How do $\pi^0$ particles exist? I have been taught that the $\pi^0$ particle contains either an up quark and an anti-up quark or a down and an anti-down. How can these exist without annihilating?
Also, it is its own antiparticle, but it doesn't make sense that the up version and down version would annihilate when they meet.
Or is what I've been taught a simplification - if so, in what state does this particle exist?
 A: Actually, the quark and antiquark do annihilate with each other. It just takes some amount of time for them to do so. The actual time that it takes for any given pion is random, and follows an exponential distribution, but the average time it takes is $8.4\times 10^{-17}\,\mathrm{s}$ according to Wikipedia, which we call the lifetime of the neutral pion.
What you've learned is a simplification, in fact (it pretty much always is in physics). The actual state of a pion is a linear combination of the up state and the down state,
$$\frac{1}{\sqrt{2}}(u\bar{u} - d\bar{d})$$
This is how it's able to be its own antiparticle: there aren't separate up and down versions of the neutral pion. Each one is a combination of both flavors.
The orthogonal linear combination,
$$\frac{1}{\sqrt{2}}(u\bar{u} + d\bar{d})$$
doesn't correspond to a real particle. (In a sense it "contributes" to the $\eta$ and $\eta'$ mesons, but I won't go into detail on that.)
A: Pions do annihilate to a pair of photons but it takes time. This is because it takes time for the quark and antiquark to come in contact. The pion with two down quarks can turn into a pion with two up quarks. This is because the down quark can emit W- boson which is absorbed by antidown to turn into antiup. So they can swap between one another. Also it depends on the case whether pions annihilate one another or not, sometimes they annihilate one another, sometimes they bounce of one another. Sometimes they can even mutually annihilate one another and create a pion with a positive and a pion with a negative charge.
