# Hans Bethe, "Solar Neutrinos” - The Worlds of Philip Morrison Symposium (Day 2)

[MUSIC PLAYING]

PRESENTER: The next speaker is Hans Beta, who's known in the physics community as a person who knows everything, and has done almost everything. I remember when I first-- no, no, that's wrong-- I remember at the early Rochester meetings on high energy physics, whenever there was a question-- "Has this experiment been done?" "Has this calculation been done?" "What is the result of this?" "What is known about this?"-- automatically, every head turned to Professor Beta. And his answer was awaited, and furthermore believed. And he always answered, too.

There probably is no field of physics to which he has not made an important contribution. He received a Nobel Prize in 1967 for the discovery of the carbon cycle as a source of energy generation in the sun. Professor Beta was born in Germany, came to the United States as a refugee from the Nazis. He arrived at Cornell in 1935, and more or less has been there ever since, with the exception, of course, of the years of the Second World War, where he worked here for a while, and then in Los Alamos, where he was central in the construction of the first nuclear weapons. He was the leader of the theoretical group there.

Ever since, while conducting-- while engaging actively in physics research and teaching, he has worked rationally and effectively to exploit and to understand and to contain the nuclear fire which he helped to invent. He was my thesis advisor 38 years ago. It was hard.

[LAUGHTER]

But I learned a lot and survived. So it's a real pleasure to welcome him back to MIT, Professor Beta. His title today is Solar Neutrinos.

[APPLAUSE]

[INTERPOSING VOICES]

PRESENTER: Yes, okay, let me show you how that works. Say something.

BETA: Okay.

PRESENTER: I think so. Let me show you how this works, so you can turn it on and off.

BETA: I think I'm following that.

PRESENTER: Are you going to use the--

BETA: Yes.

PRESENTER: Okay. It turns on by pushing the lever at the bottom.

BETA: Oh, it's practically in-- [? Friesis, ?] thanks for the welcome. I want to correct your introduction by one point. I have never worked on general relativity, and I don't--

[LAUGHTER]

--don't intend to do so.

[LAUGHTER]

I'm happy to be here to join in congratulations to [? Filmores. ?] But it is also a sad occasion for me. I'm quite used to getting old, but it is very depressing if young people get old, also.

[LAUGHTER]

When I met Phil, he was 28, and a very young man. And now, they tell me he is 70, which is hard to believe. But one thing has remained unchanged. He always gave us terribly exciting ideas, insights, and he still is at this today. So I wish, Phil, that you will remain in this situation for many years to come.

Now, I have to warn you all that this talk will be very different from Dr. Sagen's talk. It is scientific, and it will contain a number-- even a number of algebraic formulae that's terrible.

[LAUGHTER]

I am going to talk about the subject that--

[LAUGHTER]

--has been very active for the last year, namely the subject of neutrinos from the sun. For many years, this has been a great puzzle. And to talk about that, let me start by telling you the nuclear reactions which are supposed to go on in the sun. They are listed here. And many of them could use neutrinos.

The fundamental the action, of course, is the-- this doesn't seem to work-- the fundamental reaction is the top one, which is the current of two protons giving a deuteron, a positron, and a neutrino. And then, most of the deuterons which are formed in this way finally go to this reaction, making Helium 4. That gives the big energy.

But occasionally, they make Bavarium 7. which that happens maybe one time in 1,000. And Bavarium 7, then, emits another neutrino. And then again, in a very rare event, Bavarium 7 will capture another proton and give a very odd nucleus, Boron 8, which produces neutrinos of tremendous energy, 14MAB. Here, I have given the energies. C means continuous spectrum. And this is the maximum energy of the continuous spectrum. In two reactions, you get a definite energy neutrino.

Well, the question is, can you observe the neutrinos? It is very difficult to observe neutrinos of about one MAB. That is well known. And it is also well known that it gets a little easier if the neutrinos have higher energy.

