Open AccessProceeding Paper

Oscillation and Decay of Neutrinos in Matter: An Analytic Treatment^†

Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, India

Physical Research Laboratory, Navrangpura, Ahmedabad 380009, India

Author to whom correspondence should be addressed.

^†

Presented at the 23rd International Workshop on Neutrinos from Accelerators, Salt Lake City, UT, USA, 30–31 July 2022.

Phys. Sci. Forum 2023, 8(1), 66; https://doi.org/10.3390/psf2023008066

Published: 30 October 2023

(This article belongs to the Proceedings of The 23rd International Workshop on Neutrinos from Accelerators)

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Abstract

We present compact analytic expressions for neutrino propagation probabilities in matter, with effects from the invisible decay of the

ν_{3}

mass eigenstate included. These will be directly relevant for long-baseline experiments. The inclusion of decay leads to a non-Hermitian effective Hamiltonian, with the Hermitian part corresponding to oscillation, and the anti-Hermitian part representing the decay. In the presence of matter, the two components invariably become non-commuting. We employ the Cayley–Hamilton theorem to calculate the neutrino oscillation probabilities in constant density matter. The analytic results obtained provide a physical understanding of the possible effects of neutrino decay on these probabilities. Certain non-intuitive features like an increase in the survival probability

P (ν_{μ} \to ν_{μ})

at its oscillation dips may be explained using our analytic expressions.

Keywords:

neutrino oscillations; neutrino decay; long-baseline neutrino experiments

1. Introduction

Neutrino oscillation experiments have conclusively established that neutrinos have nonzero masses and that neutrino flavor eigenstates mix. The mixing parameters have been measured to a good accuracy and can explain most of the observations [1]. However, subleading effects of new physics scenarios are still allowed. One such possible scenario is the invisible decay of neutrinos [2].

In these Proceedings, we explore the effects of the invisible decay of

ν_{3}

vacuum mass eigenstate in the presence of matter effects. In matter, there will invariably be a mismatch between the effective mass eigenstates and decay eigenstates [3]. We employ the Cayley–Hamilton theorem to calculate the neutrino oscillation probabilities for long-baseline neutrino experiments like DUNE. The analytic expressions [4] help in understanding the nature of modifications to oscillation probabilities in the presence of neutrino decay and will aid in the interpretation of future data.

2. The Formalism

When the

ν_{3}

mass eigenstate decays invisibly, i.e., to particles that cannot be detected, the neutrino propagation in matter may be expressed in terms of the effective Hamiltonian

\begin{matrix} H_{f}^{(γ_{3})} = & \frac{1}{2 E_{ν}} U \cdot Diag [(0, Δ m_{21}^{2}, Δ m_{31}^{2} (1 - i γ_{3}))] \cdot U^{†} + Diag [(V_{cc}, 0, 0)], \end{matrix}

(1)

where

Δ m_{i j}^{2} \equiv m_{i}^{2} - m_{j}^{2}

, with

m_{i}

being the mass of the

ν_{i}

vacuum eigenstate. The matter potential is

V_{cc} = \sqrt{2} G_{F} N_{e}

, where

G_{F}

is the Fermi constant, and

N_{e}

is the electron number density. We define

γ_{3}

such that it is given by

γ_{3} Δ m_{31}^{2} = m_{3} / τ_{3}

, where

m_{3}

is the mass, and

τ_{3}

is the lifetime of

ν_{3}

in the rest frame. The neutrino mixing matrix is given by

U = U_{23} (θ_{23}) U_{13} (θ_{13}, δ_{CP}) U_{12} (θ_{12})

We define the dimensionless quantities

α = Δ m_{21}^{2} / Δ m_{31}^{2}

A = 2 E_{ν} V_{cc} / Δ m_{31}^{2}

and

Δ = Δ m_{31}^{2} L / 4 E_{ν}

and use

s_{i j} \equiv sin θ_{i j}

and

c_{i j} \equiv cos θ_{i j}

. Since any effect of decay must be subleading to oscillations, i.e., decay length must be larger than the oscillation length scale, we have the normalized decay width

γ_{3} ≲ 0.1

. We can express the small parameters

α, s_{13}

and

γ_{3}

, in terms of the powers of a common book-keeping parameter

λ \equiv 0.2

, as

\begin{matrix} α \approx 0.03 ≃ O (λ^{2}), s_{13} ≃ 0.14 ≃ O (λ), γ_{3} ≲ 0.1 ≃ O (λ) . \end{matrix}

