Submitted to a special issue of Symmetry
on Tachyons and Fundamental Symmetries.

# A review of searches for evidence of tachyons

Robert Ehrlich  George Mason University, Fairfax, VA 22030<EMAIL_ADDRESS>
###### Abstract

Here we review searches for empirical evidence for the existence of tachyons,
superluminal particles having $m^{2}<0.$ The review considers searches for new
particles that might be tachyons, as well as evidence that neutrinos are
tachyons from data which may have been gathered for other purposes.

##  I Introduction

This review of searches for evidence of tachyons follows the only other one
ever done a half century ago by Michael Kreisler. (Kreisler1973, ) Like that
first review this one can make no claim that “extraordinary” evidence (Sagan’s
Criterion) (Gillespie1999, ) for tachyons has been found, but unlike
Kreisler’s review, this one points to many pieces of circumstantial evidence
that tachyons actually do exist, or more specifically that some neutrinos are
tachyons. The evidence is from a wide range of fields including cosmic rays,
supernovae, cosmology, and neutrino lab experiments. Possible problems with
the tachyonic neutrino hypothesis are also discussed, but none of them appear
to be fatal. In fact, some of the often-expressed concerns about tachyons,
e.g., their lack of respect for causality, or the reversals of cause and
effect they might permit, could be viewed as an argument for their existence,
rather than the reverse, because that is the case in quantum mechanics, which
operates equally well in forward or reverse time. Furthermore, as Andrzej
Dragan and Artur Ekert have shown (Dragan2020, ), if one keeps both subluminal
and superluminal terms in the Lorentz Transformation then non-deterministic
behavior and non-classical motion of particles arise as a natural consequence,
similar to quantum theory

##  II Tachyons as new particles

In 1962 Bilaniuk, Deshpande and Sudarshan, (Bilaniuk1962, ) suggested a way
superluminal particles now known as tachyons might exist without conflicting
with Einstein’s ban on faster-than-light speeds provided they had such speeds
from the moment of their creation. Einstein himself, in fact, cautiously
restricted his ban on superluminal speeds to the case of particles that were
slowly accelerated, and had an initial speed less than light. In fact he noted
in his 1905 paper that: “For velocities greater than that of light our
deliberations become meaningless.” (Einstein1905, ) After the work by
Sudarshan and colleagues in 1962, it was Gerald Feinberg who later in a
classic paper coined the term tachyon, and proposed that tachyonic particles
could be made from excitations of a quantum field with imaginary mass.
(Feinberg1967, ).

During the next few decades after 1962 physicists searched for new particles
whose properties matched those of tachyons, specifically they either travelled
at superluminal speed or had a negative mass squared. The former category
would presumably emit Cherenkov radiation in vacuum if they were charged. In
several experiments it was assumed that tachyons of charge $Ze$ could be
produced in lead by 1.2 MeV photons from a ${}^{60}Co$ source after which they
passed between a pair of charged plates having an electric field $E$ between
them. (Alvager1968, ; Davis1969, ) It was assumed that the rate of change in
their kinetic energy with distance due to the Cherenkov effect (first term)
and the electric field (second term) is given by:

$\frac{dK}{dx}=\frac{-2\pi^{2}K^{2}Z^{2}}{h^{2}c^{2}}+ZeE$ (1)

Whatever its initial energy, the tachyon would quickly attain a kinetic energy
$K$ at which $dK/dx=0,$ Thus, experimenters looked for a peak at the predicted
energy when particles were detected in a counter. In the first experiment one
detector was used, while the second experiment used two detectors in
coincidence to reduce the background, but neither experiment found any
evidence for tachyons. The reported negative results of these experiments
meant that if tachyons were being produced their cross section was more than
$10^{8}$ times smaller than that of electron-positron pair production at the
same 1.2 MeV energy.

It is possible that tachyons might only be produced at very high energy. In
particular, they might be present in the showers created when extremely high
energy cosmic rays enter the Earth’s atmosphere from space, and could reveal
their presence if they reached ground-based detectors before the shower front
travelling at essentially light speed. Here again only upper limits were
found, although an initial 1974 report by Roger Clay and Philip Crouch
reported seeing “possible” evidence for tachyons in the form of many pulses in
the detectors in advance of the shower front. (Clay1974, ) The same group,
however, using an improved electronics system failed to reproduce their
earlier results, which was also the case in subsequent searches by other
groups conducted during the 1970’s and early 1980’s. (Prescott1975, ;
Murthy1971, ; Fegan1975, ; Emery1975, ; Hazen1975, ; Smith1977, ; Bhat1979, ;
Marini1982, ). A much more recent search in 2020 similarly found no evidence
in cosmic ray data. (Garipov2020, )

Regarding the lab experiments looking for tachyons from non-astrophysical
sources, the most familiar method used was based on the “missing mass.” For
example, consider the reaction: $K^{-}+p\rightarrow\Lambda^{0}+X^{0},$ where
$X^{0}$ is a neutral particle. Even though this neutral particle would leave
no tracks in a bubble chamber assuming it is long-lived, the neutral
$\Lambda^{0}$ does reveal that particle’s momentum vector when it decays via
$\Lambda^{0}\rightarrow p+\pi^{-},$ and so one could easily compute the unseen
$X^{0}$ mass based on track measurements for all charged particles and then
imposing energy and momentum conservation. Fig. 1 shows the results of one
such missing mass experiment by Baltay et al. (Baltay1970, ) The narrow peak
at $m_{X}^{2}=0.02GeV^{2}$ corresponds to X being a neutral pion. Most of the
rest of the events having $m_{X}^{2}>0$ have two or more neutral pions. In
principle, the small number of events having $m_{X}^{2}<0$ are tachyon
candidates, but when the tracks for these events were remeasured virtually all
of them turned out to have $m_{X}^{2}>0,$ so they just represent bad
measurements. A year after Baltay et al., Danburg et al. (Danburg1971, )
obtained the same negative result from the same reaction.

Figure 1: Plot of missing mass, that is the mass of X for the reaction
$K^{-}+p\rightarrow\Lambda^{0}+X.$ The plot is from ref (Baltay1970, ).

Another bubble chamber experiment looked for tachyons being emitted when no
incident particles were present, but where protons in the chamber underwent
decay via the reaction: $p\rightarrow p+T^{0}$ and $p\rightarrow
p+T^{0}+\bar{T^{0}}.$ These processes could only occur if the tachyon had a
negative energy, which is allowed for these particles, if they exist. The
negative result in this case was that if the process does occur, its lifetime
would be at least $2\times 10^{21}$ years.(Danburg1972, ) Further limits on
the process were later set by Ljubičic, et al. using a somewhat different
technique. (Ljubicic1975, )

A completely different type of search involved looking for tachyon-like
magnetic monopoles. The motivation for this odd-sounding possibility was based
on the fact that ordinary particles are either neutral or electrically
charged, while those that move at v = c are neutral. Therefore, symmetry might
require tachyons to be either neutral or magnetically charged. Based on this
conjecture, Bartlett and colleagues conducted several searches for tachyon
magnetic monopoles (TM) having $v>c.$ (Bartlett1972, ; Bartlett1978, ). These
searches looked for particles emitted from a radioactive source and later
cosmic ray particles that emitted Cherenkov radiation in vacuum. In principle,
the TM gains energy in a magnetic field (just like an electric charge gains
energy in an electric field) and it would rapidly attain an equilibrium energy
where the energy gained equals the energy lost due to Cherenkov radiation,
which is then detected by photomultiplier tubes. Given all the theoretical
unknowns concerning tachyon monopoles, Bartlett et al. conclude their 1972
article by listing four reasons why they might not have detected them even if
they exist. While Bartlett et al. have reported negative results for searches
for TM’s from both of their searches, some “possible” positive results have
been reported by Fredericks. (Fredericks2015, ) Those results, however, relied
on an entirely different, and somewhat subjective hard-to-quantify technique
(the shape of tracks in a nuclear emulsion).

A most unusual experiment searching for tachyons was done by V. P. Perepelitsa
in 1977 (Perepelitsa1977, ). This experiment, dubbed a “tachyon Michelson
experiment” looked for the possible tachyon reaction $e^{+}e^{-}\rightarrow
T^{+}+T^{-}$ using positrons at rest from a radioactive source. The basis of
the name of the experiment was that Perepelitsa believed that in order to
prevent a causality violation with tachyons, their emission directions would
be restricted to certain angles between the velocity of the particle and the
instantaneous velocity of the laboratory with respect to some preferred
reference frame. Under this assumption tachyons which corresponded to events
in two counters that corresponded to $v>>c$ times-of-flight would have a
distribution in time that varied due to the Earth’s rotation, and hence with
celestial time. No such variation was seen and Perepelitsa was able to set an
upper limit on the branching ratio of tachyon production relative to photons.
Of course, this negative result depended on his assumption that tachyons if
they exist could not violate causality, which is a feature of some theories of
these particles that stipulate a preferred reference frame. (Radzikowski2010,
)

One final search noted here for tachyons as new particles has also been done
by Perepelitsa and colleagues in 2019 and 2020. The group reported positive
results in a pair of unpublished papers. (Perepelitsa2019, ; Perepelitsa2020,
) Perpelitsa and colleagues searched for anomalous Cherenkov rings indicating
superluminal speed charged particles as recorded by the DELPHI detector at
CERN from high energy $e^{+}+e^{-}$ collisions. In their experiment the mass
of the charged candidate tachyon can be found from the Cherenkov ring radii
and the measured track momenta for 27 candidate events. Examining the
distribution of the magnitude of the tachyon mass ($m=\sqrt{-m^{2}}$), they
found two peaks corresponding to $m_{1}=0.29\pm.01GeV/c^{2}$ and $m_{2}=4.6\pm
0.2GeV/c^{2}.$ Given the low statistical significance of the claim
($p<10^{-3}$), however, more data is clearly warranted to increase the claim’s
credibility in the absence of other supporting data.

