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arxiv: 2605.30413 · v1 · pith:SSZIFVW4new · submitted 2026-05-28 · ✦ hep-ph · hep-th

Quark-Lepton Color-Flavor Unification

Antonio Delgado , Seth Koren This is my paper

Pith reviewed 2026-06-29 06:21 UTC · model grok-4.3

classification ✦ hep-ph hep-th
keywords SU(12) unificationquark-lepton unificationcolor-flavor symmetrystrong CP problemproton stabilitytype-I seesawgauge instantonsdiscrete gauge symmetry
0
0 comments X

The pith

An SU(12) model unifies quark color-flavor with lepton flavor, generating bottom and tau masses via instantons from a shared up-neutrino Yukawa while a discrete gauge symmetry stabilizes the proton.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds a gauge theory with SU(12) times SU(2)_L times U(1)_R that merges SU(9) quark color-flavor and SU(3) lepton flavor into one structure. It begins with only one Yukawa coupling shared by up-type quarks and neutrinos and no extra fermions. Gauged instantons from the color-flavor and lepton flavor sectors then produce the remaining Yukawas for the bottom quark and tau lepton. The construction yields the Standard Model gauge group plus a discrete gauge symmetry X equal to B minus 3 times a combination of lepton numbers, which forbids proton decay. This also implements a massless up-quark solution to the strong CP problem and a flavored type-I seesaw for neutrinos.

Core claim

The central claim is that an SU(12) × SU(2)_L × U(1)_R theory unifies SU(9) quark color-flavor with SU(3) lepton flavor starting from a single Yukawa shared by up-type quarks and neutrinos; gauged instantons dynamically generate the bottom and tau Yukawas, only two new scalar representations are required for the breaking chain that includes color-flavor deconstruction followed by infrared reunification, and the low-energy gauge group emerges as G_SM equal to the quotient of SU(3)_C × SU(2)_L × U(1)_Y × Z^X_18 by Z_3 × Γ × Z_3 where the discrete gauge symmetry X = B − 3(L_i + L_j − L_k) absolutely stabilizes the proton.

What carries the argument

The SU(12) gauge group that unifies SU(9) quark color-flavor with SU(3) lepton flavor, together with instanton-generated Yukawa splittings and the discrete gauge symmetry X that supplies proton stability.

If this is right

  • Bottom and tau Yukawas arise dynamically from the single shared up-neutrino Yukawa via instantons.
  • A massless up quark solves the strong CP problem without an axion.
  • Neutrino masses are generated through a flavored type-I seesaw.
  • Only two new scalar representations are needed to complete the full symmetry-breaking chain to the Standard Model.
  • Non-invertible chiral symmetry breaking produces a spectrum of emergent generalized symmetries and topological defects.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The model’s emphasis on color-flavor deconstruction followed by reunification suggests similar patterns could be applied to other partial unification schemes.
  • The discrete symmetry X may constrain additional baryon-number-violating operators beyond those already forbidden by the proton-stability mechanism.
  • Future collider searches for the two new scalars could directly test the minimal breaking content required by the unification.
  • The shared quotient between continuous and discrete groups at low energies may link one-form and two-form global symmetries in ways testable through lattice studies of the Standard Model.

Load-bearing premise

The assumption that gauged quark color-flavor and lepton flavor instantons dynamically generate the bottom and tau Yukawas from a single shared Yukawa for up-type quarks and neutrinos with no further new fermions.

What would settle it

Observation of proton decay at any scale would falsify the absolute stability provided by the discrete gauge symmetry X.

Figures

Figures reproduced from arXiv: 2605.30413 by Antonio Delgado, Seth Koren.

Figure 1
Figure 1. Figure 1: FIG. 1. The breaking chain from our ultraviolet theory down [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. An [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. A quiver diagram depicting the non-Abelian parts [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Flavor violation communicated to the down-type [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The up-type yukawas receive gluon loop correc [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The Dirac neutrino yukawas receive leptonic gluon [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The couplings as a function of scale in the simple [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
read the original abstract

