decays of spacelike neutrinosmerlin.fic.uni.lodz.pl/concepts/2006_2/2006_2_79.pdf · 2006-01-23 ·...

DECAYS OF SPACELIKE NEUTRINOS

Pawe l Caban, Jakub Rembielinski

and Kordian Andrzej Smolinski

Department of Theoretical Physics

University of Lodz

ul. Pomorska 149/153, 90–236 Lodz, Poland

e-mail: [email protected]



(Received 30 November 2005; accepted 6 December 2005)

Abstract

In this paper we consider the hypothesis that neutrinosare fermionic tachyons with helicity 1

2. We propose an ef-

fective model of interactions of these neutrinos and analyzedominant effects under the above hypothesis: the three-bodyneutrino decay, the radiative decay and briefly discuss β-decaywith tachyonic antineutrino. We calculate the correspondingamplitudes and the mean life time for decays, as well as thedifferential decay width.

Concepts of Physics, Vol. III (2006) 79

Pawe l Caban et al.

1 Introduction

In the last two decades one of the most interesting problems inparticle physics is the issue of neutrino masses [1]. The flavor oscilla-tions [2, 3, 4] are the indirect evidence for non-zero neutrino masses.On the other hand, the experiments devoted to direct measurementof the neutrino mass permanently yield the negative mass squaredfor the electron neutrino [5, 6] and the muon neutrino [7] (for a re-view, see also [8]). In particular, the most sensitive neutrino massmeasurement involving electron antineutrino, is based on fitting theshape of beta spectrum. An as yet not understood event excess nearthe spectrum endpoint can be explained [9] on the ground of the hy-pothesis of the tachyonic neutrinos [10]. The last Mainz results donot give any hint for a positive neutrino mass squared and also nohint for any tachyonic mass. However, as long as we do not see asignal for a non-zero neutrino mass, a tachyonic neutrino mass cannot be excluded by Mainz experiment data [11, 6]. Moreover, oneof the aims of the KATRIN experiment (under preparation) is thesearch for physics beyond the Standard Model, among other thingsfor tachyonic neutrinos [12, 13]. However, if the negative antineutrinomass squared in tritium beta decay is genuine, then it is hard to un-derstand this phenomenon using conventional particle physics ideas.Note also that the oscillation experiments cannot decide between thepositive and negative neutrino mass squared because the oscillationfrequency is the same in the both cases (see Appendix E).

In this paper we continue investigation of an unconventional possi-bility that neutrinos are fermionic tachyons. This hypothesis was pro-posed firstly by Chodos et al. [10]. In the paper by Rembielinski [14]a causal classical and quantum theory of tachyons was elaborated andnext applied by Ciborowski and Rembielinski to explanation of theresults of tritium β-decay experiments [9, 15]. In the paper [9] physi-cal consequences of the three body decay ν` → ν` + ν`′ + ν`′ was alsodiscussed. This paper is devoted to systematical calculations of decayrates for the two processes with participation of neutrinos: the radia-tive decay ν` → ν` + γ and the three body decay ν` → ν` + ν`′ + ν`′ ,both conserving the lepton flavor; we briefly discuss also the β-decay.Notice, that the first two reactions are kinematically admissible fortachyonic neutrinos only. It is remarkable that, as was stressed in[9], the emission by neutrino ν` a ν`′ ν`′ pair can be an additional,

80 Concepts of Physics, Vol. III (2006)

Decays of spacelike neutrinos

qualitatively similar process to flavor oscillations. Moreover, Ehrlich[16, 17, 18] showed that the hypothesis that the electron neutrino isa tachyon is in a good agreement with the observed cosmic ray spec-trum. As we will see (Sec. 5), the calculated mean times of life forthe aforementioned decays take reasonable values in the regions of ex-perimentally available energies. Notice however that applicability ofthese results to the description of the time evolution of a neutrino fluxis restricted at this level because both of the decays form cascades.Therefore they demand further investigation as nontrivial stochasticprocesses leading to different than Geiger–Nutall decay law. More-over, the mostly hopefull program is to involve both oscillations andthree body decay.

2 Preliminaries

In this chapter we give a brief presentation of the causal theoryof tachyons proposed in the paper [14].

It is a common opinion that existence of space-like particles is incontradiction with special relativity. This is because of the Einsteincausality violation and consequently because of a number of very se-rious difficulties like causal paradoxes, inconsistent kinematics withimpossibility of a covariant formulation of the Cauchy problem, un-bounded from below energy spectrum and many others. Even moreproblems arise on the quantum level, like the problems with construc-tion of covariant asymptotic spaces of states, vacuum instability, etc.

However, as was shown in the paper [14] it is possible to agreespecial relativity with the tachyon concept. The main idea is basedon the well known fact that the definition of the time coordinatedepends on the synchronization scheme [19, 20, 21, 22] which, inturn, is a convention related to the assumed one-way light veloc-ity (only the average of the light velocity over closed paths has anoperational meaning; in special relativity this average is frame inde-pendent). Taking into account this freedom, it is possible to realizePoincare group transformations in such a way that the constant-timehyperplane is a covariant notion [14]. In that synchronization (namedabsolute synchronization scheme) it is possible to overcome all thedifficulties which appear in the standard approach.

Let us stress main features of this nonstandard scheme. Firstly, itis fully equivalent to the standard formulation of the special relativity


Pawe l Caban et al.

if we restrict ourselves to the timelike and lightlike trajectories, i.e.,if we exclude tachyons. If we admit space-like trajectories (tachyonsare included) then one inertial frame is distinguished as a preferredframe. Thus the relativity principle is broken, however the Poincaresymmetry is still preserved in that case. The proper framework tothis construction is the bundle of Lorentzian frames [23]. For thisreason the transformation law for coordinates involves velocity of dis-tinguished frame. The preferred frame can be locally identified withthe comoving frame in the expanding universe (cosmic backgroundradiation frame), i.e., the reference frame of the privileged observersto whom the universe appears isotropic [24]. In Ref. [25] it was shownthat the preferred frame can be eventually identified by means of theEinstein–Podolsky–Rosen correlation experiment.

