# Discerning the Nature of Neutrinos: Decoherence and Geometric Phases

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Majorana and Dirac Neutrino

## 3. Total and Geometric Phases for Neutrinos in Matter

^{−3}, $\Delta {m}^{2}=7.6\times $${10}^{-3}$ eV

^{2}and a distance $z\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}100$ km. The results reported in Figure 1 could be, in principle, detected in experiments like RENO [5]. In Figure 2 we show the geometric phases associated to the flavor oscillations (Equations (9) and (10)). There we consider energies $E\sim 1$ GeV and a distance z = 300 km, which are typical of long baseline experiments, like $T2K$. We use the values $\varphi =0.3$, ${n}_{e}={10}^{24}$ cm

^{−3}and $\Delta {m}^{2}=7.6\phantom{\rule{3.33333pt}{0ex}}\times $${10}^{-3}$ eV

^{2}.

## 4. Neutrino Oscillations with Decoherence

^{2}, $\gamma =4\times {10}^{-24}$ GeV, ${\gamma}_{3}=7.9\times {10}^{-24}$ GeV, $\alpha =3.8\times {10}^{-24}$ GeV [84].

^{2}, $\gamma =1.2\times {10}^{-23}$ GeV, ${\gamma}_{3}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}2.23\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-23}$ GeV, $\alpha =1.1\times {10}^{-23}$ GeV [85].

^{−3}${N}_{A}$ and the energy range [0.3–1] GeV, compatible with the DUNE baseline parameters. It is evident from the plots that ${\Delta}_{CP}$ is different for Majorana and Dirac neutrinos in presence of decoherence, and it is also clear that the asymmetries are distinct from the case in which there is no decoherence but the matter effects are included. We note that in the energy range considered, there are negligible differences between the results obtained by employing the full procedure of Ref. [86], and those obtained by symply replacing $\Delta {m}^{2}$ and $sin2\theta $ with their counterparts in matter $\Delta {m}_{m}^{2}$ and $sin2{\theta}_{m}$.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**(Color online) Plots of the total and the geometric phases for ${\nu}_{e}$, as a function of the neutrino energy E, for a distance length z = 100 km. - The red dot dashed line is the total phase; - the blue dashed line is the geometric phase.

**Figure 2.**(Color online) Plot of the geometric phases ${\Phi}_{{\nu}_{e}\to {\nu}_{\mu}}^{g}$ (the blue dashed line) and ${\Phi}_{{\nu}_{\mu}\to {\nu}_{e}}^{g}$ (the red dot dashed line) for Majorana neutrinos as a function of the neutrino energy E, for a distance length z = 300 km. The geometric phases ${\Phi}_{{\nu}_{e}\to {\nu}_{\mu}}^{g}={\Phi}_{{\nu}_{\mu}\to {\nu}_{e}}^{g}$ for Dirac neutrinos is represented by the black solid line.

**Figure 3.**(Color online) Plots of the oscillation formulas ${P}_{{\nu}_{\mu}\to {\nu}_{\tau}}$ (the red dot dashed line) and ${P}_{{\overline{\nu}}_{\tau}\to {\overline{\nu}}_{\mu}}$ (the blue dashed line) for Majorana neutrinos and for Dirac neutrinos ($\varphi =0$, the black line), as a function of the energy E, in vacuum. The purple, dashed line is obtained for $\alpha =0$. In this case ${P}_{{\nu}_{\mu}\to {\nu}_{\tau}}={P}_{{\overline{\nu}}_{\mu}\to {\overline{\nu}}_{\tau}}$ and one has the same formula for Majorana and for Dirac neutrinos. The Pontecorvo formula is represented by the green dotted line. We consider the following values of the parameters: $\varphi =\frac{\pi}{4}$, $z=1.3\times {10}^{4}$ km, ${sin}^{2}{\theta}_{23}=0.51$, $\Delta {m}_{23}^{2}=2.5\times {10}^{-3}$ eV

^{2}, $\gamma =4\times {10}^{-24}$ GeV, ${\gamma}_{3}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}7.9\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-24}$ GeV, $\alpha =3.8\times {10}^{-24}$ GeV. Picture in the inset: plot of ${\Delta}_{CP}^{M}\left(z\right)$ for the same values of the parameters used in the main plots.

**Figure 4.**(Color online) Plots of ${P}_{{\nu}_{e}\to {\nu}_{\mu}}$ (red dot dashed line) and ${P}_{{\overline{\nu}}_{e}\to {\overline{\nu}}_{\mu}}$ (blue dashed line) for Majorana neutrinos and for Dirac neutrinos ($\varphi =0$, black line), as a function of E, in vacuum. The purple, dashed line is obtained by setting $\alpha =0$. In this case ${P}_{{\nu}_{e}\to {\nu}_{\mu}}={P}_{{\overline{\nu}}_{e}\to {\overline{\nu}}_{\mu}}$. The Pontecorvo formula is represented by the green dotted line. We use the same values of $\varphi $ and z of Figure 1, and consider: ${sin}^{2}{\theta}_{12}=0.861$, $\Delta {m}_{12}^{2}=7.56\times {10}^{-5}$ eV

^{2}, $\gamma =1.2\times {10}^{-23}$ GeV, ${\gamma}_{3}=2.3\times {10}^{-23}$ GeV, $\alpha =1.1\times {10}^{-23}$ GeV. Picture in the inset: plot of ${\Delta}_{CP}^{M}\left(z\right)$.

**Figure 5.**(Color online) Plots of ${\Delta}_{CP}={P}_{{\nu}_{e}\to {\nu}_{e}}\left(t\right)-{P}_{{\overline{\nu}}_{e}\to {\overline{\nu}}_{e}}\left(t\right)$, for Majorana neutrinos (the red dot dashed line) and Dirac neutrinos (the blue dashed line), in matter and in presence of decoherence with off-diagonal term, and for neutrinos in matter in absence of decoherence (the black dotted line). In the plots, we consider the same parameters of Figure 4 and the energy range $E\in $ (0.3–1) GeV.

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Capolupo, A.; Giampaolo, S.M.; Lambiase, G.; Quaranta, A.
Discerning the Nature of Neutrinos: Decoherence and Geometric Phases. *Universe* **2020**, *6*, 207.
https://doi.org/10.3390/universe6110207

**AMA Style**

Capolupo A, Giampaolo SM, Lambiase G, Quaranta A.
Discerning the Nature of Neutrinos: Decoherence and Geometric Phases. *Universe*. 2020; 6(11):207.
https://doi.org/10.3390/universe6110207

**Chicago/Turabian Style**

Capolupo, Antonio, Salvatore Marco Giampaolo, Gaetano Lambiase, and Aniello Quaranta.
2020. "Discerning the Nature of Neutrinos: Decoherence and Geometric Phases" *Universe* 6, no. 11: 207.
https://doi.org/10.3390/universe6110207