$E_k=\sqrt{m_k^2+p^2},$$

(5)

We assume that all mass eigenstates have the same momentum, p. when matter and neutrinos interact weakly (charged and neutral current). The sterile neutrino does not participate in any weak interactions. For mixing of four flavours of neutrinos, the effective Hamiltonian is [12].

$$H_{eff}=\frac{1}{2E}\left[U\left(\begin{array}{cccc}\hfill m_1^2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill m_2^2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill m_3^2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill m_4^2\hfill \end{array}\right)U^{†}+\left(\begin{array}{cccc}\hfill A\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill A^{{}^{\prime }}\hfill \end{array}\right)\right],$$

(6)

where U is four flavor mixing matrix and $A$ and $A^{\prime }$ is matter dependent term is given by

$\begin{array}{cccc}& & \hfill 2\sqrt{2}{G}_F{N}_e{E}_{\nu }\hfill & \hfill (neutrinos)\hfill \\ \hfill A(e{V}^2)\hfill & & & \\ & & \hfill -2\sqrt{2}{G}_F{N}_e{E}_{\nu }\hfill & \hfill (anti-neutrinos)\hfill \end{array}$

and

$\begin{array}{cccc}& & \hfill -\sqrt{2}{G}_F{N}_n{E}_{\nu }\hfill & \hfill (neutrinos)\hfill \\ \hfill A^{{}^{\prime }}(e{V}^2)\hfill & & & \\ & & \hfill \sqrt{2}{G}_F{N}_n{E}_{\nu }\hfill & \hfill (anti-neutrinos)\hfill \end{array}$

where $N_e$ and $N_n$ is the density of electron and neutron. From Eq. (4.0), we have

$\left[U\left(\begin{array}{cccc}\hfill m_1^2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill m_2^2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill m_{3^2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill m_4^2\hfill \end{array}\right)U^{†}\right]$
$\mathrm{\hspace{1em}\hspace{1em}}=U_m=\left[U\left(\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathrm{\Delta }}_{21}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\mathrm{\Delta }}_{31}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\mathrm{\Delta }}_{41}\hfill \end{array}\right)U^{†}\right]$
$\mathrm{\hspace{1em}\hspace{1em}}\mathit{\hspace{1em}}+\left[\left(\begin{array}{cccc}\hfill A\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill A^{{}^{\prime }}\hfill \end{array}\right)\right],$	(7)

where ${\mathrm{\Delta }}_{21}$,${\mathrm{\Delta }}_{31}$ and ${\mathrm{\Delta }}_{41}$ are the mass squared differences of solar, atmospheric, and sterile neutrinos.

The matrix $U_m$ can be diagonalized to determine the effective mass square difference in matter [20]. The roots of Equation (7.0) yield the matter-dependent mass squares $m_{m1}^2,m_{m2}^2,m_{m3}^2$ and $m_{m4}^2$

${\lambda }_1$	$=m_{m1}^2=-{\displaystyle \frac{b}{4}}-S-{\displaystyle \frac{1}{2}}\sqrt{-4S^2-2p+{\displaystyle \frac{q}{S}}},$
${\lambda }_2$	$=m_{m2}^2=-{\displaystyle \frac{b}{4}}-S+{\displaystyle \frac{1}{2}}\sqrt{-4S^2-2p+{\displaystyle \frac{q}{S}}},$
${\lambda }_3$	$=m_{m3}^2=-{\displaystyle \frac{b}{4}}+S-{\displaystyle \frac{1}{2}}\sqrt{-4S^2-2p-{\displaystyle \frac{q}{S}}},$
${\lambda }_4$	$=m_{m4}^2=-{\displaystyle \frac{b}{4}}+S+{\displaystyle \frac{1}{2}}\sqrt{-4S^2-2p-{\displaystyle \frac{q}{S}}}$

We may write the matter dependent mass square difference for the four flavour neutrino oscillation using the matter dependent mass squares $m_{m1}^2,m_{m2}^2,m_{m3}^2$ and $m_{m4}^2$.

${\mathrm{\Delta }}_{21}^m$	$=m_{m2}^2-m_{m1}^2=\sqrt{-4S^2-2p+{\displaystyle \frac{q}{S}}},$	(8)
${\mathrm{\Delta }}_{31}^m$	$=m_{m3}^2-m_{m1}^2=2S$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{2}}\left(\sqrt{-4S^2-2p+{\displaystyle \frac{q}{S}}}-\sqrt{-4S^2-2p-{\displaystyle \frac{q}{S}}}\right),$	(9)
${\mathrm{\Delta }}_{41}^m$	$=m_{m4}^2-m_{m1}^2=2S$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{2}}\left(\sqrt{-4S^2-2p+{\displaystyle \frac{q}{S}}}+\sqrt{-4S^2-2p-{\displaystyle \frac{q}{S}}}\right).$	(10)

We can write matter-dependent anti-neutrino mass square difference for four flavour neutrino oscillation when anti-neutrino potential term A is replaced by -A.

