aa r X i v : . [ h e p - ph ] S e p Path integral of neutrino oscillations
Kazuo Fujikawa
Interdisciplinary Theoretical and Mathematical Sciences Program,RIKEN, Wako 351-0198, Japan
Abstract
We propose an idea of the constrained Feynman amplitude for the scatter-ing of the charged lepton and the virtual W-boson, l β + W ρ → l α + W λ , fromwhich the conventional Pontecorvo oscillation formula of relativistic neutri-nos is readily obtained using plane waves for all the particles involved. In apath integral picture, the neutrino propagates forward in time between theproduction and detection vertices, which are constrained respectively on the 3-dimensional spacelike hypersurfaces separated by a macroscopic positive time τ . The covariant Feynman amplitude is formally recovered if one sums overall possible values of τ (including negative τ ). It is well-known that the formulation of neutrino oscillations [1, 2], if carefully exam-ined, has some subtleties in the fundamental aspects of quantum mechanics [3, 4].More than one neutrinos with different masses are produced or detected simultane-ously in a quantum mechanical sense and thus the energy-momentum conservationis not obvious. There appear two different kinds of neutrino fields; the mass eigen-fields ν k ( x ), which diagonalize the neutrino mass matrix, and the flavor eigenfields ν α ( x ), ( α = e, µ, τ ), are related to each other by the general mixing formula ν α ( x ) = U αk ν k ( x ) (1)where U αk stands for the PMNS mixing matrix in a natural extension of the Stan-dard Model. We mainly analyze the Dirac neutrinos in the present paper. Thistransformation preserves the form of kinetic terms of neutrinos in the Lagrangianand thus constitutes a canonical transformation which preserves the canonical anti-commutation relations. In the path integral this canonical transformation preserves1he path integral measure invariant. The canonical transformations generally alterthe mass terms and interaction terms, and thus they differ from the conventionalunitary transformations of global symmetries and Lorentz transformations in fieldtheory which preserve the form of the Lagrangian invariant. A general class ofcanonical transformations in connection with neutrinos are known as the Pauli-Gursey transformation [5] and its generalizations [6].To analyze the basic issues related to the neutrino mixing, we start with a con-crete example of the pion decay which provides an initial neutrino in an oscillationexperiment π + → µ + + ν µ . (2)It is customary to assume that the physical ν µ is produced in this decay and thenthe physical neutrino propagates toward the detector where it is detected by a weakinteraction. If one of the mass eigenstates in ν µ should be identified immediatelyafter the pion decay, for example, such a mass eigenstate due to the reduction ofquantum states would propagate without oscillation, although the charged leptonflavor change β → α will be induced by the mixing in (1). (The repeated measure-ments of flavor freedom could also suppress the oscillation.) To avoid this difficulty,Kayser [7] discussed the idea of the wave packets of particles involved, such as µ and ν µ in (2). This idea of wave packets has become the standard machinery in theanalysis of neutrino oscillations and clarified the important aspects of oscillations[3, 4, 7]. To emphasize the necessity of the wave packets, it is often stated that theplane waves are unphysical since they are spread in the entire space.If one looks at the actual neutrino oscillation experiments, however, the neutrinosare produced by the pion decay and then the neutrinos are detected by a weakinteraction inside a huge water Cerenkov detector, for example. These experimentsappear to be standard ones common in high energy experiments, and we do notsee the particular efforts of experimentalists to generate wave packets in the actualexperiments. In fact, it is very common to use the idea of Feynman diagrams definedby plane waves in almost all the analyses of the scattering of elementary particles.It is rare to use the wave packets to analyze the scattering of elementary particles.One may then wonder if it is possible to describe the neutrino oscillations in termsof the Feynman diagrams using Feynman propagators defined by plane waves for allthe particles involved. The main purpose of the present paper is to examine sucha possibility. In the explicit analysis of oscillations, we employ the field theoreticalformalism, in particular, the path integral. We do not assume the identification ofthe physical neutrino immediately after the pion decay (2), and instead the neutrinowhich appears in the decay is described by the off-shell Feynman propagator whichterminates at the weak vertex of the neutrino detector, as in the past field theoretical2ormulations [8, 9, 10, 11] and a related quantum mechanical formulation [12].We first analyze the Feynman diagram approach to neutrino oscillations andconfirm that the standard covariant Feynman amplitude, as is well known, does notproduce the conventional oscillation formula [1]. We then propose an idea of the con-strained Feynman amplitude of neutrino oscillations using the plane waves for all theparticles involved, that readily reproduces the conventional Pontecorvo oscillationformula for relativistic neutrinos. In this scheme, the neutrinos bridge the produc-tion and detection vertices located on two 3-dimensional spacelike hypersurfaces,which are defined by two fixed time-slices, fixed y and fixed x = y + τ , separatedby a macroscopic time τ >
0. The neutrinos are forced to propagate forward intime and does not propagate backward in time with negative energy; in this sense,the neutrino propagation is macroscopic and semi-classical in the measurement ofoscillations. Our proposal is summarized in (20) and (23) below.
