1.Introduction
Recently, the results obtained from the experiments dedicated to the measurement of
neutrino masses and the relative oscillation phenomena [1-9], often interpretable assuming a superluminal behavior of the
particle, have led many physicists to enhance their efforts towards the formulation
of new theories beyond the Standard Model [10-13].
In this framework, the pioneering works by Surdashan, Feinberg, and Recami [14-17] on the physics of tachyons were taken into
consideration with the aim of formulating a field theory consistent with the theory
of relativity extended to superluminal motions. One of these theories is that of
Chodos [18], whose governing
equation is obtained from the Tanaka Lagrangian [19]. This Lagrangian reads:
Lt=iψ¯tγ5γμ∂μψt-μtψ¯tψt,
(1)
and holds for half-integer spin tachyons. In Eq. (1), ψ¯t=ψt†γ0, γ5=iγ0γ1γ2γ3(γ5)2=1 is the tachyonic mass and hbar=c=1. The subscript t refers to tachyon and will be used in all quantities introduced below.
It remains clear that ψt and ψ¯t are treated as independent variables. Moreover, the Dirac matrix
γ5 is related to the fifth current, and its presence in the Lagrangian (1)
proves the chiral nature of the particles it describes. The Lagrangian (1) is
Hermitian and fulfills the classical tachyonic energy-momentum relation E2=p2c2-μ2c4. By inserting this Lagrangian in the Euler-Lagrange equation [20], the Chodos equation is
recovered:
[iγγ∂μ-μt]ψt=0,
(2)
Eq. (2) is a tachyon-like Dirac equation, one of the most used to study half-integer
spin superluminal particle. Jentschura proved that this equation is CP and T invariant, but the associated Hamiltonian operator is not Hermitian and
loses parity symmetry [21].
However, the Hamiltonian fulfills the symmetry properties of a pseudo-Hermitian
operator.
In this study, the Chodos equation is solved, in terms of Gaussian wave packets with
positive and negative frequencies, for a tachyon propagating along the z direction. In this way, we obtain the equations of the envelope
functions; once solved, show that the group velocity is always subluminal.
Therefore, the tachyonic neutrino described by relativistic quantum mechanics is a
pseudo-tachyon, namely a particle that fulfills the energy-momentum relation typical
of an imaginary mass, but which propagates at subluminal velocity, being the group
velocity equal to that of the quantum particle [22]. This result was obtained also by Salesi using a
different tachyon-like Dirac- equation and following a different approach [23]. Therefore, it is necessary to
distinguish the meaning between the group velocity, which finds its natural place in
the quantum study of the tachyonic neutrino, and the classical velocity of the
particle, which, in principle, is not upper bound. The theory developed in this
work, however, shows that the two velocities are related to each other and that the
tachyon velocity is an upper bound. This is an indirect proof that tachyonic
neutrinos, in the picture of relativistic quantum mechanics, are unstable particles
that decay following mechanisms, already investigated in other works, to return in
the subluminal realm.
As occurs for a relativistic particle with half-integer spin in the Dirac equation,
the Zitterbewegung phenomenon [24], represented by the rapid oscillation of the position of the
particle concerning the median of the Gaussian packet, also takes place for the
tachyonic neutrino. This effect is because although the wave packets with positive
and negative frequencies are orthogonal, once inserted in the integral ⟨ψ-|z|ψ+⟩, which represents the average position of the particle along z direction, show a non-vanishing overlap which leads to the typical
interference of the Zitterbewegung. It is also proved that, unlike what happens for
a relativistic Dirac particle, the oscillation velocity of Zitterbewegung is always
lower than the speed of light.
The wave packet approach is used to calculate the probability of neutrino oscillation
[25]. In this study, we apply
the obtained Gaussian packet to calculate the oscillation probability of tachyonic
neutrino. This needs the assumption that even in the tachyonic regime, the neutrino
may oscillate between possible mass states. We will show that the formula of
probability oscillation is analogous to that expected for the ordinary neutrino.
This is a confirmation that the (pseudo)-tachyonic neutrino cannot be distinguished
from the ordinary one in the experiments concerning oscillation.
