Do non-relativistic neutrinos oscillate?

Journal of High Energy Physics, Jul 2017

We study the question of whether oscillations between non-relativistic neutrinos or between relativistic and non-relativistic neutrinos are possible. The issues of neutrino production and propagation coherence and their impact on the above question are discussed in detail. It is demonstrated that no neutrino oscillations can occur when neutrinos that are non-relativistic in the laboratory frame are involved, except in a strongly mass-degenerate case. We also discuss how this analysis depends on the choice of the Lorentz frame. Our results are for the most part in agreement with Hinchliffe’s rule.

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Do non-relativistic neutrinos oscillate?

HJE Do non-relativistic neutrinos oscillate? Evgeny Akhmedov 0 1 0 Also at the National Research Centre Kurchatov Institute , Moscow , Russia 1 Saupfercheckweg 1 , 69117 Heidelberg , Germany We study the question of whether oscillations between non-relativistic neutrinos or between relativistic and non-relativistic neutrinos are possible. The issues of neutrino production and propagation coherence and their impact on the above question are discussed in detail. It is demonstrated that no neutrino oscillations can occur when neutrinos that are non-relativistic in the laboratory frame are involved, except in a strongly We also discuss how this analysis depends on the choice of the Lorentz frame. Our results are for the most part in agreement with Hinchli e's rule. Neutrino Physics; Beyond Standard Model - 3 4 2.1 2.2 3.1 3.2 1 Introduction 2 Non-relativistic neutrinos and coherence conditions Non-relativistic neutrinos and production coherence 2.2.1 2.2.2 General case Non-relativistic neutrinos 2.3 Wave packet separation and propagation coherence Lorentz boosts Usual neutrino oscillations in the rest frame of 2 Boosting non-relativistic neutrinos Summary and discussion A Decay rates B Neutrino wave packet spreading mass-eigenstate neutrino species have mass & 0:05 eV, or one of them has mass & 0:05 eV and one other & 8 10 3 eV. The present-day temperature of the cosmic neutrino background T ' 1:95 K ' 1:68 10 4 eV therefore means that at least two relic neutrino species are currently non-relativistic. Moreover, there are some indications of possible existence of (predominantly) sterile neutrinos with an eV-scale mass, and chie y sterile neutrinos with keV | MeV | GeV scale masses are also being discussed [1{3]. If exist, such neutrinos may well be non-relativistic in some situations of practical interest. The question of whether non-relativistic neutrinos can oscillate is therefore not of completely academic nature. In addition, as we shall see, it is related to some fundamental aspects of the theory of neutrino oscillations and therefore is of considerable conceptual interest. When asking whether non-relativistic neutrinos can oscillate, one should obviously specify the reference frame in which neutrinos are considered. Neutrinos that are nonrelativistic in one Lorentz frame may be highly relativistic in another, and vice versa. { 1 { If not otherwise speci ed, we shall be considering neutrinos in the laboratory frame, by which we mean the frame in which the neutrino source is at rest or is slowly moving.1 For neutrinos produced in decays, the source is just the parent particle; for neutrinos produced in collisions, the de nition of the velocity of the source is more complicated and involves a consideration of the velocities and wave packet lengths of all particles participating in neutrino production [4]. The question of whether non-relativistic neutrinos can take part in neutrino oscillations has been discussed in refs. [4{10], with varying degree of detail and di ering conclusions. In refs. [4{6] it was argued that non-relativistic neutrinos cannot participate in neutrino oscillations, whereas in refs. [7{10] the opposite conclusion has been reached (or the opposite Non-relativistic neutrinos and coherence conditions It is well known that neutrino oscillations can only be observable if neutrino production and detection processes cannot discriminate between di erent neutrino mass eigenstates. This is because only avour eigenstates undergo oscillations | mass eigenstates do not oscillate.2 Flavour eigenstates are coherent linear superpositions of mass eigenstates, and therefore the question of observability of neutrino oscillations is closely related to the question of coherence of neutrino production, propagation and detection. If at any of these stages coherence of di erent mass eigenstates is violated, oscillations will not be observable. We therefore examine here if the coherence conditions are satis ed for nonrelativistic neutrinos. The answer to this question depends on whether the neutrino mass spectrum is hierarchical or quasi-degenerate. We rst consider the latter case. 2.1 Quasi-degenerate in mass non-relativistic neutrinos Conceptual issues of neutrino oscillation theory can only be consistently studied within the quantum-mechanical (QM) wave packet framework [4, 6, 11{25]3 or in a formalism based on the quantum eld theoretic (QFT) approach [4, 26{37]. As has been mentioned 1This de nition may sometimes di er from what one would normally consider to be the laboratory frame in a neutrino experiment. For instance, in accelerator neutrino experiments the conventional de nition would likely be that this is the frame in which the neutrino detector is at rest, whereas neutrinos are produced in decays of relativistic pions. 