{'fraction_non_alphanumeric': 0.03809488205166548, 'fraction_numerical': 0.010243373584059218, 'mean_word_length': 4.171080610864102, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 17, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Flavor mixing in a Lee-type model

An exactly solvable Quantum Field Theory (QFT) model of Lee-type is constructed to study how neutrino flavor eigenstates are created through interactions and how the localization properties of neutrinos follows from the parent particle that decays. The two-particle states formed by the neutrino and the accompanying charged lepton can be calculated exactly as well as their creation probabilities. We can show that the coherent creation of neutrino flavor eigenstates follows from the common negligible contribution of neutrino masses to their creation probabilities. On the other hand, it is shown that it is not possible to associate a well defined"flavor"to mixed states of charged leptons.


I. INTRODUCTION
In recent times the neutrino oscillation phenomenon was established as an example where the coherent creation of a superposition of mass eigenstates, which is called a "flavor" eigenstate, tagged by the accompanying charged lepton, leads to an interference among those mass eigenstates inducing the flavor oscillation. We will choose to refer to those specific superpositions of mass eigenstates as neutrino flavor states because there is no well defined "flavor" charge operator to which we can associate flavor quantum numbers. Moreover, we will try to show that the concept of flavor for neutrinos is an approximate notion.
In this article a Lee-type model is deviced to investigate the neutrino flavor creation problem. The Lee model [1,2] is an example of a simple quantum field theory (QFT) which is exactly soluble but it still requires wave function and charge renormalization to make the model meaningful. Such features allied to strong conservation laws enable us to solve the model completely for each sector invariant by the conservation law. Another unique feature of this model, which will be explored here, is that it permits the study of unstable particles in an exact way [3]. The modifications performed here consist on replacing the fields N, θ, V by l i , ν j , π, where the unstable fermion V in the original model is replaced by the scalar boson π and other fields are replaced accordingly. Thus this model will be used to mimic the decay π ± → l ± α ν α where α = e, µ. The decay process π → l i ν j , i, j = 1, 2, has four kinematical decay channels available for two neutrino families. The channel π → l 1 ν (l 1 = e), with arbitrary neutrino content, is, however, very suppressed with respect to the dominant π → l 2 ν (l 2 = µ) because of helicity suppression in weak decays. The model deviced here does not account for this helicity suppression, giving comparable branching ratios for the production of both charged leptons. However, comparable branching ratios for different charged leptons accompanied by neutrinos exist in nature [4]: (a) K 0 L → π ± e ∓ ν e (40.53%) and K 0 L → π ± µ ∓ ν µ (27.02%), (b) K + → π 0 e + ν e (4.98%) and K + → π 0 µ + ν µ (3.32%), and (c) W + → l α ν α . The last process is also effectively a two body decay kinematically analogous to the pion decay.
In the aforementioned processes we could in principle observe the interference among neutrinos produced jointly with different charged leptons such as π → l 1 ν 1 and π → l 2 ν 1 .
Why is this kind of interference not observed? We will try to address this question within this Lee-type model keeping in mind a fictitious pion decay without helicity suppression, with comparable branching ratios to decay into different charged leptons. Previous discussions about the possibility of charged lepton flavor oscillations exist in the literature [5,6] but we should remark that the question raised previously is more general in the sense that we could observe the interference of the channels by observing the neutrinos and not the charged leptons. On the other hand, there is also the interesting possibility that neutrino oscillations could be suppressed by small momentum or energy uncertainties [7,8].
A throughout study of the mixing phenomenon, either in the bosonic or fermionic case, was already performed in Quantum Field Theory (QFT) by Blasone and Vitiello (BV) [9].
Despite of that, the role played by the interaction which generates the mixing was not completely carried out. Although an analysis of flavor states emerging from pion decay is shown in Ref. 10. In an almost free QFT, BV analyzes the effects of mixing and seek for what should be the flavor states for neutrinos. What emerges from that study is the possibility to define two types of unitarily inequivalent states constructed from two distinct vacua: a usual vacuum and a mixed vacuum. They use the mixed vacuum and obtain a different neutrino oscillation phenomenon. The choice of the vacuum, however, does not seem to be unique.