The long string of negative, low statistics, and ambiguous results from
searches for tachyons may have suggested to many physicists that tachyons do
not exist. However, such an inference would be misguided. Perhaps one was
looking for them in the wrong way or in the wrong places, or perhaps their
cross section was simply very small. Even more simply, perhaps they are
actually a well-known particle, rather than a new one.

##  III Neutrinos as tachyons

In 1970 Cawley was the first to raise the possibility that neutrinos might be
tachyons (Cawley1970, ), but it was not until 1985 that Chodos, Hauser, and
Kostelecký (Chodos1985, ) discussed a theoretical framework for describing
tachyonic neutrinos. While Chodos and collaborators were candid about the
difficulties of formulating a complete quantum field theory of tachyonic
neutrinos, they also noted that such difficulties cannot be used to exclude a
priori their existence. Indeed, since 1985 theoreticians including Jentschura
(Jentschura2012, ), Rembieliński and Ciborowski (Rembielinski2021, ),
Radzikowski (Radzikowski2010, ), and Schwartz (Schwartz2016, ) have found ways
to deal with many of these difficulties.

The Chodos et al. suggestion that neutrinos might be tachyons was a natural
one because unlike all other particles no neutrinos had then (or now) ever
been observed to travel slower than light or have a positive mass squared. In
fact, of the 13 experiments to measure the effective mass of neutrinos between
1990 and 2021, all but one reported $m^{2}<0$ values, two of the values being
many standard deviations below zero. (Kraus2005, ) None of the researchers
reporting those negative $m^{2}$ results, however, regarded them as
constituting evidence for electron neutrinos being tachyons, and indeed the
experimenters sometimes described their results to be “unphysical,” being due
to some unknown or suspected systematic effect. This assessment was almost
certainly correct, although I once thought otherwise.

In recent years the search for evidence of tachyonic neutrinos is very well
motivated by the increasing recognition that Lorentz invariance may be
slightly violated, and the neutrino sector is probably the most likely sector
where such violation might be observed. (Kostelecky2012, ). Superluminal
neutrinos would not be the only way to have a violation, but it is certainly
one way it could occur, so theorists open to the possibility of small
violations of Lorentz invariance should also be open to the possibility of
tachyonic neutrinos. Alan Kostelecky and colleagues have been particularly
active in formulating the Standard-Model Extension (SME) which is an effective
field theory containing all operators for different types of Lorentz
violation. They have also been very active in proposing tests to look for each
type of violation. (Diaz2014, )

Before considering experiments on the neutrino mass to look for possible
$m^{2}<0$ values, let us first consider measurements of neutrino speed. One
could, for example, demonstrate that neutrinos are superluminal if they
travelled a precisely determined distance in less time than light in vacuum.
This time-of-flight technique cannot be done using individual neutrinos, given
their low cross section, but rather with short bunches of neutrinos produced
in a pulsed accelerator, reactors, of course, being unable to produce short
pulses of neutrinos. The most memorable such neutrino velocity measurement was
made by the OPERA Group at CERN, which reported in 2011 that bunches of muon
neutrinos in a 17 GeV beam had apparently travelled a distance of 732 km in a
time that was $60.7\pm 9.9$ ns less than a photon, a $6\sigma$ result. The
corresponding excess above light speed was $\delta=v/c-1=2.48\pm 0.42\times
10^{-5}.$ Adam2011 .

The OPERA initial result was the stimulous for an impressive number of
theoretical papers, some of them very skeptical. In fact, one paper by Cohen
and Glashow showed that given the large size of $\delta$ reported, the result
would be the radiation of electrons and positrons through the process
$\nu_{\mu}\rightarrow\nu_{\mu}+e^{+}+e^{-},$ which could only occur if
neutrinos are superluminal. (Cohen2011, ) Cohen and Glashow never use the word
tachyon or consider superluminal neutrinos to have $m^{2}<0,$ but they showed
that the energy threshold for the above decay process to occur was given by
$E=2m_{e}/\sqrt{\delta}\sim 140MeV,$ $m_{e}$ being the electron mass. They
further showed that for 17 GeV neutrinos (3 orders of magnitude above
threshold), their resulting energy loss was great enough to make it impossible
for most of the neutrinos in the beam to reach the detectors, meaning that the
OPERA initial result probably had to be in error.

This OPERA neutrino anomaly was subsequently shown to be an error resulting
from a loose cable, and a clock that ticked too fast. The two errors had
opposite impacts on the measured neutrino speed, but when they were both
corrected, the excess above light speed was only $\delta=2.7\pm 5.1\times
10^{-6}.$ (Adam2012, ) Similar measurements of the speed of muon neutrinos in
pulsed accelerator beams at GeV-scale energies have been made by the ICARUS,
MINOS and T2K groups. All such speed measurements have resulted in values
consistent with that of light within experimental uncertainties.
(Antonello2012, ; Adamson2015, ; Abe2015, ) In summary, these speed
measurements neither required nor rejected the possibility of superluminal
neutrinos; they just were not sensitive enough to measure any nonzero value of
$\delta$ given the smallness of the neutrino mass. For example, if we use the
limit on $\delta=10^{-6}$ from the MINOS Collaboration, (Adamson2015, ) with
their neutrino beam energy $E=3GeV,$ we find $m\sim\sqrt{2E^{2}\delta}=4MeV,$
which is more than an order of magnitude greater than the current upper limit
on the muon neutrino mass ($m_{\nu}<0.16MeV.$) (Zyla2021, )

If tachyonic neutrino satisfy the same dispersion relation as bradyons we
expect their superluminality to become most evident the lower their energy, in
fact zero energy tachyonic neutrinos would have infinite speed . However, it
is also possible that the reverse is true if a different dispersion relation
applies for $v$ close to $c,$ in which case the degree of superluminality
might increase with the neutrino energy. The experiment observing the highest
energy neutrinos probably of extragalactic origin is IceCube, which has
observed neutrinos having $E_{\nu}>60TeV.$ If superluminal neutrinos are
present their spectrum would have a cut-off due to their creation of Cherenkov
radiation when they undergo the vacuum pair emission (VPE) process
$\nu\rightarrow\nu+e^{+}+e^{-},$ originally suggested by Cohen and Glashow, a
process which is only possible for $v>c$ particles. (Cohen2011, ). From intial
data it appeared that the spectrum of IceCube neutrinos might have a cut-off
above 2 PeV, which would correspond to $\delta\sim 0.75\pm 0.25\times
10^{-20}.$ (Stecker2014, ; Jentschura2016, ) However, more recent data through
2021 (10 years worth) showed that there appeared to be no cut-off up to an
energy of 10 PeV, the highest energy for which data was available.
(Ganster2021, )

While this indication of superluminal neutrinos in IceCube has apparently
evaporated, a more robust one may have been discovered by Yanqi Huang and Bo-
Qiang Ma in 2018 who found nine instances in which an extremely high-energy
neutrino arrived at about the same time and came from the same direction as a
gamma-ray burst (GRB) observed by other instruments. (Huang2018, ) Five of the
9 neutrinos appear to have traveled very slightly faster than c, while the
other four traveled very slightly slower, and there was a trend of linearly
increasing $\delta$ with increasing neutrino energy. After a follow-up search
published a year later they found 12 more neutrinos associated with GRB’s
satisfying a similar pattern. (Huang2019, ) Huang and Ma have suggested that
this finding represents Lorentz violation of cosmic neutrinos and also a
violation of CPT symmetry between neutrinos and antineutrinos. With such a
violation neutrinos and anti-neutrinos would have different signs for the
Lorentz Violating factor, so the neutrinos could be superluminal and
antineutrinos subluminal or the reverse. As with Cohen and Glashow, Huang and
Ma never refer to superluminal neutrinos as being tachyons having an imaginary
mass, but prefer to frame their analysis in terms of a Lorentz violating
dispersion relation. The analysis by Huang and Ma has not, however, provided
definitive evidence for superluminal neutrinos of for violations of CPT or
Lorentz invariance. Thus, Ellis et al. (Ellis2019, ) have argued that the
significance of Huang and Ma’s result is overstated because it depends on
cross-correlating sub-MeV and multi-GeV photons from GRB’s, which is risky
given the paucity of multi-GeV photons from GRB’s.