$\overline{{\mathrm{\Delta }}_{21}^m}$	$=m_{m2}^2-m_{m1}^2=\sqrt{-4S^{{}^{\prime }2}-2p^{{}^{\prime }}+{\displaystyle \frac{q^{{}^{\prime }}}{S^{{}^{\prime }}}}},$	(11)
$\overline{{\mathrm{\Delta }}_{31}^m}$	$=m_{m3}^2-m_{m1}^2=2S^{{}^{\prime }}$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{2}}\left(\sqrt{-4S^{{}^{\prime }2}-2p^{{}^{\prime }}+{\displaystyle \frac{q^{{}^{\prime }}}{S^{{}^{\prime }}}}}-\sqrt{-4S^{{}^{\prime }2}-2p^{{}^{\prime }}-{\displaystyle \frac{q^{{}^{\prime }}}{S^{{}^{\prime }}}}}\right),$	(12)
$\overline{{\mathrm{\Delta }}_{41}^m}$	$=m_{m4}^2-m_{m1}^2=2S^{{}^{\prime }}$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{2}}\left(\sqrt{-4S^{{}^{\prime }2}-2p^{{}^{\prime }}+{\displaystyle \frac{q^{{}^{\prime }}}{{}^{S^{{}^{\prime }}}}}}+\sqrt{-4S^{{}^{\prime }2}-2p^{{}^{\prime }}-{\displaystyle \frac{q^{{}^{\prime }}}{S^{{}^{\prime }}}}}\right).$	(13)

The density and neutrino oscillation parameter affect $p,p^{{}^{\prime }},q,q^{{}^{\prime }}$ $S$ and $S^{{}^{\prime }}$ [18]. The mass squared difference for neutrinos is given by Equations (8) through (10); meanwhile, Equations (11) through (13) give the mass squared difference for anti-neutrinos of matter terms A and $A^{.}$. Then, the following provides the CPT violating survival probability difference in the four flavour framework:

$\mathrm{\Delta }{P}_{\alpha \beta }^{CPT}=P({\nu }_{\alpha }\to {\nu }_{\beta })-P(\overline{{\nu }_{\alpha }}\to \overline{{\nu }_{\beta }}),$

(14)

where $\alpha ,\beta =e,\mu ,\tau ,s$

If CPT conserves in a vacuum, one has $\mathrm{\Delta }{P}_{\alpha \beta }^{CPT}=0.$

In the event that CPT violated [25, 26]

$${\mathrm{\Delta }}^m{P}_{\alpha \beta }^{CPT}\ne 0,$$
$${\mathrm{\Delta }}_{21}^m(CPT)={\mathrm{\Delta }}_{21}^m-\overline{{\mathrm{\Delta }}_{21}^m}\ne 0$$	(15)
$${\mathrm{\Delta }}_{31}^m(CPT)={\mathrm{\Delta }}_{31}^m-\overline{{\mathrm{\Delta }}_{31}^m}\ne 0$$	(16)
$${\mathrm{\Delta }}_{41}^m(CPT)={\mathrm{\Delta }}_{41}^m-\overline{{\mathrm{\Delta }}_{41}^m}\ne 0$$	(17)

4 Numerical Results

The neutrino mass square difference, a CPT violation term for four flavours of neutrino oscillation, is

${\mathrm{\Delta }}_{21}^m(CPT)$	$=\sqrt{-4S^2-2p+{\displaystyle \frac{q}{S}}}-\sqrt{-4S^{{}^{\prime }2}-2p^{{}^{\prime }}+{\displaystyle \frac{q^{{}^{\prime }}}{S^{{}^{\prime }}}}},$	(18)
${\mathrm{\Delta }}_{31}^m(CPT)$	$=2(S-S^{{}^{\prime }})$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{2}}\left(\sqrt{-4S^2-2p+{\displaystyle \frac{q}{S}}}-\sqrt{-4S^{{}^{\prime }2}-2p^{{}^{\prime }}+{\displaystyle \frac{q^{{}^{\prime }}}{S^{{}^{\prime }}}}}\right)$
	$\mathrm{\hspace{1em}}-{\displaystyle \frac{1}{2}}\left(\sqrt{-4S^2-2p-{\displaystyle \frac{q}{S}}}-\sqrt{-4S^{{}^{\prime }2}-2p^{{}^{\prime }}-{\displaystyle \frac{q^{{}^{\prime }}}{S^{{}^{\prime }}}}}\right),$	(19)
${\mathrm{\Delta }}_{41}^m(CPT)$	$=2(S-S^{{}^{\prime }})$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{2}}\left(\sqrt{-4S^2-2p+{\displaystyle \frac{q}{S}}}-\sqrt{-4S^{{}^{\prime }2}-2p^{{}^{\prime }}+{\displaystyle \frac{q^{{}^{\prime }}}{S^{{}^{\prime }}}}}\right)$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{2}}\left(\sqrt{-4S^2-2p-{\displaystyle \frac{q}{S}}}-\sqrt{-4S^{{}^{\prime }2}-2p^{{}^{\prime }}-{\displaystyle \frac{q^{{}^{\prime }}}{S^{{}^{\prime }}}}}\right),$	(20)

