Collisional Approach for Open Neutrino Systems

The neutrino as an Open Quantum System (OQS) offers a novel approach to explore physics beyond the Standard Model (SM) [1, 2, 3]. A key motivation for applying the OQS framework in neutrino physics is its ability to transform pure neutrino states into mixed states, i.e., from a single vector state into a statistical mixture. This loss of purity entails entropy production and irreversibility, unlike standard neutrino oscillation models, where purity is preserved.

The first application of the OQS framework to neutrinos dates back to the late 1990s, providing an alternative explanation for the atmospheric muon neutrino deficit [4]. That same year, Sun and Zhou [1] explored gravity effects and black hole evaporation as motivations for employing OQS in neutrino physics.

The time evolution of systems weakly coupled to a memoryless environment — where interactions are independent of previous events — is described by the Born-Markov master equation. When diagonalized, it yields the Gorini-Kossakowski-Sudarshan-Lindblad equation (Lindblad equation) [5, 6]. In 2000, Lisi et al. [2] applied the Lindblad equation to neutrino phenomenology, proposing that the Super-Kamiokande experiment could detect decoherence effects caused by unknown physics, such as quantum gravity. Lacking a physical model for a microscopic derivation, they introduced ad hoc decoherence parameters proportional to powers of the neutrino energy, assuming energy conservation and no dissipation.

While decoherence can reveal new physics, it may also affect measurements of standard oscillation parameters, including the CP-violation phase and the mixing angle $\theta_{23}$ [7, 8]. Thus, the OQS approach is crucial not only for probing new interactions but also for correctly interpreting neutrino oscillation data. In neutrino physics, an OQS framework based on scattering theory is especially appealing, as it aligns with quantum field theory—the most advanced framework for particle interactions.

Many studies link neutrino decoherence to quantum gravity [9, 10, 3, 11, 12, 13, 14, 15, 16, 17, 18], using various experiments to constrain the decoherence parameters [8, 15, 10, 19, 20]. However, a microscopic derivation requires an interaction Hamiltonian and the environmental state, which together determine the decoherence and dissipation parameters, denoted generically as $\gamma$ [5, 6]. Understanding neutrino propagation can also reveal properties of the environments they traverse, potentially altering interpretations of high-energy neutrino sources.

Different techniques exist to derive master equations. For systems with finite Hilbert spaces, the eigenoperator method is appropriate [5]. For scattering processes, however, the collisional method (or repeated interaction scheme) is often more suitable [21, 22]. We focus on two relevant approaches: i) The first approach considers the complete evolution for a short time interval, $\Delta t$, expanding the time evolution operator and tracing over the environment [23]. However, this method introduces an undesirable dependence of $\gamma$ on the arbitrary time interval $\Delta t$. ii) The second approach, developed by Hornberger and Sipe [21], applies scattering theory to model the evolution of Brownian particles interacting with their environment. They derived a cross section-dependent equation but introduced the unitary part by hand, an unexpected feature since free evolution should arise naturally from the full equation.

In this work, we develop a collisional approach to derive a Born-Markov master equation for neutrinos, termed the Collisional Approach for Open Neutrino Systems (CAONS). This method overcomes the challenges of previous approaches: i) the $\Delta t$ dependence of the $\gamma$ parameters and ii) the arbitrary inclusion of the unitary part. Additionally, unlike most of the existing methods, CAONS is not perturbative, in the sense that the master equation does not come from perturbation theory, and provides a clear interpretation of $\gamma$ in terms of scattering cross sections and decay rates. Section II presents the derivation of the CAONS framework. In Section III, we apply it to neutrino decay and scattering. Section IV discusses the implications of our results, followed by final remarks in Section V.

In this section, we develop the CAONS framework to study decoherence, particularly in neutrino systems. Although motivated by neutrino propagation, this method is general and can be applied to any particle system subject to decoherence or dissipation.

