# Entangled neutrino states in a toy model QFT

## Abstract

It has been claimed that wave packets must be covariant and also that decohered neutrino oscillations are always revived during measurement. These conjectures are supported by general arguments which are not specific to the electroweak theory, and so if they are true for neutrinos they will also be true for simplified models. In this paper we produce such a simplified model in which the neutrino wave function, including its entanglement with the source particle and the environment, can be calculated explicitly in quantum field theory. It exhibits neutrino oscillation, which is reduced at late times by decoherence due to interactions of the source with the environment. One simple lesson from this model is that only the difference between the environmental interactions before and after neutrino emission can reduce the amplitude of neutrino oscillations. The model will be used to test the conjectures in a companion paper.

## 1 Introduction

Reactor neutrino experiments report lower values of \(\theta _{13}\) than accelerator experiments. It is customary to reduce this tension by assuming the normal hierarchy and a value of the CP-violating phase \(\delta \) near \(270^\circ \). This increases the expected appearance signal at accelerator experiments, allowing the small \(\theta _{13}\) mixing reported by reactor experiments to produce almost as many electron (anti)neutrinos as are observed at muon (anti)neutrino beams. But there is another logically consistent possibility. The reactor neutrinos have lower energy, and so are expected to be more prone to decoherence than accelerator neutrinos [1]. Indeed no decoherence is expected in the case of accelerator neutrinos [2]. In this case the reactor neutrino measurement of \(\theta _{13}\), based on an analysis with no decoherence, is underestimated and the evidence for the normal hierarchy and maximal CP-violation is weakened. Furthermore, the degradation of the signal observed by JUNO would be considerable [3]. This possibility has been rejected by the Daya Bay collaboration [4]. However their study relied upon a neutrino wave packet model.

### 1.1 Wave packet models of neutrinos

The traditional view of decoherence in neutrino oscillations comes from the quantum mechanical wave packet model. Here neutrinos are produced as a flavor eigenstate wave packet, localized in space and time. The lighter mass eigenstate travels faster than the others and so the wave packets corresponding to different mass eigenstates spatially separate after travelling a distance called the coherence length. This separation leads to decoherence and therefore a decrease in amplitude of neutrino oscillations. The spatially separated mass eigenstates may nonetheless be coherently summed by the detector if the detector has a sufficiently long coherence time, leading to a restoration of neutrino oscillations [5]. The coherence length clearly depends on the spatial size of the wave packet, which is a parameter in such models. It has long been recognized [6] that this spatial size is determined by interactions of the neutrino source particles with the environment. Usually order of magnitude arguments are used to estimate this parameter [6, 7, 8, 9], and the result is substituted into the model.

In quantum mechanics, neutrino wave packets are created by hand. In quantum field theory (QFT) they are created consistently from electroweak interactions. Consistent QFT treatments necessarily create neutrinos entangled to their source particles, such as unstable nuclei or mesons, and also to charged leptons which are created simultaneously. We will refer to all of the particles involved in the interaction which produced the neutrino as source particles, including the charged leptons. Again in this case the environment plays a role. As noted, for example, in Ref. [10] the interactions of the source particles with the environment disentangle the neutrino from the rest of the state and so allow its treatment as a wave packet. This disentanglement is caused by environmental interactions which effectively measure the source particles [11]. It is customary in QFT treatments to apply this interaction by simply projecting the entangled state onto a subsector of the Hilbert space in which the source particles have some definite position or momentum wave function, as if they were actually measured. With the positions of the source particles specified, one can determine a space time region in which the neutrino is created and so the neutrino is again in a localized, flavor eigenstate wave packet. Now, just like the quantum mechanical case, the different mass eigenstates travel at different speeds and so separate, leading to decoherence.

Quite a different QFT treatment appeared in Ref. [12]. Here the different neutrino mass eigenstates were not forcibly created in the same time window. Of course modern neutrino experiments measure neutrinos in a fixed time window, in flavor eigenstates. Therefore the fact that lighter neutrinos travel faster and the travel distance is fixed implies that the lighter mass eigenstates are emitted after the heavy mass eigenstates. So instead of wave packet separation, here the wave packets coelesce, and no decoherence was reported by the authors.