##  IV Energetically forbidden $\beta-$decay

Following their 1985 paper suggesting that some neutrinos might be tachyons
Chodos and his colleagues Kostelecky, Potting, and Gates considered in 1992 a
most surprising experimental test of this possibility. (Chodos1992, ).
Consider, for example, the following two energetically forbidden examples of
beta decay:

$p\rightarrow n+e^{+}+\nu_{e}\;\;\;\;(-1.8MeV)$ (2)

${}^{4}He\rightarrow^{4}Li+e^{-}+\bar{\nu}_{e}\;\;\;\;(-22.9MeV)$ (3)

where the numbers in parenthesis indicate the deficit in energy preventing the
decay from occurring. Chodos and colleagues realized that if the electron
neutrino emitted in beta decay is a tachyon then its total energy $E$ could be
negative in some reference frames, and thereby permit the two energetically
forbidden decays if the energy of the tachyonic neutrino or antineutrino was
sufficiently negative, e.g. more negative than -1.8 MeV for reaction 1. The
idea can be easily seen using the Lorentz Transformation for the neutrino
energy in going from the lab frame (E) to the CM frame (E’):

$E^{\prime}=\gamma[E-\beta pcos(\theta)]$ (4)

If the angle of the emitted neutrino $\theta=0$ and $E>0$ we find that an
observer having a speed $\beta>E/p=E/\sqrt{E^{2}+m^{2}}<1$ (since $m^{2}<0$)
would judge the emitted neutrino energy $E^{\prime}$ to be negative. A
consequence of this fact is that any energetically forbidden beta decay
becomes allowed when the parent nucleus moves past us at sufficient speed or
energy. For example, from relativistic kinematics the thresholds for the
decays given in Eqs. 2 and 3 can be easily shown to be:

$E_{p,th}(eV)=\frac{1.8\times 10^{15}m_{p}(GeV)}{|m_{\nu}|(eV)}$ (5)

$E_{\alpha,th}(eV)=\frac{22.9\times 10^{15}m_{He}(GeV)}{|m_{\nu}|(eV)}$ (6)

Obviously, based on Eqs. 5 and 6, if the magnitude of the tachyonic neutrino
mass $|m_{\nu}|\equiv\sqrt{-m_{\nu}^{2}}$ is very small the threshold energy
for proton beta decay (Eq. 5) will be very high, while that for alpha decay
(Eq. 6) will be 50.9 times higher still. Chodos et al. suggested the beta
decay: ${}^{162}Dy\rightarrow^{162}Ho+e^{-}+\bar{\nu}_{e}$ as being
particularly promising one to look for in view of its low threshold energy,
about a third of that for proton decay. (Chodos1992, ) However, this
assessment ignores the difficulty of producing a very pure beam of high
intensity and high energy ${}^{162}Dy$ nuclei compared to protons, and the
fact that the energy required might easily exceed that of any known
accelerator, depending on the value of $|m_{\nu}|.$

##  V The cosmic ray spectrum

The cosmic rays consist of protons and other nuclei that bombard the Earth
from space with an enormous range of energies, some far higher than any
accelerator. The majority of the cosmic rays are protons and alpha particles
with small contributions from heavier nuclei. Their spectrum is approximately
given by a power law in their flux of the form: $dF/dE\propto E^{-\gamma}$
over many decades of energy E, where $\gamma\approx 3.$ There are four notable
features in the cosmic ray spectrum that have been the subject of great
interest to researchers. These four features are known as the first and second
“knees,” the “ankle,” and the GZK cut-off. The two knees are slight increases
in the value of $\gamma$ (a ”hardening” of the spectrum) that abruptly occur
at specific energies, the ankle is the reverse, a slight decrease in $\gamma$
that also occurs abruptly at a specific energy. The GZK cut-off is a
hypothesized abrupt end of the spectrum suggested independently by Greisen,
Zatsepin and Kuzmin (Greisen1966, ; Zatsepin1966, ) at an energy of about
$5\times 10^{19}$ eV. At that energy cosmic ray protons would have enough
energy to be blocked by their interaction with the cosmic background radiation
filling all space. The reaction accounting for this cut-off would be
$p+\gamma\rightarrow\Delta^{*}(1238)\rightarrow p+\pi,$ where $\gamma$ is a
background radiation photon and $\Delta^{*}(1238)$ is a well-known resonant
state of a proton and pion. If the cosmic ray particle is a nucleus of atomic
mass A instead of a proton, then the predicted cut-off would be A times higher
energy, because it is really the energy per nucleon in the nucleus that
determines the threshod of the $\Delta^{*}$ reaction.

The four features of the cosmic ray spectrum noted above have a natural
explanation if the electron neutrino is a tachyon of mass $|m_{\nu}|\approx
0.5eV$ or $m_{\nu}^{2}\approx-0.25eV^{2},$ as I discussed in a pair of 1999
papers. (Ehrlich1999, ; Ehrlich1999a, ) However, it should be noted that this
mass estimate was only an approximate value, given the uncertainty in the
energy of the knees, and in a later paper I revised it to
$m_{\nu}^{2}=-0.11eV^{2}$ based on other evidence. (Ehrlich2015, ). The
presence of the two knees and ankle show up most clearly when one displays
$E^{3}dF/dE$ rather than $dF/dE$ itself – see Fig. 2. The existence of the GZK
cut-off is not obvious in the figure, in light of the handful of data points
above $E=10^{20}eV,$ in seeming defiance of the cut-off. However, the rapid
decline of $E^{3}dF/dE$ for the more accurate data points between the dashed
line and $E=10^{20}$ clearly shows the cut-off occurs at the predicted energy
(for a proton).

Figure 2: Plot of $E^{3}J\equiv E^{3}dF/dE$ for the cosmic ray spectrum using
data from the four indicated experiments showing the locations of the two
knees, the ankle, and the GZK cut-off. The data is from ref. (Abassi2018, ),
and the three vertical bands have been added to show the approximate
uncertainty in the energies of the two knees (downturns) and the ankle (an
upturn). The dashed line shows the energy of the hypothesized GZK cut-off.

Suppose one assumes a threshold energy for proton beta decay (Eq. 2) of
$3PeV=3\times 10^{15}eV$ (the energy of the first knee). In that case, at
higher energies protons are depleted from the spectrum in increasing numbers,
which explains the occurrence of the first knee, assuming the cosmic ray
spectrum at their source were a pure power law, i.e., no knee. Based on Eq. 5,
we find for the mass of the tachyonic electron neutrino $|m_{\nu}|\approx
0.5eV,$ a fact which Alan Kostelecky had originally noted. (Kostelecky1993, )
This interpretation of the first knee is, of course, at variance with the
standard one of it being the transition from galactic to extragalactic cosmic
rays. That standard interpretation, however, remains unverified since we
cannot yet identify the locations of cosmic ray sources, and Erlykin and
Wolfendale have claimed the knee is too sharp to be explained in that manner
(Erlykin2009, ). Moreover, the tachyonic interpretation is butressed by the
observation that while the fraction of the spectrum that consists of protons
is rising before the knee, it starts declining right after that energy.
(Thoudam2016, ). The existence of the second knee constitutes further evidence
supporting the tachyonic interpretation, since as noted earlier, if the first
knee is due to proton beta decay, we would expect to find a second knee due to
alpha decay at an energy 50.9 times higher or $1.5\times 10^{17}$ eV, which is
very close to where it is found.

##  VI The n-p decay chain

Explaining the ankle in terms of the tachyonic neutrino obviously cannot be
done in the same manner as the two knees, given that it is an upturn in
$E^{3}dF/dE$ not a downturn, which could not be due to the onset of some new
decay process. The explanation of the ankle starts with the observation that
it is not possible to determine empirically whether the primary incident
cosmic ray that initiates a shower of thousands of particles is a neutron or a
proton. Furthermore, we observe that if ultra-high energy protons could beta
decay they would create neutrons having a slightly lower energy which
subsequently decay into protons, and back into protons, etc., creating a decay
neutron-proton decay chain (NPDC). The NPDC would continue until the proton
energy drops below the threshold for proton beta decay, i.e., the energy of
the first knee.

Given a neutron lifetime of about 1000 seconds, one would normally not expect
neutrons to reach us from very distant sources, but time dilation changes that
expectation if the energy is high enough. Thus at the energy $5\times
10^{18}eV$ a neutron would have its lifetime lengthened by roughly $5\times
10^{9},$ yielding a mean free path before decay of roughly $mfp\sim 0.15Mly$
for a single step in the decay chain. Given the ratio of the proton and
electron masses, an energetic neutron might give up on the order of $1/2000$
of its energy in a single step of the decay chain, so there could easily be
thousands of steps in the NPDC before the energy drops to that of the first
knee. As a result, for the full NPDC, we might expect the particle to have a
$mfp>150Mly.$ That distance is comparable to the size of the Perseus-Pisces
supercluster which stretches almost $300Mly.$ At one end of it is the Perseus
Cluster (Abell 426), which is one of the most massive galaxy clusters within
$500Mly$ of us. We therefore expect that the higher the cosmic ray energy
above $5\times 10^{18}eV$, the larger the mfp, so cosmic ray protons from more
and more distant sources would reach us. This estimate would explain why the
spectrum times $E^{3}$ abruptly turns up (softens) at this energy creating the
ankle. In addition, the idea of protons reaching us from more and more distant
sources means that the spectrum would become more proton-rich for energies
well above the ankle. Evidence for the spectrum being proton rich at the
highest energies is provided by the GZK cut-off occurring at the predicted
energy for protons rather than that for heavier nuclei, which as noted would
occur at an A times higher energy.

Most cosmic rays being charged particles have their directions randomized by
the magnetic field of the galaxy, making it difficult to know the direction to
their sources. With the hypothetical neutron-proton decay chain (NPDC),
however, for which a cosmic ray particle spends most of its time as a neutron,
its directionality would be mostly unaffected by the galactic magnetic field.
As a result we might suppose that cosmic rays that had energies close to the
knee on reaching Earth might, at least for not too distant sources, point back
to their source. If one found such a source of cosmic rays having an energy at
the knee its existence would offer evidence in support of the NPDC idea, and
hence the tachyonic neutrino hypothesis. Most cosmic ray physicists believe
that the dozen or so reports of Cygnus X-3 being such a cosmic ray source from
the observations made in the 1970-1990 era were mistaken in light of more
recent negative results with more sensitive detectors. However, in
refs.(Ehrlich1999a, ; Ehrlich1999b, ) I explained why that mainstream view may
be mistaken.