Historically, the quantum mechanical formulation of neutrino oscillations with anemphasis on the Fock space has been discussed by various authors [13, 14, 15, 16].This analysis is based on the assumption of the production of the physical neutrinoin the pion decay (2), for example, and then the physical neutrino thus producedpropagates toward the detector in the oscillation experiment. The question is thenwhat kind of vacuum one uses if one assumes the relation (1) in the form | ν α i = X k U αk | ν k i (3)which, if properly interpreted, is known to lead to the original derivation of Pon-tecorvo’s formula [1]; h ν β (0) | ν α ( t ) i ∼ P k ( U βk ) † exp[ − i p ~p + m k t ] U αk .The analyses of the Fock space in neutrino oscillations are interesting, and wehere comment on the issue of the mass eigenfields and flavor eigenfields in the Fockspace formalism from a point of view of the path integral. We show that the masseigenfields and flavor eigenfields are equivalent in defining the neutrino oscillationamplitudes in the path integral formulation. It is known that only the mass eigen-field is physically relevant in the field theoretical formulation of oscillations [8, 9].Nevertheless our simple demonstration of the equivalence of mass eigenfields andflavor eigenfields in the path integral formalism will be interesting.To analyze the neutrino oscillations, the relevant part of the Lagrangian of aminimal extension of the Standard Model by adding the right-handed components3f neutrinos and thus assuming the massive Dirac neutrinos is given by L = ν ( x )[ iγ µ ∂ µ − M ν ] ν ( x ) + l ( x )[ iγ µ ∂ µ − M l ] l ( x )+ g √ { l α γ µ W µ ( x )(1 − γ ) U αk ν k ( x ) + h.c. } (4)where the M ν and M l stand for the 3 × U stands for the 3 × ν k ( x ) correspond to the masseigenfields. When one integrates over the neutrino variables in (4) in the pathintegral, R D ν k D ν k ... exp { i R d x L} , one obtains Z d x L = Z d xl ( x )[ iγ µ ∂ µ − M l ] l ( x )+ Z d xd y ( g √ l α ( x ) γ λ W λ ( x )(1 − γ ) × U α,k h T ⋆ ν k ( x ) ν l ( y ) i ( U † ) l,β γ ρ W ρ ( y )(1 − γ ) l β ( y ) (5)where the neutrino propagator h T ⋆ ν k ( x ) ν l ( y ) i is defined for the mass eigenfields ofDirac neutrinos h T ⋆ ν k ( x ) ν l ( y ) i = Z d p (2 π ) (cid:18) i p − M ν + iǫ (cid:19) kl e − ip ( x − y ) . (6)To the accuracy of O ( g ), the effective vertex in (5)( g √ l α ( x ) γ λ W λ ( x )(1 − γ ) × U α,k h T ⋆ ν k ( x ) ν l ( y ) i ( U † ) l,β γ ρ W ρ ( y )(1 − γ ) l β ( y ) (7)generates the exact probability amplitude for the scattering of the charged leptonand the (virtual) W-boson l β + W ρ → l α + W λ (8)for the entering charged lepton l β and the W-boson at y µ to the detected outgo-ing charged lepton l α and the W-boson at x µ by exchanging the neutrinos. TheW-bosons in the above expression are usually replaced by the hadronic or leptoniccharged weak currents, but we use the above amplitude for notational simplicity.This exact amplitude describes the charged lepton flavor-changing process for spec-ified α = β since the basic Lagrangian (4) breaks the lepton flavor symmetry, al-though the fermion number is preserved in the present Dirac neutrinos. The neutrino4scillation is regarded as a very specific charged lepton flavor-changing process wherethe conversion rate of the charged leptons oscillates in time or distance between thetwo vertices y µ and x µ .On the other hand, the Lagrangian (4) is rewritten in terms of the flavor eigen-fields ν α ( x ) defined by ν α ( x ) = U αk ν k ( x ) in the form L = ν ( x )[ iγ µ ∂ µ − M ] ν ( x ) + l ( x )[ iγ µ ∂ µ − M l ] l ( x )+ g √ { l α γ µ W µ ( x )(1 − γ ) ν α ( x ) + h.c. } (9)where the 3 × M can be written in the case of the Dirac-type neutrinosas M = U M ν U † . (10)We emphasize that the Lagrangian (4) is more fundamental than (9) in the sensethat the derivation of the latter Lagrangian depends on the definition of the mixingmatrix in (4), which partly arises from the unitary matrix associated with the massdiagonalization of charged leptons . One may integrate over the flavor neutrinos in(9) in the path integral to obtain Z d x L = Z d xl ( x )[ iγ µ ∂ µ − M l ] l ( x )+ Z d xd y ( g √ l α ( x ) γ λ W λ ( x )(1 − γ ) ×h T ⋆ ν α ( x ) ν β ( y ) i γ ρ W ρ ( y )(1 − γ ) l β ( y ) . (11)If one recalls the relation (1), one has h T ⋆ ν α ( x ) ν β ( y ) i = U α,k h T ⋆ ν k ( x ) ν l ( y ) i ( U † ) l,β (12)and thus the Lagrangian (11) becomes identical to the Lagrangian (5).In the framework of the path integral, it is straightforward to derive the relation(12) from the Lagrangian (7) and thus to show the identical exact scattering ampli-tudes for two different definitions of neutrino fields. In the framework of quantummechanics with an emphasis on the structure of the Fock space [13, 14, 15, 16], The neutrino oscillations can in principle take place even without any mixing among themassive non-degenerate neutrinos in the level of the BEH mechanism. The unitary matrix arisingfrom the diagonalization of the charged lepton mass matrix, which constitutes the PMNS matrix,can still cause the massive neutrino oscillations. L ν = ν ( x )[ iγ µ ∂ µ − M ] ν ( x )= L + L int (13)with L = ν ( x )[ iγ µ ∂ µ ] ν ( x ) and L int = ν ( x )[ −M ] ν ( x ). We define the propagator forthe flavor field by summing all the Feynman diagrams defined by the free masslesspropagator h T ⋆ ν α ( x ) ν β ( y ) i = Z d p (2 π ) (cid:18) i p + iǫ (cid:19) δ αβ e − ip ( x − y ) (14)which is well-specified by the massless free Lagrangian L = ν ( x )[ iγ µ ∂ µ ] ν ( x ).We then obtain the exact propagator defined for the Lagrangian (13) by h T ⋆ ν α ( x ) ν β ( y ) i ≡ h T ⋆ ν α ( x ) ν β ( y ) i + Z d z h T ⋆ ν α ( x ) ν β ′ ( z ) i ( − i M ) β ′ α ′ h T ⋆ ν α ′ ( z ) ν β ( y ) i + ..... = Z d p (2 π ) (cid:18) i p − M + iǫ (cid:19) αβ e − ip ( x − y ) = Z d p (2 π ) (cid:18) iU ( p − M ν + iǫ ) U † (cid:19) αβ e − ip ( x − y ) = Z d p (2 π ) (cid:0) i ( U † ) − ( p − M ν + iǫ ) − U − (cid:1) αβ e − ip ( x − y ) = U αk Z d p (2 π ) (cid:18) i p − M ν + iǫ (cid:19) kl e − ip ( x − y ) ( U † ) lβ (15)that reproduces (12).This conversion of the massless propagator to a massive propagator by summinga series of (15) is sometimes called a sum of spring diagrams since it consists ofsumming the spring-like Feynman diagrams, and it has been used to formulate ahomogeneous renormalization group equation by Weinberg [17], for example.The propagator of the mass eigenfields in (4) L ν = ν ( x )[ iγ µ ∂ µ − M ν ] ν ( x ) (16)6ay also be defined by a sum of spring diagrams by defining L = ν ( x )[ iγ µ ∂ µ ] ν ( x )and L int = ν ( x )[ − M ν ] ν ( x ). In this sense, both-hand sides of (12) are on an equalfooting. In fact, the canonical transformation of fermion fields is defined by ask-ing the same form of kinetic terms before and after the transformation and thuscharacterized by the identical massless free fermion parts [5, 6]. Thus the abovederivation of both hands of (12) starting with massless fermions is natural in thespirit of canonical transformations. The equivalence of the mass eigenfields and theflavor eigenfields in the case of Majorana neutrinos shall be briefly mentioned inAppendix by taking Weinberg’s model of Majorana neutrinos [19] as an example.The conversion (15) is analogous to the use of the Bogoliubov transformation inthe manner of Nambu-Jona-Lasinio [18] in the recent paper [16], in the sense thatthe role of massless fields is emphasized in both cases. In the field theoretical