# Vacuum Structure and Confinement

Confinement in Coulomb gauge
Giuseppe Burgio, Markus Quandt, Hugo Reinhardt, Mario Schröck, Hannes Vogt
Mon, 14:00, Seminar Room E -- Parallels 1E (Slides)

We review our lattice results concerning the Gribov-Zwanziger confinement mechanism in Coulomb gauge. In particular, we verify the validity of Gribov's IR divergence condition for the Coulomb ghost form factor. We also show how the quark self energy is, like that of the transverse gluon, IR divergent, thus effectively extending the Gribov-Zwanziger scenario to full QCD.

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Coulomb gauge on the lattice: From zero to finite temperature
Hannes Vogt, Giuseppe Burgio, Markus Quandt, Hugo Reinhardt
Mon, 14:20, Seminar Room E -- Parallels 1E (Slides)

We extend our previous studies of Coulomb gauge Yang-Mills theory on the lattice by comparing the static gluon propagator at zero and finite temperature in SU(2), SU(3) and SU(4). In SU(2) we also compare the $$\langle U_0U_0\rangle$$-correlator with the $$\langle A_0A_0\rangle$$-correlator (temporal gluon propagator) and the Coulomb potential $$V_C$$ at zero temperature. To compute observables at higher temperatures we introduce anisotropic lattices, which allow lower momenta at a higher temporal resolution. On these lattices the static gluon propagator, the ghost propagator and the Coulomb potential are studied at temperatures up to $$6 T_c$$. We also investigate the effect of Gribov copies on the Coulomb potential.

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On two- and three-point functions of Landau gauge Yang-Mills theory
Markus Huber, Lorenz von Smekal
Mon, 14:40, Seminar Room E -- Parallels 1E (Slides)

Green functions of Yang-Mills theory are useful quantities whose applications range from bound state calculations to investigations of the phase diagram of quantum chromodynamics. I will present results of non-perturbative continuum calculations of the propagators and the ghost-gluon vertex which are in good, even quantitative agreement with corresponding lattice results. The mid-momentum behavior, which is affected most by truncations, can effectively be controlled via the employed three-gluon vertex model which is inspired by lattice results.

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Spectral densities from the lattice
Paulo Silva, David Dudal, Orlando Oliveira
Mon, 15:00, Seminar Room E -- Parallels 1E (Slides)

We discuss a method to extract the Kallen-Lehmann spectral density of a particle (be it elementary or bound state) propagator by means of 4d lattice data. We employ a linear regularization strategy, commonly known as the Tikhonov method with Morozov discrepancy principle. An important virtue over the popular maximum entropy method is the possibility to also probe unphysical spectral densities, as, for example, of a confined gluon. We apply our proposal to the SU(3) glue sector.

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Crossing the Gribov horizon: an unconventional study of geometric properties of gauge-configuration space in Landau gauge
Attilio Cucchieri, Tereza Mendes
Mon, 15:20, Seminar Room E -- Parallels 1E (Slides)

We present a lower bound for the smallest nonzero eigenvalue of the Landau-gauge Faddeev-Popov matrix in terms of the smallest nonzero lattice momentum and of a parameter characterizing the geometry of the first Gribov region. This allows a simple and intuitive description of the infinite-volume limit in the ghost sector. In particular, we show how nonperturbative effects may be quantified by the rate at which typical thermalized and gauge-fixed configurations approach the Gribov horizon. Our analytic results are verified numerically through an informal, "free and easy" approach.

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Gluon mass at finite temperature in Landau gauge
Pedro Bicudo, Nuno Cardoso, Orlando Oliveira, Paulo Silva
Mon, 15:40, Seminar Room E -- Parallels 1E (Slides)

Using lattice results for the Landau gauge gluon propagator at finite temperature, we investigate its interpretation as a massive type bosonic propagator. In particular, we estimate a gluon mass from Yukawa-like fits to the lattice data and study its temperature dependence.

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$$q \bar q$$-potential
Giancarlo Rossi, Massimo Testa
Mon, 16:30, Seminar Room E -- Parallels 2E (Slides)

We show how to define and compute in a non-perturbative way the potential between $$q$$ and $$\bar q$$ colour sources in the singlet and octet (adjoint) representation of the colour group.

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The colour adjoint static potential from Wilson loops with generator insertions and its physical interpretation
Marc Wagner, Owe Philipsen
Mon, 16:50, Seminar Room E -- Parallels 2E (Slides)

We discuss the non-perturbative computation and interpretation of a colour adjoint static potential based on Wilson loops with generator insertions. Numerical lattice results for SU(2) gauge theory are presented and compared to corresponding perturbative results in various gauges.

