Relations between the Gribov-horizon and center-vortex confinement scenarios with Jeff Greensite and Daniel Zwanziger Coulomb energy, vortices, and confinement, hep-lat/0302018 Coulomb energy, remnant symmetry, and the phases of non-Abelian gauge theories, hep-lat/0401003 Center vortices and the Gribov horizon, hep-lat/0407032 http://dcps.savba.sk/olejnik/seminars/villasimius04.pps
It was six men of Indostan To learning much inclined, Who went to see the Elephant (Though all of them were blind), That each by observation Might satisfy his mind […] And so these men of Indostan Disputed loud and long, Each in his own opinion Exceeding stiff and strong, Though each was partly in the right, And all were in the wrong! The Blind Men and the ElephantJohn Godfrey Saxe (1816-1887), American poet • Moral: • So oft in theologic wars, • The disputants, I ween, • Rail on in utter ignorance • Of what each other mean, • And prate about an Elephant • Not one of them has seen! • [Replace above theologic … physical?]
Outline • In this talk some connections between the center-vortex and Gribov-horizon confinement scenarios will be discussed. • I will have a look more closely on the distribution of near-zero modes of the F-P density inCoulomb gauge. I will show how the density looks like in full theory, with and without vortices. • Strong correlation between the presence of center vortices and the existence of a confining Coulomb potential. • Confining property of the color Coulomb potential is tied to the unbroken realization of the remnant gauge symmetry in CG. An order parameter for this symmetry will be introduced. • Closely related investigation in Landau gauge: • J. Gattnar, K. Langfeld, H. Reinhardt, hep-lat/0403011
Confinement scenario in Coulomb gauge • Hamiltonian of QCD in CG: • Faddeev—Popov operator:
Gribov region: set of transverse fields, for which the F-P operator is positive; local minima of I. Gribov horizon: boundary of the Gribov region. Fundamental modular region: absolute minima of I. GR and FMR are bounded and convex. Gribov horizon confinement scenario: the dimension of configuration space is large, most configurations are located close to the horizon. This enhances the energy at large separations and leads to confinement. Gribov ambiguity and Gribov copies
A confinement condition in terms of F-P eigenstates • Color Coulomb self-energy of a color charged state: • F-P operator in SU(2):
Necessary condition for divergence of e: • To zero-th order in the gauge coupling: • To ensure confinement, one needs some mechanism of enhancement of r(l) and F(l) at small l.
Center vortices in SU(2) lattice configurations • Center vortices are identified by fixing to an adjoint gauge, and then projecting link variables to the ZN subgroup of SU(N). The excitations of the projected theory are known as P-vortices. • J. Greensite, hep-lat/0301023 • M. Engelhardt, hep-lat/0409023 (Lattice 2004, plenary talk) • Direct maximal center gauge in SU(2): One fixes to the maximum of and center projects • Center dominance plus a lot of further evidence that center vortices alone reproduce much of confinement physics.
Three ensembles • Full Monte Carlo configurations: • “Vortex-only” configurations: • “Vortex-removed” configurations: • Vortex removal • removes the string tension, • eliminates chiral symmetry breaking, • sends topological charge to zero. • Philippe de Forcrand, Massimo D’Elia, hep-lat/9901020 • Each of the three ensembles will be brought to Coulomb gauge by maximizing, on each time-slice,
Full configurations • Technical details
Vortex-only configurations • Technical details
Technical details Vortex-removed configurations
Lessons • Full configurations: the eigenvalue density and F(l) at small l consistent with divergent Coulomb self-energy of a color charged state. • Vortex-only configurations: vortex content of configurations responsible for the enhancement of both the eigenvalue density and F(l) near zero. • Vortex-removed configurations: a small perturbation of the zero-field limit.
SU(2) gauge-fundamental Higgs theory • Osterwalder, Seiler ; Fradkin, Shenker, 1979; Lang, Rebbi, Virasoro, 1981 • K. Langfeld, this conference • Q for SU(2) with fundamental Higgs Vortex depercolation Vortex percolation
“Higgs-like” phase • Conclusions
Coulomb energy • Physical state in CG containing a static pair: • Correlator of two Wilson lines: • Then:
Measurement of the Coulomb energy on a lattice • Wilson-line correlator: • A. Nakamura, this conference, preliminary data for SU(3) • Questions: • Does V(R,0) rise linearly with R at large b? • Does scoul match sasympt?
scoul (2 – 3) sasymp • Overconfinement! Good news for model builders (gluon chain model). • Scaling of the Coulomb string tension?
