Mass Without Mass from a Berry--Shifted SU(3) Holonomy Rotor

We identify a local, gauge-invariant mechanism that generates a finite spectral scale in pure SU(3) Yang–Mills theory on a punctured three-ball. Fixing a $\mathbb{Z}_3$ center sector isolates a single gauge-invariant holonomy angle whose Berry shift produces a quantum rotor with strictly nonzero level spacing. Gauss law is enforced by a covariant Dirichlet Helmholtz projector built from the Dirichlet inverse of the covariant scalar Laplacian with relative boundary conditions. The slow holonomy mode is chosen variationally as the minimizer of transverse electric energy under the holonomy constraint, yielding an inertia \emph{independent of the gauge representative} with linear domain-size scaling and a controlled commutator-dominated regime. We prove projector stability and derive an adiabatic variational upper bound on the first positive Yang–Mills eigenvalue, with error controlled by the transverse vector gap of the covariant Laplacian on divergence-free one-forms. A femtometer-scale benchmark at realistic coupling gives an upper bound at a hadronic ($\sim 1,$GeV) scale. In Wilczek’s sense this realizes ``mass without mass’’: no explicit mass term or Higgs field is introduced, and the nonzero level spacing is fixed by gauge invariance, topology, and the chosen center sector. The present results are derived on a finite domain; interpreting the length $R$ in Minkowski space requires an additional physical input (e.g.\ as a local confinement length), which we make explicit.

💡 Research Summary

The paper presents a novel, fully gauge‑invariant mechanism that generates a finite spectral scale in pure SU(3) Yang–Mills theory without introducing any explicit mass term or Higgs field. The authors consider the theory on a three‑dimensional ball B_R from which a thin tubular neighbourhood of a smooth closed curve (a knot) Γ has been removed, defining a punctured domain D = B_R \ T​ub(Γ). The boundary of D consists of the outer sphere ∂B_R and an inner torus ∂T​ub(Γ). Topologically H₁(D;ℤ)≅ℤ, so there is a single non‑trivial homology class represented by a meridian loop γ that links Γ once.

By fixing a Z₃ center sector of the SU(3) gauge group, the authors isolate a single gauge‑invariant holonomy angle α associated with the Wilson line around γ. Because the center element e^{2πiJ} (with J a Cartan lattice vector) acts non‑trivially, the holonomy acquires a Berry phase (a “Berry shift”) of 2π/3 rather than the usual 2π periodicity. Consequently the slow collective coordinate α behaves as a quantum rotor on a circle with twisted boundary condition ψ(α+2π)=e^{2πiν/3}ψ(α), ν∈{0,1,2}.

Gauss’s law is enforced via a covariant Dirichlet Helmholtz projector Π(D)^T = 1 − D G D·, where D is the covariant derivative and G = (D·D)⁻¹ is the Dirichlet inverse of the covariant scalar Laplacian with relative (electric‑type) boundary conditions. Π(D)^T projects any one‑form onto its divergence‑free component and satisfies Π(D)^T = (Π(D)^T)², ‖Π(D)^T‖≤1, and an explicit distance formula that makes the projector stable under small changes of the background gauge field.

The slow holonomy mode is defined variationally: among all vector fields X that satisfy the holonomy‑variation constraint ∂_αU_γ = i(Ad V J)U_γ, the authors minimize the transverse electric energy E_E = ½ g⁻²‖Π(D)^T X‖². The constrained minimizer X_α exists and is unique modulo pure gauge directions (Im D). A convenient seed field X_seed = g⁻¹(Ad V J) ω, with ω a closed one‑form normalized by ∮_γ ω = 1, yields the same transverse component after projection: Π(D)^T X_α = Π(D)^T X_seed. Hence the effective inertia I_eff = g⁻⁴‖Π(D)^T X_seed‖² is independent of the gauge representative.

Scaling analysis shows that under the rescaling x = R y, the inertia grows linearly with the domain size, I_eff = R g⁻⁴ c₁, where c₁ is an O(1) dimensionless constant that remains bounded even in thin‑tube geometries (proved in Appendix F). In regimes where the commutator term