Chiral Oscillations in the Non-Relativistic Regime

Eur. Phys. J. C (2021) 81:411 https://doi.org/10.1140/epjc/s10052-021-09209-2

Regular Article - Theoretical Physics

Chiral oscillations in the non-relativistic regime

Victor A. S. V. Bittencourt1,a , Alex E. Bernardini2, Massimo Blasone3,4 1 Max Planck Institute for the Science of Light, Staudtstr. 2, PLZ, 91058 Erlangen, Germany 2 Departamento de Física, Universidade Federal de São Carlos, P.O. Box 676, São Carlos, São Paulo 13565-905, Brazil 3 Dipartimento di Fisica, Università degli Studi di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano, Italy 4 INFN, Sezione di Napoli, Gruppo Collegato di Salerno, 80126 Naples, Italy

Received: 14 September 2020 / Accepted: 2 May 2021 / Published online: 11 May 2021 © The Author(s) 2021

Abstract Massive Dirac particles are a superposition of ity cannot be an eigenvector of the massive Dirac Hamilto- left and right chiral components. Since chirality is not a con- nian, the free evolution of such state will induce left-right served quantity, the free Dirac Hamiltonian evolution induces chiral oscillations [5]. The degree of left and right chirality chiral quantum oscillations, a phenomenon related to the Zit- superposition in a Dirac bispinor depends on the energy to terbewegung, the trembling motion of free propagating parti- mass ratio, thus chiral oscillations are usually not relevant for cles. While not observable for particles in relativistic dynam- the description of relativistic particles, but are prominent for ical regimes, chiral oscillations become relevant when the dynamical regimes in which the momentum of the particle is particle’s rest energy is comparable to its momentum. In this comparable to (or smaller than) its mass. paper, we quantify the effect of chiral oscillations on the non- The non-relativistic regime is also interesting for explor- relativistic evolution of a particle state described as a Dirac ing the connection between the Zitterbewegung effect and bispinor and specialize our results to describe the interplay chiral oscillations. As discussed in [6], the trembling motion between chiral and flavor oscillations of non-relativistic neu- of free Dirac particles has an intimate relation to chiral oscil- trinos: we compute the time-averaged survival probability lations. Both are related to the fact that a state with initial and observe an energy-dependent depletion of the quantity definite chirality must be described as a superposition of when compared to the standard oscillation formula. In the positive and negative energy solutions of the Dirac equa- non-relativistic regime, this depletion due to chiral oscillation. Although the Zitterbewegung effect is associated with a tions can be as large as 40%. Finally, we discuss the rele- fast frequency, which is averaged out in typical high energy vance of chiral oscillations in upcoming experiments which systems, it has already been considered to explain the Dar- will probe the cosmic neutrino background. win correction term in hydrogenionic atoms [2] and it has been probed in Dirac-like systems, such as graphene [7] and trapped ions [8,9]. For example, the graphene analogous to 1 Introduction the Zitterbewegung [10] can be measured via laser excita- tions [11]. While mono-layer graphene is described by the Dirac equation has unique dynamical predictions for fermionic massless Dirac equation, bilayer graphene displays effects particles, from the Klein paradox [1,2], related to pair pro- associated with the mass term of the Dirac equation [7]plus duction in scattering problems, to the Zitterbewegung,the an effective non-minimal coupling [12], and is a system in trembling motion of free relativistic states [3]. Formally, which chiral oscillations and their relation to the Zitterbewe- the solutions of the Dirac equation are given in terms of gung could be probed. 4-component spinors, or Dirac bispinors. From a group- In this paper, we are concerned with the non-relativistic theoretical perspective, those objects belong to the irre- limit of chiral oscillations within the framework of the Dirac ducible representations of the complete Lorentz group and equation. Since the chiral oscillation amplitude depends on are constructed by combining Weyl spinors of different chi- the mass to energy ratio [13,14], the minimum survival prob- ralities [4]. Any Dirac bispinor has left and right-chirality ability of an initial state with definite chirality could be aver- components which are dynamically coupled by the mass term aged out in the non-relativistic limit. Therefore, in the non- of the Dirac equation. Since a bispinor with definite chiral- relativistic limit we specialize our discussion to (Dirac) neutrinos which indeed are ideal candidates for the study of chi- a e-mail: [email protected] (corresponding author) 123 411 Page 2 of 12 Eur. Phys. J. C (2021) 81 :411 ral oscillations due to the chiral nature of weak interaction where pˆ is the momentum operator and m is the particle’s ˆ processes in which they are produced and/or detected [15]. mass. The operators αî (i = x, y, z) and β are the 4×4Dirac The Cosmic Neutrino Background (CνB) is an important matrices satisfying the anti-commutation relations αî αˆ j + ν ˆ ˆ ˆ ˆ2 ˆ example of neutrinos in non-relativistic regime. The C B αî αˆ j = 2δijI4, αî β +ˆαi β = 0, and β = I4. is the neutrino counterpart of the Cosmic Microwave Back- In the language of group theory, Dirac equation is the ground (CMB) for photons: it is composed by decoupled dynamical equation for the irreducible representations of the relic neutrinos evolving in an expanding background [15]. complete Lorentz group,1 which are the well-known Dirac The PTOLEMY project [16] aims at detecting the CνBvia bispinors [4]. These are constructed by means of the irre- neutrino capture on tritium, a goal which should be reached ducible representations of the (proper) Lorentz group, the within the next years. In previous studies [17,18], it has been Weyl spinors. The later belong to the irreps of a group iso- noticed that the expected event rate of CνB capture on tri- morphic to the SU(2) and as such they carry an intrinsic tium will exhibit a depletion. We show that such a deple- degree of freedom - the spin. The elements of the two dis- tion for Dirac neutrinos can be understood as a manifestation connected irreps of the proper Lorentz group are labeled as of chiral oscillations and give a precise quantification of it. left and right-handed spinors. Since parity connects the left Furthermore, the capture rate depends on the nature of the and right representations, the irreps of the complete Lorentz neutrinos: Majorana or Dirac. While chiral oscillations are group are obtained by combining these representations, and present in both cases, Majorana neutrinos are subjected to thus the Dirac bispinors carry not only the spin, but also more measurable oscillation channels than Dirac neutrinos, another intrinsic discrete degree of freedom, the chirality. e.g. to right-chiral and positive helicity states. Any massive Dirac bispinor is thus a combination of left and Although the interplay between chiral and flavor oscilla- right-handed spinors. tions is very small in the ultra-relativistic regime [13,14,19, Turning our attention to the Dirac equation framework, 20], it is relevant for describing dynamical features of non- chirality is the average value of the chiral operator γˆ5 = relativistic neutrinos. We describe chiral oscillations in the −iαˆ x αˆ yαˆ z. While such