A Low Scale for Quark-Lepton Unification

Abstract

The measured fermion masses may be pointing to the scale of quark-lepton unification. Studying the renormalization-group evolution of the Standard Model Yukawa couplings, we find that $y_\tau=y_b$ and $y_\mu=3y_s$ occur at nearly the same scale, around $10^6$ GeV, while the first generation reaches the Georgi-Jarlskog relation $y_e\simeq y_d/3$ at order $10^8$ GeV and can arise from the same Clebsch structure through a simple hierarchical flavor texture. This pattern naturally suggests low-scale Pati-Salam symmetry. We show that flavor mixing and finite threshold corrections preserve this picture. If the electron mass has an independent origin, the two heavier-generation relations can be realized in theories with a unification scale as low as $(5$-$7)\times10^5$ GeV. We identify two simple theories that accommodate all three relations and in which the measured masses fix the flavor structure. The first unifies all three generations at $4.6^{+1.2}_{-0.9}\times10^6$ GeV; the second, with symmetric Yukawa matrices, at $1.5^{+0.7}_{-0.5}\times10^7$ GeV, and predicts $|V_{cb}|=0.042\pm0.002$. Their shared light-family texture fixes the relevant vector-leptoquark couplings, making $\mu\to e$ conversion in nuclei a direct test of unification: upcoming searches will probe part of the region selected by the two theories, and proposed facilities could cover it completely.