Dear List members:
I would like to give an account of some rather exciting results of
detection of cosmic ray induced Neutrinos at the Super Kamiokande site,
hereafter known as Super K.
We Know Neutrinos as the highly elusive uncharged leptons of the
standard model. Leptons come in three families (as do the associated Quarks)
therefore we have three flavors of Neutrino. These are the electron neutrino,
muon neutrino and Tau neutrino, each one associated with it's charged doublet
partner. These particles interact with other particles via the weak force and
as all particles do by a gravitational interaction. However for all practical
purpose the gravitational interaction may be ignored.
These particle have a uniquely small interaction cross section, making
them very difficult to detect. However very large detectors have been built
underground to look for proton decay, which is predicted by the Grand
Unification Theory. While that search has so far been unsuccessful these
detectors have found another use. That of course is the detection of
Neutrinos. Both the predicted proton decay and a neutrino interaction produce
what is called Cerenkov light which is a phenomena caused by a charged
particle traveling faster than the speed of light for the medium where this
event occurs. ( Please note that these particle do not exceed light speed in
a vacuum.) This makes these detectors ideal for Neutrino detection.
The most impressive of these detectors is the Super K in Japan, built
deep underground to shield it from the direct effects of cosmic rays. Of
course given the Neutrinos remarkable small interaction cross section this
does not impede in any way Neutrino detection.
Neutrinos are produced by cosmic rays interacting with the upper
atmosphere. Most Cosmic rays are protons and when they collide with the atoms
of the upper atmosphere they produce showers of Muons and electrons and
Neutrinos.
Since we have a fairly good idea of the frequency of these events
we can approximately predict the number of neutrinos to expect. In fact we
can predict the numbers to within 25 percent.
However what interest us more than the total number of Neutrinos
produced is the ratios of the different flavors of neutrinos produced.
Fortunately we have a very accurate way to determine these ratios. A careful
analysis of these cosmic ray events reveals that there are about two muon
neutrino produced for every electron neutrino. This ratio is accurate to
within 5 percent.
Of course this means that for every electron neutrino detected we
should detect two muon neutrinos. However to our surprise we detect only 1.3
to 1 muon neutrinos to every electron neutrinos. This is such a large
divergence from the predicted ratio it seems unlikely that experimental error
is responsible for this result.
One of the ways this could be explained would be if Neutrinos were
changing flavor. This is exciting because if Neutrinos had zero rest mass as
originally believed Neutrino would be unable to change their flavor. To pin
down exactly which Neutrinos were changing flavor the experimenters came up
with clever idea. They restricted the neutrinos detected to only two
direction. Directly overhead and directly underneath the detector. Then both
the Neutrinos from the down direction and the up direction would be produced
by cosmic rays hitting the upper atmosphere at the same angle. This would
insure an equal starting mix of Neutrinos produced by the up and down cosmic
ray interaction. What they found was that the upward direction neutrinos
(Which had been produced on the other side of the earth.) had about half of
the Muon neutrinos missing while the downward detected Neutrinos had just
about the expected number of Muon Neutrinos. For electron Neutrinos the
expected number were detected in both directions. This meant that over long
distances the Muon neutrinos were changing into something else while the
electron neutrinos were stable. ( At least over the distances involved in
this experiment.) What could be going on?
Theorist believed that what was being observed was neutrino mixing.
That is, the Muon Neutrino was actually in a mixed state of Muon neutrino and
Tau Neutrino. The Tau Neutrino are not detectable at Super K so they could
not be directly observed but there is good reason to suspect this is what was
happening. Because in the Quantum world all particles exhibit both wave and
particle behavior depending on how they are measured there is another way to
analysis the transport of these neutrinos.
According to QM every particle has a wavelength which is equal to
Planck's constant divided by it's momentum. This is called the De Broglie
wave named after the discoverer of this relationship. Due to the uncertainty
principal no particle has a defined momentum until measured. An accurate
description of this relationship is given by the famous Schrodinger Eigen
function equation in which the momentum would be the Eigen Value. The more
precisely the position of the neutrino was known the more uncertain the
momentum. This relationship is defined by an Eigen vector which describes a
wave whose amplitude is a probability amplitude. Now assuming that the Muon
neutrino and the Tau Neutrino have different masses then each would have a
different De Broglie wave length. Since there is a wave packard associated
with a particle both wave lengths would be present. This would put the Cosmic
ray produced Neutrino in a superposition state of two waves. Then as these
waves traveled along the wave associated with the lighter particle gets ahead
of the heavier one. These two wave would interfere in a way that would
fluctuate along the particles trajectory. This interference has a musical
analog. If these were sound waves the cause the volume to oscillate. For our
Quantum particles this will cause the Probability amplitude to oscillate for
each of the Quantum Flavor states. At the outset the Muon neutrino appears as
muon neutrino with a probability of one. After Traveling a certain distance
it looks like a Tau Neutrino with a probability of one. In between it runs
the full range of Probabilities with the both probabilities always summing to
one. So for those neutrino traveling the shorter distance we detect the
expected number of Muon Neutrinos. For those Neutrino traveling the longer
distance we are missing a substantial number of Muon Neutrino detection
events because the particle as a significant probability to be in a tau
flavor state which is not detectable.
If we take the time dependent Schrodinger equation using the
relationship between the mass and flavor Eigenstates we are able to derive
the following equation:
P(vu>vt)=sin^2(2*theta)*sin^2(1.27*dms*L/E)
Where P(vu>vt) is the probability of a muon neutrino to tau neutrino
transition, theta is the mixing angle between the flavor and mass
eigenstates, dms is the delta mass squared of the two neutrino mass
eigenstates, L is the distance traveled, and E is the neutrino energy.( This
equation only holds if the Neutrino energy is significantly greater than the
rest energy value, a very likely condition given the high energy of Cosmic
ray events and the relatively small value of the mass eigen states of the tau
and muon neutrino.) From this equation it is easy to see how the
probabilities for the muon neutrino transition oscillates from zero to some
maximum value.
This experiment raises many interesting questions. To conclude this
I would like to raise some of these question as food for thought.
First: Why doesn't the muon Neutrino engage in transitions to
Electron Neutrinos? It would seem based on the fact that dms value is not
zero for the electron neutrino and Muon Neutrino mass eigenstates this
transition should be observed. Yet this transition is not seen.
Second: What is the cosmological significance of Neutrinos having a
non zero rest mass. How does this question relate to the dark matter problem?
Third: How does this all relate to solar electron neutrinos which
seem to be detected at far too low a rate.
More experiments are planned to attempt to answer some of these
questions. The answer to the first question may possible be explained by
Gerald L Fitzpatrick's ideas concerning the "Family Problem," whose papers
are available on the LANL e-print archive.