How could QFT produce two such phenomenologically distinct paradigms? In the first case, environmental interactions were imposed by hand, with a simple projection. In the second case, environmental interactions were not included at all.

### 1.2 Wave packets from entanglement

It is our goal to understand when the wave packet treatment of neutrinos is and is not reliable, and to understand how to calculate the wave packet size. We will do this via a first principles, consistent calculation in QFT. Papers on QFT treatments of neutrinos generally calculate the S matrix for neutrino creation and detection, which is the amplitude for the creation of a given state in the asymptotic future, long after the neutrino has been absorbed. However we are interested in the state of the neutrino itself, and so are interested in intermediate states. Such information can not be directly obtained from the S matrix. It is accessible in the Schrodinger picture of QFT, in which operators are time-independent and states evolve via the action of the Hamiltonian operator. An experiment begins with a source state entangled with the environment and the Hamiltonian evolves this initial state into the future. This evolution creates neutrinos.

As was noted in Ref. [12], it is true that different neutrino mass eigenstates may be created at different times. Indeed, evolving the state of a \({}^{235}\)U nucleus for 1 year in the Schrodinger picture, neutrinos may be emitted at any time during the year and so the neutrino wave function extends for one light year. It is certainly not a localized wave packet. In the calculation of matrix elements, one must sum over each mass eigenstate and separately integrate the interaction times over the entire year.

Now the key question is, whether at a fixed time the different mass eigenstates contribute coherently to matrix elements. If they do, one expects to observe neutrino oscillations, if they do not, these oscillations will be damped. Measurements occur in a flavor basis and, in modern experiments, at a reasonably well-determined time. Therefore contributions to the relevant amplitudes come from states in which the different mass eigenstates are localized in space time at detection, meaning that the lighter neutrino was emitted later, again in agreement with [12].

The conclusion is that while Ref. [12] is correct that the times of the emissions of the various mass eigenstates need not agree, nonetheless if the difference exceeds some threshold then coherence will be lost. We claim that this threshold should be interpreted as the wave packet size in the wave packet model. In this case, decoherence will correspond to the spatial separation of the wave packets. However it is not obvious that long measurements may now restore coherence as in Ref. [5].

### 1.3 Our approach

For the questions of interest, concerning neutrino oscillations, wave packets, and decoherence, the details of the electroweak interactions do not play any essential role. Therefore, we will work in the simplest toy model which has the features of interest, a scalar field theory in 1+1 dimensions. Here we can, in the Schrodinger picture of QFT, numerically evolve the full entangled state to any desired moment in time to understand it. Thus our approach is similar to that of Ref. [13] but including environmental interactions. To simplify the situation yet further, we will not consider measurements of the neutrinos. Therefore our final states will be the neutrinos themselves and we will calculate transition amplitudes and transition probabilities from states with no neutrinos to states with a neutrino. We will see that these probabilities already have a rich phenomenology of oscillations and decoherence. Of course it means that we cannot tell whether coherence can be revived through measurement, however we feel that a robust study of coherence revival via measurement requires a characterization of the coherence before measurement, which our method provides.

We do not model interactions with the environment by projecting on to a definite state for the environment and the source particles. Instead all particles are consistently evolved in the Schrodinger picture of QFT. In the calculation of probabilities, the distinct environment and source final states are incoherently summed.

The phenomenology of wave packet models includes several potentially interesting effects, such as the revival of oscillations ruined by docoherence via long measurements in Ref. [5]. In [14] it was asserted that, presumably as a result of revival, decoherence is unobservable in neutrino oscillation experiments. Another claim [15, 16] is that neutrino wave functions are always “*covariant wavepackets*.” This means that they depend on the momentum only via Lorentz scalars. The covariant wave packet hypothesis was assumed in the experimental analysis of decoherence at Daya Bay [4]. We believe that our QFT approach will allow a robust test of these claims.