The study of a Lee type model presents two advantages with respect to the BV approach: (i) the interaction responsible by flavor mixing can be taken into account exactly and (ii) the free states are unitarily equivalent to exact states of the total Hamiltonian. The point (ii) means we do not have to choose between two unitarily inequivalent set of states but at the same time we can not study in this type of models the consequences of such phenomenon, which might be present in more realistic QFTs.
Another distinctive approach to treat neutrino oscillations in a rigorous manner was extensively reviewed in Ref. 11, where the so called external wave packet (EWP) approach was analyzed. Although, the first QFT treatment of neutrino oscillations was given in Ref. 12. In the EWP approach, the nonobservability of neutrinos through direct detection was translated into the formalism by treating them as internal Feynman propagators connecting the creation and detection processes. Some unclear aspects of such approach was clarified in Ref. 13. One of them concerned the possibility that some nonphysical contributions, such as below threshold neutrino detection, could be present in the formalism and they have to be removed by hand [13]. Such approach could account for the localization aspects of the creation and detection processes [14,15,16] but the meaning of what would be the intermediate neutrino flavor states could not be investigated. Another attempt to define neutrino flavor states in QFT can be found in Ref. 17. One can also find attempts to describe neutrino oscillations in Relativistic Quantum Mechanics [13,18].
The two goals of this article consist in providing an exactly solvable model to study (a) how neutrinos are created through interactions and (b) how the localization of neutrinos follows from the parent pion properties. As is usually adopted in rough estimates the wave packet size of the daughter particles is considered to be 1/Γ [5,7], where Γ is the decay width of the parent particle. This model provides exactly such relation. We can also calculate the two particle state composed by the charged lepton and the neutrino that is created from pion decay. This calculation is an attempt to properly define a flavor state for neutrinos. We will see that the coherent creation of neutrino flavor states is a consequence of the common negligible kinematical contribution of the different neutrino mass eigenstates to their creation probabilities (amplitudes). We will also study the meaning of defining a different "flavor" state to charged leptons as the coherent superposition that accompanies a neutrino mass eigenstate such as ν 1 , for instance. We will conclude that the concept of "flavor" associated to such superposition states can not be properly defined.
The outline of the article is as follows: in Sec. II we introduce the model and calculate the relevant eigenstates of the total Hamiltonian. In Sec. III we study how the concept of neutrino flavor emerges from the calculations and derive the localization properties of the decaying states. The discussions are made in Sec. IV and some detailed or additional calculations are shown in the appendices.

II. THE MODEL
The free Hamiltonian of the model is defined by where and A, b i and a j are respectively the annihilation operators of the fields π, l i and ν j modelling (unrealistically) the decay π → l i ν j . The summation over repeated indices are implicit.
We will restrict ourserves to two families which means l i = l 1 , l 2 (l 1 ≡ e, l 2 ≡ µ) and ν j = ν 1 , ν 2 correspond to the two neutrino mass eigenstates. Extension to three neutrino families is straightforward. The fixed bare energies of the fields π and l i corresponding to their masses µ, M i mean this model treats the static and recoilless limit of π and l i . The momentum though is conserved. The neutrinos, on the other hand, obey the relativistic energy dispersion relation (2). The creation and annihilation operators satisfy The spin indices are suppressed.
We define the Fourier transform of the fields to be where u r i (p), r = 1, 2, are Dirac spinors normalized as u r † i (p)u s i (p) = 2E i (p)δ rs while u r 0 , r = 1, 2, are the corresponding spinors for fermions at rest properly normalized to u r † 0 u s 0 = δ rs . Expression (8) corresponds to the usual Majorana type fermion expansion where η C is a phase factor appearing in the charge conjugation transformation.

