Quantum Field Theory of Space-Like Neutrino

Eur. Phys. J. C (2021) 81:716 https://doi.org/10.1140/epjc/s10052-021-09494-x

Regular Article - Theoretical Physics

Quantum ﬁeld theory of space-like neutrino

Jakub Rembieli´nski1,a , Paweł Caban1,b , Jacek Ciborowski2,c 1 Department of Theoretical Physics, Faculty of Physics and Applied Informatics, University of Łód´z, Pomorska 149/153, 90-236 Lodz, Poland 2 Department of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland

Received: 27 December 2020 / Accepted: 26 July 2021 © The Author(s) 2021

Abstract We performed a Lorentz covariant quantization spin components and consequently neutrinos should be found of the spin-1/2 fermion field assuming the space-like energy- in two helicity states. However, only the left-handed helic- momentum dispersion relation. We achieved the task in the ity component of the neutrino and the right-handed of the following steps: (i) determining the unitary realizations of antineutrino have been observed in experiments. A standard the inhomogenous Lorentz group in the preferred frame sce- but rather technical explanation of this fact makes use of the nario by means of the Wigner–Mackey induction procedure see-saw mechanism [1–3]. In contrast, we adopt a hypothesis and constructing the Fock space; (ii) formulating the the- that the neutrino is a particle satisfying the space-like disper- ory in a manifestly covariant way by constructing the field sion relation. This assumption is suggested by a repeating amplitudes according to the Weinberg method; (iii) obtain- occurrence of negative values for the neutrino mass squared ing the final constraints on the amplitudes by postulating a measured in numerous recent tritium-decay experiments [4– Dirac-like free field equation. Our theory allows to predict 9]. This observation does not make a proof that neutrino is all chiral properties of the neutrinos, preserving the Stan- a tachyon because of an insufficient level of confidence of dard Model dynamics. We discussed the form of the funda- each of these separate results, however, it encourages con- mental observables, energy and helicity, and show that non- sidering a possible theoretical descriptions of this possibil- + 1 observation of the 2 helicity state of the neutrino and the ity. Such trials have already been undertaken in the past. A − 1 hypothesis that neutrino might be a space-like particle was 2 helicity state of the antineutrino could be a direct consequence of the “tachyoneity” of neutrinos at the free level. first discussed by Chodos et al. [10]. Some arguments sup- We found that the free field theory of the space-like neutrino porting this proposition were also presented by Giannetto et is not invariant under the C and P transformations separately al. [11]. However, these attempts were unsuccessful due to but is CP-invariant. We calculated and analyzed the electron the fact that the standard relativistic quantum field theory is energy spectrum in tritium decay within the framework of inapplicable for describing space-like particles, as pointed our theory and found an excellent agreement with the recent out by Kamoi and Kamefuchi [12] and Nakanishi [13]. On measurement of KATRIN. In our formalism the questions of the other hand, it was shown [14] that an approach involving negative/imaginary energies and the causality problem does the notion of a preferred frame (PF) allows to construct a not appear. Lorentz-covariant quantum field-theoretical model of a rel- 1 ativistic helicity- 2 fermionic tachyon. In this way one can avoid the fundamental difficulties related to the lack of a 1 Introduction finite lower energy limit, appearance of infinite spin multiplets and causal problems appearing in the standard attempts After nearly 90 years of the neutrino history this particle to describe tachyons. In the space-like neutrino case the cos- is still an enigma with a number of unanswered questions mic neutrino background (CNB) frame, an artefact of the in the neutrino physics and the Standard Model. We know electroweak phase transition [15], is a natural candidate for that neutrinos oscillate so at last two neutrino generations the preferred frame. In the above framework the β decay was are massive. Therefore the neutrino field should possess two considered in [16] and the corresponding decay rate (energy spectrum) for electrons was derived and discussed in the context of the neutrino mass measurement. For other contribu- a e-mail: [email protected] tions to the tachyonic neutrino hypothesis see [17–24]. b e-mail: [email protected] (corresponding author) c e-mail: [email protected]

