FERMILAB-PUB-10-014-TNeutrino oscillations: Quantum mechanics vs. quantum field theory

# Fermilab-Pub-10-014-T Neutrino oscillations: Quantum mechanics vs. quantum field theory

Evgeny Kh. Akhmedov   and Joachim Kopp
Max-Planck-Institut für Kernphysik, Postfach 103980
D–69029 Heidelberg, Germany
National Research Centre Kurchatov Institute
Moscow, Russia

Theoretical Physics Department, Fermilab, Batavia, IL 60510, USA
email: akhmedov@mpi-hd.mpg.deemail: jkopp@fnal.gov
January 28, 2010
###### Abstract

A consistent description of neutrino oscillations requires either the quantum-mechanical (QM) wave packet approach or a quantum field theoretic (QFT) treatment. We compare these two approaches to neutrino oscillations and discuss the correspondence between them. In particular, we derive expressions for the QM neutrino wave packets from QFT and relate the free parameters of the QM framework, in particular the effective momentum uncertainty of the neutrino state, to the more fundamental parameters of the QFT approach. We include in our discussion the possibilities that some of the neutrino’s interaction partners are not detected, that the neutrino is produced in the decay of an unstable parent particle, and that the overlap of the wave packets of the particles involved in the neutrino production (or detection) process is not maximal. Finally, we demonstrate how the properly normalized oscillation probabilities can be obtained in the QFT framework without an ad hoc normalization procedure employed in the QM approach.

## 1 Introduction

It is well known by now that neutrino oscillations can be consistently described either in the quantum-mechanical (QM) wave packet approach, or within a quantum field theoretic (QFT) framework.111Although in a number of sources a plane wave approach to neutrino oscillations is employed, it is actually marred by inconsistencies and, if applied correctly, does not lead to neutrino oscillations at all Rich:1993wu [], Beuthe1 []. In the QM method Nussinov:1976uw [], Kayser:1981ye [], Giunti:1991ca [], Rich:1993wu [], Kiers [], Dolgov:1997xr [], Giunti:1997wq [], Cardall:1999ze [], Dolgov:1999sp [], Dolgov:2002wy [], Giunti:2002xg [], FarSm [], Visinelli:2008ds [], AS [], nuwp2 [], neutrinos produced in weak-interaction processes are described by propagating wave packets, the spatial length of which is related to the momentum uncertainty at neutrino production, and the detected states are also described by wave packets, centered at the detection point. The transition (oscillation) amplitude is then obtained by projecting the evolved emitted neutrino state onto the detection state. In the QFT treatment Kobzarev:1980nk [], Kobzarev:1981ra [], Giunti:1993se [], Rich:1993wu [], Grimus:1996av [], Grimus:1998uh [], Ioannisian:1998ch [], Cardall:1999ze [], Dolgov:1999sp [], Beuthe2 [], Dolgov:2002wy [], Dolgov:2004ut [], Dolgov:2005nb [], AKL1 [], one considers neutrino production, propagation and detection as a single process, described by a tree-level Feynman diagram with the neutrino in the intermediate state (see fig. 1). Neutrinos are represented in this framework by propagators rather than by wave functions. Both approaches lead to the standard formula for the probability of neutrino oscillations in vacuum in the case when the decoherence effects related to propagation of neutrinos as well as to their production and detection can be neglected. They differ, however, in the way they account for possible decoherence effects, with the QFT approach leading to a more consistent and accurate description. The QM method treats neutrino energy and momentum uncertainties responsible for these effects in a simplified way; in addition, it involves an ad hoc normalization procedure for the transition amplitude that is not properly justified.

The goal of the present paper is to compare the two approaches and establish a relationship between them, as well as to clarify some of the procedures that are employed in the QM method from the more general and consistent QFT standpoint. Some work in that direction has been done before. In Giunti:2002xg [] neutrino wave packets were derived starting from the QFT formalism (see also a discussion in Beuthe1 []). In Kopp:2009fa [], a comparison of the QM and QFT approaches was presented for the special case of Mössbauer neutrinos, i.e. neutrinos produced and detected recoillessly in hypothetical Mössbauer-type experiments (see, e.g., Raghavan:2005gn [], AKL1 [], Potzel:2008xk [] and references therein). The new results obtained in the present work include a more advanced and general study of the QFT-based derivation of the neutrino wave packets (including the possibility that some of the external particles are not detected), matching of the QFT and QM expressions for the neutrino wave packets, study of the general properties of the wave packets describing the neutrino states (including their energy uncertainties in the case when neutrinos are produced in decays of unstable particles) and clarification of the issue of normalization of the oscillation probabilities in QM and QFT.