In this connection, then, Willie Fowler and John Bacon had a very good idea. They said, well, we might use Chlorine 37. There is a state in Argon 37, which is very similar to the Chlorine 37 state. It's a so-called analog state. And therefore, the transition, the capture, of a neutrino by Chlorine 37, giving these analog states in 37, which is highly excited by six million electron watts-- this is a super-allowed transition, that is, the transition is as easy as any neutrino capture can ever be. And so they decided, let's take Chlorine 37 and observe the Argon 37 which is formed.

Ray Davis of Brookhaven, now of the University of Pennsylvania, has now tried to observe the capture of solar neutrinos for nearly 20 years. He uses a [? tanker ?] of carbon tetrachloride. He flushes the argon out by helium and argon, puts the argon into a Geiger counter, and uses the decay of Argon 37 which takes place by capture of an atomic electron in the [? cache, ?] giving Chlorine 37 and a neutrino.

Of course, you can't observe that neutrino, either. But you can observe that the resulting atom is missing a K electron. Thereupon, it emits X-rays of a few kilobytes. And those you can observe.

This is the principle of the observation. But-- and indeed, Davis did observe neutrinos, but only about a third of the predicted number. This is summarized in an article by Davis and two collaborators going back to 1981.

Well, so what's wrong? The main person who investigated this problem has been John Bacon of the Institute of Advanced Studies. And He has tried almost anything in the theory of either nuclear reactions or the constitution of the sun to try and clear up this puzzle, to no avail. It just doesn't remove the discrepancy.

The way they go about it is that maybe the temperature in the center of the sun is lower than has been generally supposed. And for that purpose, they need information about the opacity of solar material and the different conditions, density, and temperature. There is a lot of contributions by people at Los Alamos and Livermore. But the result remained the same. There is a discrepancy by a factor of 3, or maybe 4.

At the same time, Willie Fowler and his group, experimenters group, at Cal Tech have re-examined the nuclear reactions, and have found that the cross-section measurements for the reaction leading to Bavarium 7 or to Boron 8, that these measurements were correct, in spite of some people from other laboratories-- go ahead-- doubted that. Well, the whole problem was re-examined very carefully in 1982 by Bahcall and others in a very beautiful paper in the views of modern physics, in which they discuss, in particular, [? probable ?] errors of each step of the argument. Of course, one point in the argument everybody believes, namely that the proton-proton reaction makes the energy in the sun. Not only is it the logical thing to assume, but in addition, the general idea of thermonuclear reactions in stars has led to a very detailed understanding of the further evolution of stars, and in particular, all the details of the red giant formation and evolution.

I especially asked Bahcall the other day in a long telephone conversation, how sure is he of his results? And he said, well, we have gone through all the steps. We have even results with three times the standard error. I don't believe there is any way to escape the conclusion.

Well, one way was once proposed by Willie Fowler. He called it the Desperate Theory. Everybody knows that the light comes out of the sun only after some 5 or 10 million years. So what we see today is energy which was produced maybe 10 million years ago. So maybe the sun was hotter in those days, more luminous than it is today, which would be a rather sad conclusion because it would predict that we would go into a severe ice age in a few million years. That's less threatening than nuclear winter, but still not very promising.

Bahcall and others made a modified theory whose results are given here. But in the meantime, they have gone back to the theory of 1982. The theory after all corrections predicts that you should observe on Earth about 6 SNU. That's a funny unit. Admit It only-- and I won't explain it. It tells you how many counts Mr. Davis ought to observe. But this experiment gives only 2 SNU.

Now, it is somewhat frustrating to rely on that funny nucleus, Boron 8. It would be much nicer to observe directly the neutrinos which come from the combination of two photons because we know exactly how often that must happen, because that gives us the energy which the sun emits every second, the so-called solar constant. Well, to avoid the [? lions ?] on the rare side chain, it was proposed many years ago to use a different detector, namely Gallium 71. And the only thing that's wrong with that proposal is that gallium is a very rare element, and therefore terribly expensive. But it obviously was an important experiment to do, to find out the solution to this puzzle.