operator commutes with the momen- context of two-flavor neutrino propagation, including flavor tum term of the Dirac Hamiltonian, it does not commute oscillations. In particular, we compute the effects of chiral with the mass term and thus it is not a dynamically conserved conversion on the averaged flavor oscillation, that is, the sur- quantity for bispinor states describing massive particles. This vival probability of a neutrino of a given flavor averaged over can be better appreciated by considering the chiral represen- ˆ ˆ 2 one period of flavor oscillation. The chiral oscillations correc- tation of the Dirac matrices, in which γˆ5 = diag{I2, −I2} . tions are relevant when the particle’s momentum is compara- Any bispinor |ξ can be written in this representation as ble with the lightest neutrino mass and, in the non-relativistic |ξ regime, the maximum difference is 40%, a prediction consis- |ξ = R , (2) tent with the preliminary discussion pointed in [18] in con- |ξL nection with CνB detection. A more in depth discussion of chiral oscillations in the context of the CνB can be found in where ξR,L are, respective, the positive (right-handed) and [21]. To highlight the main dynamical features associated to negative (left-handed) chirality two-component spinors. The ˆ | | ˙ this phenomenon, we adopt a simple plane wave description Dirac equation HD ξ = i ξ can then be written as and leave a more realistic treatment involving wave packets [13,14,20] for a future work. Although we focus on Dirac pˆ ·ˆσ |ξR + m |ξL = i∂t |ξR , neutrinos, we discuss chiral oscillations effects in Majorana −pˆ ·ˆσ |ξL + m |ξR = i∂t |ξL , (3) neutrinos and provide detailed calculations for the general Majorana-Dirac mass term in the appendix. from which it is clear that the mass term mβˆ connects the left and right-handed components of the bispinor. For a given initial state, evolution under the free Dirac ˆ 2 Chirality and chiral oscillations in bispinor dynamics Hamiltonian HD induces left-right chiral oscillations. In order to describe this dynamical effect we first introduce the Through this paper we will describe dynamical features of positive and negative plane wave solutions of the Dirac equa- ip·x−iE , t free massive fermionic particles in the context of relativistic tion, |ψ+(x, t) = e p m |us(p, m) and |ψ−(x, t) = −ip·x+iE , t quantum mechanics. We thus consider the temporal evolution e p m |vs(p, m) which are given in terms of the as given by the Dirac equation (hereafter we adopt natural units h¯ = c = 1) 1 The complete Lorentz group is the proper Lorentz group plus Parity. σˆ ˆ ˆ ˆ 2 αˆ = 0 βˆ = 0 I2 H |ψ = pˆ ·ˆα + mβ |ψ = i∂ |ψ , (1) In the chiral representation −ˆσ and ˆ . D t 0 I2 0 123 Eur. Phys. J. C (2021) 81 :411 Page 3 of 12 411 bispinors Dirac Hamiltonian. The later is also responsible for the rela- ⎡ ⎤ tion between chiral oscillations and the Zitterbewegung effect + p·ˆσ |η ( ) E , + m 1 E , +m s p [6]. The survival probability of the initial state P(t) is given |u (p, m) = p m ⎣ p m ⎦ , s ·ˆσ by 4E p,m 1 − p |η (p) E p,m +m s ⎡ ⎤ m2 p·ˆσ P( ) =| ψ ( )ψ ( ) |2 = − 2 , 1 + |ηs(p) t m 0 m t 1 sin E p,mt (7) E p,m + m E p,m +m 2 |v ( , ) = ⎣ ⎦ , E p,m s p m ·ˆσ (4) 4E p,m p − 1 − + |ηs(p) E p,m m while the (transition) probability of being in a positive chirality state is given by P (t) = 1 − P(t), and the average 2 2 T where E p,m = p + m . In the above equations, |ηs(p) value of the chiral operator ˆγ5(t) reads is a two component spinor that depends on the spin polariza- 2 tion of the particle. We notice that the orthogonality relations 2m 2 ˆγ (t) =ψ (t)|ˆγ |ψ (t) =− + E , t . ( , )v (− , ) = v (− , ) ( , ) = 5 m 5 m 1 2 sin p m read: us p m s p m s p m us p m 0, E p,m ( , ) ( , ) = v ( , )v ( , ) = us p m us p m s p m s p m 1. For now on, (8) we describe the solutions of the Dirac equation with helicity bispinors: we assume that |ηs(p) are eigenstates of the Helic- According to (7) the minimum survival probability is p·ˆσ P = − m2 ity operator . This choice is convenient since helicity is min 1 2 , and the frequency of the chiral oscilla- p E p,m a conserved quantity and all the relevant dynamical features tions is (in natural units) E p,m. Therefore, the corresponding will be entirely related to chiral oscillations. Moreover, we period of one chiral oscillation is τCh = 2π/E p,m and as simplify our analysis by considering one-dimensional prop- the particle propagates freely, the length corresponding to e agation along the z direction, such that one chiral oscillation can be evaluated as lCh = pτ/E p,m = ⎡ ⎤ p 2π .InFig.1 we show Pmin as a function of p/m (a) p E2, + 1 ± + |± p m E p,m m ⎣ E p,m m ⎦ |u±(p, m) = , and lCh as a function of p for several mass values (b). For 4E p,m 1 ∓ p |± non-relativistic states (p ∼ m), chiral oscillations play an E p,m +m ⎡ ⎤ important role and affect significantly the probability of the ± p |± E , + m 1 E , +m state to be in its initial configuration. In fact, for p m the |v ( , ) = p m ⎣ p m ⎦ , ± p m (5) P ∼ p2 4E p,m p minimum survival probability is min 2 which vanishes − 1 ∓ + |± m E p,m m as p → 0. In this case, since the eigenspinors are a maximal where |± are the eigenstates of the Pauli matrix σˆz.For superposition of left and right chiralities, the free evolution ultra-relativistic particles, m/p → 0 and positive helicity induces a complete oscillation from left to right chiral com- bispinors have only right-handed components while negative ponents. This can also be seen in (8): for m ∼ E p,m the helicity bispinors have only left-handed components. On the oscillations of the average chirality have the maximal ampli- other hand, for p/m → 0 the left and right-handed compo- tude between the initial value −1 and 1. For p m, chiral nents are equal irrespective of the helicity. oscillations are less relevant for the state dynamics. In the We now describe chiral oscillations by considering the limit p →∞chirality and helicity coincide, thus the eigen- temporal evolution of the initial state |ψ(0) =[0, 0, 0, 1]T spinors (5) have definite chirality. In fact (8) is constant in which has negative helicity and negative chirality: γˆ5 |ψ(0) = the latter limit. The chiral oscillation length lCh is shown in − |ψ(0). As we are dealing with plane wave states propagat- Fig. 1b for several values of the mass. For masses in the range 2 ing one-dimensionally, we consider the dynamical evolution of eV/c , the expected chiral oscillation length is of the order ¯ / ∼ −6 in momentum space [14]. The time evolved state |ψm(t) is of 1hc eV 10 m. given by Our analysis has focused on Dirac particles with states given in terms of Dirac bispinors. For Majorana particles, one should have in mind that the mass term is proportional to ˆ −i HD t |ψm (t) = e |ψ(0) the charge-conjugated spinor. In fact, the Dirac equation with + the Majorana mass term, which we call Majorana equation, E p,m m p −iE , t = 1 + e p m |u−(p, m) reads [22–25] 4E p,m E p,m + m ˆ c p iE , t ∂ |ψ = ˆ ·ˆα |ψ + β |ψ , − 1 − e p m |v−(−p, m) , (6) i p m (9) E p,m + m c ˆ ∗ where |ψ = iβαˆ y |ψ [31] is the charge conjugated Chiral oscillations are generated by the massive character of bispinor. If the bispinor |ψ is its own charge conjugated, that the particle and by the fact that the initial state is a super- is, if |ψc = |ψ, then its temporal evolution is given by the position of positive and negative energy eigenstates of the “usual” Dirac equation (1). This last condition is known as the 123 411 Page 4 of 12 Eur. Phys. J. C (2021) 81 :411