##  VII SN 1987A and the $3+3$ model

The first visible supernova in or near our galaxy since the telescope was
invented in 1608 occurred on February 23, 1987. About two to three hours
before the visible light from this supernova known as SN 1987A reached us, a
burst of neutrinos was observed at three neutrino observatories. The neutrinos
arrived before the light because visible light is emitted from the supernova
only after the shock wave reaches the stellar surface. We use the term
neutrinos here to include antineutrinos as well, and in fact the detectors
observed many more $\bar{\nu}$ than $\nu.$ The three detectors included
Kamiokande II (12 events); IMB, (8 events); and Baksan, (5 events); whose
collective burst of 25 neutrinos lasted less than 13 seconds. Approximately
five hours earlier, the Mont Blanc neutrino detector saw a five-event burst.
Most physicists ignore that burst as being unrelated to SN 1987A, because of
its early arrival time, and its non-observation in the other detectors.

Shortly after the supernova was detected it was recognized that the few dozen
neutrinos observed could yield information about the neutrino mass, based on
the measured energies and arrival times. The antineutrinos detected mostly
were observed when they caused the reaction: $\bar{\nu}_{e}+p\rightarrow
n+e^{+},$ with the $e^{+}$ detected based on the Cherenkov radiation it emits.
The recorded quantities were the event time, approximate arrival direction,
and the visible ($e^{+}$) energy, from which the neutrino energy can be found
based on the previous reaction using: $E_{\nu}=E_{vis}+1.3MeV.$ The usual
analysis of this data yields only an upper limit on the electron neutrino
mass, because it assumes the spread in neutrino arrival times reflects mainly
a spread in their emission times from the supernova rather than a spread in
their travel times, which would be negligible, if the three neutrino mass
states were nearly degenerate as normally assumed.

However, Huzita Huzita1987  and a year later Cowsik (Cowsik1988, ) made the
opposite assumption, and they showed that if the emissions were near
simultaneous from SN 1987A, the 25 observed neutrinos were all consistent with
having one of two masses, which Cowsik cited as $m_{1}=4\pm 1eV$ and
$m_{2}=24\pm 7eV.$ I rediscovered this result in a paper I wrote in 2012,
(Ehrlich2012, ) although my values were a bit different than Cowsik:
$m_{1}=4.0\pm 0.5eV$ and $m_{2}=21.4\pm 1.2eV.$ The main reason for the
difference between Cowsik’s values and mine for $m_{2}$ was the result of my
omission of the Baksan data in its computation on somewhat questionable
grounds. This omission not only resulted in a smaller $m_{2},$ but an
uncertainty that was probably unjustifiably small (1.2 eV versus his 7 eV).
Recall that in the simultaneous emission analysis, the neutrino arrival times
are assumed to be a function of their travel time from the supernova, and the
question of whether multiple masses are present is left for the data to
decide, rather than some favored model of the neutrino masses.

To see how the analysis works, we define the arrival time $t$ relative to that
of light, that is, the neutrino travel time is: $t_{trav}=t+T,$ where $T$ is
the light travel time, approximately 168,000 years for SN 1987A. Since the
detectors were not precisely synchronized, following the usual practice we set
$t=0$ for the earliest arriving neutrino in each detector. Under these
assumptions one can then compute individual neutrino masses from relativistic
kinematics based on their arrival time $t$ and energy $E.$ We begin by
observing that:

$v=\frac{L}{T+t}\approx\frac{L}{T}(1-t/T)=c(1-t/T)$ (7)

and further that

$v=c\sqrt{1-1/\gamma^{2}}\approx c\big{(}1-\frac{m^{2}c^{4}}{2E^{2}}\big{)}$
(8)

Equating the two right hand sides yields

$m^{2}c^{4}=\frac{2E^{2}t}{T}$ (9)

Eq. 9 means that in a plot of $1/E^{2}$ versus $t$ neutrinos of a given mass
$mc^{2}$ will cluster about a straight line through the origin of slope
$M=2/Tm^{2}c^{4}.$ The mass for each cluster can then be found from the slope
of the best fit straight line – see Fig. 3. Note that the fact that the two
straight lines passing through the origin give good fits to the data verify
that the first arriving neutrino in each detector did have speeds very close
to light, as was assumed. If there were any neutrinos having a specific value
of $m^{2}<0$ they would cluster about a straight line through the origin with
negative slope, which in fact is the case for the five Mont Blanc neutrinos,
as will be discussed later. It is difficult to assign a probability of finding
such a result of two distict masses by chance, given the small amount of data.
However, it is noteworthy that Loredo and Lamb who did a Bayesian analysis of
this data without any suggestion of two distinct masses reported that two-
component models for the neutrino signal are 100 times more probable than
single-component models. (Loredo2002, )

Figure 3: Plot of $1/E^{2}$ versus neutrino arrival time t for 25 neutrinos
from SN 1987A providing evidence for two masses, based on the clustering of
neutrino events near two straight lines from ref. (Ehrlich2012, ).

In 1987, before neutrino oscillations were discovered, and the neutrino mass
hierarchy became established, such large widely separated neutrino masses as
the above analysis implied could not be ruled out. However, by 2012 those
large widely separated masses were obviously in direct conflict with the
standard neutrino mass hierarchy. In both the normal and inverted hierarchy
the $m^{2}$ values, while unknown, have values that are separated by no more
than $dm^{2}=0.0024eV^{2},$ that is, the sum of the atmospheric and solar
$dm^{2}$ values found from neutrino oscillation data. The whole basis for this
standard neutrino model, however, was that there is observed to be no third
$dm^{2}$ value observed. In other words in both the normal and inverted
hierarchy the oscillations corresponding to $dm^{2}=dm^{2}_{atm}+dm^{2}_{sol}$
are experimentally indistinguishable from those for $dm^{2}=dm^{2}_{atm}.$

Continued adherence to the conventional hierarchies, however, ignores evidence
that has been accumulating for a third oscillation frequency corresponding to
$dm^{2}\sim 1eV^{2}.$ Moreover, the usual $3+1$ way of incorporating a fourth
sterile neutrino assumed to be heavier than the three active ones still does
not give a satisfactory global fit in light of a tension between appearance
and disappearance data, (Kopp2013, ) and the conflict remains even with two or
three sterile neutrinos. (Maltoni2007, ) In 2013, I therefore, proposed an
alternative to the widely accepted view, namely the $3+3$ model. (Ehrlich2013,
) Note that the model depicted in Fig. 4 allows for having three very unequal
neutrino masses since it assumes the atmospheric and solar oscillations are
within the two $m^{2}>0$ active-sterile doublets.

The basis of the third doublet being tachyonic was so that the effective mass
of the electron neutrino defined by $m_{\nu}^{2}=\Sigma|U_{ij}|^{2}m_{j}^{2},$
could have the small negative value inferred from the cosmic ray data, as
discussed earlier. The specific tachyonic mass value indicated in Fig.4 was
based on a remarkable numerical coincidence, namely that the fractional
separation of the two $m^{2}>0$ doublets turn out to be identical, namely
$dm_{1}^{2}/m_{1}^{2}=dm_{2}^{2}/m_{2}^{2}=5.0\times 10^{-6}$ to within 4%.
Therefore, it is reasonable to suppose the same value held for the third
$m^{2}<0$ doublet. Given the increasingly clear evidence for a third
oscillation frequency corresponding to a $dm_{3}^{2}\sim 1eV^{2}$ the value of
the tachyonic mass would then be $m_{3}^{2}=dm_{3}^{2}/5\times
10^{-6}=-450^{2}eV^{2}\sim-0.2keV^{2}$ to within a factor of two, given the
uncertainty in $dm_{3}^{2}.$

Figure 4: $m^{2}$ values in units of $eV^{2}$ for the three active-sterile
doublets in the $3+3$ model and their splittings, $dm^{2}.$ The choice of
$m_{3}$ was made so that the three doublets have a common fractional
splitting, $dm^{2}/m^{2}.$ The plot is not to scale.

##  VIII Evidence for the $3+3$ model

Most of the rest of this paper consists of a description of evidence for the
$3+3$ model. The extensive attention to this evidence is justified in light
of: (a) its magnitude and variety, (b) the lack of irrefutable conflicts with
observations, (c) the new information since a previous summary of the evidence
in ref. (Ehrlich2019, ), (d) the absence of any other tachyonic neutrino model
that makes specific predictions, and (e) the paucity of any other predictions
of specific tachyon properties that are even testable. Aside from the
previously discussed prediction of the tachyonic mass value for the electron
neutrino $(m^{2}~{}\sim-0.11eV^{2}),$ (Ehrlich2015, ), the one other specific
testable prediction has been advanced by Chodos. This prediction based on a
new proposed symmetry principle, known as Light Cone Reflection, postulates
the existence of pairs of tachyonic and bradyonic neutrinos having the same
magnitude of their mass $m=\sqrt{|m^{2}|}.$ (Chodos2012, ).