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References to the notion of a PF in the context of the quan- the Wigner–Mackey induction procedure [44]. Secondly, in tum theory are owed to several authors. The historical term of Sect. 4, we construct the Fock space by generation of the mul- “aether” used in the field-theoretical context by Dirac [25,26] tiparticle basis states from the Lorentz invariant vacuum state was superseded by that of the PF, as, e.g., in de Broglie–Bohm by using the corresponding algebra of creation/annihilation formulation of quantum mechanics [27,28]. Bell suggested operators. This quantization procedure is formulated for all that it would have been helpful to consider a PF at the fun- inertial frames. Thirdly, in Sect. 5, we formulate the theory in damental level for resolving incompatibilities between spe- a manifestly covariant way by constructing the field ampli- cial relativity theory and nonlocality of quantum mechan- tudes according to the Weinberg method [45]. Fourthly, in ics [29] – an opinion also shared by other authors [30–35]. Sect. 6, we obtain the final constraints on the amplitudes by Let us mention in particular: Lorentz-violating extensions postulating a Dirac-like free field equation determining the of the Standard Model [36–39], approaches to classical and free space-like neutrino field completely. Next, in Sect. 7 quantum gravity like the Einstein’s aether [40] and Hoˇrava– we show that the amplitudes of the space-like neutrino field Lifshitz theories of gravity [41] (including vacuum solutions have properties which are consistent with the high energy in this model [42]), and the so-called doubly special relativ- chiral properties of the neutrino observed in reality. On the ity theories [43], characterized by modified dispersion rela- other hand, we predict that at very low neutrino energies, both tions, common for the Lorentz violating models. In almost helicities should be observed. We discuss the form of the fun- all of the above theories specific physical effects are pre- damental observables, namely energy and helicity. Sect. 8 dicted, of magnitude usually suppressed by a power of the is devoted to the discussion of the propagator of the space- Planck scale, like, e.g., vacuum birefringence [36–40,43]. like particles. In Sect. 9 we find that the free field theory of This brief outlook demonstrates that the concept of a PF has the space-like neutrino is not invariant under the parity and been frequently referred to in the context of Lorentz symme- charge conjugation transformations separately, however, it is try violation within numerous contemporary theories. invariant under the CP transformation. Finally, in Sect. 11 we It is important to stress that if tachyons exist, one could, construct the anomaly free variant of the electroweak model in principle, consider synchronizing distant clocks in almost and calculate and analyze the β decay rate (electron energy absolute way (instantaneous synchronization) in the limit of spectrum) in this context. zero energy (infinite velocity) if they interact with matter Because the tachyon kinematics in the preferred frame with finite probability in these conditions. In consequence, scenario as well as the construction of the corresponding one has a possibility of introducing absolute time between quantum field theory is not widely known, we present our observers. The only way to reconcile this implication with approach with necessary details, restricting ourselves to one the Lorentz symmetry lies in an assumption of existence of neutrino generation for simplicity. a preferred frame [14]. We discuss this point in Sect. 2. In the present paper we formulate a fully consistent quantum field theory of the space-like neutrino with both one- half helicity components under the assumption of the exis- 2 Tachyon kinematics in the preferred frame scenario tence of the preferred frame. In our approach the states of μ tachyonic field excitations observed from an arbitrary iner- Tachyon is a particle with a space-like four-momentum k , tial frame should depend on the PF four-velocity as seen from satisfying the following Lorentz-covariant dispersion rela- the observer frame. All other physical fields/states are sim- tion ply unaffected by the PF. This means that the PF is related μ 02 2 2 with the tachyonic sector only. As we will see the preferred k kμ = k − k =−κ , (1) frame concept is crucial for the successful construction of a viable theory of fermionic tachyons enabling to elimi- where κ denotes the “tachyonic mass” (κ>0). However, κ nate the issue of negative/imaginary energies and causality can be also viewed as a residual momentum of the tachyon problems as well as to perform the field quantization proce- in the limit of zero energy. Eq. (1) defines a one sheet hyper- dure in arbitrary inertial frames. At the interaction level we boloid where energy k0 takes the values from minus to plus obtain the anomaly free, perturbatively renormalizable vari- infinity. Lorentz transformations relating inertial frames can ant of the electroweak model. We construct this theory in few transform positive k0 to any negative value. Hence, this steps. Firstly, in Sect. 2 we construct the one-particle phase causes the vacuum instability on the quantum level, i.e., pos- space and discuss its Lorentz covariance as well as the issue sibility of a spontaneous creation from the vacuum of pairs of of imaginary energies, stability of the theory and causality. particles with the total four-momentum equal to zero. Below Next, in Sect. 3, we determine the unitary realizations of the we show that this problem can be resolved in the preferred inhomogenous Lorentz group under the condition of exis- frame scenario. Let us denote the inertial observer’s frame tence of an inertial preferred frame. We do this by means of by Σu. From his point of view the PF, ΣPF, moves with a 123 Eur. Phys. J. C (2021) 81:716 Page 3 of 14 716 constant four-velocity uμ satisfying 2.1 Problem of negative energies 02 − 2 = . u u 1 (2) Negative energies can formally appear in the standard Thus for observers stationary in the PF, its four-velocity is description of tachyons for the reason that covariant lower bound of energy does not exist in this case. In contrast, exis- given by u PF = (1, 0, 0, 0). The PF three-velocity (in units tence of a preferred frame allows to define the Lorentz covari- of c) is given by V = u/u0 so u0 = 1/ 1 − V2 is the μ ant condition (4), constituting the lower bound of energy in Lorentz factor of the PF. Now, using the four-momentum k μ each inertial frame, eliminating possibility of the kinematical of the tachyon and the four-velocity u we can construct a instability. Indeed, if the four-momentum kμ belongs to the Lorentz invariant upper half of the one-sheet hyperboloid, i.e., it satisfies the μ 0 0 μ q = uk = u kμ = u k − u · k (3) inequality q = uk > 0 then −k must belong to the lower part of hyperboloid because (−uk) =−q < 0 does not equal to the tachyonic energy measured in the preferred satisfy the condition (4). Therefore, the condition (4) elim- frame. Hence, the physically acceptable four-momenta kμ inates the possibility of the kinematical instability. We note are bounded by the Lorentz covariant condition that this condition is analogous to choosing the upper energy- q > 0, (4) momentum cone for massless particles as the physical one. guaranting nonnegativity of energy in the preferred frame. 2.2 Problem of imaginary energies We see from Eq. (3) that only in the preferred frame, ΣPF, tachyons have always non negative energies. Indeed, applying the Lorentz boost transformation one can also obtain This is rather fictitious problem in the case of particles sat- kinematical states of the tachyon with negative energies in isfying the dispersion relation (1). It is rooted to the fact 0 ∼ 2 − κ2 0 other inertial frames (but still with a lower bound set by that Eq. (1) implies k k so k can be in princi- 2 <κ2 Eq. (4)). The explicit form of the energy and the value of ple imaginary for k and we can expect exponentially the momentum obtained from the relations (1,2,3,4)inan divergent trajectories in that case. However this is impossible for tachyons on the mass shell, i.e., satisfying the dispersion arbitrary inertial frame Σu is the following relation (1) because in that case the inequality |k| >κmust k0 hold for physical momenta. In our formalism this is guaran- qu0 + cos θ (u0)2 − 1 q2 + κ2[cos2 θ + (u0)2 sin2 θ] teed by the choice of the invariant measure (8) which vanishes = , (5) inside the sphere defined by inequality k2 = κ2. Absence of cos2 θ + (u0)2 sin2 θ |k|≡ω(q, u,θ) the imaginary eigenvalues of the energy operator is also evi- dent in the explicit formulation of our theory in Sects. 4–6. q cos θ (u0)2 − 1 + u0 q2 + κ2[cos2 θ + (u0)2 sin2 θ] = , (6) cos2 θ + (u0)2 sin2 θ 2.3 Problem of non-causal behavior where q > 0, u0 = 1/ 1 − V2, θ is the angle between the momentum, k, and the PF velocity, V. Thus, the energy The notion of causality is inseparably related to the definition bound in an arbitrary frame Σu has the following form of the coordinate time. In the standard relativistic theories time is identified with the zeroth coordinate in the Minkowski κ|V| cos θ k0 > . (7) space-time. Consequently, for a space-like separation of − (| | θ)2 1 V cos events, the Lorentz transformations can change their time We introduce an invariant measure respecting the disper- ordering. However, existence of a preferred frame provides a sion relation (1) and the covariant condition (4): solution of this problem. Namely, we can introduce another, μ Lorentz invariant, dynamical parameter, T := uμx . Notice, dμ(k, u) = d4kδ(k2 + κ2)Θ(uk) that this parameter allows a Lorentz invariant T -time order- = 0 1 ing is in this case Lorentz invariant. In the PF, T xPF, = |k| Θ(q) dq dΩ, (8) i.e., t = T so it is equal to the Einsteinian time (in the 2u0 PF c units). In an arbitrary frame Σu from the definition of T , Ω τ ≡ tPF = t − V · x V (see Appendix A), where d is the solid angle differential we obtain u0 where is the defined related to the neutrino momentum k, |k| is given in Eq. (6) above velocity of the preferred frame, i.e., V = u/u0.The and Θ is the Heaviside Theta function. coordinate time redefinition τ = t − V · x between the Ein- Now, we comment common objections against the tachy- steinian time t and the time τ is simply the admissible change onic theories in the context of the formalism introduced of clock synchronization [14]. For a stationary observer, in above. each fixed point (dx = 0) dt = dτ, i.e., the flow of the time 123 716 Page 4 of 14 Eur. Phys. J. C (2021) 81:716 is the same. However, it follows from the time redefinition at his disposal the Hilbert space Hu of neutrino states. The dτ = − · ϑ ϑ = dx H formula that dt 1 V , where the velocity dt . family of Hilbert spaces u form a bundle corresponding to Notice that for subluminal or luminal motion (|ϑ|≤1) the the bundle of inertial frames Σu, i.e., to the quotient mani- 3 derivative dτ/dt is always nonnegative, i.e., the arrows of fold SO(1, 3)/SO(3) ∼ R as the base space and with Hu the Einsteinian time t and the time τ are the same (causality, as fibers. To apply the Wigner-Mackey construction to this i.e., sequence of events is the same) . However, for super- case we should find the little group of the pair of four-vectors luminal motion (|ϑ| > 1), the derivative dτ/dt can change (k, u) determining the neutrino state. To do this we transform sign to negative because the scalar product V·ϑ can, in some the pair (k, u) to the preferred frame by action of the boost τ −1 ( −1 = , −1 = ) velocity configurations, exceed 1. Because d is always pos- Lu to obtain the pair Lu k k Lu u u PF .Next −1 itive (the arrow of time tPF is fixed) then dt must change the we rotate k with the help of the rotation Rn to the z-axis ˜ −1 2 2 sign, which corresponds to the indeterminacy of the causal in order to obtain k = (Lu Rn) k = (q; 0, 0, q + κ ) relation in the Einsteinian synchronization scheme for veloc- while u PF is left unchanged (for the explicit form of Lu and ities higher than the light velocity while, in terms of the time Rn see Appendix B). Thus, the pair (k, u) can be obtained ˜ τ, it is determined for all velocities. Moreover, the time- from the standard pair (k, u PF) by a sequence of Lorentz τ synchronization satisfies the crucial physical requirement transformations Lu Rn, i.e., – the average value of the light velocity over closed paths (k, u) = L R (k˜, u ) (9) is frame independent and equal to c. For more information u n PF on the issue of clock synchronization in the special relativ- provided the unit vector n is equal to ity see [46–48]. Summarizing, only the τ synchronization 1 q + k0 is adequate to determine uniquely the sequence of events n = n(k, u) = k − u , (10) + 0 (causality) for phenomena with participation of tachyons in q2 + κ2 1 u the presence of the PF or equivalently, causality should be where, by means of Eqs. (3) and (4), k0 = (q + u · k)/u0 and referred to observers stationary in the PF. q > 0. It is obvious that the orthogonal group SO(2) is the ˜ stability group of the pair (k, u PF). Therefore, irreducible unitary orbits of the inhomogeneous Lorentz group should 3 Space of states of the space-like neutrino be induced from the unitary irreducible representations of SO(2), i.e., from the one-dimensional representation of the The condition sine qua non to apply the tachyonic hypothesis U(1) group. Applying the Wigner–Mackey procedure to the to neutrino physics is to construct a quantum field theory of basis vectors |k, u,λ, λ ∈ R, in the manifold of Hilbert space-like fields. Indeed, the standard relativistic field theory spaces Hu we obtain a result that the unitary action of the is inapplicable to this case because of two reasons: Firstly, the Lorentz group is of the form Wigner little group of the space-like four-momentum of the λϕ(Λ, , ) tachyon is the noncompact SO(2, 1) group, so its unitary rep- U(Λ)|k, u,λ=ei k u |Λk,Λu,λ, (11) resentations are scalar or infinite-dimensional. Consequently, where eiϕ(Λ,k,u) is the phase factor corresponding to the the spin multiplets are either one-dimensional or infinite- 1 Wigner rotation dimensional. Therefore, a spin- 2 neutrino cannot be iden- −1 tified with such representations. Secondly, if we go around W(k, u,Λ)= (LΛu Rn(Λk,Λu)) ΛLu Rn(k,u). (12) the above problem and try describing the tachyonic neutrino with the help of the bi-spinor nonunitary representation by The standard arguments lead to integer or half-integer values λ identifying the Lagrange density appropriate to the space- of . In the case of the tachyonic neutrino we restrict further λ =±1 like dispersion relation as is done, e.g., in [10], we evidently considerations to the values 2 . H loose unitarity, i.e., the probabilistic character of the quan- Moreover, in each fixed space u we adopt the following tum description. However, in the scenario with a PF, these Lorentz covariant normalization for the momentum eigen- difficulties do not arise. states for neutrinos and antineutrinos As was stated above, a reasonable description of tachyons , ,λ| , ,λ= 1 0δ δ( − )δ(ˆ − ˆ ), k u k u |k| 2u λλ q q k k (13) needs the presence of a preferred frame. Consequently, the tachyonic neutrino basis states should be dependent not only where kˆ denotes the unit vector k/|k| and δ(kˆ − kˆ ) is the on the space-like four-momentum, kμ, but also on the PF spherical Dirac delta (see Appendix A). Notice that it is con- four-velocity, uμ. Our aim is to apply the Wigner–Mackey venient to choose the z-axis in the direction of the preferred induction procedure in this case. Hereafter we will denote frame velocity V, the angle θ is the polar angle of kˆ in such eigenvectors of the four-momentum operator as |k, u,λ, a case. where λ is identified with the neutrino/antineutrino helic- Therefore, the one-particle space of tachyonic states is a ity. Now, an observer in an arbitrary reference frame Σu has direct sum of one particle tachyonic neutrino ⊕ antineutrino 123 Eur. Phys. J. C (2021) 81:716 Page 5 of 14 716 space (of course neutrino and antineutrino spaces are mutu- fundamental observables. The helicity operator λˆ is defined ally orthogonal). The completeness relation in this space has as the form † λ(ˆ u) = dμ(k, u)λ a (k, u)aλ(k, u) λ λ dμ(k, u) |k, u,λν k, u,λ|+|k, u,λν¯ k, u,λ| = I, (14) λ † +bλ(k, u)bλ(k, u) (22) where the Lorentz invariant measure dμ is defined in Eq. (8). We prove this relation explicitly in Appendix A. and by means of (18–19) it satisfies