The interaction Hamiltonian is chosen to be
where {U ij } corresponds to the mixing matrix, ν (−) j refers to the creation part of expansion (8) and Using the Dirac representation for gamma matrices, C = iγ 2 γ 0 , u r 0 = e r [19] and u s j (p) = The function f represents a "form factor" necessary to regularize the expressions in the Lee model [2]. For consistency reasons to be discussed later it also has to be smooth, analytic, approximately flat for energies much lower than the cutoff scale, but falling off not so rapidly above the cutoff scale. In addition, the Lee model may violate unitarity if the coupling constant is above a critical value [2]. We assume the coupling constant is below the critical value thus avoiding such situation. Nevertheless, the case of an unstable parent particle can be properly described in the unitary regime [3].
There are two charges Q 1 , Q 2 that are conserved and within those sectors of fixed charges we can find the explicit eigenstates of the whole Hamiltonian The two charges are In the sector containing one π [(Q 1 , Q 2 ) = (1, 0)] and one pair of we can calculate the exact eigenstates of H in terms of the following eigenstates of H 0 where and |0 is the vacuum state of H 0 and of H as well [20]. The function ψ π is an arbitrary function controlling the pion momentum distribution. The subscript 0 indicates the states correspond to eigenstates of H 0 , i.e., bare states. Then we find the eigenvalue equations The eigenstate of H with energy (mass) E π corresponding to a dressed state of |π 0 can be found if we discover the function χ in From the eigenvalue equation we obtain and The function φ 0 is defined as We assume there is only one root for Eq. (25) which defines the pion mass M π , i.e., the energy of the dressed state |π . Therefore we could have written the bare pion mass µ as where is the mass counterterm.
We can simplify where The intermediate steps to reach Eq. (29) can be found in appendix A. We should emphasize, however, that the principal value is understood in Eq. (31) if x > m j [3].
To have an unstable pion we should have [3] h 0 (M 1 + m 1 ) < 0 (32) for M 1 = min(M i ) and m 1 = min(m j ). This condition implies that with h ′ 0 (x) = dh 0 (x)/dx. For stable pion, the norm (34) is used to (re)normalize the pion state since the expression in Eq. (34) diverges when there is no cutoff, i.e., f (E) = 1. It means there is no stable dressed pion state and the only stable eigenstates of H in this sector (one π or one pair of l i , ν j ) are the scattering states containing one l i and one ν j . These states complete the Hilbert space in this sector. This is proved in appendix B.
The scattering states, eigenstates of H, can be calculated from the expansion if we find the functions α and Z 1 2 2 β. The eigenvalue equation leads to Inverting Eq. (37) using the incoming (+) or outgoing (-) wave boundary conditions we Notice that the domain of the function h 0 was extended to the complex numbers, hence satisfying the property where Therefore Despite the divergence in Eq. (34), we can still define the approximate pion state We calculate, for outgoing states, with superscript (−) suppressed, which yield For small λ, i.e., |λ| ≪ Γ 0 (M π )/2φ ′ 0 (M π ), we can approximate We have defined the renormalized coupling constant which defines the "renormalized" functions Γ, φ obtained by replacing γ 0 by γ in Γ 0 , φ 0 , thus multiplicatively, while The renormalization constant is Since the scattering states |l i (r), ν j (p, s) are eigenstates of the total Hamiltonian H with where |π λ (t) = e −iHt |π λ , φ (2) Eq. (42) were used.
If the functions Γ(E), φ(E) vary more slowly than E near E = M π we can approximate Γ(E) ≈ Γ(M π ) and if M π ≫ M i + m j we can replace the lower limit of the integral in Eq. (53) by −∞: where Equation (55) expresses the usual exponential decay law with decay rate Γ π . This result is an indication that the approximate state |π λ in Eq. (45) represents appropriately a ressonant pion state that decays into |l i (r), ν j (p, s) states independently of the arbitrary parameter λ.