2.Tachyonic Wave Packets
Let us consider a superluminal neutrino propagating along z direction. For classical physics, the particle velocity ut can take any value higher than the speed of light. Before proceeding, we
clarify that the mass μt and the tachyonic Lorentz factor γt=[(1-ut2/c2)]-1/2 are pure imaginary, being γt=-i|γt|. Therefore, the product μtγt is always real, as well as the momentum p and the energy E [26]. The mass-energy
μtc2 is instead pure imaginary. The Chodos equation for this model reads:
[iℏγ5γ0∂t-iℏcγ5γ3∂z-μtc2]ψt=0.
(3)
Using the gamma Dirac matrices, the operators γ5γ0 and γ5γ3 are:
γ5γ0=(001¯00001¯10000100),γ5γ3=(1¯000010010100101¯),
(4)
where 1¯ means -1. The bispinor ψt has two components, ψ+ and ψ-, each of which is associated with the positive and negative frequencies
(energies):
*20cψ+=*20cu1+u2+exp{i(kz-ω+t)}ψ-=*20cu1-u2-exp{-i(kz-ω+t)}
(5)
where k=p/ℏ and ω+=E+/ℏ, while p is the z-components of four-momentum. Introducing the bispinor ψt in the Eq. (3) we get a system of four linear differential
equations:
*20ciℏc∂z-μtc20-iℏ∂t00-iℏc∂z-μtc20-iℏ∂tiℏ∂t0-iℏc∂z-μtc200iℏ∂t0iℏc∂z-μtc2*20cu1+u2+u1-u2-e±i(kz-ω+t)=0.
We note that the matrix on the left-side of the equation is anti-Hermitian. As it is
known, anti-Hermitian operators are the infinitesimal generators of unitary
transformations and in quantum mechanics are associated with imaginary eigenvalues
[27]. The system can be easily
solved giving us all the spinor components:
{u1+=-E/(pc+μtc2);u2+=-E/(pc-μtc2);u1-=-E/(pc-μtc2);u2-=-E/(pc+μtc2).
(6)
Considering that E=γtμtc2 and p=γtμtc, Eqs. (6) can be written as functions of the dimensionless factor
γt:
{u1+=|γt|/(|γt|+1);u2+=-|γt|/(|γt|-1);u1-=|γt|/(|γt|-1);u2-=|γt|/(|γt|+1).
(7)
We see that the two-component spinors are real and orthogonal. The obtained solutions
must be normalized by applying the usual normalization procedure:
∫ψ†ψ=1⇒N=2(|γt|2+1)/(|γt|2-1),
where N is the normalization factor. Therefore, the plane waves solutions of the
Chodos equation can be summarized as:
ψ±=2γt2+1γ2-1γtγt±1∓γtγt∓1×exp±ikz-ω+t
(8)
The trend of the real spinor components u1+ and u2+ is shown in Fig. 1.
As can be seen, the two components converge to zero as ut→c, while as ut→2c that tend to separate and diverge towards infinity with different
slopes.
For the models we are developing, we want the solutions to be Gaussian wave packets.
Therefore, we need to find an envelope function that multiplied by the tachyonic
plane wave that provides the expected wave packet. This function is a smooth curve
outlining the extremes in the amplitude of the rapidly varying single wavefunction
that spreads in space and time. Its profile must be that type of a Gaussian
function. To do this, we have to set a given value of the classicalvelocity of the
tachyonic neutrino, denoted by u0, to which correspond the wave vector k0 and the angular frequency ω0. These values represent the center of the Gaussian packet. The Gaussian
spinor can be written as:
ψG±=2γt2+1γ2-1f±(t,z)γtγt±1∓γtγt∓1×exp±ik0z-ω0+t
(9)
where f±(t,z) are the envelop functions for positive and negative frequencies.
Introducing function (9) in the Eq. (2) we get the two differential equations that,
once solved, provide the explicit form of the envelope functions:
*20c(∂∂t-cu1+u1-|γ0∂∂z+Λ0+iℏu1+|γ0)f+(t,z)=0(∂∂t+cu2-u2+|γ0∂∂z+Λ0-iℏu2+|γ0)f-(t,z)=0,
(10)
where γ0 is the module of the tachyonic Lorentz factor corresponding to the
velocity u0 and Λ0±=2μtc2u1±|γ0. The numerical coefficient of the second term in Eqs. (10) is the
propagation velocity, which coincides with the group velocity of the wave packet.