ˆ † † [λ, aλ(k, u)]=λaλ(k, u), (23) ˆ † † 4 Fock space of the space-like neutrino [λ, bλ(k, u)]=λbλ(k, u). (24)

We construct a free field theory of the tachyonic neutrino in Similarly, we define the covariant four-momentum operator ˆ the preferred frame scenario, in close analogy with the stan- Pμ by the standard formula dard formalism, restricted to a single neutrino generation for ˆ † simplicity. Because an irreducible realization of the Lorentz Pμ(u) = dμ(k, u)kμ aλ(k, u)aλ(k, u) λ group is fixed in our case by the choice of , the basis vec- λ tors |k, u,λ are obtained by the action of the neutrino cre- † † +bλ(k, u)bλ(k, u) (25) ation operators aλ(k, u) and antineutrino creation operators † bλ(k, u) on the normalized to the unity vacuum vector |0 which implies, that defined by the standard conditions ˆ † † [Pμ, aλ(k, u)]=kμaλ(k, u), (26) aλ(k, u)|0=bλ(k, u)|0=0. (15) ˆ † † [Pμ, b (k, u)]=kμb (k, u). (27) Thus λ λ

† |k, u,λν = aλ(k, u)|0, (16) 5 Manifestly covariant formulation † |k, u,λν¯ = bλ(k, u)|0. (17) The neutrino field ν(x, u) is defined as the Dirac bispinor The above vectors respect the normalization (13) provided operator of the form the creation and annihilation operators satisfy, for each fixed 1 ikx † u, the following anti-commutation canonical relations να(x, u) = dμ(k, u) e vαλ(k, u)b (k, u) ( π)3/2 λ 2 λ { ( , ), † ( , )} aλ k u aσ k u −ikx +e uαλ(k, u)aλ(k, u) , (28) 1 0 ˆ ˆ = 2u δ(q − q )δ(k − k )δλσ , (18) ω(q, u,θ) with the standard manifestly covariant transformation rule { ( , ), † ( , )} − bλ k u bσ k u U(Λ)ν(x, u)U(Λ)† = S(Λ 1)ν(Λx,Λu), (29)

1 0 ˆ ˆ 1 1 = 2u δ(q − q )δ(k − k )δλσ . (19) (Λ) ( ,0) ⊕ (0, ) ω(q, u,θ) where S belongs to the representation D 2 D 2 of the homogenous Lorentz group. The remaining anti-commutators vanish. In order to fulﬁll the transformation law (29), the ampli- To reproduce formula (11), creation operators should tudes vλ(k, u) and uλ(k, u), must satisfy the Weinberg con- transform under the action of the Lorentz group according to sistency conditions obtained with the use of Eqs. (20, 21, 28), the law namely

† † iλϕ(Λ,k,u) † −iλϕ(Λ,k,u) U(Λ)aλ(k, u)U(Λ) = e aλ(Λk,Λu), (20) uλ(Λk,Λu) = S(Λ)uλ(k, u)e , (30) † † iλϕ(Λ,k,u) † iλϕ(Λ,k,u) U(Λ)bλ(k, u)U(Λ) = e bλ(Λk,Λu). (21) vλ(Λk,Λu) = S(Λ)vλ(k, u)e . (31)

