Anomalous cyclic in the neutrino oscillations

The neutrino oscillations between mass and flavour eigenstates in the Pontecorvo’s oscillation scheme is the periodic and Markovian for non-relativistic and relativistic neutrinos. The transition probability between neutrino flavours in Eq. (11) satisfies both the periodic and Markovian condition for the oscillation dynamics. It is well know in stochastic theory that the nature of the translation or oscillation dynamics are very strikingly depend on the structure of the space-time. The smallest change in the space-time structure significantly affects the dynamics of motion since dynamics is coupled to the space-time metric. In our case, there are many sources, such as gravitational waves and the mass distribution in the universe or decoherence and deformation of the Hilbert space at the quantum level³³, that can cause the space-time deformations at both the local and general cosmological scales. Here, we consider the neutrino oscillation dynamics coupled to the space-time in the stochastic framework, however, the models where the oscillation dynamics explicitly coupled to gravity may also be interesting.

As a results, if we turn to the mathematical notation we say that in the classical and quantum levels the deformation in the space-time leads to divergent characteristic waiting time $T=\int _{0}^{\infty }dt w(t)t$ while the jump length variance $\Sigma ^{2}=\int _{-\infty }^{\infty } dx \lambda (x)x^{2}$ is finite where $w(t)=\int _{-\infty }^{\infty } dx \psi (x,t)$ and $\lambda (x)=\int _{0}^{\infty } dt \psi (x,t)$ are waiting time and the jump length probability distribution for $\psi (x,t)$ of particle wave function^34,35. The non-Markovian motion has memory and leads to fractional dynamics^34,35,36,37. Based on non-Markovian framework, Dirac equation in (6) for the deformed Minkowski space-time can be written^38,39,40 as

$$\begin{aligned} i \hbar D_{t} ^{\eta } \psi _{j} (t, \textbf{x}) = – \left[ |\textbf{p}| c + \frac{m^{2}_{j} c^{3}}{2|\textbf{p}|} \right] \frac{ \mathbf {\sigma }\cdot \textbf{p} }{| \textbf{p} |} \psi _{j} (t, \textbf{x}) \end{aligned}$$

(14)

where $D_{t} ^{\eta }$ denotes the Caputo fractional derivative operator of order $\eta $ and $0<\eta <1$. Here with the aim of retaining the dimensional coherence on both sides of Eq. (14), the energy eigenvalue E varies with power $\eta $. For $\eta =1$, the fractional Schrödinger equation reduces to the standard one. The solution of Eq. (14) is the same form with Eq. (7). Therefore, Eq. (7) can be written as

$$\begin{aligned} \psi _{j} (t, \textbf{x}) = \chi ^{(0)}_{j} (t, \textbf{x}) \psi ^{(0)}_{j}(t,\textbf{x}) \end{aligned}$$

(15)

where

$$\begin{aligned} \chi ^{(0)}_{j} (t, \textbf{x}) = e^{-i \frac{m^{2}_{j} c^{3}}{2 \hbar |\textbf{p}|} t}, \qquad \psi ^{(0)}_{j}(t,\textbf{x}) = e^{-i\frac{pc}{\hbar } t}. \end{aligned}$$

(16)

Given wave functions in Eq. (16) satisfies

$$\begin{aligned} i D_{t} ^{\eta } \chi ^{(0)}_{j}(t,\textbf{x}) = – \lambda ^{\eta }_{1} \chi ^{(0)}_{j}(t,\textbf{x}), \quad \lambda _{1}= \frac{ c \mathbf {\sigma }\cdot \textbf{p} }{\hbar } \end{aligned}$$

(17)

and

$$\begin{aligned} i D_{t} ^{\eta } \psi ^{(0)}_{j}(t,\textbf{x}) = – \lambda ^{\eta }_{2} \psi ^{(0)}_{j}(t,\textbf{x}), \quad \lambda _{2} = \frac{m^{2}_{j} c^{3}}{2 \hbar |\textbf{p}|} \end{aligned}$$

(18)

In the Riemann–Liouville formalism, the fractional integral of order $\eta $ is given by the definition^34,35,36,37

$$\begin{aligned} D_{t}^{\eta } * f(t) = \frac{1}{\Gamma (\eta )} \int _{0}^{t} \left( t-\tau \right) ^{\eta -1} f (\tau ) d\tau , \quad t>0 \end{aligned}$$

(19)

where $\eta $ is any positive number and $\Gamma (\eta )$ is Gamma function.

