February 27, 1999

This Week's Finds in Mathematical Physics (Week 130)

John Baez

All sorts of cool stuff is happening in physics - and I don't mean mathematical physics, I mean real live experimental physics! I feel slightly guilty for not mentioning it on This Week's Finds. Let me atone.

Here's the big news in a nutshell: we may have been wrong about four fundamental constants of nature. We thought they were zero, but maybe they're not! I'm talking about the masses of the neutrinos and the cosmological constant.

Let's start with neutrinos.

There are three kinds of neutrinos: electron, muon, and τ neutrinos. They are closely akin to the charged particles whose names they borrow - the electron, muon and τ - but unlike those particles they are electrically neutral and very light. They are rather elusive, since they interact only via the weak force and gravity. I'm sure you've all heard how a neutrino can easily make it through hundreds of light years of lead without being absorbed.

But despite their ghostly nature, neutrinos play a very real role in physics, since radioactive decay often involves a neutron turning into a proton while releasing an electron and an electron antineutrino. (In fact, Pauli proposed the existence of neutrinos in 1930 to account for a little energy that went missing in this process. They were only directly observed in 1956.) Similarly, in nuclear fusion, a proton may become a neutron while releasing a positron and an electron neutrino. For example, when a type II supernova goes off, it emits so many neutrinos that if you're anywhere nearby, they'll kill you before anything else gets to you! Indeed, in 1987 a supernova in the Large Magellanic Cloud, about 100,000 light years away, was detected by four separate neutrino detectors.

I said neutrinos were "very light", but just how light? So far most work has only given upper bounds. In the 1980s, the Russian ITEP group claimed to have found a nonzero mass for the electron neutrino, but this was subsequently blamed on problems with their apparatus. As of now, laboratory experiments give upper bounds of 4.4 eV for the electron neutrino mass, .17 MeV for the muon neutrino, and 18 MeV for the τ neutrino. By contrast, the electron's mass is .511 MeV, the muon's is 106 MeV, and the τ's is a whopping 1771 MeV.

For this reason, the conventional wisdom used to be that neutrinos were massless. After all, the electron neutrino is definitely far lighter than any known particle except the photon - which is massless. The larger upper bounds on the other neutrino's masses are mainly due to the greater difficulty in doing the experiments.

Having neutrinos be massless would also nicely explain their most stunning characteristic, namely that they're only found in a left-handed form. What I mean by this is that they spin counterclockwise when viewed head-on as they come towards you. It turns out that this violation of left-right symmetry comes fairly easily to massless particles, but only with more difficulty to massive ones. The reason is simple: massless particles move at the speed of light, so you can't outrun them. Thus everyone, regardless of their velocity, agrees on what it means for such a particle to be spinning one way or another as it comes towards them. This is not the case for a massive particle!

There was, however, a fly in the ointment. Since the sun is powered by fusion, it should emit lots of neutrinos. In fact, the standard solar model predicts that here on earth we are bombarded by 60 billion solar neutrinos per square centimeter per second! So in the late 1960s, a team led by Ray Davis set out to detect these neutrinos by putting a tank of 100,000 gallons of perchloroethylene down into a gold mine in Homestake, South Dakota. Lots of different nuclear reactions are going on in the sun, producing neutrinos of different energies. The Homestake experiment can only detect the most energetic ones - those produced when boron-8 decays into beryllium-8. These neutrinos have enough energy to turn chlorine-37 in the tank into argon-37. Being a noble gas, the argon can be separated out and measured. This is not easy - one only expects about 4 atoms of argon a day! So the experiment required extreme care and went on for decades.

They only saw about a quarter as many neutrinos as expected.

Of course, with an experiment as delicate as this, there are always many possibilities for error, including errors in the standard solar model. So a Japanese group decided to use a tank of 2,000 tons of water in a mine in Kamioka to look for solar neutrinos. This "Kamiokande" experiment used photomultiplier tubes to detect the Cherenkov radiation formed by electrons that happen to be hit by neutrinos. Again it was sensitive only to high-energy neutrinos.

After 5 years, they started seeing signs of a correlation between sunspot activity and their neutrino count. Interesting. But more interesting still, they didn't see as many neutrinos as expected. Only about half as many, in fact.