The corresponding Fock space of multiparticle states can In the following we choose the Weyl bi-spinor representation be now defined in a standard way by a successive action of of the Lorentz group in the form creation operators on the vacuum state. At this stage, given the Lorentz invariant measure, the space of states and the cor- A 0 S(Λ(A)) = −1 , (32) responding transformation rules, we are ready to define the 0 A† 123 716 Page 6 of 14 Eur. Phys. J. C (2021) 81:716 where the matrices A belong to the SL(2, C) group. The cor- ing conditions for the chiral amplitudes responding representation of the γ matrices is the following ˜ −iλφ ˜ U (φ)u˜ λ (k, u )e = u˜ λ (k, u ), (44) 0 I 0 σ k z L/R PF L/R PF γ 0 = ,γk = , λφ k (33) U (φ)˜ (˜, ) i = ˜ (˜, ), I 0 −σ 0 z vλL/R k u PF e vλL/R k u PF (45) −I 0 γ 5 = γ = iγ 0γ 1γ 2γ 3 = , (34) leading to 5 0 I where σ k are the standard Pauli matrices. Note that the Pauli 0 1 u˜ − / = c / , u˜ / = c , (46) 1 2L/R L R 1 1 2L/R L/R 0 matrices are contravariant. In the Weyl representation the ( , ) =[ ( , )] neutrino field and the amplitudes v k u vαλ k u and 0 1 v˜ / = d / , v˜− / = d . (47) u(k, u) =[uαλ(k, u)] admit the following chiral decompo- 1 2L/R L R 1 1 2L/R L/R 0 sitions u (k, u) v (k, u) In order to determine coefficients cL/R, c / , dL/R, d / ,we u(k, u) = L , v(k, u) = L (35) L R L R uR(k, u) vR(k, u) must use a manifestly covariant first order differential equation connecting the left and right handed chiral components = 1 ( ∓γ 5) corresponding to the chiral projections PL/R 2 I . of the neutrino field. This equation must be consistent with By means of Eqs. (9, 12, 30, 31, 32) we obtain the above field construction procedure.

˜ u(k, u) = S(Lu Rn)u(k, u PF), (36) ˜ 6 Dirac-like equation v(k, u) = S(Lu Rn)v(k, u PF), (37) where Now, we introduce an analog of the Dirac equation in our formalism to conclude defining the free dynamics of the tachy- L U u n 0 onic neutrino field. A variety of the Lorentz covariant Dirac- S(Lu Rn) = , (38) 0 Luπ Un 1 like equations for spin- 2 tachyons was discussed in Ref. [14], however, only two equations satisfy the CPT invariance. Here and Lu is the Lorentz boost matrix representing Lu in the we adopt the simplest of them, namely left-handed spinor space, Luπ is the boost acting in the right- π 5 μ handed spinor space, denotes the parity operation on four- (γ γ i∂μ − κ)ν(x, u) = 0. (48) vectors whereas Un represents the rotation Rn. Explicitly This equation was first introduced by Tanaka [49] while the 1 0 corresponding Lagrangian was proposed by Chodos et al. Lu = (1 + u )I − u · σ , (39) 2(1 + u0) [10]. Equation (48) implies the space-like dispersion relation − 1 L π = L 1 = ( + 0) + · σ , u u 1 u I u (40) for kμ and fixes interrelations between left and right com- 2(1 + u0) ponents of the neutrino field. We use this equation for the 3 1 2 1 1 + n −n + in free field ν(x, u) introduced in Eq. (28) in the context of Un = (41) ( + 3) n1 + in2 1 + n3 2 1 n our approach to determine the coefficients cL/R, cL/R, dL/R, dL/R of field amplitudes in Eqs. (46, 47). Thus we obtain the and the unit vector n(k, u) is defined in Eq. (10). following equations for the amplitudes ˜ Now, choosing k = k, u = u PF in (30, 31) and Λ as the Λ = (φ) 5 μ stability group element of this pair, i.e., Rz , where (γ γ kμ − κ)uλ(k, u) = 0, (49) (φ) Rz is rotation around the third axis represented, according 5 μ (γ γ kμ + κ)vλ(k, u) = 0, (50) to (32), by the matrix Uz(φ) 0 where kμ satisfies the covariant conditions (1) and (4). S(Rz(φ)) = (42) 0 Uz(φ) For the chiral amplitudes, the above equations imply with 1 0 uλL (k, u) =−κ (k I − k · σ )uλR(k, u), (51) iφ/2 e 0 1 0 U (φ) = (43) vλ (k, u) = (k I − k · σ )vλ (k, u), (52) z 0 e−iφ/2 L κ R and noting that from (12) the Wigner phase is given by and analogously for the space inverted pair. These equations ϕ( (φ), ˜, ) = φ Rz k u PF in this case, we obtain the follow- allow to determine the values of the coefficients cL/R, cL/R, 123 Eur. Phys. J. C (2021) 81:716 Page 7 of 14 716

= ˜ = ∼ dL/R, dL/R in Eqs. (46, 47)fork k and u u PF. Taking fulfilled already for q tens of eV so we reproduce the prop- into account (36) and (37), we obtain the final form of the erty observed experimentally in the MeV and GeV energy normalized amplitudes in an arbitrary frame Σu: range since ever. On the other hand, if q ∼ κ then the dependence of the ⎛ ⎞ amplitudes (54, 53, 56, 55)onq indicates that both neu- 2 2 1 trino helicities are present and in consequence one can expect ⎜ −q + q + κ LuUn ⎟ ⎜ 0 ⎟ √1 ⎜ ⎟ specific predictions to be different from those based on the u1/2(k, u) = ⎜ ⎟ , (53) 2 ⎝ ⎠ 2 2 1 SM. This regards in particular the electron energy spectrum q + q + κ Luπ Un 0 near the endpoint in β decay with the tachyonic neutrino ⎛ ⎞ (see Sect. 11). The q-dependence of the tachyonic neutrino 2 2 0 ⎜ q + q + κ LuUn ⎟ amplitudes was discussed firstly in Ref. [14] in the context ⎜ 1 ⎟ √1 ⎜ ⎟ 1 u−1/2(k, u) = ⎜ ⎟ , (54) of the reduced model with only one helicity component (− 2 ⎝ 2 2 2 0 ⎠ 1 − −q + q + κ Luπ Un for neutrino and for antineutrino, respectively). 1 2 ⎛ ⎞ 2 2 0 ⎜ q + q + κ LuUn ⎟ ⎜ 1 ⎟ √1 ⎜ ⎟ 7 Helicity observable v1/2(k, u) = ⎜ ⎟ , (55) 2 ⎝ 2 2 0 ⎠ −q + q + κ Luπ Un 1 The helicity operator λˆ given in Eq. (22) is realized in the ⎛ ⎞ Λˆ ∼ μ μ =−1 εμντσ bispinor space as uμW , where W 2 Pν Sτσ 2 2 1 ⎜ −q + q + κ LuUn ⎟ is the Pauli–Lubanski pseudo-vector. In the coordinate rep- ⎜ 0 ⎟ √1 ⎜ ⎟ v−1/2(k, u) = ⎜ ⎟ , (56) resentation it reads 2 ⎝ 1 ⎠ − + 2 + κ2 L π U q q u n ˆ 5 ν μ 0 Λ(u) ∼ γ [γ (i∂ν), uμγ ]. (61) After normalizing in the momentum representation it takes where Lu and Un are given by Eqs. (39) and (41), respectively, while q and n are functions of k and u, defined in the form Eqs. (3) and (10). ∓ ˆ 1 5 μ ν Using (51, 52) and the appropriate formulas found in Λ(k, u) = γ [kμγ , uνγ ] (62) 4 q2 + κ2 Appendix B one can easily check by means of Eqs. (1, 2, σ 3 3, 4), that the amplitudes (53, 54, 55, 56) fulfill Eqs. (49, 50), =∓ ( )1 0 ( ), S Lu Rn σ 3 S Lu Rn (63) i.e., the field ν(x, u) satisfies Eq. (48). The scalar products 2 0 of amplitudes and the polarization operators are collected in − + Appendix C. where the sign or is related to the negative (v)orpositive (u) frequencies in Eq. (28). In the PF the Weyl representation Dependence of the amplitudes (53, 54, 55, 56)onq is such ˆ κ of Λ has a simple form that for q the following chiral amplitudes vanish to high accuracy for each uμ k · σ 0 Λ(ˆ k, u ) =∓1 |k| . PF 2 k · σ (64) 0 |k| u(1/2)L (k, u) → 0, u(−1/2)R(k, u) → 0, (57) Λ(ˆ , ) v(1/2)R(k, u) → 0, v(−1/2)L (k, u) → 0, (58) The action of u k on the amplitudes can be obtained from (62, 63) and (53, 54, 55, 56). from which the following limiting forms are obtained for q κ 8 Propagator 1 5 1 5 u− / → (1 − γ )u− / , v / → (1 − γ )v / , (59) 1 2 2 1 2 1 2 2 1 2 In the standard field theory the propagator is identified with → 1 ( + γ 5) , → 1 ( + γ 5) . u1/2 2 1 u1/2 v−1/2 2 1 v−1/2 (60) the corresponding Green function (Feynman function) or defined as the vacuum expectation value of time ordered Thus, anticipating the Standard Model dynamics where only product of field operators. The equivalence of these two the left-handed chirality of the neutrino and the right-handed approaches is guaranteed by fundamental axioms of rela- chirality of the antineutrino participate, we conclude that one tivistic quantum field theory (Wightman axiomatic formula- can effectively observe exactly only neutrinos with helicity tion and LSZ formalism), in particular by the locality and − 1 1 2 and antineutrinos with helicity 2 . If the mass of a space- positive definiteness of the Hilbert space, see, e.g., [50,51]. like neutrino is about or less than 1 eV then this condition is However, one cannot expect commutativity of observables 123 716 Page 8 of 14 Eur. Phys. J. C (2021) 81:716