Well, the National Science Foundation unfortunately rejected the proposal several times to carry out this experiment. The Europeans are somewhat more open to suggestions, so this experiment is going to be done by a collaboration of Italian, French, and German physicists, which will be carried out in Gran Sasso in Italy, and is being started now. This gallium experiment measures directly the neutrinos which are made in the proton-proton reaction.

And I have written down here the capture rates in these funny units, SNU, for three detectors. Don't pay any attention to the lithium. The chlorine which has been used observes practically nothing from the proton-proton reaction. And the observation comes mostly from Bavarium 7 and Boron 8.

By contrast, gallium would observe mostly proton-proton neutrinos, some, Bavarium 7, and very few, Boron 8. It also would give a lot more SNU. In gallium, the neutrino capture leads to germanium. And germanium being an analog of carbon in the Periodic Table, lends itself to a very ingenious detection scheme proposed by Davis. He's a good chemist. You make germanium.

So the problem remains, why are there only one third as many neutrinos as there ought to be? It was suggested fairly early that maybe neutrinos transform into each other. Namely, we know there are different kinds of neutrinos, electron neutrinos, and muon neutrinos, and tau neutrinos.

And it was suggested that maybe these neutrinos are not the ones which propagate in free space. And that's a very standard occurrence in particle physics. You know about it probably in the case of K mesons. And there are many other examples.

The particles which interact with other particles are usually not the same as the particles which propagate in free space. So then, you assume that free neutrinos are not equal to these. And if that is so, then neutrinos which I should call Neu-1 and Neu-2 will propagate. And that will have the consequence that in free space, neutrinos oscillate between electron neutrinos and mu neutrinos.

[? Linus ?] claimed at one time that he had observed such oscillations using the neutrinos from the Savannah River Reactor. But later experiments did not confirm his results. In fact, experiments recently done at CERN find a certain result which involves two quantities, namely the difference in mass of these neutrinos and the angle of coupling.

I go into that a little more later. Namely, the electron neutrino is not quite the neutrino Neu-1, but they are off by a certain angle, theta. And the CERN experiments show that the square of the mass difference times square of twice that angle must be less than 0.03 electron watts square. This is the dimension because of the mass. The direct experiments on the mass of the electron neutrino show probably by now that this mass is less than 10 electron watts, very far from this number.

Now, if oscillations were to exist, then the electron neutrinos from the sun might be converted into either mu or tau neutrinos on their way to the Earth. For this, you need a mass difference greater than 10 to minus 13. But you would need desperately that the coupling angle would be very large, namely that the free neutrinos are maximally different from the neutrinos interacting with electrons or mus. That is very unlikely. But it can-- *a* *priori,* it cannot be ruled out completely.

So this was the situation until recently. And recently, two Russian scientists [? Michraiv ?] and [? Smirnov ?] proposed a theory which I believe is the correct theory. And they use the mixing of neutrinos, but in a much more intelligent original way. Then I'll discuss in the last slide.

I assume only two neutrino types, electron and mu neutrinos. And in vacuum, they are coupled. In terms of the free neutrinos, Neu-1 and Neu-2, N-1 commonly [? writes ?] that the electron neutrino is cross of a certain coupling [? and ?] with theta times the free neutrino Neu-1 plus sine theta, times Neu-2. And the mu neutrino, of course, is orthogonal to that.

Then if the free neutrinos have masses, M2 and M1, you can write the mass matrix in this form. There is a diagonal term which is of no relevance, plus another term which contains the difference of the two masses, and times this very simple matrix, which is non diagonal. In terms, the [? first term ?] refers to electron neutrinos, the second one to mu neutrinos.