The emission spectrum of l i , ν j is given by where is the expansion coefficient in Eq. (49). We can see The function W ij (E, λ) is the emission probability for π → l i ν j , per unit energy, with neutrino energy E, summed over spins: The total emission probability is which approximates the usual Breit-Wigner distribution in the λ ≫ Γ π limit, within the From probability conservation we have The energy E in W λ (E) refers to the total energy of the pair l + ν.
We can see the pion state is a ressonant state by calculating the scattering cross section of l i ν j → lν, irrespective of the final product flavors and spins and averaged over initial where E = E j + M i corresponds to the total energy of the incident particles. The kinematics is totally determined because the energy of l i and π are fixed. The difference between the initial and final momenta (velocities) of neutrinos has to be taken into account to obtain the appropriate cross section obeying The presence of W (E) guarantees the resonance for E = M π .

III. FLAVOR (EIGEN)STATES
In the previous section we have seen, from the QFT point of view, it makes more sense to calculate transition probabilities (amplitudes) with respect to the state |l i (r), ν j (p, s) that has definite energy E j (p) + M i and thus are mass eigenstates. What wee see in the actual pion decay, however, is not π → l i ν j , i, j = 1, 2, but π → l α ν α , with a definite flavor Let us define flavor states as |l α (r), ν β (p, s) ≡ δ αi U βj |l i (r), ν j (p, s) , α, β = e, µ .
Let us then calculate already in the λ ≫ Γ π limit. If we take the m j → 0 limit (E j → E ν = |p|) in the terms except in the exponent we arrive at where and W α (E) is W αj (E) in the m j → 0 limit and with |U ij | 2 factored out. We see the probability of finding the flavor state (66) from pion decay decouples into the usual oscillation probability times the emission probability π → l α ν irrespective of the neutrino flavor.
The concept of neutrino flavor states emerges from Eq. (69) taking t = 0 (or t ≪ oscillation period), for which we obtain Therefore only a neutrino of flavor α is produced jointly with the charged lepton of flavor α. If we try to calculate the creation probability for the equally possible superposition state, corresponding to a neutrino mass eigenstate accompanied by a superposition state of charged leptons, where This "wave function" squared (|Ψ rs ij (x, t)| 2 ) would give us the probability density of finding the neutrino ν j in the position x at time t jointly with a charged lepton l i . The exact information of Ψ rs ij (x, t) allow us to expand Equation (75) characterizes completely how the neutrinos and charged leptons arise from pion decay. Although the spatial information of the charged lepton is not determined or calculable in this model.
The exact Fourier transform necessary to compute Eq. (73) is very difficult to be performed analytically but a rough approximate calculation can be performed. We find, in the r = |x| and the bar inv j ,p j andĒ j denote respectively the velocity, momentum and energy of ν j whenĒ j = M π − M i , which corresponds to the kinematics of π → l i ν j with π, l i at rest. The detailed calculation of Eq. (77) is carried out in appendix C.
We can verify the normalization of Eq. (77): for t ≫ 1/Γ π or Γπ p j ≪ 1. Therefore, the total probability is conserved by the approximation. One can not, however, recover the flavor oscillation phenomenon from the approximate wave function (77) because the phase is common to all states l i ν j with fixed i. Numerical studies can be performed to test the accuracy of the approximation, although appropriate care should be taken with the enormous difference between the scales of m j and M π , M i .
To visualize the localization aspects of the neutrinos created, we can calculate the radial probability density for the creation of neutrinos ν j at radius r and time t jointly with l i , For t ≫ 1/Γ π , we see only the first term containing τ − contributes. Thus ρ ij (r, t) has a triangular shape with an abrupt peak in r =v j t and a exponential tail for r <v j t, being negligible for r >v j t. The neutrino is then roughly localized in the regionv j (t − 1/Γ π ) r ≤v j t. The size of the wave packet is roughlyv j /Γ π as is usually assumed in rough estimates [5,7].
