# On Mikheyev–Smirnov–Wolfenstein Resonance Widths

## Abstract

**:**

## 1. Introduction

## 2. Framework and Definitions

## 3. Half-Widths with Respect to the Variable $\mathit{y}$

## 4. Full Consideration of the Resonance Width

## 5. Applications

^{3}with an electron fraction number ${Y}_{e}\approx 0.5$ and Avogadro constant, ${N}_{A}$, as ${N}_{e}={Y}_{e}{\rho}_{m}{N}_{A}$. From Equation (4), the resonance energy is

## 6. Discussion

## Funding

## Conflicts of Interest

## Abbreviations

PMNS | Pontecorvo—Maki—Nakagawa—Sakata |

MSW | Mikheyev—Smirnov—Wolfenstein |

## References

- Lehnert, R. CPT Symmetry and Its Violation. Symmetry
**2016**, 8, 114. [Google Scholar] [CrossRef] [Green Version] - Antonelli, V.; Miramonti, L.; Torri, M.D.C. Phenomenological Effects of CPT and Lorentz Invariance Violation in Particle and Astroparticle Physics. Symmetry
**2020**, 12, 1821. [Google Scholar] [CrossRef] - Wolfenstein, L. Neutrino oscillations in matter. Phys. Rev. D
**1978**, 17, 2369–2374. [Google Scholar] [CrossRef] - Mikheyev, S.P.; Smirnov, A.Y. Resonance enhancement of oscillations in matter and solar neutrino spectroscopy. Sov. J. Nucl. Phys.
**1985**, 42, 913–917. [Google Scholar] - Mikheyev, S.P.; Smirnov, A.Y. Resonant Amplification of ν Oscillations in Matter and Solar-Neutrino Spectroscopy. Nuovo Cim. C
**1986**, 9, 17–26. [Google Scholar] [CrossRef] - Lunardini, C.; Smirnov, A.Y. The minimum width condition for neutrino conversion in matter. Nucl. Phys. B
**2000**, 583, 260–290. [Google Scholar] [CrossRef] [Green Version] - Chizhov, M.V.; Petcov, S.T. Enhancing mechanisms of neutrino transitions in a medium of nonperiodic constant-density layers and in the Earth. Phys. Rev. D
**2001**, 63, 073003. [Google Scholar] [CrossRef] [Green Version]

**Figure 2.**The difference between the upper and lower curves presents the absolute half-width dependence, with respect to $tan2\theta $.

**Figure 3.**The total width dependence, with respect to the parameter $\theta $ for the variable y (solid line) and maximal width for the variable x (dashed line).

**Figure 4.**Resonance shapes (solid lines for ${P}_{2}=1/2$) for the mixing parameters $\pi /100$ (

**a**), $2\pi /25$ (

**b**), and $\pi /4$ (

**c**). Dashed lines correspond to Equation (16), and the dots show the absolute maxima.

**Figure 5.**Comparison of the minimal width of the medium in units of ${d}_{0}$ from Equation (21) (dashed line) and direct calculations (solid line).

**Figure 6.**Resonance shape of the probability distribution (solid lines for ${P}_{2}=1/2$) for the atmospheric neutrinos in Earth. Dashed line corresponds to Equation (16) and dot shows absolute maximum.

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**MDPI and ACS Style**

Chizhov, M.
On Mikheyev–Smirnov–Wolfenstein Resonance Widths. *Symmetry* **2022**, *14*, 176.
https://doi.org/10.3390/sym14010176

**AMA Style**

Chizhov M.
On Mikheyev–Smirnov–Wolfenstein Resonance Widths. *Symmetry*. 2022; 14(1):176.
https://doi.org/10.3390/sym14010176

**Chicago/Turabian Style**

Chizhov, Mihail.
2022. "On Mikheyev–Smirnov–Wolfenstein Resonance Widths" *Symmetry* 14, no. 1: 176.
https://doi.org/10.3390/sym14010176