2In this paper we shall only deal with neutrino oscillations in vacuum. Mass eigenstates can oscillate in also to the case of non-relativistic but highly degenerate in mass neutrinos, i.e. when j m2j 2E E ; where m2 m2 i m2k and E are the neutrino mass squared di erence and the average energy of the neutrino mass eigenstates composing a produced neutrino avour state, respectively. As an example, in [22] it has been demonstrated that the conditions for neutrino oscillations to occur and to be described by the standard oscillation probability are that (i ) the energy di erence of the neutrino mass eigenstates composing the produced avour eigenstate, E Eik = qpi2 + m2 i qp2k + m2 ; k HJEP07(21) (2.1) (2.2) m2 and (2.3) is related to their momentum di erence p pi pk, mass squared di erence average group velocity vg by E 2E ; respectively: j Ej which yields eq. (2.3). E, j p j and (ii ) the coherence conditions for neutrino production, propagation and detection (to be discussed in detail in sections 2.2 and 2.3) are satis ed. The relation in eq. (2.3) is satis ed with very good accuracy for ultra-relativistic neutrinos; however, it is easy to see that for its validity it is su cient that the neutrino mass eigenstates be quasi-degenerate in mass, i.e. condition (2.1) be satis ed. In this case di erent mass eigenstates are produced under essentially the same kinematic conditions, i.e. their energy di erence E and momentum di erence p are small compared to their average energy E and average momentum p, p. This means that one can expand E in p and m2, The conditions of coherent neutrino production, propagation and detection put upper limits on m2=E2 and m2=(2E) and actually do not require neutrinos to be ultrarelativistic; they can also be satis ed if neutrinos are su ciently degenerate in mass. Thus, the standard formalism of neutrino oscillations developed for ultra-relativistic neutrinos applies without any modi cations also to non-relativistic but highly degenerate in mass neutrinos. Such neutrinos undergo the usual avour oscillations provided that the standard coherence conditions are satis ed. In the rest of this paper we shall be assuming that neutrinos are not quasi-degenerate in mass. 2.2 Non-relativistic neutrinos and production coherence Let us now consider neutrino production coherence in the case when non-relativistic neutrinos are produced. We start with a general discussion of neutrino production coherence. { 3 { 2.2.1 Di erent neutrino mass eigenstates can be emitted coherently and compose a avour state only if the intrinsic QM uncertainties of their energies and momenta, E and p, are su ciently large to accommodate their di ering energies in momenta. Assuming absence of certain cancellations (as will be explained towards the end of this subsection), this condition can be written as j Ej E 1 ; j p j p p are very small, then by measuring 4-momenta of the other particles involved in neutrino production and using energymomentum conservation, one could in principle determine the energy and momentum of the produced neutrino state with high accuracy. This would allow one to accurately infer the neutrino mass, which would mean that a mass (rather than avour) eigenstate has been emitted, and neutrino oscillations would not take place. Indeed, assume that by measuring the energies and momenta of all particles taking part in neutrino production we determined neutrino energy and momentum with some accuracy. According to QM uncertainty relations, the uncertainties of the determined neutrino energy and momentum cannot be smaller than the intrinsic energy and momentum uncertainties E and p related to localization of the production process in nite space-time region. Assuming E and p to be independent, from the on-shell dispersion relation E2 = p2 + m2 one can then nd the minimum error in the determination of the squared neutrino mass [ 12 ]: m2 = (2E E)2 + (2p p)2 1=2: (2.5) If m2 satis es m2 j m2j = jmi2 m2kj, one cannot kinematically distinguish between the mass eigenstates i and k in the production process, i.e. they can be emitted coherently. Conversely, for m2 . j m2j one can nd out which neutrino eigenstate has actually been produced; this means that i and k cannot be produced coherently. Since neutrino oscillations are a result of interference of amplitudes corresponding to di erent mass eigenstates, absence of their coherence means that no oscillations will take place. The avour transition probabilities would then correspond to averaged neutrino oscillations. This situation is quite similar to that with the electron interference in double slit experiments. If there is no way to nd out which slit the detected electron has passed through, the detection probability will exhibit an interference pattern; however if such a determination is possible, the interference pattern will be washed out. In the general case when more than two neutrino species are involved, the mass eigenstates i and k can be produced coherently if their mass di erence satis es j even if the other pairs of mass eigenstates do not satisfy such a condition and therefore m2 , mi2kj cannot be produced coherently. In this case partial decoherence takes place. As mentioned above, intrinsic energy and momentum uncertainties E and p characterizing a produced neutrino state are related to space-time localization of the production process: the better the localization, the larger the E and p, and the easier it is to satisfy { 4 { the production coherence condition. It is instructive to formulate this condition in con guration space [ 12, 20, 22 ]; this will also allow us to nd out when eq. (2.4) represents the coherent production condition. To this end, consider the oscillation phase acquired over the distance x and the time interval t from the space-time point at which neutrino was produced:4 osc = E t p~ ~x : The 4-coordinate of the neutrino production point has an intrinsic uncertainty related to the nite space-time extension of the production process; this leads to uctuations of the oscillation phase HJEP07(21) osc = E t p~ ~x ; where t and j ~xj are limited by the duration of the neutrino production process t and its spatial extension X : t . t, j ~xj . of the oscillation phase must satisfy j oscj X . For oscillations to be observable, the uctuations 1 | otherwise oscillations will be washed out upon averaging of the phase over the 4-coordinate of neutrino production. That is, observability of neutrino oscillations requires that the condition (2.6) (2.7) (2.8) j E t p~ ~x j 1 be satis ed. Barring accidental cancellations between the two terms in (2.8) and taking into account that t 1 E , X p 1, we arrive at eq. (2.4). Therefore, di erent neutrino mass eigenstates are produced coherently and hence neutrino oscillations may be observable only if the oscillation phase acquired over the space-time extension of the production region is much smaller than unity. This condition essentially coincides with the obvious requirement that the size of the neutrino production region be much smaller than the oscillation length (which corresponds to osc = 2 ). It should be noted that coherent neutrino production is necessary for observability of neutrino oscillations, but it is not by itself su cient: for the oscillations to take place, also the propagation and detection coherence conditions must be satis ed. Detection coherence can be considered quite similarly to the production one; propagation coherence will be discussed in section 2.3. 