Our study has three advantages over most quantum field theory (QFT) approaches to neutrino oscillations and decoherence. First, we calculate the full, entangled state consisting of the source, the neutrinos and the environment^{1} at arbitrary times and not just the asymptotic S-matrix. This will allow a robust test of the covariant wave packet proposal. Second, we explicitly consider interactions between the source and the environment.^{2} Third, we integrate our transition probability over the possible final states of the source and the environment. It is this integration which leads to decoherence, reducing the amplitude of neutrino oscillations in the transition probability.

Perhaps one of the most serious attempts at the determination of the wave packet size, in the case of solar neutrinos, was Ref. [6]. Unlike later estimates, it includes an estimate of the phase angle variation resulting from each interaction instead of merely assuming that an interaction automatically results in decoherence. However, in the case of reactor neutrinos, unlike solar neutrinos, the source nuclei are large and so the Coulomb interactions in some cases are hardly affected by a beta decay. We will see in our example that the decoherence is not determined by the total phase induced by an interaction, but rather by the difference in the phase that would be acquired before and after the beta decay. This difference, in the case of reactor neutrinos, may be one or two orders of magnitude smaller than the total phase, and thus the wave packet size may be expected to be an order or magnitude or two larger than may be expected by simply adapting the argument of Ref. [6] to the case of reactor neutrinos. This is one immediate lesson that may be drawn from our simple model.

We begin in Sect. 2 with a simplified model in which the neutrinos are created from a classical source. This model exhibits oscillations. However the neutrinos are always off-shell and also, because the source is classical, it cannot be entangled with the environment and so there is no decoherence. Next in Sect. 3 we introduce our full model. We include both source fields and also environment states. Our analysis of this model is presented in Sect. 4.

## 2 Warm up: a classical source

### 2.1 The model, fields and states

We do not believe that spin plays a key role in a qualitative understanding of decoherence in neutrino oscillations. Therefore our model will involve only real scalar fields. Similarly, we will restrict our attention to one space and one time dimension. So long as our fields are massive, this assumption leads to only a modest reduction in computational complexity. Finally, as our most significant assumption, we will consider one-body and two-body decays instead of three-body decays. Therefore the scalar fields which we will call “neutrinos” will carry no conserved lepton charge. Nonetheless we will introduce two flavors of neutrinos, so that there will be oscillations.

*i*labels the mass eigenstates \(\psi _1\) and \(\psi _2\). The conjugate momenta are

^{3}

*x*. While it is straightforward to define an orthogonal position basis for the 1-particle states, this does not reflect the basis in which neutrinos are usually measured in modern experiments. Usually one measures both a neutrino’s momentum and also position. Clearly the uncertainty principle implies that these are each measured with a finite resolution. Let \(\sigma \) be the momentum resolution of a given detector. For simplicity, we will consider a detector which is only sensitive to neutrinos of momentum \(p_0\), although this can easily be generalized to a multichannel detector. Then the relevant basis of 1-neutrino states will be

### 2.2 Evolution

*t*one obtains the state

*p*. No such covariance is manifest in Eq. (2.11). In a sequel, we will investigate whether the wave packets in our models possess the covariance property demanded in these references and assumed by the Daya Bay collaboration in their analysis [4].

*x*at time

*t*

*t*.

*t*and position

*x*, given that the system began in the \(H_0\) ground state at time \(t=0\). Naively the transition probability would be

*x*is continuous one might expect the probability of finding a neutrino at any given

*x*to vanish, implying that \(\lambda =0\). For a continuous

*x*one is interested instead in the probability density.

*x*it has a nonzero probability to also be observed at

*y*. Therefore one cannot define a normalized probability distribution function (PDF) for

*x*. However, such double-valued position probabilities are exponentially suppressed at distances larger than the de Broglie wavelength corresponding to the momentum resolution. The position resolution of any neutrino detector is much larger than this distance, and so for all practical purposes (2.15) is a PDF.

