paradoxes of neutrino oscillations...it relies on the energy-momentum conservation and the...

ISSN 1063-7788, Physics of Atomic Nuclei, 2009, Vol. 72, No. 8, pp. 1363–1381. c© Pleiades Publishing, Ltd., 2009.

ELEMENTARY PARTICLES AND FIELDSTheory

Paradoxes of Neutrino Oscillations*

E. Kh. Akhmedov1), 2)** and A. Yu. Smirnov3), 4)***

Received January 19, 2009

Abstract—Despite the theory of neutrino oscillations being rather old, some of its basic issues arestill being debated in the literature. We discuss a number of such issues, including the relevance ofthe “same energy” and “same momentum” assumptions, the role of quantum-mechanical uncertaintyrelations in neutrino oscillations, the dependence of the coherence and localization conditions that ensurethe observability of neutrino oscillations on neutrino energy and momentum uncertainties, the questionof (in)dependence of the oscillation probabilities on the neutrino production and detection processes,and the applicability limits of the stationary-source approximation. We also develop a novel approach tocalculation of the oscillation probability in the wave-packet approach, based on the summation/integrationconventions different from the standard one, which allows a new insight into the “same energy” vs. “samemomentum” problem. We also discuss a number of apparently paradoxical features of the theory of neutrinooscillations.

PACS numbers: 14.60.PqDOI: 10.1134/S1063778809080122

1. INTRODUCTION

More than 50 years have already passed since theidea of neutrino oscillations was put forward [1, 2],and over 10 years have passed since the experimentaldiscovery of this phenomenon (see, e.g., [3] for re-views). However, surprisingly enough, a number ofbasic issues of the neutrino oscillation theory are stillbeing debated. Moreover, some features of the theoryappear rather paradoxical. Among the issues that arestill under discussion are:

(1) Why do the often used same-energy andsame-momentum assumptions for neutrino masseigenstates composing a given flavor state, which areknown to be both wrong, lead to the correct result forthe oscillation probability?

(2) What is the role of quantum-mechanical un-certainty relations in neutrino oscillations?

(3) What determines the size of the neutrino wavepackets?

∗The text was submitted by the authors in English.1)Max-Planck-Institut fur Kernphysik, Heidelberg, Germany.2)Russian Research Centre “Kurchatov Institute,” Moscow,

Russia.3)The Abdus Salam International Centre for Theoretical

Physics, Trieste, Italy.4)Institute for Nuclear Research, Russian Academy of Sci-

ences, Moscow, Russia.**E-mail: [email protected]

***E-mail: [email protected]

(4) How do the coherence and localization con-ditions that ensure the observability of neutrino os-cillations depend on neutrino energy and momentumuncertainties?

(5) Are wave packets actually necessary for a con-sistent description of neutrino oscillations?

(6) When can the oscillations be described by auniversal (i.e., production and detection process in-dependent) probability?

(7) When is the stationary-source approximationvalid?

(8) Would recoillessly emitted and absorbed neu-trinos (produced and detected in Mossbauer-type ex-periments) oscillate?

(9) Are oscillations of charged leptons possible?In the present paper we consider the first seven is-

sues listed above, trying to look at them from differentperspectives. We hope that our discussion will helpclarify these points and finally put them to rest. Forthe last two issues, we refer the reader to the recentdiscussions in [4–6] (for oscillations of Mossbauerneutrinos) and [7] (for oscillations of charged lep-tons).

2. SAME ENERGY OR SAME MOMENTUM?

In most derivations of the so-called standard for-mula for the probability of neutrino oscillations invacuum (see Eq. (6) below), usually the assumptionsthat the neutrino mass eigenstates composing a given

1363

1364 AKHMEDOV, SMIRNOV

flavor eigenstate either have the same momentum [8–11] or the same energy [12–15] are made. The deriva-tion typically proceeds as follows.

First, recall that in the basis in which the massmatrix of charged leptons has been diagonalized thefields describing the massive neutrinos νi and flavor-eigenstate neutrinos νa and the corresponding states|νi〉 and |νa〉 are related by

νa =∑

i

Uaiνi, |νa〉 =∑

i

U∗ai|νi〉, (1)

where U is the leptonic mixing matrix. If one nowassumes that all the mass eigenstates composing theinitially produced flavor state |ν(0)〉 = |νa〉 have thesame momentum, then, after time t has elapsed, theith mass eigenstate will simply pick up the phasefactor exp(−iEit), and the evolved state |ν(t)〉 will begiven by

|ν(t)〉 =∑

i

U∗aie

−iEit|νi〉. (2)

Projecting this state onto the flavor state |νb〉 andtaking the squared modulus of the resulting transitionamplitude, one gets the probability of the neutrinoflavor transition νa → νb after the time interval t:

P (νa → νb; t) =

∣∣∣∣∣∑

i

Ubie−iEitU∗

ai

∣∣∣∣∣

2

. (3)

Next, taking into account that the energy Ei of arelativistic neutrino of mass mi and momentum p is

Ei =√

p2 + m2i � p +

m2i

2p, (4)

and that for relativistic pointlike particles the distanceL they propagate during the time interval t satisfies

L � t, (5)

one finally finds

P (νa → νb;L) (6)

=

∣∣∣∣∣∑

i

Ubi exp

(−i

∆m2ij

2pL

)U∗

ai

∣∣∣∣∣

2

,

where ∆m2ij = m2

i − m2j and the index j corresponds

to any of the mass eigenstates. This is the standardformula describing neutrino oscillations in vacuum.Note that in this approach neutrino states actuallyevolve only in time (see Eq. (3)); the usual coordinatedependence of the oscillation probability (6) is onlyobtained by invoking the additional “time-to-spaceconversion” assumption (5). Without this conversion,one would come to a paradoxical conclusion thatneutrino oscillations could be observed by just putting

the neutrino detector immediately next to the sourceand waiting long enough.

Likewise, one could assume that all the mass-eigenstate neutrinos composing the initially producedflavor state |νa〉 have the same energy. Using the factthat the spatial propagation of the ith mass eigenstateis described by the phase factor eipix and that for arelativistic neutrino of mass mi and energy E

pi =√

E2 − m2i � E − m2

i /2E, (7)

one again comes to the same standard formula (6) forthe oscillation probability. Note that in this case theneutrino flavor evolution occurs in space and it is notnecessary to invoke the “time-to-space conversion”relation (5) to obtain the standard oscillation formula.

The above two alternative derivations of the oscil-lation probability are very simple and transparent, andthey allow one to arrive very quickly at the desiredresult. The trouble with them is that they are bothwrong.

In general, there is no reason whatsoever to as-sume that different mass eigenstates composing aflavor neutrino state emitted or absorbed in a weak-interaction process have either the same energy or thesame momentum.5) Indeed, the energies and momen-ta of particles emitted in any process are dictated bythe kinematics of the process and by the experimentalconditions. Direct analysis of, e.g., 2-body decayswith simple kinematics, such as π± → l± + νl(νl),where l = e, µ, allows to find the 4-momenta of theemitted particles and shows that neither energies normomenta of the different neutrino mass eigenstatescomposing the flavor state νl are the same [16, 17].One might question this argument on the basis thatit relies on the energy-momentum conservation andthe assumption that the energies and momenta of theemitted mass eigenstates have well-defined (sharp)values, whereas in reality these quantities have in-trinsic quantum-mechanical uncertainties (see thediscussion in Section 5.1). However, the inexactnessof the neutrino energies and momenta does not in-validate our argument that the “same energy” and“same momentum” assumptions are unjustified, andin fact only strengthens it. It should be also noted thatthe “same energy” assumption actually contradictsLorentz invariance: even if it were satisfied in somereference frame (which is possible for two neutrinomass eigenstates, but not in the 3-species case), itwould be violated in different Lorentz frames [17, 18].The same applies to the “same momentum” assump-tion.

5)The only exception we are aware of are neutrinos produced ordetected in hypothetical Mossbauer-type experiments, sincefor them the “same energy” assumption is indeed justifiedfairly well.

PHYSICS OF ATOMIC NUCLEI Vol. 72 No. 8 2009

PARADOXES OF NEUTRINO OSCILLATIONS 1365

3. WHY WRONG ASSUMPTIONS LEADTO THE CORRECT RESULT?

One may naturally wonder why two completelydifferent and wrong assumptions (“same energy” and“same momentum”) lead to exactly the same andcorrect result—the standard oscillation formula. Tounderstand that, it is necessary to consider the wave-packet picture of neutrino oscillations.

3.1. Wave-Packets Approach to NeutrinoOscillations

In the discussion in the previous section we wereactually considering neutrinos as plane waves or sta-tionary states; strictly speaking, this description wasinconsistent because such states are in fact non-propagating. Indeed, the probability of finding a par-ticle described by a plane wave does not depend onthe coordinate, while for stationary states this prob-ability does not depend on time. In quantum theorypropagating particles must be described by movingwave packets (see, e.g., [19]). Let us recall the mainfeatures of this approach [20–30].