approach, the wave packets in a broad sense have been usedto formulate the oscillation formula and to clarify some of the important physicalaspects of neutrino oscillations [3, 4, 8, 9, 10, 11, 12]. We here instead propose asimple scheme which works in the description of the neutrino oscillations using onlythe plane waves for both internal and external particles.We re-examine the effective vertex in (7). We have derived this effective vertexby integrating out the neutrino fields in (4), but it is more common to encounterthis effective vertex in the second order perturbation in weak interactions when oneanalyzes the Feynman amplitudes. Depending on the final states, the contributionsof neutral current couplings leads to a slightly more involved formula [9], but weforgo the analysis of the complications here.If one integrates over the four-dimensional space-time both at x µ and y µ , theenergy-momentum conservation is imposed at both x µ and y µ and thus one has the off-shell massive neutrino propagators in (7) in general and thus it is not clear ifoscillations occur, although the flavor change of charged leptons such as µ → e willgenerally take place due to the lepton flavor violation (such as the muon numberviolation) in (4). This is the well-known fact.To be more explicit, we have the amplitude after the amputation of external legsof Feynman diagrams for the charged lepton flavor changing process l β → l α as in78), Z d xd ye iP f x u α ( p f ) ǫ ( q f ) U αk h T ⋆ ν Lk ( x ) ν Ll ( y ) i ( U † ) lβ ǫ ( q i ) u β ( p i ) e − iP i y + ( ǫ ( q i ) ↔6 ǫ ( q f ))= Z d xd ye iP f x u α ( p f ) ǫ ( q f ) U αk ( 1 − γ Z d p (2 π ) (cid:18) i pp − M ν + iǫ (cid:19) kl e − ip ( x − y ) × ( U † ) lβ ǫ ( q i ) u β ( p i ) e − iP i y + ( ǫ ( q i ) ↔6 ǫ ( q f ))= (2 π ) δ ( P f − P i ) { u α ( p f ) ǫ ( q f )( 1 − γ X k U αk i pp − m k + iǫ ( U † ) kβ | p µ = P µi × 6 ǫ ( q i ) u β ( p i )+ u α ( p f ) ǫ ( q i )( 1 − γ X k U αk i pp − m k + iǫ ( U † ) kβ | p µ = p µi − q µf ǫ ( q f ) u β ( p i ) } (17)where m k is the diagonalized mass of the k-th neutrino, and P i = p i + q i and P f = p f + q f are the entering and the outgoing external total four-momenta, respectively,which are carried by the charged leptons and (virtual) W-bosons. Note that weassume the plane waves for all the particles involved and take into account the Bosestatistics of two virtual W-bosons. The neutrinos provide a kind of potential forcebetween the scattering charged leptons. We have the kinematical constraint of thefour-momentum conservation p ν µ = p i + q i = P i , or p ν µ = p i − q f (18)with the common four-momentum p ν µ for all the neutrino mass eigenstates, whichgenerally imply the off-shell neutrinos. We see no clear indication of the oscillatingbehavior in (17), although we expect the enhanced behavior near p ν µ = m k (19)with k=1, 2, 3. It is important that we do not have constraints which would ariseif one should constrain the neutrinos on-shell as in (19) [4]. We emphasize thatthe configurations of the initial and final states consisting of the charged leptonsand the (virtual) W-bosons in the present process can be very close to those of theoscillation experiments. But the distance or time scale which characterizes the neu-trino oscillation is missing in the formula (17), and thus the conventional covariantFeynman amplitude does not describe the phenomenon of neutrino oscillations, asis well known. 