The paper is organized as follows. To make the presentation self-contained, in Secs. 2 and 3 we review, respectively, the QM wave packet formalism and the QFT approach to neutrino oscillations. Sections 46 contain our main results. In Sec. 4 we discuss how the neutrino wave packets, which are a necessary ingredient of the QM approach, can be derived starting from the QFT formalism. Next, we consider some general properties of the neutrino wave packets and discuss the conditions under which they can be approximated by Gaussian wave packets. Using the case of Gaussian wave packets as an example, we then discuss how the QFT-derived wave packets can be represented in the form usually adopted in the QM treatment. We also find expressions for the effective parameters describing the QM wave packets in terms of the more fundamental input parameters of the QFT framework. Next, we discuss the neutrino energy uncertainty in the case when neutrinos are produced in decays of unstable particles. In Sec. 5 we consider the problem of normalization of the neutrino wave packets in the QM framework and show how the normalization problem is solved in a natural way in the QFT-based approach. In Sec. 6 we discuss how one can relax some assumptions usually adopted in the QM and QFT approaches. Those include the assumption that the maxima of wave packets of all particles involved in the neutrino production (or detection) process meet at one space-time point, as well as the assumption that the mean momenta of the emitted and detected neutrino wave packets coincide. We summarize our results and conclude in Sec. 7. Some technical material is included in Appendices A and B.

## 2 Review of the QM wave packet formalism

We start with some generalities that are common to QFT and QM and then move on to review the QM wave packet approach to neutrino oscillations. We shall use the natural units throughout the paper.

In quantum theory, one-particle states of particles of type can be written as

 |A⟩=∫[dp]fA(p,P)|A,p⟩, (1)

where is the one-particle momentum eigenstate corresponding to momentum and energy (for free particles, , being the mass of the particle), is the momentum distribution function with the mean momentum , and we use the shorthand notation

 [dp]≡d3p(2π)3√2EA(p). (2)

For particles with spin, the states and depend also on a spin variable, which we suppress to simplify the notation. We will also often omit the second argument of where this cannot cause confusion.

We choose the Lorentz invariant normalization condition for the plane wave states :

 ⟨A,p′|A,p⟩=2EA(p)(2π)3δ(3)(p−p′). (3)

The standard normalization of the states then implies

 ∫d3p(2π)3|fA(p)|2=1. (4)

The quantity is actually the momentum-space wave function of : . The time dependent wave function is , where and is the free Hamiltonian of . The coordinate-space wave function is the Lorentz-invariant Fourier transform of :222 Recall that the Fourier transformation is based on the completeness condition for 1-particle momentum eigenstates, which for our normalization convention reads (see, e.g., PeskSchr [], eq. (2.39)). Here the right hand side is the unit operator in the subspace of 1-particle states and zero in the rest of the Hilbert space. Note that the integration measure is Lorentz invariant.

 ΨA(t,x)≡⟨x|A(t)⟩=∫d3p(2π)32EA(p)⟨p|A(t)⟩eipx, (5)

or

 ΨA(t,x)=∫[dp]fA(p)e−iEA(p)t+ipx. (6)

In the QFT framework, it can be written as

 ΨA(x)=⟨0|^ΨA(x)|A⟩, (7)

where and is the second-quantized field operator of . Using the standard decomposition of the field in terms of production and annihilation operators, one can readily obtain (5) from (7) and (1). Note that expressions (5) and (6) can describe both bound states and propagating wave packets (in the case of bound states or particles propagating in a potential, the relation simply has to be replaced by the proper dispersion relation). A wave packet is obtained when the momentum distribution function is sharply peaked at or close to a nonzero mean momentum ,333The peak momentum coincides with the mean momentum for symmetric wave packets. i.e. when the momentum dispersion satisfies ; for the rest of this section we will assume this to be the case. The wave function (6) then describes a wave packet whose maximum of amplitude is located at at time . A wave packet that is peaked at coordinate at time is obtained by acting on the state by the space-time translation operator , where is the 4-momentum operator. For the coordinate-space wave function this yields

 ΨA(x)=∫[dp]fA(p)e−iEA(p)(t−t0)+ip(x−x0) (8)

Eqs. (6) and (8) represent wave packets that propagate with the group velocity and in general spread with time both in the longitudinal direction and in the directions transverse to their mean momentum. The spreading is due to the fact that different momentum components of the wave packet have slightly different velocities .