We present an $SU(12) \times SU(2)_L \times U(1)_R$ model unifying $SU(9)$ quark color-flavor with $SU(3)$ lepton flavor as a flavorful alternative to conventional theories of unification. We begin in the ultraviolet with a single yukawa shared by the unified up-type quarks and neutrinos, and no further new fermions. We show that gauged quark color-flavor and lepton flavor instantons dynamically generate the bottom and tau yukawas, which implements a massless quark solution to the strong CP problem and sets up a flavored type-I seesaw mechanism. Only two new scalar irreps are needed for the symmetry-breaking steps, which include quark color-flavor deconstruction and then infrared reunification, and the Standard Model gauge group in this theory emerges as \[G_{\rm SM} = \frac{SU(3)_C \times SU(2)_L \times U(1)_Y \times \mathbb{Z}^X_{18}}{\mathbb{Z}_{3} \times \Gamma \times \mathbb{Z}_3},\] where $\Gamma \in \lbrace 1, \mathbb{Z}_2 \rbrace$ is the electroweak global structure and there is a discrete gauge symmetry $X = B - 3(L_i + L_j - L_k)$ which brings additional $\mathbb{Z}_3$ global structure to the SM. This gauge symmetry acts as a flavorful upgrade of the $\mathbb{Z}^{B+L}_{18}$ anomaly-free global symmetry of the SM and stabilizes the proton absolutely. Non-invertible chiral symmetry-breaking is crucial to our model, and we discuss the rich spectrum of emergent generalized symmetries and topological defects appearing at various stages. In the infrared, the novel shared quotient between continuous and discrete groups links the one-form and two-form global symmetries of the Standard Model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper presents an SU(12) × SU(2)_L × U(1)_R model that unifies SU(9) quark color-flavor with SU(3) lepton flavor. It begins with a single UV Yukawa coupling shared by up-type quarks and neutrinos (no additional fermions), relies on gauged instantons in the color-flavor and lepton-flavor sectors to dynamically generate the bottom and tau Yukawas, implements a massless up-quark solution to strong CP, and sets up a flavored type-I seesaw. Symmetry breaking uses only two new scalar irreps and proceeds via quark color-flavor deconstruction followed by infrared reunification. The SM gauge group emerges as the quotient G_SM = (SU(3)_C × SU(2)_L × U(1)_Y × Z^X_18) / (Z_3 × Γ × Z_3) with discrete gauge symmetry X = B − 3(L_i + L_j − L_k) that stabilizes the proton absolutely; the construction also invokes non-invertible chiral symmetry breaking and emergent generalized symmetries.

Significance. If the central instanton mechanism is shown to produce operators of the correct flavor structure and magnitude, the construction would offer a minimal-fermion flavorful unification alternative with absolute proton stability, a strong-CP solution via a massless up quark, and a novel link between continuous/discrete quotients and one-/two-form global symmetries. The restriction to two new scalars and the explicit discrete gauge symmetry X are concrete strengths.

major comments (1)
  1. [Abstract] Abstract, paragraph 2: the claim that gauged SU(9) color-flavor and SU(3) lepton-flavor instantons dynamically generate the bottom and tau Yukawas from the single shared up/neutrino Yukawa (without further new fermions) is load-bearing for minimality, the strong-CP solution, and the subsequent seesaw; no explicit instanton-induced operators, suppression factors, or texture-matching calculation is supplied to confirm that the required down-type and charged-lepton structures arise at the correct scale while leaving the up quark massless.
minor comments (2)
  1. The quotient expression for G_SM is written with a mixture of continuous and discrete factors; the precise embedding of Z^X_18 and the action of Γ should be stated explicitly in a dedicated section on the infrared gauge group.
  2. Notation for the discrete symmetry X = B − 3(L_i + L_j − L_k) should include a brief table or paragraph clarifying its action on the three lepton generations to make the proton-stabilization argument self-contained.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting both the potential strengths of the construction and the need for further substantiation of the central instanton mechanism. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph 2: the claim that gauged SU(9) color-flavor and SU(3) lepton-flavor instantons dynamically generate the bottom and tau Yukawas from the single shared up/neutrino Yukawa (without further new fermions) is load-bearing for minimality, the strong-CP solution, and the subsequent seesaw; no explicit instanton-induced operators, suppression factors, or texture-matching calculation is supplied to confirm that the required down-type and charged-lepton structures arise at the correct scale while leaving the up quark massless.

    Authors: We agree that the explicit form of the instanton-induced operators, their suppression factors, and the resulting flavor textures are essential to validate the load-bearing claims. While the manuscript outlines the qualitative mechanism and its consequences for minimality and the strong-CP solution, it does not supply the detailed operator derivations or numerical matching. In the revised manuscript we will add a dedicated subsection that computes the leading instanton-generated effective operators in the SU(9) color-flavor and SU(3) lepton-flavor sectors, estimates the exponential suppression, and verifies that the resulting down-type and charged-lepton textures are consistent with observation while leaving the up quark massless. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper constructs an SU(12) × SU(2)_L × U(1)_R UV model starting from a single shared Yukawa for up-type quarks and neutrinos, then claims to demonstrate via gauged instanton effects that bottom and tau Yukawas are generated dynamically. This is presented as a non-perturbative derivation rather than a fit or self-definition. The emergence of G_SM with the discrete X symmetry and the two-scalar breaking chain are stated as consequences of the gauge structure and symmetry breaking steps. No equations or claims in the abstract reduce a prediction to a fitted input by construction, invoke self-citations as load-bearing uniqueness theorems, or rename known results. The instanton step is a dynamical claim whose validity is external to the construction itself. The derivation chain therefore remains independent of its target outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Review based on abstract only; full text required for complete ledger. The model postulates a new gauge structure and instanton effects without independent evidence supplied.

axioms (1)
  • domain assumption Gauged instantons from the new color-flavor and lepton flavor groups dynamically generate the bottom and tau Yukawas
    Invoked to produce the down-type and tau masses from the single UV Yukawa.
invented entities (1)
  • SU(12) gauge group with two new scalar irreps no independent evidence
    purpose: Unify quark color-flavor and lepton flavor and perform symmetry breaking
    New gauge structure and scalars introduced to realize the unification and breaking pattern.

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