In this computation, we’re assuming that the mass order is normal. We presummate matter densities for electron ${\rho }_e=3g/c{m}^3$ and ${\rho }_n=3g/c{m}^3$ for calculation of neutrino mass square differences in matter. ${\mathrm{\Delta }}_{21}^m,{\mathrm{\Delta }}_{31}^m$ and ${\mathrm{\Delta }}_{41}^m$ are modified mass square differences in matter that depend on the matter density, the six neutrino mixing angles, and the Dirac phases ${\delta }_{34},{\delta }_{13}$, and ${\delta }_{14}$. We select the following mixing angles: ${\theta }_{12}={34}^{\circ }$, ${\theta }_{23}={45}^{\circ }$, ${\theta }_{13}={10}^{\circ }$ abandon majorana stages, etc. We examine the following value for sterile neutrino mixing angles in this paper [22], where ${\theta }_{14}={3.6}^{\circ }$, ${\theta }_{24}=4^{\circ }$, ${\theta }_{34}={18.5}^{\circ }$ and ${\mathrm{\Delta }}_{31}=0.002e{V}^2$ [23], ${\mathrm{\Delta }}_{41}=1.7e{V}^2$ [22] and ${\mathrm{\Delta }}_{21}=0.00008e{V}^2$ [2] have been taken. Table (1.0) lists the maximum differences in neutrino and anti-neutrino mass square differences in matter for four flavour frames in the region of 1 to 3 GeV for various values of Dirac phases ${\delta }_{34}=0^{\circ }<{\delta }_{34}<{180}^{\circ },$ ${\delta }_{13}=0^{\circ }<{\delta }_{13}<{180}^{\circ }$, ${\delta }_{14}=0^{\circ }<{\delta }_{14}<{180}^{\circ }$.

Table 1 For the matter effect-related neutrino mass square difference and anti-neutrino mass square difference. Having taken ${\mathrm{\Delta }}_{31}=2.0\times {10}^{-3}e{V}^2$, ${\mathrm{\Delta }}_{21}=8.0\times {10}^{-5}e{V}^2$, ${\mathrm{\Delta }}_{41}=1.7e{V}^2$ and mixing angles ${\theta }_{13}={10}^{\circ },{\theta }_{23}={45}^{\circ },{\theta }_{12}={34}^{\circ }$ ,${\theta }_{34}={18.5}^{\circ },{\theta }_{24}=4^{\circ },{\theta }_{14}={3.6}^{\circ }$

${\mathrm{\Delta }}_{ij}^m$	Energy	${\mathrm{\Delta }}_{ij}^m(CPT)=|{\mathrm{\Delta }}_{ij}^m$-$\overline{{\mathrm{\Delta }}_{ij}^m}|$
${\mathrm{\Delta }}_{21}^m$	1	3.44 $\times$ 10${}^{-3}$
${\mathrm{\Delta }}_{31}^m$	1	1.52$\times$ 10${}^{-4}$
${\mathrm{\Delta }}_{41}^m$	1	4.47 $\times$ 10${}^{-5}$
${\mathrm{\Delta }}_{21}^m$	2	7.76$\times$ 10${}^{-3}$
${\mathrm{\Delta }}_{31}^m$	2	3.69$\times$10${}^{-4}$
${\mathrm{\Delta }}_{41}^m$	2	2.66$\times$ 10${}^{-5}$
${\mathrm{\Delta }}_{21}^m$	3	1.21$\times$ 10${}^{-3}$
${\mathrm{\Delta }}_{31}^m$	3	5.94$\times$ 10${}^{-4}$
${\mathrm{\Delta }}_{41}^m$	3	3.73$\times$ 10${}^{-6}$

5 Conclusions

We have taken into account potential CPT violations in four different frame works before coming to this result. We started by calculating the squared mass differences of all four flavours of neutrinos in matter. We found that the four flavour frame work neutrino and antineutrino mass square differences are not the same. The maximum change of solar mass square difference and anti-neutrino solar mass square difference in matter is predicted by our numerical study. ${\mathrm{\Delta }}_{21}^m-\overline{{\mathrm{\Delta }}_{21}^m}$=$(1.21-3.44)\times {10}^{-3})eV{}^2$ for ${\delta }_{34}={\delta }_{13}={\delta }_{14}=0,45,90,135,180$ degrees in region 1 to 3 GeV energy region.