Let us now consider neutrino oscillations in the framework of the QM wave packet formalism, sometimes also called the “intermediate wave packet” approach. Neutrinos produced or absorbed in charged-current weak interaction processes are considered to be flavour eigenstates (, which are coherent linear superpositions of mass eigenstates ( with coefficients given by the elements of the leptonic mixing matrix . The mass eigenstates are represented by the corresponding wave packets. If a neutrino of flavour was produced at time at a source centered at , its momentum-space wave function at a time is

 ⟨p|ναP(t)⟩ =∑jU∗αj√2EA(p)fjP(p,P)e−iEj(p)(t−tP)−ipxP. (9)

Here the subscript shows that the wave packet corresponds to a neutrino produced at the source. Note that the index at simply indicates that the emitted neutrino was of flavour at its production time ; it is, of course, no longer so for . The shape of the wave packet of the th mass-eigenstate neutrino is given by the momentum distribution function , which is determined by the mechanism and conditions of neutrino production. In the QM framework, however, the neutrino production and detection processes are not explicitly taken into account; therefore the functions are postulated rather than determined, with the corresponding momentum widths estimated from the localization properties of the production process. Usually, the wave packets are taken to be of the Gaussian form

 (10)

where characterizes the momentum uncertainty of the produced neutrino state, and similarly for the state of the detected neutrino. The advantage of Gaussian wave packets is that they allow most calculations to be done analytically (the same is also true for Lorentzian wave packets, see ref. Kopp:2009fa []).

The state of the detected neutrino is described by a wave packet peaked at the detection coordinate . In the momentum-space representation it is given by

 ⟨p|νβD⟩ =∑kU∗βk√2EA(p)fkD(p,P′)e−ipxD, (11)

where the subscript stands for detection. The momentum distribution functions are governed by the properties of the detection process; however, just as for neutrino production, in the QM approach these functions are postulated rather than determined. Note that, although the assumption is adopted in most studies, in general there is no reason to expect the mean momenta of the produced and detected wave packets to coincide. We will discuss this point in more detail in Sec. 6.

The amplitude for the transition is obtained by projecting the evolved neutrino production state onto the detection state:

 Aαβ(T,L)=⟨νβD|ναP(tD)⟩, (12)

where is the detection time, and . Performing the projection in momentum space, we obtain from (9) and (11) 444Projection in momentum space will turn out to be convenient for our subsequent discussion. The coordinate-space projection yields, of course, the same result.

 Aαβ(T,L) =∑jU∗αjUβj∫d3p(2π)3fjP(p,P)f∗jD(p,P′)e−iEj(p)T+ipL. (13)

For future reference, we shall also write this as a superposition of the amplitudes corresponding to the contributions of different neutrino mass eigenstates:

 Aαβ(T,L)=∑jU∗αjUβjAj(T,L) (14)

with

 Aj(T,L)=∫d3p(2π)3fjP(p,P)f∗jD(p,P′)e−iEj(p)T+ipL. (15)

The oscillation probability is given by the squared modulus of the transition amplitude: . Since in most experiments the neutrino emission and detection times are not measured, the standard procedure is then to integrate over . This gives

 P(να→νβ,L)≡Pαβ(L)=∫dT|Aαβ(T,L)|2. (16)

Substituting here the transition amplitude (13) yields, up to a normalization factor, the standard probability of neutrino oscillations in vacuum provided that all decoherence effects are negligible. The normalization factor can then be fixed by requiring that the oscillation probability satisfy the unitarity condition (see Sec. 5 for a more detailed discussion).