The most faithful state that we can construct to describe the neutrino flavor states created from pion decay jointly with a charged lepton l i with momentum q and spin r is where ψ π is the function appearing in the definitions (16) and (17), corresponding to the pion momentum wave function. In the limit t ≫ Γ π , we can approximate m j → 0 in all factors except in the exponent since m j ≪ |p|, M i , M π . Equation (83) becomes We could then write, for t = 0, where l i = l α , M i = M α and If we impose ψ π (p) = δ 3 (p) we obtain exactly q = −p for the momenta of the charged leptons (q) and neutrinos (p). In general, the presence of the function ψ π ensures momentum conservation and forces the amplitude (83) to be appreciable only around q = −p where p has magnitude satisfying E j (p) ≈ M π − M i . Equation (83) also clarifies the contribution of the intrinsic momentum uncertainty ∆p (intrinsic of ψ π ) of the parent particle that is inherited by the daughter particles, apart from the contribution of the decay width Γ π : the smallest between Γ π and ∆p dominates. Other transition amplitudes of |π(t) with respect to the free states |π 0 and |l i (r), ν j (p, s) 0 can be found in appendix E. One can identify some similarity with the Wigner-Weisskopf approximation used in systems with couplings between discrete and continuum energy levels.

IV. DISCUSSIONS
One knows neutrino flavor oscillation ceases when quantum coherence is lost due to the lack of spatial overlap among the mass eigenstates that compounds the flavor state neutrinos ν α [21,22]. The characteristic time (distance) scale for such phenomenon to occur is usually very large for neutrinos and is given by δx/∆v ij , where δx is the characteristic spatial size of the neutrino wave packets and ∆v ij is the velocity difference of the ν i and ν j that compound the flavor state. In this model, in the pion restframe, Neutrino flavor oscillation occurs because the neutrinos produced in channels π → l 2 ν 1 and π → l 2 ν 2 interfere coherently and remain interfering in the same region of space as long as But what prevents the channels π → l 1 ν 1 and π → l 2 ν 1 from interfering?
For real pion decays such interference is not observable even in principle since these two channels have very different probabilities to occur because of helicity suppression in weak decays. In this model, however, the branching ratios are comparable and may mimic other real decays.
One should notice that the neutrinos created in channels π → l 1 ν 1 and π → l 2 ν 1 , for example, may have significant spatial overlap. We see that ∆ M v(m j ) defined as may be of order of unless the neutrino masses are nearly degenerate ∆m 2 ij ≪ m 2 i , m 2 j . We use the notation v j (M i ) instead ofv j throughout this discussion to avoid ambiguities. It means that the non interference of these channels is probably due to the charged leptons, but its detailed account deserves further study. Simple calculations show [5] that the big mass difference between charged leptons make them lose spatial coherence over a distance of atomic length or less for most decaying processes. Is this lost of spatial coherence responsible for the incoherence of the channels with definite flavor l 1 = e or l 2 = µ? Another possibility is that the detection of charged leptons is what makes the coherent superposition of the four channels π → l i ν j , i, j = 1, 2, reduce into two incoherent π → l 1 ν and π → l 2 ν quantum processes. These two explanations are, however, very distinct since the former is detection independent while the latter is detection dependent.
Concerning the absence of the interference between the channels π → l 1 ν and π → l 2 ν, there is one ingredient that could play an important role, i.e., the spatial entanglement.
This question, however, can not be discussed within this model due to the lack of spatial entanglement between l i , in the static limit, and ν j .
