- Open Access
Lorentzian Gribov no-pole condition for Yang-Mills theory
Phys. Rev. D 114, 045024 – Published 24 August, 2026
DOI: https://doi.org/10.1103/8jz8-v392
Abstract
For over four decades, Gribov’s no-pole condition has been almost exclusively explored in Euclidean space, where the elliptic nature of the Faddeev-Popov operator provides a clear spectral boundary. In physical Minkowski spacetime, however, this operator becomes a hyperbolic wave operator, and the Euclidean positivity criterion collapses. We show that the natural Lorentzian replacement is a real-time boundary-value problem: a gauge configuration remains inside the first Gribov region as long as the Faddeev-Popov wave equation admits no source-free solutions obeying the Feynman boundary condition. For backgrounds localized in time, this condition translates to the injectivity of the negative-frequency block of the classical ghost scattering map. For stationary backgrounds, it becomes a spatial bound-state or threshold-resonance problem under Fourier transformation. Using a Wronskian identity, we prove that pure frequency mixing in stable self-adjoint time-dependent channels is structurally protected and cannot by itself produce the obstruction. While static chromomagnetic backgrounds reproduce the familiar zero-frequency horizon crossing, static chromoelectric potentials reach the horizon at finite, nonvanishing frequency—a uniquely Lorentzian phenomenon arising because couples directly to the ghost time derivative. We also cast the condition in Fredholm form and show that the exact restriction is a functional determinant, which the naive local continuation of the Gribov-Zwanziger action fails to reproduce, leaving the construction of a genuine local real-time action as the central open problem.
Physics Subject Headings (PhySH)
Article Text
References (28)
- L. D. Faddeev and V. N. Popov, Feynman diagrams for the Yang–Mills field, Phys. Lett. B 25, 29 (1967).
- I. M. Singer, Some remarks on the Gribov ambiguity, Commun. Math. Phys. 60, 7 (1978).
- V. N. Gribov, Quantization of non-Abelian gauge theories, Nucl. Phys. B139, 1 (1978).
- A. J. Gomez, M. S. Guimaraes, R. F. Sobreiro, and S. P. Sorella, Equivalence between Zwanziger’s horizon function and Gribov’s no-pole ghost form factor, Phys. Lett. B 683, 217 (2010).
- M. A. L. Capri, D. Dudal, M. S. Guimaraes, L. F. Palhares, and S. P. Sorella, An all-order proof of the equivalence between Gribov’s no-pole and Zwanziger’s horizon conditions, Phys. Lett. B 719, 448 (2013).
- D. Zwanziger, Action from the Gribov horizon, Nucl. Phys. B321, 591 (1989).
- D. Zwanziger, Local and renormalizable action from the Gribov horizon, Nucl. Phys. B323, 513 (1989).
- D. Zwanziger, Renormalization in the Coulomb gauge and order parameter for confinement in QCD, Nucl. Phys. B518, 237 (1998).
- C. Feuchter and H. Reinhardt, Variational solution of the Yang–Mills Schrödinger equation in Coulomb gauge, Phys. Rev. D 70, 105021 (2004).
- A. A. Slavnov, A Lorentz invariant formulation of the Yang-Mills theory with gauge invariant ghost field Lagrangian, J. High Energy Phys. 08 (2008) 047.
- L. Baulieu, D. Dudal, M. S. Guimaraes, M. Q. Huber, S. P. Sorella, N. Vandersickel, and D. Zwanziger, Gribov horizon and i-particles: About a toy model and the construction of physical operators, Phys. Rev. D 82, 025021 (2010).
- N. Vandersickel and D. Zwanziger, The Gribov problem and QCD dynamics, Phys. Rep. 520, 175 (2012).
- K.-I. Kondo, M. Watanabe, Y. Hayashi, R. Matsudo, and Y. Suda, Reflection positivity and complex analysis of the Yang–Mills theory from a viewpoint of gluon confinement, Eur. Phys. J. C 80, 84 (2020).
- M. Reed and B. Simon, Methods of Modern Mathematical Physics. III: Scattering Theory (Academic Press, New York, 1979).
- D. R. Yafaev, Mathematical Scattering Theory: General Theory, Translations of Mathematical Monographs Vol. 105 (American Mathematical Society, Providence, 1992).
- B. Simon, Notes on infinite determinants of Hilbert space operators, Adv. Math. 24, 244 (1977).
- E. S. Fradkin, D. M. Gitman, and S. M. Shvartsman, Quantum Electrodynamics with Unstable Vacuum, Springer Series in Nuclear and Particle Physics (Springer, Berlin, 1991).
- G. Pöschl and E. Teller, Bemerkungen zur Quantenmechanik des anharmonischen Oszillators, Z. Phys. 83, 143 (1933).
- F. Sauter, Über das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs, Z. Phys. 69, 742 (1931).
- N. Rosen and P. M. Morse, On the vibrations of polyatomic molecules, Phys. Rev. 42, 210 (1932).
- J. S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82, 664 (1951).
- G. Dell’Antonio and D. Zwanziger, Every gauge orbit passes inside the Gribov horizon, Commun. Math. Phys. 138, 291 (1991).
- P. van Baal, More (thoughts on) Gribov copies, Nucl. Phys. B369, 259 (1992).
- J. Greensite, The confinement problem in lattice gauge theory, Prog. Part. Nucl. Phys. 51, 1 (2003).
- G. ’t Hooft, Topology of the gauge condition and new confinement phases in non-Abelian gauge theories, Nucl. Phys. B190, 455 (1981).
- H. Reinhardt and W. Schleifenbaum, Hamiltonian approach to dimensional Yang–Mills theory in Coulomb gauge, Ann. Phys. (Amsterdam) 324, 735 (2009).
- J. E. Hetrick, Some non-perturbative aspects of gauge fixing in two dimensional Yang–Mills theory, Nucl. Phys. B, Proc. Suppl. 34, 805 (1994).
- D. Dudal, C. P. Felix, L. F. Palhares, F. Rondeau, and D. Vercauteren, The BRST-invariant vacuum state of the Gribov–Zwanziger theory, Eur. Phys. J. C 79, 731 (2019).