(2)

3. Neutrino Oscillation Probabilities

We employ the Cayley–Hamilton theorem to calculate the oscillation probabilities with exact dependence on the matter term A. Using the Cayley–Hamilton theorem, any function

g (M)

of a matrix

M

can be expressed as

g (M) = \sum_{i = 1}^{k} M_{i} g (Λ_{i}), with M_{i} \equiv \prod_{j = 1, j \neq i}^{k} \frac{1}{Λ_{i} - Λ_{j}} (M - Λ_{j} I),

(3)

where values of

Λ_{i}

are distinct eigenvalues of the matrix

M

. Taking

M = - i H_{f}^{(γ_{3})} L

, the probability amplitude matrix

A_{f}

in the flavor basis may be calculated. The neutrino oscillation probabilities are then obtained as

P_{α β} \equiv P (ν_{α} \to ν_{β}) = {| {[A_{f}]}_{β α} |}^{2}

Expanding in terms of the small parameters

s_{13}

α

and

γ_{3}

(and hence in terms of powers of

λ

) and expressing this as

P_{α β} \equiv P_{α β}^{(0)} + P_{α β}^{(γ_{3})}

, the survival and conversion probabilities are

\begin{array}{l} P_{μ μ}^{(0)} & = 1 - {sin}^{2} 2 θ_{23} {sin}^{2} Δ - 4 s_{13}^{2} s_{23}^{2} \frac{{sin}^{2} [(A - 1) Δ]}{{(A - 1)}^{2}} + α c_{12}^{2} {sin}^{2} 2 θ_{23} Δ sin 2 Δ \\ - \frac{2}{A - 1} s_{13}^{2} {sin}^{2} 2 θ_{23} (sin Δ cos A Δ \frac{sin [(A - 1) Δ]}{A - 1} - \frac{A}{2} Δ sin 2 Δ) + O (λ^{3}), \end{array}

(4)

\begin{matrix} P_{μ μ}^{(γ_{3})} = & - γ_{3} Δ ({sin}^{2} 2 θ_{23} cos 2 Δ + 4 s_{23}^{4}) + γ_{3}^{2} Δ^{2} ({sin}^{2} 2 θ_{23} cos 2 Δ + 8 s_{23}^{4}) + O (λ^{3}), \end{matrix}

(5)

\begin{array}{l} P_{μ e}^{(0)} = & 4 s_{13}^{2} s_{23}^{2} \frac{{sin}^{2} [(A - 1) Δ]}{{(A - 1)}^{2}} \\ + 2 α s_{13} sin 2 θ_{12} sin 2 θ_{23} cos (Δ + δ_{CP}) \frac{sin [(A - 1) Δ]}{A - 1} \frac{sin A Δ}{A} + O (λ^{4}), \end{array}

(6)

\begin{matrix} P_{μ e}^{(γ_{3})} = & - 8 γ_{3} s_{13}^{2} s_{23}^{2} Δ \frac{{sin}^{2} [(A - 1) Δ]}{{(A - 1)}^{2}} + O (λ^{4}) . \end{matrix}

(7)

The probability

P_{e μ}

is obtained from

P_{e μ} = P_{μ e} (δ_{CP} \to - δ_{CP})

and the antineutrino oscillation probabilities are obtained using

P_{\bar{α} \bar{β}} = P_{α β} (δ_{CP} \to - δ_{CP}, A \to - A)

. In the vacuum limit (

A \to 0

), the probabilities given above match those given in [5] to appropriate orders. The perturbative expansions in Equations (4)–(7) remain valid as long as

α Δ ≲ 1

and

γ_{3} Δ ≲ 1

We have also obtained the probability expressions with exact dependence on

γ_{3}

[4]. In addition to the naively expected

e^{- γ_{3} Δ}

behavior, analytic expressions also involve additional terms with non-trivial dependence on

γ_{3}

. Taking into account the exact dependence on

γ_{3}

improves the accuracy and expands the region of validity to lower energies.

4. Results

Let us now compare the accuracy of our analytic expressions against the exact numerical results. We take

γ_{3} = 0.1

, and the neutrino mixing parameters as

\begin{matrix} θ_{12} = 33^{\circ}, θ_{23} = 45^{\circ}, & θ_{13} = 8 . 5^{\circ}, δ_{CP} = 0^{\circ}, \\ Δ m_{21}^{2} = 7.37 \times 10^{- 5} {eV}^{2}, & Δ m_{31}^{2} = 2.56 \times 10^{- 3} {eV}^{2} . \end{matrix}

(8)

The values chosen agree with the global fit [1] within

3 σ

for normal mass ordering.

