On the Dynamics of Non-Relativistic Flavor-Mixed Particles

On the Dynamics of Non-Relativistic Flavor-Mixed Particles


Evolution of a system of interacting non-relativistic quantum flavor-mixed particles is considered both theoretically and numerically. It was shown that collisions of mixed particles not only scatter them elastically, but can also change their mass eigenstates thus affecting particles’ flavor composition and kinetic energy. The mass eigenstate conversions and elastic scattering are related but different processes, hence the conversion -matrix elements can be arbitrarily large even when the elastic scattering -matrix elements vanish. The conversions are efficient when the mass eigenstates are well-separated in space but suppressed if their wave-packets overlap; the suppression is most severe for mass-degenerate eigenstates in flat space-time. The mass eigenstate conversions can lead to an interesting process, called ‘quantum evaporation,’ in which mixed particles, initially confined deep inside a gravitational potential well and scattering only off each other, can escape from it without extra energy supply leaving nothing behind inside the potential at . Implications for the cosmic neutrino background and the two-component dark matter model are discussed and a prediction for the direct detection dark matter experiments is made.

particle physics - cosmology connection, dark matter theory, cosmological neutrinos

I Introduction

A number of known and hypothetic particles are flavor-mixed, e.g., neutrinos, kaons, quarks, a neutralino, an axion (can be mixed with a photon), to name a few. How these particles behave in the non-relativistic limit has not been carefully studied for years. This paper addresses some important aspects of this profound physical problem.

Mass (propagation) and flavor (interaction) eigenstates are the vectors obtained by diagonalizing the propagation and interaction parts of particle’s Hamiltonian, respectively, and they can generally be not identical but related through a unitary transformation


where and denote the flavor and mass eigenstates, and is a unitary matrix. Hence, a mixed particle produced in a reaction has a specific flavor eigenstate, , described by a wave-function being a superposition of several mass eigenstates Pontecorvo (1958). When a mixed particle is propagating, the mass eigenstates move with different velocities, which causes time-dependent interference known as flavor oscillations.

An interesting and rather counter-intuitive property of non-relativistic flavor-mixed particles has been found, which is illustrated in the following example. Let us create a non-relativistic electron neutrino in a gravitational potential well. One should expect that if the neutrino is initially confined in the potential, it will remain confined forever (flavor oscillations do not change the picture). However, this is not so if the neutrino scatters elastically off other non-mixed particles from time to time. It has been shown that there is a non-vanishing probability to detect this electron neutrino outside the potential at a later time, although no extra energy has been supplied to it Medvedev (2010). This effect, referred to as the “quantum evaporation”, is associated with mass eigenstate conversions — another process discussed in Ref. Medvedev (2010) — which we will often refer to as the “-conversion” or “-process”, for brevity. In our example here, a conversion of a heavier mass eigenstate yields a lighter one with a larger velocity. If this velocity exceeds the escape velocity, the light mass eigenstate is unbound and escapes to infinity. Of course, the time scale for scattering has to be less than that for the eigenstate separation to allow this cycle to proceed. Note, however, that evaporation in such a thought experiment is not complete: only the heavy eigenstate can be converted into the escaping lighter eigenstate, whereas the initially created least massive eigenstate remains always bound if it was bound initially. We underscore that such quantum evaporation and -conversions, proposed in (Medvedev, 2010) and further elaborated here, have no relation to vacuum flavor oscillations or oscillations in matter whatsoever.

In this paper, we explore elementary processes in an ensemble of flavor-mixed particles. Specifically, we consider a system of two non-relativistic flavor-mixed particles confined inside the gravitational potential well which can scatter off each other. We demonstrate that complete evaporation of both these particles is possible in this case. Indeed, when a bound mixed particle scatters off normal matter, only the heavy eigenstate can be -converted with the increase of its kinetic energy (a lá an exothermic reaction) and ultimately escape. Conversions of the least heavy eigenstates are always “endothermic”, hence the trapped ones will never get enough speed to escape. However, if conversions occur in interaction of two mass eigenstates of two flavor-mixed particles, then the trapped lightest state can get substantial recoil velocity and escape. It is this process that opens up a possibility of a complete evaporation of an ensemble of mass-eigenstates. We also show in the paper that the scattering and conversion transition amplitudes are fairly independent, so it is possible to have conversions even when the scattering -matrix elements vanish. Finally, we demonstrate that the -conversion amplitude (and hence its cross-section) in Minkowsky space (without gravity) is strongly suppressed in the mass-degenerate case. These results are important for better understanding of the properties of mixed particle in general, as well as have interesting implications for the cosmic neutrino background and, possibly, dark matter physics and cosmology.