2.2.2 Non-relativistic neutrinos Let us now discuss the production coherence condition in the case when one or more neutrino mass eigenstates are non-relativistic in the frame where the neutrino source is at rest or is slowly moving. In this case di erent neutrino mass eigenstates are produced under very di erent kinematic conditions, and therefore have vastly di ering energies and momenta. We shall demonstrate that large energy and momentum di erences will then prevent coherent neutrino production. 4The phase osc (as well as the energy di erence E and the momentum di erence p~) is actually de ned for each pair of neutrino mass eigenstates i and k and should carry the indices ik. We suppress them to simplify the notation. { 5 { For illustration, we start with a concrete example. Consider neutrino production in 2-body decays at rest X ! l i, where l denotes a charged lepton, i is the ith neutrino mass eigenstate, and X is either a charged pseudoscalar meson ( , K, . . . ), or W -boson, or a charged scalar particle. The energies and momenta of the produced neutrino mass eigenstates are Ei = m2X 2mX ml2 + m2 i ; pi = [m2X (ml2 + mi2)]2 distribution anomaly [38], which has been interpreted as a production in ! decay of a non-relativistic neutrino with mass m ' 33:9 MeV and velocity v ' 0:02. Assuming that the heaviest of the neutrino mass eigenstates produced in X-boson decay is non-relativistic and barring near degeneracy of the charged lepton and X-boson masses, from eq. (2.10) we then nd thus reduces to Consider now the energy uncertainty E of the produced neutrino state. For neutrinos born in decay of a free particle at rest, E is given by the energy uncertainty of the parent particle, i.e. by its decay width X . The rst of the two coherence conditions in eq. (2.4) j Ej mX : j Ej X : X mX : { 6 { The decay rates (X ! l i) for the processes under discussion are given in appendix A. Their common feature is that they can be written as X = X mX , where mX is the mass of the X-boson and X 1. The smallness of X is due to the fact that it contains a product of small numerical and dynamical factors; in the cases when non-relativistic neutrinos are produced, the coe cients X are additionally suppressed by a small kinematic factor which comes from the suppression of the phase space volume available to the nal-state particles. We thus have From eq. (2.11) it then follows that the coherent production condition (2.12) is strongly violated, i.e. di erent neutrino mass eigenstates cannot be produced coherently. Note that (2.13) is a general property of all unstable particles | their decay width is small compared to their mass. The only exception are decays of very broad resonances, for which X mX ; however, even in this case the coherent production condition (2.12) is violated if a non-relativistic neutrino is involved. This result is actually quite general and holds also when neutrinos are produced in more complicated decays or in reactions. To see this, recall that the produced neutrino state is described by a wave packet, whose energy dispersion E is determined by the (2.11) (2.12) (2.13) temporal localization of the production process. The mean energy of the neutrino state E can be much larger than E or of the order of E, but can never be much smaller than the energy dispersion E. For processes with production of a non-relativistic neutrino, the di erences E between its energy and the energies of the other neutrino mass eigenstates are of the order of the corresponding mean energies. Therefore, which means that the rst of the two coherent production conditions in eq. (2.4) is not met. What about the second condition in eq. (2.4)? When a non-relativistic neutrino is p between its momentum and momenta of the other mass eigenstates satisfy j pj & p, where p is the mean momentum, similarly to what we found for neutrino energy di erences and energy uncertainty. However, since momentum is a vector whose projections on coordinate axes can be of either sign, one cannot in general claim that the modulus of its mean value satis es p & p . In particular, for a wave packet describing neutrino at rest, p =0 while p is nite. The momentum dispersion of the produced neutrino state is determined by the momentum uncertainty inherent in the production process, which in turn depends on the spatial localization of this process. The latter depends on how the source particles were created and on other features of neutrino production [4], and there are no simple and general arguments that would allow one to tell if the condition p p is satis ed for non-relativistic neutrinos, in contrast to the situation with the requirement E E. However, for slow neutrino sources the temporal duration and spatial localization of the neutrino production process are not directly related. This means that in general no cancellations between the two terms in (2.8) occur, and for neutrino production to be coherent both conditions in eq. (2.4) must be separately satis ed. Hence, violation of the rst of these two conditions is su cient to prevent coherent neutrino emission and thus neutrino oscillations. One might naturally wonder what happens if we consider the usual neutrino oscillations (such as e.g. oscillations of reactor, accelerator or atmospheric neutrinos)5 in a reference frame where one of the neutrino mass eigenstates is slowly moving or at rest. Indeed, in that case the relations in eq. (2.14) should also be valid, and yet neutrinos must be oscillating: the answer to the question of whether neutrinos oscillate cannot depend on the choice of the reference frame in which neutrinos are considered. We shall discuss this issue in section 3.1. 