*x*,

*t*) in the flavor basis is

*p*comes from the stationary point of the phase

### 2.3 Numerical results

The amplitudes \(\mathcal {A}_i(x,100)\) defined in Eq. (2.13) are shown in Fig. 2. Three peculiar features are evident in the left panel. First, the maximum amplitude occurs near \(x=0\). This is a consequence of the fact that the initial energy of the system is equal to zero, since \(H_0\) annihilates the initial state \(|\Omega \rangle \). The final energy is therefore also equal to zero, as *H* is time-independent and so time evolution conserves energy. However the neutrinos are massive, and so they will always be off-shell. This is reflected in the \(\omega \) in the denominator, which vanishes only if \(\omega =0\), as is never the case. The smallest \(\omega \) is the least off-shell, and therefore the highest amplitude. As a result the highest amplitude arises for the neutrinos with the smallest momentum, which cannot travel far.

The second peculiar feature is the peak near \(x=t\) corresponding to neutrinos created at \(t_0=0\). Recall from Eq. (2.12) that one integrates over \(t_0\), and so why should most of the neutrinos observed arise from \(t_0\sim 0\)? This is another consequence of the fact that the neutrinos are off-shell. As \(\omega \ne 0\), the phase \(e^{-i\omega t}\) in Eq. (2.12) always oscillates, damping the integral and so the amplitude. This damping is reduced at \(t_0=0\) just because this is a boundary of the domain of integration, and so there is no oscillation at \(t_0<0\). In this sense, the peak is a consequence of the fact that our classical source is suddenly turned on at \(t=0\), or equivalently our initial condition that there are no neutrinos at \(t=0\).

The third peculiar feature is the small tail at \(x>t\). One may attribute this tail to the finite size \(1/(2\sqrt{\alpha })\) of the classical source. However the tail is too large to be created by this alone. It is also a consequence of the fact that \(\mathcal {A}\) is essentially the Feynman propagator \(\langle \Omega |\psi (t)\psi (t_0)|\Omega \rangle \), albeit with some additional factors. Recall that in quantum field theory only the retarded propagator is causal. The causality of the retarded propagator results from the presence of a commutator term \(-\langle \Omega |\psi (t_0)\psi (t)|\Omega \rangle \). However no such term is present in \(\mathcal {A}\). The physical explanation for the lack of causality of the Feynman propagator is that a particle of mass *m* cannot be kept in a box of size beneath 1 / *m*, and so a leaking of order 1 / *m* is inevitable [19]. Despite the small mass of the neutrino, the length scale 1 / *m* is far smaller than the position resolution of any experiment and so this tail is irrelevant in neutrino physics.

## 3 The model

We are interested in decoherence resulting from interactions of the source particle with the environment, together with quantum entanglement between the neutrino, the source and the environment. The source above was classical and so could not be entangled. Therefore, to incorporate decoherence in our model we must introduce quantum source fields \(\phi _I\) and environment fields \(E_\alpha \).

### 3.1 The fields and their interactions

*i*which labels neutrino mass eigenstates. Thus the simplest model with oscillations contains four real scalar fields \(\phi _H\), \(\phi _L\), \(\psi _1\) and \(\psi _2\) with masses \(M_H>M_L>m_i\) together with the interaction Hamiltonian

### 3.2 The states

*I*runs over the indices \(\{H,L\}\). The decomposition of the environment fields will not be needed as our nonrelativistic approximation (3.2) is sufficient to characterize their interactions.

*i*and momentum

*q*and a source particle of flavor

*I*and momentum

*p*, together with the states \(|\alpha ;I,p\rangle \) which contain no neutrino. The free particle ground states, with an environment field, may be written as simply \(|\alpha \rangle \). These are annihilated by all operators

*a*and

*A*and are orthonormal. The normalizations of the other states are fixed by

*H*is the total Hamiltonian and

*t*is the time to which the system evolves. This matrix element is the amplitude, calculated in the Schrodinger picture, for the initial state \(|0\rangle \) to evolve into the final state \(|\alpha ;L,p;i,q\rangle \).

