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$\pi^0$ particles are either up-antiup or down-antidown (or strange-antistrange?) They must be opposite colors to preserve neutrality. Why don't the opposite quarks annihilate?

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It's a metastable state--another example of a metastablestate made up of a particle antiparticle pair like this is positronium. –  Jerry Schirmer Jun 13 '13 at 23:15
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Pions in general and the neutral pion is not an stable particle, its mean lifetime is $\tau=8.4 \cdot10^{-18}$ s Pion - Wikipedia

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They do annihilate. Neutral pions aren't stable. Their half-life is something on the order of $10^{-17}$ seconds, which is about a millionth of the amount of time your computer takes to execute a single processor instruction.

The more fun question is 'annihilate via what reaction'? Something has to carry away the energy and angular momentum. It turns out that there are actually lots of pion decay modes, meaning lots of final states into which a pion can decay. (A slight subtlety: neutral pions are usually defined to be linear superpositions of quark antiquark pairs, roughly $|u\bar{u}\rangle - | d\bar{d}\rangle$. This is done to make them isospin eigenstates.)

The dominant decay process is into two photons, 98% of the probability mass. Understanding the details of how this happens is a fun odyssey into quantum field theory. If you want me to write more about that, you're going to have write a question with more than three sentences in it.

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