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I have a limited understanding of antiparticles, so this may be why I am unable to explain why certain mesons are their own antiparticles, while others are not. My understanding is that antiparticles have properties (ie. spin, charge, mass, etc) which are of the same magnitude, but opposite signs, of their elementary particle.

My question is, why is the charmed eta meson its own antiparticle, while the neutral kaon is not its own antiparticle?

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Mesons are not elementary, they are composed of quarks. So take a look at their quark content.

The charmed eta meson consists of a charm and an anti-charm quark, denoted $c\overline{c}$. An anti charmed eta meson would therefore be an anti-charm and an anti-anti-charm (which is just a charm) quark, i.e. $\overline{c}c$, which is obviously the same as $c\overline{c}$ and so it is its own anti-particle.

The neutral kaon, on the other hand, is composed of a down quark and an anti-strange quark, denoted $d\overline{s}$. The anti of this composite is an anti-down quark and a strange quark, $\overline{d}s$, which obviously is not the same as $d\overline{s}$. Therefore it is not its own anti-particle.

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