κ 2i 0 0 0 0 T separated by space-like intervals in a tachyonic theory. Con- = θ(x − y )WT (x, y) + θ(y − x )W (y, x) . (72) ( π)3 T sequently, the locality axiom cannot be fulfilled in such a 2 case. Therefore, the equivalence of a corresponding Green By means of the standard procedure and taking into account function with the definition of the propagator as the vacuum the form of the measure dμ(k, u) given in Eq. (8), integral expectation value of the time ordered product of quantum representation of the theta function (Eq. (A.7)), we obtain fields does not hold. Fortunately, in our case it is possible to from Eqs. (70, 71, 72) redefine the two-point Wightman functions to obtain a propagator equivalent to the Feynman function. The Wightman ST (x − y, u) ∞ 5 μ function W(x, y) =0|ν(x)ν(¯ x)|0 calculated with the help 1 − ( − ) κ + γ (γ pμ) = dμ(p, u)E dp0e ip x y , (73) of Eqs. (15, 18, 19, 28), has the following, seemingly stan- 2(2π)4 −∞ p2 + κ2 + iη dard, form where the energy E = E(q,θ,u) is explicitly given by

W(x, y) =0|ν(x)ν(¯ x)|0 Eq. (5) while the functional dependence of the measure dμ(p, u) = dμ(q,θ,u) = 1 |p| dΩ dq θ(q), |p|= 1 −ik(x−y) u0 = dμ(k, u)e uλ(k, u)uλ(k, u), (65) ( π)3 ω(q,θ,u) with ω given by Eq. (6) and the infinitesimal 2 λ parameter η ∼ Eε, where ε>0. Notice that, as is evi- but using equations from Appendix C we obtain dent from Eq. (5), the energy E can take negative values, too. In that case η also takes negative values and thus the ( , ) ( , ) = κΠ (Π − Π ), uλ k u uλ k u + 1/2 −1/2 (66) propagator changes the character from Feynman (causal) to λ Dyson (anticausal) with respect to the coordinate time t.As where the projector Π+ has the form it follows from Eq. (7), taking the upper limit for the neutrino 5 μ mass as κ = 1 eV and identifying PF with the CMBR frame, κ + γ (γ kμ) − Π+ = , (67) this can hold for energies between 0 eV and −10 3 eV, i.e., 2κ anticausal effects with respect to the coordinate time t are while extremely small. The propagator ST (x, u) also can be expressed in the form 1 λ μ ν 1 Πλ = I + γ 5[γ kμ,γ uν ] ,λ=± (68) 2 q2 + κ2 2 5 μ ST (x, u) = (κ + iγ γ ∂μ)Δ (x, u), (74) project on the helicity states (compare with Eq. (62)). As it T was expected, the above form does not lead to an accept- where able T -ordered function (it does not lead to a proper Green ΔT (x, u) − function). However, if we replace the sum (66)in(65)bythe 1 ∞ e ipx = dμ(q,θ,u)E(q,θ,u) dp0 (75) following one 2(2π)4 −∞ p2 + κ2 + iη κ is the Green function of the tachyonic Klein–Gordon equa- λuλ(k, u)uλ(k, u) = Π+, (69) λ 2 tion. Recall that the integration measure used by us preserves the structure of the tachyon physical momentum manifold we obtain an acceptable modification of the Wightman func- (topologically equivalent to R3 \ Bκ where Bκ is the open tion ball of the radius κ). Notice also that for this reason tachyons ( , ) cannot be sharply localized due to the Heisenberg uncertainty WT x y principle. 1 −ik(x−y) 5 μ = dμ(k, u)e (κ + γ (γ kμ)). (70) 4(2π)3 T ( , ) The corresponding function WT y x , related to the Wight- 9 Discrete symmetries man function 0|¯ν(y)T ν(x)T |0, where the superscript T means transposition, is equal to It is easy to see that the standard space inversion of a bispinor W T (y, x) 0 π π T νP (x, u) = γ ν(x , u ), (76) 1 ik(x−y) 5 μ = dμ(k, u)e (κ − γ (γ kμ)). (71) π = ( 0, )π = ( 0, − ) 4(2π)3 where x x x x x and similarly for other four-vectors, does not preserve the form of the Dirac-like The propagator can be defined analogously to the standard Eq. (48); instead, this equation is form-invariant with respect one, i.e., to the parity transformation defined as

5 0 π π ST (x − y, u) νP (x, u) = iγ γ ν(x , u ). (77) 123 Eur. Phys. J. C (2021) 81:716 Page 9 of 14 716

