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Let us assume that the standard model is correct up to Planck mass. (Yes, I know, this is a big assumption.)

If we continue the running of quark masses with energy (due to renormalization), what are the mass values we get for the six quarks at Planck energy? Is the sequence of mass values the same at Planck energy or do some quarks "catch" up with others?

Is there some literature on this issue?

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Have a look at Updated Values of Running Quark and Lepton Masses by Zhi-zhong Xing, He Zhang and Shun Zhou. In TABLE IV the authors list the various fermion masses, for example, at the GUT scale $\approx 10^{16}$ GeV, which is quite close to the Planck scale.

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Looking at the literature, and please tell me if I am wrong, it seems that the yukawa couplings fail to unify.

This seems to counter the intuition that all the particles in a multiplet should have the same mass, but surely it can be argued that the mass of the multiplet is zero until the higgs mechanism is activated. (Still, comments are welcome about this; even yukawa "charge" could be expected to be unified in GUT, should it?).

At most, you can get some intersection between the yukawa of the charged lepton and the one of the down-type quark. Note that most work on GUT unification was done still in the age of empirically massless neutrinos, so no yukawa coupling for them, or in any case not enough experimental input to try to match them to the up-type quarks.

Sometimes a reverse approach is applied, assuming that some "texture" happens in the GUT scale and then running down the yukawa couplings it a way that the one of the top approaches to unity. I guess that one could also try with a texture where only the top has mass, and some custodial protects the other yukawas so they are zero in first order.

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I think that your idea of mass is a little wrong. The quark mass is given in a renormalization scheme, if you change it you would have different masses for quarks. But for example the pair production is a physical process, in fact, if you do a pair production with all the radiative corrections you will find the same energy with whatever renormalization you will use. The same for the energy of bounded states.

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