Ii Interacting mixed particles

In this paper we are interested in interactions of individual mass eigenstates. A mixed particle is created in a flavor state, but it consists of several mass eigenstates. Although all of these ‘pieces’ comprise a single particle, it is possible to visualize their kinematics as if they were normal particles having different masses. In general, these mass eigenstates propagate with different velocities and in the non-relativistic limit they separate from each other rapidly. (This is quite different form the relativistic case in which all eigenstates propagate nearly at the speed of light and it may take a while for them to separate, hence the plane wave approximation is commonly used.) Therefore, fairly soon, a mixed particle becomes a collection of spatially separated mass eigenstates which can interact with other particles independently. Apparently, an ensemble of non-relativistic flavor-mixed particles is, in most cases, an ensemble of individual mass eigenstates. Therefore, it is very natural to investigate the evolution of such an ensemble in the mass basis rather than the flavor basis, which is usually used. The interaction matrix, however, is non-diagonal in the mass basis and off-diagonal terms represent transitions between different mass eigenstates. Should the mass eigenstates overlap to represent a particular flavor, these off-diagonal couplings ‘balance’ transitions of mass eigenstates into each other precisely to produce the scattered particle in a flavor eigenstate again. In contrast, if an individual mass eigenstate interacts, there is no such a ‘balance’, so new (absent) mass eigenstates are produced. Thus, one mass eigenstate can be converted into others. Such a process of a -conversion is of primary interest to us.

Here we consider a simple model of stable two-flavor particles. We will interchangeably denote flavor eigenstates as and or as just and , whenever it’s not confusing. Since masses of the mass eigenstates are different, we refer to them as heavy and light eigenstates, hence . Thus, similarly, the mass eigenstates are denoted as and or as just and . A two-component flavor-mixed particle is described by a two-component wave-function, which representations in the flavor and mass bases are related via a rotation matrix, , where is the mixing angle, i.e.,


Fig. 1 illustrates such a particle. The bold red and blue curves represent heavy and light mass eigenstates assumed to have gaussian wave-packets, as in Eq. (36), and thin cyan and magenta curves are the corresponding flavor eigenstates; flavor oscillations occur where mass eigenstates overlap, see Appendix A for technical details.

Figure 1: Probability distributions of different eigenstates in space. The bold curves (red and blue) represent heavy and light mass eigenstates and thin curves (cyan and magenta) are flavor eigenstates. Flavor oscillations are seen in the overlap region.

Because each interaction involves two flavor-mixed particles, the system is described by a two-particle wave-function, which has four components in the flavor and mass bases, namely




respectively, where the subscripts denote particle 1 and particle 2. Note that when the particles 1 and 2 are far apart (before or after an interaction), a two-particle wave-function is separable, being a direct product of one-particle ones: , where ,   and , where ,  . The two-particle flavor and mass eigenstates are related as before,


where the unitary matrix is


in which and . For simplicity, we will restrict further study to one-dimensional motion of particles.

The evolution of the system at hand is described by the two-particle two-component Schrödinger equation. In the mass basis, it reads


Here the free particle Hamiltonian


satisfies energy conservation, where

and . Gravity enters via



and is an arbitrary gravitational potential.

The interaction matrix is diagonal in the flavor basis,


where for indistinguishable particles. In the mass basis, we have


where the hermitian conjugate for the real-valued unitary matrix and

If the particles are distinguishable, one should make the substitution in the above equations. Since trace is invariant under a unitary transformation, ; also useful is .