The Lagrangian leading to equivalent massless Dirac equations 5 μ μ 5 μ L = ν(x, u) γ (γ i∂μ) − κ ν(x, u) (78) (γ i∂μ)ψ = 0 and γ (γ i∂μ)ψ = 0. (86) related to Eq. (48) is not invariant under the standard inver- However, mass generation via the Yukawa coupling with the sion (76) and changes sign under the parity transformation Higgs fields leads to inequivalent free field Lagrangians (77) so both operations lead to parity nonconservation. ψ(γ¯ μ ∂ )ψ − ψψ¯ ψγ¯ 5(γ μ ∂ )ψ − κψψ,¯ Now, if we introduce the charge conjugation according to i μ m and i μ (87) the standard procedure (see also Ref. [10]) as respectively. The first Lagrangian results in the standard T Dirac equation while the second Lagrangian leads to the free νC (x, u) = Cν(x, u) , (79) field Eq. (48) describing a tachyonic fermion. μ where the unitary matrix C satisfies the relation Cγ T CT = On the other hand as was pointed out in [17], before the μ γ in this case (in the Weyl representation C = iγ 5γ 2γ 0 = onset of the spontaneous symmetry breaking process, mass- γ 3γ 1), the Lagrangian L is also non invariant. Besides the less Lagrangians for the neutrino field ν and the charged fact that neither the parity nor charge conjugation opera- LI lepton field l had the form 0 in the standard case (leading tions can be realized as symmetries of the tachyonic neu- ν LII to two Dirac leptons, l and ) and 0 in the mixed case trino Lagrangian even on the free level, their composition is (leading to a Dirac lepton and a tachyonic neutrino) a symmetry. Indeed, it is possible to define a combination of the parity and charge conjugation, CP LI = ¯(γ μ ∂ ) +¯ν(γ μ ∂ )ν, 0 l i μ l i μ (88) ν ( , ) = γ 5γ 0ν ( π , π ). LII = ¯(γ μ ∂ ) +¯νγ 5(γ μ ∂ )ν. CP x u C x u (80) 0 l i μ l i μ (89) Thus Now, in the chiral representation of the fields ν and l, 1 ikx LI ( ( ) × ( ) ) × ν (x, u) = dμ(k, u) e vλ(k, u) Lagrangian 0 is invariant under SU 2 L U 1 L CP (2π)3/2 λ (SU(2)R × U(1)R) transformations of the chiral left doublet † π π −ikx π π L and the right doublet R. On the other hand, Lagrangian ×a (k , u ) + e uλ(k, u)bλ(k , u ) ,(81) λ LII ( ( ) × ( ) ) 0 is invariant under SU 2 L U 1 L transformation of ( π , π ) = γ 5γ 0 ( , ) ( ( ) × ( ) ) wherewehaveusedtherelationsuλ k u uλ k u the left doublet L and under U 1 I3 R U 1 R subgroup of ( π , π ) =−γ 5γ 0 ( , ) ( ( ) × ( ) ) LII and vλ k u vλ k u (see Appendix C). the right SU 2 R U 1 R group. Therefore, 0 admits Next, we connect νCP(x, u) with the action of the unitary only a doublet L and two singlets, νR and lR, exactly as one operator Q representing the CP transformation needs for a formulation of the electroweak sector of the Stan- dard Model. Concluding, the mixed Lagrangian (89)fixes ζν (x, u) ≡ Qν(x, u)Q†, (82) CP exactly the weak group and its representation without addi- where ζ is a phase factor. Consequently, by means of Eq. (81) tional requirements. By means of the standard procedure of gauging and the † π π Qaλ(k, u)Q = ζ bλ(k , u ), (83) spontaneous symmetry breaking up to the electromagnetic † ∗ π π gauge group U(1)QED we obtain a Lagrangian which differs Qbλ(k, u)Q = ζ aλ(k , u ) (84) from the SM Lagrangian by the neutrino kinetic term only. In the unitary gauge the leptonic part of this Lagrangian takes and the remaining relations are obtained by the Hermitian the usual form conjugation and exchange (k, u) with (kπ , uπ ). Concluding, ¯ μ 5 μ in the case of a tachyonic neutrino one cannot define separate Llepton = l(γ i∂μ − ml )l +¯ν(γ γ i∂μ − κ)ν μ μ μ μ discrete symmetries C or P; but only the CP symmetry can √g − + g + (Wμ j+ + Wμ j−) + θ Zμ( j + jν ) be realized. 2 2 2cos W l L − μ − ml ¯ − κ ν¯ ν, eAμ je v lHl v H (90) √ where g = e/ sin θ , v2 = 1/( 2G ), e is the elec- 10 Interactions W F tric charge, G F – the Fermi constant, θW – the Weinberg angle, ml – mass of the lepton l. Here, H denotes the Let us start with an observation made in the paper by Chodos ± Higgs field, W and Zμ are the charged and neutral weak et al. [10] that the Higgs mechanism can, in principle, lead to μ bosons fields, respectively, and Aμ the electromagnetic four- a tachyonic fermion as well as to a massive fermion. Indeed, potential. The electroweak currents take also the standard the kinetic part of the massless bispinor field Lagrangian μ ¯ μ 5 μ μ 5 μ ¯ μ form j+ = lγ (1 − γ )ν, j− =¯νγ (1 − γ )l, je = lγ l, could have two different forms μ = ¯γ μ( − γ 5) μ = 1 νγ¯ μ( − γ 5)ν μ = jl l gV gA l, jνL 2 1 , jν R ψ(γ¯ μ ∂ )ψ ψγ¯ 5(γ μ ∂ )ψ 1 νγ¯ μ( + γ 5)ν = 2 θ − 1 =−1 i μ or i μ (85) 2 1 , where gV 2sin W 2 , gA 2 . 123 716 Page 10 of 14 Eur. Phys. J. C (2021) 81:716

Taking into account the form of the neutrino amplitudes discussed in Sect. 6, Lagrangian (90), at least in the tree approx- imation, leads to the SM results for energies signiﬁcantly higher than the neutrino mass κ (as well as to an agreement with the model reduced to the one helicity component dis- cussedin[14,16]). However, for energies close to the tachyonic neutrino mass κ, where both the left and right handed chiral components are present, one could expect new effects.

11 Beta decay of 3H

Fig. 1 Theoretical electron energy spectrum (differential decay rate) Tritium decay with a space-like neutrino was discussed by us near the endpoint in tritium decay at rest; E0 = 18573.7eVisthe within the framework of the reduced model [16]; the corre- endpoint energy fitted by KATRIN. The curves demonstrate the cases sponding differential decay rate (electron energy spectrum) for: massive neutrino with mν = 1 eV (the kinematical limit in the decay with a massive neutrino is E0 − mν ); massless (Weyl) neutrino; shows an anomalous energy dependence close to the end- 2 massive neutrino with inverted sign of mν illustrating the outcome of the point. Earlier calculations of the Kurie plot [10] also indicated KATRIN fit to their data (dashed line); complete model (solid line) and differences compared to the prediction for a massive neu- the reduced model (dotted line) for a tachyonic neutrino with κ = 1eV. trino. Below we derive the corresponding expressions for the The curves for the complete model and the KATRIN fit overlap in reality complete model, i.e., with the lepton interaction Lagrangian but have been infinitesimally shifted for the purpose of illustration. Inset: figurative visualization of the range covered on the main plot w.r.t. the given by (90), taking into account the contributions of all full electron energy spectrum within bounds [0, E0] two chiral and helicity components of the tachyonic neutrino field. 2 The amplitude squared of this process, |M| , is given at The above differential decay rate dΓ/dE, as a function the tree level by the formula of the electron kinetic energy, is presented in Fig. 1 together μ with the corresponding curve predicted in the reduced model |M|2 ∼ G2 Tr (lγ + m )γ (1 − γ 5)(γ 5kγ + κ) F e [16]. We also show curves for a massive neutrino and mass- ν 5 5 ×γ (1 − γ ) Tr (pγ + m P )γμ(I − gAγ ) less Weyl neutrino. The predictions of the complete and the 5 ×(rγ + m N )γν(I − gAγ ) , (91) reduced models are slightly different near the endpoint which allows (under a working hypothesis that the neutrino is a i.e., tachyonic fermion) to test both possibilities experimentally. Recently the KATRIN Collaboration delivered the most | |2 ∼ [( )( ) − ( )( )]−( + 2 )[( )( ) M 2gA lr kp lp kr 1 gA lp kr precise measurement of the neutrino mass squared in tritium 2 2 +0.9 2 +(lr)(kp)]+(1 − g )m N m P (kl), (92) m =−. A decay: νe 1 0−1.1 eV [4,5]. The prediction of the complete model with κ2 = 1eV2 and the recent KATRIN where m , m , m , κ and r, p, l, k are the masses and four- N P e fit representing the above value are indistinguishable. The momenta of tritium 3H, helium 3He, electron and tachyonic KATRIN result, despite its yet insufficient statistical signifi- anti-neutrino, respectively. By means of the above formulas cance, is a subsequent one in a row yielding negative central one can obtain the differential decay rate in terms of the values for the fitted neutrino mass squared, following earlier outgoing electron energy, E, in the form measurements at Mainz [6,8] and Troitsk [7,9]. dΓ 1 k+ = |M|2 dq, (93) π 3 dE 128 m N max{0,k−} 12 Conclusions where, the limiting values of the neutrino energy following from the tachyonic kinematics in the preferred frame are We have presented for the first time a fully consistent formal- = ( − )(Δ 2 − 0 ) ± ( 02 − 2) ism of quantization of space-like fermions with helicity ± 1 . k± m N l0 m 2l m N l me 2 In our formalism all common concerns have been resolved × (Δ 2 − 0 )2 + κ2( 2 + 2 − 0 ) m 2l m N 4 me m N 2l m N and explained. Specifically, (i) negative energy problem is

