1. Gribov horizon: a pathway to confinement
Silvio Paolo Sorella, UERJ -- Univ. Rio de Janeiro
D. Dudal
N. Vandersickel
H. Verschelde
L. Baulieu
D. Zwanziger
J. Gracey
UERJ – collab.
Gomez
M. Capri
M. Huber
M. Guimaraes
V. Lemes
R. Sobreiro
Faddeev-Popov action and BRST symmetry
Gribov horizon: the Gribov-Zwanziger action and its soft BRST breaking
Gluons get confined by the horizon
A possible path to construct composite meaningful operators: introducing i-particles
Conclusion

2. Faddeev-Popov action and BRST symmetry

3. The Faddeev-Popov action enjoys exact BRST invariance:
BRST invariance is at the origin of the Slavnov-Taylor identities, which enable us to prove the renormalizability
The BRST charge allows us to define the sub-space of the physical states and to establish the unitarity of the S matrix
The physical operators of the theory are identified by the cohomology classes of the nilpotent BRST operator s

4. Example: the scalar glueball operator
Observation: BRST symmetry is an exact unbroken invariance of the Faddeev-Popov action, s I0> = 0. The BRST exact term sΛ(x) is irrelevant in the evaluation of the correlation functions, namely
Moreover

5. Gribov horizon: the Gribov-Zwanziger action and its soft BRST breaking
The Landau gauge is plagued by the existence of the Gribov copies
There still exist equivalent configurations obeying the Landau condition

6. The existence of Gribov copies implies, for example, that the Faddeev-Popov operator has zero modes, i.e.
In order to get rid of the Gribov copies, the domain of integration in the Feynman path integral has to be restricted to a smaller region, called the Gribov region

8. Properties of the Gribov region in
the Landau gauge
Ω = { ∂A=0 ; - ∂2 + A∂ > 0 }
The origin { A= 0 } in field space belongs to Ω
Ω is bounded in every direction in field space
Ω is convex
Every gauge orbit passes through Ω
Warning: Ω is still plagued by Gribov copies

9. Thus, for the partition function in the Landau gauge, we have
D, Zwanziger, Nucl. Phys. B323 (1989) 513

10. SH is called the Zwanziger horizon term and is given by
In spite of its apparent nonlocality, Zwanziger’s horizon function can be cast in local form by means of the introduction of additional fields.
The parameter Ɣ is called the Gribov parameter. It has the dimension of a mass. It is not a free parameter, being determined in a self-consistent way by the gap equation:

11. Local formulation of Zwanziger’s action

12. Thus, we get the following local action

13. BRST soft breaking
After the localization of the horizon term, we end up with a quantized local action

14. Gluons get confined by the horizon
Despite the presence of the soft breaking term Δ, the resulting action enjoys renormalizability. This is due to the existence of a rich set of Ward identities which allow us to control the breaking Δ at the quantum level, while ensuring the renormalizability of the theory. Only two renormalization constants are needed, ZA, Zg. In particular
Gluons get confined by the horizon
It has complex poles
Its Fourier transformation shows positivity violation
It cannot be interpreted as describing physical excitations

15. From T. Mendes talk at “Confinement 8”,

16. Positivity violation of the gluon prop. , from P
Positivity violation of the gluon prop., from P. Bowman et al, hep-lat/

17. A possible way to construct the physical operators: introducing i-particles
Due to the presence of the soft breaking, it follows that the vev of BRST exact quantities is not necessarily vanishing
Dudal et al. PRD 78:065047,2008
How one can construct the physical observables in the presence of the Gribov horizon, and thus of the BRST breaking?

18. Gphys displays a physical cut at
An interesting observation has been done by D. Zwanziger, Nucl. Phys. B323 (1989) 513
Gphys displays a physical cut at
p2 =-2Ɣ2
Gunphys displays unphysical cuts at p2= ±4i Ɣ2
How to get rid of the unphysical cuts present in Gunphys ?
We could argue that, due to the presence of the horizon, the construction of the physical operators gets deformed by the horizon.

19. < Ophys(x) Ophys(y) > has only phys. cuts
RƔ stands for the deformation coming from the horizon. The term RƔ should account for the unphysical cuts, so that
< Ophys(x) Ophys(y) > has only phys. cuts
How one can find RƔ ?
Observe that a Gribov propagator describes two unphysical modes of complex imaginary mass, which we call i-particles
Recently, we have been able to construct toy models of i-particles which allow for the introduction of composite operators exhibiting only real cuts and displaying a nice spectral representation with positive spectral functions.

20. A toy model for i-particles
L. Baulieu, D.Dudal,
M.S. Guimaraes, M. Huber,
S.P. Sorella, N.Vandersickel, D. Zwanziger
In preparation

21. Conclusion
The issue of the Gribov copies leads to a modification of the Faddeev-Popov formula
Gluons turn out to be confined by the Gribov horizon. The gluon propagator is suppressed in the infrared, while displaying positivity violation.
The BRST invariance of the Faddeev-Popov action turns out to be softly broken. The understanding of the physical consequences of this breaking represents a big challenge
A Gribov type propagator is associated with the propagation of i-particles, i.e. a pair of modes with complex conjugate imaginary masses. The study of i-particles looks very interesting. Hopefully, it might shed some light on the unitarity issue and on the construction physical operators in the continuum, in the presence of the Gribov horizon.

22. Thank you for
your attention