Starting in the 1990s, various people began to build detectors that could detect lower-energy neutrinos - including those produced in the dominant fusion reactions powering the sun. For this it's good to use gallium-71, which turns to germanium-71 when bombarded by neutrinos. The GALLEX detector in Italy uses 30 tons of gallium in the form of gallium chloride dissolved in water. The SAGE detector, located in a tunnel in the Caucasus mountains, uses 60 tons of molten metallic gallium. This isn't quite as scary as it sounds, because gallium has a very low melting point - it melts in your hand! But still, of course, these experiments are very difficult.

Again, these experiments didn't see as many neutrinos as expected.

By this point, the theorists had worked themselves into a full head of steam trying to account for the missing neutrinos. Currently the most popular theory is that some of the electron neutrinos have turned into muon and τ neutrinos by the time they reach earth. These other neutrinos would be not be registered by our detectors.

Folks call this hypothetical process "neutrino oscillation". For it to happen, the neutrinos need to have a nonzero mass. After all, a massless particle moves at the speed of light, so it doesn't experience any passage of time - thanks to relativistic time dilation. Only particles with mass can become something else while they are whizzing along minding their own business.

If in fact you posit a small mass for the neutrinos, oscillations happen automatically as long as the "mass eigenstates" are different from the "flavor eigenstates". By "flavor" we mean whether the neutrino is an electron, muon or τ neutrino. For simplicity, imagine that the state of a neutrino at rest is given by a vector whose 3 components are the amplitudes for it to be one of the three different flavors. If all but one of these components are zero we have a neutrino with a definite flavor - a "flavor eigenstate". On the other hand, the energy of a particle at rest is basically just its mass. Thus in the present context the energy of the neutrino is described by a 3 x 3 self-adjoint matrix H, the "Hamiltonian", whose eigenvectors are called "mass eigenstates". These may or may not be the same as the flavor eigenstates! Schroedinger's equation says that any state ψ of the neutrino evolves as follows:

dψ/dt = -iH ψ

Thus if ψ starts out being a mass eigenstate it stays a mass eigenstate. But if it starts out being a flavor eigenstate, it won't stay a flavor eigenstate - unless the mass and flavor eigenstates coincide! Instead, it will oscillate.

I bet you were wondering when the math would start. Don't worry, there won't be much this time.

Anyway, for other particles, like quarks, it's well-known that the mass and flavor eigenstates don't coincide. So we shouldn't be surprised at neutrino oscillations, at least if neutrinos actually have nonzero mass.

Actually things are more complicated than I'm letting on. In addition to oscillating in empty space, it's possible that neutrinos oscillate more as they are passing through the sun itself, thanks to something called the MSW effect - named after Mikheyev, Smirnov and Wolfenstein. And there are two different ways for neutrinos to have mass, depending on whether they are Dirac spinors or Majorana spinors (see "week93").

But I don't want to get caught up in theoretical nuances here! I want to talk about experiments, and I haven't even gotten to the new stuff yet - the stuff that's getting everybody really confused!

First of all, there's now some laboratory evidence for neutrino oscillations coming from the Liquid Scintillator Neutrino Detector at Los Alamos. What these folks do is let positively charged pions decay into antimuons and muon neutrinos. Then they check to see if any muon neutrinos become electron neutrinos. They claim that they do! They also claim to see evidence of muon antineutrinos becoming electron antineutrinos.

Secondly, and more intriguing still, there are a bunch of experiments involving atmospheric neutrinos: Super-Kamiokande, Soudan 2, IMB, and MACRO. You see, when cosmic rays smack into the upper atmosphere, they produce all sorts of particles, including electron and muon neutrinos and their corresponding antineutrinos. Cosmic ray experts think they know how many of each sort of neutrino should be produced. But the experimenters down on the ground are seeing different numbers!

Again, this could be due to neutrino oscillations. But what's REALLY cool is that the numbers seem to depend on where the neutrinos are coming from: from the sky right above the detector, from right below the detector - in which case they must have come all the way through the earth - or whatever. Neutrinos coming from different directions take different amounts of time to get from the upper atmosphere to the detector. Thus an obvious explanation for the experimental results is that we're actually seeing the oscillation process AS IT TAKES PLACE.

If this is true, we can try to get detailed information about the neutrino mass matrix from the numbers these experiments are measuring!

And this is exactly what people have been doing. But they're finding something very strange. If all the experiments are right, and nobody is making any mistakes, it seems that NO choice of neutrino mass matrix really fits all the data! To fit all the data, folks need to do something drastic - like posit a 4th kind of neutrino!

