C, P, T (And Their Combinations)

During the 20th century, particle physicists learned that it is important to consider all the possible symmetries that the laws of nature governing elementary particles might exhibit.  The presence or absence of symmetries can reveal aspects of nature that aren’t otherwise obvious.

Of the many possible symmetries to consider, there are three simple ones that play a unique role: charge conjugation (C), parity (P), and time-reversal (T). These three transformations, which affect particles, space, and time, involve

• C: changing all particles to particles of opposite charge (including electric charge, but also other less familiar charges; even some neutral particles get switched.  For instance, neutrinos switch to anti-neutrinos, and neutrons switch to anti-neutrons.)
• P: putting the world in a mirror (more precisely, flipping the orientation of the three directions of space)
• T: running the world backwards in time (more precisely, flipping the direction in which time evolves.)

Each of these transformations has the feature that if you do it twice, you get back to where you started. In jargon, we say P² = P × P = 1 (i.e., if you put a mirror in a mirror, what you see looks the same as if there were no mirrors at all), and similarly C² = 1 and T² = 1.

Also, you can do two of these transformations together.  For instance you can do C and then P, which we simply write “CP” (or you can do PC, which is the same — for these transformations, the order doesn’t matter) in which you put the world in a mirror and flip particles’ charge. You can also consider CT, PT, or even CPT.  Like C, P and T themselves, any one of these combined transformations, performed twice, gives back the world you started with.

Now what should you do, now that you’re thinking about these transformations? The question which you should ask is this: if I imagine a world which is created from ours by making one of these transformations, do the laws of nature that govern elementary particles and forces work the same way in the transformed world as in our own?

If the answer is “yes”, then everything that can happen in the new, transformed world can also happen in our own; and in this case we say this transformation is a symmetry of our world.  More precisely, it is a symmetry of our world’s laws of nature.  If not, well, then you can still do the transformation, but it’s not a symmetry of our world, because the world you get after the transformation differs from our own.

It’s not hard to get a feel for how parity (P) works.  A particular object may or may not have symmetry under parity.  As shown in Figure 1, reflecting a simple triangle in a mirror gives back a triangle which looks identical to the first one, so the triangle is symmetric under P. But the more complicated shape shown at the bottom of figure 1 does not look the same in a mirror, so it is not symmetric under P.

Obviously the world around us is not symmetric in a mirror, as you can see in any natural photograph (see Figure 2, top.) However, we have to distinguish between the symmetry of an object and the symmetry of the laws of nature that govern all possible objects.  The underlying processes of particle physics could be symmetric, which would mean that for any process that can happen in nature, the mirror image of that process could also happen  (Figure 2, bottom).

But in fact, the underlying processes of nature are not symmetric under P!

The remarkable thing is that neither C, nor P, nor T, nor CP, nor CT, nor PC, is a symmetry of nature. The basic processes that physicists knew about up through the early 1900s — in particular, those involving the gravitational and electromagnetic forces, and therefore those that hold the earth together and in orbit round the sun, and those that govern the physics of atoms and molecules and all of chemistry — are in fact C, P and T symmetric. So it was quite surprising to physicists when, in the 1950s and 1960s, it was learned that the weak nuclear force violates all of these symmetries.  The only one of these transformations that is still widely believed to be a symmetry of nature (for profound theoretical reasons) is CPT.

Note that if CPT is a symmetry, then CP and T must have the same effect. Since it is a symmetry, doing CPT gives you back the world you started with, but we also know that doing T twice gives you back the world you started with, so CP must be do the same thing as T.  The same is true for CT and P, and for PT and C.

CPT transforms particles and their interactions of our world to anti-particles and their interactions of the transformed world, and vice versa.  And since in our world, every type of particle has an anti-particle [possibly itself again], and since every interaction involving various particles has an anti-interaction involving their anti-particles (so to speak), this is believed to be an exact symmetry. More specifically, in any world whose particles are governed by  quantum field theory, the math used in the equations of the Standard Model, which describes all the known particles and forces, one can prove that CPT must be a symmetry.  (Whether this is true of a fully unified theory [such as string theory] that combines a quantum theory of gravity with the non-gravitational forces isn’t clear; but experimentally no violations of CPT are known.)

C and P Aren’t Symmetries, Because of the Weak Nuclear Force

Up to around 1950, everything physicists knew — all of chemistry and atomic physics, all the effects of gravitational and electromagnetic forces, light waves and the basics of atomic nuclei — was consistent with the world being symmetric under P.  But it turns out that C and P aren’t even close to being symmetries of the laws of nature. They are violated about as much as they possibly could be, by the weak nuclear force.