*i*corresponding to a disappearance channel experiment. Also, since a measurement will also measure, with some resolution \(\sigma \), the neutrino momentum

*q*and will find a value \(q_0\), we will calculate the matrix elements (shown in Fig. 5)

*p*is integrated over. This is reasonable as the final state of the source particle is never measured, and as a result the neutrino is never in a localized wave packet.

These amplitudes and probability densities correspond to transitions from the heavy particle to the light particle plus a neutrino. These are the usual transition amplitudes and transition probabilities in quantum field theory. These are not equal to the amplitudes or probabilities for neutrino measurement, which would require an additional interaction in which the neutrino is absorbed. Nonetheless, these amplitudes and probability densities are interesting because they already manifest neutrino oscillations and decoherence and therefore provide a simple setting in which these pheneomena may be studied.

## 4 Results

### 4.1 Analytical calculation

*x*,

*t*) would allow a determination of \(t_0\) to within some uncertainty. However no measurement is implied here and so all values of \(t_0\in [0,t]\) contribute to the amplitudes.

The Hamiltonian is again time-independent and so evolution conserves energy. \(E_0\) and \(E_1\) are not precisely the energies of the initial and final states, but rather the energies that they would have were they on-shell. Therefore if the particles are all on-shell then \(E_0=E_1\). The phase in (4.8) oscillates rapidly in \(t_0\) unless \(E_0=E_1\). Therefore the \(t_0\) integral will be dominated by the stationary phase corresponding to the case in which the particles are on-shell. In this way we naturally recover the fact that particles are on-shell when *t* is large. This is also apparent in Eq. (4.7), where the \((E_1-E_0)\) in the denominator favors \(E_0\sim E_1\). Note that there is no pole as the numerator vanishes when \(E_0=E_1\).

*t*

*p*is the final momentum of the source and

*q*is the momentum of the neutrino. In neutrino measurements, often both the position and the momentum of the neutrino are determined with some known uncertainty. This motivates us to consider a transition amplitude in which both the momentum and the position of the neutrino are fixed, as in Eq. (3.7)

*x*with momentum \(q_0\)

*P*(

*x*,

*t*) as this only depends on the absolute value of the amplitude. Therefore in this simple model we see that it is not the total interaction of the source with the environment which contributes to decoherence, as has been assumed in many calculations of decoherence such as Refs. [6, 7], but rather the difference between the interaction with the source state before and after the neutrino production. In the case of a Coulomb interaction with a nucleus that produces a neutrino via \(\beta \) decay, this would correspond to the difference in the Coulomb interaction caused by a shift in the charge

*Z*by one unit and the creation of a positron. We claim that this factorization argument is quite general, and not a specific feature of our model.

### 4.2 Numerical results: amplitudes

*p*of the source particles are set to \(p=-2\) and \(p=-3\). As we have assumed that the measured neutrino momentum is equal to 2, the amplitudes are in general supported at \(x>0\). However the source particle momentum \(p+q\) is, within \(\sigma \), equal to \(p+q=0\ (-1)\) when \(p=-2\) \((p=-3)\). Therefore in the later case the \(\phi _H\) moved left and so the measured position of the neutrino tends to lower values of

*x*in the lower panels. The phases oscillate quite rapidly, as can be seen in the right panels, but it is the beating of the phases which leads to neutrino oscillations. Note that interference is only possible between final states with identical quantum numbers, including the recoil momenta. Therefore it is the beating at fixed

*p*which yields neutrino oscillations. On the other hand, one sees from the difference between the red and the green curves that the large environmental energy shifts \(\epsilon _\alpha \) considered here have a visible effect on the spectra already at \(t=50\). As the environmental state is not measured, the corresponding probabilities \(P_\alpha \) will be incoherently summed, degrading the oscillation signal.

*x*, and in fact vanishingly small at \(p=-2\). This is easy to understand. Recall that the neutrino momentum is \(q=2.0\pm 0.2\). When \(p=2\), then \(p+q=0.0\pm 0.2\) and so

*x*and near the light cone are artifacts of the boundary conditions, as in the classical source case considered in Sect. 2.