Now, it's no light matter to posit another neutrino. The known neutrinos couple to the weak force in almost identical ways. This allows one to create equal amounts of neutrino-antineutrino pairs of all 3 flavors by letting Z bosons decay - the Z being the neutral carrier of the weak force. When a Z boson seemingly decays into "nothing", we can safely bet that it has decayed into a neutrino- antineutrino pair. In 1989, an elegant and famous experiment at CERN showed that Z bosons decay into "nothing" at exactly the rate one would expect if there were 3 flavors of neutrino. Thus there can only be extra flavors of neutrino if they are very massive, if they couple very differently to the weak force, or if some other funny business is going on.

Now, electron or muon neutrinos are unlikely to oscillate into a very massive sort of neutrino - basically because of energy conservation. So if we want an extra neutrino to explain the experimental results we find ourselves stuck with, it'll have to be one that couples to the weak force very differently from the ones we know. A simple, but drastic, possibility is that it not interact via the weak force at all! Folks call this a "sterile" neutrino.

Now, sterile neutrinos would blow a big hole in the Standard Model, much more so than plain old massive neutrinos. So things are getting very interesting.

Wilczek recently wrote a nice easy-to-read paper describing arguments that massive neutrinos fit in quite nicely with the possibility that the Standard Model is just part of a bigger, better theory - a "Grand Unified Theory". I sketched the basic ideas of the SU(5) and SO(10) grand unified theories in "week119". Recall that in the SU(5) theory, the left-handed parts of all fermions of a given generation fit into two irreducible representations of SU(5) - a 5-dimensional rep and a 10-dimensional rep. For example, for the first generation, the 5-dimensional rep consists of the left-handed down antiquark (which comes in 3 colors), the left-handed electron, and the left-handed electron neutrino. The 10-dimensional rep consists of the left-handed up quark, down quark, and up antiquark (which come in 3 colors each), together with the left-handed positron.

In the SO(10) theory, all these particles AND ONE MORE fit into a single 16-dimensional irreducible representation of SO(10). What could this extra particle be?

Well, since this extra particle transforms trivially under SU(5), it must not feel the electromagnetic, weak or strong force! Thus it's tempting to take this missing particle to be the left-handed electron antineutrino. Of course, we don't see such a particle - we only see antineutrinos that spin clockwise. But if neutrinos are massive Dirac spinors there must be such a particle, and having it not feel the electromagnetic, weak or strong force would nicely explain why we don't see it.

Grotz and Klapdor consider this possibility in their book on the weak interaction (see below), but unfortunately, it seems this theory would make the electron neutrino have a mass of about 5 MeV - much too big! Sigh. So Wilczek, following the conventional wisdom, assumes the missing particle is very massive - he calls it the "N". And he summarizes some arguments that this massive particle could help give the neutrinos very small masses, via something called the "seesaw mechanism". Unfortunately I don't have the energy to describe this now, so for more you should look at his paper (referred to below).

To wrap up, let me just say one final thing about the cosmic significance of the neutrino. Massive neutrinos could account for some of the "missing mass" that cosmologists are worrying about. So there's an indirect connection between the neutrino mass and the cosmological constant! The cosmological constant is essentially the energy density of the vacuum. It was long assumed to be zero, but now there are some glimmerings of evidence that it's not. In fact, some people are quite convinced that it's not. The fate of the universe hangs in the balance....

Unfortunately I am too tired now to say much more about this. So let me just give you a nice easy starting-point:

1) Special Report: Revolution in Cosmology, Scientific American, January 1999. Includes the articles "Surveying space-time with supernovae" by Craig J. Horgan, Robert P. Kirschner and Nicholoas B. Suntzeff, "Cosmological antigravity" by Lawrence M. Krauss, and "Inflation in a low-density universe" by Martin A. Bucher and David N. Spergel.

How can you learn more about neutrinos? It can't hurt to start here:

2) Nikolas Solomey, The Elusive Neutrino, Scientific American Library, 1997.

If you want to dig in deeper, you need to learn about the weak force, since we've only seen neutrinos via their weak interaction with other particles. The following book is a great place to start:

3) K. Grotz and H. V. Klapdor, The Weak Interaction in Nuclear, Particle and Astrophysics, Adam Hilger, Bristol, 1990.