2.3 Wave packet separation and propagation coherence In addition to neutrino production and detection coherence, there is another important coherence condition that has to be satis ed for neutrino oscillations to be observable: propagation coherence. Coherence may be lost on the way between the neutrino source and detector because the wave packets of di erent neutrino mass eigenstates propagate with di erent group velocities. After long enough time (coherence time) they will separate 5In what follows by the `usual neutrino oscillations' we shall always mean oscillations of neutrinos which are ultra-relativistic in the rest frame of their source. { 7 { by a distance exceeding the spatial length x of the wave packets, which then cease to overlap. The coherence time can therefore be found from the relation where vg is the di erence of the group velocities of di erent neutrino mass eigenstates. The corresponding coherence distance is given by j vgj tcoh ' x ; Lcoh ' vg tcoh ' vg j vgj x ; (2.15) (2.16) where vg is the average group velocity of di erent neutrino mass eigenstates. Since in the case of ultra-relativistic or quasi-degenerate in mass neutrinos di erent mass eigenstates are produced under essentially the same kinematic conditions, the lengths of their wave packets xi are practically the same. For processes with emission of non-relativistic neutrinos, the lengths of the wave packets of di erent mass eigenstates may di er; the quantity x in eqs. (2.15) and (2.16) should then be understood as the largest among xi. In all known cases the lengths of the neutrino wave packets are tiny (microscopic);6 still, in the case of the usual neutrino oscillations (with relativistic or highly degenerate in mass neutrinos) the coherence distance Lcoh is macroscopic and very long because vg=j vgj ' 2E2= m2 is extremely large. The situation is quite di erent when one or more of the produced neutrinos are non-relativistic in the laboratory frame, and neutrinos are not quasi-degenerate in mass. The velocity di erences between di erent non-relativistic neutrino mass eigenstates (or between relativistic and non-relativistic states) in that case are j vgj 1; the coherence distance is therefore microscopic, Lcoh x. This means that, even if non-relativistic neutrinos were produced coherently, they would have lost their coherence due to wave packet separation practically immediately, before getting a chance of being detected. 3 Lorentz boosts We have found that in the case when one or more of the produced neutrino mass eigenstates are non-relativistic in the reference frame where their source is at rest or is slowly moving, the production coherence condition is violated and therefore neutrino oscillations cannot take place. It is interesting to see how this analysis changes and what prevents neutrinos from oscillating if we go to a frame where all neutrinos are ultra-relativistic. A di erent but related question is this: how would the usual neutrino oscillations (such as oscillations of reactor, accelerator or atmospheric neutrinos) look like in a reference frame where one of the neutrino mass eigenstates is at rest? Neutrinos must obviously oscillate in that frame as well, but it is very instructive to see where our previous arguments against oscillations of non-relativistic neutrinos fail in this case. We study this issue rst. 6A possible exception is the hypothetical recoilless neutrino emission from crystals in Mossbauer-type experiments, in which x could actually be as long as a few meters [37]. It is not, however, clear if it will ever be possible to realize such experiments. { 8 { Consider for simplicity 2- avour neutrino oscillations in 1-dimensional approach, i.e. assuming that p~ k ~x. For de niteness, we shall assume that neutrinos are produced in pion decays at rest. Extensions to the cases of more then two avours and of moving neutrino source are straightforward; extension to the full 3-dimensional picture of neutrino oscillations is somewhat more involved but does not pose any problems [4]. Consider rst the neutrino oscillation phase in the frame where the neutrino source and detector are at rest. By making use of eq. (2.3) valid for ultra-relativistic neutrinos one can rewrite eq. (2.6) as 1 vg osc ' (x vgt) E + m2 2p x : The distance x and the time t between neutrino production and detection may both be very large, but the di erence x vgt is always small. It vanishes for pointlike neutrinos; in the case when neutrinos are described by nite-size wave packets, it is less than or of the order of the spatial length of the neutrino wave packet x: jx vgtj . x. The quantity x is, in turn, determined by the space-time extension of the neutrino production region and is typically dominated by its temporal localization or, equivalently, by the energy uncertainty E inherent in the neutrino production process [4, 22, 34]. In particular, for neutrinos produced by non-relativistic sources x ' vg= E . The rst term on the right hand side of eq. (3.1) is therefore . xj Ej=vg ' j Ej= E . If the rst of the coherent production conditions in eq. (2.4) is satis ed, this term can be neglected, and we obtain the standard expression for the oscillation phase osc ' [ m2=(2p)]x. Let us demonstrate that under very general assumptions the second condition in eq. (2.4) actually follows from the rst one. From eq. (2.3) we nd that the condition j Ej= E 1 is equivalent to Barring accidental cancellations, this gives vg j p j E j m2j 2E E ' j m2j x 2p p; taking also into account that vg ' 1, we nd that the rst strong inequality in (3.3) yields the second condition in (2.4), as advertised. Note that the second condition in (3.3), [j m2j=(2p)] x meaning: the size of the neutrino wave packet x should be much smaller than the neutrino 1, has a simple oscillation length losc = 4 p= m2. In the example we consider (free pion decay at rest), we have E ' ' 2:5 10 8 eV, E ' p ' 29:8 MeV, and for m2 = m2atm ' 2:5 10 3 eV2 we nd that the coherent production conditions (2.4) are satis ed with a very large margin. Let us now go to a frame where the heavier of the two neutrino mass eigenstates (which we choose to be 2) is at rest. This is certainly not the best frame to consider neutrino { 9 { oscillations, as the whole setup will look rather weird in it! Indeed, assume that in the initial frame where the neutrino source and detector are at rest neutrinos are moving in the positive direction of the x-axis. Then in the new frame 2 will be at rest, 1 will still be moving in the positive direction of x (though with a smaller velocity), the parent pion will be moving in the negative x-direction, and the detector will also be moving in the negative direction of x towards the neutrinos. On top of that, the wave packet describing the state of 2 will be fast spreading. Indeed, being in the rest frame of 2 means that the mean momentum of its wave packet vanishes. Still, the neutrino wave packets are characterized by a nite momentum spread, which means that in its rest frame 2 will have both positive and negative momentum components along the x-axis, i.e. its wave packet will quickly spread. Even though this will