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I'm having trouble putting the pieces together. In SM, neutral kaon oscillation is heavily constrained. This means, roughly, that the squark mass matrices have to be diagonal. And this is called universality of soft parameters.

What exactly is universality and why do we have universality in this situation? Furthermore, is having diagonal squark mass matrices sufficient? In a lot of SUSY theories, they go one step further and assume the scalar masses are the same for the first and second generations (pMSSM for instance). Why is this additional step necessary?

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By universality, we usually mean not just the diagonal form; we mean a diagonal form proportional to the unit matrix. Universality is flavor-blindness.

It could be sufficient to have diagonal squark matrices – in the basis of superpartners of the energy eigenstates of quarks – for the unwanted new effects to be suppressed. However, it is extremely unlikely – unnatural – for both quark and squark matrices to be diagonal in the "same" basis. Such an accident would mean a fine-tuning that would require an extra explanation. Assuming we don't want to look for such an explanation, the only principle that may guarantee that the squark matrix is diagonal in a seemingly arbitrary basis is to assume that it is proportional to the unit matrix – it is universal, flavor-blind. This assumption is legitimate as far as naturalness goes because it increases a symmetry, the unitary rotation symmetry acting on the squarks.

The proportionality to the unit matrix is especially important for the first two generations. The third generation may fail to be completely universal and it causes smaller problems because the effects involving the third generation are suppressed due to the heavy top quark mass.

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