At time \(t=50\) there are not yet any oscillations and certainly no decoherence. The amplitudes at \(t=2000\) are shown in Fig. 8. These are qualitatively similar to the \(t=50\) case. However the off-shell contribution at the boundary has become thinner. Note that while the integral of the off-shell region is greatly reduced at later time, as expected, nonetheless in the small region of *x*-space where it is visible due to boundary effects, the amplitudes at \(t=50\) and \(t=2000\) are similar.

### 4.3 Numerical results: probabilities

Let us return to the large splitting case \(m_2=0.4\), \(\sigma =0.1\), \(c_\alpha =2^{3\alpha /2}\). The (partial) PDFs are shown in Fig. 9. Note that these PDFs are not localized in *x* as one would expect from wave packets. This is because all values of \(t_0\in [0,t]\) are considered. If the source particles were measured, this would fix \(t_0\) to within some precision and the resulting PDFs would be localized in *x*. Also a measurement of the neutrino would allow an approximate determination of \(t_0\).

The fractional amplitude of the oscillations does appreciably decrease with time, as expected. However this decrease is mostly present already in the partial probabilities. It therefore does not result from the environmental interaction, which is not present at all in \(P_0(x,t)\). Rather this is the kinematic decoherence resulting from the fact that the higher mass neutrino has less phase space and so a lower amplitude, as was seen in Fig. 6.

To observe a clear signature of decoherence resulting from environmental interactions, we return to the small splitting case \(m_2=0.35\), \(\sigma =0.2\), \(c_\alpha =2^{3\alpha /4}\). The corresponding (partial) PDFs are shown in Fig. 10. In this figure, convergence of numerical integration over *q* required that *q* only be integrated from \(q_0-3\) to \(q_0+3\) instead of all values, which reduces some of the partial probability densities by up to 10% and the total probability densities by up to 5%. Now the difference in the amplitudes of the two neutrino mass eigenstates is smaller, as was seen in Fig. 8. Thus while the amplitude of the partial PDF oscillation does clearly shrink with time, this effect is less pronounced than it was in the large splitting case.

In both cases one may observe that at lower values of *x* the oscillation phases differ for the various partial probabilities \(P_\alpha \). By \(x\sim 0\) this difference is about \(60^\circ \). Therefore the total probability *P*, which is an incoherent sum of these partial probabilities, has a smaller oscillation amplitude at small *x* than the partial probabilities. This is the decoherence arising from destructive interference between the various environmental interaction eigenstates. One may observe in Fig. 10 that by \(x\sim 0\), at \(t=3000\), it nearly removes the oscillation minimum.

*t*. However in this example our results appear to be consistent with the thesis that for the first few oscillations \(\epsilon \) should be of the same order as the neutrino momentum. It is also clear that decoherence has a large effect on the positions where the neutrinos have oscillated more times. In our figures this corresponds to the low values of

*x*, but at JUNO it would correspond to the lower energy part of the spectrum.

## 5 Conclusions

In this note we have introduced a simple model of neutrino production, oscillation and decoherence due to environmental interactions of the source particle. This model was treated consistently in quantum field theory and is sufficiently simple that the various wave functions have been calculated explicitly, albeit numerically. Interactions between the source particle(s) and the environment yield a characteristic coherence time. The usual approach is to consider a Gaussian neutrino wave packet with width equal to this coherence time but then to neglect the entanglement with the environment, and often also the entanglement with the source. Following the suggestion of [13], our approach is different. We have kept the full entangled state consisting of the neutrino, source particle and also the environment. Our first principles calculation of the neutrino wave function can be used to test various conjectures in literature, such as the covariant wave packet conjecture of Refs. [15, 16]. We have not yet included a model of measurement, but to do so in the future will be straightforward. A consistent treatment of entanglement and measurement will allow us to test the revival mechanism of Refs. [5, 14].