###  VIII.1 The $m^{2}>0$ masses in the $3+3$ model

A confirmation of the two $m^{2}>0$ masses in the model comes from fits
involving astrophysical data. Sterile neutrinos have long been considered a
plausible possibility for dark matter, which holds galaxies and their clusters
together. In 2014 Man Ho Chan and I showed, using the dark matter radial
distribution for the Mlky Way that was inferred from the radial velocity curve
could be fit very well using a nearly degenerate gas of neutrinos having a
mass close to 21.4 eV. (Chan2014, ) We further showed that for four clusters
of galaxies the dark matter radial distributions holding them together could
be fit using neutrinos having a mass close to 4.0 eV, these being the two
$m^{2}>0$ masses in the $3+3$ model. (Chan2014, )

Additional evidence for the existence for one of the $m^{2}>0$ neutrino masses
comes from work by Mohapatra and Sciama. In a 1998 unpublished manuscript
(Mohapatra1998, ) these physicists build on earlier work by Sciama
(Sciama1993, ). They noted that the interstellar medium of the Milky Way
galaxy is known to contain ionized hydrogen gas with properties which seem to
defy common astrophysical explanations, and that the same is true for other
galaxies. They therefore proposed that relic neutrinos which are supposed to
pervade the cosmic background could explain the observed diffuse ionization if
they had a mass of 27.4 eV and decayed into a lighter neutrino ($\nu_{L}$)
plus a photon: $\nu\rightarrow\nu_{L}+\gamma$ with a lifetime $10^{22}$ sec.
In that case the photons would have an energy of 13.7 eV and could provide
enough ionizing photons to produce the observed diffuse ionization.

Since 1998, the puzzle of the observed diffuse ionization in the galaxy
remains unexplained by conventional mechanisms. Thus, for example, Phan,
Morlino and Gabici (Phan2018, ) have noted that, while cosmic rays are usually
assumed to be the main ionization agent for the interior of molecular clouds,
where UVand X-ray photons cannot penetrate, a calculation of the amount of
ionization produced by cosmic rays is more than ten times less than what is
observed. Mohapatra and Sciama’s inferred neutrino mass for the sterile
neutrinos in the galaxy is, of course, consistent with the mass that Cowsik
found ($m_{2}=24\pm 7$) eV from his analysis of the SN 1987A data, although it
is slightly too high to match the mass I had used in developing the $3+3$
model derived from that same data, owing to my very likely too small error bar
on $m_{2}=21.4\pm 1.2eV.$

One further place where evidence for the presence of the two $m^{2}>0$ masses
in the $3+3$ model might be found would be in oscillation data where one might
look for oscillations corresponding to $dm^{2}=21.4^{2}-4.0^{2}\sim
450eV^{2}.$ As is well known, neutrino oscillations corresponding to a
particular $dm^{2}$ result in a variation in probability of neutrino
appearance or disappearance of the form:

$P\sim\alpha\times sin^{2}\big{[}1.27dm^{2}(eV^{2})L(km)/E(GeV)\big{]}$ (10)

Thus, to see oscillations having as high a frequency as implied by $dm^{2}\sim
450eV^{2}$ would require both a short baseline $L$ and a high energy $E.$ As
of 2022, only two experiments have so far been sensitive to this high
frequency range. The MINOS group in 2019 reported seeing no evidence for
oscillations with $dm^{2}<1000eV^{2}$ whose amplitude larger than $\alpha\sim
0.02.$ (Adamson2019, ) However, that negative result was based on the $3+1$
model (a single sterile neutrino oscillation), and it very well might not have
picked up evidence for a double oscillation, namely both $dm^{2}\sim 1eV^{2}$
and $dm^{2}\sim 450eV^{2}.$ In any case, the MINOS data would not have
revealed oscillations with amplitude $\alpha<0.02.$

A second recent experiment sensitive to oscillations with a
$dm^{2}<1000eV^{2}$ was done by the MicroBooNE group. (Denton2021, ) This
experiment did observe a hint of oscillations corresponding to $dm^{2}\sim
1.4eV^{2},$$(2.8\sigma)$ but for $dm^{2}\sim 450eV^{2}$ the situation was less
clear. They had analyzed the data by different methods, and according to one
method (the Pandora analysis with 0 protons), their data was consistent with
oscillations anywhere in the range $10<dm^{2}<1000eV^{2}.$ Clearly, more
oscillation experiments need to explore the little-explored high $dm^{2}$
realm. While $m^{2}<0$ neutrinos can participate in oscillations too,
(Caban2006, ) searching for a $dm^{2}\sim-200,000eV^{2}$ due to the $m_{3}$
mass in the model is probably out of the question. Fortunately, there are
other ways to look for tachyonic neutrinos.

###  VIII.2 The $m^{2}<0$ mass in the $3+3$ model

When the $3+3$ model was first proposed in 2013 I accepted the conventional
view that the 5-hour early Mont Blanc burst (5 neutrinos in 7 seconds with
average energy $E_{avg}=8\pm 0.5MeV$) was not connected to SN 1987A, even
though I had been looking for signs of the tachyonic mass postulated in the
model. I had even been aware that since the burst was 282 min= 16,900 sec
early, then the inferred tachyonic mass using Eq. 9 ($m^{2}=-0.38keV^{2}$) was
within a factor of two equal to that in the $3+3$ model. Finally, I was also
aware that there was an explanation why that early burst was not seen in the
other three detectors, namely that the Mont Blanc detector had a much lower
energy threshold, which allowed it to detect the neutrinos in this burst,
which all had much lower energies than those in the main burst.

One other potential problem with the Mont Blanc neutrinos being superluminal,
with an excess above light speed $\delta=(v-c)/c=3.2\times 10^{-9}$ was the
VPE Cherenkov radiation postulated by Cohen and Glashow. (Cohen2011, )
However, the threshold for the process would be $E=m_{e}/\sqrt{\delta}=9GeV,$
so neutrinos of 8 MeV energy whould be well below the threshold, and they
could reach Earth. Despite all the above, I still initially rejected the idea
that the Mont Blanc burst was due to tachyonic neutrinos, a claim initially
made by Giani in 1999. (Giani1999, ) My initial rejection of the Mont Blanc
burst being the tachyons in the $3+3$ model was based on the requirement from
Eq. 9 that the spread in their energies would need to be:

$\frac{\Delta E}{E}=\frac{2\Delta t}{t}=2(7sec)/16,900sec\approx 0.02\%$ (11)

which essentially required the Mont Blanc neutrinos constituted a line in the
SN 1987A spectrum of energy 8 MeV, their average value. Incidentally, the 5
Mont Blanc neutrinos strangely all had energies consistent with their average
value, which means they are at least consistent with being monoenergetic.
Still, in 2013 despite all the above, I regarded the notion of an 8 MeV line
in the SN 1987A spectrum as being “inconceivable.” (Ehrlich2013, ) By 2018,
however, I understood how such an “inconceivable” neutrino line could be
created during the supernova core collapse, and I provided evidence for its
existence. (Ehrlich2018, ).

In that 2018 paper I explained how the $Z^{\prime}$ particle (sometimes called
X17) (Krasznahorkay2016, ; Krasznahorkay2018, ) of mass 16.8 MeV could produce
monoenergetic neutrinos (and antineutrinos) if there existed hypothetical cold
dark matter X-particles of mass $m_{X}\sim 8MeV,$ which annihilated in the
stellar core just prior to core collapse. The $XX$ annihilation essentially
opened a portal between the dark matter and standard model matter worlds,
which occurred when the supernova core reached a sufficiently high
temperature, i.e., the threshold of the “Z’-mediated reactions”:

$X+X\rightarrow Z^{\prime}\rightarrow\nu\;+\;\bar{\nu}$ (12) $X+X\rightarrow
Z^{\prime}\rightarrow e^{+}+e^{-}$ (13)

Note that if the dark matter $X$ particles are cold ($v<<c$) the $Z^{\prime}$
are nearly at rest, and the leptons in Eqs. 12 and 13 are essentially
monoenergetic having $E\approx 8MeV.$ Having proposed a model for producing an
8 MeV neutrino line, it was essential to find corroborating evidence for it.
One supporting piece of evidence provided in my 2018 paper involved gamma-ray
data from the galactic center which was a plausible place to find dark matter
$X$ particles in addition to stellar cores. If indeed $X$ particles of mass
$m_{X}$ resided there, this would lead to a broad enhancement to the spectrum
up to an energy $E_{\gamma}=m_{X}$ through the annhilation of $e^{+}$ created
by the reaction in Eq. 13. A fit to the gamma ray spectrum showed that in fact
the data gave a best fit for $m_{X}=10^{+5}_{-1.5}MeV,$ which is consistent
with $X$ particles of mass $m_{X}\sim 8MeV$ being partly responsible for the
galactic center $\gamma-$rays, which in turn supported the Z’-mediated
reaction as the cause of an 8 MeV neutrino line.

Of course, the best way to confirm such an 8 MeV neutrino line would be direct
evidence for it in the SN 1987A neutrino data themselves, particularly the
data from the Kamiokande II detector, the largest of the four recording data
on the day of the supernova. Fortunately, in publishing their data for the
neutrinos observed on that day the Kamiokande II Collaboration showed plots of
data taken during eight 17-min long intervals before, and after the 12-event
burst. That data consisted of 997 events in the form of dot plots of arrival
time versus number of “hits” or photomultipliers struck during a short time
interval, which was a measure of the neutrino energy. The Collaboration
considered this data merely detector background since it showed no obvious
signs of events above 20 hits at any time besides the main 12-event burst,
which was the way supernova neutrinos could be distinguished from the
background consisting of radioactivity and cosmic rays.