Using Eq. (7) we obtain the explicit form of these velocities:
cu1+u1-|γ0=-cu2-u2+|γ0=γ0-1γ0+1c≤c ∀ u0>c,
(11)
Eq. (11) proves that the neutrino described by Chodos equation behaves like a
pseudo-tachyon, namely a particle propagating with subluminal velocity v=c(γ0-1)/(γ0+1) but fulfilling the energy-momentum relation of a tachyon. This result,
which may seem surprising and unexpected, was also obtained by Salesi following a
different approach [23]. It must
be clear that the velocity v in Eq. (11) is of quantum mechanics nature and is obtained through the
operator iℏc∂z, which is conserved under the action of Lorentz transformations.
Equation (11) gives the relation between the pseudo-tachyon velocity and the
classical velocity u0.
To solve Eqs. (10) it is sufficient to impose that the envelope functions f± are Gaussian. As a basic function, we can take the following Gaussian
envelope, widely used in quantum optics [28-29]:
f±t,z=12πσ+iAt1/2×exp-z∓v0t22σσ+iAt
(12)
Where v0 is the pseudo-tachyon velocity given by Eq. (11), σ is the wave packet dispersion coefficient along z direction, and A is a numerical constant that must be found. Introducing function (12) in
the first equation of (10), and calculating its value in the point (z,t)=(0,0), we get an algebraical equation from which constant A is easily
obtained:
A+=-ω0θ0whereθ0=Λ0+/(ℏω0u-1-|γ0),
(13)
Eq. (13) holds for positive frequencies. Repeating the same procedure for the second
equation of (10) we obtain the value of the constant A for the negative frequencies:
A-=ω0θ0whereθ0=Λ0+/(ℏω0u-1-|γ0).
(14)
Therefore, the Gaussian envelope function for positive and negative frequencies
is:
f±t,z=12πσ∓iω0θ0t1/2×exp-z∓v0t22σσ∓iω0θ0t
(15)
The term ω0θ0 is the tachyonic correction to the dispersion of the wave packet. Figure 2 shows the real and imaginary components
of the wave packet:
Let us analyze in detail this term replacing to ω0 and θ0 their explicit forms:
ω0θ0=E-0ℏ2|μt|c2ℏω0u1+u1-|γ0=3ωPlankγ0-1γ0+1,
(16)
where ωPlank is the angular frequency given by |μt|c2/ℏ. For the factor (γ0-1)/(γ0+1), which is simply the relativistic factor β=v/c of the pseudo-tachyon, the following limits hold:
limu0→c(γ0-1γ0+1)=1;limu0→2c(γ0-1γ0+1)=0:limu-0→∞(γ0-1γ0+1)=-1.
(17)
Since the pseudo-tachyon has a subluminal velocity, the factor (γ0-1)/(γ0+1) must range between [0,1]. This means that the classical velocity of the
neutrino is upper bound to u0=2c. This is a further confirmation of how quantum physics can lead to
completely different results from those obtained applying classical physics, even in
the tachyon field. We conclude this section noting that as u0 increases, the tachyonic dispersion of the wave packet progressively
decreases up to a minimum value of zero, corresponding to the classical velocity
2c. This behavior is similar to that of a wave packet associated with an
ordinary relativistic particle, where the dispersion correction factor is
proportional to 1/γ3[29].
3.Zitterbewegung of Pseudo-tachyon Neutrino
A fermion that obeys the Dirac equation presents a rapid oscillation of the position
along the direction of propagation, known as Zitterbewegung [24]. This happens for the interference between
states with positive and negative energy and occurs with a frequency of 2ωPlank. Assuming that the motion takes place in the z direction, the equation of the position of the ordinary particle around
the median is:
zt=pc2Et+12iℏcEv0-pcE×exp-2iωt-1
(18)
The first term of Eq. (18) is the particle position along the direction of
propagation, while the second one is the oscillation due to the partial overlap of
the positive and negative frequency wave packets. We want to investigate whether
this behavior also takes place for the pseudo-tachyon wave packet. To do this, we
follow the same approach used by Park to study the Zitterbewegung of a relativistic
electron wave packet [29]. In this
regard, we consider the wave function ψ(t,z) given by the linear combination of the two wave packets given by Eq.