not a ect observability of neutrino oscillations because the neutrino detector will \collide" with neutrinos before a signi cant spreading occurs (see appendix B),7 this adds weirdness to the whole picture. Still, considering neutrino oscillations in the rest frame of one of the mass eigenstates is very instructive for understanding when and why non-relativistic neutrinos can actually oscillate. Let us go to a reference frame in which the whole neutrino source | detector setup is boosted with velocity u along the x-axis. The standard Lorentz transformations read x0 = u(x + ut) ; Ei0 = u(Ei + upi) ; t0 = u(t + ux) ; p0i = u(pi + uEi) ; where u = (1 u2) 1=2 is the Lorentz factor of the boost and the prime refers to the quantities in the new frame. To go to the rest frame of 2 we choose u = vg2 = (p2=E2), which gives u = E2=m2. In the new frame we then have:8 E20 = m2 ; p02 = 0 ; E10 ' m22 + m21 ; 2m2 p01 ' m22 2m2 m21 : For E0 E0 2 E10 and p0 p0 2 p01 this gives E0 ' p0 ' m22 2m2 m21 : where v0 = (1 transforms as Next, we consider the transformation laws for neutrino energy and momentum uncertainties. For neutrinos produced in pion decay at rest, the energy uncertainty is given by the pion decay width: E = . In a moving frame in which the parent pion has velocity v0 , the energy uncertainty is given by the pion decay width in that frame, 0 = v02) 1=2 is the Lorentz factor of the boost from the pion's rest frame. That is, upon going from the pion rest frame to a moving frame the neutrino energy uncertainty E0 = E = v0 u ; where we have taken into account that v0 coincides with the boost velocity u. 7The spreading of the wave packets of neutrinos that are ultra-relativistic in the rest frame of their source has negligible e ect on their oscillations. Obviously, the same should also be true in any other frame, including the rest frame of one of the neutrino mass eigenstates. We discuss these points in appendix B. 8Here and below we take into account that neutrinos are ultra-relativistic in the original frame, with E1 ' E2 and p1 ' p2. (3.4) (3.5) (3.6) (3.7) = v0 , (3.8) Let us consider now the neutrino momentum uncertainty p . By the coordinate | momentum uncertainty relation, it is the reciprocal of the neutrino coordinate uncertainty. The latter essentially coincides with the length x of the wave packet of the produced neutrino. It has been demonstrated in [20, 22] that the quantity xj Ej is invariant under Lorentz boosts, i.e. where we have used eq. (3.5). For the momentum uncertainty pj ' 1= xj we therefore have i.e. the neutrino momentum uncertainty transforms in the same way as the neutrino energy. To go from the rest frame of the parent pion to the 2 rest frame we choose u = vg2, and eqs. (3.9) and (3.10) give x0j = xj E0 Ej = j xj u(1 + uvgj ) ; p0j = pj u(1 + uvgj ) ; are ultra-relativistic) all the neutrino mass eigenstates composing the produced avour eigenstate have essentially the same momentum uncertainty p and their wave packets have the same length x, this is no longer true in reference frames where some of the neutrino mass eigenstates are non-relativistic. In particular, in the rest frame of 2 its wave packet is the longest one and therefore it is characterized by the smallest momentum uncertainty, p0min = p02. Note that it is actually the smallest momentum uncertainty that is of interest to us from the viewpoint of possible violation of the production coherence condition. Combining eqs. (3.7), (3.8) and (3.11), we nd that in the rest frame of 2 j E0j 0 E m2 u ' 2m2 ' 2E m2 2 u ; j p0j p0min m2 ' 2m2 vg2 u ' 2E m2 2 u ; (3.12) where E ' l is the mean neutrino energy in the pion rest frame and we have taken into account that u = E2=m2 ' E=m2. From eq. (3.12) it follows that both j E0j= E0 and j p0j= p0min scale as u2. Therefore, even though conditions (2.4) are satis ed in the original frame where the parent pion is at rest, they may be badly violated in the rest frame of 2 provided that the boost factor u is large enough, i.e. that the group velocity of the second neutrino mass eigenstate in the pion rest frame vg2 is su ciently close to 1. So, something went wrong here. To understand the root of the problem, let us note that the primary condition of coherent neutrino production is the requirement (2.8) that the variation of the oscillation phase with varying 4-coordinate of the neutrino emission point be small. Condition (2.4) is secondary and obtains from eq. (2.8) only under the assumption that the two terms in (2.8) are uncorrelated and do not cancel (or approximately cancel) each other. It is easy to see that it is actually this seemingly innocent assumption that led to the above problem. To show this, let us note that the Lorentz transformation (3.4) with u = vg2 ' 1 gives t0 ' u( t x) ; x0 ' u( x t) ; (3.13) (3.9) (3.11) (3.10) i.e. t0 ' x0. Thus, even if in the original frame t and x are completely independent, the corresponding quantities in the rest frame of 2 are highly correlated. In addition, eq. (3.7) tells us that E0 ' in (2.7) approximately cancel each other: p0.9 Therefore, in the rest frame of 2 the two terms 0osc = E0 t0 p0 x0 ' E0 ( t0 + x0) ' 0 : (3.14) This shows that (i) eq. (2.8) does not lead to the conditions in eq. (2.4) in this case and (ii) no enhancement of 0osc actually occurs. More accurate calculation taking into account the small deviation of u = vg2 from 1 yields 0osc = osc neutrino production condition is satis ed in both frames. 1, so that the coherent This is exactly as it must be: both the oscillation phase and its variation, being products of two 4-vectors, are Lorentz invariant. So must be the coherence conditions: the answer to the question of whether di erent mass eigenstates are emitted coherently cannot depend on the choice of the Lorentz frame in which we look at neutrinos. The conditions in eq. (2.4), which are often used in the literature as the coherent production conditions, are not Lorentz invariant; they follow from the Lorentz invariant condition (2.8) only in reference frames where the neutrino source is non-relativistic. Obviously, they cannot be automatically extrapolated from one Lorentz frame to another. So, we can now answer the question posed at the end of section 2.2.2. In the reference frame in which neutrino source is at rest or is slowly moving the two terms in the expression osc = E t p x do not in general cancel, and the coherent production condition (2.8) reduces to (2.4). Since the usual neutrinos produced in pion decay at rest are highly relativistic with very small energy and momentum di erences of their mass eigenstates, the coherence conditions (2.4) are very well satis ed for them. In the frame where one of the produced neutrino mass eigenstates is at rest, the energy and momentum di erences of neutrino mass eigenstates become large, and conditions (2.4) are no longer satis ed, as discussed in section 2.2.2. However, in this case eq. (2.4) does not represents the coherent production condition and is actually irrelevant. This happens because the boost with a very large Lorentz factor which is necessary to go to the new frame leads to near cancellation of the two terms in eq. (2.8) in that frame. As a result, conditions (2.4) no longer follow from the coherent production condition (2.8). 