Let a flavor eigenstate νa be born at time t = 0 in asource centered at x = 0. The wave packet describingthe evolved neutrino state at a point with the coordi-nates (t,x) is then

|νa(x, t)〉 =∑

i

U∗aiΨi(x, t)|νi〉. (8)

Here Ψi(x, t) is wave packet describing a free propa-gating neutrino of mass mi:

Ψi(x, t) =∫

d3p

(2π)3/2fS

i (p − pi)eipx−iEi(p)t, (9)

where fSi (p− pi) is the momentum distribution

function with pi being the mean momentum, and

Ei(p) =√

p2 + m2i . The superscript “S” at fS

i (p −pi) indicates that the wave packet corresponds tothe neutrino produced in the source. We will assumethe function fS

i (p − pi) to be sharply peaked at orvery close to zero of its argument (p = pi), with thewidth of the peak σp � pi.6) No further propertiesof fS

i (p − pi) need to be specified. If fSi (p − pi) is

normalized according to∫

d3p|fSi (p − pi)|2 = 1, (10)

6)For symmetric wave packets (i.e., when fSi (p − pi) is an

even function of its argument), the position of the centerof the peak coincides with the mean momentum pi. Forasymmetric wave packets, it may be displaced from pi.

then the wave function Ψi(x, t) has the standardnormalization

∫d3x|Ψi(x, t)|2 = 1. We will, however,

use a different normalization convention, which yieldsthe correct normalization of the oscillation probability(see Eq. (20) below). Expanding Ei(p) around themean momentum,

Ei(p) = Ei(pi) +∂Ei(p)

∂pj

∣∣∣∣pi

(p − pi)j (11)

+12

∂2Ei(p)∂pj∂pk

∣∣∣∣pi

(p − pi)j(p − pi)k + . . . ,

one can rewrite Eq. (9) as

Ψi(x, t) � eipix−iEi(pi)tgSi (x − vgit), (12)

where

gSi (x − vgit) =

∫d3p

(2π)3/2fS

i (p)eip(x−vgit) (13)

is the shape factor and

vgi =∂Ei

∂p

∣∣∣∣pi

=pEi

∣∣∣pi

(14)

is the group velocity of the wave packet. Here we haveretained only the first and the second terms in the ex-pansion (11), since the higher order terms are of sec-ond and higher order in the small neutrino mass andso can be safely neglected in practically all situationsof interest. This approximation actually preserves theshape of the wave packets (and, in particular, neglectstheir spread). Indeed, the shape factor (13) dependson time and coordinate only through the combination(x − vgit); this means that the wave packet prop-agates with the velocity vgi without changing itsshape.

If the momentum dispersion corresponding to themomentum distribution function fS

i (p− p0) is σp,then, according to the Heisenberg uncertainty rela-tion, the length of the wave packet in the coordinatespace σx satisfies σx � σ−1

p ; the shape-factor func-tion gS

i (x − vgit) decreases rapidly when |x − vgit|exceeds σx. Equation (12) actually justifies and cor-rects the plane-wave approach: the wave function of apropagating mass-eigenstate neutrino is described bythe plane wave corresponding to the mean momen-tum pi, multiplied by the shape function gS

i (x− vgit)which makes sure that the wave is strongly sup-pressed outside a finite space–time region of width σx

around the point x = vgit.

In the approximation, where the wave packetspread is neglected, the evolved neutrino state is givenby Eq. (8) with Ψi(x, t) from Eq. (12). Note thatthe wave packets corresponding to different neutrino



mass eigenstates νi are in general described by dif-ferent momentum distribution functions fS

i (p − pi)and therefore by different shape factors gS

i (x − vgit).

Next, we define the state of the detected flavorneutrino νb as a wave packet peaked at the coordinateL of the detector:

|νb(x − L)〉 =∑

i

U∗biΨ

Di (x − L)|νi〉. (15)

It will be convenient for us to rewrite ΨDi (x − L)

pulling out the factor eipi(x−L):

|νb(x − L)〉 (16)

=∑

i

U∗bi[e

ipi(x−L)gDi (x − L)]|νi〉.

Here gDi (x− L) is the shape factor of the wave packet

corresponding to the detection of the ith mass eigen-state. The transition amplitude Aab(L, t) is obtainedby projecting the evolved state (8) onto (15):

Aab(L, t) =∫

d3x〈νb(x − L)|νa(x, t)〉 (17)

=∑

i

U∗aiUbi

∫d3xΨD∗

i (x− L)Ψi(x, t).

Substituting here expressions (12) and (16) yields

Aab(L, t) (18)

=∑

i

U∗aiUbiGi(L − vgit)e−iEi(pi)t+ipiL,

where

Gi(L − vgit) (19)

=∫

d3xgSi (x − vgit)gD∗

i (x − L).

That this integral indeed depends on L and vgit onlythrough the combination (L − vgit) can be easilyshown by shifting the integration variable in (19).Gi(L − vgit) is an effective shape factor whose widthσx depends on the widths of both the productionand detection wave packets σxS and σxD. Indeed,since the moduli of the shape-factor functions gS,D

iquickly decrease when the argument exceeds the cor-responding wave-packet widths σxS or σxD, fromEq. (19) it follows that the function Gi(L − vgit)decreases when |L − vgit| becomes large comparedto max{σxS , σxD}. Actually, since σx characterizesthe overlap of the wave packets describing the pro-duction and detection states, it exceeds both σxS andσxD. In particular, for Gaussian and Lorentzian (inthe coordinate space) wave packets one has σx =√

σ2xS + σ2

xD and σx = σxS + σxD, respectively. If the

production and detection wave packets are symmet-ric in the coordinate space, i.e., the shape factorsgSi (x − vgit) and gD

i (x − L) are even functions oftheir arguments, then so is Gi(L − vgit). In that casethe modulus of Gi(L − vgit) reaches its maximum atL = vgit; otherwise the maximum may be displacedfrom this point.

The probability Pab(L, t) ≡ P (νa → νb;L, t) offinding a flavor eigenstate neutrino νb at the detectorsite at time t is given by the squared modulus ofthe amplitude Aab(L, t) defined in Eq. (17). Since inmost experiments the neutrino emission and arrivaltimes are not measured, the standard procedure inthe wave-packet approach to neutrino oscillationsis then to integrate Pab(L, t) over time. In doing soone has to introduce a normalization factor which isusually not calculated,7) and in fact is determinedby imposing “by hand” the requirement that theprobabilities Pab(L) satisfy the unitarity condition.This is an ad hoc procedure which is not entirelyconsistent; the proper treatment would require toconsider the temporal response function of the de-tector and would automatically lead to the correctnormalization of the oscillation probabilities. Sincewe are primarily interested here in the oscillationphase which is practically insensitive to the detectorresponse function, we follow the same procedure ofintegration over time. The proper normalization ofthe oscillation probability is achieved by imposing thenormalization condition

∞∫

−∞

dt|Gi(L − vgit)|2 = 1. (20)

For simplicity, from now on we neglect the trans-verse components of the neutrino momentum, i.e.,the components orthogonal to the line connecting thecenters of the neutrino source and detector; this isa very good approximation for neutrinos propagatingmacroscopic distances. The probability of finding a νb

at the detector site, provided that a νa was emitted bythe source at the distance L from the detector, is then

Pab(L) =

∞∫

−∞

dt|Aab(L, t)|2 (21)

=∑

i,k

U∗aiUbiUakU∗

bkIik(L),

where

Iik(L) ≡∞∫

−∞

dtGi(L − vgit) (22)

7)For an exception, see [26].



× G∗k(L − vgkt)e−i∆φik(L,t).

Here, Gi(L − vgit) is the effective shape factor corre-sponding to the ith neutrino mass eigenstate definedin (the 1-dimensional version of) Eq. (19). The quan-tity ∆φik(L, t) is the phase differences between theith and kth mass eigenstates:

∆φik = (Ei − Ek)t − (pi − pk)L (23)

≡ ∆Eikt − ∆pikL,

where

Ei =√

p2i + m2

i . (24)

Note that this phase difference is Lorentz invariant.To calculate the observable quantities—the num-

bers of the neutrino detection events—one has tointegrate the oscillation probability (folded with thesource spectrum, detection cross section and detectorefficiency and energy resolution functions) over theneutrino spectrum and over the macroscopic sizes ofthe neutrino source and detector.

3.2. Oscillation Phase: the Answer to the Question

We are now in a position to present our argument.To simplify the notation, we suppress the indices iand k, where it cannot cause a confusion, so that∆E ≡ ∆Eik, ∆m2 ≡ ∆m2

ik, etc. Consider the case∆E � E which corresponds to relativistic or quasi-degenerate neutrinos. In this case one can expand thedifference of the energies of two mass eigenstates inthe differences of their momenta and masses. Retain-ing only the leading terms in this expansion, one gets

∆E =∂E

∂p∆p +

∂E

∂m2∆m2 (25)

= vg∆p +1

2E∆m2,

where vg is the average group velocity of the two masseigenstates and E is the average energy. Substitutingthis into Eq. (23) yields [31]

∆φ =∆m2

2Et − (L − vgt)∆p. (26)

Note that our use of the mean group velocity andmean energy of the two mass eigenstates in Eq. (26)is fully legitimate. Indeed, going beyond this approx-imation would mean retaining terms of second andhigher order in ∆m2 in the expression for ∆φ. Theseterms are small compared to the leading O(∆m2)terms; moreover, though their contribution to ∆φ canbecome of order one at extremely long distances, theleading contribution to ∆φ is then much greater thanone, which means that neutrino oscillations are in

the averaging regime and the precise value of theoscillation phase is irrelevant.

Let us now consider expression (26) for the phasedifference ∆φ. If one adopts the same-momentumassumption for the mean momenta of the wave pack-ets, ∆p = 0, the second term on the right-hand sidedisappears, which leads to the standard oscillationphase in the “evolution in time” picture. If, in addition,one assumes the “time-to-space conversion” rela-tion (5), the standard formula for neutrino oscillationsis obtained.