## 3 Neutrino oscillations in QFT

In the QFT approach (which is sometimes also called the “external wave packet” formalism), neutrino production, propagation, and detection are considered as a single process, described by the Feynman diagram shown in fig. 1, with the neutrino in the intermediate state. In our overview of the QFT formalism we will mostly follow ref. Beuthe1 []. Assume that the neutrino production process involves one initial state and one final state particle (besides the neutrino). Likewise, we will assume that the detection process involves only one particle besides the neutrino in the initial state and one particle in the final state. The generalization to the case of an arbitrary number of particles involved in the neutrino production and detection processes is straightforward and would just complicate the formulas without providing further physical insight. All external particles will be assumed to be on their respective mass shells.555 Since only one particle is assumed to be in the initial state of the production process, it must be unstable. This will be of no importance for us here because, as was already mentioned, the results are easily generalized to the case of an arbitrary number of external particles. Possible instability of the parent particle will be taken into account in Sec. 4.4.

The states describing the particles accompanying neutrino production and detection (“external particles”) can be represented in the form (1). For the initial and final states at neutrino production we can write

 |Pi⟩=∫[dq]fPi(q,Q)|Pi,q⟩,|Pf⟩=∫[dk]fPf(k,K)|Pf,k⟩, (17)

and similarly for the states accompanying neutrino detection:

 |Di⟩=∫[dq′]fDi(q′,Q′)|Di,q′⟩,|Df⟩=∫[dk′]fDf(k′,K′)|Df,k′⟩. (18)

We assume these states to fulfill the normalization condition (4). Some (or all) of the mean momenta of the external particles , , and may vanish, i.e. the states in eqs. (17) and (18) can describe bound states at rest as well as wave packets.

The amplitude of the neutrino production - propagation - detection process is given by the matrix element

 iAαβ=⟨PfDf|^Texp[−i∫d4xHI(x)]−\mathbbm1|PiDi⟩, (19)

where is the time ordering operator and is the charged-current weak interaction Hamiltonian.666We consider neutrino production and detection at energies well below the -boson mass, so that is the effective 4-fermion Hamiltonian of weak interactions. Note that no neutrino flavour eigenstates have to be introduced in the QFT framework, and the indices and simply refer here to the flavour of the charged leptons participating in the production and detection processes.

From eq. (19) it is easy to calculate the transition amplitude in the lowest nontrivial (i.e. second) order in using the standard QFT methods. The resulting expression corresponds to the Feynman diagram of fig. 1 and can be written as

 iAαβ = ∑jU∗αjUβj∫[dq]fPi(q,Q)∫[dk]f∗Pf(k,K) (20) ×∫[dq′]fDi(q′,Q′)∫[dk′]f∗Df(k′,K′)iAp.w.j(q,k;q′,k′).

Here the sum runs over all intermediate states (i.e. different neutrino mass eigenstates), and the quantity is the plane-wave amplitude of the process with the th neutrino mass eigenstate propagating between the source and the detector:

 iAp.w.j(q,k;q′,k′)= ∫d4x1∫d4x2~MD(q′,k′)e−i(q′−k′)(x2−xD) ×i∫d4p(2π)4⧸p+mjp2−m2j+iϵe−ip(x2−x1)⋅~MP(q,k)e−i(q−k)(x1−xP). (21)

Here and are the 4-coordinates of the neutrino production and detection points, the quantities and are the plane-wave amplitudes of the processes and , respectively, with the neutrino spinors and excluded. The choice of the 4-coordinate dependent phase factors corresponds to the assumption that the peaks of the wave packets of particles involved in the production process are all located at at the time , whereas for the detection process the corresponding peaks are all situated at at the time (we will discuss in Sec. 6 how this assumption can be relaxed). The integral in the second line of eq. (21) gives the coordinate-space propagator of the th neutrino mass eigenstate.

It is convenient to switch to shifted 4-coordinate variables , , defined according to , . Taking into account that , one can then rewrite eq. (21) as

 iAp.w.j(q,k;q′,k′)=i∫d4p(2π)4e−ip(xD−xP)p2−m2j+iϵ∫d4 x′1√2p0MjP(q,k)e−i(q−k−p)x′1 ×∫d4 x′2√2p0MjD(q′,k′)e−i(q′+p−k′)x′2, (22)

where

 MjP(q,k)≡¯ujL(p)√2p0~MP(q,k)andMjD(q′,k′)≡~MD(q′,k′)ujL(p)√2p0 (23)

are the full amplitudes (with the neutrino spinors included) of the processes and , respectively, and we have taken into account that the matrix elements and involve the left-handed chirality projection, so that only the left-handed spinors and contribute to the sum over the neutrino spin variable .