8he neutrino oscillation phenomenon may be regarded as a macroscopic quantumeffect, and thus we propose to generalize the notion of the effective vertex whichgenerates Feynman amplitudes by Z d xd yδ ( x − y − τ )( g √ l α ( x ) γ λ W λ ( x ) × U αk h T ⋆ ν Lk ( x ) ν Ll ( y ) i ( U † ) lβ γ ρ W ρ ( y ) l β ( y ) (20)with an extra δ -functional constraint δ ( x − y − τ ) using a fixed positive macroscopic τ . We call this amplitude with x − y = τ > constrained Feynman amplitude . From a point of view of path integral, we sumthose paths of neutrinos starting on the spacelike hypersurface defined by the fixed y and ending at the spacelike hypersurface defined by the fixed x = y + τ with aseparation by a macroscopic constant τ , and at the end we sum over y . The vertex x µ of the neutrino propagator is always after the vertex y µ by a time lapse τ > ; in this sense our prescription is macroscopic and semi-classical.The path integral with the constraint x − y = τ does not spoil the symmetryunder the simultaneous constant shifts of all the time variables, namely, x → x + δt, y → y + δt. (22)Thus the overall energy-conservation of the observed systems consisting of the en-tering charged lepton and (virtual) W-boson and the outgoing charged lepton and(virtual) W-boson in (20) is ensured after integration over y (or time-averagingover y ) but the energy conservation on the neutrinos described by the Feynmanpropagator in the intermediate states is not imposed. The momentum conservationis imposed at all the vertices since we integrate or sum over the spatial coordinates x k and y k inside the spacelike hypersurfaces. We sum x k and y k in (20) over allthe 3-dimensional spaces but in the physical interpretation we still regard that thecovered spaces in the path integral are “localized” seen from a macroscopic scale τ . For example, a gigantic water Cerenkov counter at Kamioka, for example, which The definition of a particle and an antiparticle is by convention. The particle in our terminologycorresponds to the neutrino which we naively identify as propagating in the oscillation experiments.
9s very large by a microscopic scale, is still very small compared to the oscillationlength. Our premise is that we can formulate Feynman amplitudes with idealizedplane waves and we can incorporate the possible momentum spread arising from alarge but finite detector, for example, by smearing the external states of the chargedlepton and the virtual W-boson when we define the final scattering amplitude, ifnecessary. We assume that these corrections are small and forgo this refinement inthe present paper.To be more explicit, we have the amputated oscillation amplitude generatedby (20) when written as a matrix element between the states | W ( q i ) i i ⊗ | i f and h l α ( p f ) , W ( q f ) | ⊗ h ¯ l β ( p i ) | , which correspond to the case of the pion decay (2), Z d xd yδ ( x − y − τ )( g √ × e iP f x u α ( p f ) ǫ ( q f ) U αk h T ⋆ ν Lk ( x ) ν Ll ( y ) i ( U † ) lβ ǫ ( q i ) v β ( p i ) e − iP i y = Z d xd yδ ( x − y − τ )( g √ e iP f x u α ( p f ) ǫ ( q f )( 1 − γ × U αk Z d p (2 π ) (cid:18) i pp − M ν + iǫ (cid:19) kl e − ip ( x − y ) ( U † ) lβ ǫ ( q i ) v β ( p i ) e − iP i y = (2 π ) δ ( P f − P i )( g √ u α ( p f ) ǫ ( q f )( 1 − γ × U αk Z dp π (cid:18) i pp − M ν + iǫ (cid:19) kl e − ip τ + iP i τ ( U † ) lβ | ~p = ~P i ǫ ( q i ) v β ( p i )= (2 π ) δ ( P f − P i )( g √ u α ( p f ) ǫ ( q f )( 1 − γ ×{ X k U αk p p e − ip τ + iP i τ ( U † ) kβ | p = √ ~p + m k , ~p = ~P i } 6 ǫ ( q i ) v β ( p i ) (23)where P i = q i − p i is the entering four-momentum in the case of the pion decay,for example, and P f = q f + p f is the outgoing four-momentum which is carried bythe charged lepton and (virtual) W-boson. We are assuming that all the particlesare expressed by plane waves. The wave functions v β ( p i ) and ¯ u α ( p f ) stand for theexternal charged leptons such as µ + and e , respectively, and the wave functions ǫ µ ( q i )and ǫ µ ( q f ) stand for the (virtual) W-bosons; the initial ǫ µ ( q i ) is actually proportionalto the