Fig. 1 a Minimum survival (a) (b) probability as a function of the 1.0 6 momentum p in units of mass; b 5 2 0.75 eV mass (eV/c )

Chiral oscillation length as a / 4 0.25 c function of the momentum for 0.5 3 0.5 different values of the mass 2 0.25 1.0 Probability 1 2.0 length ( )

0 Chiral oscillation 0 Minimum Survival 0 246810 0 246810 Momentum (units of mass) Momentum (eV/c)

Majorana condition. It can be shown (see the Appendix) that bispinor with negative helicity of the last section. The tem- for any bispinor satisfying the Majorana condition ˆγ5=0. poral evolved flavor state is therefore In other words, any bispinor that is its own self conjugate has ∗ |να(t) = Uα, U ψ (t) ⊗ νβ , (12) zero average chirality. Since the Majorana condition is pre- i β,i mi i β served under the time evolution, there is no chiral oscillations ψ ( ) in bispinors satisfying the Majorana condition, irrespective if where the mi t are given by eq. (6) with the substitution the bispinor is an eigenstate of the Hamiltonian. Notice that m → mi . The survival probability, i.e. the probability of the this is in stark contrast with the Dirac bispinor case: a Dirac state |να(t) to be a left-handed state of α flavor, reads bispinor is an unconstrained object whose time evolution is 2 given by the usual Dirac equation. The average chirality of 2 2 Pα→α =| να(0) να(t) | = |Uα, | ψ(0) ψ (t) . i mi a Dirac bispinor is only constant if it is an eigenstate of the i Dirac Hamiltonian. (13) Nevertheless, given the intrinsic handedness of the weak interaction, we can consider a bispinor that is initially in a left For two flavors mixing, the time evolution of an initial handed and negative helicity state but whose time evolution electron neutrino state is [14] is given by the Majorana equation. In this case, we follow the 2 2 |νe(t) = cos (θ) ψm (t) + sin (θ) ψm (t) ⊗ |νe formalism of [26,27] and obtain that the survival probability 1 2 + ψ ( ) − ψ ( ) (θ) (θ) ν , of such state is the same as the one given in (7) while there is m1 t m2 t sin cos μ (14) a transition probability to a right-handed component, asso- and the survival probability can be decomposed as ciated with the charge conjugation of the left-handed initial ( ) = P S ( ) + A ( ) + B ( ). state. This fact was also briefly quoted in [21]. A more general mathcalPe→e t e→e t e t e t (15) situation includes both Dirac and Majorana mass terms with S In this formula P → (t) is the standard flavor oscillation for- two non-degenerate Majorana masses, which we describe in e e mula the appendix. − S 2 2 E p,m2 E p,m1 P → (t) = 1 − sin (2θ)sin t (16) e e 2 3 Flavor mixing and chiral oscillations in and non-relativistic regime m A ( ) =− 1 2(θ) We now study the effects of chiral oscillations in non- e t cos sin E p,m1 t E p,m 1 relativistic neutrino mixing. The state of a neutrino of flavor 2 α + m2 2(θ) , at a given t is given by the superposition of mass states: sin sin E p,m2 t E p,m 2 2 |ν ( ) = ψ ( ) ⊗ |ν , 1 p + m1m2 α t Uα,i mi t i (10) B ( ) = 2( θ) ( ) ( ) − , e t sin 2 sin E p,m1 t sin E p,m2 t 1 i 2 E p,m1 E p,m2 (17) ψ ( ) where U is the mixing matrix, and mi t are the bispinors describing the temporal evolution of the mass eigenstate |νi are correction terms due to the bispinor structure. Those cor- with mass mi [13,14]. The state at t = 0 reads rections include an interplay between chiral and flavor oscillations effects and are in agreement with results presented in |να(0) = |ψ(0) ⊗ Uα, |ν = |ψ(0) ⊗ |να , (11) i i the literature [19]. i While the standard flavor oscillations have a time scale |ν ≡ |ν ψ ( = ) = |ψ( ) − with α i Uα,i i , and thus mi t 0 0 . set by the energy difference E p,m2 E p,m1 , chiral oscil- Since weak interaction processes only create left handed lations depend roughly on the mass-momentum ratio (see neutrinos, for now own, we take |ψ(0) as the left handed the previous section). The terms Ae(t) and Be(t) depend 123 Eur. Phys. J. C (2021) 81 :411 Page 5 of 12 411