We have worked in a basis in which the environmental interactions \(H^\prime \) are diagonal. As the Hamiltonian is Hermitian, it may always be diagonalized in principle. While in the case of accelerator neutrinos, the interactions may be relatively simple [2] and so such a diagonalization is straightforward, in the case of reactor neutrinos there are a number of distinct interactions contributing to \(H^\prime \) and an explicit diagonalization would be difficult. However, our analysis suggests that the environmental interaction is appreciable only if the eigenvalues of \(\epsilon _\alpha \) are not too far beneath the neutrino energy, or perhaps the neutrino energy divided by the number of oscillations. In the case of reactor neutrinos, interactions within the nucleus itself after a \(\beta \) decay may be expected to have characteristic energies of hundreds of keV, which would be sufficient. The inner electrons have binding energies of 10s of keV, and so interactions with these electrons may also cause noticeable coherence, at least in experiments such as JUNO that are sensitive to many oscillations. On the other hand interatomic interactions, which are commonly used to set the coherence scale [8, 9], have energy scales of eV, and so are unlikely to have noticeable decoherence effects in any proposed reactor neutrino experiment. We have seen that only the difference between the interaction strength before and after the neutrino emission contributes to decoherence, further reducing the impact of interatomic interactions.

## Footnotes

- 1.
The key role played by the entanglement of the neutrino and the source particles in a QFT treatment has been stressed in Ref. [13]. In Ref. [17] it is claimed that the full entangled QFT treatment leads to the same amplitudes as a wave packet treatment. However neither study included interactions of the source with the environment.

- 2.
Such interactions were included in Ref. [18] by including a phenomenological smearing of energies. We instead consistently treat the interactions in QFT.

- 3.
While the Hamiltonian

*H*could be rewritten as a free Hamiltonian via a momentum-dependent coordinate transformation, such a transformation would not be convenient for our purposes as we will consider states in the*n*-particle Fock space of \(H_0\).

## Notes

### Acknowledgements

We are greatful to Carlo Giunti for comments on this draft. JE is supported by the CAS Key Research Program of Frontier Sciences Grant QYZDY-SSW-SLH006 and the NSFC MianShang grants 11875296 and 11675223. EC is supported by NSFC Grant No. 11605247, and by the Chinese Academy of Sciences Presidents International Fellowship Initiative Grant No. 2015PM063. JE and EC also thank the Recruitment Program of High-end Foreign Experts for support.