Alternatively, instead of expanding the energy dif-ference of two mass eigenstates in the differencesof their momenta and masses, one can expand themomentum difference of these states in the differencesof their energies and masses:

∆p =∂p

∂E∆E +

∂p

∂m2∆m2 (27)

=1vg

∆E − 12p

∆m2,

where p is the average momentum. Substituting thisinto Eq. (23) yields [32]

∆φ =∆m2

2pL − 1

vg(L − vgt)∆E. (28)

Note that this relation could also be obtained directlyfrom Eq. (26) by making use of Eq. (25). If one nowadopts the same energy assumption for the meanenergies of the wave packets, ∆E = 0, the secondterm on the right-hand side vanishes, and one arrivesat the standard oscillation formula.

However, as we shall show now, Eqs. (26) and (28)actually lead to the standard oscillation phase evenwithout the same-energy or same-momentum as-sumptions. For this purpose, let us generically writeEqs. (26) and (28) in the form

∆φ = ∆φst + ∆φ′, (29)

where ∆φst is the standard oscillation phase eitherin “evolution in time” or in “evolution in space” ap-proach, and ∆φ′ is the additional term (the secondterm in Eq. (26) or (28)). The first thing to notice isthat ∆φ′ vanishes not only when ∆p = 0 (in Eq. (26))or ∆E = 0 (in Eq. (28)), but also at the center of thewave packet, where L = vgt. Away from the center,the quantity L − vgt does not vanish, but it never ex-ceeds substantially the length of the wave packet σx,since otherwise the shape factors would strongly sup-press the neutrino wave function; thus, |L − vgt| �σx. The physical meaning of the two terms in Eq. (29)is then clear: ∆φst is the oscillation phase acquiredover the distance L between the centers of the neu-trino emitter and absorber wave functions, whereas∆φ′ is the additional phase variation along of the wave



packet. Note that in the wave-packet approach theeffective spatial length of the neutrino wave packetis determined by the sizes of the neutrino productionand detection regions8), so that the phase ∆φ′ isrelated to the condition of localization of the neutrinoemitter and absorber. In what follows we shall showthat explicitly.

Since ∆E and ∆p are the differences of, respec-tively, the mean energies and mean momenta of differ-ent neutrino mass eigenstates, for relativistic neutri-nos they are both of the order of ∆m2/2E or smaller;recalling that the neutrino oscillation length is losc =4πE/∆m2, we conclude that the additional phase∆φ′ can be safely neglected (and the oscillation phasetakes its standard value) when the effective length ofthe wave packets is small compared to the neutrinooscillation length, i.e., σx � losc.

We shall show now that under very general as-sumptions the oscillation phase takes the standardvalue. For this we need the wave-packet treatment ofneutrino oscillations. We start with the expression for∆φ in Eq. (28). Substituting it into (22), we find

Iik(L) = exp(−i

∆m2ik

2pL

)(30)

×∞∫

−∞

dtGi(L − vgit)G∗k(L − vgkt)

× exp(

i1vg

∆Eik(L − vgt))

.

Let us first neglect the difference between the groupvelocities of the wave packets describing differentmass eigenstates, i.e., take vgi = vgk = vg and Gk =Gi. In this approximation (which neglects the deco-herence effects due to the wave-packet separation,see Appendix), the integral in (30) does not dependon L; this can be readily shown by changing the in-tegration variable according to t → (L − vgt). Thus,in the case when the difference of the group veloc-ities of different mass eigenstates is negligible, wearrive at the standard oscillation phase. The integralin Eq. (30) is then just the Fourier transform of thesquared modulus of the shape factor:

1vg

∞∫

−∞

dx′|Gi(x′)|2 exp(

i1vg

∆Eikx′)

. (31)

8)In the more general quantum field theoretic framework thelength of the neutrino wave packet is related to the timescales of the neutrino emission and detection processesrather than to the localization properties of the neutrinoemitter and absorber, see Section 5.4.

It is essentially a localization factor, which takes intoaccount effects of suppression of oscillations in thecase when the localization regions of the neutrinoemitter and/or absorber are not small compared to theneutrino oscillation length (see Section 5.2 and Ap-pendix for a more detailed discussion). If localizationcondition (A.8) is fulfilled, one can replace the oscil-lating phase factor in the integrand by unity; the re-sulting integral is then simply the normalization fac-tor, which has to be set to 1 in order for the oscillationprobability to be properly normalized (see Eq. (20)).If, on the contrary, the condition opposite to the local-ization condition (A.8) is satisfied, the integral (31) isstrongly suppressed due to the fast oscillations of thefactor exp(i∆Eikx′/vg) in the integrand, leading tothe suppression of the oscillations. In the borderlinecase ∆EikσX/vg ∼ 1, a partial decoherence due tothe lack of localization occurs.

If one now allows for vgi �= vgk, then, as shown inAppendix, the dependence of the integral in Eq. (30)on L is still negligible provided that L∆vg/vg � σx,or

L � lcoh = σxvg

∆vg, (32)

where ∆vg = |vgi − vgk|. This is nothing but the con-dition of the absence of decoherence due to the wave-packet separation: the distance traveled by neutrinosshould be smaller than the distance over which thewave packets corresponding to different mass eigen-states separate, due to the difference of their groupvelocities, to such an extent that they can no longerinterfere in the detector9). If the condition opposite tothat in Eq. (32) is satisfied, the integral in Eq. (30)is strongly suppressed because of the lack of overlapbetween the factors Gi(L − vgit) and G∗

k(L − vgkt)in the integrand (if Iik is written as a momentum-space integral, the suppression is due to the fast os-cillations of the integrand, see Eq. (A.7) of Appendix).The L independence of the integral in Eq. (30) inthe absence of decoherence due to the wave packetseparation means that the neutrino oscillation phasetakes its standard value in that case.

Thus we conclude that the standard oscillationphase is obtained if neutrinos are relativistic or quasi-degenerate in mass and the decoherence effects dueto the wave-packet separation or lack of the emitteror absorber localization are negligible. No unjustified“same energy” or “same momentum” assumptionsare necessary to arrive at this result.

9)We reiterate that σx is an effective spatial width of the wavepacket, which depends on the widths of both the productionand detection wave packets σxS and σxD . Condition (32)therefore automatically takes into account possible restora-tion of coherence at detection, as discussed in Section 5.3.



One may wonder why these assumptions are ac-tually so popular in the literature and even made theirway to some textbooks, though there is no good rea-son to believe that the different mass eigenstates haveeither same momentum or same energy. One possiblereason could be the simplicity of the derivation ofthe formula for the oscillation probability under theseassumptions. However, we believe that simplicity isno justification for using a wrong argument to arriveat the correct result.

4. ANOTHER STANDPOINT

In this section we outline a more general approachto the calculation of the oscillation probability, whichgives an additional insight into the issues discussed inthe present paper. Here we just illustrate some pointsrelevant to our discussion rather than presenting acomplete formalism, which is beyond the scope of ourpaper.

The flavor transition amplitude is given by (the1-dimensional version of) Eq. (17). The functionΨS

i (x, t) can be represented as the integral overmomenta of Eq. (9). Inserting this expression and asimilar formula for ΨD

i (x − L) into the expression forthe amplitude and performing the integral over thecoordinate, we obtain

Aab(L, t) (33)

=∑

i

U∗aiUbi

∫dqihi(qi)eiqiL−iEi(qi)t,

where

hi(q) ≡ fSi (q − pi)fD∗

i (q − pi). (34)

Here, the momenta qi are the integration variables,whereas pi are, as before, the mean momenta of thecorresponding wave packets.

The amplitude (33) is the sum of plane waves withdifferent momenta and different masses. The integra-tion over momenta can be formally substituted by asummation to make this point clearer. The oscillationprobability is then

Pab(L, t) ≡ |Aab(L, t)|2 (35)

=

∣∣∣∣∣∑

i

∑

qi

U∗aiUbihi(qi)eiqiL−iEi(qi)t

∣∣∣∣∣

2

.

The waves with all momenta and masses should besummed up; the resulting expression for the oscil-lation probability includes the interference of thesewaves.

The standard approach to the calculation of theoscillation amplitude (33) (or probability (35)) is tosum up first the waves with different momenta but

the same mass, and then sum over the mass eigen-states. In this way first the wave packets correspond-ing to different mass eigenstates are formed, and thenthe interference of these wave packets is considered.Since ∫

dqihi(qi)eiqiL−iEi(qi)t (36)

= Gi(L − vgit)eipiL−iEi(pi)t,

Eq. (33) then directly reproduces the results of Sec-tion 3.

Another possibility is to sum up first the planewaves with different masses but equal (or related)momenta and then perform the integration over themomenta. In particular, one can select the waves withequal energies. Clearly, the final result should not de-pend on the order of summation if no approximationsare made, and should be almost independent of thisorder if the approximations are well justified. How-ever, different summation conventions allow differentphysical interpretations of the result. In what fol-lows we will perform computations using the “equalmomenta” and “equal energy” summation rules andidentify the conditions under which they lead to thestandard result for the oscillation probability.

Let us first consider the “equal momenta” sum-mation. Setting qi = p for all i, one can write theamplitude (33) as

Aab(L, t) =∫

dp∑

i

U∗aiUbihi(p)eipL−iEi(p)t. (37)

Next, we note that, for ultrarelativistic neutrinos, toan extremely good approximation

hi(p) = h(p,Ei(p)), (38)

i.e., the functions hi(p) depend on the index i onlythrough their dependence on the neutrino energy

Ei(p) =√

p2 + m2i . Indeed, hi(p) depend on i

through the neutrino mass dependence of the phase-space volumes and amplitudes of the neutrino emis-sion and detection processes. The phase-space vol-umes depend on mi only through Ei(p); the pro-duction and detection amplitudes can in principledepend directly on mi due to the chiral suppression,as, e.g., in the case of π± or K± decays. However, thecorresponding contributions to the total amplitudesare completely negligible compared to the maincontributions, which in this case are proportional tothe masses of charged leptons.