Now, the velocity of the Solar System deduced from the dipoleanisotropy of the background radiation is about 350 km/sec [26, 27],so it is almost at rest relatively to the preferred frame. Therefore,with a good approximation, we can perform calculations in the pre-ferred frame.

To be concrete, the interrelation with coordinates in the Einsteinsynchronization (xµ

E) is given by

x0E = x0 + u0~u · ~x , ~xE = ~x . (1)

Here uµ is the four-velocity of the preferred frame with respect to theobserver. However the corresponding interrelations between velocities~vE and ~v obtained from (1) are singular for superluminal velocities.Furthermore, the Lorentz group transformation law for xE (x′E =ΛxE) implies by means of (1) the corresponding transformation lawfor x [23].

Now, on the quantum level, the corresponding quantum mechan-ics and quantum field theory can be formulated with help of thebundle of Hilbert spaces associated with the aforementioned bundleof frames [23]. The Fock construction can be done in this frameworkand unitary orbits can be classified [14]. For usual particles theycoincide with the standard unitary representations of the Poincaregroup. For tachyons they are induced from SO(2) group (insteadof SO(2, 1)) and labelled by the helicity. The following two facts,true only in absolute synchronization, are extremely important forquantization of tachyons:



• invariance of the sign of the time component of the space-likefour-momentum, i.e., ε(k0) = inv, which follows from the trans-formation law (see [14, 23]);

• existence of a covariant lower energy bound; in terms of thecontravariant space-like four-momentum kµ (k2 < 0) this lowerbound is exactly zero, i.e., k0 ≥ 0 as in the timelike and lightlikecase.

This is the reason why an invariant Fock construction with a stablevacuum can be done in this case on the quantum level [14]. Noticethat the measure

dµ(k, κ) = d4k θ(k0) δ(k2 + κ2) (2)

is Poincare covariant in the absolute synchronization scheme.

3 Fermionic tachyons with helicity λ = ±12

To construct tachyonic field theory describing field excitationswith the helicity ± 1

2 , we assume that our field transforms underPoincare group like bispinor (for discussion of transformation rulesfor local fields in the absolute synchronization see [14, 28]); namely

U(Λ)ψ(x, u)U(Λ−1) = S(Λ−1)ψ(x′, u′) , (3)

where S(Λ) belongs to the representation D12 0 ⊕D0 1

2 of the Lorentzgroup. Because we are working in the absolute synchronization, it isconvenient to introduce an appropriate (absolute-covariant) base inthe algebra of Dirac matrices as

γµ = T (u)µνγ

νE , (4)

where γµE are standard γ-matrices, while T (u) is determined by means

of the relation xµ = Tµν(u)xν

E and the Eq. (1). Therefore

γµ, γν = 2gµν(u)I , (5)

where the metric tensor is

gµν(u) = [TT (u)ηT (u)]µν = T (u)µαηαβT (u)β

ν , (6)


Pawe l Caban et al.

with η = diag(+,−,−,−).Notice, however, that the Dirac conjugate bispinor ψ = ψ†γ0

E.Furthermore γ5 = −iεµνσλγ

µγνγσγλ/4! = γ5E. Let us consider the

Dirac-like equation1 [14](γ5(iγ∂)− κ

)ψ = 0 , (7)

which can be derived from the Lagrangian density

L = ψ(γ5(iγ∂)− κ

)ψ . (8)

Dirac equation (7) implies the Klein–Gordon equation(gµν(u)∂µ∂ν − κ2

)ψ = 0 , (9)

related to the space-like dispersion relation k2 = −κ2. The equa-tion (7) is analogous to the one of the Chodos et al. [10] Dirac-likeequation for tachyonic fermion. However, contrary to the standardapproach, it can be consistently quantized in the absolute synchro-nization scheme if it is supplemented by the covariant helicity condi-tion [14]

λ(u)ψ(u, k) = λψ(u, k), (10)

with λ given by

λ(u) = − Wµuµ√(Pu)2 − P 2

, (11)

whereWµ = 1

2εµσλτJσλPτ

is the Pauli–Lubanski four-vector. This condition is quite analogousto the condition for the left (right) bispinor in the Weyl’s theory of themassless field. It implies that particles described by ψ have helicity−λ, while antiparticles have helicity λ. For the obvious reason in thefollowing we will concentrate on the case λ = 1

2 .Notice that the pair of equations (7) and (10) is not invariant

under the P or C inversions separately.

1Hereafter uγ = uµγµ, u∂ = uµ∂µ, γ∂ = γµ∂µ.



Now, in the bispinor realization the helicity operator λ has thefollowing explicit form [14]

λ(u) =γ5[iγ∂, uγ]

4√

(iu∂)2 + , (12)

where the integral operator((iu∂)2 +

)− 12 in the coordinate repre-

sentation is given by the well behaving distribution

1√(−iu∂)2 +

=1

(2π)4

∫d4p ε(up)eipx

((up)2 − p2

)− 12 . (13)

The free field ψ(x, u) has the following Fourier decomposition

ψ(x, u) =1

(2π)32

∫d4k δ(k2 + κ2)θ(k0)

[w(k, u)eikxb†(k) +

+v(k, u)e−ikxa(k)]. (14)

The creation and annihilation operators of a tachyonic fermion (a)with helicity − 1

2 and an antifermion (b) with helicity 12 satisfy almost

standard canonical anticommutation relations; the nonzero ones are[a(k), a†(p)

]+

= 2ω~kδ(~k − ~p) , (15a)[b(k), b†(p)

]+

= 2ω~kδ(~k − ~p) . (15b)

Here ω~k = k0 > 0 is the positive root of the dispersion relationk2 = −κ2 and ~k denotes covariant components ki, i = 1, 2, 3 of k.The amplitudes w and v satisfy(

κ+ γ5kγ)w(k, u) = 0 , (16a)(

1 +γ5[kγ, uγ]

2√q2 + κ2

)w(k, u) = 0 , (16b)(

κ− γ5kγ)v(k, u) = 0 , (16c)(

1 +γ5[kγ, uγ]

2√q2 + κ2

)v(k, u) = 0 . (16d)


Pawe l Caban et al.