## References

- 1.D. Boyanovsky, Short baseline neutrino oscillations: when entanglement suppresses coherence. Phys. Rev. D
**84**, 065001 (2011). https://doi.org/10.1103/PhysRevD.84.065001. arXiv:1106.6248 [hep-ph]ADSCrossRefGoogle Scholar - 2.B.J.P. Jones, Dynamical pion collapse and the coherence of conventional neutrino beams. Phys. Rev. D
**91**(5), 053002 (2015). https://doi.org/10.1103/PhysRevD.91.053002. arXiv:1412.2264 [hep-ph]ADSCrossRefGoogle Scholar - 3.Y.L. Chan, M.-C. Chu, K.M. Tsui, C.F. Wong, J. Xu, Wave-packet treatment of reactor neutrino oscillation experiments and its implications on determining the neutrino mass hierarchy. Eur. Phys. J. C
**76**(6), 310 (2016). https://doi.org/10.1140/epjc/s10052-016-4143-4. arXiv:1507.06421 [hep-ph]ADSCrossRefGoogle Scholar - 4.F.P. An et al., [Daya Bay Collaboration], Study of the wave packet treatment of neutrino oscillation at Daya Bay. Eur. Phys. J. C
**77**(9), 606 (2017). https://doi.org/10.1140/epjc/s10052-017-4970-y. arXiv:1608.01661 [hep-ex] - 5.K. Kiers, N. Weiss, Neutrino oscillations in a model with a source and detector. Phys. Rev. D
**57**, 3091 (1998). https://doi.org/10.1103/PhysRevD.57.3091. arXiv:hep-ph/9710289 ADSCrossRefGoogle Scholar - 6.S. Nussinov, Solar neutrinos and neutrino mixing. Phys. Lett. B
**63**, 201 (1976). https://doi.org/10.1016/0370-2693(76)90648-1 ADSCrossRefGoogle Scholar - 7.L. Krauss, F. Wilczek, Solar neutrino oscillations. Phys. Rev. Lett.
**55**, 122 (1985). https://doi.org/10.1103/PhysRevLett.55.122 ADSCrossRefGoogle Scholar - 8.J. Rich, The quantum mechanics of neutrino oscillations. Phys. Rev. D
**48**, 4318 (1993). https://doi.org/10.1103/PhysRevD.48.4318 ADSCrossRefGoogle Scholar - 9.B. Kayser, J. Kopp, Testing the wave packet approach to neutrino oscillations in future experiments. arXiv:1005.4081 [hep-ph]
- 10.C. Giunti, Neutrino wave packets in quantum field theory. JHEP
**0211**, 017 (2002). https://doi.org/10.1088/1126-6708/2002/11/017. arXiv:hep-ph/0205014 ADSCrossRefGoogle Scholar - 11.W.H. Zurek, Environment induced superselection rules. Phys. Rev. D
**26**, 1862 (1982). https://doi.org/10.1103/PhysRevD.26.1862 ADSMathSciNetCrossRefGoogle Scholar - 12.A. Kobach, A.V. Manohar, J. McGreevy, Neutrino oscillation measurements computed in quantum field theory. Phys. Lett. B
**783**, 59 (2018). https://doi.org/10.1016/j.physletb.2018.06.021. arXiv:1711.07491 [hep-ph]ADSMathSciNetCrossRefGoogle Scholar - 13.A.G. Cohen, S.L. Glashow, Z. Ligeti, Disentangling neutrino oscillations. Phys. Lett. B
**678**, 191 (2009). https://doi.org/10.1016/j.physletb.2009.06.020. arXiv:0810.4602 [hep-ph]ADSCrossRefGoogle Scholar - 14.K.T. McDonald, Oscillations and decoherence. In: IHEP, Beijing, August 19–24, 2013. http://nufact2013.ihep.ac.cn/. https://indico.ihep.ac.cn/event/2996/session/4/contribution/198/material/slides/0.pdf
- 15.D.V. Naumov, V.A. Naumov, A diagrammatic treatment of neutrino oscillations. J. Phys. G
**37**, 105014 (2010). https://doi.org/10.1088/0954-3899/37/10/105014. arXiv:1008.0306 [hep-ph]ADSCrossRefGoogle Scholar - 16.D.V. Naumov, On the theory of wave packets. Phys. Part. Nucl. Lett.
**10**, 642 (2013). https://doi.org/10.1134/S1547477113070145. arXiv:1309.1717 [quant-ph]CrossRefGoogle Scholar - 17.E.K. Akhmedov, A.Y. Smirnov, Neutrino oscillations: entanglement, energy-momentum conservation and QFT. Found. Phys.
**41**, 1279 (2011). https://doi.org/10.1007/s10701-011-9545-4. arXiv:1008.2077 [hep-ph]ADSCrossRefzbMATHGoogle Scholar - 18.E.K. Akhmedov, J. Kopp, M. Lindner, Oscillations of mossbauer neutrinos. JHEP
**0805**, 005 (2008). https://doi.org/10.1088/1126-6708/2008/05/005. arXiv:0802.2513 [hep-ph]ADSCrossRefGoogle Scholar - 19.B.G.G. Chen, D. Derbes, D. Griffiths, B. Hill, R. Sohn, Y.S. Ting, Lectures of sidney coleman on quantum field theory. https://doi.org/10.1142/9371 Google Scholar
- 20.M. Beuthe, Oscillations of neutrinos and mesons in quantum field theory. Phys. Rept.
**375**, 105 (2003). https://doi.org/10.1016/S0370-1573(02)00538-0. arXiv:hep-ph/0109119 ADSMathSciNetCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}.