4.1. Accuracy of the Analytic Approximations

In Figure 1, we plot the different analytic approximations and the exact numerical probabilities (calculated within the constant density approximation) to check the accuracy of our results. We plot the absolute accuracy

| Δ P_{α β} |

, defined by

Δ P_{α β} \equiv P_{α β} (analytic) - P_{α β} (numerical) .

(9)

We observe that the salient features like the positions and heights of oscillation dips and peaks are predicted accurately by the analytic expressions. Note that the analytic approximations are very accurate, with the absolute accuracy

| Δ P_{α β} | \sim 0.001

in the 2–4 GeV regime, especially in the conversion channel

P_{μ e}

. Curiously, the oscillation dips for the survival and conversion probability do not reach zero in spite of satisfying the maximal mixing condition of

θ_{23} = π / 4

. We elaborate upon this below.

4.2. Increase in the Survival Probability at Oscillation Dips

From the probability expression with exact dependence on

γ_{3}

in [4], we obtain the leading contribution to the survival probability

P_{μ μ}

due to

ν_{3}

decay as

\begin{matrix} P_{μ μ}^{leading} = & c_{23}^{4} + s_{23}^{4} e^{- 4 γ_{3} Δ} + 2 s_{23}^{2} c_{23}^{2} cos (2 Δ) e^{- 2 γ_{3} Δ} . \end{matrix}

(10)

For a value of

γ_{3} \sim O (λ)

, the above expression suggests significant deviations from the standard neutrino oscillation probabilities. We focus on these deviations at the first and second oscillation dips. Our leading order analytic approximation in Equation (10) predicts

\begin{matrix} P_{μ μ} (first dip) ≃ \frac{1}{4} {(1 - e^{- π γ_{3}})}^{2} \geq 0, P_{μ μ} (\sec ond dip) ≃ \frac{1}{4} {(1 - e^{- 3 π γ_{3}})}^{2} \geq 0 \end{matrix}

(11)

θ_{23} = π / 4

. In the absence of decay, we would have expected

P_{μ μ} = 0

. Such a increase in the probabilities due to

ν_{3}

decay is a non-intuitive feature of our analytic prediction.

For

ν_{3}

decay with

γ_{3} = 0.1

, Equation (11) predicts an increase of ∼0.02 at the first oscillation dip and ∼0.1 at the second oscillation dip for

P_{μ μ}

. In Figure 2, we test this prediction by plotting the numerical probabilities

P_{μ μ}

at these dips, in scenarios with and without decay. It is observed that the increase in the probability at these dips matches our estimates, even at large baselines where matter effects are important.

The increase in the probability at the oscillation dip may be used as a novel signature of neutrino decay. At a long-baseline experiment like DUNE, the first (second) dip is expected at ∼2.7 GeV (∼1 GeV), where identifying this signature may be possible.

5. Concluding Remarks

In these Proceedings, we present the modifications to the neutrino probabilities due to the possible invisible decay of

ν_{3}

in matter, in a compact analytic form. Furthermore, we show that our expressions are accurate enough to be of use for long-baseline neutrino experiments like DUNE. The accuracy of our analytic expressions ensures that the salient features of the modifications to the oscillation probabilities due to neutrino decay are captured.

As long as the constant matter density approximation is valid, the neutrino oscillation probabilities given in these Proceedings can be used to probe the physics of the invisible decay of

ν_{3}

for any long-baseline and atmospheric neutrino experiment.

Author Contributions

All authors contributed to the analytic and numerical calculations in the work described in these Proceedings. All authors have read and agreed to the published version of the manuscript.

Funding

The work of D.S.C. and A.D. was supported by the Department of Atomic Energy (DAE), Government of India, under Project Identification No. RTI4002. The work of S.G. was supported by the J.C Bose Fellowship (JCB/2020/000011) of the Science and Engineering Research Board of the Department of Science and Technology, Government of India.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

The authors thank S. M. Lakshmi for her inputs in the initial stages of this project.

Conflicts of Interest

The authors declare no conflict of interest.