Now so far, it is no different from the last slide. But now, they make use of a theory which was developed by Wolfenstein of Carnegie Tech and published in *Physics* *Review* in 1978. And he says as follows. Neutrinos-- when neutrinos are in matter, then all neutrinos interact with matter by means of the neutral weak current. But that's of no interest because it's the same for all neutrinos. It doesn't help convert one into the other.

However, the electrons also will interact with the electron neutrinos, and none other, by the charge weak current in which an electron is absorbed by a proton and becomes a neutrino plus a neutron. And this has the consequence that there is effectively, in matter, a potential energy for neutrinos which is the universal Faraday constant, g multiplied by the electron density, a number of electrons, for aggregate volume.

So far, the Wolfenstein theory, it says then that the relation between momentum and energy of a neutrino goes like this. The momentum square plus mass square is not equal to E square, the energy square, but energy minus the potential energy squared. And this potential energy, which only exists for electron neutrinos-- I expand this-- is equivalent to having an extra mass for electron neutrinos, which is equal to 2E, twice the neutrino energy, times that potential energy.

And so now, my mass matrix gets added to it, an interaction term, which is this 2EV. And that can be written in this way. It's proportional to the density of the matter, and to the energy of the neutrino. And I call that A.

Now, I add this to the mass matrix. And this will add to just to the electron part of the mass matrix, electron neutrino part. It will add a term here. And I wrote it in such a way that I added one half A here and minus one half A for the mu neutrino. And delta here is the difference in mass, the mass of the, essentially, mu neutrino minus the mass of the electron neutrino. And remember, you always have to use the square of the masses.

Now, therefore, obviously, when you would look at this equation, you see that the difference in masses becomes smallest when the A, which is proportional to the density, is equal to delta times cosine 2 theta. And then, you have a minimum in the difference of the two masses. And I will show you a curve of what that-- of the consequence of that.

Namely, there are two possible values for the mass square of a neutrino. If you have no density of electrons, then you have the electron neutrino here at the low mass, no neutrino at a higher mass. But when you get to the point where this condition is satisfied, that A is equal to the difference of mass square, that's this point, then at this point-- you can probably not see that faint line-- at this point, the electron neutrino would get a bigger mass than the mu neutrino. But there is the non-diagonal term in the matrix, which means that, actually, the curves do not cross. But this curve which starts as the electron neutrino ends up at high electron density as the mu neutrino, and vise versa.

And so, this means, then, a very interesting thing. And that's now the theory of [? Michraiv ?] and [? Smirnov. ?] If you start-- well, let me first say that for our given neutrino energy, there is a certain density at which the two curves cross. And that density is determined by the difference in mass of mu neutrino and electron neutrino, and by the universal [? Faraday ?] constant, which we know very well.

So at present, the difference of the masses of the two neutrinos is not known. But it surely is a constant of nature. And therefore, this lambda is a certain number, which we don't know. But it's a number.

And now, let's assume that neutrinos are produced at a certain density, one near the center of the sun. Then as they travel out through the sun, they will go through this resonance if their energy is greater than the energy-- than a certain critical energy which is equal to that lambda divided by the density at the center of the sun. If, however, the-- so if this is the case, then they were automatically run down the green curve, starting out as electron neutrinos running down the green curve.

They become mu neutrinos. And they change thereby into mu neutrinos. And that means they become unobservable because mu neutrinos can only be changed into-- in matter, they could only be changed into mu mesons, which have a mass of 100 million electron watts. And no neutrino, which is produced in the sun, has anything like that energy.

So those neutrinos, therefore, which are created at a density above the crossing point, go down the green curve and end up as unobservable mu neutrinos once they leave the sun and come to Earth. And that would explain how it comes that the neutrinos from the sun are not observable on Earth. That clearly is going too far, because after all, Mr. Davis did observe some neutrinos-- not enough, but some. And this is an important point.

So conversely, if the neutrino is made-- has a lesser energy than the density at the center of the sun, it would be somewhere on this curve. Remember, what matters is the product of the density times the electron-- times the energy of the neutrino. So neutrinos of lesser energy than might be produced here. And [? SU ?] would use the density as the neutrino goes out of the sun. It goes down the periphery curve and ends up as an electron neutrino.