2 non-trivially both on the energy difference and on the indi- f21 4π m + ( + f ) 1 4 θ 2 sin 1 21 2 cos 4π(1 − f ) f E , vidual energies E p,m2 and E p,m1 . In the limit p m1,2, 21 21 p m1 − ∼ − + O[ 2/ ] E p,m2 E p,m1 m2 m1 p m and if the mass m2 +(1 − f ) 2 sin4 θ (19) difference is |m2 − m1|m2,1 or m2,1, there will 21 2 E p,m be no interference effects between flavor and chiral oscil- 2 lations. Those phenomena can be individually identified in with f = (E , − E , )/(E , + E , ). Considering | − | 21 p m2 p m1 p m2 p m1 the survival probability. If m2 m1 is comparable to one that the average of the standard oscillation formula (16)is of the masses, there will be an interference between flavor given by and chiral oscillations. We furthermore notice that the term 2 ( θ) Be(t) is equivalent to corrections obtained via a quantum ¯ S sin 2 P → = 1 − , (20) field description of flavor oscillations [28,29]. In fact, after e e 2 P S ( ) + B ( ) some algebra we obtain that e→e t e t reproduces eq. ¯ S ¯ one concludes that the quantity P → − Pe→e contains all (30)of[29](seealso[30]). In the quantum field framework, e e the chiral oscillation effects on the averaged survival proba- the corrections have a similar origin to the usual Dirac Zitter- bility, and therefore properly quantifies the effects of chiral bewegung: the flavor ladder operator has contributions from oscillations on flavor oscillations. both particle and antiparticle operators of the massive fields. ¯ S ¯ In Fig. 3 we show the difference P → − Pe→e as a func- Since those corrections do not take into account chiral oscil- e e tion of the neutrino momentum and the squared mass dif- lations, we interpret Ae(t) as the corrections due to chiral ference and as a function of p/m1 for fixed values of the oscillations. squared mass difference. The difference between the aver- We show the survival probability (15)inFig.2 for several aged standard probability and the full one becomes 0.1 values of p/m. As already anticipated in the previous sec- for p m1, indicating the influence of chiral oscillations on tion and in [13,14,20], the chiral oscillations induce small flavor oscillations. In the non-relativistic regime, the differ- corrections to the state’s dynamics in the ultra-relativistic ence between the probabilities can be as big as ∼ 40% for regime p m. For the non-relativistic regime p m , , 1 2 typical values of the mixing angle. Moreover, we notice that the full oscillation formula differs significantly from the stan- for bigger values of the mass difference, oscillations in the dard result due to high amplitude chiral oscillations. For the probability (with the momentum) are observed. This is due parameters of Fig. 2, the mass difference is very small, and to the terms ∝ f21 sin (4π/f21) in (19), which depend on the the flavor oscillations have a time scale much longer than the sum of the energies, and oscillate with the momentum when chiral oscillations. m1 and m2 are well separated. In the ultra-relativistic regime p m2,1, we can expand the full averaged formula (19) with respect to m2,1/p as to have 3.1 Chiral oscillations effect on the average flavor Δ2 oscillation ¯ UR ¯ ¯ S 21 P → = Pe→e(p m , ) = P → + cos(2θ) e e 2 1 e e 4p2 Σ2 2( θ) Given the different time scales of the flavor and chiral oscil- 21 sin 2 4 4 − 1 − + O[m , /p ], (21) lations, we consider the averaged survival probability over 4p2 2 1 2 one flavor oscillation, defined as The corrections to the standard result are thus proportional to both the squared mass difference Δ2 = m2 − m2 and to τ 21 2 1 ¯ 1 12 the squared mass sum Σ2 = m2 + m2.TheinsetofFig.3b P → = P → (t), (18) 21 2 1 e e τ e e P¯ S − P¯ UR 21 0 depicts the corrections e→e e→e in logarithmic scale.

3.2 Chiral oscillations and tests for Dirac and Majorana τ = 4π where 12 , − , is the period of one ﬂavor oscil- E p m2 E p m1 neutrinos lation according with the standard formula (16). The time integration of (15) leads to Throughout the main text, we have assumed that the initial neutrino state is a left-handed chiral eigenstate, as a conse- quence of the properties of weak interactions, in which neu- 2 ( θ) π 2 trinos are created. We do not address here questions related ¯ sin 2 f21 4 p Pe→e = 1 − 2 + sin − 1 to the production of the initial state (such as its localization 4 4π f E , E , 21 p m2 p m1 features). m2 m2 − 1 1 4 θ + 2 4 θ In proposals for non-relativistic neutrinos detection [16– 2 cos 2 sin 2 E , E , p m1 p m2 18], it was found that the nature of neutrinos (Dirac or Majo- 123 411 Page 6 of 12 Eur. Phys. J. C (2021) 81 :411

Fig. 2 Survival probability Pe→e(t) as a function of time. The black curves indicate the standard survival probability formula (16) and the red curves depict the full formula including the fast chiral oscillations (depicted in the insets). Parameters: sin2 θ = 0.306, 2 = Δ2 + 2 m2 21 m1,with Δ2 / 2 = . 21 m1 0 01 (a) (b)

Δ2 / 2 = . . Fig. 3 Difference between the averaged standard survival probability of the momentum p for 21 m1 0 01 (black) and 0 5 (red). The ¯ S sin2(2θ) P → = 1 − and the full survival probability including chiral inset shows the chiral oscillations corrections for the ultra-relativistic e e 4 Δ2 / 2 = . oscillation effects (19) a as a function of the state’s momentum p and regime as given by (21)for 21 m1 0 01 in logarithmic scale. Other Δ2 = 2 − 2 parameter as in Fig. 2 of the squared mass difference 21 m2 m1 and b as a function

