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

Abstract

We present compact analytic expressions for neutrino propagation probabilities in matter, with effects from the invisible decay of the ${\nu }_3$ 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 $P({\nu }_{\mu }\to {\nu }_{\mu })$ at its oscillation dips may be explained using our analytic expressions.

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 ${\nu }_3$ 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 ${\nu }_3$ 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

$$\begin{array}{cc}\hfill {\mathcal{H}}_f^{\left({\gamma }_3\right)}=& \frac{1}{2E_{\nu }}U·\mathrm{Diag}\left[\left(0,\Delta m_{21}^2,\Delta m_{31}^2(1-i{\gamma }_3)\right)\right]·U^{†}+\mathrm{Diag}\left[\left(V_{\mathrm{cc}},0,0\right)\right],\hfill \end{array}$$

(1)

where $\Delta m_{ij}^2\equiv m_i^2-m_j^2$, with $m_i$ being the mass of the ${\nu }_i$ vacuum eigenstate. The matter potential is $V_{\mathrm{cc}}=\sqrt{2}{G}_F{N}_e$, where $G_F$ is the Fermi constant, and $N_e$ is the electron number density. We define ${\gamma }_3$ such that it is given by ${\gamma }_3\mathsf{\Delta }{m}_{31}^2=m_3/{\tau }_3$, where $m_3$ is the mass, and ${\tau }_3$ is the lifetime of ${\nu }_3$ in the rest frame. The neutrino mixing matrix is given by $U=U_{23}\left({\theta }_{23}\right)U_{13}({\theta }_{13},{\delta }_{\mathrm{CP}})U_{12}\left({\theta }_{12}\right)$.

We define the dimensionless quantities $\alpha =\mathsf{\Delta }{m}_{21}^2/\mathsf{\Delta }{m}_{31}^2$, $A=2E_{\nu }{V}_{\mathrm{cc}}/\mathsf{\Delta }{m}_{31}^2$ and $\mathsf{\Delta }=\mathsf{\Delta }{m}_{31}^{2}L/4E_{\nu }$ and use $s_{ij}\equiv sin{\theta }_{ij}$ and $c_{ij}\equiv cos{\theta }_{ij}$. 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 ${\gamma }_3\lesssim 0.1$. We can express the small parameters $\alpha ,s_{13}$ and ${\gamma }_3$, in terms of the powers of a common book-keeping parameter $\lambda \equiv 0.2$, as

$$\begin{array}{c}\hfill \alpha \approx 0.03\simeq O\left({\lambda }^2\right),s_{13}\simeq 0.14\simeq O\left(\lambda \right),{\gamma }_3\lesssim 0.1\simeq O\left(\lambda \right).\end{array}$$

(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 $g\left(\mathbb{M}\right)$ of a matrix $\mathbb{M}$ can be expressed as

$$g\left(\mathbb{M}\right)=\sum _{i=1}^k{M}_{i}g\left({\mathsf{\Lambda }}_i\right),\mathrm{with}{M}_i\equiv \prod _{j=1,j\ne i}^k\frac{1}{{\mathsf{\Lambda }}_i-{\mathsf{\Lambda }}_j}(\mathbb{M}-{\mathsf{\Lambda }}_j\mathbb{I}),$$

(3)

where values of ${\mathsf{\Lambda }}_i$ are distinct eigenvalues of the matrix $\mathbb{M}$. Taking $\mathbb{M}=-i{\mathcal{H}}_f^{\left({\gamma }_3\right)}L$, the probability amplitude matrix ${\mathcal{A}}_f$ in the flavor basis may be calculated. The neutrino oscillation probabilities are then obtained as $P_{\alpha \beta }\equiv P({\nu }_{\alpha }\to {\nu }_{\beta })={\left|{\left[{\mathcal{A}}_f\right]}_{\beta \alpha }\right|}^2$.

Expanding in terms of the small parameters $s_{13}$, $\alpha$ and ${\gamma }_3$ (and hence in terms of powers of $\lambda$) and expressing this as $P_{\alpha \beta }\equiv P_{\alpha \beta }^{\left(0\right)}+P_{\alpha \beta }^{\left({\gamma }_3\right)}$, the survival and conversion probabilities are

$$\begin{array}{ll}{P}_{\mu \mu }^{\left(0\right)}& =1-{sin}^{2}2{\theta }_{23}{sin}^2\mathsf{\Delta }-4s_{13}^2{s}_{23}^2\frac{{sin}^2[(A-1)\mathsf{\Delta }]}{{(A-1)}^2}+\alpha c_{12}^2{sin}^{2}2{\theta }_{23}\mathsf{\Delta }sin2\mathsf{\Delta }\\ & -\frac{2}{A-1}{s}_{13}^2{sin}^{2}2{\theta }_{23}\left(sin\mathsf{\Delta }cosA\mathsf{\Delta }\frac{sin\left[\right(A-1)\mathsf{\Delta }]}{A-1}-\frac{A}{2}\mathsf{\Delta }sin2\mathsf{\Delta }\right)+O\left({\lambda }^3\right),\end{array}$$

(4)

$$\begin{array}{cc}\hfill P_{\mu \mu }^{\left({\gamma }_3\right)}=& -{\gamma }_3\mathsf{\Delta }\left({sin}^{2}2{\theta }_{23}cos2\mathsf{\Delta }+4s_{23}^4\right)+{\gamma }_3^2{\mathsf{\Delta }}^2\left({sin}^{2}2{\theta }_{23}cos2\mathsf{\Delta }+8s_{23}^4\right)+O\left({\lambda }^3\right),\hfill \end{array}$$