Then you'll be ready for this book, which examines every aspect of neutrinos in detail - complete with copies of historical papers:

4) Klaus Winter, ed., Neutrino Physics, Cambridge U. Press, Cambridge, 1991.

And then, if you want to study the possibility of massive neutrinos, you should try this:

5) Felix Boehm and Petr Vogel, Physics of Massive Neutrinos, Cambridge U. Press, Cambridge, 1987.

But neutrino physics is moving fast, and lots of the new stuff hasn't made its way into books yet, so you should also look at other stuff. For links to lots of great neutrino websites, including websites for most of the experiments I mentioned, try:

6) The neutrino oscillation industry, http://www.hep.anl.gov/NDK/hypertext/nu_industry.html

For some recent general overviews, try these:

7) Paul Langacker, Implications of neutrino mass, http://dept.physics.upenn.edu/neutrino/jhu/jhu.html

8) Boris Kayser, Neutrino mass: where do we stand, and where are we going?, preprint available as hep-ph/9810513.

For information on various experiments, try these:

9) GALLEX collaboration, GALLEX solar neutrino observations: complete results for GALLEX II, Phys. Lett. B357 (1995), 237-247.

Final results of the CR-51 neutrino source experiments in GALLEX, Phys. Lett. B420 (1998), 114-126.

GALLEX solar neutrino observations: results for GALLEX IV, Phys. Lett. B447 (1999), 127-133.

11) SAGE collaboration, Results from SAGE, Phys. Lett B328 (1994), 234-248.

The Russian-American gallium experiment (SAGE) CR neutrino source measurement, Phys. Rev. Lett. 77 (1996), 4708-4711.

12) LSND collaboration, Evidence for neutrino oscillations from muon decay at rest, Phys. Rev. C54 (1996) 2685-2708, preprint available as nucl-ex/9605001.

Evidence for anti-muon-neutrino → anti-electron-neutrino oscillations from the LSND experiment at LAMPF, Phys. Rev. Lett 77 (1996), 3082-3085, preprint available as nucl-ex/9605003.

Evidence for muon-neutrino → electron-neutrino oscillations from LSND, Phys. Rev. Lett. 81 (1998), 1774-1777, preprint available as nucl-ex/9709006.

Results on muon-neutrino → electron-neutrino oscillations from pion decay in flight, Phys. Rev. C58 (1998), 2489-2511.

13) Super-Kamiokande collaboration, Evidence for oscillation of atmospheric neutrinos, Phys. Rev. Lett 81 (1998), 1562-1567, preprint available as hep-ex/9807003.

14) MACRO collaboration, Measurement of the atmospheric neutrino-induced upgoing muon flux, Phys. Lett. B434 (1998), 451-457, preprint available as hep-ex/9807005.

15) IMB collaboration, A search for muon-neutrino oscillations with the IMB detector, Phys. Rev. Lett 69 (1992), 1010-1013.

For a fairly model-independent attempt to figure out something about neutrino masses from the latest crop of experiments, see:

16) V. Barger, T. J. Weiler, and K. Whisnant, Inferred 4.4 eV upper limits on the muon- and tau-neutrino masses, preprint available as hep-ph/9808367.

For a nice summary of the data, and an argument that it's evidence for the existence of a sterile neutrino, see:

17) David O. Caldwell, The status of neutrino mass, preprint available as hep-ph/9804367.

For a very readable argument that massive neutrinos are evidence for a supersymmetric SO(10) grand unified theory, see

18) Frank Wilczek, Beyond the Standard Model: this time for real, preprint available as hep-ph/9809509.

Finally, with all these cracks developing in the Standard Model, it's nice to think again about the rise of the Standard Model. The following book is packed with the reminiscences of many theorists and experimentalists involved in developing this wonderful theory of particles and forces, including Bjorken, 't Hooft, Veltman, Susskind, Polyakov, Richter, Iliopoulos, Gell-Mann, Weinberg, Lederman, Goldhaber, Cronin, and Kobayashi:

19) Lilian Hoddeson, Laurie Brown, Michael Riordan and Max Dresden, eds., The Rise of the Standard Model: Particle Physics in the 1960s and 1970s.

It's a must for anyone with an interest in the history of physics!

By the way, here's a cool picture of the sun, taken using neutrinos rather than light:

20) LSU Super-Kamiokande group homepage, http://beavis.phys.lsu.edu/~superk/

Thanks go to Jim Carr for pointing this out.

© 1999 John Baez