Substituting (22) into eq. (20), we finally obtain

 iAαβ=i∑jU∗αjUβj∫d4p(2π)4ΦjP(p0,p)ΦjD(p0,p)2p0e−ip0T+ipLp2−m2j+iϵ. (24)

Here the so-called overlap functions and are defined as

 ΦjP(p0,p)= ∫d4x′1eipx′1∫[dq]∫[dk]fPi(q,Q)f∗Pf(k,K)e−i(q−k)x′1MjP(q,k), (25) ΦjD(p0,p)= ∫d4x′2e−ipx′2∫[dq′]∫[dk′]fDi(q′,Q′)f∗Df(k′,K′)e−i(q′−k′)x′2MjD(q′,k′).

Note that they are independent of and . Expressions (24) and (25) are the main results of the QFT-based approach to neutrino oscillations Giunti:2002xg [], Beuthe1 [].

## 4 Comparing the QM and QFT approaches to neutrino oscillations

Let us now compare the results of the QM and QFT approaches to neutrino oscillations. Consider first the transition amplitude (24) obtained in the QFT formalism. The integration over the neutrino 4-momentum in this expression can be done in different order. Here it will be more convenient for us to integrate first over and then over (the opposite order will be used in Sec. 5). Since the distance between the neutrino source and detector is macroscopic, the phase factor in the integrand of eq. (24) undergoes fast oscillations and the integral is strongly suppressed except when the intermediate neutrino is on the mass shell. Thus, the dominant contribution to the integral is given by the residue at the pole of the neutrino propagator at ,777The contribution of the pole at is strongly suppressed due to an approximate conservation of mean energies at production and the fact that . where

 Ej(p)=√p2+m2j. (26)

Eq. (24) can therefore be rewritten as

 iAαβ=Θ(T)∑jU∗αjUβj∫d3p(2π)3ΦjP(Ej(p),p)ΦjD(Ej(p),p)e−iEj(p)T+ipL. (27)

where is the Heaviside step function.

### 4.1 Deriving neutrino wave packets in the QFT-based approach

Let us now compare eqs. (27) and (13). We see that the two equations are of the same form and actually coincide if we identify the QM wave packets as

 fjP(p)=ΦjP(Ej(p),p),fjD(p)=Φ∗jD(Ej(p),p), (28)

where the functions and were defined in eq. (25).

The obtained result can be easily understood. Indeed, as follows from the definition of , for (i.e. on the mass shell of ) this quantity is the probability amplitude of the production process in which the th mass eigenstate neutrino is emitted with momentum ; but this is nothing but the momentum distribution function of the produced neutrino, i.e. the momentum-state wave packet . A similar argument applies to the neutrino detection process and . The wave packets and in eq. (28) are not normalized according to (4), though they can be easily modified to satisfy this condition. However, as we shall see in Sec. 5, this is not necessary and actually would be misleading.

An alternative method of deriving neutrino wave packets in the QFT framework, based on the S-matrix approach, was suggested in Giunti:2002xg []; the obtained results are equivalent to those in eqs. (28) and (25).

Let us now consider the wave packet describing the produced neutrino state in more detail (the state of the detected neutrino can be studied quite analogously). According to (28), the momentum distribution function characterizing the state of the emitted neutrino of mass is essentially given by the on-shell function . Since the matrix element is a smooth function of the on-shell 4-momenta and , whereas the wave packets of the external states are assumed to be sharply peaked at or near the corresponding mean momenta, one can replace by its value at the mean momenta and pull it out of the integral. Eqs. (25) and (28) then yield

 fjP(p)≃MjP(Q,K)∫d4xeiEj(p)t−ipx∫[dq]∫[dk]fPi(q,Q)f∗Pf(k,K)e−i(q−k)x, (29)

where the 4-momenta and are defined as

 Q=(EPi(Q),Q),K=(EPf(K),K). (30)

From eq. (29) (or eqs. (25) and (28)) one can draw some important conclusions about the properties of the neutrino momentum distribution functions which determine the emitted neutrino wave packets:

• Since the quantities and depend only on the properties of the external particles, and the -dependence of the matrix elements comes through the on-shell neutrino spinor factors , which depend on only through the neutrino energy, the functions depend on the index solely through the neutrino energy . This, in particular, means that for ultra-relativistic or quasi-degenerate in mass neutrinos the momentum distribution functions of all neutrino mass eigenstates are practically the same (provided that their energy differences are small compared to the energy uncertainty ).