(9):
ψ(t,z)=c-1ψG++c2ψG- c1,c-2∈R.
(19)
The mean value of particle position is given by ⟨ψ|z|ψ⟩. The real coefficients c1 and c2 are such as to ensure that the function ψ(t,z) is normalized:
{c1,c2∈R:⟨ψ|ψ⟩=1}.
Although the wave packets ψG+ and ψG- are orthogonal, the integral ⟨ψ|z|ψ⟩ is non-vanishing. The solution of this integral that for the
relativistic electron has been already obtained by Park [29], is:
z=votc-12-c22+1+14σ2ℏ2γ02μt2-1×ℏ2γ02μtcσ2σ2+ω0θ0t1/4×exp-v0t22σ2+ω0θ0t2sin2ω0t-ω0θ0t2σ2+ω0θ0t2γ02-1ω0t+φ22c1c2
where φ=arctan(ω0θ0t) and v0=c(γ0-1)/(γ0+1). The first term in the right-side is the median position of the particle
modulated by the coefficient (c12-c22), which can be positive or negative. Therefore, the median position does
not coincide with the center of the wave packet. The product between the first and
the second factor in the right-size represents the maximum oscillation amplitude,
while the product between the third factor and the exponential represents the
oscillation damping term. Since it has been shown that the term ω0θ0 ranges within [0,2ωPlank], the damping ∑ is maximum when u0=2c:
∑|uo=2c=σ2σ2+2ωPlamk21/4×exp-v0t22[σ2+2ωPlank2
(20)
We note also that when ut→c, i.e. γ→∞, the maximum oscillation amplitude I goes quickly to zero, while it increases progressively as ut→2c up to the upper limit given by:
I|u0=2c=λPlank21+λPlank216σ2-1,
(21)
whereλPlank is the reduced Plank wavelength of pseudo-tachyon neutrino. Finally, we
observe that when ut→2c the coefficient ω0θ0 vanish and ⟨z⟩ becomes:
z|uo=2c=v0tc12-c22+21+14σ2ℏ2μt2-1×ℏ2μtcexp-v0t22σ2sin2ω0t2c1c2
The oscillation frequency around the median position is the argument of sin function; Taylor expanding this function around t→0 and truncating the sum in the first term we get:
ωZB≅ω02-(ω0θ0t)2σ2(γ0-1).
(22)
Using Eqs. (21) and (22) we obtain the Zitterbewegung velocity,
vZB=I|u-0=2cωZB=C1-2ωPlankt2σ2×γ0-1γ0-1γ02-1
(23)
From Eq. (23) we see that, in the range [c,2c], the velocity vZB is always lower than the speed of light.
4.Oscillation of Pseudo-tachyon Neutrino
Recently, Caban et al. have shown that the hypothesis of tachyonic
neutrino leads to the same oscillation phenomenon of ordinary neutrino [30]. This result can be used to
validate the theory presented in Sec. 2. To do this we use the wave packet approach
[25], considering that we have
to write down the function ψG± for each mass eigenstate. By limiting the attention to only the positive
frequency and assuming that there are only three mass eigenstates, the evolved state
of the pseudo-tachyon neutrino produced in the initial state vα is:
|v(t,z)⟩=∑i=13Uαi*ψGi+|vi(t,z)⟩,
(24)
where Uαi is the leptonic mixing matrix that we suppose holds also for tachyonic
particles. The oscillation probability from a state of imaginary mass μi to a state with mass μβ is:
Pva→vβ=vβ|vt,z2=vβ|∑i=33Uai*ψGi+|vit,z2
(25)
To solve this integral one must know the phase difference between the IN and OUT
states:
ΔΦ=ΔE⋅t-Δp⋅z.
(26)
Considering that we are in an ultra-relativistic regime (where ut→c and (γ0-1)/(γ0+1)≅1) and that we are dealing with a wave packet, the following approximations
hold:
ΔE≪E and ΔP≅σp=ℏ/σ.