References

Esteban, I.; Gonzalez-Garcia, M.C.; Maltoni, M.; Schwetz, T.; Zhou, A. The fate of hints: Updated global analysis of three-flavor neutrino oscillations. J. High Energy Phys. 2020, 9, 178. [Google Scholar] [CrossRef]
Bahcall, J.N.; Cabibbo, N.; Yahil, A. Are neutrinos stable particles? Phys. Rev. Lett. 1972, 28, 316–318. [Google Scholar] [CrossRef]
Chattopadhyay, D.S.; Chakraborty, K.; Dighe, A.; Goswami, S.; Lakshmi, S.M. Neutrino Propagation When Mass Eigenstates and Decay Eigenstates Mismatch. Phys. Rev. Lett. 2022, 129, 011802. [Google Scholar] [CrossRef] [PubMed]
Chattopadhyay, D.S.; Chakraborty, K.; Dighe, A.; Goswami, S. Analytic treatment of 3-flavor neutrino oscillation and decay in matter. J. High Energy Phys. 2023, 1, 51. [Google Scholar] [CrossRef]
Akhmedov, E.K.; Johansson, R.; Lindner, M.; Ohlsson, T.; Schwetz, T. Series expansions for three flavor neutrino oscillation probabilities in matter. J. High Energy Phys. 2004, 4, 78. [Google Scholar] [CrossRef]

Figure 1. The top panels show the probabilities

P_{μ e}

and

P_{μ μ}

with

γ_{3} = 0.1

, for

L = 1300

km, for the analytic expressions mentioned in these Proceedings, as well as for the One Mass Scale Dominance (OMSD) approximation [4]. The bottom panels show the absolute accuracy

| Δ P_{α β} |

of these approximations. The thick (thin) curves indicate positive (negative) signs of

Δ P_{α β}

. The figure is taken from [4].

Figure 1. The top panels show the probabilities

P_{μ e}

and

P_{μ μ}

with

γ_{3} = 0.1

, for

L = 1300

km, for the analytic expressions mentioned in these Proceedings, as well as for the One Mass Scale Dominance (OMSD) approximation [4]. The bottom panels show the absolute accuracy

| Δ P_{α β} |

of these approximations. The thick (thin) curves indicate positive (negative) signs of

Δ P_{α β}

. The figure is taken from [4].

Figure 2. The survival probability

P_{μ μ}

at the first (left) and the second (right) oscillation dips

(Δ = \frac{π}{2}, \frac{3 π}{2})

for a range of baselines L, with

θ_{23} = 45^{\circ}

and

γ_{3} = 0.1

. The figure is taken from [4].

Figure 2. The survival probability

P_{μ μ}

at the first (left) and the second (right) oscillation dips

(Δ = \frac{π}{2}, \frac{3 π}{2})

for a range of baselines L, with

θ_{23} = 45^{\circ}

and

γ_{3} = 0.1

. The figure is taken from [4].

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© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

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MDPI and ACS Style

Chattopadhyay, D.S.; Chakraborty, K.; Dighe, A.; Goswami, S. Oscillation and Decay of Neutrinos in Matter: An Analytic Treatment. Phys. Sci. Forum 2023, 8, 66. https://doi.org/10.3390/psf2023008066

AMA Style

Chattopadhyay DS, Chakraborty K, Dighe A, Goswami S. Oscillation and Decay of Neutrinos in Matter: An Analytic Treatment. Physical Sciences Forum. 2023; 8(1):66. https://doi.org/10.3390/psf2023008066

Chicago/Turabian Style

Chattopadhyay, Dibya S., Kaustav Chakraborty, Amol Dighe, and Srubabati Goswami. 2023. "Oscillation and Decay of Neutrinos in Matter: An Analytic Treatment" Physical Sciences Forum 8, no. 1: 66. https://doi.org/10.3390/psf2023008066

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Oscillation and Decay of Neutrinos in Matter: An Analytic Treatment^†

Abstract

1. Introduction

2. The Formalism

3. Neutrino Oscillation Probabilities

4. Results

4.1. Accuracy of the Analytic Approximations

4.2. Increase in the Survival Probability at Oscillation Dips

5. Concluding Remarks

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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Oscillation and Decay of Neutrinos in Matter: An Analytic Treatment †

Abstract

1. Introduction

2. The Formalism

3. Neutrino Oscillation Probabilities

4. Results

4.1. Accuracy of the Analytic Approximations

4.2. Increase in the Survival Probability at Oscillation Dips

5. Concluding Remarks

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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Article Access Statistics

Further Information

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Oscillation and Decay of Neutrinos in Matter: An Analytic Treatment^†