However, as I described in ref. (Ehrlich2018, ), these 997 events were
consistent with an 8 MeV neutrino line sitting atop a background, which if
true, would lend strong support to the Mont Blanc neutrinos being the tachyons
in the $3+3$ model, since as already noted, that is possible only if they were
monochromatic with energy 8 MeV. This claimed 8 MeV neutrino line was deduced
by comparing the energy distribution of the events recorded on the day of the
supernova with published background data from the Kamiokande detector recorded
in the months before and after SN 1987 A. (Hirata1989, ). Despite finding a
very large excess of events above a background with a shape, width and central
value all being consistent with an 8 MeV neutrino line, this claim was
controversial (and certainly not endorsed by the Kamiokande II Collaboration),
since the excess occurred very close to the peak of the background. In a
subsequent publication, however, I presented various reasons to believe the
claimed line was genuine despite its falling close to the peak of the
background, and explained how various objections to its existence could be
rebutted. (Ehrlich2021, ).

##  IX Neutrino mass experiments

In a previous (2019) summary of evidence for the $3+3$ model in ref.
(Ehrlich2019, ), I had noted that it is supported by fits to the observed
tritium beta spectra that had been reported in three pre-KATRIN direct
neutrino mass experiments. That claim turns out to have been a mistake on my
part since those experiments simply were not sensitive enough to test the
model. Nevertheless, it is worthwhile to explain how that mistake occurred,
since it has bearing on the KATRIN experiment. Most neutrino mass experiments
are based on observing the spectrum of tritium beta decay very close to the
endpoint energy, and seeing what value of the electron neutrino mass gives a
best fit. Apart from various corrections, the phase space factor or the square
of the Kurie function can be used to describe the differential spectrum,
assuming the three neutrinos have virtually indistinguishable masses, and
$m_{\nu}$ is their effective mass.

$K^{2}(E)=(E_{0}-E)\sqrt{R((E_{0}-E)^{2}-m_{\nu}^{2}})$ (14)

Here $R$ is the ramp function $R(x)=x$ for $x>0$ and $R(x)=0$ otherwise.
However, when fitting the spectrum to multiple distinguishable masses as in
the $3+3$ model one cannot use a single effective mass but instead must use:

$K^{2}(E)=\sum|U_{ej}|^{2}(E_{0}-E)\sqrt{R((E_{0}-E)^{2}-m_{j}^{2}})$ (15)

which may be thought of a weighted average of three spectra with weighting
factors $|U_{ej}|^{2}$ for each of the three masses $m_{1},m_{2},m_{3}.$ In
ref. (Ehrlich2019, ) I noted that the results from three experiments showed a
“kink” in their spectra at around 20 eV before the spectrum endpoint, and I
was able to obtain a good fit to the $3+3$ model masses to those three
experiments’ spectra when the weights for $m_{1}$ and $m_{2}$ were both $\sim
0.5$ with only a tiny weight for the tachyonic $m_{3},$ which kept the
effective mass very close to zero. These fits with equal weights for $m_{1}$
and $m_{2}$ resulted in a maximum amount of “kinkiness” in the differential
spectrum near 20 eV and they fit the data from the three experiments quite
well. Regrettably, I later learned that those spectral anomalies in the data
or kinks near 20 eV could be explained in terms of molecular tritium final
states, (Bodine1995, ) and they probably had nothing to do with my model. On
the other hand, the anomalies never would have appeared in my fits had I
simply chose a smaller spectral contribution from the $m_{2}=21.4eV$ mass, so
there was at least no conflict with the $3+3$ model and those pre-KATRIN
experiments, which simply were not sensitive enough to test it.

##  X The KATRIN experiment

The KATRIN experiment has been taking data since 2019 and it has published two
values for the electron neutrino effective mass, based on the total amount of
data so far accumulated at the time. In 2019 it reported an effective neutrino
mass square value of $m^{2}=-1.0^{+0.9}_{-1.1}eV^{2}$ (tachyonic by $1\sigma$)
(Aker2019, ), and in 2022 it reported $m_{\nu}=0.26\pm 0.34)eV^{2},$ which was
also given in terms of an upper limit (at 90% CL) of $m_{\nu}<0.8eV$ (90%CL).
(Aker2022, ) Unlike most previous experiments which recorded a differential
spectrum, KATRIN records its data as an integral spectrum, meaning that it
records the integrated number of counts above a series of energies at a number
of non-equally spaced set points chosen for maximum sensitivity.

The signature of the presence of the two $m^{2}>0$ masses in the integral
spectrum is not spectral kinks (which would be the case in the differential
spectrum), but rather slight bumps or excess numbers of counts at a distance
before the spectral endpoint $E_{0}$ equal to the two masses $m_{1}$ and
$m_{2}.$ The height of each predicted bump depends on the weight of the
spectral contributions from the two $m^{2}>0$ masses, and these predicted
deviations are shown in Fig. 5. The $m^{2}<0$ mass in the model influences the
spectrum more subtly (because of its very small contribution to the overall
spectrum), but it does modify the spectrum shape before the endpoint.

In reporting its data KATRIN shows both the actual spectrum as well as the
residuals (in standard deviations) to their best fit. Those residuals could
reveal systematic departures from a single mass spectrum if they were large
and not randomly distributed. In fact, the residuals in the KATRIN data from
2019 appear relatively random, but they also do give a strong hint of matching
the predicted $3+3$ bumps, as I noted in a 2019 publication. (Ehrlich2019, )
The agreement with the data KATRIN published in 2022 from their second
campaign fails to show those same hints. (Ehrlich2021, ) However, the
alternate explanation of the data offered by the $3+3$ model still remains
viable, particularly in light of the very large number (37) of free parameters
KATRIN uses in their fit in ref. (Aker2022, )

That large number of free parameters occurs because KATRIN assumes different
values of the signal, background, and endpoint energies for each of 12
detector rings, and a single common neutrino mass, yielding $12\times 3+1=37$
free parameters, unlike what was done in 2019. (Aker2019, ; Aker2022, ) The
use of so many free parameters by KATRIN could mask any $O(0.5\%)$ departures
from the single mass spectrum that would occur if the $3+3$ model were a valid
description of the data. That possible masking could occur not because KATRIN
actually uses 37 parameters to fit their spectrum. Rather, it could happen
because departures from the single mass spectrum become less statistically
significant when the spectral fits are done for separate rings rather than all
rings at once. The KATRIN data analysis team has expressed their commitment to
doing their own fit to the $3+3$ model at some point during the course of the
experiment, hopefully with a reduced number of free parameters. (Parno2022, )

Figure 5:  Predicted percentage deviations from the single mass $m=0$
integrated spectrum for the $3+3$ model, for three choices of the contribution
from the $m_{1}=4eV$ mass. The choice of $94\%$ results in the smallest
departure from the single mass spectrum, namely $<0.8\%$ at all energies.

##  XI $3+3$ Model and Cosmology

The most challenging apparent conflict with the $3+3$ model masses is posed by
cosmological constraints. As of 2022, the most constraining neutrino mass
bounds yields: $\sum m_{\nu}<0.09eV$ based on five different data sets,
(Valentino2021, ) although one other 2022 assessment claims evidence for an
actual value at a $4\sigma$ level: $\sum m_{\nu}=0.23\pm
0.06eV.$(Chudaykin2022, ) The basis for a constraint on $\sum m$ from
cosmological observations, comes from events occurring in the very young hot
universe. The ability of the relic neutrino energy density to influence CMB
anisotropy, and play a role in the formation of galaxies and their structures
is the basis of the cosmological constraint on the neutrino mass sum.

At about one second after the big bang, neutrinos decouple and form the cosmic
neutrino background (CNB). During that early hot ($T\sim MeV$) epoch neutrinos
and antineutrinos would be produced in the decay of the $Z^{0}$ and $W^{\pm}$
bosons. In these decays, the three neutrino flavors would be produced in equal
numbers, i.e., $n=n_{e}=n_{\mu}=n_{\tau},$ and the overall neutrino energy
density $\rho$ and the number density $n$ in the early universe would be
related through:

$\rho=n_{e}m_{e}+n_{\mu}m_{\mu}+n_{\tau}m_{\tau}=n\sum m_{\nu}$ (16)

where $\sum m_{\nu}$ is the sum of the three neutrino masses. However, notice
Eq. 16 is the sum of the flavor (effective) masses, as Stecker (Stecker2022, )
has explicitly stated, not the sum of the mass eigenstate masses, as is almost
always assumed. Of course, for the conventional neutrino picture where the
three neutrino masses are nearly degenerate this distinction makes very little
difference, but this is not the case for the $3+3$ model, especially since one
mass state is tachyonic.

Various authors, including Jentschura, (Jentschura2013, ) Davies (Davies2012,
) and Schwartz (Schwartz2018, ) have considered tachyonic neutrinos as a
candidate for dark energy, essentially a form of gravitational repulsion. The
simplest way to understand why tachyonic neutrinos in the CNB might provide an
effective gravitational repulsion is that the classical Newtonian formula for
the force between any pair of neutrinos involves the product between two
imaginary mass values, which of course, is negative, meaning repulsion. Thus,
if we have a mixture of tachyonic and bradyonic neutrinos making up the CNB,
any concentrations of the latter promotes the subsequent gravitational
coalescence of matter to form structures, while the former has the opposite
effect. Clearly, in $\sum m_{\nu}$ with both tachyonic and bradyonic flavors
present, the former will need to appear with a negative mass, not an imaginary
one, to allow for a partial cancellation of the two effects to occur.