Let us expand hi in power series in ∆m2i3:

hi(p) = h(p,Ei) = h(p,E3) (39)

+ Ei∂hi

∂Ei

∣∣∣∣mi=m3

∆m2i3

2E23

+ . . .



Inserting this expression into (37), we obtain

Aab(L, t) =∫

dph(p,E3(p))eipL−iE3(p)t (40)

×∑

i

U∗aiUbie

−i[Ei(p)−E3(p)]t + A∆ab(L, t),

where

A∆ab(L, t) ≡

∫dpeipL−iE3(p)t (41)

×∑

i

U∗aiUbie

−i[Ei(p)−E3(p)]t Ei∂hi

∂Ei

∣∣∣∣mi=m3

∆m2i3

2E23(p)

.

It is easy to see that the termA∆ab(L, t) is typically very

small:

A∆ab(L, t) ∼ ∆m2

i3

EσEA1, (42)

where A1 is the first term on the right hand side ofEq. (40). Indeed, if the width of the effective momen-tum distribution function h(p) is σp, one has ∂h/∂p ∼h/σp, so that

E∂h

∂E=

E

vg

∂h

∂p∼ Eh

vgσp=

Eh

σE, (43)

which immediately leads to (42). Thus, if

σE ∆m2

2E, (44)

A∆ab(L, t) can safely be neglected.

Consider now the first term in Eq. (40). For a fixedmomentum the phase difference is ∆φi3 = (Ei −E3)t ≈ ∆m2

i3t/2E310). If the width of the effective

momentum distribution function h(p,E3) is smallenough, so that the change of the phase within thewave packet is small, we can pull the oscillatoryfactor out of the integral at some effective momentum(corresponding to an energy E):

Aab(L, t) �[∑

i

U∗aiUbi exp

(−i

∆m2i3

2Et

)](45)

×∫

dph(p,E3(p))eipL−iE3(p)t

=

[∑

i

U∗aiUbi exp

(−i

∆m2i3

2Et

)]

× G3(L − vg3t)eip3L−iE3(p3)t.

10)This approximation breaks down at very small momenta.Note, however, that the small-p contribution to the inte-gral in (40) is strongly suppressed because of the effectivemomentum distribution function h(p, E3), which is stronglypeaked at a relativistic momentum p = p3. This justifiesusing the approximation for ∆φi3 in (40).

Here the factor in the square brackets gives the stan-dard oscillation amplitude in the “evolution in time”approach. Integrating the squared modulus of ampli-tude (45) over time and using once again Eq. (44),one arrives at the standard expression for the oscilla-tion probability.

Thus, we obtain the standard oscillation formulaby first summing up the waves with equal momentaand different masses and then integrating over themomenta provided that the following two conditionsare satisfied:

(i) The variation of the oscillation phase withinthe wave packet due to the energy spread is small:σE � (2πE2/∆m2

ik)L−1; this condition allows one

to pull the oscillatory factor out of the integral over themomenta, as discussed above. Note that it is actuallyequivalent to the condition of no wave-packet sep-aration, Eq. (32) (recall that ∆vg � ∆m2/2E2 andσx � vg/σE).

(ii) The momentum distribution functions hi(p)are not too narrow: σE ∆m2/2E. This condition,in particular, allows one to neglect the contributionA∆

ab(L, t) in Eq. (40). It is essentially the localizationcondition in the wave-packet picture, as it actuallyensures that the neutrino wave-packet length σx �vg/σE is small compared to the neutrino oscillationlength.11)

These conditions for obtaining the standard oscil-lation formula coincide with the conditions found in adifferent framework in Section 3.2.

To describe the possible decoherence effects due tothe separation of the wave packets and reproduce thelocalization factor in the oscillation probability explic-itly, one should lift conditions (i) and (ii) and considerthe corresponding corrections to the oscillation am-plitude. This is discussed in detail in Section 5.

Similarly, we can consider summation of waveswith equal energies and different masses, with thesubsequent integration over energies (or momenta).

Requiring Ei(qi) = E3(p) yields qi = ±√

p2 + ∆m23i,

q3 = p. Taking into account that dqi =

±dp/√

1 + ∆m23i/p

2, we obtain

Aab(L, t) (46)

=∫

dp∑

i

U∗aiUbi

hi(qi(p), E3(p))√1 + ∆m2

3i/p2eiqi(p)L−iE3(p)t.

Here, it is assumed that ν3 is the heaviest masseigenstate, so that ∆m2

3i ≥ 0 and no singularities

11)Note that the oscillation probability may take the standardform even if this condition is violated, as it is the case, e.g.,for Mossbauer neutrinos [4, 6].



appear in the integrand. Note that our change of theintegration variables qi → p excludes, for i = 1, 2, thesmall regions of momenta qi around zero; this, how-ever, introduces only a tiny error, because the maincontributions to the integral come from the regionsaround the points qi = pi, where the functions hi(qi)are strongly peaked.

Expanding hi(qi(p), E3(p)) as

hi(qi(p), E3(p))√1 + ∆m2

3i/p2� h(p,E3(p)) (47)

+(

h(p,E3(p)) − p∂h(p,E3(p))

∂p

)∆m2

3i

2p2

and inserting this expression into (46), we obtain

Aab(L, t) =∫

dph(p,E3(p))eipL−iE3(p)t (48)

×∑

i

U∗aiUbie

i[qi(p)−p]L + A∆ab,

where

A∆ab(L, t) ≡

∫dpeipL−iE3(p)t (49)

×∑

i

U∗aiUbie

i[qi(p)−p]L

(h − p∂h

∂p

)∆m2

3i

2p2.

Just as in the previous case, one can show that theamplitude A∆

ab(L, t) can be neglected if condition (44)is satisfied. Assuming this to be the case and thatthe variation of the oscillation phase within the wavepacket due to the momentum spread is small, andtaking into account that (qi − p)L = (qi − q3)L �∆m2

3iL/2p, one finds

Aab(L, t) �[∑

i

U∗aiUbi exp

(i∆m2

3i

2pL

)](50)

× G3(L − vg3t)eip3L−iE3(p3)t,

where p is the average neutrino momentum. Thisis the standard oscillation amplitude multiplied bythe effective shape factor of the wave packet of ν3

(note that in our current approximation we actuallyneglect the difference between the wave packets ofdifferent mass eigenstates). Integrating the squaredmodulus of the amplitude in Eq. (50) over time andusing normalization condition (20), we again arrive atthe standard expression for the oscillation probability,just as in the previous case when we first summedthe terms with equal momenta and different massesand then integrated over the momenta. In derivingthis result we once again used conditions (i) and (ii)discussed above.

Our discussion of the new summation rules forcalculating the oscillation probability presented here

leads to an alternative explanation of why the “sameenergy” and “same momentum” assumptions even-tually lead to the correct physical observables, asdiscussed in Section 6.

5. QUANTUM-MECHANICAL UNCERTAINTYRELATIONS AND NEUTRINO

OSCILLATIONS

Neutrino oscillations, being a quantum-mecha-nical interference phenomenon, owe their very exis-tence to quantum-mechanical uncertainty relations.The coordinate–momentum and time–energy uncer-tainty relations are implicated in the oscillations phe-nomenon in a number of ways. First, it is the energyand momentum uncertainties of the emitted neutrinostate that allow it to be a coherent superposition ofthe states of well-defined and different mass. Thesame applies to the detection process—for neutrinodetection to be coherent, the energy and momen-tum uncertainties inherent in the detection processshould be large enough to prevent a determinationof the absorbed neutrino’s mass in this process. Theuncertainty relations also determine the size of theneutrino wave packets and therefore are crucial to theissue of the loss of coherence due to the wave-packetseparation. In addition, these relations are importantfor understanding how the produced and detectedneutrino states are disentangled from the accompa-nying particles. Let us now discuss these issues inmore detail.

5.1. Uncertainty Relations and Disentanglementof Neutrino States

In the majority of analyses of elementary-particleprocesses it is assumed that the energies and mo-menta of all the involved particles have well-defined(sharp) values and obey the exact conservation laws.However, for this description to be exact, the consid-ered processes (and the particles involved) should becompletely delocalized in space and in time, whereasin reality these processes occur in finite and relativelysmall spatial volumes and during finite time intervals.For this reason, the energy and momenta of all theparticipating particles have intrinsic quantum me-chanical uncertainties, and the particles should be de-scribed by wave packets rather than states of definitemomentum—plane waves (see, e.g., [19]). The con-servation of energy and momentum for these particlesis also fulfilled up to these small uncertainties.

This does not, of course, mean that the energy–momentum conservation, which is a fundamental lawof nature, is violated: it is satisfied exactly when oneapplies it to all particles in the system, including thosewhose interactions with the particles directly involved



in the process localize the latter in a given space–timeregion. Schematically speaking, if we consider theprocess as occuring in a box, the interactions with thewalls of the box and the contributions of these wallsto the energy–momentum balance have to be takeninto account. In reality, this is never done; however,the resulting inaccuracy of the energy and momen-tum conservation as well as the intrinsic quantum-mechanical uncertainties of the energies and momen-ta of the involved particles are usually completelynegligible compared to their energies and momentathemselves, and therefore can be safely ignored inmost processes.