So the simplest conclusion, which you may draw, is the high energy neutrinos will be created here and will be unobservable. The lower energy neutrinos will be created here, and will remain electron neutrinos, and remain, therefore, observable. That is the theory of [? Michraiv ?] and [? Smirnov. ?] Some are translated into different languages, quite apart from being translated from Russian into English.

And so, let's see what we can conclude. For that, I go to the paper which I mentioned before, which says that from Boron 8, you get 4 SNU. From all other nuclei, you get 1 and 1/2. Davis has observed 2.1. And so, I say, these neutrinos are low energy neutrinos, and therefore are green-- no, are purple-- and are therefore observable. And also, a few of the Boron 8 neutrinos must be observable, maybe 0.5, plus or minus 0.5, not clear, whether any of them need to be observable.

I evaluated this wrongly. Fowler then corrected me in a letter. And the result is that the Boron 8 neutrinos are observable if they have less than 8MAB. And that leads to a difference in mass square between the two types of neutrinos of 10 to the minus 4 electron watts square.

It means, by the way, very importantly, this works only if the mu neutrino has a higher mass than the electron neutrino. And so, if we assume tentatively the electron neutrino has no mass, then the mu neutrino would have a mass of one hundredth of an electron watt, totally unobservable by other means.

Now, that's not the only condition. You also need to have-- as the neutrinos go out of the sun, you have to have an adiabatic transition. Well, that gives a certain condition. And I will not bother you with the details. It tells you that the coupling angle-- you'll remember, electron neutrinos are cosine theta times Neu-1 plus sine theta, times Neu-2.

But this coupling angle must be greater than 0.01 radian, which is 0.6 degrees. It's almost certain if there is a coupling angle at all that it will be greater than this number. For instance, those of you who would know particle physics will remember that the Cabibbo angle is about 13 degrees, much, much larger than 0.6 degrees. So it is essentially certain that the angle is big enough.

However, it says one very important thing, namely that both neutrinos must have mass. And that is not at all obvious. When we started talking about neutrinos back 50 years ago, we thought, I think, practically all of us, of neutrinos without any mass. And they have great attraction, aesthetically. But I have no preconceived notions, so I am perfectly satisfied. If neutrinos have some mass, it will be very, very small.

So if all this theory is correct, then it says that we don't observe those Boron 8 neutrinos. And only the Boron 8 neutrinos are converted into mu neutrinos.

And if you remember the table I showed you at one time, if you use a gallium detector, then you do not preferentially observe Boron 8 neutrinos, but instead, you observe all neutrinos, and particularly those coming directly from the proton-proton reaction.

So with a gallium detector, you should observe essentially the same as the theorists have always predicted. You should get the theoretical result, which is about 110 SNU. So we have to hope for the Europeans to get experiments with the gallium detector.

Well, I have shown you the qualitative analytic discussion, which I have made. But there are people who are much more industrious than I, and know how to use a computer, which I don't. And they found, then, not only one solution of the problem, but two. And the first people who did that were the Russians themselves. And then with somewhat improved technique, it was done by Rosen and Gelb at Los Alamos, and got these results.

Now, if you believe that Davis observes one third of the expected neutrinos, you should use the diamonds. The more recent ideas of Bahcall are that he observed only a quarter. And then, you should look at the [? circus. ?]

And so what is plotted here is delta m square in electron watts square, and 10 to the minus 4 is my solution. Constant mass, arbitrary coupling angle, theta. But there is a second solution, which is shown here with much smaller mass difference. You would have thought 10 to the minus 4 is small enough, but they get even less, but with a definite relation between the mass difference and the coupling angle.

And in fact, the relation is that the product of these two is a certain number. It is delta m squared times sine square of the coupling angle, is 4 times 10 to the minus 8 electron watts square, small enough. They have confirmed the solution which I've published in addition to this.