3.2 Boosting non-relativistic neutrinos the fact that in the new frame transformation (3.5) with u ' 1. E0 ' After we have studied in great detail coherence of the usual neutrino oscillations in the rest frame of one of the neutrino mass eigenstates, it is easy to understand what happens when neutrinos produced as non-relativistic in the rest frame of their source are boosted to become relativistic. In the original (laboratory) frame, the variations of temporal and spatial coordinates of the neutrino production point within the production region are not correlated, and neither are the energy and momentum di erences of the neutrino mass 9While eq. (3.7) is speci c to neutrinos produced in pion decays (or more generally in 2-body decays), p0 is actually quite general. It directly follows from the Lorentz eigenstates. Under these circumstances the coherent production condition (2.8) leads to eq. (2.4). Violation of the rst of the constraints in (2.4), j Ej E, which, as discussed in section 2.2.2, takes place in this case, therefore means that the coherent production condition (2.8) is not met. Assume now we go to a fast moving frame in which all neutrino mass eigenstates are highly relativistic and have nearly the same energies and momenta. Because of Lorentz invariance of osc, the production coherence condition will be violated in the new frame as well. Thus, boosting neutrinos that were non-relativistic in the laboratory frame to make them relativistic will not let them oscillate, as expected. 4 Summary and discussion We have studied in detail the question of whether neutrinos that are non-relativistic in a reference frame in which their source is at rest or is slowly moving can oscillate. The answer to this question depends on the neutrino mass spectrum. If neutrinos are highly degenerate in mass, the standard formalism of neutrino oscillations applies to them without any modi cations, and they do oscillate provided that the standard coherence conditions are satis ed. This also answers the question why non-relativistic neutral K, B and D mesons oscillate: this is because their corresponding mass eigenstates are highly degenerate in mass. If, however, non-relativistic neutrinos are not quasi-degenerate in mass, their large energy and momentum di erences prevent di erent mass eigenstates from being produced coherently. As a result, no oscillations with participation of non-relativistic neutrinos are possible. The avour transition probabilities would correspond to averaged-out oscillations in that case, and in particular survival probabilities would exhibit a constant suppression.10 We have also shown that even if non-relativistic neutrinos were produced coherently, they would have lost coherence due to their wave packet separation practically immediately, at microscopic distances from their birthplace. Although propagation decoherence may in principle be undone by a very coherent neutrino detection [14], in the case of non-relativistic neutrinos this would require a completely unrealistic degree of coherence of the detection process. In addition, even though in general detection may restore neutrino coherence if it was lost on the way between the source and the detector, the coherence can never be restored if it was violated at neutrino production. We have also considered in detail how the choice of the Lorentz frame in uences our arguments and explicitly demonstrated that the coherence conditions are Lorentz invariant, as they should be. In particular, since neutrinos which are non-relativistic in the rest frame of their source are produced incoherently and do not oscillate in that frame, they will also be incoherent and will not oscillate upon a boost to a reference frame where they are all ultrarelativistic. On the other hand, the usual neutrinos that are ultra-relativistic and oscillate 10When the production (or detection) coherence conditions are violated, the probability of the overall neutrino production | propagation | detection process does not factorize into the production rate, oscillation probability and detection cross section, so that the very notion of the oscillation probability loses its sense. In that case one could still, in principle, study the oscillatory behaviour of the overall probability (with or without lepton avour change) as a function of the distance between the neutrino production and detection points. Decoherence, however, means that no such oscillatory behaviour will take place. in the frame where their source is at rest or is slowly moving will maintain their coherence and will be oscillating also in the rest frame of any of the neutrino mass eigenstates. Our discussion demonstrated that the conditions j Ej= E 1, j pj= p 1 that are often employed as criteria of neutrino production coherence are not Lorentz invariant and should be used with caution. They can only serve as the coherent production conditions in the case of non-relativistic neutrino sources, and in general should be replaced by the Lorentz-invariant constraint on the variation of the oscillation phase over the neutrino production region (2.8). The main reason why neutrinos that are non-relativistic in the frame where their source is at rest or is slowly moving do not oscillate is their very large energy and momentum di erences, which signi cantly exceed the corresponding energy and momentum uncertainties inherent in the neutrino production process. This is very similar to the reason why charged leptons do not oscillate [39]. The results of our study are for the most part in agreement with Hinchli e's rule [40, 41]. A Decay rates We present here the rates of 2-body decays X ! l i, where l denotes