The physics represented by the -matrix is easy to understand. There are four different interaction combinations (input channels): , , , and in a statistical ensemble of indistinguishable particles interacting with each other, labeling of one particle to be the “first” one and the other to be the “second” is completely arbitrary, hence the states and describe the same statistical representation; nevertheless, we treat them separately for the sake of generality). In each of these interactions, there are four different outcomes (output channels): , , , and . Thus, the -matrix ‘sandwiched’ between initial and final states gives all 16 -matrix elements, , where take the values . More explicitly, for a given target particle and a projectile particle of species , one has with and being the initial and final states of the projectile particle, and being those of the target particle. For example and corresponds to .

There are two types of processes: (i) elastic scatterings in which the system composition does not change (e.g., , etc.) and (ii) mass eigenstate conversions in which the composition changes (e.g., , etc.). The diagonal elements correspond to pure elastic scattering. Two off-diagonal elements, 23 and 32, describe ‘mass exchange’, but they contribute to scattering as well if the particles are indistinguishable. All other elements represent conversion of one or two mass eigenstates. The total energy and momentum must be conserved in all processes. The energy-momentum conservation in elastic scattering is trivial, so we skip it. Conversions are different. Transitions in which a heavy eigenstate is converted into a light one go with the increase of kinetic energy and thus have no threshold. The opposite ones, where is converted into , have a threshold and can only occur if kinematically allowed, i.e., if the initial kinetic energy of the interacting eigenstates is greater than the threshold.

Interestingly, there is a set of parameters, for which the -matrix elements for elastic interactions vanish identically but the conversion amplitudes (off-diagonal elements) do not. Indeed, (i) the diagonal matrix elements, Eq. (11), namely contribute to the total elastic scattering cross-section, ; (ii) the off-diagonal ‘mass exchange’ matrix elements also contribute to scattering in a statistical ensemble sense, if particles are indistinguishable; and (iii) the remaining elements and contribute to the total conversion cross-section, . It is easy to see that one can have simultaneously with . First, scatterings like and vanish if , which requires that , i.e., different flavors do not interact with each other, and also that . Second, scattering channels and vanish if , which additionally requires maximal mixing, . Thus, the matrix becomes


and is the only independent matrix element. Thus, (diagonal terms) and (i.e., and terms, which play a role of scattering in a statistical ensemble) identically and , i.e., conversions can occur even if the gas of mixed particles has vanishing elastic scattering -matrix elements.

The -matrix elements are used to compute interaction cross-sections in the usual way (LL, ). Appendix B briefly discusses the scattering standard theory and presents some useful results. The scattered wave function can be expanded in angular momentum (or, equivalently, the impact parameter) as


where are Legendre polynomials, are radial functions being the solution of the radial part of the Schrödinger equation with a given scattering potential and are partial -matrix amplitudes of the processes for a given . The elastic scattering [i.e, ] cross-sections and the conversion [i.e., , where ] cross-sections are, see Eqs. (68), (68),


where is the initial wave-number in the center of mass frame.

In general, the cross-sections depend on the shape of the scattering potential, as well as particle momentum, and angular momentum, . However, for slow particles and sufficiently well-localized potentials , being the characteristic size of the potential, the partial amplitudes with large angular momentum are small compared to the term. Thus, it is enough to keep the leading term in Eqs. (16), (17). Therefore


Two asymptotic cases worth noting. First, if the amplitude of conversions are much smaller than that of elastic scattering, then the cross-sections scale as follows, Eqs. (73), (74):


which are standard for -wave scattering. Second, more interesting is the case of very efficient conversions (i.e., when the elastic -matrix elements vanish). Then all the the cross-sections scale in the same way, Eqs. (75), (76):


with an additional constraint that the elastic scattering cross-section is equal to the total cross-section of all conversions:


For completeness, we also present a useful parameterization of the cross-section matrix, which can easily be implemented in numerical models:


where and are common normalization parameters and the elements of the -matrix are proportional to the squares of the properly normalized matrix elements of the -matrix, Eq. (11) together with Eqs. (16) and (17), and simply enforce relative strengths of interaction channels that can occur. For instance, for the conversion-dominated interactions, the -matrix follows from Eqs. (12), (20), (21) to be


where is the Heaviside function which ensures that the process is kinematically allowed (i.e., negative final kinetic energy, , means the process cannot occur), where is equal to the initial kinetic energy of the particles in their center of mass frame plus , which is for each conversion, or for each conversion; thus, for example, for , and for in the process , .