In order to understand what the sum over neutrino flavor state masses might
predict for the $3+3$ model we need to discuss the PMNS mixing matrix relating
mass and flavor states. The PMNS matrix for the usual $3+0$ neutrino model is
a $3\times 3$ unitary matrix characterized by three mixing angles and a CP-
violating phase, while the $3+3$ model with its three active-sterile masses
requires a $6\times 6$ unitary matrix having 15 mixing angles and 10 phases.
Given the difficulty of fitting 25 free parameters to obtain a good fit to all
experiments, finding a $6\times 6$ PMNS matrix required by the $3+3$ model has
so far not been possible. While the exact form of this PMNS matrix connecting
mass and flavor states remains to be described, we at least be assured from
its unitarity that the sums of the mass-squared values for the mass and flavor
states are identical, yielding:

$m_{e}^{2}+m_{\mu}^{2}+m_{\tau}^{2}=4.0^{2}+21.4^{2}-200,000eV^{2}$ (17)

Finally, since the electron neutrino mass is so close to zero, Eq. 17 leads to
the $3+3$ model prediction that $m_{\mu}^{2}+m_{\tau}^{2}=-0.2keV^{2}.$
Regrettably, this prediction of the model is many orders of magnitude from
being testable, given the current upper limits on these two flavor masses.
(Zyla2021, ) Nevertheless, the main points of the preceding discussion are
that given that the near zero constraint of the sum of the neutrino masses
from cosmology is a sum over flavor states, and that tachyon masses enter the
sum with a negative value, so that the apparent conflict of the $3+3$ model
with the cosmological constraint on $\sum m_{\nu}$ can be resolved provided
that the tau and muon neutrino effective masses have the stipulated
relationship.

There is one other cosmological constraint on the relic neutrinos in the CNB,
namely the effective number of neutrino species, which would include sterile
neutrinos. Data from the Planck spacecraft has published the tightest bound to
date on the this quantity: $N_{eff}=3.15\pm 0.23,$ (Ade2016, ) which seems to
be in conflict with a model having three light sterile neutrinos. However,
Tang has shown that this contradiction can be resolved if the sterile
neutrinos are self-interacting, in which case they can actually lower not
raise the effective number of neutrinos. (Tang2015, ) That is because as the
Universe cools down, flavor equilibrium between active and sterile species can
be reached after big bang nucleosynthesis (BBN) epoch, but it causes a
decrease of $N_{eff}.$ In fact, Tang noted that based on his analysis at least
three eV-scale sterile neutrino species are needed to be consistent with the
cosmological data then available (in 2015), which would offer further support
for the $3+3$ model.

##  XII Summary and Future Tests

This article has described the searches for tachyons, particles having $v>c$
and $m^{2}<0.$ Initial searches were made for new particles having one of
those properties, and apart from some false initial claims, they have yielded
negative results. Following the pioneering work by Chodos et al. (Chodos1985,
), most of the attention shifted to neutrinos as possible tachyon candidates.
Here the negative results of searches could better be described as
inconclusive rather than negative, at least as long as no neutrinos having
either $m^{2}>0$ or $v<c$ have been found. Even if some neutrino flavor is
found to have $m^{2}>0$ or $v<c$, there would remain the possibility that one
of the other flavors is a tachyon. Much of the second half of this review
paper describes the evidence for neutrinos as tachyons, and most especially
for the $3+3$ model, which I proposed in 2013 Ehrlich2013 . This model
postulated three specific neutrino masses, one of which is a tachyon. These
masses, are, of course, in conflict with the conventional neutrino paradigm,
which requires three nearly degenerate masses, but nevertheless a variety of
observations have been cited in support of the three masses in the model.

Apart from the $3+3$ model, a separate earlier prediction was discussed for a
specific electron neutrino effective mass from cosmic ray and other data.
(Ehrlich2015, ) Both predictions are so far consistent with KATRIN
data,(Ehrlich2019, ; Ehrlich2021, ), and that experiment by its conclusion
should be able to prove the $3+3$ model and the predicted $\nu_{e}$ effective
mass right or wrong. If not, a new generation of direct mass experiments
including Project 8 may be able to reach the necessary sensitivity.
(Esfahani2022, ) Project 8 uses atomic not molecular tritium, and it has the
potential to surmount the systematic limitations of current-generation
methods, because atomic tritium avoids an irreducible systematic uncertainty
associated with the final states populated by the decay of molecular tritium.
Project 8 hopes to achieve a $0.040eV,$ neutrino-mass sensitivity (about 5
times better than KATRIN). Another way of testing the $3+3$ model would be to
look for evidence of the $dm^{2}=m_{2}^{2}-m_{1}^{2}\sim 450^{2}eV^{2},$ in
future oscillation experiments having sufficiently high energy and/or short
baselines, but this would be not as impressive a result as finding evidence
for all three masses in the model in a single direct neutrino mass experiment.

One final suggested way to test the $3+3$ model would be to look for neutrinos
associated in time with gamma-ray bursts, as was done by Huang and Ma in
connection with the IceCube data. (Huang2018, ; Huang2019, ) Recall that they
interpreted the apparent early arrival of some neutrinos relative to that of
recorded gamma ray bursts, not as evidence for some neutrinos being tachyons,
but rather their satisfying a Lorentz violating (LV) dispersion relation
between velocity and energy, which Huang and Ma assumed to be:

$v(E)=c\big{[}1-s_{n}\frac{n+1}{2}(E/E_{LV,n})^{n}\big{]}$ (18)

Here $n=1,2$ corresponds to linear or quadratic dependence of the velocity on
energy, $s_{n}=\pm 1$ is the sign factor of LV correction (+1 for superluminal
and -1 for subluminal neutrinos), and $E_{LV,n}$ is the nth-order LV energy
scale to be determined by fits to the data. Instead of the above interpretaion
of the IceCube neutrino data, if the GRB’s come from sources having a known
distance, $L$, one could test the $3+3$ model seeing if the data satisfied Eq.
9, given the neutrino measured energies and the source distance for each of
the three $3+3$ model masses. Specifically, on a plot of $1/E^{2}$ versus
$t=L/c-t_{trav}$ one would look for data points falling near near of three
lines of slopes corresponding to the three neutrino masses in the model,
similar to the analysis done for SN 1987A. (Ehrlich2012, )

Aside from neutrinos associated with gamma ray bursts, it might seem promising
to use this same search method with the few dozen extragalactic supernovae
that have been recorded since SN 1987A, but the number of neutrinos expected
to reach Earth from them would be miniscule. Thus, for example, the nearest of
these occurring in 1994 was 64 times further away than SN 1987A, and given the
inverse square law, even a detector the size of Super-Kamiokande (25 times
bigger than its descendant that observed 12 $\bar{\nu}$ from SN 1987A) would
observe only $\sim 25\times 12/64^{2}=.075$ neutrinos from SN 1994. Of course,
a new supernova in our galaxy would be quite another matter. The current
generation of neutrino detectors is capable of detecting perhaps 10,000
neutrinos for a supernova at the Galactic Center in contrast to the mere few
dozen observed for SN 1987A. Moreover, the next generation of detectors will
increase that yield by another order of magnitude. Regrettably, given that
only about three supernovae occur per century in the galaxy, the chances of
this 84 year-old physicist witnessing such a marvelous event are not great.
Nevertheless, the neutrino data from that next supernova will leave no doubt
as to the truth or falsity of the $3+3$ model with its three specific masses
inferred from SN 1987A, including the one with $m^{2}<0.$

A more extensive account of the “hunt for the tachyon” discussed in this
review can be found in my recent popular-level book. (Ehrlich2022, )

## Acknowledgments

The author wishes to thank Alan Chodos and Michael Kreisler for helpful
comments on this article.