• Because the integral over the 3-coordinate in eq. (29) yields , and the momentum distribution functions and are sharply peaked at or near their respective mean momenta and , the neutrino momentum distribution functions are sharply peaked at or close to the momentum , with the width of the peak dominated by the largest between the momentum uncertainties of the states of and .

Taking into account eq. (5), eq. (29) can be rewritten as

 fjP(p)≃MjP(Q,K)∫d4xeipxΨPi(x)Ψ∗Pf(x)∣∣p0=Ej(p). (31)

Thus, the momentum distribution function that determines the wave packet of the emitted neutrino is essentially the 4-dimensional Fourier transform of the product of the coordinate-space wave functions of the external particles participating in the neutrino production process, taken under the condition that the components of the neutrino 4-momentum are on the mass shell. Eq. (31) can be readily generalized to the case when more than two external particles participate in the neutrino production process: the expression in the integrand of (31) should simply be replaced by the product of the wave functions of all particles in the initial state of the production process and of complex conjugates of the wave functions of all particles in the final state (except the neutrino).

The neutrino wave packet in coordinate space is obtained from eq. (31) by performing the Fourier transformation over the 3-momentum variable according to the transformation law (6), which gives

 ψjP(x)≃ ¯uj(P,s)~MjP(Q,K)∫d4x′ΨPi(x′)Ψ∗Pf(x′) ×Θ(t−t′)|x−x′|−i2(2π)2∫∞0dpp√p2+m2je−i√p2+m2j(t−t′)[eip|x−x′|−e−ip|x−x′|]. (32)

Here and we have used eq. (23) to extract the -dependent factor from . The integral over in eq. (32) can be expressed in terms of the modified Bessel function Gradshtein [], giving

 ψjP(x)≃¯uj(P,s)~MjP(Q,K)(2π)2∫d4x′ΨPi(x′)Ψ∗Pf(x′)mjΘ(t−t′)√−(x−x′)2K1(mj√−(x−x′)2), (33)

where . The integral in the second line of eq. (32) is greatly simplified in the limit of vanishing neutrino mass:

 ψjP(x)≃¯uj(P,s)~MjP(Q,K)−1(2π)2∫d4x′ΨPi(x′)Ψ∗Pf(x′)Θ(t−t′)(x−x′)2. (34)

Note, however, that in this limit all neutrino species travel with the same speed and therefore the wave functions in eq. (34) cannot describe decoherence due to the separation of wave packets. In order to take possible wave packet separation effects into account the more accurate expression (33) has to be used. Alternatively, one can employ the momentum-representation wave function (31).

Expression (33) for the wave function of the produced neutrino state allows a simple interpretation. Note that is the scalar retarded propagator in the coordinate representation. Therefore the neutrino wave packet (33) is essentially the convolution of the neutrino source (the role of which is played by the neutrino production amplitude ) with the retarded neutrino propagator, in full agreement with the well known result of QFT. Note that only the scalar part of the propagator contributes to ; this is because the coordinate space and momentum space neutrino wave functions and are scalars in our formalism. The spinor factors are included in the matrix elements and (note that these quantities are also scalar, whereas the amputated matrix elements and , i.e. those with the neutrino spinors removed, have spinorial indices).

In our discussion of the wave packets of the emitted neutrino states, we were assuming that the momentum distribution functions of all the external particles accompanying neutrino production are known. This implies, in particular, that all particles in the final state of the production process are “measured”, either by direct detection or through their interaction with the medium in the process of neutrino production. It is quite possible, however, that some of the particles accompanying neutrino production escape undetected; this is, e.g., the case for atmospheric or accelerator neutrinos born in the process , in which the final state muon is normally not detected. It is also possible that some of the particles accompanying neutrino detection are “unmeasured”. How can one determine the neutrino wave packets in those cases?

