constants

How Many Fundamental Constants Are There?

John Baez

April 22, 2011

You might at first think that the speed of light, Planck's constant and Newton's gravitational constant are great examples of fundamental physical constants.

But in fundamental physics, these constants are so important that lots of people use units where they all equal 1! The point is that we can choose units of length, time and mass however we want. That's three independent choices, so with a little luck we can use them to get our favorite three constants to equal 1. Planck was the first to notice this, so these units are called "Planck units".

Planck units are great for quantum gravity. They are not so convenient for other purposes, however. The Planck length, for example, is ridiculously small: about 2 × 10-35 meters. The Planck time looks even worse: about 5 × 10-44 seconds. The Planck mass is 2 × 10-8 kilograms. In ordinary life, and even in nuclear physics, Planck units can be a real nuisance.

But in the grand scheme of things, units are not very important. They are arbitrary human conventions. As long as you stick with some choice or other you will do okay.

Many constants involves units of length, time, mass, temperature, charge and so on. The numerical value of these constants depend on the units we use. The numbers would change if we used different units. Thus, though they certainly tell us something about nature, to some extent they are human artifacts.

On the other hand, certain constants don't depend on the units we use - these are called "dimensionless" constants. Some of them are numbers like pi, e, and the golden ratio - purely mathematical constants, which anyone with a computer can calculate to as many decimal places as they want. But others - at present - can only be determined by experiment. These tell us facts about nature that are completely independent of our choices of units.

The most famous example is the "fine structure constant", e2/ℏc. Here e is the electron charge, ℏ is Planck's constant, and c is the speed of light. If you work out the units involved you'll see it's dimensionless, and experiments show that it's about 1/137.03599. Nobody knows why it equals this. At present, it's a completely mysterious raw fact about the universe!

Constants that aren't dimensionless can be regarded as relating one sort of unit to another. For example, the speed of light has units of length over time, so it can be used to turn units of time (like years) into units of length (like light-years), or vice versa. People who are interested in fundamental physical constants usually start by doing this as much as possible - leaving the dimensionless constants, which are the really interesting ones.

How many of these dimensionless fundamental constants are there? This depends on your opinion on some new developments, but my best guess is 26. All other dimensionless constants (aside from those built into the initial conditions) can in principle be derived from these, if our best theories of physics are correct - by which I mean general relativity, which covers gravity, and the Standard Model, which covers all the other forces. Of course, "in principle" means "not necessarily by any simpler method than by simulating the whole universe"!

General relativity and pure quantum mechanics have no dimensionless constants, because the speed of light, the gravitational constant, and Planck's constant merely suffice to set units of mass, length and time. Thus, all the dimensionless constants come in from our wonderful, baroque theory of all the forces other than gravity: the Standard Model.

For starters, we have a bunch of masses. There are 6 kinds of quarks, one positively charged and one negatively charged of each generation: up, down; charmed, strange; top, and bottom. The masses of these quarks, divided by the Planck mass, give 6 dimensionless constants. We also have 3 kinds of massive leptons --- electron, muon, tau. The W and Z bosons also have their masses. Then there is the Higgs, which while still not detected, is very much part of the theory, so we get another mass.

This gives us 6 + 3 + 2 + 1 = 12 dimensionless constants so far.

Then we have two coupling constants: the electromagnetic coupling constant and the strong coupling constant. The electromagnetic coupling constant is just another name for the fine structure constant; it describes the strength of the electromagnetic field. Similarly, the strong coupling constant describes the strength of the strong force - the force transmitted by gluons, which binds quarks together into baryons and mesons.

You may wonder why I'm not listing a coupling constant for the weak force here. The reason is that you can calculate this from the numbers I've already listed.

I should warn you here: there are different ways of slicing the pie. Instead of the electromagnetic coupling constant together with the masses of the W, Z, and Higgs, we could have used 4 other constants: the U(1) coupling constant, the SU(2) coupling constant, the mass of the Higgs, and the expectation value of the Higgs field. These are the numbers that actually show up in the fundamental equations of the Standard Model. The idea is that the photon, the W and the Z are described by an U(1) × SU(2) gauge theory, which involves two coupling constants. The beautiful symmetry of this theory is hidden by the way it interacts with the Higgs particle. The details of this involve two further constants - the Higgs mass and the expectation value of the Higgs field - for a total of 4. If we know these 4 numbers we can calculate the numbers that are easier to measure in experiments: the masses of the W and Z, the electromagnetic coupling constant, and the mass of the Higgs. In practice, we go back backwards and use the constants that are easy to measure to determine the theoretically more basic ones.