This is, however, not justified when neutrino os-cillations are considered, since the neutrino energyand momentum uncertainties, as tiny as they are, arecrucially important for the oscillation phenomenon. Inthis respect, we believe that the attempts to use theexact energy–momentum conservation in the analy-ses of neutrino oscillations are inconsistent. In someanalyses the exact energy–momentum conservationis assumed for the neutrino production and detectionprocesses in order to describe neutrinos as beingentangled with accompanying particles. The subse-quent disentanglement, which is necessary for neu-trino oscillations to occur, is assumed to be due tothe interaction of these accompanying particles (suchas, e.g., electrons or muons produced in decays ofcharged pions) with medium. This localizes thoseparticles and creates the necessary energy and mo-mentum uncertainties for the neutrino state. The de-scribed approach misses the fact that the parent par-ticles are already localized in the neutrino productionand detection processes, and so no additional disen-tanglement through the interaction of the accompa-nying particles with medium is necessary. Indeed, itis clear that neutrinos produced, for example, in π±

decays oscillate even if the accompanying chargedleptons do not interact with medium, i.e., are not“measured”. The measurement of the flavor of thesecharged leptons that discriminates between e± andµ± and makes neutrino oscillations possible is ac-tually provided by the decoherence of the chargedleptons due to their very large mass difference [7].

5.2. Coherence of the Produced and DetectedNeutrino States

In order for a neutrino state produced in a charged-current weak interaction process to be a coherent su-perposition of different neutrino mass eigenstates, itshould be in principle impossible to determine whichmass eigenstate has been emitted. This means thatthe intrinsic quantum-mechanical uncertainty of thesquared mass of the emitted neutrino state σm2 mustbe larger than the difference ∆m2 of the squared

masses of different neutrino mass eigenstates [21,23]: σm2 � ∆m2. Conversely, if σm2 � ∆m2, one candetermine which mass eigenstate has been emitted,i.e., the coherence of different mass eigenstates isdestroyed. This situation is quite similar to that withthe electron interference in double-slit experiments:If there is no way to find out which slit the detectedelectron has passed through, the detection probabil-ity will exhibit an interference pattern, but if such adetermination is possible, the interference pattern willbe washed out.

Assume that by measuring energies and momentaof the other particles involved in the production pro-cess we can determine the energy E and momentump of the emitted neutrino state, and that the intrinsicquantum-mechanical uncertainties of these quanti-ties are σE and σp. From the energy–momentumrelation E2 = p2 + m2 we can then infer the squaredmass of the neutrino state with the uncertainty σm2 =[(2EσE)2 + (2pσp)2]1/2, where it is assumed that σE

and σp are uncorrelated. Therefore the condition thatthe neutrino state be emitted as a coherent superpo-sition of different mass eigenstates is [21, 23]

σm2 ≡[(2EσE)2 + (2pσp)2

]1/2 ∆m2. (51)

This condition has a simple physical meaning. Notefirst that in many cases the two terms in the squarebrackets in Eq. (51) are of the same order of mag-nitude: EσE ∼ pσp (more generally, EσE ≤ pσp, seeSection 5.4). Therefore the condition in Eq. (51) es-sentially reduces to 2pσp ∆m2, or σp ∆m2/2p =2π/losc. Since the momentum uncertainty σp isrelated to the coordinate uncertainty of the neutrinosource σxS by σp ∼ σ−1

xS , we finally get

σxS � losc. (52)

This is nothing but the obvious requirement that theneutrino production be localized in a spatial regionthat is small compared to the neutrino oscillationlength; if it is not satisfied, then neutrino oscillationswill be averaged out upon the integration over theneutrino emission coordinate in the source, which isequivalent to decoherence. For this reason the condi-tion in Eq. (51) is often called the localization condi-tion. Similar argument applies to the detection pro-cess, i.e., the spatial size of the neutrino absorber σxD

should satisfy

σxD � losc. (53)

Since both (52) and (53) have to be fulfilled for neu-trino oscillations to be observable, one can refor-mulate the localization condition as the requirementthat the oscillation length losc be large compared toσX ≡ max{σxS , σxD}, i.e., essentially compared to



the effective size of the wave packet σx which is oforder of σX :

σx � losc. (54)

That this directly follows from the wave packet for-malism is shown in Appendix.

In real situations one always deals with large en-sembles of neutrino emitters, and the detectors alsoconsist of a large number of particles. Therefore, incalculating the observable quantities—the numbersof the neutrino detection events—one always has tointegrate over the macroscopic volumes of the neu-trino source and detector. In some situations (e.g.,for solar or reactor neutrinos) the source and detectorare much larger than the localization domains of thewave functions of individual neutrino emitters andabsorbers. In these cases the integration over thesource and detector volumes modifies the localizationconditions: instead of depending on the spatial sizesof the individual neutrino emitter and absorber, σxSand σxD, they contain the macroscopic lengths of thesource and detector in the direction of the neutrinobeam, LS , and LD. In other words, a necessary con-dition for the observability of neutrino oscillations is

LS,D � losc, (55)

which is much more restrictive than the conditionsin Eqs. (52) and (53). If this condition is violated,neutrino oscillations are averaged out.

5.3. Wave Packet Separation and Restorationof Coherence at Detection

Let us now assume that a neutrino flavor statewas produced coherently in a weak interaction pro-cess and consider its propagation. The wave packetdescribing a flavor state is a superposition of the wavepackets corresponding to different mass eigenstates.Since the latter propagate with different group veloc-ities, after some time tcoh they will separate in spaceand will no longer overlap. If the spatial width ofthe wave packet is σx, this time is tcoh � σx/∆vg .The distance lcoh that the neutrino state travels dur-ing this time is lcoh � vg(σx/∆vg). If the distance Lbetween the neutrino emission and detection pointsis small compared to the coherence length, i.e., ifcondition (32) is satisfied, then the coherence of theneutrino state is preserved, and neutrino oscillationscan be observed. If, on the contrary, L ∼ lcoh or L lcoh, partial or full decoherence should take place.

If the wave-packet width σx in the above discus-sion is assumed to be fully determined by the emissionprocess, then this is not the full story yet: even ifthe coherence is lost on the way from the source tothe detector, it still may be restored in the detector ifthe detection process is sufficiently coherent, i.e., is

characterized by high enough energy resolution [33].According to the quantum-mechanical time–energyuncertainty relation, high energy resolution requiresthe detection process to last sufficiently long; in thatcase, the different wave packets may arrive during thedetection time interval and interfere in the detectoreven if they have spatially separated on their wayfrom the source to the detector. One can take intoaccount this possible restoration of coherence in thedetector by considering σx to be an effective widthof the wave packet, which exceeds the width of theemitted wave packet and takes the coherence of thedetection process into account. This effective wave-packet width is actually the width that character-izes the shape factors Gi(L − vgit), as discussed inSection 3.1. With σx being the effective wave-packetwidth, coherence condition (32) includes the effects ofpossible coherence restoration at detection.

It is well known that coherence plays a crucial rolein observability of neutrino oscillations. It is inter-esting to note, however, that even non-observationof neutrino oscillations at baselines that are muchshorter than the oscillation length is a consequenceof and a firm evidence for coherence of the neu-trino emission and detection processes: if it were bro-ken (i.e., if the different neutrino mass eigenstateswere emitted and absorbed incoherently), the survivalprobability of neutrinos of a given flavor, instead ofbeing practically equal to 1, would correspond to av-eraged neutrino oscillations.

5.4. What Determines the Size of the Wave Packet?

According to the quantum-mechanical uncer-tainty relations, the energy and momentum uncer-tainties of a neutrino produced in some process aredetermined by, correspondingly, the time scale of theprocess and spatial localization of the emitter. Thesetwo quantities are in general independent; on theother hand, for a free on-shell particle of definite massthe dispersion relation E2 = p2 + m2 immediatelyleads to

EσE = pσp. (56)

Since this relation is satisfied for each mass eigen-state component of the emitted flavor state, it mustalso be satisfied for the state as a whole (provided thatthe energies and momenta of different componentsas well as their uncertainties are nearly the same,which is the case for relativistic or quasi-degenerateneutrinos). Thus, we have an apparently paradoxicalsituation: on the one hand, σE and σp should beessentially independent, while on the other hand theymust satisfy Eq. (56).

The resolution of this paradox comes from the ob-servation that at the time of their production neutrinos



are actually not on the mass shell and therefore donot satisfy the standard dispersion relation. Thereforetheir energy and momentum uncertainties need notsatisfy (56). However, as soon as neutrinos moveaway from their production point and propagate dis-tances x such that px 1, they actually go on themass shell, and their energy and momentum uncer-tainties start obeying Eq. (56). This happens becausethe bigger of the two uncertainties shrinks towardsthe smaller one, so that Eq. (56) gets fulfilled. Indeed,when the neutrinos go on the mass shell, the standardrelativistic dispersion relation connecting the ener-gies and momenta of their mass eigenstate compo-nents allows to determine the less certain of thesetwo quantities through the more certain one, thusreducing the uncertainty of the former. As a result, thetwo uncertainties get related by Eq. (56), with the onethat was smaller at production retaining its value alsoon the mass shell. Note that for neutrino energies inthe MeV range neutrinos go on the mass shell as soonas they propagate distances x � 10−10 cm from theirbirthplace.

From the above discussion it follows that the spa-tial width of the neutrino wave packet, which de-termines the neutrino coherence length lcoh throughEq. (32), is determined by the smaller of the twouncertainties at production, σE and σp. This comesabout because lcoh is a characteristic of neutrinosthat propagate macroscopic distances and thereforeare on the mass shell. At the same time, localizationcondition (54) always depends on the spatial localiza-tion properties of the neutrino emitter and detector,which are related to the corresponding momentumuncertainties at production and detection.