derivative of the pion field in the case of the pion decay.Since the phase e iP i τ is common to all the massive neutrinos (for example, P i = We are assuming that the Hilbert spaces at the production vertex and at the detection vertexare effectively factored for large τ since the weak interactions are short ranged. π − p µ in the pion decay (2)), we have the essential part of the amplitude from (23) X k U αk i p p e − ip τ ( U † ) kβ | p = √ ~p + m k , ~p = ~P i (24)which is a field theoretical version of the oscillation amplitude [8, 9, 10, 11] if onereplaces τ = L (25)where L is the spatial distance of two vertices [4]. For the relativistic neutrinos,it may be natural to assume that the neutrinos mainly propagate along the light-cone between two hypersurfaces separated by τ , which implies (25). For the ultra-relativistic neutrinos, the factor p p = 12 [ γ + γ l p l p ] (26)in (24) is regarded to be independent of the neutrino masses since p l /p = [ p l / | ~p | ](1 − (1 / m k / | ~p | + ... ) ≃ p l / | ~p | and thus not essential for the oscillation (this statementis valid also for ~p = 0). The amplitude (24) then contains the well-known oscillatingfactor in quantum mechanics [1]12 sin 2 θ (cid:16) e − i √ ~p + m L − e − i √ ~p + m L (cid:17) = − i sin 2 θe [ − i ( √ ~p + m + √ ~p + m ) L/ sin { (cid:18)q ~p + m − q ~p + m (cid:19) L/ } (27)for the specific two-flavor case µ → e , for example.The minimum length L to measure the oscillation is then specified by (cid:12)(cid:12)(cid:12)(cid:12)q ~p + m − q ~p + m (cid:12)(cid:12)(cid:12)(cid:12) L/ ≃ (cid:12)(cid:12)(cid:12)(cid:12) m − m | ~p | (cid:12)(cid:12)(cid:12)(cid:12) L ∼ ~p of the virtualneutrinos, which is determined by the measured charged lepton and W-boson system.From a physics point of view, our prescription (20) probes a tiny energy-splittingcontained in the neutrino propagator by varying the macroscopic time τ , which isat least of the order of the energy-time uncertainty limit τ × (cid:12)(cid:12)(cid:12)(cid:12)q ~p + m − q ~p + m (cid:12)(cid:12)(cid:12)(cid:12) / ≥ ~ / , (29)11n a notation with explicit ~ . It is notable that, if one sums or integrates over all τ (including negative τ ) in (20), one would formally recover the original covarianteffective vertex (7) . In the present formulation (20), it might be more appropriateto say that we determine the oscillation time or length by measuring τ or L ratherthan predicting the oscillation time or length.To clarify the physical picture of our prescription, we repeat the analysis of theenergy-momentum balance in (20) and (23). In the context of the explicit exampleof the pion decay (2) at the production vertex, we have the momentum conservation ~p ν µ = ~p π − ~p µ = ~P i (30)with the common momentum ~p ν µ for all the neutrino mass eigenstates . We do notimpose the energy-conservation at the decay vertex (nor at the detection vertex) p ν µ = p π − p µ = P i , (31)but instead we have the on-shell constraints p ν µ | k = q ~p ν µ + m k (32)arising from the iǫ -prescription with τ > k = 1 , ,
3. We ensure the overallconservation of the observed energy-momentum by the factor (2 π ) δ ( P f − P i ) in(23). In our picture, the assembled neutrino mass eigenstates appearing in the Feyn-man propagator are in the virtual states, somewhat analogously to the old fashionedperturbation theory, and consistent with the energy-time uncertainty relation (29). We have shown that the use of the mass eigenfields or the flavor eigenfields is amatter of canonical transformation of field variables in the path integral and thuscauses no essential differences in the definition of the exact charged lepton and(virtual) W-boson scattering amplitudes.We then proposed an idea of the constrained