Randy Poe:

Helluvus wrote:

What is the latest mass of a neutrino. During my Ph.D it was

be zero but I read somewhere that the Japanes found a very exact

for it.

This site seems to suggest the question is connected to
neutrino oscillation (there is non-zero mass if and only
if there is oscillation).

No. There are oscillations only if the masses are not all the same.
For simplicity, consider two neutrinos only, \nu_e and \nu_u, which
are the weak interaction eigenstates. Define two different neutrinos
by,

|\nu_1> = a(t) |\nu_e> + b(t)|\nu_u>

|\nu_2> = c(t) |\nu_u> - d(t)|\nu_e>

Normalization and the orthogonality of \nu_1, \nu_2, \nu_e and

give you the conditions on the coefficients. For example, sines and
cosines work.

The states evolve as \Psi(t) = \Psi(x)\exp(-iwt), with w =

so, |\nu_e> evolves as, \exp(-iw_et)|\nu_e> and |\nu_u> evolves as
\exp(-iw_u t)|\nu_u> and likewise, |\nu_1> evolves as \exp(-iw_1

and similarly for for |\nu_2>.

You can invert the definitions to obtain \nu_e and \nu_u in terms

\nu_1 and \nu_2, which is more convienient initially, since \nu_e and
\mu_u are eigenstates of the weak hamiltonian. That implies that the
weak eigenstate, \nu_e, for example, is a linear combination of \nu_1
and \nu_2 and will oscillate between those two states.

If the masses of \nu_1 and \nu_2 are identical, then at most \nu_1
and \nu_2 differ by a phase, which is constant. Since that phase is
not measurable, you can redefine \nu_1 and \nu_2 to be equal to
\nu_e and \nu_u. This is equivalent to saying that the weak

and the mass eigenstates can be simultaneously diagonalized by the

hamiltonian.


On the other hand, suppose the masses of \nu_1 and \nu_2 are

In that case, the mass eigenstates and weak eigenstates cannot be
simultaneously diagonalized by the weak hamiltonian. Then, \nu_e will
consist of a mixture of \nu_1 and \nu_2 each of which will have a
relative amplitude that varies in time. So, call \nu_1 and \nu_2 your
mass eigenstates and invert that back to obtain \nu_1 and \nu_2 in
terms of \nu_e and \nu_u.

You then see that \nu_1 and \nu_2 each oscillate between \nu_e
and \nu_u. So, if a \nu_e is emitted in a positron decay, then
assuming the mass alone is an eigenstate of some hamiltonian,
like say, gravity, then the hamiltonian under which the neutrinos
propagate through space will be either \nu_1 or \nu_2. Both of those
oscillate between \nu_e and \nu_u.

http://www.ps.uci.edu/~superk/nuosc.html

Is neutrino oscillation now considered pretty much established? I
heard that was the accepted explanation for the "solar neutrino
problem", the apparently missing solar neutrinos.

The ``solar neutrino problem'' is a somewhat vague problem in that

problem applies to neutrinos from a rather select set of reactions.

appears to solve the problem associated with neutrinos emitted in p-p

8Be reactions (i.e., the ones to which our detectors are most

It's also possible that the disovery of oscillations coupled with

has resulted in declaring victory a bit soon. For example, the

masses have obviously not been determined, especially in any way that

independent of the problem the oscillations are supposed to solve. I

the most that could be said for certain, is that neutrinos oscillate

the oscillations have a reasonable chance of clearing up a few

at the expense of introducing a few new ones.

Anyway, all I get from this site is the apparent answer
"greater than zero", no actual bound.

Hence my caveat regarding what precisely is solved and what will
be solved once more data are available and things get pinned down.
(Recall the discovery of the muon left a few people rather perplexed,
since what was being sought was the pion and the expected mass was
135 MeV, not the 105 MeV of the particle actually discovered.)

Here's another page with a lot of info:
http://cupp.oulu.fi/neutrino/nd-mass.html

I see a lot of upper bounds, no lower bounds.

Welcome to the world of weak interaction physics, where easy exper-
iments are technologically impossible and experiments which are very
near impossible only have a slight edge on technological feasibility.

Welcome to the world of weak interaction physics, where easy exper-
iments are technologically impossible and experiments which are very
near impossible only have a slight edge on technological feasibility.

Randy Poe wrote:

Is anyone proposing a lower-bound experiment?

Not that I know of.

A friend
told me many years ago (in 1987, actually) that he thought
the data from Supernova 1987a might be able to establish
a lower bound (based on the velocity spectrum as I recall)
but I gather that never panned out.

SPIRES is your friend. Go to
http://www.slac.stanford.edu/spires/hep/
and enter
find k 1987a neutrino

Is there any other experiment, on the drawing board or
in progress, which is hoped to establish a lower bound for
any neutrino masses?

There are lots of experiments investigating various aspects of neutrino
mixing. Most of them can be interpreted as establishing lower bounds on
at least some of the mass eigenstates....

Unfortunately none of these are a direct measurmeent, and the mass
difference is highly correlated with various mixing angles....


Tom Roberts tjroberts@lucent.com