Which of the two uncertainties, σp or σE , is actu-ally the smaller one at production and so determinesthe spatial width of the neutrino wave packet? Quitegenerally, this happens to be the energy uncertaintyσE . Indeed, consider, e.g., an unstable particle, thedecay of which produces a neutrino. In reality, suchparticles are always localized in space, so one canconsider them to be confined in a box of a linearsize LS . The localizing “box” is actually created bythe interactions of the particle in question with someother particles. Assume first that the time interval TS

between two subsequent collisions of the decayingparticle with the walls of the box (more precisely, theinterval between its collisions with the surroundingparticles) is shorter than its lifetime τ = Γ−1. Thenthe energy width of the state produced in the decayis given by the so-called collisional broadening andis actually �T−1

S . This width directly gives the neu-trino energy uncertainty, i.e., σE � T−1

S . On the otherhand, the neutrino momentum uncertainty is σp �

L−1S . Since TS is related to LS through the velocity

of the parent particle v as TS � LS/v, one finds

σE < σp, (57)

which is actually a consequence of v < 1.Consider now the situation when the lifetime of

the parent particle is shorter than the interval betweentwo nearest collisions with the walls of the box. In thiscase the decaying particle can be considered as quasi-free, and the energy uncertainty of the produced neu-trino is given by the decay width of the parent particle:σE � Γ.12) The momentum uncertainty of neutrinois then the reciprocal of its coordinate uncertaintyσx, which in turn is just the distance traveled byneutrino during the decay process: σx � (p/E)τ =(p/E)Γ−1 � (p/E)σ−1

E . Thus we find pσp � EσE ,i.e., Eq. (56) is approximately satisfied in this case.Once again condition (57) is fulfilled. It can be shownthat this inequality is also satisfied when neutrinosare produced in collisions rather than in decays ofunstable paricles [34].

Thus, we conclude that the spatial width σx ofthe wave packets describing the flavor neutrino statesis always determined by the energy uncertainty ofthe state as the smaller one between σp and σE ,i.e., σx ∼ vg/σE . This is in accord with the knownfact that for stationary neutrino sources (for whichσE = 0) the neutrino coherence length is infinite [33,35]. On the other hand, the localization conditions forneutrino production and detection are determined bythe corresponding momentum uncertainties.

It then follows that the parameters σx that enterinto the coherence condition and into the localizationcondition, discussed in Sections. 3.2 and 5.2 and inAppendix (see Eqs. (32) and (54)), are in general dif-ferent; they only coincide (or nearly coincide) when atproduction σp ∼ σE , so that Eq. (56) is approximatelysatisfied from the very beginning. As we discussedabove, such a situation is not uncommon, but it is notthe most general one.

This observation underlines a shortcoming of thesimple wave-packet approach to neutrino oscilla-tions: it does not take into account the neutrino pro-duction and detection processes, except by assigningto the neutrino state a momentum uncertainty, whichis supposed to be determined by these processes. Inparticular, the neutrino wave-packet picture assumesthe mass eigenstate components of the flavor neutrinostates to be always on the mass shell, so that their en-ergy and momentum uncertainties are always related

12)This only holds for slow parent particles. In the case of rela-tivistic decaying particles, σE depends on the angle betweenthe momenta of the parent particle and of neutrino [28]. Still,condition (57) holds in that case as well.



by Eq. (56). Obviously, this approach is adequate inthe cases when at production and detection σp � σE ,but is not satisfactory when this condition is stronglyviolated. In the latter cases one has to resort to themore consistent quantum field theoretical (QFT)treatment of neutrino oscillations, in which neutrinoproduction, propagation and detection are consideredas one single process. For QFT approach to neutrinooscillations see, e.g., [4, 23, 26, 34–38].

An interesting case in which the condition σp �σE is strongly violated both at production and de-tection is the proposed Mossbauer neutrino exper-iment, for which σE ∼ 10−11 eV and σp ∼ 10 keVare expected [4, 6], so that σE ∼ 10−15σp. This casealso gives a very interesting example of coherencerestoration at detection, which resolves another para-dox of neutrino oscillations. In a Mossbauer neutrinoexperiment neutrinos are produced coherently due totheir large momentum uncertainty [4]. However, assoon as the emitted neutrino goes on the mass shell,its momentum uncertainty shrinks to satisfy Eq. (56),i.e., essentially becomes equal to the tiny energy un-certainty. Therefore, for on-shell Mossbauer neutri-nos σp ∼ 10−11 eV � ∆m2/2E � 10−7 eV, i.e., themomentum uncertainty is much smaller than the dif-ference of the momenta of different mass eigenstates.This means that coherence of different mass eigen-states in momentum space is lost. However, the factthat in this case both the energy and momentum un-certainties of the propagating neutrino state are muchsmaller than ∆m2/2E does not mean that oscilla-tions cannot be observed. In fact, it has been shownin [4] that the Mossbauer neutrinos should exhibitthe usual oscillations. The resolution of the paradoxlies in the detection process: the large momentumuncertainty at detection, σpD ∼ 10 keV ∆m2/2E,restores the coherence by allowing the different masseigenstates composing the flavor neutrino state to beabsorbed coherently.

Our final comment in this section is on the casewhen σp ∼ σE at production, in which the coherenceand localization conditions depend on the same pa-rameter σx ∼ σ−1

p . While the localization conditionrequires relatively large σp (σp ∆m2/2p) for theemitted and detected neutrino states to be coherentsuperpositions of mass eigenstates, the condition ofno decoherence due to the wave-packet separation,on the contrary, requires long wave packets, i.e., rel-atively small σp. Is there any clash between these tworequirements? By combining the two conditions wefind ∆m2/2p � σp � (vg/∆vg)L−1, which can onlybe satisfied if

∆m2/2p � (vg/∆vg)L−1. (58)

This can be rewritten as the following condition onthe baseline L:

2πL

losc� vg

∆vg. (59)

Since vg/∆vg 1, this condition is expected to besatisfied with a large margin in any experiment whichintends to detect neutrino oscillations: if it were vio-lated, neutrino oscillations would have been averagedout because of the very large oscillation phase (exceptfor unrealistically good experimental energy resolu-tion δE/E < ∆vg/vg ∼ ∆m2/2E2).

6. WHEN IS THE STATIONARY-SOURCEAPPROXIMATION JUSTIFIED?

It has been pointed out in [23, 33] and greatly elab-orated and exploited in [14] that for stationary neu-trino sources the following two situations are physi-cally indistinguishable:

(a) a beam of plane-wave neutrinos, each with adefinite energy E and with an overall energy spectrumΦ(E);

(b) a beam of neutrinos represented by wave pack-ets, each of them having the energy shape factor h(E)such that |h(E)|2 = Φ(E).

As was stressed in [14], this actually follows fromthe fact that in stationary situations the spectrumΦ(E) fully determines the neutrino density matrix andtherefore contains the complete information on theneutrino system.

This, in fact, gives an alternative explanation ofwhy the “same energy” approach, though based onan incorrect assumption, leads to the correct result. Ithas been shown in Section 4 that, within the properwave-packet formalism, one can choose to sum upfirst the states of different mass but the same energy,and then integrate over the energy (or momentum)distributions described by the effective energy or mo-mentum shape factors of the wave packets, h(E) orh(p). In the light of the physical equivalence of situa-tions (a) and (b), it is obvious that the integration overthe spectrum of neutrinos, which is inherent in anycalculation of the event numbers, leads to the sameresult as the integration over the energy spread withinthe wave packets (provided that the correspondingenergy distributions coincide). Thus, the “same en-ergy” assumption, though by itself incorrect, leadsto the correct number of events upon the integrationover the neutrino energy spectrum. The same is truefor the “same momentum” assumption. This actuallymeans that the wave-packet description becomes un-necessary in stationary situations, when the temporalstructure of the neutrino emission and detection pro-cesses is irrelevant and the complete information on



neutrinos is contained in their spectrum Φ(E), as wasfirst pointed out in [14].

Let us derive the results of stationary source ap-proximation in terms of the wave-packet picture de-scribed in this paper. We start with Eq. (17) for theoscillation amplitude and Eq. (9) for the evolved wavefunction ΨS

i (x, t). Notice that the shape factor fSi (p−

pi) in (9) does not depend on time, and furthermorethis expression is determined for all moments t from−∞ to +∞. The only time dependence in (9) is in theform of the plane waves in the integrand. This is pre-cisely what corresponds to the stationarity condition:the source has no special time feature, and there is notagging of neutrino emission and detection times.

Substituting Eqs. (9) and (15) into Eq. (17) weobtain, upon neglecting the transverse components ofthe neutrino momenta,

Aab(L, t) (60)

=∑

i

U∗aiUbi

∫dpfS

i (p − pi)fDi (p)eipL−iEi(p)t,

where

fDi (p) ≡

∫dxΨD∗

i (x − L)eip(x−L) (61)

is the Fourier transform of the detection state. Notethat Ri ≡ |fD

i (p)|2 characterizes the momentum (en-ergy) resolution of the detector.

The oscillation probability is obtained by integrat-ing the squared modulus of the amplitude over time.We have

|Aab(L, t)|2 =∑

i,k

U∗aiUbiU

∗bkUak (62)

×∫

dp

∫dp′fS

i (p − pi)fS∗k (p′ − pk)fD

i (p)

× fD∗k (p′)ei(p−p′)Le−i[Ei(p)−Ek(p′)]t.