effective vertex which generatesFeynman amplitudes for the neutrino oscillation process. This scheme is based onthe neglect of the neutrino propagating backward in time relative to the neutrinopropagating forward in time, namely, the contribution of the neutrino with negativeenergy is neglected, besides assuming the macroscopic time separation by τ . This This fact may imply that the breaking of the Lorentz symmetry in (20) and (23) is not fatal. τ , andthe covariant Feynman amplitude is formally recovered if one sums over all possible τ (including negative τ ). It is hoped that a very simple prescription of the presentformulation may lead to a new insight into neutrino oscillations.I thank A. Tureanu for informing me the recent interesting development in theFock space formalism of neutrino oscillations. The present work is supported in partby JSPS KAKENHI (Grant No.18K03633). A Ma jorana neutrinos
In this appendix we briefly comment on the connection of the mass and flavoreigenfields using Weinberg’s model of Majorana neutrinos [19]. The relevant part ofthe Lagrangian to analyze the neutrino oscillations is given by L = L ν + l ( x )[ iγ µ ∂ µ − M l ] l ( x )+ g √ { l α γ µ W µ ( x ) U αk ν Lk ( x ) + h.c. } (33)where M l stands for the 3 × U standsfor the 3 × ν k ( x ) are described by the model Lagrangian L ν of Majorana neutrinos,for which we adopt Weinbergs model that is known to describe the essential aspectsof various seesaw models of Majorana neutrinos. The model is defined by an effectivehermitian Lagrangian [19] L ν = ν L ( x ) iγ µ ∂ µ ν L ( x ) − (1 / { ν TL ( x ) CM ν ν L ( x ) + h.c. } = (1 / { ¯ ψiγ µ ∂ µ ψ ( x ) − ¯ ψ ( x ) M ν ψ ( x ) } (34)where M ν stands for the 3 × ψ ( x ) ≡ ν L ( x ) + Cν LT ( x ) . (35)13he field ψ ( x ) satisfies the classical Majorana condition identically regardless of thechoice of ν L , ψ ( x ) = Cψ ( x ) T . (36)When the charge conjugation operation defined for a chiral fermion by ν L ( x ) → Cν RT ( x ) is not a good symmetry, we define the Majorana fermion by (36) togetherwith the Dirac equation [ iγ µ ∂ µ − M ν ] ψ ( x ) = 0. Following the recent analysis in[20], for example, one can then confirm the equivalence of the oscillation amplitudeunder a canonical transformation ν Lα ( x ) = U αk ν Lk ( x ) in the model (33) and (34).Our proposed formula (20) for the process (8) is valid for the Majorana neutrinosalso with a due care when neutral current effects are included [9]. References [1] V. Gribov and B. Pontecorvo, Phys. Lett. B , 293 (1969).S.M. Bilenky and B. Pontecorvo, Phys. Rep. , 225 (1978).[2] Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. Phys. (1962) 870.[3] M. Beuthe, Phys.Rept. (2003) 105-218, and references therein.[4] E. K. Akhmedov, Quantum mechanics aspects and subtleties of neutrino oscil-lations, arXiv:1901.05232v1[hep-ph], and references therein.[5] W. Pauli, Nuovo Cimento , 204 (1957).F. Gursey, Nuovo Cimento , 411 (1958).[6] K. Fujikawa, Phys. Lett. B , 76 (2019).[7] B. Kayser, Phys. Rev. D (1981) 1275.[8] C. Giunti, C. W. Kim, J. A. Lee and U. W. Lee, Phys. Rev. D , 4310 (1993).[9] W. Grimus and P. Stockinger, Phys. Rev. D , 3414 (1996).[10] C. Giunti, C. W. Kim and U. W. Lee, Phys. Lett. B (1998)237.[11] W. Grimus, P. Stockinger and S. Mohanty, Phys.Rev.D (1999) 013011.[12] J. Rich, Phys. Rev. D , 4318 (1993).[13] M. Blasone and G. Vitiello, Ann. Phys. , 283 (1995).1414] K. Fujii, C. Habe and T. Yabuki, Phys. Rev. D , 013011 (2001).[15] C. Giunti, J. Phys. G (2007) R93, and references therein.[16] A. Tureanu, Eur. Phys. J. C (2020) 68.[17] S. Weinberg, Phys. Rev. D (1973) 3497.[18] Y. Nambu and G. Jona-Lasinio, Phys. Rev. , 345 (1961).[19] S. Weinberg, Phys. Rev. Lett. , 1566 (1979).[20] K. Fujikawa, Eur. Phys. J. C80