Let $\rho_{tot}(0)$ denote the initial state of the composite neutrino-environment system in the Schrödinger picture, with $\nu$ representing the neutrino. Assuming the two systems are initially uncorrelated, we write

where $\rho(0)$ and $\rho^{E}(0)$ are the initial density operators of the neutrino and environment, respectively. Given the weak interaction between neutrinos and the environment, the macroscopic state of the environment remains unchanged, and the composite system can be approximated by a product state throughout the evolution—an assumption known as the Born approximation [6]. We focus on free particles and vacuum environments, so $\rho^{E}$ commutes with the free Hamiltonian.

After evolving for a time interval $\Delta t$, the state becomes

where $U(\Delta t)$ is the time-evolution operator. In the interaction picture, Eq. (2) transforms into

where $S_{+}$ is the scattering matrix for non-negative times, defined as

with $\mathcal{T}$ denoting the time-ordering operator and $\mathcal{H}_{int}(x)$ the interaction Hamiltonian coupling the system and environment.

Expanding $S_{+}=\mathds{1}+iT_{+}$, where $T_{+}$ is the non-negative time transfer matrix, we obtain

It is important to note that we choose $\Delta t$ in Eqs. (2)–(5) to be small compared to the system’s evolution timescale but large compared to the interaction time. This ensures that the upper limit of the time integral in $S_{+}$, and consequently in $T_{+}$, can extend to $\Delta t\to\infty$, introducing the Markov approximation. This approximation enables us to compute the matrix elements of $T_{+}$ similarly to the standard transfer matrix $T$. Specifically, instead of

we obtain

where P.V. denotes the Cauchy principal value.

Since $S_{+}$ is unitary, i.e., $S_{+}^{\dagger}S_{+}=\mathds{1}$, it follows that

Applying these results to Eq. (5) and dividing by $\Delta t$, we obtain

Taking the limit $\Delta t\to 0$ and reverting to the Schrödinger picture, then performing a partial trace over the environment, we find

where $H_{0}$ is the free Hamiltonian of the neutrino, and

with $U_{0}$ being the time-evolution operator for the free neutrino and $\langle\cdot\rangle_{E}$ denoting the average over the environment states. In Eq. (11),

Although $\mathcal{D}(\rho)$ is defined as in Eq. (13), it requires corrections to ensure unitarity. Specifically, the imaginary part must satisfy $\text{Im}\,\mathcal{D}(\rho)=-[H_{LS},\rho]$, where $H_{LS}$ is the Lamb-shift Hamiltonian. Thus, we redefine $\mathcal{D}(\rho)\rightarrow\text{Re}\,\mathcal{D}(\rho)$ and include $H_{LS}$ in the unitary part of the evolution.

There are three key points regarding the master equation: i) The initial time $t=0$ is arbitrary, so the equation must hold for any $t\geq 0$. ii) Although $\Delta t$ in $V$ and $\mathcal{D}(\rho)$ appears divergent, it is regularized in $V$ and governs the emergence of probability rates in the dissipator. iii) Unlike standard Born-Markov equations, our result is not perturbative by default, though perturbation theory can be applied to the dissipation and decoherence parameters derived later.

To express the master equation using operators acting solely on neutrino states, we solve the trace over the environment. Let $\{\ket{A}\}$ be a complete set of orthonormal states for the environment, with labels $A$ and $B$ for environment states, and $I,J,N,$ and $M$ for neutrino states. These labels fully specify the system; for instance, $\ket{I}\equiv\ket{i,\mathbf{p}}$ denotes a neutrino with momentum $\mathbf{p}$ and state $i$, whether in the flavor or mass basis. This notation applies to states with any number of particles.

Assuming a diagonal density operator for the environment:

we solve the partial trace and apply the rotating wave approximation [5, 6] to eliminate fast oscillatory terms. The resulting dissipator is:

where the operators and parameters are defined as:

The $\gamma_{NM}(t)$ parameters are probability rates averaged over the environment, with $P(N,A\to M,B)$ representing the transition probabilities.

We emphasize that $\gamma$ parameters can be interpreted in terms of decay and scattering rates due to their connection with transition probabilities. CAONS fully determines the master equation structure, requiring only the probability rates to compute the neutrino time evolution within this framework.