Furthermore, the projectors ww and vv read

w(k, u)w(k, u) = (κ− γ5kγ)12

(1− γ5[kγ, uγ]

2√q2 + κ2

), (17a)

v(k, u)v(k, u) = −(κ+ γ5kγ)12

(1− γ5[kγ, uγ]

2√q2 + κ2

). (17b)

The above amplitudes fulfill the covariant normalization conditions

w(k, u)γ5uγw(k, u) = v(k, u)γ5uγv(k, u) = 2uk , (18a)

w(kπ, u)γ5uγv(k, u) = 0 , (18b)

where the superscript π denotes the space inversion of k.It is easy to see that in the massless limit κ → 0 Eqs. (16) give

the Weyl equations

kγw = kγv = 0 , γ5w = −w , γ5v = −v .

Now, the normalization conditions (18) together with the canon-ical commutation relations (15) guarantee the proper work of thecanonical formalism. In particular, starting from the Lagrangian den-sity (8) we can derive the translation generators

Pµ =∫

d3~k

2ω~k

kµ

(a†(k)a(k) + b†(k)b(k)

). (19)

Thus we have defined a consistent Poincare covariant free field theoryfor a fermionic tachyon with helicity − 1

2 ; the Weyl’s theory for aleft spinor is obtained as the κ → 0 limit. For more details of thisconstruction see [14].

4 Dynamics

To formulate a dynamical model with tachyonic neutrino we canuse, on the tree level, the phenomenological form of the weak interac-tion Lagrangian, namely the current-current Fermi Lagrangian. Theonly condition we take into account is that the massless limit κ→ 0of our model should coincide with the standard four-fermion inter-action. This leads to three natural possibilities for the part of thelepton charged current containing tachyonic neutrino:



¯(x)γµ 12 (1− γ5)ν(x) + h.c.— chiral coupling,

¯(x)γµν(x) + h.c. — helicity coupling,¯(x)γ5γµν(x) + h.c. — γ5 coupling,

and similarly for the neutral currents.Recall that ν(x) satisfies the helicity condition (10) with λ = 1

2so in the κ→ 0 limit ν → νL.

However, if we consider higher order processes (like ν → νγ)it is necessary to have a more sophisticated model like the Salam–Weinberg one. Let us notice firstly, that the Lagrangian for the mass-less Dirac field, under assumption of CP conservation, can be writtenin two different forms ψiγ∂ψ or ψγ5iγ∂ψ. Both forms lead to thesame massless Dirac equation iγ∂ψ = 0 because of the invertibility ofγ5 matrix. Therefore generation of a mass via Yukawa coupling withGoldstone fields can lead to massive fermions or fermionic tachyons,because of creation in the Lagrangian of Dirac ψ (iγ∂ −m)ψ or ψ

(γ5iγ∂ −m

)ψ

term respectively. Thus, because we have decided to treat neutrinosas tachyons, we start with the following free massless Lagrangian fora fixed lepton generation (ν, `)

L0 = νγ5iγ∂ν + ¯iγ∂` . (20)

After splitting ν and ` into left-handed and right-handed pairsνL,R, `L,R, it is easy to see that the (compact) invariance group of theLagrangian L0 is exactly the

(SU(2)×U(1)

)L×U(1)R group! There-

fore, taking into account the electric charges of ν and `, we concludethat the symmetry group of L0 is U(1)lepton number×(U(1)Y × SU(2)I)and the left-handed fields νL and `L must form the weak dubletwhile the right-handed νR and `R are singlets of the weak group.Therefore L0 leads uniquely to the standard Salam–Weinberg choiceof the weak symmetry group and their realization on the lepton fields(recall also that in the κ → 0 limit ν(κ) → ν(0) = νL(0), i.e.,νR(κ) → νR(0) = 0). Now, under the standard choice of the Gold-stone fields (as the U(1)Y × SU(2)I dublet) as well as the Higgspotential and Yukawa coupling (for neutrinos like for neutral quarks)and after gauging the weak group we obtain the final Lagrangian

L = Lfermions+int. + (Lgauge bosons + LHiggs+int. + · · · ) . (21)


Pawe l Caban et al.

The part of L in the parentheses has the standard form (we chooseconvention as in [29]) while the Lfermions+int. reads

Lfermions+int. = νγ5iγ∂ν − κνν + ¯iγ∂`−m ¯ + eAµ¯γµ`

+g

2 cos θWZµ

¯γµ(gV − gAγ

5)`− g

2 cos θWZµνγ

µ 1− γ5

2ν

− g√2W+

µ νγµ 1− γ5

2`− g√

2W−

µ¯γµ 1− γ5

2ν

− g

2√

2mW

ν((m− κ)I + (m+ κ)γ5

)`G+

− g

2√

2mW

¯((m− κ)I − (m+ κ)γ5)νG−

+ig

2mW

(κνγ5ν −m¯γ5`

)G0

− g

2mW

(κνν +m ¯ )H . (22)

Here gV = 12 − 2 sin2 θW , gA = 1

2 , g—weak coupling constant, θW —Weinberg angle, G± and G0—Goldstone bosons and H—Higgs field.The corresponding Feynman rules are listed in the Appendix A.