Iii Kinematics of interactions

Let us consider an illustrative example of interaction of and belonging to two different particles. As we mentioned before, we consider one-dimensional motion, for simplicity. Let us consider conversion and we assume here that the inverse process is kinematically forbidden. Before the interaction, the mass eigenstates propagate along geodesics which are different (because the eigenstates have different velocities) and localized in space (because the eigenstates are trapped). In the center of mass frame the momentum and energy conservations are and , where ‘prime’ means ‘after scattering’, that is


and we remind here that the incoming particles are non-relativistic, . If , the outgoing mass eigenstates are relativistic


Alternatively, if the masses are degenerate and , then


where we used that the velocities of and are also comparable, because in the center of mass frame; hence and . Here we also introduced the “kick velocity”,  — this is the velocity a heavy eigenstate at rest gets upon conversion into a light eigenstate, provided the recoil velocity of the scatterer is vanishing. Thus, after the interaction, the mass eigenstates propagate along new geodesics, and if is greater than the escape velocity of the potential, , then both -eigenstates escape from the potential well. Alternatively, if elastic scattering occurs, or , then the kinetic energy does not change and the eigenstates remain trapped.

Therefore, upon any interaction involving the process the amplitude of the heavy eigenstate decreased irreversibly and both eigenstates can become unbound. Though the total probability remains unity, the probability to detect the particle (an electron neutrino, for example) inside the potential has decreased and the probability of its detection somewhere outside has become larger. Of course, the overall energy is conserved: the light eigenstate climbs up the potential and loses energy (e.g., a massless particle is redshifted). By repeating this cycle, one can further decrease the amplitude of the trapped eigenstates. Colloquially speaking, the particles “evaporate” from the potential well.

Iv Evolution of a two-particle system

Here we show how two stable flavor-mixed particles, which are trapped in a gravitational potential and scatter off each other from time to time, gradually escape — or “evaporate” — from it. More precisely, the probability to detect the particles inside the potential decreases with time and the probability of their detection elsewhere increases. Such “evaporation” is a result of mass eigenstate conversions in which a heavier eigenstate converts into a lighter one, thus adding kinetic energy to the scattered particles. We emphasize that the phenomena of -conversion and quantum evaporation are not related in any way to particle decays or other reactions, quantum tunneling and such.

To illustrate the evaporation effect, we numerically solve the two-particle two-component Schrödinger equation, Eq. (7). To ease numerical computations, we chose a model potential with strong screening, , where (meaning that the potential is attractive) determines its depth and sets its size ( in computational units). The interaction potential is given by Eq. (11). Interactions of particles occur via a -function potential, i.e., , which is numerically represented by , where and ; the actual shape of does not significantly affect the results so long as is small enough. The relative strengths are chosen to be , the mixing angle is and the masses are chosen to be degenerate, . The initial wave-function components are taken to be gaussian wave-packets.

Now we present exact numerical solutions of the Schrödinger equation for a pair of mixed particles. In order to simplify representation of the four-component two-dimensional time-dependent wave-function, Eq. (4), we compute probability densities of mass eigenstates for each particle (denoted by a subscript) as follows


and similarly for other components. Since the particles are indistinguishable, we define the total probability density of the heavy and the light mass eigenstates as

Figure 2: Space-time diagram of two interacting mass eigenstates in a gravitational potential well showing the effect of -conversions and particle evaporation, obtained by direct numerical solution of a two-particle two-component Schrödinger equation. The probability densities of light (cyan) and heavy (orange) mass eigenstates belonging to different particles are shown; yellow color originates from color blending of cyan and orange. The potential is localized between and (in computation units). At the system consists of and and both are trapped. During the evolution, each collision produces forward and reflected wave-packets of all possible mass eigenstates; those corresponding to conversions escape to infinity in the form of light mass eigenstates.
Figure 3: Space-time diagrams, as in Fig. (2), but for the separate wave-function components: (left panel), (second panel), (third panel) and (right panel). Colors denote different particles: orange is particle 1 and blue is particle two. Initially, only component of the 2-particle wave-function is non-zero, meaning the presence of mass eigenstate of particle 1 and mass eigenstate of particle 2. Other mass eigenstates are produced upon interactions, which are seen as vertexes. See text for more details and discussion.