Due to matter effects, maximum change of atmospheric mass square difference in matter ${\mathrm{\Delta }}_{31}^m-\overline{{\mathrm{\Delta }}_{31}^m}=(1.68-6.41)\times {10}^{-4})eV{}^2$ for ${\delta }_{34}={\delta }_{13}={\delta }_{14}=0,45,90,135,180$ degrees in energy region 1 to 3 GeV and maximum change of sterile neutrino mass square difference in matter ${\mathrm{\Delta }}_{41}^m-\overline{{\mathrm{\Delta }}_{41}^m}=(1.06-4.60)\times {10}^{-5})eV{}^2$ for ${\delta }_{34}={\delta }_{13}={\delta }_{14}=0,45,90,135,180$ degrees in energy region 1 to 3 GeV . We note that ${\mathrm{\Delta }}_{21}^m(CPT)={\mathrm{\Delta }}_{21}^m-\overline{{\mathrm{\Delta }}_{21}^m}$ contributes significantly to CPT violation in four flavour neutrino oscilation in matter.

Data Availability Statement

No Data associated in the manuscript.

References

[1] John N. Bahcall et al., Phys. Rev. Lett. 40, 1351–1354 (1978).

[2] John M. LoSecco, Phys. Rev. D 59, 117302 (1999).

[3] Zeno Dixon Greenwood et al., Phys. Rev. D 53, 6054–6064 (1996).

[4] G.J. Feldman et al., Adv. High Energy Phys, 475749 (2013).

[5] Chiaki Yanagisawa et al., Phys. Rev. D 83, 072002 (2011).

[6] G. Conforto et al., Phys. Lett. B 447, 122–126 (1999).

[7] James E. Hill, Phys. Rev. Lett. 75, 2654–2657 (1995).

[8] MiniBoone Collaboration, A.A Aguilar-Arevalo et al., Phys. Rev D 103, 5 (2021).

[9] S. Esposito and G. Salesi, arXiv:hep-ph/0906.5542.

[10] J.N. Bahcall et al., JHEP 0408, 016 (2004).

[11] A. de Gouvea and C. Pena-Garag, Phys. Rev D 71, 093002 (2005).

[12] H. Murayama et al., Phys. Rev. LettB 520, 263 (2001).

[13] G. Barenboim et al., J. High Energy Phys, 10, 001 (2002).

[14] G. Barenboim et al., Phys. Lett. B 534, 106 (2002).

[15] G. Barenboim et al., Phys. Lett. B 537, 227 (2002).

[16] J.N. Bahcall et al., Phys. Lett. B 534, 120 (2002).

[17] S. Skadhauge, Nucl. Lett. B 639, 281 (2002).

[18] Bipin Singh Koranga and Vivek Nautiyal, Int. J. Theor. Phys. 60, 976–981 (2021).

[19] Bipin Singh Koranga and Vivek Nautiyal, Int. J. Theor. Phys. 60, 781–792 (2021).

[20] S. Dev et al., Nuclear.Phys B 491, 401 (2019).

[21] Wei Li, Jiajie Ling, Fanrong Xu and Baobiao Yue, JHEP, 21 (2018).

[22] M. Pandey et al., hep-ph/2003.02102.

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

[24] KamLAND Collaboration, J. Hosaka et al., Phys. Rev. Lett. 94, 08180 (2005).

[25] Bipin Singh Koranga and P. Khurana, Int. J. Theor. Phys. 53 (2014) 11, 3737–3743.

[26] Bipin Singh Koranga et al., arXiv:hep-ph/2106.15945v1.

Biographies

Bipin Singh Koranga is a Professor in the Department of Physics, Kirorimal College, University of Delhi. He joined the Theoretical Physics Group at the Indian Institute of Technology Bombay in 2006 and earned his Ph.D Physics (Neutrino Masses and Mixings) in 2007. For over 17 years, he has been teaching graduate-level courses in foundational physics and mathematical physics. His research interests include: the origin of the universe, physics beyond the Standard Model, theoretical nuclear physics, quantum mechanical neutrino oscillations, selected topics in astrophysics. He has published more than 65 research papers in reputed international journals. His current research focuses on neutrino mass models and their phenomenological implications.

Seema Rawat is a Professor of Physics at Zakir Husain College, University of Delhi, where she teaches Quantum Mechanics and Theoretical Physics to undergraduate students. She received both her Master’s degree and Ph.D. in Physics from Kumaun University, Nainital. She has published more than 30 research papers in reputed international journals. Her current research focuses on high-energy particle physics.

Journal of Graphic Era University, Vol. 13_2, 481–492.
doi: 10.13052/jgeu0975-1416.13210
© 2025 River Publishers