Now, if delta m squared is less than my 10 to the minus 4, then also, neutrinos other than the Boron neutrinos will be converted into mu neutrinos. But we know that we must have some neutrinos left over, because after all, Davis did observe one third or one quarter of the expected number. So there must be some other condition which prevents the conversion of the electron neutrinos into mu neutrinos. And that's the condition that the transition be adiabatic.

This was pointed out not by Rosen and Gelb, but by Haxton of the University of Washington. And he has shown that the Rosen-Gelb solution essentially corresponds to the adiabatic condition, which I mentioned before. And assuming that Haxton finds analytically that this product should be 6, 10 to the minus 8, close enough to the 4 10 to the minus 8 that Rosen and Gelb had found.

Now, because the low energy neutrinos are now lost, the gallium experiment, which measures mostly low energy neutrinos, will now also give a response less than the original theory, if the second solution is correct. I'll show you here a rather nice curve of Rosen and Gelb, which tells you the probability of reaching Earth as a function of the energy in MAB. And there is the stars, which is my solution. You have practically no probability for the high energy neutrino, but all the low energy neutrinos will remain electron neutrinos and be observable. And contrary-wise, from the Rosen-Gelb solution, it's the other way around.

So the gallium experiment therefore has become even more important. It will make it possible to distinguish the two solutions. A question was raised, is there any upper limit to sine square theta? This was investigated by Suckley group, who found, yes, there is. The upper limit is something like a half-- well, two thirds, actually-- essentially independent of the mass. difference.

The best calculation so far has been made by two people at [? Feremy ?] Lab, Park and Walker, who plot here some beautiful curves. It all uses the same general idea, namely conversion of neutrinos from electron neutrinos to mu neutrinos in the sun due to high density of the matter in the sun. Plotted here is, again, the difference in mass between the two types of neutrinos. Plotted here is essentially the sine square of 2 theta.

And the contour lines correspond to various possible results of the Davis experiment. What he gives is 2.1 SNU. That's this curve. But he says plus or minus 0.3. That's three times more than error. So it can be anything between 1.8 and 2.4.

All right. If you follow the 2.1 curve, there's first a horizontal [? sledge, ?] which is my attempt at a minus 4 mass difference. Then there is the slant line, which is essentially the Rosen-Gelb solution. It has a wiggle because different types of neutrinos undergo the non-adiabatic transition, which goes to smaller mass difference and higher sine square theta. And finally, there is the vertical line, which is essentially the Suckley limit at high coupling angle. So this tells you, then, you have essentially two solutions to the problem. Either the mass difference is 10 to the minus 4, and then the coupling angle can be anything, or else, the mass difference is much less than 10 to the minus 4, and then the coupling angle must be directly related to the mass difference.

Well, the [? Feremy ?] Lab people, Park and Walker, they went on and discussed what should be the result of the gallium experiment, once it is done. And that now is particularly interesting. This, again, is mass difference, angle, and the dotted lines are the two limits of the Davis experiment with chlorine.

The solid lines are the contour lines for the gallium experiment to be done. And so, you see, if the solution is along this line, or along this line, then the gallium experiment should give, oh, somewhere between 100-120 SNU. But if the solution is the type preferred by Rosen and Gelb, then the gallium experiment will give you a greatly reduced count, maybe as low as 10 SNU, 10 percent of the theoretically predicted number. So this is extremely interesting, and makes, of course, the gallium experiment more and more important.

And so what can we conclude? If the gallium experiment gives the high value, about 100 SNU, then there are two alternatives. Either my original interpretation of the two-- of the Russian suggestion is the right one, the mass difference is 10 to the minus 4 and the angle is undetermined, or there is some totally unknown other explanation.

Now, Bahcall swears that his theory of the temperature distribution of the sun is correct. So it's very unlikely that there is another explanation. However, one would have to say, if gallium gives 100 SNU or so, then the [? Michraiv-Smirnov ?] explanation is merely likely. It is not absolutely proved.