−1 solved and no imaginary energies appear, (ii) causality para- × ( 2 + 2 − 0 ) , 2 me m N 2l m N (94) doxes do not appear, (iii) the proper quantization procedure has been described in detail (Hilbert space with suitable 0 = + Δ 2 = ( 2 − 2 ) + ( 2 − κ2) where l E me and m m N m P me . Poincare properties indicated), (iv) interactions are unitary 123 Eur. Phys. J. C (2021) 81:716 Page 11 of 14 716 and theory is perturbatively renormalizable (anomaly free), It is useful to discuss shortly similarities and differences (v) since the presented description is Lorentz covariant, all between our construction of the tachyonic propagator and properties apply to arbitrary reference frames (and not only the approach given in the recent paper by Jentschura and the preferred frame). In order to achieve that and to avoid Wundt [22]. Their construction assumes an indefinite-norm causal problems with tachyons, existence of a preferred frame Hilbert space equipped with unconventional canonical anti- must be assumed; it is natural to suppose that the PF coin- commutation relations (CACR) dependent on neutrino chi- cides with the Cosmic Neutrino Background frame. We again ralities. We adopt the usual CACR, so from the beginning we indicate that the preferred frame, if exists in nature, would be deal with a positive-norm Hilbert space. Next, the authors uniquely a part of the tachyonic sector and not that of conven- of [22] use a formalism analogous to the Gupta–Bleuler tional physics that describes light and slower than light (mas- approach in the quantum electrodynamics. The unobserv- sive) objects. For those the preferred frame is but an ordinary able neutrino/antineutrino polarizations are related to the inertial frame respecting the principles of the Einsteinian rel- states with the negative norm and thus are unphysical. In our ativity. The preferred frame cannot be discovered with the use approach these polarizations are physical, however rapidly of light or massive particles owing to the intrinsic properties decreasing with growing energy which leads to the same of the Einsteinian relativity, in particular the corresponding effect. In contrast to our work, the authors of [22] do not clock synchronisation procedure. We determined the unitary assume a preferred frame scenario. Their propagator is con- realizations of the inhomogenous Lorentz group by means of structed with a modification of the sum of products of ampli- the Wigner-Mackey induction procedure and constructed the tudes similar to ours (Eq. (69)) but with relations to chirality corresponding Fock space. In the preferred frame scenario, while in our approach a similar role is played by helicity. the irreducible unitary realizations of the Lorentz group for However, there is an essential difference between both for- tachyons are labeled by particle helicity (not by spin). The malisms. Namely, as indicated in Sect. 8, the integration mea- ultimate, manifestly Lorentz covariant formalism was devel- sure used by us preserves the structure of the tachyon phys- oped by way of constructing the field amplitudes according to ical momentum manifold. In [22] the momentum manifold the Weinberg method. Since our theoretical findings regard- includes also unphysical momenta from Bκ , i.e., is identified ± 1 R3 ing the space-like helicity 2 fermions suggest connotations with the whole space . This forced appearance of resonant with neutrinos, we show that indeed the following facts and states as discussed in [22]. For the above reasons the propa- observations from the field of neutrino physics can be inter- gators in our approach and in [22] are different, irrespectively preted within the presented approach at the level of funda- of their superficial similarity. mental properties. According to our formalism, only neutri- As was mentioned in Sect. 10, the mixed free field − 1 + 1 nos with helicity 2 and antineutrinos with helicity 2 can Lagrangian (89) yet before of the spontaneous symme- be effectively observed at asymptotic energies (significantly try breaking admits only the Weinberg–Salam weak group exceeding the neutrino mass, κ, amounting to or a fraction and its representation without additional requirements. After of a single eV) which is a well established knowledge based gauging and the electroweak symmetry breaking in the lep- on the lack of experimental evidence for the complementary ton sector one obtains finally the Lagrangian (90), identical to cases at MeV–and higher energies. It should be stressed that the SM Lagrangian except of the neutrino kinetic term. The in our formalism this fact would be a consequence solely resulting model is anomaly free and perturbatively renormal- of the “tachyoneity” of the neutrino, i.e., its intrinsic prop- izable. erty at the free level and not exclusively due to the observed Tachyonic neutrinos also can oscillate according to the character of weak interactions. Likewise would be the chiral same pattern as massive neutrinos; oscillations in the present asymmetry of weak interactions, thus far introduced in an formulation do not distinguish between the massive and explicit way to the structure of the neutrino current. These tachyonic forms of the dispersion relation [19]. inherent features of space-like fermions follow directly from In the low energy neutrino limit, the negative value of the the energy dependence of the amplitudes for asymptotic ener- neutrino mass squared, determined in several recent tritium gies q κ, described by Eqs. (57, 58, 59, 60). This allows decay experiments, makes an interesting observation, though to explain the mysterious chiral properties of the neutrinos as not yet conclusive but pointing to the hypothesis of a space- − 1 well as occurrence of only 2 helicity neutrino and its con- like nature of the neutrino. As we have shown in Fig. 1,the jugate in nature, both age-old experimental observations. In conventional fit, recently confirmed by KATRIN, coincides addition, we have also shown that neither C nor P symmetry with the prediction of our complete model under the assump- ± 1 κ = holds for space-like helicity 2 fermions at the free level but tion that the tachyonic neutrino mass amounts to 1eV. rather the combined CP symmetry is conserved. The nature On the other hand, tachyonic neutrinos are the Dirac parti- of this property is intrinsic if neutrinos are tachyons and not cles in our approach. Thus one way of falsifying the hypoth- arising due to their interactions. esis that neutrinos are tachyons would be to observe the neu- trinoless double beta decay indicating that neutrinos were the 123 716 Page 12 of 14 Eur. Phys. J. C (2021) 81:716