The following remarks are in order. First of all, the derivationof the Lagrangian (21) with Lfermions+int. given by (22) is ratherheuristic one; the point is that the right component of ν, νR goesto zero with κ → 0. This means that the mass generation mecha-nism for tachyons is not well defined by Yukawa coupling if we startwith the massless Lagrangian. Intuitively, it seems that the mecha-nism of mass generation for the tachyonic neutrinos should be causedrather by the interaction with gravitational field. This is related toour natural assumption about coincidence of the tachyon preferredframe with the comoving frame. The next problem is the helicitycondition (10) which is necessary for consistency of the free tachyontheory. Of course it can be gauged by introduction of the covari-ant derivatives and causes additional relations between multipointGreen functions of the theory. In such a case perturbation consis-tency and renormalizability of this model should be proved. We shiftthese unanswered questions to further investigations and here we willtreat the Lagrangian (21) as an effective reasonable approximation ofa more realistic theory. Because in the following we do not analyze



ν

ν′

ν

Z ν′+

ν

ν′

ν

G0 ν′

Figure 1: Neutrino three-body decay ν` → ν` + ν`′ + ν`′

processes with intrinsic neutrino lines, this approximation seems tobe correct.

Note that the Feynman rules derived from (21) and (22) (Appen-dix A), lead to the effective chiral coupling in the Fermi Lagrangian.Therefore, in the following all the calculations on the tree level willbe done for this coupling. However, we will comment results obtainedwith the help of the helicity and γ5 couplings too.

Tachyonic neutrino is in general unstable even if the leptonic fla-vors are conserved; namely decays of the form tachyon → tachyon +sth. are kinematically admissible. If we take into account the ex-perimental upper bounds on the neutrinos masses [8] we can se-lect two dominant processes of this type, namely: three body decayν` → ν` + ν`′ + ν`′ and radiative decay ν` → ν` + γ.

5 Neutrino decay ν` → ν` + ν`′ + ν`′

This process can be analyzed on the tree level. The correspondingamplitude is the sum of the graphs given in Fig. 1 and leads to theeffective four-fermion vertex in Fig. 2. Consequently

M =√

2GF v`(p)Γµv`(k)v`′(l)Γµw`′(r) , (23)

where the momenta k, p, l and r correspond to the incoming neutrinoν` and to the outgoing neutrinos ν`, ν`′ and ν`′ respectively. Themasses of ν and ν`′ are denoted by κ and µ respectively. In ourcase the Feynman rules (Appendix A) lead to the chiral couplingΓµ = (1+γ5)γµ/2. Thus, by means of the polarization relations (17),


Pawe l Caban et al.

ν

ν′

ν

ν′

k

l

p

r

Figure 2: Effective four-fermion vertex in neutrino three-body decayν` → ν` + ν`′ + ν`′

we obtain in the preferred frame

|M |2 = 2G2F

(|~k|+ k0

) (|~p|+ p0

) (|~l|+ l0

) (|~r|+ r0

)|~k||~p||~l||~r|

×(|~p||~l| − ~p ·~l

)(|~k||~r| − ~k · ~r

), (24)

where from dispersion relations |~k| =√κ2 + (k0)2, |~p| =

√κ2 + (p0)2,

|~l| =√µ2 + (l0)2, |~r| =

√µ2 + (r0)2.

The general forms of the square of the amplitude for all couplingsare given in the Appendix B. The qualitative behavior of the decayrate

Γ``′ =((2π)52k0

)−1∫d4p d4l d4r |M |2 θ(p0)θ(l0)θ(r0)

× δ(p2 + κ2)δ(l2 + µ2)δ(r2 + µ2)δ4(k − p− l − r) (25)

is rather similar for all three couplings. Here we present in the Fig. 3the numerical calculations for the mean life time in the case of chi-rality coupling (according to our gauge model) done in the preferredframe (u = (1,~0)). For the helicity coupling the corresponding resultswere given in [9]. The mean life time of the tachyonic neutrino in thisprocess is determined by its energy E = k0, as measured in the pre-ferred frame, and masses of all three neutrino species. As we see fromthe Fig. 3 the life time τe` corresponding to the partial width Γe`, is



= 3 eV = 019 MeV = 182 MeV

(sec

)

(MeV)

106 105105

00001 0001 001 01 1

1

10 100 1000

100000

1010

1015

1020

1025

1030

1035

Figure 3: The mean time life τe` of electron neutrino in three-bodydecay into electron neutrino and neutrino–antineutrino pair of type` for κ = 3 eV and masses µ` = 3 eV, µ` = 0.19 MeV, and µ` =18.2 MeV


Pawe l Caban et al.

a decreasing function of the initial energy and mass of the neutrinoν`. Note that the dashed and dotted lines in the Fig. 3 correspondto the upper bounds on the νµ and ντ masses [30]. Since the ∆(m2)for the diffrent neutrino flavors does not exceed a few eV, the meanlife times τee, τeµ and τeτ are almost equal and their dependence onneutrino energy is given by the most upper (solid) line in the Fig. 3.

It is important to stress that:

• The decay ν` → ν` + ν`′ + ν`′ has the character of an emissionby ν` a neutrino–antineutrino pair ν`′ ν`′ ; as a consequence theresulting neutrino flux contains neutrinos and antineutrinos ofall possible flavors. Therefore this process can simulate the neu-trino flavor oscillations. Notice that for massive and masslessneutrinos such decays are kinematically forbidden.

• The above process is repeated and forms a cascade. Therefore,as was mentioned in the Introduction, it demands more sophis-ticated treatment as a nontrivial stochastic process. Howeverafter the neutrino energy degradation it is slowing down withsubsequent decays; notice that, for low energies in the preferredframe, neutrinos are almost stable (Fig. 3).

Finally, let us analyse the influence of the neutrino three bodydecay on the neutrino flux. The effect, which we predict below, can becomplementary effect on the intensity of solar neutrino flux togetherwith neutrino oscillations and the MSW mechanism [31, 32, 33].