As a first case, we consider the interaction of a heavy and a light mass eigenstates belonging to two different particles, , which illustrates the effect of quantum evaporation. Fig. 2 shows the space-time diagram of the probability density of a heavy (orange) and a light (cyan) mass eigenstates given by Eqs. (28a), (28b); yellow color originates from color blending in regions where both mass eigenstates propagate along very similar paths. Initially, there is only the heavy mass eigenstate of particle 1 located at (in computational units) and the light mass eigenstate of particle 2 located at , and both are moving toward each other. Both eigenstates are initially gaussian wave-packets with momenta small enough to be trapped in the gravitational potential well. In each collision, forward and reflected wave-packets of all possible mass eigenstates are produced and light mass eigenstates participating in and/or resulting from conversions escape to infinity.

To further elucidate the dynamics of the interactions, we show in Fig. (3) the wave-function components, Eqs. (27a), (27b), namely, (first panel), (second panel), (third panel) and (last panel), where . Here we use different color coding: orange represents particle 1 (i.e., ) and blue represents particle 2. As one can see, at only the state (second panel) is non-vanishing; orange shows the wave-packet (i.e., the heavy eigenstate of particle 1 – the only heavy eigenstate initially present in the system) and blue is the wave-packet (i.e., the light eigenstate of particle 2 – the only light eigenstate initially present in the system). The first interaction occurs at and the second at . Note that several processes occur at each interaction. First, no propagating wave-packets form, as is seen from the first panel, because such conversions are kinematically forbidden. Second, standard elastic collisions occur in which both forward- and back-scattered wave-packets are produced, as is seen in the second panel. Third, elastic “exchange” also occurs, as is seen in the third panel (as we mentioned earlier, if the particles are indistinguishable, this process is equivalent to elastic scattering). Here the wave-function -component, which was initially absent, appears at as a vertex because both forward- and backward-scattered wave-packets appear. After that, the wave-packets of both particles propagate but remain trapped in the potential, so they meet each other again at (blue, particle 2) and (orange, particle 1). Note that although the wave-packet paths intersect, no interactions occur: the wave-packets belonging to the same particle do not self-interact but can only interfere. Finally, the fourth panel shows that light eigenstates are produced in conversions seen as vertexes at and . The velocities of these wave-packets exceed the escape velocity (controlled by the potential depth) so they leave the gravitational potential.

Figure 4: Normalized expectation values of the number of particles confined inside the gravitational potential as a function of time, see Eqs. (29a)–(29c). A heavy mass eigenstate is converted into a light mass eigenstate in each collision, at and . Soon after each collision/conversion, the light mass eigenstate escapes, thus decreasing the total confined mass.

Finally, Fig. 4 shows the expectation value of the number of particles inside the gravitational potential. To simplify comparison, we normalize them to the initial value as follows:


where Eqs. (28a), (28b) were used. One sees that a light mass eigenstate is produced in each collision ( and ) at the expense of the heavy eigenstate. Later, the light mass eigenstate escapes, thus decreasing the total mass inside.

Figure 5: Space-time diagram of two interacting flavor-mixed particles in a gravitational potential, similar to Fig. 2. The probability densities of (cyan) and (yellow) mass eigenstates of both particles are shown. At the system consists of two particles of a particular flavor , each being the superposition of and , and all the eigenstates are trapped inside a gravitational potential. At , each collision produces forward and reflected wave-packets of all possible mass eigenstates and those with escape to infinity.
Figure 6: Evolution of mass eigenstates shown which are overlaid in Fig. 5. Here, left panel (red) represents the eigenstate given by Eq. (28a) and right panel (blue) is the , Eq. (28b).