The simplest (but by no means only) example of this involves neutrinos.  When a neutrino is created in a particle physics process, it is always produced via the weak nuclear force.  And when it is produced, it always spins counter-clockwise, seen from the point of view of someone at its departure point.  (Neutrinos, like electrons and protons and many other types of particles, are always, in some sense, spinning; more precisely they have angular momentum that is always present.) In other words, it spins like a left-handed screw (see Figure 3). [The jargon is that it has negative helicity — helicity as in “helix”, appropriate for a screw.] But a neutrino produced via the weak nuclear force never spins like a right-handed screw. Since P would exchange right-handed and left-handed (as you’d expect for a mirror), this means that the weak nuclear force violates P.

As a more specific example (Figure 3), when a positively-charged pion (a hadron made from an up quark, an anti-down quark, and many gluons and quark/anti-quark pairs) decays to an anti-muon and a neutrino, the neutrino is always left-handed and never right-handed.  That violates P.   And meanwhile, when a negatively-charged pion decays to a muon and an anti-neutrino, the anti-neutrino is always right-handed. This difference between the processes involving negatively and positively charged pions violates C.

This type of P and C violation is now very well-understood.  The Standard Model (the equations we use to describe all the known particles and forces) incorporates it very naturally (see here for some discussion), and the details of its equations have been tested very thoroughly in experiments.  So while the violation of P and C was a big surprise in the 1950s, today it is now a standard part of particle physics.

However, if we simply look at the particles themselves (and not in detail at how they interact with one another), CP (which is the same as PC) does at first appear to be a symmetry. That’s because P flips the spin of the neutrino from left-handed to right-handed, but C flips the charge of the pion particle, turns the anti-muon into a muon, and replaces the neutrino with an anti-neutrino; and the resulting process does occur in our world (see Figure 4). So for a brief period, physicists thought the weak nuclear force would preserve CP, even though it maximally violates C and P separately.

[Another way to see this is to look at my article on what the particles would be like if the Higgs field were zero. There you see that there are, for instance, electron-left and neutrino-left particles which come together in a pair, and are affected by the weak isospin force, while the electron-right particle comes separately from the neutrino-right particle, and neither is affected by the weak isospin force. Meanwhile what is true for the electron-left is true for the positron-right, and what is true for the positron-right is true for the electron-left. But P exchanges the electron-left and the electron-right, so clearly it is not a symmetry; C exchanges the electron-left and the positron-left, and since the positron-left is not affected by the weak-force, C is also not a symmetry. Note CP, however, exchanges the electron-left and the positron-right, both of which are affected by the weak nuclear force.]

CP Also is Not a Symmetry

But it turned out, as was learned in the 1960s, that CP is also violated in the weak nuclear interactions.   Again this was a surprise, one that we understand well but are still studying today.  Here’s the basic story.

Most hadrons [particles made from quarks, anti-quarks and gluons] decay almost instantaneously via the strong nuclear force, in times shorter than a trillionth of a trillionth of a second. One hadron, the proton, is stable; the neutron, on its own, lives about 15 minutes. (Atomic nuclei, made from protons and neutrons, are themselves sometimes called hadrons, but I personally prefer to call them “collections of hadrons.”) But a number of hadrons, historically and even practically of great importance, have short but not so short lifetimes — anywhere from a billionth of a trillionth of a second to a billionth of a second — and for most of them, their decay is induced by the weak nuclear force (while a few others decay via the electromagnetic force.) As I’ll describe elsewhere, some of them  — especially mesons that contain one bottom quark or one bottom anti-quark — have one or more decays that have been measured to violate CP. (There are other signs of CP violation in oscillations between two hadrons, similar to oscillations that happen in neutrinos.)

This type of CP violation is very interesting because it occurs naturally if there are three or more “flavors” or “generations” of up-type quarks (up, charm and top) and three flavors of down-type quarks (down, strange and bottom). As Kobayashi and Maskawa pointed out, a version of the Standard Model with only two generations could not have this type of CP violation; there would need to be some entirely new source for it.  Since they observed this back before any particles from the third generation were discovered, they essentially predicted there should be a third generation, and for this they were consequently awarded the 2008 Nobel Prize in Physics (along with Nambu, for his extensive work on other subjects.)

So far, there are no signs of CP violation from other sources than the one Kobayashi and Maskawa identified.  But if there are particles and forces beyond those we know in the Standard Model, there may well be more places to see effects of CP violation.

However, even within the Standard Model, there’s still one very big puzzle…

The Strong Nuclear Force and CP

Very surprisingly, CP is not significantly violated by the strong nuclear force, and no one knows why.  We know the strong nuclear force does not violate CP symmetry very much because of a certain property of the neutron, called an “electric dipole moment”.

The neutron is an electrically neutral hadron, very similar to a proton.  The quarks, anti-quarks and gluons which make up the neutron are held together by the strong nuclear force. Now, an interesting question you can ask about any electrically neutral object is whether it has an electric dipole.

A magnet such as you played with as a child is a magnetic dipole, with a north pole and a south pole (see Figure 5.)  A magnetic monopole would be either a north pole or a south pole, on its own; you’ve never seen one, and neither has anyone else.  Meanwhile, an electric dipole has total electric charge zero but has a positively charge on one side and a negatively charge on the other. This could be as simple as a hydrogen atom, with an electron as a negative charge and a proton as a positive charge.