(5)

$$\begin{array}{ll}{P}_{\mu e}^{\left(0\right)}=& 4s_{13}^2{s}_{23}^2\frac{{sin}^2[(A-1)\mathsf{\Delta }]}{{(A-1)}^2}\\ & +2\alpha s_{13}sin2{\theta }_{12}sin2{\theta }_{23}cos\left(\mathsf{\Delta }+{\delta }_{\mathrm{CP}}\right)\frac{sin\left[\right(A-1)\mathsf{\Delta }]}{A-1}\frac{sinA\mathsf{\Delta }}{A}+O\left({\lambda }^4\right),\end{array}$$

(6)

$$\begin{array}{cc}\hfill P_{\mu e}^{\left({\gamma }_3\right)}=& -8{\gamma }_3{s}_{13}^2{s}_{23}^2\mathsf{\Delta }\frac{{sin}^2[(A-1)\mathsf{\Delta }]}{{(A-1)}^2}+O\left({\lambda }^4\right).\hfill \end{array}$$

(7)

The probability $P_{e\mu }$ is obtained from $P_{e\mu }=P_{\mu e}({\delta }_{\mathrm{CP}}\to -{\delta }_{\mathrm{CP}})$ and the antineutrino oscillation probabilities are obtained using $P_{\overline{\alpha }\overline{\beta }}=P_{\alpha \beta }({\delta }_{\mathrm{CP}}\to -{\delta }_{\mathrm{CP}},A\to -A)$. 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 $\alpha \mathsf{\Delta }\lesssim 1$ and ${\gamma }_3\mathsf{\Delta }\lesssim 1$.

We have also obtained the probability expressions with exact dependence on ${\gamma }_3$ [4]. In addition to the naively expected $e^{-{\gamma }_3\mathsf{\Delta }}$ behavior, analytic expressions also involve additional terms with non-trivial dependence on ${\gamma }_3$. Taking into account the exact dependence on ${\gamma }_3$ 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 ${\gamma }_3=0.1$, and the neutrino mixing parameters as

$$\begin{array}{cc}\hfill {\theta }_{12}={33}^{\circ },{\theta }_{23}={45}^{\circ },& {\theta }_{13}=8.5^{\circ },{\delta }_{\mathrm{CP}}=0^{\circ },\hfill \\ \hfill \mathsf{\Delta }{m}_{21}^2=7.37\times {10}^{-5}{\mathrm{eV}}^2,& \mathsf{\Delta }{m}_{31}^2=2.56\times {10}^{-3}{\mathrm{eV}}^2.\hfill \end{array}$$

(8)

The values chosen agree with the global fit [1] within $3\sigma$ 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 $|\mathsf{\Delta }{P}_{\alpha \beta }|$, defined by

$$\mathsf{\Delta }{P}_{\alpha \beta }\equiv P_{\alpha \beta }\left(\mathrm{analytic}\right)-P_{\alpha \beta }\left(\mathrm{numerical}\right).$$

(9)

Figure 1. The top panels show the probabilities $P_{\mu e}$ and $P_{\mu \mu }$ with ${\gamma }_3=0.1$, 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 $|\mathsf{\Delta }{P}_{\alpha \beta }|$ of these approximations. The thick (thin) curves indicate positive (negative) signs of $\mathsf{\Delta }{P}_{\alpha \beta }$. The figure is taken from [4].

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 $|\mathsf{\Delta }{P}_{\alpha \beta }|\sim 0.001$ in the 2–4 GeV regime, especially in the conversion channel $P_{\mu e}$. Curiously, the oscillation dips for the survival and conversion probability do not reach zero in spite of satisfying the maximal mixing condition of ${\theta }_{23}=\pi /4$. We elaborate upon this below.

4.2. Increase in the Survival Probability at Oscillation Dips

From the probability expression with exact dependence on ${\gamma }_3$ in [4], we obtain the leading contribution to the survival probability $P_{\mu \mu }$ due to ${\nu }_3$ decay as

$$\begin{array}{cc}\hfill P_{\mu \mu }^{\mathrm{leading}}=& c_{23}^4+s_{23}^4{e}^{-4{\gamma }_3\mathsf{\Delta }}+2s_{23}^2{c}_{23}^{2}cos(2\mathsf{\Delta })e^{-2{\gamma }_3\mathsf{\Delta }}.\hfill \end{array}$$

(10)

For a value of ${\gamma }_3\sim O\left(\lambda \right)$, 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

$$\begin{array}{c}\hfill P_{\mu \mu }\left(\mathrm{first}\mathrm{dip}\right)\simeq \frac{1}{4}{\left(1-e^{-\pi {\gamma }_3}\right)}^2\ge 0,P_{\mu \mu }\left(\sec \mathrm{ond}\mathrm{dip}\right)\simeq \frac{1}{4}{\left(1-e^{-3\pi {\gamma }_3}\right)}^2\ge 0\end{array}$$

(11)

at ${\theta }_{23}=\pi /4$. In the absence of decay, we would have expected $P_{\mu \mu }=0$. Such a increase in the probabilities due to ${\nu }_3$ decay is a non-intuitive feature of our analytic prediction.

For ${\nu }_3$ decay with ${\gamma }_3=0.1$, Equation (11) predicts an increase of ∼0.02 at the first oscillation dip and ∼0.1 at the second oscillation dip for $P_{\mu \mu }$. In Figure 2, we test this prediction by plotting the numerical probabilities $P_{\mu \mu }$ 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.

Figure 2. The survival probability $P_{\mu \mu }$ at the first (left) and the second (right) oscillation dips $(\mathsf{\Delta }=\frac{\pi }{2},\frac{3\pi }{2})$ for a range of baselines L, with ${\theta }_{23}={45}^{\circ }$ and ${\gamma }_3=0.1$. The figure is taken from [4].

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 ${\nu }_3$ 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 ${\nu }_3$ for any long-baseline and atmospheric neutrino experiment.