rana) influences directly the experiment’s measurement rate. oscillations behavior, as discussed in the appendix. In the At low energies Dirac neutrinos can be detected as left-chiral general case of mixed Dirac-Majorana masses, there will and negative helicity, while Majorana neutrinos can also be be interferences between the different chiral components, detected as right-chiral and positive helicity [21]. When chi- which will lead to further imprints in the survival probabil- ral oscillations are considered, the overall flux of detected ity. This most general case is also discussed in the appendix. neutrinos is halved due to chiral oscillations. As we show, Furthermore, in the presence of an external magnetic field, Dirac neutrinos chiral oscillations exhibit a strong effect in the coupled flavor-chiral oscillations are modified due to the the non-relativistic regime: oscillations from left-chiral to non-minimal coupling to the field [32]. Majorana neutrinos right-chiral become more prominent. In the case of Majo- exhibit different oscillations properties in the presence of a rana neutrinos, the possible transition probabilities (cf. in the magnetic field, which can also be used as a way to distinguish appendix) have the same energy dependence as in the Dirac Dirac from Majorana particles [23,34]. Works within this case, and the survival/oscillation probabilities are equal to framework have focused on relativistic neutrinos, an inter- those for Dirac particles. Therefore, the measurement of non- esting question is whether such effects are more prominent relativistic neutrinos will unavoidably be affected by chiral in the non-relativistic regime. oscillations and are a probe of those effects. Since both Majo- As for the observability of such an effect we note that, rana and Dirac neutrinos are subjected to chiral oscillations, in connection to the aforementioned proposal [16–18], neu- in the standard scenario, chiral oscillations and their impact trinos from the CνB are expected to be extremely non- in other relevant quantities, such as absorption rates, can be relativistic and therefore subject to chiral oscillation effects. computed once the nature of the neutrinos is known (see the The current temperature of the CνB TCνB is related to the discussion in [21]). temperature of the cosmic microwave background (CMB) 1/3 The mass term for Majorana particles can be generalized TCMB via TCνB (4/11) TCMB ∼ 0.168 meV. Since neu- such that left and right-handed components of the neutrino trinos maintain a Fermi-Dirac like distribution after decou- have different masses [31]. In the case of a vanishing Dirac pling [35], one can obtain the root mean square momentum mass term, but for different left and right Majorana mass as p¯0 ∼ 0.603 meV [17]. Considering the lightest mass as terms, we recover the same survival probability and chiral m1 ∼ 0.1 eV, these values would give the maximum effect

123 Eur. Phys. J. C (2021) 81 :411 Page 7 of 12 411 persisting for m1 down to the meV order. A more complete While our results were derived for two flavors, the exten- recent study of the effects of chiral oscillations in the CνB sion of the formalism to three flavors is straightforward with can be found in [21]. qualitatively similar conclusions. Moreover, wave packets can be readily considered [14]. In these case, there would be additional effects which also have a non-trivial influence 4 Conclusions in flavor oscillation. Nevertheless, since chiral oscillations depend on the individual energies of the mass eigenstates, In this paper we have described dynamical features of chiral decoherence effects due to a wave-packet treatment, which oscillations in the non-relativistic regime, both from a gen- depend on the difference of neutrino masses, do not influence eral perspective including their relation with Zitterbewegung, chiral oscillations. In fact, the latter are intrinsic to the Dirac to the study of flavor oscillations for non-relativistic neutri- bispinors, and thus to each mass eigenstate. Finally we would nos. Such framework is especially interesting for upcoming like to point that the formalism adopted here is an effective experiments that will be capable to measure the CνB[16], for description: flavor oscillations are correctly described in the which chiral oscillations should have a prominent influence. framework of quantum field theory [28,29], to which we plan We considered the free plane wave propagation of an ini- to extend our treatment in the future. tial left-chiral state, as those generated, for example, via weak interactions. The left-right chiral oscillations are induced by Acknowledgements We thank L. Smaldone for useful discussions. the mass term of the Dirac equation, and we computed the VASVB acknowledges financial support from the Max Planck Gesellschaft through an Independent Max Planck Research Group. characteristic oscillation length in the non-relativistic limit: in this regime, the survival probability of the initial state could Data Availability Statement This manuscript has no associated data vanish. For masses of the order of eV, the chiral oscillation or the data will not be deposited. [Authors’ comment: All the plots ∼ −6 correspond to the formulas derived in the paper with parameters given length is 10 m. in the text.] We then specialized our results to describe chiral oscillations in neutrino propagation by considering a two-flavor Open Access This article is licensed under a Creative Commons Attri- problem described within the Dirac bispinor framework. bution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you While, as discussed in [13,14,19,20], such corrections are give appropriate credit to the original author(s) and the source, pro- negligible for ultra-relativistic particles, we verified that chi- vide a link to the Creative Commons licence, and indicate if changes ral oscillations are relevant in the non-relativistic dynami- were made. The images or other third party material in this article cal regime. We obtained a modified flavor oscillation for- are included in the article’s Creative Commons licence, unless indi- cated otherwise in a credit line to the material. If material is not mula including the chiral oscillations effects, compatible included in the article’s Creative Commons licence and your intended with known results in the literature [19]. While the standard use is not permitted by statutory regulation or exceeds the permit- flavor oscillation term goes with the mass eigenstate energy ted use, you will need to obtain permission directly from the copy- difference, chiral oscillations depend on the individual ener- right holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. gies of the mass eigenstates. Funded by SCOAP3. To quantify the effects of chiral oscillations, we computed the averaged survival probability and observed that, for non- relativistic neutrinos, chiral oscillations can affect averaged Appendix: Majorana bispinors and chiral oscillations flavor oscillations up to ∼ 40% of the standard value. This is ν especially relevant for upcoming C B tests which will probe The Lagrangian for a massive fermion field Ψ is: non-relativistic neutrinos. Furthermore, Majorana neutrinos ¯ μ exhibit the same chiral oscillations properties of Dirac neu- L = Ψ γˆ pˆμΨ + Lmass, (22) trinos, although the framework in the Majorana case is more where the mass term L depends on the nature of field: enhanced. In particular, we show in the appendix that any mass Dirac or Majorana. For Dirac fermions, the mass term reads bispinor that is its own charge-conjugate (that satisfies the Majorana equation), must have vanishing average chirality. L(D) =− ΨΨ,¯ mass mD (23) Nevertheless, for a neutrino state initially in a left-handed configuration evolving under the Majorana equation, one which can be rewritten in terms of the left and right-handed Ψ = ( −ˆγ )Ψ/ Ψ = ( + gets a survival probability compatible with the Dirac case. A components of the field L 1 5 2, R 1 γˆ )Ψ/ further investigations of those effects in the non-relativistic 5 2as dynamical regime for a mixed Dirac-Majorana mass deserve (D) ¯ ¯ L =−mD(ΨRΨL + ΨL ΨR). (24) a more detailed analysis and can shed a light on other pos- mass sible distinctions to the usual scenario based on oscillation The equation of motion for the total Lagrangian in this case is experiments (see for example [33]). the Dirac equation whose solutions are given in terms of the 123 411 Page 8 of 12 Eur. Phys. J. C (2021) 81 :411

Dirac bispinors presented in the main text. The mass term is We notice that (30) decouples into two independent sec- responsible for coupling the left-handed to the right-handed tors: components of the bispinor, yielding to chiral oscillations. ˆ i∂ |ψ± = pˆ ·ˆα |ψ± ± m β |ψ± , (31) Otherwise, the Majorana mass terms reads t M