Another known situation when neutrino oscillations can be suppressed is the case when it is possible to probe the neutrino kinematical quantities with enough precision to single out one neutrino mass eigenstate that is being created in the process [7]. In the language of this article it means all the four pion decay channels proceeds incoherently. Such suppression, of course, can be induced by localization properties of the source or detector that produces and detects the neutrinos [14]. Conditions necessary for neutrino oscillation can be analyzed, for instance, within a realistic QFT description [16]. In that case, the condition for neutrino oscillation requires [Eq. (4.7) of Ref. [16]] if the intrinsic momentum spread of the decaying particle, apart from the decay width Γ, is small compared to its mass. Condition (90) is exactly the condition necessary to prevent one from knowing which neutrino mass eigenstate is being created [7], when the momentum uncertainty is of the order of Γ, which is also the case of this work. Notice, however, that To be extremely precise, the notion of neutrino flavor states rely on the approximation of negligible neutrino masses, as could be seen in Eq. (69). For that reason, there is a negligible probability to create the wrong flavor ν e , for example, jointly with the charged lepton l 2 = µ.
Such probability is calculable but negligible because of the proportionality to the neutrino masses. Such flavor indefiniteness due to the neutrino mass differences can be also seen in other attempts to exactly define what is a neutrino flavor state such as in Refs. 9 or 17.
The exact definition is always blurred by the differences in the kinematical or dynamical factors that depend, strictly speaking, on the neutrino masses, that in turn, can not be factored out as a common factor. For example, the spinorial character of neutrinos prevent even a description as simple as a mixed free second quantized theory to provide exact initial neutrino flavor definition [13]. Another simple treatment that leads to initial flavor violation is the adoption of general kinematical conditions when scalar wave packets are used [23].
To summarize, we could confirm two aspects, within a Lee-type model: (a) the spatial size of the neutrinos produced through a decaying particle with decay rate Γ is v/Γ for the parent particle at rest. This relation is commonly used in estimates of the neutrino wave packet sizes but this model provides an explicit example satisfying such relation. (b) The coherent creation of neutrino flavor states is possible because the kinematical contribution of neutrinos to their creation probabilities are negligible, allowing to define the flavor state as a superposition of mass eigenstates in the expected way. We should remark that the second conclusion is, in principle, applicable to more realistic decays because the main ingredient is universal: neutrino mass differences, compared to the charged lepton mass differences, contribute negligibly to their creation probabilities.
The spin indices are suppressed. We get The first coefficient yields The contour P is a path along the real axis coming from ∞ to m 1 + M 1 below the real axis then going from m 1 + M 1 to ∞ over the real axis. If the contour is closed with the aid of a very large circle, the integral over the closed contour is zero due to the absence of poles.
The integral over the circle is equal to 1 due to h 0 (z) ∼ z.

We obtain
Analogously one can prove B ij (q) = 0 .
In Eq. (C4) we approximated E = M π − M i in the whole integrand, except in the exponentials and in the term h −1 . It is assumed that these functions vary slowly in the interval E ≈ M π ± Γ π . For p j in the cosine we approximated where E is obtained from the shift E → E + M π compared to Eq. (C3). It is also assumed that M π − Γ π m j + M i , which allows us to replace the lower limit of the integral by −∞ without changing the integral appreciably. At last, Eq. (77) is recovered by using the integral computed in the complex plane closing the contour on the upper or lower half plane.

APPENDIX D: APPROXIMATE EIGENSTATE
We can see the state in Eq. (45) can not be an exact eigenstate of the total Hamiltonian H by calculating the eigenvalue equation, analogous to Eq. (25), with the complex eigenvalue We thus obtain M π + iλ + φ 0 (M π + iλ) = 0 .
The real part is just Eq. (25). The imaginary part yields which can never be satisfied because of the minus sign. Therefore if both |λ| and Γ π are small, the state (45) represents an approximate eigenstate of H. approximation [24] employed in atomic physics.
For the transition amplitude of the approximate pion state to the free l i ν j states we obtain 0 l i (r), ν j (p, s)|π(t) = C rs ij (p) e −i(M i +E j )t Equations (E2) and (E4) illustrate clearly the irreversible probability flow from the free state |π 0 Z − 1 2 to the free states |l i (r), ν j (p, s) 0 .

ACKNOWLEDGMENTS
This work was partially supported by Fundação de Amparoà Pesquisa do Estado de São Paulo (Fapesp) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).