But if the gallium experiment gives a much lower number-- let us say, something less than 60, perhaps as low as 10 SNU-- then that can only be explained by the Russian suggestion, because there is nothing that we can do about the distribution of temperature in the sun, which would reduce the number of quote, unquote, [? "unreactions," ?] because those reactions give us our light. We know exactly how much energy the sun emits. And so, we know exactly how many proton-proton correlations occur. And therefore, no change of the solar temperature distribution could explain the gallium result anywhere in that region.

In that case, then, the Michraiv-Smirnov answer theory is definitely proved. And we must then assume that, indeed, neutrinos have a mass, that the mass of the mu neutrino is greater than that of the electron neutrino. And how much it is can be determined, then, from the gallium experiment.

If the gallium experiment gives the high value, then one probably has to do still another experiment showing that it is, indeed, the Russian suggestion and not something funny about the sun that gives the-- that is the cause of the Davis result. If gallium is as low as about 10 SNU, then delta m square is of order 10 to the minus 6, and sine square of 2-theta is of the order of 0.1. And that's very close to the Cabibbo angle, which would be 0.16. There is no good reason why it should be just the Cabibbo angle.

Now, the most exciting part of this whole story [INAUDIBLE] is that it might permit us a glimpse into the gut. And by this, I don't mean the intestines. But I mean the grand unified theory that particle theorists are fond of, and of which I understand nothing.

[LAUGHTER]

However, all particle theorists tell me that in gut, there should be right handed neutrinos in addition to the ones we know, which are left handed neutrinos. And these right handed neutrinos should have a mass about equal to the grand unification mass, which is 10 to the 14 GEV, or 10 to the 23 EV. So we are talking at the same time about terribly small masses and terribly big masses.

There is a particular theory by Gell-Mann and two others, simultaneously obtained by Yanagida, '78 and '79, according to which, the mass of neutrinos of a given [? flavor ?] [? I ?] should be the mass squared of the corresponding quark divided by some constant and the mass of the grand unification theory. Of course, that mass nobody knows. But it is about that number. Why the K appears there, I haven't any idea.

[LAUGHTER]

The Michraivs neutrino would have 10 to the minus 6 EV, and the tau neutrino 10 to the minus 3 EV, with the right handed neutrinos having a mass like the gut mass. It is obvious that the mu neutrino would then not be enough to satisfy any solution of the solar neutrino puzzle. Only the tau neutrino would. The difference of the square of the tau neutrino and E neutrino mass would be 10 to the minus 6 EV square, which falls right into the range of the Rosen-Gelb solution. And it falls right at the place where gallium would give the minimum response of about 10 SNU.

So we must then assume that, in the sun, the electron neutrino converts into a tau neutrino, not a mu neutrino. At present, I don't know any experiment which would distinguish between tau and mu neutrino as the product. But you can see that it will be most exciting to have the result of the gallium experiment.

This can, first of all, show whether or not the [? Michraiv-Smirnov ?] suggestion is correct, and whether this is the solution of the solar neutrino puzzle. And secondly, it may give us an insight into grand unified theory, and insight into things which occur at energies of 10 to the 14 GEV, energies which surely are beyond the most ambitious plans of accelerators, and surely-- and which also are beyond the range of cosmic ray neutrinos. They go up to about 10 to the 20 EV, whereas the gut is about 10 to the 23 EV.

So there is a good chance, I believe, that the neutrinos will give us a look into energies so high that we have no reasonable expectation of reaching them by accelerators or other observations. And if that is so, then it was most fortunate that Ray Davis did not observe the neutrinos that were expected, but observed only a small fraction of them. And no matter how much the theorists pressed him, he still stuck to his small number. And this may have shown us a very interesting feature of neutrino physics, and possibly, the best [? hand ?] at knowing something about the very highest energy particle physics. Thank you.

[APPLAUSE]