To answer this question, let us recall that the momentum uncertainty characterizing the emitted neutrino depends in general on the momentum uncertainties of all the external particles at neutrino production and is dominated by the largest among them (see the discussion after eqs. (29) and (30)). In particular, in the case of Gaussian wave packets, one has Giunti:2002xg [], Beuthe1 []

 σ2pP=σ2pPi+σ2pPf. (35)

For more than two external particles at production, the sum on the right-hand side of this relation would contain the contributions of the squared momentum uncertainties of all these particles. Now, if a particle goes “unmeasured” in the neutrino production process, its momentum uncertainty cannot affect the momentum uncertainty of the emitted neutrino state and therefore can be neglected. To put it differently, undetected particles are completely delocalized, and therefore, according to Heisenberg’s uncertainty relation, have vanishing momentum uncertainty. This means that undetected particles can be represented by states of definite momenta, i.e. by plane waves. If, for example, the particle at production is undetected, one has to replace in eq. (29) the momentum distribution function by where is the normalization volume, and in eq. (31) the coordinate-space wave function by , with eqs. (32) - (34) modified accordingly. The mean momentum of the neutrino state depends, of course, on the momentum of the undetected particle; if the latter can take values in some range, the same will be true for the mean momentum of the emitted neutrino state. In this case the flux of emitted neutrinos will be characterized by a continuous spectrum.

In most of our discussion in this subsection we concentrated on the wave packets of the produced neutrino states. Our consideration, however, applies practically without changes to the detected neutrino states; the corresponding formulas can be obtained from eqs. (29) and (31)-(34) by replacing, where appropriate, , and .

### 4.2 General properties of neutrino wave packets

We have already considered some of the general properties of the neutrino wave packets in the previous subsection. In particular, we have found that the momentum distribution functions of mass-eigenstate neutrinos depend on the index only through the neutrino energy , and that the functions are sharply peaked at or near the momentum , with the width of the peak dominated by the largest between the widths of the functions and . Further insight into the general properties of the neutrino wave packets can be gained by comparing expressions (25) with their plane-wave limits. If the external particles were described by plane waves, the quantities and , which determine the neutrino wave packets, would be just equal to the matrix elements of the neutrino production or detection processes divided by the factor for each external particle and multiplied, correspondingly, by . The latter factors represent energy-momentum conservation at the production and detection vertices. As follows from (25), in the case when the external particles are described by wave packets, the quantities and (and therefore the momentum distribution functions of the neutrino wave packets) correspond to “smeared -functions”, representing approximate conservation of the mean energies and mean momenta of the participating particles. How exactly this smearing occurs will depend on the form of the wave packets of the external particles, and to move ahead one has to specify this form.

A particularly useful and illuminating example of a specific form of the external wave packets, and the one most often used in the literature, is the case of Gaussian wave packets. We will employ this example to illustrate the general properties of the neutrino wave packets.

Let us discuss first the conditions under which an arbitrary wave packet can be accurately approximated by a Gaussian one. For simplicity, we will consider here 1-dimensional wave packets. This is a good approximation in the case when the distance between the neutrino source and detector is very large compared to their sizes, so that the neutrino momentum is practically collinear with (the generalization to the 3-dimensional case is straightforward). Consider a wave packet described by a momentum distribution function , sharply peaked at some value of the momentum. We can write this function in the exponential form as

 f(p)=e−g(p),whereg(p)=−ln[f(p)]. (36)

The Gaussian approximation corresponds to the case when the integral over of the function multiplied by any function of that is smooth in the vicinity of can be evaluated in the saddle point approximation. Indeed, in this approach one expands the function around its minimum at and keeps the terms up to and including the quadratic one:

 g(p)≃g(P0)+12g′′(P0)(p−P0)2. (37)

This precisely means the wave packet is approximated by the Gaussian one. The validity condition for this approximation is given in terms of the derivatives of the function at :

 14!|g(IV)(P0)|≪12|g′′(P0)|2. (38)

It can be satisfied for a wide range of functions . However, it is easy to construct wave packets for which it is not satisfied. Consider, e.g. a class of wave packets

 f(p)=Cn[(p−P0)2+γ2]n (39)

with integer and a constant, which can be found from the normalization condition for . It is easy to check that condition (38) is equivalent to . Thus, the momentum distribution functions (39) can be accurately approximated by the Gaussian ones only when . This condition, in particular, is not satisfied for Lorentzian wave packets.