The solution of Eqs. (17) and (18) can be obtained by taking the Laplace transform. If we perform Laplace transformation on Eq. (17)

$$\begin{aligned} i \mathscr{L} \{ D_{t} ^{\eta } \psi ^{(0)}_{j}(t,\textbf{x}) \}= \mathscr{L} \{ \lambda ^ {\eta }_{1,2} \psi ^{(0)}_{j}(t,\textbf{x}) \} \end{aligned}$$

(20)

which yields

$$\begin{aligned} i s^{\eta } \tilde{\psi }^{(0)}_{j}(t,\textbf{x}) – i s^{\eta -1} \tilde{\psi }^{(0)}_{j}(0) = \lambda ^{\eta }_{1,2} \tilde{\psi }^{(0)}_{j}(t,\textbf{x}) \end{aligned}$$

(21)

where $ \tilde{\psi }^{(0)}_{j}(t,\textbf{x})$ is the Laplace transform of the $| \nu _{k} (t) \rangle $, and it can be obtained as

$$\begin{aligned} \tilde{\psi }^{(0)}_{j}(t,\textbf{x}) = \tilde{\psi }^{(0)}_{j}(0) \frac{i s^{\eta -1} }{is^{\eta } – \lambda ^{\eta }_{1,2} } \end{aligned}$$

(22)

By employing the inverse Laplace transform one can obtain two independent solutions as

$$\begin{aligned} \chi ^{(0)}_{j}(t,\textbf{x}) = \chi ^{(0)}_{j}(0) E_{\eta } (-i \lambda ^{\eta }_{1} t^{\eta } ) \end{aligned}$$

(23)

and

$$\begin{aligned} \psi ^{(0)}_{j}(t,\textbf{x}) = \psi ^{(0)}_{j}(0) E_{\eta } (-i \lambda ^{\eta }_{2} t^{\eta } ) \end{aligned}$$

(24)

where $E_{\eta }$ denotes the Mittag–Leffler function⁴¹. It should be noted that the Mittag–Leffler function can be considered a generalization of the natural exponential one.

Combing Eqs. (23) and (24) into Eq. (15), general solution of fractional Dirac equation can be written as

$$\begin{aligned} \psi _{j}(t,\textbf{x}) = \psi ^{(0)}_{j}(0) E_{\eta } (-i \lambda ^{\eta }_{1} t^{\eta } ) E_{\eta } (-i \lambda ^{\eta }_{2} t^{\eta } ) \end{aligned}$$

(25)

As a consequence of Eq. (3), we obtain

$$\begin{aligned} | \nu _{j} (t) \rangle = \sum _{j=1}^{3} U_{\alpha j} E_{\eta } (-i \lambda ^ {\eta }_{1} t^{\eta } ) E_{\eta } (-i \lambda ^ {\eta }_{2} t^{\eta } ) | \nu _{j} \rangle \end{aligned}$$

(26)

The series expression of the Mittag–Leffler function is given as

$$\begin{aligned} E_{\eta } (-i \lambda ^ {\eta } t^{\eta } ) = \sum _{j=0}^{\infty } \frac{(-i \lambda ^ {\eta } t^{\eta })^{j} }{\Gamma (1+\eta j)} \end{aligned}$$

(27)

For $\eta =1$, Eq. (27) reduces to the standard exponential form, whereas for $0<\eta < 1$, initial stretched exponential behaviour

$$\begin{aligned} E_{\eta } (-i \lambda ^ {\eta } t^{\eta } ) \sim \exp \left( – \frac{i \lambda ^ {\eta } t^{\eta } }{\Gamma (1+\eta )} \right) \end{aligned}$$

(28)

turns over to the power-law long-time behaviour

$$\begin{aligned} E_{\eta } (-i \lambda ^ {\eta } t^{\eta } ) \sim \frac{1 }{i \Gamma (1-\eta )\lambda ^ {\eta } t^{\eta }}. \end{aligned}$$

(29)

By using the above expression, we can give Eq. (26) in the simplified form

$$\begin{aligned} | \nu _{\alpha } (t) \rangle = \sum _{j=1}^{3} U_{\alpha j} e^{-i \left( \frac{m^{2}_{j} c^{3}}{2 \hbar p} \right) ^{\eta } t^{\eta } } e^{ -i \left( \frac{pc}{\hbar }\right) ^{\eta } t^{\eta } } | \nu _{j} \rangle \end{aligned}$$

(30)

In this case, the probability amplitude becomes

$$\begin{aligned} \langle \nu _{\beta } |\nu _{\alpha } (t) \rangle = \sum _{j} U_{\alpha j} U^{*}_{\beta j} e^{-i \left( \frac{m^{2}_{j} c^{3}}{2 \hbar p} \right) ^{\eta } t^{\eta } } e^{ -i \left( \frac{pc}{\hbar }\right) ^{\eta } t^{\eta } } \end{aligned}$$

(31)

Hence, the probability for a transition $| \nu _{\alpha } \rangle \rightarrow | \nu _{\beta } \rangle $ under time evolution is

$$\begin{aligned} P_{\nu _{\alpha } \rightarrow \nu _{\beta } } (t) = \sum _{jk} U_{\alpha j} U^{*}_{\beta j} U^{*}_{\alpha k} U_{\beta k} e^{-i \left( \frac{(m^{2}_{j} – m^{2}_{k} ) c^{3}}{2 \hbar p}\right) ^{\eta } t^{\eta } } \end{aligned}$$

(32)

where we will set $t=L$. One can see that the transition probability has the stretched exponential form which is often called as the Kohlrausch–Williams–Watts function^42,43. It should be noted that for $\eta =1$, Eq. (32) reduces to Eq. (11).