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The static quark self-energy and the Plaquette at large orders in perturbation theory
Antonio Pineda, Gunnar Bali, Clemens Bauer
Mon, 17:10, Seminar Room E -- Parallels 2E (Slides)

We compute the coefficients of the perturbative expansions of the plaquette, and of the self-energy of static sources in the triplet and octet representation, up to very high orders in perturbation theory. We use numerical sthocastic perturbation theory and lattice regularization. We explore if the results obtained comply with expectations from renormalon dominance, and what they may say for a model independent and nonperturbative determination of the value of the gluon condensate.

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Recent progress in the effective string theory description of LGTs.
Michele Caselle, Marco Billò, Ferdinando Gliozzi, Marco Meineri, Davide Fioravanti, Roberto Pellegrini, Roberto Tateo
Mon, 17:30, Seminar Room E -- Parallels 2E (Slides)

In presence of a static pair of sources, the spectrum of low-lying states of whatever confining gauge theory in D space-time dimensions is described, at large source separations, by an effective string theory. Recently two important advances improved our understanding of this effective theory. First, it was realized that the form of the effective action is strongly constrained by the requirement of the Lorentz invariance of the gauge theory, which is spontaneously broken by the formation of a long confining flux tube in the vacuum. This constraint is strong enough to fix uniquely the first few subleading terms of the action. Second, it has been realized that the first of these allowed terms - a quartic polynomial in the field derivatives - is exactly the composite field $$T\bar{T}$$, built with the chiral components, $$T$$ and $$\bar{T}$$, of the energy-momentum tensor of the 2d QFT describing the infrared limit of the effective string. This irrelevant perturbation is quantum integrable and yields, through the Thermodynamic Bethe Ansatz (TBA),the energy levels of the string which exactly coincide with the Nambu-Goto spectrum. In this talk we first review the general implications of these two results and then, as a test of the power of these methods, use them to construct the first few boundary corrections to the effective string action.

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Fine structure of the confining string in an analytically solvable 3D model
Davide Vadacchino, Michele Caselle, Marco Panero, Roberto Pellegrini
Mon, 17:50, Seminar Room E -- Parallels 2E (Slides)

In the U(1) lattice gauge theory in three spacetime dimensions, confinement can be analytically shown to persist at all values of the coupling. Furthermore, the explicit predictions for the dependence of string tension and mass gap on the coupling allow one to tune their ratio at will. These features, and the possibility of obtaining high-precision numerical results via an exact duality map to a spin model, make this theory an ideal framework to test the effective string description of confining flux tubes. In this contribution, we discuss our investigation of boundary- and next-to-leading-order corrections to the confining potential, and of the finite-temperature behavior of the flux tube width. Our data represent a very stringent test of the theoretical predictions for these quantities.

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Analytical relation between the Polyakov loop and the Dirac eigenvalues in temporally odd-number lattice QCD
Hideo Suganuma, Takahiro Doi, Takumi Iritani
Mon, 18:10, Seminar Room E -- Parallels 2E (Slides)

We analytically derive an identity between the Polyakov loop and Dirac eigenvalues in temporally odd-number lattice QCD. For the temporal lattice with an odd-number N, the Polyakov loop $$L_P$$ is expressed with the Dirac eigenvalues $$\lambda_n$$: $$L_P={\rm const} \sum_n \lambda_n^{N-1} (n|U_4|n)$$. Thus, for the Polyakov loop, the contribution from the low-lying Dirac eigenvalues is found to be negligibly small in this sum. Since the low-lying Dirac modes are essential for chiral symmetry breaking, this identity indicates no one-to-one correspondence between confinement and chiral symmetry breaking, as was numerically shown in a different manner [1][2]. [1] S. Gongyo, T. Iritani and H. Suganuma, Phys. Rev. D86 (2012) 034510. [2] T. Iritani and H. Suganuma, arXiv:1305.4049[hep-lat].

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A direct relation between confinement and chiral symmetry breaking in temporally odd-number lattice QCD
Takahiro Doi, Hideo Suganuma, Takumi Iritani
Mon, 18:30, Seminar Room E -- Parallels 2E (Slides)

We derive an identity connecting Polyakov loop and Dirac modes in temporally odd-number lattice, where the temporal length is odd in lattice unit. The Polyakov loop is an order parameter for quark confinement. On the other hand, according to Banks-Casher relation, low-lying Dirac modes are important for chiral symmetry breaking. Thus, the above identity describes the relation between confinement and chiral symmetry breaking. From this identity, we conclude that there is no one-to-one correspondence between confinement and chiral symmetry breaking. We have numerically confirmed this identity. Moreover, modifying Kogut-Susskind formalism for even lattice, we develop the method for spin-diagonalizing Dirac operator in the temporally odd-number lattice.