Either way we slice it, we're now up to 12 + 2 = 14 fundamental constants.

But alas, it ain't that simple. The W particle interacts with quarks in a complicated way that depends on a bunch of parameters called the Cabibbo-Kobayashi-Maskawa matrix. The point is that the W is charged, and any positively charged quark can emit a W+ and turn into any negatively charged quark, not necessarily of the same generation. (Thus while a top can turn into a bottom, it can also turn into a strange or a down; this is how funky exotic hadrons are able to decay into the usual boring stuff we see around us, which is made of ups and downs.) We need a 3 × 3 matrix of numbers to describe the amplitude for any positively charged quark to turn into any negatively charged one by this mechanism. There is however some room to simplify this matrix by multiplying the quark fields by phases, and there are some constraints this matrix has to satisfy, too, so there are not really 9 independent numbers but only 4.

That's 4 more, for a total of 18.

Now we get to the new stuff: neutrinos. In the old days, the Standard Model said that neutrinos were massless, and the three different kinds - electron neutrino, muon neutrino and tau neutrino - couldn't turn into each other. But there was a big problem with this theory. Namely, we see only about one third as many electron neutrinos coming from the sun as we should! Recent experiments are making it ever more certain that neutrinos do have mass and do turn into each other. As I write these words, it's still not proven that they all have mass: since the experiments mainly measure mass differences, the mass of one could still be zero. If we assume that's not true, then we get at least 3 more fundamental constants, from the 3 kinds of neutrinos. That would give 21 constants.

But actually, most people seem to think that neutrinos get their mass just like the quarks do - from interaction with the Higgs. If this is true, we need another 3 × 3 matrix for neutrinos, just like we do for quarks. People call this the Pontecorvo-Maki-Nakagawa-Sakata matrix, and they're busy measuring its entries. Again, it has 4 independent numbers in it.

If this new extension of the Standard Model holds up, and all the neutrinos have nonzero mass, this brings the total of fundamental constants to 25!

There is another parameter in the Standard Model which measures how much the strong force violates parity — the symmetry between right and left. It's sometimes called "theta". However, as far as experiments can tell so far, this parameter is zero. As I said, I'm not counting "zero" or any other number you can crank out on a computer as a fundamental physical constant. So until we get a glimmering of evidence that this parameter might be nonzero, I won't count this one.

So far I've been talking about constants that people can measure using particle accelerators. But recent astronomical observations have suggested that there are a few other fundamental constants. For example, it seems that the universe is expanding at an ever-faster rate, and the most conservative explanation for this is that the vacuum has a nonzero energy density. This energy density is called the "cosmological constant", and it brings the total of fundamental constants up to 26.

There is also astronomical evidence that the universe is full of mysterious "dark matter". If this consists of new particles, it will probably take new fundamental constants to describe their properties. But so far we don't know enough about dark matter to start talking about new fundamental constants that describe it.

26 constants is not too many — but most physicists would prefer to have none. The goal is to come up with a theory that lets you calculate all these constants, so they wouldn't be "fundamental" any more. However, right now this is merely a dream.

So, what are the fundamental physical constants? We have 26. If we use the ones that theorists like best, they are:

the mass of the up quark

the mass of the down quark

the mass of the charmed quark

the mass of the strange quark

the mass of the top quark

the mass of the bottom quark

4 numbers for the Kobayashi-Maskawa matrix

the mass of the electron

the mass of the electron neutrino

the mass of the muon

the mass of the mu neutrino

the mass of the tau

the mass of the tau neutrino

4 numbers for the Pontecorvo-Maki-Nakagawa-Sakata matrix

the mass of the Higgs boson

the expectation value of the Higgs field

the U(1) coupling constant

the SU(2) coupling constant

the strong coupling constant

the cosmological constant

Most of these are masses, so clearly we need to understand how particles get their mass! In the Standard Model, they get it from interacting with the Higgs boson, so all the masses listed above — and also the Cabbibo-Kobayashi-Maskawa and Pontecorvo-Maki-Nakagawa-Sakata matrices — really show up when we describe how particles interact with the Higgs.

The Higgs has not yet been seen — at least not with any certainty — but of the 26 fundamental constants of nature, 22 describe it or its interactions with other particles! Isn't that weird???

I suspect we're in for some big surprises here....

But you're probably wondering: what do these constants equal? For that, check out this great table:

David Black, The fundamental constants of nature, PDF or DOC format.

© 2011 John Baez

baez@math.removethis.ucr.andthis.edu