### 4.3 Matching the QFT and QM neutrino wave packets

Let us now discuss how one can match the QFT and QM wave packets of neutrinos. Using the case of Gaussian wave packets as an example, we shall find out how the effective parameters describing the QM wave packets can be expressed in terms of the more fundamental parameters entering into the QFT approach.

We start by introducing some notation (we mostly follow Giunti:2002xg [], Beuthe1 [] here). The coordinate uncertainty characterizing the wave function of the initial state particle in the neutrino production process is related to its momentum uncertainty by

 σxPiσpPi=12, (40)

and similarly for all other external particles. One can also introduce the effective coordinate uncertainty of the production process , which is connected to the effective momentum uncertainty of this process defined in eq. (35) by a relation similar to (40), or equivalently

 1σ2xP=1σ2xPi+1σ2xPf. (41)

This formula has a simple physical interpretation: since the neutrino production process requires an overlap of the wave functions of all the participating particles, the effective uncertainty of the coordinate of the production point is determined by the particle with the smallest coordinate uncertainty. This is in accord with the already discussed fact that the effective momentum uncertainty at production , which determines the momentum uncertainty of the produced neutrino, is dominated by the largest among the momentum uncertainties of all the external particles involved in neutrino production.

Next, we define the effective velocity of the neutrino source and its effective squared velocity as

 vP≡σ2xP⎛⎝vPiσ2xPi+vPfσ2xPf⎞⎠,ΣP≡σ2xP⎛⎝v2Piσ2xPi+v2Pfσ2xPf⎞⎠. (42)

If , they are approximately equal to, respectively, the velocity and squared velocity of the particle with the smallest coordinate uncertainty. We will also need the quantity defined through

 σ2eP=σ2pP(ΣP−v2P)≡σ2pPλP. (43)

This quantity can be interpreted as the effective energy uncertainty at neutrino production Beuthe1 []. It can be also shown that , i.e. .

We can now discuss the results obtained in the QFT framework in the case when the external particles are represented by Gaussian wave packets. The function , which coincides with the momentum distribution function of the emitted mass-eigenstate neutrino , can be written as Giunti:2002xg [], Beuthe1 []

 ΦjP(Ej(p),p)=NPMjP(Q,K)1σePσ3pPexp[−gP(Ej(p),p)], (44)

where

 NP=π2(2πσ2xPi)3/4(2πσ2xPf)3/4[2EPi(Q)⋅2EPf(K)]1/2 (45)

is the normalization factor and

 gP(Ej(p),p)=(p−P)24σ2pP+[Ej(p)−EP−vP(p−P)]24σ2eP. (46)

Here

 P≡Q−K,EP≡EPi(Q)−EPf(K), (47)

and was defined in eq. (26). Note that in the limit when the external particles are represented by plane waves (, ), the first equation in (25) yields

 ΦjP(Ej(p),p)=(2π)4δ[EPi(Q)−EPf(K)−Ej(P)]δ(3)(Q−K−P) ×MjP(Q,K)√2EPi(Q)V⋅2EPf(K)V, (48)

as discussed in the previous subsection. From eq. (35) and the fact that it follows that in this limit the momentum uncertainty and energy uncertainty of the produced neutrino state vanish as well; thus, if the external particles are described by plane waves, then so is the produced neutrino. As follows from eq. (48), the plane wave limit corresponds to exact energy and momentum conservation at production.888Since the neutrino energy and momentum are completely determined by those of the external particles in this case, only one neutrino mass eigenstate can be produced in any given interaction process, and therefore no oscillations are possible in the plane wave limit Beuthe1 []. This can also be seen from eqs. (44) and (46): indeed, in the limit the right hand side of (44) is proportional to the product of the energy and momentum conserving -functions. For finite values of and , eqs. (44) and (46) yield Gaussian-type “smeared delta functions”, i.e. describe approximate conservation laws for the mean momenta and mean energies of the wave packets, for which are responsible, respectively, the first term and the second term in (46).

Let us now try to cast expressions (44) and (46) into the form usually adopted in the QM wave packet approach to neutrino oscillations. We want to reduce to an expression similar to that in eq. (10). To this end, we expand the neutrino energy around the point and keep terms up to the second order:

 Ej(p)≃Ej+vj(p−P)+12Ej(δkl−vkjvlj)(p−P)k(p−P)l. (49)

Here