a charged lepton, i stands for ith neutrino mass eigenstate, and X is either a charged pseudoscalar meson ( , K, . . . ), or W -boson, or a charged scalar particle. All the rates are given in the rest frame of the parent particle and are calculated to leading order in electroweak interaction, initially without neglecting any masses of the involved particles. We start with the rate of the charged pion decays ! l i. Direct calculation yields m2 Here Uli is the element of the leptonic mixing matrix, g ' 0:65 is the SU(2)L gauge coupling constant, mW is the W -boson mass, f ' 130 MeV is the pion decay constant, the rest of notation being self-explanatory. Note that the pion decay rate is usually expressed through the Fermi constant GF = p2g2=(8m2W ); we prefer to express it here through the dimensionless gauge coupling constant g. For other charged pseudoscalar bosons (X = K, B, . . . ) the decay rates (X ! l i) can be obtained from (A.1) by the obvious substitution m ! mX , f ! fX . Usually, the decay rates of charged pseudoscalar mesons are calculated under the assumption that all neutrino masses are very small and can be neglected from the outset. The X ! l l decay rates are then obtained by summing over all the neutrino mass eigenstates. The resulting expressions are independent of Uli due to unitarity of the leptonic mixing matrix. Such an approximation is not applicable if neutrinos with mass mi mX exist. For small lepton masses the factor ml2m+2mi2 in (A.1) describes chiral suppression of the X-meson decay; however, for decays with production of non-relativistic neutrinos this factor is not small, i.e. there is no chiral suppression. The factor in the curly brackets in eq. (A.1) (and similar factors in eqs. (A.2) and (A.4) below) is of kinematic origin; it is just the magnitude of the momentum of the produced neutrino (and of equal in magnitude but opposite in sign momentum of the charged lepton) in units of the mass of the parent particle. It vanishes when ml + mi approaches the parent particle's mass. Consider next leptonic decays of W boson. The leading order W ! l i decay rate reads (W ! l i) = g 2 48 mW jUlij2 1 ml2 + mi2 2m2W (ml2 mi2)2 2m4W 1 Finally, we consider the rate of decay of a charged scalar caused by the Yukawa-type where y is the Yukawa coupling constant. Note that such charged scalars exist in many extensions of the Standard Model, e.g. in 2 Higgs doublet models. Direct calculation to the leading order in the Yukawa coupling y yields ( ! l i) = jyj2 m 8 1 (ml + mi)2 m2 1 The production coherence condition (2.12) is more easily satis ed for larger values of X ; the latter are generally increased with decreasing mass of the produced charged lepton ml (because this increases the phase space volume available to the nal-state particles). It therefore may be useful to consider the decay widths of X-bosons also in an (unrealistic) limit ml ! 0. The rates in eqs. (A.1), (A.2) and (A.4) then simplify to m2 Neutrino wave packet spreading Consider the spreading of the neutrino wave packets in the case of the usual neutrino oscillations. We shall discuss how the e ects of this spreading change when going from the rest frame of the neutrino source (where all neutrinos are ultra-relativistic) to the rest frame of one of the neutrino mass eigenstates. The wave packet spreading is caused by the velocity dispersion, i.e. by the dependence of the group velocity ~vgi of the neutrino mass eigenstate i on its momentum. Indeed, from Ei = (p~ 2 + m2)1=2 it follows that for mi 6= 0 the group velocity ~vgi = @Ei=@p~ = p~=Ei is i a function of p~. Therefore, the momentum spread within the neutrino wave packet means that its di erent momentum components propagate with di erent velocities, leading with time to its spreading. The spreading velocity is thus11 vj ' k X 1 E k jk vgj vgk k p ; where pk is the neutrino momentum dispersion in the kth direction. For spreading of the wave packet of the ith neutrino mass eigenstate in the direction of the neutrino propagation (longitudinal spreading) we obtain HJEP07(21) vik = m2 E 3i p ; i tspr i ' mi2 p2 i : E3 tspr i tosc ' 4 E2 m2 p2 m2 : i where p is the momentum dispersion in the longitudinal direction and we have taken into account that for neutrinos that are ultra-relativistic in the rest frame of their source the momentum dispersion of the di erent neutrino mass eigenstates is practically the same. As follows from eq. (B.2), for ultra-relativistic neutrinos the longitudinal spreading velocity is very small. As a result, the spreading of their wave packets is of no relevance to neutrino oscillations. To show this, let us de ne the characteristic spreading time tspr i as the time over which the wave packet of i spreads to about twice its initial length, x 1= p: viktspr i ' 1= p. This gives Let us compare this time with the oscillation time tosc (which for ultra-relativistic neutrinos coincides with the oscillation length losc = 4 E= m2): (B.1) (B.2) (B.3) (B.4) Barring quasi-degeneracy of the neutrino masses, from the oscillation data it follows that m2=mi2 & m221= m231 1=30. Next, we note that in realistic situations neutrino energy is always very large compared to the energy uncertainty: E ! decay at rest E ' (m2 m2 )=(2m ) ' 29:8 MeV and E ' that E= E ' 1:2 1015. Since for ultra-relativistic neutrinos p ' E. As an example, for = 2:5 10 8 eV, so E, the ratio in eq. (B.4) is extremely large.12 Thus, it takes a much longer time for the neutrino wave packet to spread by about a factor of two than for neutrino oscillation probability to reach its rst maximum. This means that the e ects of wave packet spreading on neutrino oscillations can be safely neglected in all realistic situations. 11We use superscripts to label the Cartesian components of the vectors, whereas lower indices are used to mark the mass eigenstates. In eq. (B.1) the latter are omitted in order not to overload the notation. 