As it is evident from the Fig. 3, in the case of electron neu-trino (κe ∼ 3 eV) decay (for energies 1–10 MeV and time taken asSun–Earth distance/c ∼ 500 sec) to obtain the significant effect, themass of the resulting ν`′ should of the order of 1 MeV. On the otherhand the neutrino oscillation experiments suggest that the masses ofat least of two neutrino flavors must lie very near each other. There-fore let us consider the contribution of the one νe → νe + νh + νh

decay only, where νh denotes a possible heavy neutrino. Our aim is



to analyze the cascade process

νe → νe + νh + νh

↓νe + νh + νh

↓νe + νh + νh

↓...

taking into account that probabilities of the possible subsequent de-cays νh → νh + νe + νe and νh → νh + νe + νe are not significant inthe investigated energy regime.

This greatly simplifies treatment of this stochastic process leadingto the following evolution equation for the electron neutrino flux:

~dΦt(E)dt

= −ΓEΦt(E) +∫ ∞

E

dE′dΓE′

dEΦt(E′) . (26)

Here Φt(E) denotes the electron neutrino flux of energy E after timet, ΓE is the total decay rate for the initial neutrino of energy E anddΓE′/dE is the differential decay rate of the decaying neutrino νe

with energy E′ into outgoing electron neutrino with energy E. Theequation (26) has the solution as a formal power series

Φt = etF/~Φ0 (27)

with F (E,E′) = −ΓEδ(E − E′) + θ(E′ − E) dΓE′/dE.

6 Radiative decay of the tachyonic neutrino ν` →ν` + γ

This process can play an important role in cosmology; in particu-lar, as was noticed in [34, 35, 36], it can affect the cosmic backgroundradiation in the submilimeter region, leading to its distortion.

Let us denote the four-momenta of the initial and final neutrinosby k and p respectively, their masses by κ, while photon has thefour-momentum q = k − p. The kinematics of this process is verysimple; namely k2 = −κ2, p2 = −κ2 and for the photon on the mass


Pawe l Caban et al.

shell q2 = (k − p)2 = 0 so kp = −κ2, qk = qp = 0. Let us considerthis kinematics in the preferred frame u = (1,~0). It is immediate tosee that cos θ = ~k · ~q/(|~k||~q|) = k0

((k0)2 + κ2

)−1/2. Therefore theoutgoing photons are emitted on the cone determined by the angle2θ. Notice that for k0 κ, cos θ ≈ 1 so the process is approximatelycollinear in that case.

The amplitude has the form

Mλ(p, k) = Ωµ(p, k)eλµ(q) , (28)

where eλµ is the polarization vector for the photon with helicity λ.

The Ωµ(p, k) should satisfy transversality and neutrality conditions.The first one follows from the electromagnetic current conservationand has the form

qµΩµ(p, k) = 0 , (29)

while neutrality condition means that the neutrino charge is zero andconsequently the corresponding matrix element vanishes:

Ωµ(k, k) = 0 . (30)

By means of the equations of motion for the spinors v and v (16c),the most general form of the Ωµ(p, k) reads

Ωµ(p, k) = v(p)[(FV + FAγ

5)γµ + (G1V + iG1Aγ5)kµ

+ (G2V + iG2Aγ5)pµ

]v(k) , (31)

where the form factors F and G are in general functions of q2. No-tice that the form factors GiA control CP non-invariant terms in theEq. (31). Now, taking into account the transversality (29) and neu-trality (30) we obtain the following constraints

FV = 0 , (32a)FA = κ(G1V +G2V ) , (32b)

(G2A −G1A)q2 = 0 , (32c)

(G2V −G1V )q2 = 0 . (32d)

Therefore, because of the analyticity of GiV and GiA in the pointq2 = 0, we obtain

Ωµ(p, k) = −12v(p) [γµ, γν ] qν(GV + iGAγ

5)v(k) , (33)



where we denoted GV ≡ G1V = G2V , GA ≡ G1A = G2A and theequations of motion (16c) was used again. Therefore the square ofthe amplitude Mλ, after summation over final polarizations of thephoton (Appendix D), takes the form:

|M |2 = 4κ4(|GV |2 + |GA|2

)× (uq)2√

(u(k − q))2 + κ2√

(uk)2 + κ2(34)

where we have used the form of the polarization operator (17b).In the framework of our SU(2)×U(1) gauge model, the diagrams

contributing to the process ν` → ν` +γ, to one-loop order, are shownin the Fig. 4.

Now, the transversality and neutrality conditions (29) and (30)can be treated as renormalization conditions. By means of the Feyn-man rules listed in the Appendix A we can easily calculate the formfactors GA and GV :

GA = 0 , (35)

as we expected because of the CP invariance of this model, while

GV =eg2κ

64π2[Ia + Ib + Ic + Id + Ie + If ] , (36)

where (see Appendix C), under the experimental conditions for leptonmasses m mW and for corresponding neutrino masses κ mW ,

Ia =4

3m2W

,

Ib = − 76m2

W

,

Ic =2m2

m4W

ln(mW

m

)2

,

Id =5m2 + κ2

6m4W

,

Ie + If =1

2m2W

.


Pawe l Caban et al.