As a second case, we consider the full evolution of two flavor-mixed particles, each being a composition of both mass eigenstates. The essential difference of this case from the previous one is that all mass eigenstates of both particles are present. The initial state of the system is two flavor-mixed particles produced as flavor eigenstates at and for particle 1 and 2, respectively. These particles are a coherent mixture of mass eigenstates propagating with different velocities: the heavy eigenstates move toward each other and the light ones move initially away from each other. This initial setup allows us to separate the mass eigenstate interaction locations thus simplifying the analysis of the dynamics. Fig. 5 is analogous to Fig. 2 and shows the conversion of heavy mass eigenstates into light ones and their escape from the gravitational potential. Cyan and yellow colors here denote and mass eigenstates. To elucidate the dynamics, we also separate the mass eigenstates into different panels in Fig. 6, which is otherwise identical to Fig. 5.

Figure 7: Space-time diagrams, as in Fig. (3) but for the two-particle case. Shown are wave-function components: (left panel), (second panel), (third panel) and (right panel); colors denote particles 1 (orange) and particle 2 (blue). Mass eigenstates conversions are seen as vertexes in these graphs (see text for detail).
Figure 8: Normalized expectation values of the number of particles confined inside the gravitational potential as a function of time, similar to Fig. 4. Any time a heavy mass eigenstate is converted into a light mass eigenstate in a collision, soon after that the light mass eigenstate escapes to infinity.

Figures 7 and 8 are similar to 3 and 4. They show the evolution of the wave-function components and the number of particles inside the potential. From Fig. 7 one can see that mass eigenstates interact first (at , first panel) to produce elastically scattered trapped -states (panel one) and the outgoing and -states via and (panels two and three, respectively), both have large enough velocities to escape ( gets large by recoil). These escaping -states interact on their way out (at ) with the scattered trapped -states to further produce escaping - and -states via the processes of conversion , and “exchange” , ; trapped -states are also produced at this time via inverse processes , . Such processes repeat later as well, e.g., at . The amplitude of the direct conversions is rather small for the chosen mixing angle and the values of -matrix, so they are not visible in this figure. However, they are seen in Fig. 8 as the decrease of the mass of the heavy eigenstate at , when only collision had occurred. The recoil velocity is larger in this process, hence the light eigenstate escape is fast. Overall, one can see from Fig. 8 that the particle evaporation is rapid and efficient in this case.

We also note that the above examples are one-dimensional for illustration purposes. Whereas they captures all essential physics of the mixed-particle interactions, they cannot be used to evaluate interaction cross-sections for real three-dimensional world. The three-dimensional cross-sections are generally much smaller than the one-dimensional ones because the colliding particles have a huge phase space to miss each other.

V Asymptotic state,

We demonstrated that evaporation of both light and heavy eigenstate can occur, which opens up a possibility of complete evaporation of both particles, which were initially trapped. What conditions are needed for this to occur? Here we present some general estimates; a dedicated analysis may be needed for a specific system. Let the initial composition of the trapped particle population be and . For a single two-component particle of flavor , these are and , and for a particle of flavor , they are and , as follows from Eq. (2). Note that in both cases , i.e., there is exactly one particle in the system. If we consider a system of many particles, and must be multiplied by the number of particles.

Let us also assume that the system is “optically thin”, i.e., probability of particle interaction during one bounce is very small, so if a conversion occurred, the escaping eigenstate experiences no further interactions and just leaves the system for good. We also assume that only forward conversions () can occur; inverse processes () are kinematically forbidden. We consider indistinguishable particles and also assume that . These assumptions are very natural for non-relativistic mixed particles such as neutrinos (e.g., relic neutrinos from big bang) and some dark matter candidates because of their very small interaction cross-sections.

