Emergent Gribov horizon from replica symmetry breaking in Yang--Mills theories

We show that the Serreau–Tissier (ST) replica sector can dynamically generate a Gribov–Zwanziger (GZ)–type horizon functional in Yang–Mills (YM) theories. After integrating out the replica superfields, the expansion of the determinant of the Faddeev–Popov (FP) operator in the regulator $ζ$ produces, at linear order in $ζ$, a nonlocal kernel with the same color and Lorentz structure as the Gribov horizon functional, thereby defining an induced Gribov scale. Depending on the replica phase selected by the dynamics, the ST sector yields either (i) a local Curci–Ferrari (CF) screening mass (replica-symmetric phase) or (ii) an induced horizon-like interaction (replica-broken phase). In the latter case, the resulting BRST-invariant local formulation leads to a tree-level gluon propagator of the refined Gribov-Zwanziger (RGZ) decoupling type, whereas in the former it reduces to the massive FP/CF form, avoiding double counting of infrared scales by construction. A superspace derivation confirms that the induced horizon term originates from the ST superdeterminant, providing a microscopic mechanism for the emergence of the Gribov scale within the replica framework.

💡 Research Summary

The paper investigates the dynamical origin of the Gribov horizon within Yang‑Mills theory by exploiting the replica sector introduced by Serreau and Tissier (ST). In the ST construction, the gauge‑fixing weight for each Gribov copy is proportional to det(F + ζ 1)/|det F| · exp(−β f), where F is the Faddeev‑Popov (FP) operator, β and ζ are mass‑dimension parameters, and f is the Landau functional. By applying the replica trick (originally used in disordered systems) the non‑linear sigma superfields V_k are localized, yielding a perturbatively renormalizable local action.

Two distinct phases of the replica sector are identified. In the replica‑symmetric phase (χ̂ > 0) the integration over the sigma fields produces only a local mass term β; the linear term in the regulator ζ contributes solely to a local screening mass. Consequently the gluon two‑point function reduces to the massive Faddeev‑Popov (or Curci‑Ferrari) propagator D(p) ∝ (p² + β)⁻¹, and no non‑local horizon term appears.

In the replica‑broken phase (χ̂ = 0) the situation changes dramatically. After integrating out the sigma superfields, the effective action contains the term −ζ Tr M⁻¹(A_h), where M(A_h)=−∂·D(A_h) is the FP operator evaluated on the transverse, BRST‑invariant composite field A_h. Expanding the FP operator as M = M₀ + V