The integration over time is trivial:+∞∫

−∞

dt e−i[Ei(p)−Ej(p′)]t (63)

= 2πδ[Ei(p) − Ek(p′)],

which means that only the waves with equal energiesinterfere. We stress once again that this is a con-sequence of the fact that no time structure appearsin the detection and production processes, which isreflected in the time independence of the momentumdistribution functions fS

i (p − pi) and fDi (p), and in

the integration over the infinite interval of time. Theequality Ei(p) = Ek(p′) leads to

p − p′ = ∆m2ki/2p (64)

to the leading order in the momentum difference. Theδ function (63) can be used to remove one of themomentum integrations in (62), so that we finallyobtain

Pab(L) =

+∞∫

−∞

dt|Aab(L, t)|2 (65)

= 2π∫

dp|fS(p − p)|2|fD(p)|2

×∑

i,k

U∗aiUbiU

∗bkUak exp

(−i

∆m2ik

2pL

).

Here we have neglected the dependence of the shapefactors on the neutrino mass. Replacing the integra-tion over momenta by the integration over energies,we can rewrite the oscillation probability as

Pab(L) =

+∞∫

−∞

dt|Aab(L, t)|2 (66)

=2πvg

∫dEΦ(E)R(E)Pab(E,L),

wherePab(E,L) (67)

=∑

i,k

U∗aiUbiU

∗bkUak exp

(−i

∆m2ik

2EL

)

is the standard expression for the oscillation proba-bility, Φ(E) ≡ |fS(E − E)|2 is the energy spectrumof the source, and R(E) ≡ |fD(E)|2 is the resolutionfunction of the detector (note that we have substitutedthe momentum dependence of these quantities bythe energy dependence using the standard on-shelldispersion relation).

The expression in Eq. (66) corresponds to thestationary-source approximation: the oscillation prob-ability is calculated as an incoherent sum of theoscillation probabilities, computed for the same-energy plane waves, over all energies.

Notice that we have performed integration over thespatial coordinate at the level of the amplitude andover time at the probability level. Apparently, suchan asymmetry of space and time integrations is notjustified from the QFT point of view. In QFT com-putations the integration over time is performed inthe amplitude, and this leads (in the standard setup)to the δ function which expresses the conservationof energy in the interaction process. To match ourpicture with that of QFT we need to consider thedetection process and take into account the energiesof all the particles that participate in the process.Suppose that the algebraic sum of the energies of



all the accompanying particles (taken with the “−”sign for all incoming paricles and the “+” sign for theoutgoing ones) is ED. Then instead of (63) we willhave in the probability

+∞∫

−∞

dte−i[Ei(p)−ED ]t

+∞∫

−∞

dt′ei[Ek(p′)−ED ]t′ (68)

= 4π2δ[Ei(p) − Ek(p′)]δ[Ei(p) − ED],

where the second δ function on the right-hand sidereflects the energy conservation in the detection pro-cess. Using (68) we again obtain the “same energy”interference, as before.

In the above calculation we have not introducedany time structure at detection and performed theintegration over t from −∞ to +∞. In reality, cer-tain time scales are always involved in the detectionprocesses (even if we do not perform any time tag-ging in the emission process). For example, we canmeasure with some accuracy the appearance time ofa charged lepton produced by the neutrino capture inthe detection process. In this case, the neutrino stateof detection will have a time dependence:

ΨDi = ΨD

i (x − L, t − t0), (69)

where ΨDi has a peak at t0 with a width σt that is

determined by the accuracy of the measurement ofthe time of neutrino detection. Since in practice thespatial characteristics of neutrino detection do notchange with time, the dependences of ΨD

i on x andt factorize: ΨD

i = ΨDxi(x − L)ΨD

ti (t − t0). Integratingover time in the amplitude, we will have

∫dx

∫dtΨD∗

i (x − L, t − t0)eipx−iEit (70)

= eipL−iEit0 fDi (p)fD

ti (E),

where fDi (p) was defined in (61) and

fDti (E) ≡

∫dtΨD∗

ti (t − t0)e−iEit (71)

is the Fourier transform of ΨD∗ti (t − t0). As we have

mentioned, ΨDti (t − t0) has a peak of width σt at t =

t0. Taking for σt the value σt ∼ 10−9 s (which is prob-ably the best currently achievable time resolution), weobtain δE ∼ σ−1

t ∼ 10−6 eV. This is many orders ofmagnitude smaller than the typical energy resolutionin the oscillation experiments. Therefore one can sub-stitute fD

ti (E) → δ(E −ED), which brings us back toour previous consideration.

7. WHEN CAN NEUTRINO OSCILLATIONSBE DESCRIBED BY PRODUCTIONAND DETECTION INDEPENDENT

PROBABILITIES?

In most analyses of neutrino oscillations it is as-sumed that the oscillations can be described by uni-versal, i.e., production and detection process inde-pendent probabilities. In other words, it is assumedthat by specifying the flavor of the initially producedneutrino state, its energy and the distance betweenthe neutrino source and detector, one fully determinesthe probability of finding neutrinos of all flavor atthe detector site (for known neutrino mass squareddifferences and leptonic mixing matrix). The standardformula for neutrino oscillations in vacuum, Eq. (6), isactually based on this assumption. Such an approachis very often well justified, but certainly not in allcases. It is, therefore, interesting to study the appli-cability limits and the accuracy of this approximation.

A natural framework for this is that of QFT, whichprovides the most consistent approach to neutrinooscillations. In this approach the neutrino production,propagation, and detection are considered as a sin-gle process with neutrinos in the intermediate state.This allows one to avoid any discussion of the prop-erties of the neutrino wave packets since neutrinosare actually described by propagators rather than bywave functions. The properties of neutrinos in theintermediate state are fully determined by those of the“external” particles, i.e., of all the other particles thatare involved in the neutrino production and detectionprocesses. The wave functions of these external par-ticles have to be specified. Usually, these particles areassumed to be described by wave packets; for thisreason the QFT-based treatment is often called the“external wave packets” approach [30], as opposed tothe usual, or “internal wave packets” one, which wasdiscussed in Sections 3, 4, 6 and Appendix and whichdoes not include neutrino production and detectionprocesses. The results of the QFT-based approachturn out to be similar, but not identical, to thoseof a simple wave-packet one; in particular, possibleviolations of the on-shell relation (56) between theneutrino energy and momentum uncertainties is nowautomatically taken into account. Moreover, the val-ues of these uncertainties, which specify the proper-ties of the neutrino wave packet in the “internal wavepackets” approach and which have to be estimatedin that approach, are now directly derived from theproperties of the external particles.

The results of the QFT approach can be summa-rized as follows. For neutrinos propagating macro-scopic distances the overall probability of theproduction—propagation—detection process for rel-ativistic or quasi-degenerate neutrinos can to a verygood accuracy be represented as a product of the



individual probabilities of neutrino production, propa-gation (including oscillations), and detection13). Theoscillation probability, however, is not in general inde-pendent of production and detection processes, whichmeans that the factorizability of the probability of theentire production–propagation–detection processand the universality of the oscillation probability (orlack thereof) are in general independent issues. Theoscillation probability can be generically representedas

P (νa → νb;L) =∑

i,k

U∗aiUbiUakU∗

bk (72)

× exp(−i

∆m2ik

2pL

)Scoh(L/lcoh

ik )Sloc(σlocx /losc

ik ).

Here, Scoh(L/lcohik ) and Sloc(σloc

x /loscik ) are, respec-

tively, the coherence and localization factors, whichaccount for possible suppression of the oscillationsdue to wave-packet separation and violation of the lo-calization conditions. They both are equal to unity atzero argument and quickly decrease (typically expo-nentially) when their arguments become large. Notethat these factors depend on the indices i and kbecause so do the partial oscillation and coherencelengths losc

ik and lcohik . The simple “internal wave pack-

ets” approach leads to an expression for the oscil-lation probability that is similar in form to that inEq. (72); however, unlike in that simple approach,in the QFT-based framework the coherence and lo-calization lengths entering into Eq. (72) depend ingeneral on different length parameters, σcoh

x and σlocx .

This is related to the fact that the energy and mo-mentum uncertainties at production and detectionneed not satisfy (56), as discussed in Section 5.4. Inaddition, the form of the coherence and localizationfactors Scoh(L/lcoh

ik ) and Sloc(σlocx /losc

ik ) in Eq. (72),rather than being postulated, is derived from the prop-erties of the external particles and of the detection andproduction processes.

For the oscillation probability to be independent ofthe production and detection processes, the followingconditions have to be fulfilled:

(i) Decoherence effects due to wave-packet sepa-ration and due to violation of the localization condi-tions should be negligible;

(ii) The energy release in the production and de-tection reactions should be large compared to theneutrino mass (or compared to mass differences).

13)A notable exception from this rule is the case of theMossbauer effect with neutrinos, in which the probabilitiesof neutrino production and detection do not factorize but areinstead entangled with each other [4]. Still, even in this case,the oscillation (actually, νe survival) probability can to a verygood accuracy be factored out of the expression for the overallprobability of the process.

The necessity of (i) is clear from the discussionabove: if this condition is fulfilled, the coherence andlocalization factors in Eq. (72) are both equal tounity, and the standard neutrino-oscillation formulais recovered. If, on the contrary, (i) is violated, theoscillations will suffer from the production and detec-tion dependent decoherence effects. As to the condi-tion (ii), it ensures that the production and detectionprobabilities are essentially the same for all masseigenstate components of the emitted or detectedflavor neutrino states (modulo the different values of|Uai|2); if this condition is violated, the phase spaceavailable in the production or detection process willdepend on the mass of the participating neutrino masseigenstate, and the mass eigenstate composition ofthe flavor eigenstates will no longer be given by simpleformula (1).

