It is said that a CP violation would mean that the behaviour of the particle is different from the behaviour of antiparticle. Why is C violation not good/enough?
Answer
The operation that maps particles to antiparticles is just C. (This is somewhat of a simplification. A better thing to say is that in theories with C symmetry, you can pair particle states with the same spacetime quantum numbers but the opposite internal quantum numbers. When C is violated, there may exist no pairing that gets the quantum numbers right. In extreme cases, there may not be any way to define a C-like operator at all, no matter how you modify the quantum numbers; an example is a theory with a single Weyl spinor. In such cases you can still define a pairing using CP, if it exists, or failing that using CPT, which always exists and is conserved, but these pairings don't have the familiar properties you would expect. For much much more, see here and here.)
Why do people focus on CP violation? The issue is that C violation is ubiquitous in the Standard Model; in fact, in a certain sense it is as strong as possible in the charged current weak interactions. However, there are interesting phenomena that require both C violation and CP violation. So since CP violation is the hard part, we talk about it a lot more.
One key example is the creation of a matter/antimatter imbalance in baryogenesis. For simplicity, suppose that C and CP are both defined, though they may not be obeyed. For any particle states i and f, there are four related processes: i→f,ˉi→ˉf,iP→fP,ˉiP→ˉfP where a bar denotes the antiparticle, defined by the action of C. If these processes have rates a, b, c, and d, and the states have different baryon number, then the rate of baryon number violation is proportional to a−b+c−d. If C symmetry is obeyed, then a=b and c=d, giving a rate of zero. If CP symmetry is obeyed, then a=d and b=c, again giving a rate of zero. One needs both C and CP violation to get baryogenesis.
Unfortunately, in popular science these statements are sometimes oversimplified to just "CP distinguishes matter from antimatter", which is confusing.
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