(a) (b)

ν−

W+

−

γ

ν

ν

−

W+

ν

W+

γ

(c) (d)

ν−

G+

−

γ

ν

ν

−

G+

ν

G+

γ

(e) (f)

ν

−

W+

ν

G+

γ

ν

−

G+

ν

W+

γ

Figure 4: Contributions to the radiative decay ν` → ν`+γ to one-looporder



Thus in this approximation Ic + Id can be omitted and the final formof GV reads

GV =eg2κ

96π2m2W

=κGF

6π

√α

2π. (37)

Consequently

Ωµ = −κGF

12π

√α

2πv(p) [γµ, γν ] qνv(k) (38)

and

|M |2 =ακ6G2

F

18π3

(uq)2√(uk)2 + κ2

√(u(k − q))2 + κ2

. (39)

Notice, that (38) implies that the magnetic moment µν of the neu-trino ν is given by

µν =κGF

3π

√α

2π=GFmeκ

3√

2π2µB , (40)

where µB is the Bohr magneton and me the electron mass. (N.B.the value of µν is of the same form as for the massive neutrino; thedifference lies in the numerical factor only [37]). To calculate thedecay rate we integrate |M |2 over the phase space:

Γ = (8π2k0)−1

∫d4q d4p |M |2 θ(q0)θ(p0)

× δ(q2)δ(p2 + κ2)δ(k − p− q) . (41)

We shall do it in the preferred frame, that is we put u = (1,~0).Inserting |M |2 given by (39), we obtain the energy spectrum of theemitted photons as:

dΓdq0

=αG2

Fκ6(q0)2θ(k0 − q0)θ(q0)

288π4k0 ((k0)2 + κ2)√

(k0 − q0)2 + κ2. (42)

Finally, the total decay rate reads:

Γ =ακ6G2

F

576π4k0 ((k0)2 + κ2)

(4κk0 − 3k0

√(k0)2 + κ2

+(2(k0)2 − κ2

)ln

√(k0)2 + κ2 + k0

κ

). (43)


Pawe l Caban et al.

Table 1: Upper bounds on the neutrino magnetic moment

from (40) from [8]

µe < 4.27× 10−19µB < 1.0× 10−10µB

µµ < 2.70× 10−14µB < 6.8× 10−10µB

µτ < 2.59× 10−12µB < 3.9× 10−7µB

Note that the experimental upper bounds on the neutrino massesapplied to the magnetic moment formula (40) give results which arein agreement with the experimental data [8] (cf. Table 1).

Now, for illustration we present the mean life time τ = ~/Γ forthis process as a function of the energy k0 in Fig. 5. From (43) it iseasy to see that for k0 = ξκ, with ξ ≈ 2.6899 the mean time of lifehas a minimum. The minimal value τmin, as the function of the massκ, reads

τmin(κ) ' 902π4~αG2

F

κ−5 (44)

and is presented in Fig. 6. Finally, in Fig. 7 differential rate dΓ/dq0

as the function of the photon energy q0 and the initial energy k0 isgiven.

7 β-decay with tachyonic antineutrino

The tritium β-decay with tachyonic antineutrino was discussed inthe papers by Ciborowski and Rembielinski [9, 15]. Here for a com-pleteness we present only a brief view on results obtained in [15]. Asin the standard case, the two graphs shown in the Fig. 8 contribute tothis process on the tree level, where dominant is the first one. Effec-tively, for energies much less than mW , it reduces to the four-fermioninteraction with the amplitude square given by chiral coupling

|Mch|2 = 2G2F Tr

[ueueγ

µ 1− γ5

2wwγν 1− γ5

2

]× Tr

[upupγµ(1− gAγ

5)ununγν(1− gAγ5)], (45)



τ(κ,k

0)

τmin(κ)

κ ξκ 5κ 10κ 15κ 20κ

k0

Figure 5: Mean life time τ in the radiative decay ν → νγ as thefunction of the neutrino mass κ and the initial energy k0. Here ξ ≈2.6899


Pawe l Caban et al.

κ (MeV)

τ min

(κ)

(sec

)

10−6 10−5 10−4 0.001 0.01 0.1 1

1

10 10010−10

1010

1020

1030

1040

Figure 6: Minimal mean time of life τmin for the radiative decayν → νγ as the function of the neutrino mass κ



q00.5κ κ ξκ 5κ 10κ

1.15

5×

10−

29M

eV−

4κ

4

dΓ/dq

0

k0 = 0.5κκξκ5κ

10κ

Figure 7: Differential rate dΓ/dq0 for the radiative decay ν → νγ asthe function of the photon energy q0 close to the minimum of themean time of life

n

νe

p+

W−e−

+n

νe

p+

G− e−

Figure 8: Graphs contributing to the β-decay


Pawe l Caban et al.

where the axial coupling gA ≈ 1.25; in the case of helicity coupling,factors (1 − γ5)/2 are replaced by 1. The explicit calculation of theelectron spectrum for chiral and helicity couplings as well as com-parision with the exeprimental data are given in [15]. Let us stressonly that the mysterious bump near the end point is reproduced onthe theoretical plot; also the second anomaly problem in the tritiumβ-decay can be solved with help of the tachyonic hypothesis [15].

8 Conclusions

In this paper we have analyzed the introductory results related totachyonic neutrino hypothesis. We do not pretend to explain all theproblems with neutrino physics by means of this hypothesis. How-ever, from the behaviour of the calculated decay rates for three dom-inant processes: emission of a νν pair by a neutrino, radiative decayν → νγ and β-decay, it follows that this possibility cannot be ruledout of considerations. Moreover such a hypothesis provides us withan additional mechanism simulating flavor oscillations.

Acknowledgement

One of us (J.R.) thanks to Jacek Ciborowski for interesting dis-cussions.

A Vertices in our gauge model

The vertices necessary for calculation of amplitudes for threeprocesses considered here in our gauge model are shown in the Ta-ble 2. The rest of the Feynman rules in the convention adopted inthis paper is given, e.g., in the Ref. [29].