8. DISCUSSION AND SUMMARY

In the present paper we discussed a number ofsubtle issues of the theory of neutrino oscillationswhich are still currently under debate or have notbeen sufficiently studied yet. For each problem wediscussed, we were trying to present our analysis fromdifferent perspectives and obtained consistent results.We have also developed a new approach to calculatingthe oscillation probability in the wave-packet picture,in which we changed the usual order of integrationover the momenta (or energies) and summation overthe mass eigenstate components of the wave packetsrepresenting the flavor neutrino states. This alloweda new insight into the question why the unjustified“same energy” and “same momentum” assumptionslead to the correct result for the oscillation probability.

We have also presented an alternative derivationof the equivalence between the results of the sharp-energy plane-wave formalism and of the wave-packetapproach in the case of stationary neutrino sources,as well as discussed the applicability conditions forthe stationary-source approximation.

Below we give a short summary of our answers tothe first seven questions listed in Introduction.

(1) The standard formula for the probability of neu-trino oscillations is obtained if the decoherence effectsdue to the wave-packet separation are negligible andthe neutrino emitter and absorber are sufficiently welllocalized. Under these conditions the additional os-cillation phase ∆φ′ which is acquired in the neutrinoproduction and detection regions is negligible. The“same energy” and “same momentum” assumptions,which allow one to nullify this additional phase, arethen unnecessary. They still lead to the correct resultbecause their main effect is essentially just to removethis extra phase. An alternative explanation of the factthat the “same energy” assumption gives the correct



result comes from the observation that in going fromthe oscillation probability to the observables such asevent numbers, one has to integrate over the neutrinospectra. As discussed in [14, 33] and in Section 6,for stationary neutrino sources this is equivalent tointegration over the energy distribution within wavepackets (provided that this energy distribution coin-cides with the spectrum of plane-wave neutrinos). InSection 4 we have shown that the integration overthe spectrum of plane-wave neutrinos is just a cal-culational convention in the wave-packet approach,which does not involve any additional approxima-tions.

(2) Quantum-mechanical uncertainty relationsare at the heart of the phenomenon of neutrinooscillations. For neutrino production and detection tobe coherent, its energy and momentum uncertaintiesmust be large enough to prevent a determination ofthe neutrino’s mass in these processes. These un-certainties are governed by the quantum-mechanicaluncertainty relations, which also determine the sizeof the neutrino wave packets and therefore are pivotalfor the issue of the coherence loss due to the wave-packet separation.

(3) The spatial size of the neutrino wave packetsis always determined by their energy uncertainty σE .(4) The coherence condition ensures that the wavepackets corresponding to the different mass eigen-states do not separate to such an extent that theycan no longer interfere in the detector. This conditionis therefore related to the spatial size of the neutrinowave packets, which is determined by the neutrinoenergy uncertainty σE . Note that this is an effectiveuncertainty which depends on the energy uncertain-ties both at neutrino production and detection. Atthe same time, the localization conditions are de-termined by the effective momentum uncertainty σp,which depends on the momentum uncertainties atproduction and detection. In the simple wave-packetapproach, the neutrino energy and momentum un-certainties are related to each other due to the on-shellness of the propagating neutrino, whereas in amore general quantum field theoretic framework theyare in general unrelated.

(5) Wave-packet approach (or a superior QFTone) are necessary for a consistent derivation of theexpression for the oscillation probability. Once thishas been done, wave packets can be forgotten in allsituations except when the decoherence effects dueto the wave-packet separation or due to the lack oflocalization of the neutrino source or detector be-come important. Even in those cases, though, thedecoherence effects can in most situations be reliablyestimated basing on the standard oscillation formulaand simple physical considerations. In addition, the

wave packets are unnecessary in the case of sta-tionary problems [14]. Thus, the wave-packet ap-proach is mainly of pedagogical value. It is also usefulfor analysing certain subtle issues of the neutrino-oscillation theory.

(6) The oscillation probability is independent of theproduction and detection processes provided the fol-lowing conditions are satisfied: (i) decoherence effectsdue to wave-packet separation and due to violation ofthe localization conditions are negligible, and (ii) theenergy release in the production and detection reac-tions is large compared to the neutrino mass (or com-pared to mass differences). Note that if the conditionopposite to (i) is realized, the probabilities of flavortransitions also take a universal form, as in that casethey simply correspond to averaged oscillations.

(7) The stationary-source approximation is validwhen the time-dependent features of the neutrinoemission and absorption processes are either absentor irrelevant, so that one essentially deals with steadyneutrino fluxes. Integration over the neutrino detec-tion time then results in the equivalence of the oscil-lation picture to that in the “same energy” approxi-mation.

APPENDIX

INTEGRAL Iik AND ITS PROPERTIES

Let us consider the properties of the integral Iik(L)defined in Eq. (22). Expressing the shape factorsgS,Di (x) of the wave packets through the correspond-

ing momentum distribution functions according to(the 1-dimensional version of) Eq. (13) and substi-tuting the result into (19), we find the following rep-resentation for Gi(L − vgit):

Gi(L − vgit) (A.1)

=

∞∫

−∞

dpfSi (p)fD∗

i (p)eip(L−vgit).

Consider now the integral

Iik(L) ≡∞∫

−∞

dtGi(L − vgit)G∗k(L − vgkt) (A.2)

=

∞∫

−∞

dt

∞∫

−∞

dp1

∞∫

−∞

dp2fSi (p1)fD∗

k (p1)fS∗i (p2)

× fDk (p2)eip1(L−vgit)−ip2(L−vgkt).



Performing first the integration over time and makinguse of the standard integral representation of Dirac’sδ-function, we obtain

Iik(L) =2πvgk

∞∫

−∞

dpfSi (p)fD∗

i (p)fS∗k (rp) (A.3)

× fDk (rp))eip(1−r)L,

where

r ≡ vgi

vgk� 1. (A.4)

In the limit when the group velocities of the wavepackets corresponding to different mass eigenstatesare exactly equal to each other, r = 1, Iik does notdepend on the distance L. Since the dominant contri-bution to the integral in (A.3) comes from the region|p| � σP ≡ min{σpS , σpD} of the integration interval,Iik is practically independent of L, provided that |1 −r|LσP � 1, or

L � lcoh = σXvg

∆vg, (A.5)

where ∆vg = |vgi − vgk| and σX = 1/σp. This ismerely the condition of the absence of the wave-packet separation: the distance traveled by neutrinosshould be smaller than the distance over which thewave packets corresponding to different mass eigen-states separate due to the difference of their groupvelocities and cease to overlap. If the condition oppo-site to that in Eq. (A.5) is satisfied, the integral Iik isstrongly suppressed because of the fast oscillations ofthe integrand. Iik is actually the overlap integral thatindicates how well the wave packets correspondingto the ith and kth neutrino mass eigenstates overlapwith each other upon propagation the distance L fromthe source.

Consider now the integral Iik that enters into ex-pression (21) for the oscillation probability P (νa →νb;L) and is given by Eq. (30). The calculation similarto that of the integral Iik in Eq. (A.2) yields

Iik(L) =2πvgk

e−i∆Eik[(vgi−vgk)/2vgvgk ]L (A.6)

× exp(−i

∆m2ik

2pL

) ∞∫

−∞

dpfSi (p)fD∗

i (p)

× fS∗k (rp + ∆Eik/vgk)fD

k (rp + ∆Eik/vgk)eip(1−r)L.

The first exponential factor here must be replacedby unity because the exponent contains the prod-uct of the factors ∆Eik and (1/vgk − 1/vg) � [(vgi −

vgk)/2vgvgk], both of which are ∝∆m2ik, and hence, is

of fourth order in neutrino mass. Thus, we finally get

Iik(L) = exp(−i

∆m2ik

2pL

)2πvgk

(A.7)

×∞∫

−∞

dpfSi (p)fD∗

i (p)fS∗k (rp + ∆Eik/vgk)

× fDk (rp + ∆Eik/vgk)eip(1−r)L.

Just as in the case of Iik, the integral on the righthand side of Eq. (A.7) is practically independent of Lwhen the condition (A.5) is satisfied and is stronglysuppressed in the opposite case. In addition, it isquenched if the split of the arguments of fS,D

i and

fS,Dk in the integrand exceeds σP ; thus, a necessary

condition for unsuppressed Iik is

∆EikσX/vg � 1. (A.8)

This is often called the localization condition for thefollowing reason. Since ∆Eik/vg ∼ ∆m2

ik/2p ∼ l−1osc

and σX , being the inverse of min{σpS , σpD}, is thelargest of the sizes of the two spatial localization re-gions, of the emitter and detector, the condition (A.8)is actually equivalent to the obvious requirement thatthe neutrino source and detector be localized withinspatial regions that are small compared to the oscil-lation length losc. If this condition is violated, inte-gration over the coordinates of the neutrino emissionand detection points within the source and detectorresults in neutrino oscillations being averaged out.

From the above consideration it follows that thefactor Iik in the expression for the oscillation prob-ability yields the standard oscillation phase factor

exp(−i

∆m2ik

2pL

)multiplied by the integral which

accounts for possible suppression of the oscillatingterms due to decoherence caused by wave-packetseparation and/or lack of localization of the neu-trino source and detector. Note that both decoher-ence mechanisms lead to exponential suppressionof the interference terms in the oscillation probabil-ities since they come from the infinite-limits inte-grals of fast oscillating functions. The exact form ofthese suppression factors depends on the shape ofthe wave packets, i.e., is model dependent; in par-ticular, for Gaussian and Lorentzian wave packets,these factors are ∼ exp[−(L/lcoh)2] exp[−(σX/losc)2]and exp(−L/lcoh) exp(−σX/losc), respectively.

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