(M) ¯ c ¯ c where L =−mR(Ψ ΨR + h.c.) − mL(Ψ ΨL + h.c.), (25) R L |ψ ± |ψ(c) Ψ c = βˆαˆ Ψ ∗ |ψ± = . (32) where i y denotes the charge-conjugated field 2 and the left and right handed components have different The |ψ± are eigenstates of charge conjugation, and follow- masses. The above Lagrangian term can be rewritten as [31] ing the literature we call them Majorana bispinors. We have (M) now the following scenario: a Dirac bispinor is an uncon- L =−mRωω¯ − mLχχ,¯ (26) strained bispinor with dynamical evolution given by the Dirac where equation; a Majorana bispinor satisfies the Majorana condi- c tion (is an eigenstate of charge conjugation) AND has the χ = ΨL + Ψ , L (27) ω = Ψ + Ψ c . dynamics given by the Dirac equation (31). It is common R R to discuss the properties of Majorana bispinors in the so- Notice that χ c = χ and ωc = ω. A more general mass term called Majorana representation of the Dirac matrices, where includes both Dirac an Majorana mass terms3: the Dirac matrices are all imaginary and charge conjugation is just a complex conjugation operation. To better compare (DM) mD ¯ ¯ c c L =− (ΨRΨL + Ψ Ψ + h.c.) Majorana and Dirac bispinors, we keep with the chiral repre- 2 R L sentation of the Dirac matrices. The single particle relativis- mR ¯ c − (Ψ ΨR + h.c.) 2 R tic quantum mechanics for Majorana particles is obtained by |ψ mL ¯ c considering the sector related to + . − (Ψ ΨL + h.c.). (28) 2 L The Majorana condition imposes a constrain in the form of a bispinor. In particular it relates the upper to the lower While in the main text we have considered the “usual” Dirac (or, in our language the left to the right-handed components) equation (the one obtained via the mass term (24)), we now of the bispinor. Let ψ(M) be a Majorana bispinor satisfying consider briefly quantum oscillations under the Majorana ψ(M) c = ψ(M) mass term and under the general Majorana-Dirac mass term. the condition . We can readily show that, if we write ⎡ ⎤ ( ) Degenerate Majorana mass term in the bispinor formalism ξ M (M) ⎣ R ⎦ ψ = , (33) ξ (M) While Majorana fermions are usually described within the L framework of quantum field theory, we will consider from ( ) the charge conjugation condition implies that ξ M = now own the framework of single particle relativistic quan- L ∗ tum mechanics and, as in the main text, consider the dynam- σˆ ξ (M) i y R and a general bispinor satisfying the Majorana ics of the bispinors. The single particle relativistic quantum condition must be of the form ⎡ ⎤ mechanics of the Dirac equation with a Majorana mass term ξ (M) has its own peculiarities which we will not discuss here, see (M) ⎣ R ⎦ ψ = ∗ , (34) for example [23–25,36]. R σˆ ξ (M) i y R In this section we consider mL = mR = mM,forwhich the equation of motion reads which is often called a right-handed Majorana bispinor, or of the form i∂ Ψ(x, t) =−iαˆ ·∇Ψ(x, t) + m βΨˆ c(x, t), (29) ⎡ ⎤ t M ( ) ∗ −iσˆ ξ M (M) ⎣ y L ⎦ or in the language used through the text ψ = , (35) L ξ (M) L ˆ c i∂t |ψ = pˆ ·ˆα |ψ + mMβ |ψ , (30) called a left-handed Majorana bispinor. Those bispinors cor- c ˆ ∗ where |ψ = iβαˆ y |ψ is the charge-conjugated bispinor. respond to (27), as can be seen by writing ψR,L in a two com- For now own, we call the Dirac equation with a Majorana ponent notation and using the explicitly form of the charge mass term as the Majorana equation. conjugation operator. We conclude that any bispinor satisfying the Majorana condition must have both left and right- 3 For convenience we have included factors 2 in the definition of the handed components, this is a well known result and com- masses. mented through the literature (see e.g. [22]). 123 Eur. Phys. J. C (2021) 81 :411 Page 9 of 12 411

The average chirality of a generic bispinor is given by thus, for our calculations in the momentum space, the charge conjugation has the effect ˆγ5=ψ|ˆγ5|ψ. (36) c |u±(p, m) =±|v∓(−p, m) . Since this is a real quantity, we have Adopting the shorthand notation ∗ ∗ ψ|ˆγ5|ψ= ψ|ˆγ5|ψ = (ψ|) γˆ5 (|ψ) , (37) p f p,m = 1 − , where in the last equality we have explicitly assumed that we E p,m + m are working in the chiral representation (thus γˆ5 is real). By p γˆ 2 =−ˆ gp,m = 1 + , recalling then that y I4 we can write E , + m p m ∗ ∗ E p,m + m ψ|ˆγ |ψ=(ψ|) γˆ (|ψ) N , = , (43) 5 5 p m ∗ ∗ 4E p,m = (ψ|) γˆ5iγˆy iγˆy (|ψ) =−(ψ|)∗ iγˆ γˆ |ψc, y 5 (38) notice that f−p,m = gp,m. We can write the plane wave Dirac bispinor describing the evolution of a Dirac state initially in where in the last line we have used that {ˆγy, γˆ5}=0. Finally, left-handed and negative helicity, see Eq. (6), as ψ|c = (ψ|)∗ γˆ γˆ † =−ˆγ since i y (because y y)wehave ( ) − |ψ ( ) ≡ ψ D ( ) = N iEp,m t | ( , ) m t m t p,m gp,me u− p m c c ˆγ5=ψ|ˆγ5|ψ=−ψ| γˆ5|ψ =−ˆγ5c, (39) iEp,m t − f p,me |v−(−p, m) . (44) c c where we have defined ˆγ5c =ψ| γˆ5|ψ . The above equations brings no new information: the average chirality of the The corresponding Majorana bispinor state is given by charge conjugated spinor is opposite to the average chiral- |ψ ity of the spinor. For example, if is right-handed, then ( ) Np,m − ψ M ( ) = √ iEp,m t (| ( , ) − | ( , )) ˆγ =1, and thus ˆγ =−1, in other words, |ψc is a m t gp,m e u− p m u+ p m 5 5 c 2 left-handed spinor. Charge conjugation flips the chirality of iEp,m t − f p,m e (|v−(−p, m) + |v+(−p, m)) . (45) a spinor. |ψc =|ψ ˆγ =ˆγ For a Majorana bispinor , thus 5 c 5 Here we have used that the charge conjugation includes a and from (39)wehave complex conjugation, which has to be taken into account in ˆγ =−ˆγ →ˆγ = . the spatial dependence of the state, which can be grouped 5 5 5 0 (40) (D) with the terms from ψm (t) by making p →−p,ina Thus, any bispinor that is its own charge conjugate must have similar fashion to what is done to group the negative-energy zero average chirality. Notice that these calculations were part of the wave function. performed for generic bispinors, and the only assumption is At t = 0 the state reads the Majorana condition. Therefore, if a bispinor satisfy the 1 − |+ Majorana condition it has zero chirality at any time, inde- ψ(M)( ) ≡ ψ(M)( ) = √ , m 0 0 |− (46) pendent of the specific time evolution it follows. This con- 2 ˆγ clusion can also be reached by computing 5 explicitly for while for the Dirac case we had a bispinor satisfying the Majorana condition, that’s it, for a bispinor of the forms (34,35). ψ(D)( ) ≡ ψ(D)( ) = 0 . m 0 0 |− (47) We now illustrate this point by considering the time evolution of a Majorana bispinor constructed with the Dirac ( ) ( ) ( ) ψ D ( ) ψ M ( ) bispinor of the main text. The Majorana bispinor ψ M is Whether m 0 or m 0 really represents the initial thus written as state of a mass eigenstate depends on the specific weak processes that generates the particle, and will not be discussed (D) (D) c ( ) ψ + ψ ψ M = √ . (41) here. 2 The survival probability of (46) is given by In what follows we use the following relations for the ( ) ( ) ( ) 2 P M (t) = ψ M (0)ψ M (t) bispinors (see Eq. (5)): S m m 2 c m 2 ip·x −ip·x = 1 − sin (E , t) = P(t). (48) e |u±(p, m) =±e |v∓(p, m) , (42) 2 p m E p,m 123 411 Page 10 of 12 Eur. Phys. J. C (2021) 81 :411