## References

  * (1) M. Kreisler, Am. Scientist , 61, (2) 201-208 (1973)
  * (2) C. C. Gillispie, I. Gratton-Guinness, and R. Fox (1999). Pierre Simon Laplace, A Life in Exact Science. Princetion, NJ: Princeton University Press.
  * (3) A. Dragan and A. Ekert 2020 New J. Phys. 22, 033038
  * (4) O.M.P. Bilaniuk, V. K. Deshpande, and E. C. G. Sudarshan, Am. J. of Phys. 30, 718 (1962)
  * (5) A. Einstein, Annalen der Physik, 17 (1905), https://einsteinpapers.press.princeton.edu/vol2-doc/311
  * (6) G. Feinberg, (1967), Phys. Rev., 159 (5): 1089–1105.
  * (7) T. Alvager and M. N. Kreisler, 1968, Phys. Rev. 171:1357
  * (8) M. B. Davis, M. N. Kreisler, and T. Alväger, Phys. Rev. 183, 1132
  * (9) Clay R W and Crouch P C, 1974, Nature 248 28
  * (10) Emery M W, Fenten A G, Fenton K B, Greenhill J G and Humble J, 1975, Proc. 14th Int. Conf on Cosmic Rays, Munich 7 2486
  * (11) Fegan D J, OBrien D P, OBrien S and Porter N A, 1975, Proc. 14th Inc. Conf on Cosmic Rays, Munich
  * (12) Hazen W E, Green B R, Hodson A L and Kass J R, 1975, Nucl. Phys. B 96 401
  * (13) Prescott J R 1975, Proc. 14th Int. Conf on Cosmic Rays, Munich 7 2474
  * (14) Murthy P V R 1971, Nuouo Cim. Lett. 1 YO8
  * (15) Smith G R and Standi1 S 1977, Can. J. Phvs. 55 1280
  * (16) P N Bhat et al 1979, J. Phys. G: Nucl. Phys. 5 L13
  * (17) A. Marini et al., Phys. Rev. D 26, 1777 (1982).
  * (18) G. K. Garipov and A. A. Silaev Phys. of Atomic Nuclei, 83, 442–452 (2020)
  * (19) G. Baltay, G. Feinberg, N. Yeh, and R. Linsker. 1970. Phys. Rev. D 1:759.
  * (20) J. S. Danburg et al., Phys. Rev. D 4, 53 (1971)
  * (21) J. S. Danburg and G. R. Kalbfleish. 1972. Phys. Rev. D 5:1575.
  * (22) A. Ljubičic, Ž. Pavlović, K. Pisk, and B. A. Logan, Phys. Rev. D 11, 696 (1975)
  * (23) D. F. Bartlett, and M. D. Lahana, Phys. Rev. D, 1972, 6:1817
  * (24) D. F. Bartlett, D. Soo, and M. G. White, Phys. Rev. D 18, 2253 (1978)
  * (25) K. A. Fredericks, J. of Condensed Matter Nuclear Science 15 (2015) 203–230
  * (26) V. P. Perepelitsa, Phys. Lett., 67B (4) 471-473 (1977)
  * (27) V. P. Perepelitsa, T. Ekelof, A. Ferrer, and B. R. French, arXiv:hep-ex/1912.11839 (2019)
  * (28) V. P. Perepelitsa, T. Ekelof, A. Ferrer, and B. R. French, arXiv:hep-ex/2001.08576
  * (29) M. Radzikowski, CPT and Lorentz Symmetry, 224-228 (2010) World Scientific.
  * (30) R. G. Cawley, Bull. Am. Phys. Soc.,15, 25 (1970); R. G. Cawley, Lettere al Nuovo Cimento, 3, 523 - 525 (1972)
  * (31) A. Chodos, A. I. Hauser, and V. A. Kostelecký, Phys. Lett. B, 1985, 150B, (6)
  * (32) U. D. Jentschura & B. J. Wundt , Eur. Phys. J. C 72, 1894 (2012)
  * (33) J. Rembieliński, P. Caban & Jacek Ciborowski, Eur. Phys. J. C , 81, 716 (2021)
  * (34) C. Schwartz, Int. J. of Mod. Phys. A, 31, 09, 1650041 (2016)
  * (35) Ch. Kraus, Eur. Phys. J. C, 40, 447–468 (2005)
  * (36) V. Alan Kostelecký and Matthew Mewes Phys. Rev. D 85, 096005 (2012
  * (37) J S. Díaz, V. Kostelecký, and M. Mewes, D 89, 043005 (2014)
  * (38) T. Adam et al. (OPERA Collaboration), https://arxiv.org/vc/arxiv/papers/1109/1109.4897v1.pdf
  * (39) A. G. Cohen, S. L. Glashow, (2011), Phys. Rev. Lett., 107 (18): 181803,
  * (40) T. Adam et al. (OPERA Collaboration), J. of High En. Phys., 2012, Article number: 93 (2012)
  * (41) M. Antonello, et al., (ICARUS), JHEP 11 (2012) 049
  * (42) P. Adamson et al. (MINOS), Phys. Rev. D92 (2015) 052005.
  * (43) K. Abe et al. (T2K), Phys. Rev. D93 (2016) 012006.
  * (44) F. W. Stecker and Sean T. Scully, Phys. D 90, 043012 (2014)
  * (45) E. Ganster, R. Naab, and Z. Zhang (for the IceCube Collaboration), the 37th International Cosmic Ray Conference (ICRC 2021)
  * (46) Y. Huang and Bo-Qiang Ma, Communications Physics, 1, 62 (2018)
  * (47) Y. Huang, H. Li, and Bo-Qiang Ma, Phys. Rev. D 99, 123018, 2019
  * (48) J. Ellis et al., Phys. Rev. D 99, (8) 083009 (2019)
  * (49) U. Jentschura and R. Ehrlich, Adv. in High En. Phys., 2016, Article ID 4764981
  * (50) P.A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020) and 2021 update.
  * (51) A. Chodos, V. A. Kostelecky, R. Potting, and E. Gates, Mod. Phys. Lett. A, 07, (6) 467-476 (1992)
  * (52) K. Greisen, Phys. Rev. Lett. 16 (17): 748–750
  * (53) G. T. Zatsepin and V. A. Kuz’min (1966), J. of Exp. and Theor. Phys. Lett. 4: 78–80
  * (54) R. Ehrlich, Phys. Rev. D, 60, 17302 (1999)
  * (55) R. Ehrlich, Phys. Rev. D, 60, 73005 (1999)
  * (56) R. Ehrlich, Astropart. Phys. 66, 11 (2015)
  * (57) R. U. Abassi, et al., Ap. J., 865:74 , 2018
  * (58) V. A. Kostelecky´, in Topics on Quantum Gravity and Beyond, edited by F. Mansouri and J. J. Scanio World Scientific, Singapore, 1993
  * (59) A. Erlykin and A. Wolfendale, Proc. of the 31st ICRC, Lodz (2009)
  * (60) S. Thoudam et al., Astron. & Astrophys., 595, A33 (2016)
  * (61) J. Becker and M. Lukas, Phys. Reports, 872, (2020)
  * (62) http://arxiv.org/abs/1307.3944
  * (63) H. Huzita, Mod. Phys. Lett. A 2, 905 (1987)
  * (64) R. Cowsik, Phys. Rev. D, 37, 16851687 (1988)
  * (65) R. Ehrlich, Astropart. Phys., 35, 625-628 (2012)
  * (66) T. J. Loredo and D. Q. Lamb, Phys. Rev. D 65, 063002 (2002)
  * (67) J. Kopp, P. A. N. Machado, M. Maltoni & T. Schwetz, J. of High En. Physi. 2013, 50 (2013)
  * (68) M. Maltoni and T. Schwetz, Phys. Rev. D 76, 093005 (2007)
  * (69) R. Ehrlich, Astropart. Phys., 41 1-6 (2013)
  * (70) R. Ehrlich, Adv. in Astron. 2019, Article ID 2820492
  * (71) A. Chodos, https://arxiv.org/abs/1206.5974
  * (72) M. H. Chan, and R. Ehrlich, Astrophys. and Space Sci., 349, 1, 407, (2014)
  * (73) R. N. Mohapatra, and D. W. Sciama, arXiv:hep-ph/9811446v2
  * (74) D. W. Sciama, Modern Cosmology and the Dark Matter Problem, Cambridge University Press, 1993.
  * (75) V. H. N. Phan, G. Morlino, and S. Gabici, MNRAS 480, 5167–5174 (2018)
  * (76) P. Adamson, Phys. Rev. Lett. 122, 091803 (2019)
  * (77) P. Denton, arXiv:2111.05793, 2021
  * (78) P. Caban, J. Rembieliński, K. A. Smoliński & Z. Walczak Found. of Phys. Lett. 19, 619–623 (2006)
  * (79) S. Giani, AIP Conference Proceedings 458, 881 (1999)
  * (80) R. Ehrlich, Astropart. Phys.,99, 2018, 21-29
  * (81) Krasznahorkay, A.J.; et al. (2016), Phys. Rev. Lett. 116, 042501.
  * (82) Krasznahorkay, A.J.; et al. (2018) J. Phys.: Conf. Series. 1056, 012028
  * (83) K. S. Hirata, et al. (Kamiokande collaboration), Phys. Rev. Lett., 63, 1 (1989)
  * (84) R. Ehrlich, Lett. in High En. Phys., LHEP-199, 2021; arXiv:2106.00681v4
  * (85) L. I. Bodine, D. S. Parno, and R. G. H. Robertson, Phys. Rev. C 91, 035505 (2015)
  * (86) M. Aker et al. (KATRIN Collaboration), Phys. Rev. Lett. 123, 221802 (2019).
  * (87) M. Aker et al. (KATRIN Collaboration), Nature Physics, 18 (2) (2022).
  * (88) Private communication with Dr. Diana Parno.
  * (89) E. Di Valentino, S. Gariazzo and O. Mena, Phys. Rev. D, 104, 083504 (2021)
  * (90) A. Chudaykin, D. Gorbunov, and N. Nedelko, arXiv:2203.03666
  * (91) F. W. Stecker, ”Neutrino Physics and Astrophysics”, Stecker, in Encyclopedia of Cosmology II, edited by G. G. Fazio, World Scientific Publishing Company, Singapore, 2022; arxiv.org/abs/2203.17223
  * (92) U. D. Jentschura andB. J. Wundt, ISRN High Energy Physics, 2013, 374612
  * (93) P.C.W. Davies and I. G. Moss, Astropart. Phys. 35, 10, 609-684 (2012)
  * (94) C. Schwartz, Int. J. of Mod. Phys. A, 33, (10), 1850056 (2018)
  * (95) P. A. R. Ade, et al. (2016), Astron. & Astrophys., 594, A13
  * (96) Y. Tang, Phys.Lett. B750 (2015) 201-208
  * (97) A. Ashtari Esfahani et al., Proceedings of the US Community Study on the Future of Particle Physics (Snowmass 2021).
  * (98) R. Ehrlich, “Hunt for the Faster than Light Tachyon, and Finding Three Unicorns and a Herd of Elephants,” CRC Press, Taylor & Francis Group, Oxfordshire, UK (2022).

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