Figure 2

As in Fig. 1, neutrino oscillation probabilities assuming the initial state is an electron neutrino. The blue curve denotes the survival probability $P(\nu _{e} \rightarrow \nu _{e})$, the red curve corresponds to the transition probability from electron neutrino to a muon neutrino $P(\nu _{e} \rightarrow \nu _{\mu })$. In (a), the survival and oscillation probability in the Minkowski space-time for $\eta =1$ indicates the ordinary Minkowski space-time. The oscillation probabilities in the deformed Minkowski space-time are given in (b–d) sub-figures. The deformation parameters of the Minkowski space-time are $\eta =0.9$ in (b) $\eta =0.8$ in (c) and $\eta =0.7$ in (d).

Here, we analytically obtained neutrino oscillation probability for the deformed Minkowski space-time above. To see the effect of the fractional order on the survival and oscillation probability, assuming the initial flavour as electron neutrino, we numerically solved Eq. (32) for various $\eta $ values and plotted all numerical results in Fig. 2. As in Fig. 2 as, in the numerical procedure we set $\theta _{12}=33^{0}$, $\Delta m^{2}_{12}=7.37\times 10^{-5}$ eV$^{2}$ and $c=3\times 10^{8}$ m/s. As mentioned above, these parameters are obtained by fitting experimental results between electron and muon oscillations³².

In Fig. 2a–d, the blue curve denotes the survival probability of the electron neutrino $P(\nu _{e} \rightarrow \nu _{e})$ and the red line indicates the oscillation probability transition from electron neutrino to the muon neutrino $P(\nu _{e} \rightarrow \nu _{\mu })$.

In Fig. 2a, oscillation probability in the Minkowski space-time for $\eta =1$ which indicates ordinary Minkowski space-time. Therefore, the solution in this figure are the same with Fig. 2. However, the survival probability of electron neutrino and oscillation probabilities electron neutrino to the muon neutrino for the fractional Minkowski space-time are given in Fig. 2b–d sub-figures. The fractional parameters of Minkowski space-time are set as $\eta =0.9$ in (b) $\eta =0.8$ in (c) and $\eta =0.7$ in (d).

As can be seen from Fig. 2, the survival probability of the electron neutrino $P(\nu _{e} \rightarrow \nu _{e})$ has maximum value at the points where the oscillation probabilities $P(\nu _{e} \rightarrow \nu _{\mu })$ are minimum. It is seen that almost nowhere the probability of oscillation and survival does not take values of zero or one as well in Fig. 1.

As mentioned above the survival probability and oscillation probability remain the same without change for $\eta =1.0$ which corresponds to the Minkowski space-time where the oscillations are quite smooth and appear subsequently at the transition points $T_{p} = \pi , 3\pi , 5\pi ,\ldots $ as seen from Fig. 2a or Table 1. However, the neutrino survival and oscillation probability for the deformed Minkowski space-time is quite different as can be seen from Fig. 2b–d unlike for $\eta =1$ as well as in Fig. 1 or Fig. 2b. However, for $\eta < 1$ the survival and oscillation probabilities in the deformed Minkowski space leads to two main and important results:

Firstly, as it can be seen from Fig. 2a or Table 1 the survival probability of electron neutrino $P(\nu _{e} \rightarrow \nu _{e})$ and oscillation probability transition from electron neutrino to the muon neutrino $P(\nu _{e} \rightarrow \nu _{\mu })$ takes the maximum and minimum values $T_{p} = \pi , 3\pi , 5\pi ,\ldots $ for $\eta =1$, respectively. However, for $\eta < 1$, the survival and transition points slide to the right and these increments regularly increase for the subsequent oscillations as seen in Fig. 2b–d. This means that when $\eta $ decreases, the period of oscillation shifts. For $\eta < 1$, the amounts of the increments in the survival and transition points are given in Table 1. One can see that the shifts in the survival and transition points increase when $\eta $ values decrease. These results strongly provide that the fractional dynamics between mass and flavour eigenstates lead to anomalous behaviour in neutrino survival and oscillation probabilities.

Secondly, the period length of the oscillation between two neutrino flavour, for instance electron and muon neutrinos, i.e., oscillation time, regularly increases for each $\eta $ value depending on the shifts of the transition points. This behaviour may be interpreted as gravitational red and blue shifts⁴⁴. This behaviour can also be seen in Fig. 2b–d. One can see that the period lengths dramatically increase when $\eta $ decreases. The increments in the period length are given Table 2. These results clearly indicate that deformation in space-time extends the survival probabilities of the flavours. On the other hand, we expect that if a particle whose period length changes in a deformed Minkowski space-time continues to move in a homogeneous Minkowski space-time, it is expected that the particle will return to its original period-length state and can continue to oscillate in its original form. Furthermore, our results provide that the oscillation length in the deformed Minkowski space-time is bigger than $L^{osc}$ i.e., $L>>L^{osc}$.

Table 1 Oscillation points for various $\eta $ values.

Table 2 The period lengths of the oscillation for various $\eta $ values.

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