B Amplitudes for neutrino decay ν` → ν`ν`′ ν`′ forall couplings

The square of the amplitude for the chiral coupling is

|M |2 = 2G2Fm

µνnµν , (46)



Table 2: Vertices in our gauge model used in this work

W+

γ

W+ie [(r − q)λgµν + (q − p)νgλµ + (p− r)µgνλ]

γ

ieγµ

G+

γ

G+−ie(p + q)µ

G+

γ

W+iemW gµν

Z

igγµ

2 cos θW(gV − gAγ5)

ν

Z

ν− igγµ

2 cos θW

1−γ5

2

−

W+

ν− igγµ

√2

1−γ5

2

ν

W−

−− igγµ

√2

1−γ5

2

H

− igm

2mW

ν

H

ν− igκ

2mW

G0

gmγ5

2mW

−

G+

ν− ig

2√

2mW

[(m− κ)I + (m + κ)γ5

]

ν

G0

ν−gκγ5

2mW

ν

G−

−− ig

2√

2mW

[(m− κ)I − (m + κ)γ5

]


Pawe l Caban et al.

where

mµν =12

(1 +

(uk)√(uk)2 + κ2

)(1 +

(up)√(up)2 + κ2

)×(kµpν + kνpµ − (kp)gµν − iεµναβkαpβ

)+

κ2

2√

(up)2 + κ2

(1 +

(uk)√(uk)2 + κ2

)(kµuν

+ kνuµ − (uk)gµν − iεµναβkαuβ

)+

κ2

2√

(uk)2 + κ2

(1 +

(up)√(up)2 + κ2

)(pµuν

+ pνuµ − (up)gµν + iεµναβpαuβ

)+

κ4

2√

(uk)2 + κ2√

(up)2 + κ2(2uµuν − gµν)

and

nµν =12

(1 +

(ul)√(ul)2 + µ2

)(1 +

(ur)√(ur)2 + µ2

)×(lµrν + lνrµ − (lr)gµν + iεµναβl

αrβ)

+µ2

2√

(ur)2 + µ2

(1 +

(ul)√(ul)2 + µ2

)(lµuν

+ lνuµ − (ul)gµν + iεµναβlαuβ

)+

µ2

2√

(ul)2 + µ2

(1 +

(ur)√(ur)2 + µ2

)(rµuν

+ rνuµ − (ur)gµν − iεµναβrαuβ

)+

µ4

2√

(ul)2 + µ2√

(ur)2 + µ2(2uµuν − gµν) .

Squares of the amplitudes for helicity (s = 0) and γ5 (s = 1)couplings read

|M |2 = 2G2Fm

µνs nsµν , (47)



where

mµνs = gµν(u)

[((−1)sκ2 − kp

)+

1√(uk)2 + κ2

√(up)2 + κ2

((−1)sκ2 ((kp)

−(uk)(up))− κ4 − κ2((up)2 + (uk)2

)−(uk)(up)(kp))]

+ uµuν 2κ2(κ2 − (−1)s(kp)

)√(uk)2 + κ2

√(up)2 + κ2

+ (kµpν + kνpµ)

[(uk)(up)− (−1)sκ2√(uk)2 + κ2

√(up)2 + κ2

+ 1

]

+ (uµkν + uνkµ)κ2 ((−1)s(up) + (uk))√(uk)2 + κ2

√(up)2 + κ2

+ (uµpν + uνpµ)κ2 ((−1)s(uk) + (up))√(uk)2 + κ2

√(up)2 + κ2

+ iεµναβ

[κ2 ((−1)skα + pα)uβ − (uk)kαpβ√

(uk)2 + κ2

−κ2 ((−1)spα + kα)uβ + (up)kαpβ√

(up)2 + κ2

]

and

− nsµν = gµν(u)[(

(−1)sµ2 + lr)

+1√

(ul)2 + µ2√

(ur)2 + µ2

((−1)sµ2 ((lr)

−(ul)(ur)) + µ4 + µ2((ul)2 + (ur)2

)+(ul)(ur)(lr))]

+ uµuν

2µ2(−µ2 − (−1)s(lr)

)√(ul)2 + µ2

√(ur)2 + µ2

+ (lµrν + lνrµ)

[− (ul)(ur)− (−1)sµ2√

(ul)2 + µ2√

(ur)2 + µ2− 1

]


Pawe l Caban et al.

+ (uµlν + uν lµ)µ2 ((−1)s(ur)− (ul))√(ul)2 + µ2

√(ur)2 + µ2

+ (uµrν + uνrµ)µ2 ((−1)s(ul)− (ur))√(ul)2 + µ2

√(ur)2 + µ2

+ iεµναβ

[µ2 ((−1)srα − lα)uβ + (ur)rαlβ√

(ur)2 + µ2

−µ2 ((−1)slα − rα)uβ − (ul)rαlβ√

(ul)2 + µ2

].

C Amplitudes of processes contributing to the ra-diative decay

Amplitudes of processes contributing to the radiative decay de-scribed by diagrams in the Fig. 4 (a)–(f) read

Ωµi = v(p)Mµ

i v(k), (48)

where i = a, b, c, d, e, f and:

Mµa =

g2e

2

∫d4r

(2π)41 + γ5

2γν

× [(p− r)γ +m]γα[(k − r)γ +m][r2 −m2

W ][(p− r)2 −m2][(k − r)2 −m2]

× γν1− γ5

2, (49a)

Mµb = −g

2e

2

∫d4r

(2π)41 + γ5

2γβ

× [rγ +m][r2 −m2][(p− r)2 −m2

W ][(k − r)2 −m2W ]

×[(2p− k − r)λg

µβ + (2r − k − p)µgλβ

+ (2k − r − p)βgµλ

]γλ 1− γ5

2, (49b)



Mµc = − g2e

8m2W

∫d4r

(2π)4[(m− κ)I + (m+ κ)γ5

]× [(p− r)γ +m]γµ[(k − r)γ +m]

[r2 −m2W ][(p− r)2 −m2][(k − r)2 −m2]

×[(m− κ)I − (m+ κ)γ5

], (49c)

Mµd =

g2e

8m2W

∫d4r

(2π)4[(m− κ)I + (m+ κ)γ5

]× [(p− r)γ +m](k + p− 2r)µ[(k − r)γ +m]

