Strangeness oscillations

Hello again! On the off chance that you're not sick of the neutral kaons yet, here's yet another post on how awesome they are. Aside from hinting at parity violation in the weak interactions, providing the first evidence of CP violation, and generally confusing physicists, they also have this really neat behavior called strangeness oscillations.

The first hint of these is that by second-order weak interactions (a Feynman diagram for one of these is shown below. Feynman diagrams are awesome and certainly deserve a future post, but I won't go into how to read them at the moment. Note, however, that I use the convention used in my current quantum course, with time on the vertical axis), a $K^0$ can transform into a $\overline{K}^0$, and vice versa. We can identify a particle as either a $K^0$ or its antiparticle based on a couple of decay modes. For example, the $K^0$ has quark content $d\overline{s}$, so it's easy for it to decay into a negative pion and a positive lepton (like the positron or the positive muon), along with the correspondingly flavored neutrino. On the other hand, the $\overline{K}^0$ has quark content $\overline{d}s$, so it preferentially decays into a positive pion and a negative lepton, along with an antineutrino. These allow us to experimentally determine the $K^0$ content of a beam of neutral kaons.

One of the two lowest order
Feynman diagrams in the
$K^0\leftrightarrow \overline{K}^0$ oscillation.

As I've previously mentioned, neutral kaon decays are determined by the CP eigenstates $K_1$ and $K_2$. What I hadn't mentioned is that because these two states are their own antiparticles, the CPT theorem (which states that the combined operation of C, P, and T transformations should yield a universe identical to our own) allows them to have distinct lifetimes and masses. The lifetime effect we've already seen, in the form of the quick $2\pi$ decay of the $K_1$ state and the longer-lifetime $K_2$, which decays instead into $3\pi$. The mass difference, even though it ends up being absolutely tiny, still turns out to be really important to the system. It means that the two states have slightly different energies, which means that they have slightly different time evolutions (the time evolution of a system is a complex phase determined by its energy. This can be easily derived from the Schrödinger equation). If we start off with a pure $K^0$ beam, such as by interacting $K^+$ mesons with matter (the interactions are strong, so they preserve strangeness and generate only $K^0$), we can view it as a combination of $K_1$ and $K_2$. These evolve slightly differently in time, so if we switch back to the $K^0$/$\overline{K}^0$ states, we find that the probability of detecting a $K^0$ at a time $t$ oscillates a bit. This oscillation (with appropriate experimental constants like lifetime and mass differences) is shown in the graph below, and it matches experiment beautifully. The strangeness of a neutral kaon beam can change over time, and it's all predicted by elementary quantum mechanics. Neat, eh?

The strangeness evolution of an originally pure $K^0$ beam. The black
shows the probability that a given member of the beam is a $K^0$, and the
gray shows the probability that it's a $\overline{K}^0$. The units on the
time axis are $10^{-10}$ seconds.