Unified Flavor: Lattice Quantization, Chain Locality, and a Dynamical Origin of Hierarchical Yukawas

Abstract

We present Unified Flavor (UF), a framework that synthesizes the $B$-lattice flavor hierarchy with a dynamical realization based on TeV-scale vectorlike fermion (VLF) chains. Hierarchical Yukawa couplings arise from discrete ninths-quantized lattice exponents enforced by a single flavon $Φ$ with $ε\equiv\langleΦ\rangle/Λ=1/B$, $B=75/14$. Effective Yukawa entries are generated as algebraic path sums along nearest-neighbor chains of vectorlike quarks (VLQs), factorizing into entry, chain-propagation, and exit amplitudes controlled by the discrete gauge charges. A multi-messenger structure -- in which each Yukawa entry receives coherent contributions from several chain configurations -- generates O(1) complex coefficients whose phases are the physical origin of CP violation. We derive a general chain-inversion theorem, perform systematic perturbative diagonalization of both up- and down-type Yukawa textures, and show that the Cabibbo--Kobayashi--Maskawa (CKM) mixing hierarchy and CP-phase structure emerge naturally from the lattice exponent algebra and multi-messenger interference. All six quark masses are reproduced with O(1) coefficients that are essentially unity. The chain locality simultaneously suppresses dangerous flavor-changing neutral currents (FCNCs) and satisfies electroweak precision constraints, while requiring VLQs with masses in the multi-TeV range accessible at the High-Luminosity Large Hadron Collider (HL-LHC). The same discrete gauge symmetry that enforces the lattice structure also protects the Peccei--Quinn axion quality, unifying flavor, CP violation, and the strong CP problem. The framework extends to the lepton sector, reproducing charged-lepton mass hierarchies, the normal-ordered neutrino spectrum, and PMNS mixing with a predictive two-branch octant--$δ$ correlation testable at DUNE and Hyper-Kamiokande.

Figures (3)

• Figure 1: Schematic of the four-site down-type VLQ chain. Solid arrows denote nearest-neighbor hops with the indicated $\epsilon^{h_a/9}$ suppression; dashed arrows show the endpoint couplings to SM fields. The $Z_9$ charges are shown below each site. The effective Yukawa exponent for entry $(i,j)$ is $(\Sigma+A_i+B_j)/9$, an exact result of the chain-inversion theorem (Sec. \ref{['sec:chain-theorem']}), not a perturbative approximation.
• Figure 2: Combined down-type (blue, upper) and up-type (red, lower) VLQ chains. Both chains share the same internal hop structure $\{1,2,4\}/9$ and the same left-handed endpoint dressings $A_i=(27,18,0)$, enforced by SU(2)$_L$ gauge invariance acting on the shared quark doublet $q_{L,i}$. The right-handed dressings differ: $B_j^d=(10,3,0)$ for the down sector and $B_j^u=(37,12,0)$ for the up sector. The $\epsilon^{h/9}$ suppressions and the resulting Yukawa exponents are exact consequences of the upper-triangular chain mass matrix (Sec. \ref{['sec:convergence']}); they do not rely on $\kappa_a\epsilon^{h_a/9}\ll 1$. The CKM matrix arises from the mismatch between the left-handed rotations $U_{dL}$ and $U_{uL}$ that diagonalize the two Yukawa textures.
• Figure 3: Approximate VLQ production cross sections at $\sqrt{s}=14$ TeV for pair production (solid) and single production with $\theta_L^b\sim 0.1$ (dashed). The horizontal dotted line indicates the approximate HL-LHC discovery threshold.