12A possible exception are supernova neutrinos, for which E= E can be as small as 10. However, in this case the spreading of the neutrino wave packets is not relevant to neutrino oscillations either [24]. Note that the requirement tspr i losc as a condition for neglecting the wave packet spreading e ects is actually a very conservative one. Indeed, for the usual neutrino oscillations, the coherent production condition is satis ed with a large margin, which, in particular, means that the initial length of the neutrino wave packets satis es x losc (see section 3.1). In these circumstances the value of x has no e ect on neutrino oscillations,13 and neither will have its doubling. One naturally expects that, if the wave packet spreading e ect on neutrino oscillations is negligible in the rest frame of the neutrino source, the same will hold in all other Lorentz frames. We shall now demonstrate this explicitly. Let us go to the frame where the neutrino source moves with a velocity u in the direction of neutrino emission. In the new frame eq. (B.2) yields HJEP07(21) vik0 = m2 E0i3 p0 ' vik u2(1 + uvgi)2 i ; 1 (B.5) (B.6) (B.7) where we have used eqs. (3.5) and (3.10). Note that in the rest frame of the neutrino mass eigenstate i (i.e. for u = vgi) the longitudinal spreading velocity of its wave packet is a factor u2 larger than it is in the rest frame of the parent pion. Next, let us nd the characteristic spreading time t0spr i in the moving frame. Eqs. (3.9) and (B.5) yield 0 vk0 i t0spr i ' x = tspr i u(1 + uvgi) : Let us now compare the spreading times tspr i and t0spr i with, respectively, the time intervals between the i production and detection in the original frame and in the new frame, ti and t0 .14 Taking into account that i xi ' vgi ti, from the Lorentz transformation law (3.4) one nds t0i = ti u(1 + uvgi). Together with eq. (B.6) this gives Thus, if the wave packet spreading time is much larger than the neutrino ight time in the rest frame of the neutrino source, the same will be true in any other Lorentz frame, including the rest frame of one of the neutrino mass eigenstates. Therefore, the relative e ects of the wave packet spreading on neutrino oscillations is frame independent, as expected. 13Except for neutrino propagation decoherence, for which the nite length of the neutrino wave packet x is crucial, see section 2.3. However, as was shown in [24], the spreading of the neutrino wave packets does not a ect the coherence length, which is therefore de ned by the initial value of x. 14Note that if we choose the new frame to be the rest frame of the neutrino mass eigenstate i, the quantity t0i will actually be the interval between the neutrino production time and the time when the detector \collides" with the resting i. t0 i t0spr i = tspr i : t i Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 1330019 [arXiv:1303.6912] [INSPIRE]. HJEP07(21) [7] D.V. Ahluwalia, Notes on the kinematic structure of the three avor neutrino oscillation framework, Int. J. Mod. Phys. A 12 (1997) 5081 [hep-ph/9612471] [INSPIRE]. [8] D.V. Ahluwalia and J.T. Goldman, Interplay of nonrelativistic and relativistic e ects in neutrino oscillations, Phys. Rev. D 56 (1997) 1698 [hep-ph/9702308] [INSPIRE]. [9] J.-M. Levy, Exercises with the neutrino oscillation length, hep-ph/0004221 [INSPIRE]. [10] H.T. He, Z.Y. Law, A.H. Chan and C.H. Oh, Non-relativistic neutrino oscillation in dense medium, in Proceedings of the Conference in Honour of Murray Gell-Mann's 80th Birthday, H. Fritzsch and K.K. Phua eds., World Scienti c (2012), pg. 480{487. [INSPIRE]. [13] C. Giunti, C.W. Kim and U.W. Lee, When do neutrinos really oscillate?: Quantum mechanics of neutrino oscillations, Phys. Rev. D 44 (1991) 3635 [INSPIRE]. [14] K. Kiers, S. Nussinov and N. Weiss, Coherence e ects in neutrino oscillations, Phys. Rev. D 53 (1996) 537 [hep-ph/9506271] [INSPIRE]. Phys. B 502 (1997) 3 [hep-ph/9703241] [INSPIRE]. [15] A.D. Dolgov, A.Yu. Morozov, L.B. Okun and M.G. Shchepkin, Do muons oscillate?, Nucl. [16] C. Giunti and C.W. Kim, Coherence of neutrino oscillations in the wave packet approach, Phys. Rev. D 58 (1998) 017301 [hep-ph/9711363] [INSPIRE]. [17] A.D. Dolgov, Neutrino oscillations and cosmology, hep-ph/0004032 [INSPIRE]. [18] A.D. Dolgov, Neutrinos in cosmology, Phys. Rept. 370 (2002) 333 [hep-ph/0202122] B 805 (2008) 356 [arXiv:0803.0495] [INSPIRE]. [19] C. Giunti, Coherence and wave packets in neutrino oscillations, Found. Phys. Lett. 17 (2004) [20] Y. Farzan and A.Yu. Smirnov, Coherence and oscillations of cosmic neutrinos, Nucl. Phys. [arXiv:1309.1717] [INSPIRE]. avor mixing in quantum eld theory, Phys. Rev. D 61 (2000) 073006 [hep-ph/9909332] [INSPIRE]. (2002) 013003 [hep-ph/0202068] [INSPIRE]. [hep-ph/0205014] [INSPIRE]. [1] K.N. Abazajian et al., Light Sterile Neutrinos: A White Paper , arXiv: 1204 .5379 [INSPIRE]. [2] M. Drewes , The Phenomenology of Right Handed Neutrinos, Int. J. Mod. Phys. E 22 ( 2013 ) [3] M. Drewes et al., A White Paper on keV Sterile Neutrino Dark Matter, JCAP 01 ( 2017 ) 025 [4] M. Beuthe , Oscillations of neutrinos and mesons in quantum eld theory , Phys. Rept . 375 [5] C.W. Kim , C. Giunti and U.W. Lee , Oscillations of non-relativistic neutrinos , Nucl. Phys. [6] J. Rich , The quantum mechanics of neutrino oscillations , Phys. Rev. D 48 ( 1993 ) 4318 [11] S. Nussinov , Solar Neutrinos and Neutrino Mixing, Phys. Lett. B 63 ( 1976 ) 201 [INSPIRE]. [12] B. Kayser , On the Quantum Mechanics of Neutrino Oscillation, Phys. Rev. D 24 ( 1981 ) 110 [21] L. Visinelli and P. Gondolo , Neutrino Oscillations and Decoherence, arXiv: 0810 .4132 [22] E.K. Akhmedov and A. Yu . Smirnov, Paradoxes of neutrino oscillations , Phys. Atom . Nucl. [23] D.V. Naumov , On the Theory of Wave Packets, Phys. Part. Nucl. Lett . 10 ( 2013 ) 642 [24] J. Kersten and A. Yu . Smirnov, Decoherence and oscillations of supernova neutrinos , Eur. Phys. J. C 76 ( 2016 ) 339 [arXiv: 1512 .09068] [INSPIRE]. Polon. B 29 ( 1998 ) 3925 [ hep -ph/9810543] [INSPIRE]. [25] M. Zralek , From kaons to neutrinos: Quantum mechanics of particle oscillations , Acta Phys. [26] I. Yu. Kobzarev , B.V. Martemyanov , L.B. Okun and M.G. Shchepkin, The phenomenology of neutrino oscillations , Sov. J. Nucl. Phys . 32 ( 1980 ) 823 [INSPIRE]. [27] I. Yu. Kobzarev , B.V. Martemyanov , L.B. Okun and M.G. Shchepkin , Sum Rules for Neutrino Oscillations, Sov . J. Nucl. Phys . 35 ( 1982 ) 708 [INSPIRE]. [28] C. Giunti , C.W. Kim , J.A. Lee and U.W. Lee , On the treatment of neutrino oscillations without resort to weak eigenstates , Phys. Rev. D 48 ( 1993 ) 4310 [ hep -ph/9305276] [INSPIRE]. [29] W. Grimus and P. St ockinger, Real oscillations of virtual neutrinos , Phys. Rev. D 54 ( 1996 ) [40] S. Carroll , Guest Blogger: Joe Polchinski on the string debates , ( 2006 ), [41] B. Peon , Is Hinchli e's rule true?, ( 1988 ),


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Evgeny Akhmedov. Do non-relativistic neutrinos oscillate?, Journal of High Energy Physics, 2017, 70, DOI: 10.1007/JHEP07(2017)070