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It seems like the values of the CKM matrix are somewhat related to the masses of the quarks. For example, the heavier the quark the less likely it is to change into another quark in a weak interaction.

For example if $m_i$ are the masses of the three generations of the down quarks and $M_i$ are the masses of the up quarks,

when $i\neq j$ a quite good approximation of the CKM matrix is:

$$CKM(i,j) \approx 70\left(\frac{m_i}{M_j} - \frac{M_j}{m_i} +\frac{M_i}{m_j} - \frac{m_j}{M_i}\right)^{-1}$$

We know for example the weak mixing angle is related to the ratio of the $W^+$ and $Z^0$ masses which comes from $SU(2)$

Is there a theory behind why the CKM angles may be related to the quark masses? Perhaps utilising a $SU(3)$ family symmetry?

  • You've completely suppressed phases. Your are making a gambit of quark mass relations implied by the just-so Wolfenstein parameterization. What is it, and do you believe it is implicit in any understanding of the officially arbitrary quark mass matrices? Textures? – Cosmas Zachos Aug 28 '23 at 15:40
  • Cf. this. – Cosmas Zachos Aug 28 '23 at 15:47
  • What makes something officially arbitrary? –  Aug 28 '23 at 19:43
  • Absence of any good or credible theories, as opposed to Koide-style numerology. – Cosmas Zachos Aug 28 '23 at 20:00
  • I don't think I mentioned Wolfenstein parameterisation. Is that relevant here? BTW. "Textures" seems like numerology too - I never understood what they are about. –  Aug 28 '23 at 22:26
  • No, it is the other way round! Your formula snags at the phenomenological regularity of the λ versus λ² , versus λ³ elements in the Wolfenstein parameterization!! Agreement on "textures". It is all numerology. You seem to not have experienced the shameful Fritsch / Stech matrix fiasco... – Cosmas Zachos Aug 28 '23 at 22:27
  • By snags do you mean contradicts? I would like to know more about the regularity. I am here to learn. If there is some regularity in the CKM matrix. I would like to learn more about it. BTW, I only noted the formula is approximate. Just a guess at trying to relate mass ratios. –  Aug 28 '23 at 22:34
  • It appears to fail miserably for 1,2... Are you using correct quark masses as per the PDG? – Cosmas Zachos Aug 29 '23 at 14:16

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