Majorana particles. This process has not been discovered to Below we prove explicitly the completeness relation (14). date despite several decades of experimenting. Since neutrino and antineutrino spaces are mutually orthogonal, it is enough to prove this relation in one of those spaces, Data Availability Statement This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This is a theoretical say neutrino space of states. Therefore, we are to prove that paper and does not require the use of any data other then these contained for any vector in standard tables and cited references.] |ψ, u= dμ(p, u)ψ(p, u,λ)|p, u,λ (A.4) Open Access This article is licensed under a Creative Commons Attri- λ bution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you from one-particle neutrino space it holds give appropriate credit to the original author(s) and the source, pro- vide a link to the Creative Commons licence, and indicate if changes dμ(k, u) |k, u,λk, u,λ| |ψ, u=|ψ, u. (A.5) were made. The images or other third party material in this article λ are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not Indeed, we have included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copy- (|k, u,λk, u,λ|)|ψ, u right holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. = dμ(p, u)ψ(p, u,λ)k, u,λ|p, u,λ|k, u,λ 3 Funded by SCOAP . λ ω(qp, u,θp) = dqp dΩ(θp,ϕp) Θ(qp) 2u0 Appendix A: Invariant measure, the spherical Dirac delta λ 0 and completeness relation 2u ˆ × δλλ δ(qk − qp)δ(k − pˆ) ω(qk, u,θk) The Lorentz invariant measure taking into account existence × ψ(p, u,λ)|k, u,λ of the preferred frame has the following form = ψ(k, u,λ)|k, u,λ, (A.6) dμ(k, u) = d4k δ(k2 + κ2)Θ(uk) whereweusedEqs.(8, 13, A.3). This last equation imme- = 0| |2 | | ΩΘ( )δ(| |2 − ω2) dk k d k d uk k diately implies Eq. (A.5). When calculating the propagator δ(|k|−ω) = dk0|k|2 d|k| dΩΘ(uk) we use the following representation of the theta function ω 2 ∞ χ ω( , ,θ) 1 eit = q u Θ( ) Ω. θ(t) = lim dχ . (A.7) q dq d (A.1) π + χ − ε 2u0 2 i ε→0 −∞ i Here θ is the angle between u and k, ω(q, u,θ)is given in Eq. (6). Furthermore Θ is the Heaviside Theta and Appendix B: Rotations, boosts etc. d4k = dk0 ∧ d3k = 1 d(q + u · k) ∧ d3k u0 The rotation Rn reduced to the space sector has the form 1 3 ⎛ ⎞ = dq ∧ d k ( 1)2 1 2 u0 − n − n n 1 1 + 3 + 3 n 1 3 ⎜ 1 n 1 n ⎟ = dqd k. (A.2) = 1 2 (n2)2 u0 Rn ⎝ − n n 1 − n2⎠ (B.1) 1+n3 1+n3 −n1 −n2 n3 The spherical Dirac delta is the angular part of the three (θ, ϕ) dimensional Dirac delta. It is defined as follows: Let n while its spinor counterpart Un denote a unit vector depending on the spherical coordinates + 3 − 1 + 2 (angles) θ and ϕ. Then the spherical Dirac delta is defined U = 1 1 n n in . n 1 + 2 + 3 (B.2) by the formula 2(1 + n3) n in 1 n Here the unit vector n(k, u) is given by Eq. (10). Boost and dΩδ(n − n0) f (n) = f (n0), (A.3) S2 its spinor representative have the form where n0 = n(θ0,ϕ0) and dΩ is the solid angle differential. 0 T In terms of the angles θ and ϕ dΩ = sin θ dθ dϕ and δ(n − u u δ(θ−θ )δ(ϕ−ϕ ) Lu = ⊗ T , (B.3) ) = 0 0 u I + u u n0 sin θ . 1+u0 123 Eur. Phys. J. C (2021) 81:716 Page 13 of 14 716 1 0 The scalar products of the amplitudes take the form Lu = (1 + u )I − u · σ . (B.4) 2(1 + u0) uλ(k, u)uσ (k, u) = 2λδλσ , (C.4) One can also show that the following relations hold vλ(k, u)vσ (k, u) = 2λδλσ , (C.5) − L† = L , L π = L 1 (B.5) u u u u uλ(k, u)vσ (k, u) = vλ(k, u)uσ (k, u) = 0, (C.6) L U ( ) = u n 0 , ( , )γ 5γ 0 ( , ) =− λ 0δ , S Lu Rn L−1U (B.6) uλ k u uσ k u 2 k λσ (C.7) 0 u n 5 0 0 −1 vλ(k, u)γ γ vσ (k, u) = 2λk δλσ , (C.8) S (Lu Rn) = S(Lu Rn). (B.7) 5 0 π π uλ(k, u)γ γ vσ (k , u ) = 0, (C.9) Let kσ = k0 I − k · σ, kπ σ = k0 I + k · σ, and k˜σ = 5 0 π π − 2 + κ2 σ 3 vλ(k, u)γ γ uσ (k , u ) = 0. (C.10) qI q . Then 5 μ κ + γ (γ kμ) = 2λuλ(k, u)uλ(k, u) ˜ † kσ = LuUn(kσ)U Lu λ n = ( ) κ + γ 5(γ μ ˜ ) ( )−1, 2 2 S Lu Rn kμ S Lu Rn = Lu qI − q + κ n · σ Lu, (B.8) 5 μ κ − γ (γ kμ) = 2λvλ(k, u)vλ(k, u) π −1 ˜π † −1 k σ = L Un(k σ)U L λ u n u = ( ) κ − γ 5(γ μ ˜ ) ( )−1, = L−1 + 2 + κ2 · σ L−1, S Lu Rn kμ S Lu Rn (C.11) u qI q n u (B.9) Defining projectors and Π = 1 κ + γ 5(γ μ ) ,Π= 1 κ − γ 5(γ μ ) L σ 0L = L2 = 0 − · σ, + 2κ kμ − 2κ kμ (C.12) u u u u I u (B.10) where we obtain n3 n1 − in2 n · σ = = U σ 3U† (B.11) Π + Π = , n1 + in2 −n3 n n + − I (C.13) Π+uλ(k, u) = uλ(k, u), Π+vλ(k, u) = 0, (C.14) and Π−vλ(k, u) = vλ(k, u), Π−uλ(k, u) = 0. (C.15) 1 q + k0 n = n(k, u) = k − u . (B.12) + 0 q2 + κ2 1 u Projectors on helicity states have the following form Moreover λ 1 5 μ ν 1 σ 2U∗σ 2 = U ,σ2LT σ 2 = L−1. Πλ = I + γ [γ kμ,γ uν ] ,λ=± (C.16) n n u u (B.13) 2 q2 + κ2 2

and the following holds Appendix C: Relations between amplitudes: Space- [Π±,Πλ]=0. (C.17) inverted amplitudes, scalar products and polarization operators Moreover, we use the relations

The amplitudes given in (53, 54, 55, 56) satisfy a number of uλ(k, u)uλ(k, u) = 2κλΠ+Πλ, (C.18) relations listed in this Appendix. Taking into account the vλ(k, u)vλ(k, u) = 2κλΠ−Π−λ, (C.19) fact that in the case of the space inverted pair (kπ , uπ ), π π ˜π ˜π we have (k , u ) = Luπ Rn(k , u PF), where k = uλ(k, u)uλ(k, u) = κΠ+(Π1/2 − Π−1/2), (C.20) (q, 0, 0, − q2 + κ2), we can obtain space inverted ampli- λ π ˜ ˜ L L π tudes. To do this, k should be replaced by k and u by u vλ(k, u)vλ(k, u) =−κΠ−(Π1/2 − Π−1/2), (C.21) in the formulas (36, 37). Notice, that the following relations λ hold κ λuλ(k, u)uλ(k, u) = Π+, (C.22) λ 2 π π 5 0 uλ(k , u ) = γ γ uλ(k, u), (C.1) κ λvλ(k, u)vλ(k, u) = Π−. (C.23) ( π , π ) =−γ 5γ 0 ( , ) =−γ 0 ( , ), vλ k u vλ k u u−λ k u (C.2) λ 2 5 uλ(k, u) =−γ v−λ(k, u). (C.3)

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