The survival probability of the considered initial state is the we are interested in the dynamics of |ψL (t) under the general same as in the Dirac case. This is consistent with the con- mass term (28).4 clusions drawn in [21] regarding chiral oscillations: Majo- This Lagrangian can be rewritten as rana neutrinos also exhibit quantum oscillations. In this case, ( ) L DM =−m η¯ η − m η¯ η , (55) the upper right-handed and positive helicity component can 1 1 1 2 2 2 oscillate to a left-handed positive helicity component, while where [31] the lower left-handed and negative helicity oscillates to a right-handed negative helicity component. These two oscil- + ± ( + )2 + 2 mL mR mL mR 4mD lation channels yield the survival probability obtained in the m1,2 = , above equation. This can be further understood by consider- 4 ( ) P M [ , , , ]T η1 = cos (φ)χ − sin (φ)ω, ing the probability L,− of measuring 0 0 0 1 (a left handed and negative helicity state), which is given by η2 = sin (φ)χ + cos (φ)ω, 2m tan (2φ) = D . (56) ( ) P(t) − P M = . (49) mL mR L,− 2 We then write ηi (i = 1, 2) in terms of their right and This probability is half of the survival probability for the left-chiral components Dirac bispinor state (7). Since (45) is a superposition of both η positive and negative helicities, the probability that the state η = i,R , i η (57) is in a right-handed and positive helicity state is not null. In i,L fact, such probability is given by and considering negative helicity eigenstates, we get from the equations of motion: (M) P(t) P (t) = . (50) E , − p −m η , R,+ p mi i i R = , 2 − + η 0 (58) mi E p,mi p i,L We therefore notice that (±) =± , = which gives the eigenenergies E p,mi E p mi P(M)( ) + P(M)( ) = P( ). ± p2 + m2 and the eigenvectors R,+ t L,− t t (51) i (+) (θ ) (θ ) η Finally, the average chirality of (45) vanishes: as discussed in fi cos i sin i i,R (−) = , the beginning of this section, the Majorana condition implies − sin (θ ) cos (θ ) η , fi i i i L that the bispinor is a superposition of left and right chirality m tan (2θ ) = i . (59) components, thus its average chirality vanishes. i p

The eigenstates of the Hamiltonian time evolution are given Quantum oscillations of left-handed chiral eigenstates by

(±) ∓ , (±) | ( )= iEp mi t | . The calculations above are an illustrative example of the fi t e fi (60) dynamics of a Majorana bispinor obtained with the corre- |ψ (t) sponding Dirac bispinors. We can instead turn our atten- To recover the dynamics of L , we backtrack the prob- tion to the formalism of [26,27] to evaluate the time evolu- lem by using the relations (56) and then Eqs. (27), such that tion of an initially left-handed state.We generalize the result − , + |ψ ( )= (φ) (θ ) iEp m1 t | in the aforementioned references and consider through this L t cos sin 1 e f1 section the general Dirac-Majorana mass term (28), which , − + (θ ) iEp m1 t | χ = Ψ + Ψ c cos 1 e f1 can be written in terms of the bispinors L L and ω = Ψ + Ψ c − , + R as + (φ) (θ ) iEp m2 t | R sin sin 2 e f2