For a simple electric dipole consisting of two charges a distance D apart, one with charge q and one with charge -q, the electric dipole moment is simply defined to be q ×D.  Notice that if the positive and negative charges sit right on top of each other, then this object has no dipole moment; the charges have to be separated in space to be “polarized”.  A hydrogen atom normally isn’t polarized.  But many molecules have a dipole moment, even though they are electrically neutral. For example, a water molecule H₂O has a dipole moment equal to 3.9 × 10^-8 e cm, where “e” is the charge of a proton (-e the charge of an electron), and “cm” is 1 centimeter.  For comparison, this is just a little bit smaller than you’d get if you separated an electron and a proton by a distance that is about the size of a water molecule. (If you did that, the resulting dipole would have a dipole moment of about 9× 10^-8 e cm).  This is telling you that the electrons on the two hydrogen atoms in H₂O are spending a lot of their time over with the oxygen atom.

Now, how big would you expect the dipole moment of a neutron to be? Well, the neutron has a radius of about 10^-13 cm, so you’d expect D should be about that size. And it consists of quarks, anti-quarks and gluons; the gluons are electrically neutral, but the quarks and anti-quarks have electric charges: 2/3 e (up quarks), -1/3 e (down quarks), -2/3 e (up anti-quarks) and +1/3 e (down anti-quarks). So you might expect q to be about that size.  So you’d expect the neutron to have an electric dipole moment with a size in the vicinity of 10^-13 e cm. That’s about a million times smaller than the dipole moment of a water molecule, mainly since the radius of a neutron is a million times smaller.

Actually there are some subtle effects which make a more accurate estimate a little smaller.  The real expectation is about 10^-15 e cm.

But if the neutron had an electric dipole moment, this would violate T, and therefore CP, if CPT is even an approximate symmetry. (It also violates P.) So if CP and CPT were exact symmetries, then the electric dipole of the neutron would have to be exactly zero.

Of course we already know that CP is not an exact symmetry; it’s violated by the weak nuclear force. But the weak force is so weak (at least as far as it affects neutrons, anyway) that it can only give the neutron an electric dipole moment of about 10^-32 e cm. That’s far smaller than anyone can measure!  So it might as well, for current purposes, be zero.

But if the strong nuclear force, which holds the neutron together, violates CP, then we’d expect to see an electric dipole moment of 10^-15 e cm or so.  Yet experiment shows that the neutron’s electric dipole moment is less than 3 × 10^-26 e cm!!  That’s over ten thousand million times smaller than expected.  And so the strong nuclear force does not violate CP as much as naively anticipated.

Why is it so much smaller than expected? No one knows, though there have been various speculations. This puzzle is called the strong CP problem, and it is one of the three greatest problems plaguing the general realm of particle physics, the others being the hierarchy problem and the cosmological constant problem.

Specifically, the problem is this. When one writes down the theory of the strong nuclear force — the equations for gluons, quarks and anti-quarks called “QCD” — the equations have various parameters, including

• the coupling overall strength of the strong nuclear force
• the masses of the various quarks
• the theta angle, which does not affect any Feynman diagrams, but nevertheless determines the effects of certain subtle processes [quantum tunneling, with the buzzwords “instantons” or “pseudoparticles”] in the physics of gluons

Huh?! What’s that last one?? Well, this additional parameter of QCD was discovered in the 1970s (and is one of the contexts in which Polyakov, who won a pri$e recently, is famous.) The issue is too technical to explain here, but suffice it to say that if the theta angle isn’t equal to 0 or π, then the strong nuclear force violates CP. More precisely, and more disturbingly, it is a certain combination of the theta angle and the masses of the various quarks [specifically, the product of the complex phases of the masses] that violates CP. And these two things (the theta angle and the quark masses) are not obviously related to each other — so how can they combine to cancel perfectly?  Yet for some reason this combination is zero, or at least ten billion times smaller than it could have been. There’s no obvious reason why.

Solutions to this 35-year-old puzzle include the following

• maybe the up quark is massless (this is a very difficult thing to check, because there’s no direct way to measure its mass; indirect methods have long suggested it has a mass a few times larger than that of the electron, but there are subtleties that make these methods difficult to interpret with complete confidence.)
• maybe there is a field called the axion field, which removes this effect; a prediction of this idea is the existence of an axion particle, which has been sought for over 30 years but hasn’t been found up to now. The axion could also serve as the universe’s dark matter, by the way.

There are a couple of other possible known solutions, but I won’t cover them here; generally they don’t have a near-term experimental consequence, unfortunately.

Addenda:

What does CP not being a symmetry have to do with the fact that there is more matter than anti-matter in the universe?

Do right-handed neutrinos and left-handed anti-neutrinos exist at all?

Coming soon…