Coulomb energy and remnant symmetry • Maximizing R does not fix the gauge completely: • Under these transformations: • Both L and Tr[L] are non-invariant, their expectation values must vanish in the unbroken symmetry regime. • The confining phase is therefore a phase of unbroken remnant gauge symmetry; i.e. unbroken remnant symmetry is a necessary condition for confinement.
An order parameter for remnant symmetry in CG • Define • Order parameter (Marinari et al., 1993): • Relation to the Coulomb energy:
Compact QED4 • SU(2) gauge-fundamental Higgs theory
A surprise: SU(2) in the deconfined phase • Does remnant and center symmetry breaking always go together? NO!
SU(2) in the deconfined phase: an explanation • Spacelike links are a confining ensemble even in the deconfinement phase: spacelike Wilson loops have an area law behaviour. ( cf. Quandt, this conf.) • Removing vortices removes the rise of the Coulomb potential.
Conclusions – Coulomb energy • Coulomb energy rises linearly with quark separation. • Coulomb energy overconfines, scoul ¼3s. Overconfinement is essential to the gluon chain scenario. • Center symmetry breaking (= 0) does not necessarily imply remnant symmetry breaking (coul=0). In particular: • coul > 0in the high-T deconfined phase. • coul > 0in the confinement-like phase of gauge-Higgs theory. • The transition to the Higgs phase in gauge–fundamental Higgs system is a remnant-symmetry breaking, vortex depercolation transition.
Conclusions – Numerical study of F-P eigenvalues • Support for the Gribov-horizon scenario: Low-lying eigenvalues of the F-P operator tend towards zero as the lattice volume increases; the density of eigenvalues and F(l) go as small power of l near zero, leading to infrared divergence of the energy of an unscreened color charge. • Firm connection between center-vortex and Gribov-horizon scenarios: The enhanced density of low-lying F-P eigenvalues can be attributed to the vortex component of lattice configurations. The eigenvalue density of the vortex-removed component can be interpreted as a small perturbation of the zero-field result, and is identical in form to the (non-confining) eigenvalue density of lattice configurations in the Higgs phase of a gauge-Higgs theory.
Some analytical results • Center configurations lie on the Gribov horizon: When a thin center vortex configuration is gauge transformed into minimal Coulomb gauge it is mapped onto a configuration that lies on the boundary of the Gribov region. Moreover its F-P operator has a non-trivial null space that is (N2-1)-dimensional. • (Restricted) Gribov region (and restricted FMR) is a convex manifold in lattice configuration space. • Thin vortices are located at conical or wedge singularities on the Gribov horizon. • The Coulomb gaugehas a special status; itis an attractive fixed-point of a more general gauge condition, interpolating between the Coulomb and Landau gauges. • hep-lat/0407032
Center configurations lie on the Gribov horizon • Assertion: When a center configuration is gauge-transformed to minimal Coulomb gauge it lies on the boundary ¶L of the fundamental modular region L. • Proof: Take a lattice configuration Zi(x) of elements of the center, ZN. It is invariant under global gauge transformations: Now take h(x) to be the gauge transformation that brings the center configuration into the minimal Coulomb gauge: The transformed configuration Vi(x) is still invariant:
Now g’(x) can be parametrized through N2-1 linearly independent elements wn(x) of the Lie algebra of SU(N), and Vi(x) through Ai(x), then A lies at a point where the boundaries of the Gribov region and FMR touch. F-P operator of a center configuration has a non-trivial null space that is (N2-1)-dimensional. • Similar argument applies to abelian configurations. The F-P operator of an abelian configuration gauge-transformed into minimal Coulomb gauge has only an R-dimensional null space, with R being the rank of the group.
Convexity of FMR and GR in SU(2) lattice gauge theory • If A1 and A2 are configurations in L (or W), then so is A=aA1+bA2, where 0<a<1, and b=1-a. • M. Semenov—Tyan-Shanskii, V. Franke, 1982 • A slightly weaker statement holds in SU(2) LGT. We parametrize SU(2) configurations by Take the northern hemisphere only: One can quite easily prove the convexity of