(DM) mD mR mL L =− (ωχ¯ +¯χω)− ωω¯ − χχ.¯ (52) 4 2 2 2 Notice that the connection with the notation used in Eq. (28)is: 0 ψR ΨL = ,ΨR = , In other words, considering ψL 0 (54) ψc Ψ c = 0 ,Ψc = R . ψ L ψc R R L 0 Ψ = ΨL + ΨR = , (53) ψL 123 Eur. Phys. J. C (2021) 81 :411 Page 11 of 12 411 (D) (D) (M) (M) iEp,m t − ψ ( )|ˆγ |ψ ( )=ψ ( )|ˆγ |ψ ( ) + cos (θ2)e 2 | f , (61) t 5 t t 5 t , thus chiral 2 L L L L ψ(M)( ) ψ(D)( ) oscillations of the states L t and L t are equal. and therefore Those dynamical features are also the same in the more general case of mD = 0 and mR = mL. This is expected: for |ψ ( )=A ( )|ψ +A ( )|ψ +Ac ( )|ψc +Ac ( )|ψc . a vanishing Dirac mass, the equations of motion are fully L t L t L R t R L t L R t R (62) decoupled in terms of the bispinor χ and ω, and the dynamics of a state initially in a left-handed chirality is determined The time dependent coefficients are given explicitly by entirely by the sector related to χ, which only induces oscil- c lations between |ψL and |ψ . In the case mD = mL = 0, R 2 AL (t) = cos (φ) cos (E p,m t) + i cos (2θ ) sin (E p,m t) there are no quantum oscillations. 1 1 1 2 When both Dirac and Majorana masses are present, the + sin (φ) cos (E p,m t) + i cos (2θ2) sin (E p,m t) , 2 2 survival probability and the average chirality display a more i sin (2φ) A ( ) = ( θ ) ( ) R t sin 2 1 sin E p,m1 t intricate dynamics. In this case, there will be interferences 2 − ( θ ) ( ) , due to the different values of the masses m1,2 which drive sin 2 2 sin E p,m2 t different oscillation channels. Further investigation of such c 1 A (t) =− sin(2φ) cos(E p,m t) + i cos(2θ ) sin(E p,m t) L 2 1 1 1 effects will be addressed in a future work. + 1 ( φ) ( ) + ( θ ) ( ) , sin 2 cos E p,m2 t i cos 2 2 sin E p,m2 t 2 Ac ( ) =− 2(φ) ( θ ) ( ) R t i cos sin 2 1 sin E p,m1 t References + 2(φ) ( θ ) ( ) . sin sin 2 2 sin E p,m2 t (63) 1. O. Klein, Z. Phys. 53, 157 (1929) The survival probability of the state is given by 2. C. Itzykson, J.B. Zuber, Quantum Field Theory (Mc Graw-Hill Inc., New York, 2006) P(G)( ) =|A ( )|2 S t L t (64) 3. E. Schrödinger, Sitzungsber. Preuss. Akad. Wiss. Berlin 28, 418 (1930) while the average chirality is calculated by using that 4. W.K. Tung, Group Theory (World Scientific, London, 2003) ψ |ˆγ |ψ =ψc |ˆγ |ψc =− ψ |ˆγ |ψ = 37 L 5 L L 5 L 1 and R 5 R 5. S. De Leo, P. Rotelli, Int. J. Theor. Phys. , 2193 (1998) c c 6. A.E. Bernardini, Eur. Phys. J. C 50, 673 (2007) ψ |ˆγ5|ψ =1, such that R R 7. A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. ˆγ ( )=ψ ( )|ˆγ |ψ ( )=−|A ( )|2 −|Ac ( )|2 Geim, Rev. Mod. Phys. 81, 109 (2009) 5 t L t 5 L t L t L t 98 +|A ( )|2 +|Ac ( )|2. 8. L. Lamata, J. León, T. Schätz, E. Solano, Phys. Rev. Lett. , R t R t (65) 253005 (2007) = = 9. L.Lamata,J.Casanova,R.Gerritsma,C.F.Roos,J.J.García-Ripoll, For a pure Dirac mass term, mR mL 0, the explicit E. Solano, New J. Phys. 13, 095003 (2011) temporal evolution (62) simplifies to 10. T.M. Rusin, W. Zawadzki, Phys. Rev. B 78, 125419 (2008) 11. T.M. Rusin, W. Zawadzki, Phys. Rev. B 80, 045416 (2009) ψ(D)( ) = A( ) |ψ + B( ) |ψ , L t t L t R (66) 12. V.A.S.V. Bittencourt, A.E. Bernardini, Phys. Rev. B 95, 195145 (2017) where 13. A.E. Bernardini, S. De Leo, Eur. Phys. J. C 37, 471 (2005) 14. A.E. Bernardini, S. De Leo, Phys. Rev. D 71, 076008 (2005) A( ) = ( ) + p ( ), 15. C. Giunti, C.W. Kim, Fundamentals of Neutrino Physics and Astro- t cos E p,mD t i sin E p,mD t E p,m physics (Oxford University Press Inc., New York, 2007) 16. E. Baracchini et al. (PTOLEMY Collaboration), JCAP 07, 047 B( ) =− mD ( ). t i sin E p,mD t (67) (2019) E p,m 17. A.J. Long, C. Lunardini, E. Sabancilar, JCAP 1408, 038 (2014) 18. E. Roulet, F. Vissani, JCAP 1810, 049 (2018) This dynamics is in complete agreement with the one derived 19. C. Nishi, Phys. Rev. D 73, 053013 (2006) in the main text within the bispinor formalism. For a pure 20. A.E. Bernardini, M.M. Guzzo, C.C. Nishi, Fortschr. Phys. 59(5), Majorana mass term with degenerate masses, m = m = 372 (2011) R L 21. S.-F. Ge, P. Pasquini, Phys. Lett. B 811, 135961 (2020) m , and m = 0, we have M D 22. P.B.Pal,Am.J.Phys.79, 485 (2011) 23. M. Dvornikov, J. Maalampi, Phys. Rev. D 79, 113015 (2009) ψ(M)( ) = A(M)( ) |ψ + B(M)( ) ψc , L t t L t R (68) 24. H. Arod´z, Acta Phys. Pol. B 50, 2165 (2019) 25. M.H. Al-Hashimi, A.M. Shalaby, U.-J. Wiese, Phys. Rev. D 95, ( ) ( ) where A M (t) and B M (t) are given by (67) with the substi- 065007 (2017) ( ) tution m → m . The survival probability of ψ M (t) is 26. S. Esposito, N. Tancredi, Mod. Phys. Lett. A 12(25), 1829 (1997) D M L 27. S. Esposito, Mod. Phys. Lett. A 13(29), 5023 (1998) ( ) the same as the one obtained for the state ψ D (t) ,butthe 28. M. Blasone, G. Vitiello, Ann. Phys. 244, 283 (1995) L 29. M. Blasone, P.A. Henning, G. Vitiello, Phys. Lett. B 451, 140 ψc transition probability it to R . Moreover, we conclude that (1999) 123 411 Page 12 of 12 Eur. Phys. J. C (2021) 81 :411

30. M. Blasone, L. Smaldone, G. Vitiello, J. Phys. Conf. Ser. 1275, 33. S.M. Bilenky, B. Pontecorvo, Phys. Lett. 102B(1), 32 (1981) 012023 (2019) 34. M. Dvornikov, (2010). arXiv:1011.4300 [hep-th] 31. T.-P.Cheng, L.-F. Li, Gauge Theory of Elementary Particle Physics 35. J. Lesgourgues, S. Pastor, Phys. Rep. 429, 307 (2006) (Oxford Science publication, Oxford, 1995) 36. H. Arod´z, Phys. Lett. A 383(12), 1242 (2019) 32. A.E. Bernardini, Eur. Phys. J. C 46, 113 (2006)

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