Therefore, ΔE can be Taylor expanded obtaining:
ΔE≅∂E∂pσp+∂E∂μ2Δμ2=vσp-c42EΔμ2,
(27)
where v is the pseudo-tachyon velocity obtained in Sec. 2, and E is the tachyonic energy-momentum relation. Supposing that the error
σp affecting the momentum is of the order of p0, then the wave packet dispersion σ can be reworked as follows:
∆p≅σp=ℏσ⟹σ=ℏp0=ℏμtu0=ℏμtcγ0-1γ0+1=λPlankγ0-1γ0+1
Substituting this result in Eq. (27) we get:
ΔE≅ℏcλ{Plank}(γ0-1γ0+1)2-Δμ2c42E,
(28)
and substituting Eq. (28) in Eq. (26) we obtain the explicit form of ΔΦ:
∆Φ=-L-γ0-1γ0+1ctγ0-1γ0+1×ℏλPlank-∆μ2c42Et
(29)
For the oscillation to take place there must be interference between the mass states
and this is possible only if the term L-(γ0-1)/(γ0-1) is of the order of dispersion σ. But this means that the first term of Eq. (29) is ≪1 and can be neglected. Therefore:
ΔΦ≅-Δμ2c42Et.
(30)
Considering that t≅L/c (since we are in the ultra-relativistic limit) we arrive at the final
result:
ΔΦ≅-Δμ2c42ELc=-Δμ222pL.
(31)
Therefore, the oscillation probability for a tachyonic neutrino in the Chodos
equation is:
P(vα→vβ)=|∑i=33UαiexpiΔμ222pLUαi*|2,
(32)
Eq. (32) is analogous, except for the sign of the square mass, to the oscillation
probability expected for ordinary neutrino [31]. Therefore, in the state of the art of current
experiments concerning the phenomenon of oscillation, is not possible to distinguish
the bradyonic or tachyon nature of the neutrino. This confirms the result obtained
by Caban et al. [30] and proves the correctness of the theory developed on the
Chodos equation.
5.Discussion
In this study, the Chodos equation for the tachyonic neutrino has been solved,
obtaining Gaussian wave packets, with positive and negative frequencies, analogous
to those obtained by solving the ordinary Dirac equation [32]. The equations obtained for the envelope
functions, which guarantee the Gaussian shape of the wave packet, show that the
group velocity is always subluminal. Since the group velocity coincides with that of
neutrino propagation, one concludes that a particle with a half-integer spin with a
classical velocity greater than the speed of light, in the framework of quantum
mechanics behaves like a subluminal fermion with imaginary mass. Furthermore, the
theory shows that the results retain their physical meaning if the classical analog
of the tachyonic velocity is upper bound by 2c. This suggests that such particles are theoretically possible but are
unstable and do not decrease their energy by increasing their velocity, as the
classical tachyon theory would predict [26]. This hypothesis has been deeply investigated by
Jentschura [33], who proposed
possible mechanisms of decay.
The first validation test of the obtained solutions is represented by the study of
the Zitterbewegung effect. The theory shows that this effect also occurs for the
Chodos. It highlights the typical oscillation of the position of the particle around
the median. However, the oscillation velocity always remains lower than the speed of
light, unlike what was predicted for the electron by the Dirac equation, where this
velocity results equal to the speed of light [34].
The second validation test of the theory is represented by the calculation of the
oscillation probability of tachyonic neutrino, assuming that the modules of the
imaginary masses are identical to those of the three ordinary neutrinos and that the
leptonic mixing matrix is the same as the current model. Following the wave packet
approach, it is obtained a probability formula having the same form of that used for
ordinary neutrino, confirming the same result achieved by other authors [30] using a different approach,
which provides for the impossibility of distinguishing tachyonic and ordinary
neutrinos in the oscillation phenomena.
This work proves that the equation proposed by Chodos for the description of a
superluminal neutrino is consistent with what is expected from a theory that has its
foundation in the Dirac equation. Many physical-mathematical aspects of the Dirac
equation also recur for that of Chodos [21], and its application to real problems reproduce
results obtained by following other approaches [30]. The most evident result, however, is that which
proves that in the Chodos equation, the neutrino behaves like a subluminal particle
that obeys the energy-momentum relationship typical of classical tachyons. This
could be one of the reasons why, to date, there is no experimental evidence that
proves with certainty the possible tachyonic nature of neutrinos and that the
efforts to detect them must be oriented towards the precision measurement of the
value of their square masses. Only negative values of this quantity can confirm
whether or not the neutrinos can have imaginary mass states.
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