Parity violation is an important effect in particle physics. It is a characteristic feature of the weak nuclear force and is one of the cornerstones of the modern theory of weak interactions (the Glashow-Salam-Weinberg electroweak theory). It was a completely unexpected development when it was first discovered, as it undermines an intuitively-held assumption about the nature of physical laws, this being the idea that the physicality of a system does not change if it is replaced by its mirror image.

### Understanding Parity

A *parity transformation* is a physical transformation that changes the sign of all spatial coordinates. This means that the entire universe is replaced by its mirror image: left becomes right, up becomes down, and forward becomes backward. Intuitively, the laws of physics should remain the same under such a transformation; this is the concept of *parity invariance*. Classical mechanics is clearly parity invariant; the activities on a pool table seem equally reasonable whether viewed directly or through a mirror, and the mirror image of a human is only biologically odd, not physically odd. Note that, even in a parity invariant universe, the objects themselves may be different under a parity transformation. A parity-transformed human has his or her heart on the right-hand side, but this does not contradict the principle of parity invariance.

Some objects are invariant under parity transformations, though, such as basketballs or frisbees. A parity transformed basketball is identical to a non-parity transformed basketball, whereas a parity transformed frisbee is actually an *upside-down* frisbee. The upside-down frisbee is otherwise identical to the original frisbee; if flipped over physically it will be in the same configuration as the original. This is different from the case of the human, as no amount of physical rotation will restore it to its original configuration.

In quantum mechanics, parity becomes an operator that can be applied to quantum mechanical states which does a parity transformation on that state. Like any quantum mechanical operator, it has eigenstates which can be said to have a definite 'parity' value. Since the parity operator is its own inverse, the mathematics of eigenstates requires that the only possible parity values are 1 and -1. Since subatomic particles are represented as quantum mechanical states, they can be (and generally are) parity eigenstates and thus can be said to carry a quantum number called 'parity'. When multiple particles are combined to form a compound state, the total parity is just the product of the individual parities. (Technical note: if they have any relative angular momentum, an additional factor of -1 per unit angular momentum is added)

If all physics is parity invariant, Noether's theorem requires that there be a conservation law corresponding to this symmetry. This conservation law is conservation of parity; the final state of any physical process must have the same parity as the initial state. Since parity invariance was seen as intuitively necessary, it was never explicitly tested. However, as of 1956 the extant data on the strong nuclear force and the electromagnetic force showed those forces to be parity invariant.

### The Test

The road to an explicit test of parity invariance began with two theorists, Lee and Yang, studying an odd case of two almost identical particles. The θ^{+} meson and the τ^{+} meson (as they were then called) had the same mass, same charge, and were in fact identical in all respects but one, their parity. The θ^{+}decayed into two pions, a state with parity +1, and the τ^{+} decayed into three pions, which have total parity -1. (A pion has a parity of -1) Since all mesons are parity eigenstates, this meant that the θ^{+} had parity +1 and the τ^{+} had parity -1. Hence they must be different particles, if parity is conserved.

Lee and Yang wondered if maybe parity just wasn't conserved and these two particles were in fact the same. Noticing that this particular decay proceeds through the weak nuclear force they searched for experimental confirmation that the weak force conserves parity and found none. Thus they proposed that such a test be made. The experimentalist who followed this proposal was C.S. Wu, better known as Madame Wu, who designed an elegant experiment to test the parity invariance of the weak force.

Beta decay is the most prominent process that occurs through the weak nuclear force. Cobalt 60 is one of the most common beta emitting isotopes. Madame Wu's experiment involved aligning the spins of a group of cobalt 60 nuclei and examining the direction in which the beta particles were emitted. A parity transform reverses the spin of the cobalt 60 nucleus relative to the direction of the emitted beta particle, since momentum is a *vector* quantity which reverses direction under a parity transformation, and angular momentum is a *pseudovector* which remains the same. Thus, if the weak interaction is parity invariant the probability that the beta particle comes out along the direction of the spin should be the same as the probability that the beta particle comes out opposite the direction of the spin.

This, of course, was not the case. The vast majority of the beta particles came out in the direction of the nuclear spin. Thus, the weak interaction cannot conserve parity. This made Lee and Yang happy, but infuriated many physicists who found this lack of symmetry deeply unsettling. The θ^{+} and τ^{+} actually were the same particle, called the K^{+} or kaon.

### Consequences of Parity Violation

Violation of parity means that the weak force distinguishes between different "handednesses" of particles. Of course, we need to define what it means for a particle to be "handed". The particles on which the weak force acts are all fermions with spin 1/2, meaning that they only have two possible spin orientations relative to some axis. We can call particles whose spin points along their direction of motion "right-handed" and particles whose spin points opposite their direction of motion "left-handed". The technical term for this is helicity.

This comes into play in the decay of the pion. A positive pion decays into a positive muon and a muon antineutrino through the weak force. If the pion is at rest, the particles must come out in opposite directions, and they also must have opposite spins from conservation of angular momentum. Thus the two decay products must have the same helicity. If the weak force did not distinguish between helicities, there would be an equal chance that this helicity would be right- or left-handed.

What happens in reality is that the particles are *always* right-handed. In fact, in all observations to date, only left-handed neutrinos and right-handed antineutrinos have been observed. Parity violation in weak interactions is 'maximal', meaning that parity violating interactions have equal strength to parity conserving interactions.

Parity violation serves as a good method for identifying interactions as occuring through the weak nuclear force, as it is unique to the weak interaction and is a large effect. Any process which occurs despite violating parity must be mediated by the weak force, although the converse is not true. Physicists attempted to 'fix' parity invariance by introducing "CP invariance", where a parity transformation is followed by the replacement of all particles with their corresponding antiparticles. The presumption here is that the correct "parity transform" of a left-handed neutrino, for example, is actually a right-handed antineutrino, not a right-handed neutrino as it would intuitively be. However, with the discovery of CP violation this, too, failed to preserve a sense of parity invariance.

### Conclusion

Parity violation has been fully integrated into modern particle physics. It is the signature of the weak nuclear force and is responsible for the nonexistence or invisibility of right-handed neutrinos and left-handed antineutrinos (collectively, "wrong-handed neutrinos"). The effect is maximal, meaning that the weak force ignores parity completely while at the same time discriminating based on helicity. In general, despite being one of the most surprising parts of modern particle physics, parity violation is a useful effect for probing the nature of matter at the fundamental level.

Sources: David Griffiths, Introduction to Elementary Particles (Wiley, 1987), and my undergraduate and graduate particle physics courses.

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This writeup is copyright 2005 D.G. Roberge and is released under the Creative Commons Attribution-NoDerivs-NonCommercial licence. Details can be found at http://creativecommons.org/licenses/by-nd-nc/2.0/ .