The paper claims that the mass matrices of leptons, quarks, and neutrinos are Fourier coefficient matrices of SL(2,Z(√2)) modular forms, with parameter-free agreement for charged leptons and fitted agreement for quarks and neutrinos.
Universal Evolution of CKM Matrix Elements
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abstract
We derive the two-loop evolution equations for the Cabibbo-Kobayashi-Maskawa matrix. We show that to leading order in the mass and CKM hierarchies the scaling of the mixings $|V_{ub}|^2$, $|V_{cb}|^2$, $|V_{td}|^2$, $|V_{ts}|^2$ and of the rephase-invariant CP-violating parameter $J$ is universal to all orders in perturbation theory. In leading order the other CKM elements do not scale. Imposing the constraint $\lambda _b=\lambda _{\tau}$ at the GUT scale determines the CKM scaling factor to be $\simeq 0.58$ in the MSSM.
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The paper claims that the mass matrices of leptons, quarks, and neutrinos are Fourier coefficient matrices of SL(2,Z(√2)) modular forms, with parameter-free agreement for charged leptons and fitted agreement for quarks and neutrinos.