[r2 −m2][(p− r)2 −m2W ][(k − r)2 −m2

W ]

×[(m− κ)I − (m+ κ)γ5

], (49d)

Mµe =

g2e

8

∫d4r

(2π)4γµ(1− γ5)

× [rγ +m][r2 −m2][(p− r)2 −m2

W ][(k − r)2 −m2W ]

×[(m− κ)I − (m+ κ)γ5

], (49e)

Mµf =

g2e

8

∫d4r

(2π)4[(m− κ)I + (m+ κ)γ5

]× [rγ +m]

[r2 −m2][(p− r)2 −m2W ][(k − r)2 −m2

W ]

× (1 + γ5)γµ . (49f)

After calculations and renormalization, the integrals contributing tothe form factor GV in the Eq. (36) read

Ia = −2∫ 1

0

dxx(x2 − 3x+ 2)

x2κ2 + x(m2W − κ2 −m2)−m2

W

' 43m2

W

,

Ib =∫ 1

0

dxx2(2x+ 1)

x2κ2 + x(m2 − κ2 −m2W )−m2


Pawe l Caban et al.

' − 76m2

W

,

Ic =1m2

W

∫ 1

0

dxx2[x(κ2 −m2)− (m2 + κ2)]

x2κ2 + x(m2W − κ2 −m2)−m2

W

' 2m2

m4W

ln(mW

m

)2

,

Id =1m2

W

∫ 1

0

dxx(x− 1)[x(κ2 −m2) + 2m2]

x2κ2 + x(m2 − κ2 −m2W )−m2

' 5m2 + κ2

6m4W

,

Ie + If = −∫ 1

0

dxx2

x2κ2 + x(m2 − κ2 −m2W )−m2

' 12m2

W

.

D Sum over photon polarization vectors

Sum over photon polarization vectors (on the mass shell q2 = 0)in the absolute synchronization reads:

Πµν(q) =1∑

λ=−1λ6=0

eµλe

νλ = − 1

(uq)2[(uq)2gµν(u) + qµqν

−(uq)(uµqν + uνqµ)] . (50)

Notice, that qµΠµν = uµΠµν = 0, so eµλ, qµ and uµ form an pseudo-

orthogonal basis.

E Oscillations of tachyonics vs. massive neutrinos

Here we compare the oscillation effects for the massive and tachy-onic neutrinos. Calculations will be performed in the preferred frame,in which the metric tensor has the standard Minkowskian form.



Let us consider three flavors of neutrinos. We shall denote theneutrino flavor states by

|νe〉 =

100

|~p〉, |νµ〉 =

010

|~p〉, |ντ 〉 =

001

|~p〉. (51)

The |~p〉 denotes the vector from the representation space of the Poincaregroup, corresponding to the momentum ~p.

In the basis (51) the free Hamiltonian describing neutrinos can bewritten as

H = U†H0U, (52)

where U is a 3× 3 unitary mixing matrix, acting in each subspace ofthe Hilbert space determined by momentum ~p, and the HamiltonianH0 in the mass eigenstates basis is of the form

H0 =

√~p2 ±m2

1 0 00

√~p2 ±m2

2 00 0

√~p2 ±m2

3

. (53)

Herem2j (j = 1, 2, 3) are the absolute values of the squares of neutrino

four-momenta and the choice of the signs corresponds to the massive(+) and tachyonic (−) case; moreover |~p| > maxj mj in the tachyoniccase.

Following the standard procedure, we consider the time evolutionof the neutrino state. Assume that the initial state |νi〉 (i = e, µ, τ)is an eigenstate of the fractional lepton number operator. Thus, aftertime t the probability that we get the neutrino νk is given by

Pνi→νk(t) =

∣∣〈νk|U†e−itH0U |νi〉∣∣2 . (54)

Now, taking (54) in the limit |~p| mj , we can expand the matrixelements of the Hamiltonian as follows√

~p2 ±m2i ' p± m2

i

2p. (55)

Using (55) we obtain from (54)

Pνi→νk(t) =

∣∣∣∣∣∣3∑

j=1

u∗jkujie∓ it

2p m2j

∣∣∣∣∣∣2

. (56)


Pawe l Caban et al.

On introducing the polar representation such that

u∗2ku2iu3ku∗3i = r1(i, k)eiθ1(i,k), (57a)

u∗3ku3iu1ku∗1i = r2(i, k)eiθ2(i,k), (57b)

u∗1ku1iu2ku∗2i = r3(i, k)eiθ3(i,k), (57c)

and

ω1 =m2

2 −m23

2p, ω2 =

m23 −m2

1

2p, ω3 =

m21 −m2

2

2p, (58)

we can write (56) in the following form:

Pνi→νk(t) =

3∑j=1

|ujk|2|uji|2

+ 23∑

j=1

rj(i, k) cos[ωjt± θj(i, k)], (59)

where, as before, the signs + and − correspond to massive and tachy-onic neutrinos, respectively. In particular, if k = i then θj(i, i) = 0(j = 1, 2, 3) and

Pνi→νi(t) =3∑

j=1

|uji|4 + 23∑

j=1

rj(i, i) cos(ωjt), (60)

so this probability is the same for massive and tachyonic neutrinos.We have shown that the only difference between the neutrino os-

cillations in the massive and tachyonic case lies in the initial phaseof oscillations θj(i, k). Moreover, the initial phases for tachyonic caseare obtained by taking the complex conjugations of the elements ofmixing matrix for the massive case (see (57a)–(57c)). In the oscilla-tion experiments we cannot decide whether we should take the mixingmatrix U or its complex conjugation U∗ without taking into accountthe nature of the neutrinos. More precisely, we can explain the os-cillations for massive or tachyonic neutrinos simply by the differentchoice of the mixing matrix (U or U∗). Therefore, the experimentalevidence of neutrino oscillations does not distinguish between massiveand tachyonic neutrinos.



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