Next, we briefly discuss $H_{LS}$ and $V$. The Lamb-shift Hamiltonian, $H_{LS}$, introduces a small correction to the system’s energy levels [5]. It commutes with the free Hamiltonian and has no significant influence on the evolution a priori. Thus, we absorb $H_{LS}$ into the free Hamiltonian in this study.

Focusing on $V$, we expand $T_{+}(T_{+}^{\dagger})$ to first order using Eq. (12):

Thus, $V$ corresponds to the matter potential in neutrino propagation, similar to the case of solar neutrinos [24].

This section presents basic applications of the collisional approach, focusing on three examples: i) Neutrino invisible decay: The simplest case, where the neutrino decays to its vacuum state. ii) Neutrino visible decay: The decay results in less energetic neutrinos due to energy-momentum conservation, with final states tending to be less massive. iii) Neutrino scattering on particles at rest: Similar to visible decay, with the neutrino interacting with stationary particles.

For simplicity, we consider a two-family neutrino scenario. The flavor eigenstates, $\ket{\alpha,\mathbf{k}}$ and $\ket{\beta,\mathbf{k}}$, are related to the mass eigenstates $\ket{1,\mathbf{k}}$ and $\ket{2,\mathbf{k}}$ by:

The transition probabilities are computed using:

where $\rho^{(\alpha,\mathbf{p})}(t)$ is the density matrix of a neutrino created in state $\ket{\alpha,\mathbf{p}}$.

This section discusses indirect implications of our results, including estimates of weak interaction-induced decoherence, limits on neutrino-dark matter coupling constants, and the connection between CAONS and the non-Hermitian Hamiltonian approach.

As noted in Sec. I, early studies modeled decoherence parameters as proportional to some power of the neutrino energy, with bounds determined for different spectral indices. However, as shown in Eq. (17), $\gamma_{N,M}$ corresponds to environment-averaged probability rates. For simple environments, such as vacuum or particles at rest, $\gamma_{N,M}$ can be expressed in terms of decay rates and cross sections, which often scale with the neutrino energy.

For the analyses in Secs. IV.1 and IV.2, we follow Ref. [20], which offers a detailed study of Earth’s matter density profile. In their model, neutrino dynamics depend on three decoherence parameters expressed as $\gamma=\gamma_{0}(E/E_{0})^{n}$, where $n$ is an integer in the range $[-2,2]$.

Using quantum scattering theory, we derived a general Born-Markov master equation, linking the $\gamma$ parameters to decay rates and cross sections. The resulting equation is not perturbative, setting it apart from most methods in the literature. A key advantage of the CAONS framework is its flexibility: $\gamma$ can be computed from interaction models and environmental particle profiles or inferred from phenomenological transition rates, making it especially useful when kinematics play a central role.

The CAONS framework enables precise determination of neutrino time evolution, decay rates, and energy dissipation through scattering processes. Additionally, the relation between $\gamma$ and cross sections provides valuable insights into neutrino-environment interactions. One important finding is that decoherence caused by hadronic matter lies within established limits, highlighting the need to account for it when searching for non-standard neutrino interactions. If neutrino decoherence is observed, contributions from hadronic matter must be considered to accurately estimate the effect of non-standard interactions.

We also applied the CAONS framework to constrain the coupling between neutrinos and ultra-light dark matter. Current limits on $\gamma$ yield a much stricter constraint on $g_{\nu\phi}/M_{I}$, with $g_{\nu\phi}/M_{I}\approx 10^{-16}\,(10^{-20})\,\text{GeV}^{-1}$ for complex (real) scalar dark matter—tightening the bound by at least 15 orders of magnitude compared to previous studies. This demonstrates the framework’s power in refining our understanding of neutrino interactions.

Finally, we showed how CAONS connects to the non-Hermitian Hamiltonian method, useful for studying irreversible particle disappearance. This connection underscores CAONS’s potential to enhance the application of non-Hermitian approaches across various contexts.

We would like to thank A. de Gouvêa for his comments and careful reading of the paper. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.