The composition at is described by and , which are governed by equations


where we also assumed, for simplicity, that the particle density is uniform throughout the system. Here is the relative velocity of two interacting eigenstates which are comparable for heavy and light eigenstates if . Here also is the total cross-section of the processes and is the total cross-section of the processes , hence and , see Eqs. (11), (17). Whereas the general solution to these equations has no simple analytical solution, the asymptotic state can be found as follows. From Eqs. (30a), (30b):


This equation has a solution:

where , and

if . We still do not know and , but we note that conversions will occur as long as . Therefore, asymptotically, when , – some constant value:

which is valid for both and , and

if .

We now conclude that when the initial composition satisfies the inequality


complete evaporation of mixed particles occurs, that is no particles will be left inside the gravitational well, . Of course, the particles will be outside and traveling to infinity as light mass eigenstates only. This means that the flavor composition will be .

Vi Conversions in Minkovsky space

It is also important to investigate interactions of the particles in free space when gravity is negligible. This regime is relevant, for example, for the flavor-mixed dark matter in the early universe before structure formation starts, and for the relic cosmological neutrinos when they eventually become non-relativistic but still too hot to be confined by the gravitational attraction of the the large scale structure.

As before, mass eigenstates of a mixed particle move as if they are normal particles with certain (unequal) velocities and masses. The key difference between free and gravitationally confined particles is how their wave-packets spread with time. Depending on the shape of the potential, the wave-packet of a trapped particle, generally, spreads slower than in free space or even contracts (e.g., near the turning points). In this case, the separation of mass eigenstates occurs rapidly and can be nearly perfect as , so one can treat these eigenstates independently. In contrast, the wave-packets widths of free particles grow linearly with time and so does the separation between them. Therefore, the wave-packets of the two mass eigenstates can remain partially overlapped as , and the effect may be very significant depending on particle masses. Particle interactions in this case will involve both mass eigenstates leading to suppression of mass-conversion amplitudes. For example, when mass eigenstate wave-packets perfectly overlap, each particle is in a specific flavor eigenstate, and interactions do not change particle flavors (and hence mass eigenstate composition) by definition of an eigenstate.

Let us consider a non-relativistic mixed particle created at some moment of time at a position in a certain flavor eigenstate. It is a coherent superposition of mass eigenstates and each is described by a wave-packet, which we assume here to be gaussian:


where , and are the wave-packet width, mass and velocity and . The first term describes a gaussian shape and the second term is simply the phase . Note that is the same for all mass eigenstates because the wave-packets must overlap completely at  — the particle is created in a well-defined flavor eigenstate everywhere (i.e., at any ). Here we consider a one-dimensional case. At any time the wave-packet is given by the solution of the Schrödinger equation (WPspread, ) for an initial state , i.e.,


The generalization of this result to three dimensions is straightforward: wave packet spreading occurs independently in each orthogonal Cartesian direction . This can be seen from that the gaussian wave-packet in three dimensions is separable into a product of three one-dimensional gaussians, the Hamiltonian of a non-relativistic free particle is quadratic in momentum , and the orthogonal components of and commute, if . Thus the triple integral in breaks down into three single integrals. The result is: the coordinates and velocities in Eq. (36) become vector quantities.

This wave-function describes motion of -th eigenstate with velocity and the wave-packet spreading due to the momentum uncertainty, . In general, the velocities are different so the wave-packets of different mass eigenstates tend to separate in time: the gaussian centroids separate as . On the other hand, the widths of the wave-packets also grow in time as as . Since both grow linearly in time at late times, there will always be a non-zero overlap of the mass eigenstates.

Interactions of mass eigenstates occur as follows. First, if the mass eigenstate wave-packets overlap completely, they both interact simultaneously as a flavor wave-function. This results in elastic scatterings only (flavor is conserved in interactions), because the interaction hamiltonian, , is diagonal is flavor basis, and no -process can occur. Second, in the opposite case of completely separated mass eigenstates, as in the case of trapping in a gravitational field discussed earlier, the interaction matrix is non-diagonal, so both elastic scattering and conversions do occur. Finally, if the mass eigenstates partially overlap, there are non-zero chances for the particle to interact along both scenarios. In particular, interactions as flavor eigenstates (i.e., non-separated mass eigenstates) is proportional to the overlap integral of the mass wave-packets. We calculate the overlap integral now.

A wave-packet given by Eq. (36) can be written as