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Analysis of topological structure of the QCD vacuum with overlap-Dirac operator eigenmode
Takumi Iritani, Shoji Hashimoto, Guido Cossu
Mon, 18:50, Seminar Room E -- Parallels 2E (Slides)

We investigate topological structure of the QCD vacuum using eigenmodes of the overlap-Dirac operator, which keeps exact chiral symmetry on lattice. Based on a Dirac eigenmode expression of field-strength tensor $$F_{\mu\nu}$$, we analyse instanton-like behavior of the QCD vacuum, topological charge distribution and topological susceptibility. We also analyse contributions of the overlap-Dirac eigenmode to color-flux tube, which leads to confining force between quark and anti-quark.

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Non-Abelian dual Meissner effect and confinement/deconfinement phase transition in SU(3) Yang-Mills theory
Akihiro Shibata, Kei-Ichi Kondo, Seikou Kato, Toru Shinohara
Tue, 14:00, Seminar Room E -- Parallels 3E (Slides)

The dual superconductivity is a promising mechanism for quark confinement. We presented the non-Abelian dual superconductivity picture for SU(3) Yang-Mills theory, which demonstrated the restricted field dominance, called conventionally Abelian'' dominance and non-Abelian monopole dominance in the string tension. In the last conference, we have demonstrated by measuring the chromoelectric flux that the non-Abelian dual Meissner effect exists and determined that the type of vacuum for SU(3) case is type I, which is in sharp contrast to the SU(2) case: the border of type I and type II. In this talk, we focus on the confinement/deconfinemen phase transition and the non-Abelian dual superconductivity at finte temperature: We measure the hromoelectric flux between a pair of static quark and antiquark at finte temperature, and investigate its relevance to the phase transition and the non-Abelian dual Meissner effect.

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Study of thermal monopoles in lattice QCD
Vitaly Bornyakov
Tue, 14:20, Seminar Room E -- Parallels 3E (Slides)

We present results of the ongoing study of the properties of the thermal Abelian color-magnetic monopoles in the maximally Abelian gauge in the deconfinement phase of the lattice SU(3) gluodynamics and $$N_f=2$$ lattice QCD. We compute density, correlators, interaction parameters of the thermal Abelian monopoles in these two theories.

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Surface operator study in an SU(2) gauge field theory
Tue, 14:40, Seminar Room E -- Parallels 3E (Slides)

The most important probes for the phase states of a four-dimensional gauge field theory are the Wilson and t'Hooft line operatros that are difined on one-dimensional curves in the space-time. For example, these line-operators define order parameters for the confinemnet-deconfiment phase transition of the QCD vacuum. However, for more detail understanding of four-dimensional gauge field theory dynamics and vacuum topology we need additional probes expressed by operators defined on the subspaces with higher dimensions. Possible candidates are operators that are defined on the two-dimensional surface in the four-dimensional space-time. In the present work the surface operator in an SU(2) non-Abelian gauge field theory is studied. We analyze abelian projection of the SU(2) symmetry to the U(1) group calculating the Witten parameter using multilevel and multi-hit algorithms for the sake of statistical confidence. The Witten parameter dependence on the surface area and volume studied in confinement and deconfinement phases. It is shown that at the deconfinement phase the spatial surface operator exhibits nontrivial area dependence. In the confinement phase the operator is trivial with no area and volume dependence. It is shown also that the temporal surface operator exhibits the same phase behavior.

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Effective theta term by CP-odd electromagnetic background fields.
Marco Mariti, Claudio Bonati, Guido Cossu, Massimo D'Elia, Francesco Negro
Tue, 15:00, Seminar Room E -- Parallels 3E (Slides)

We present our study of QCD in the presence of CP-odd electromagnetic (em) background fields. We investigate the propagation of the CP-odd term from the em sector to the strong sector, inducing an effective theta term. We discuss the method we have used in our lattice QCD simulations, and the results of our analysis, which are relevant to the determination of the effective pseudoscalar QED-QCD interactions. We also explore the distribution of the fermion zero modes and of the topological charge in presence of external em fields.

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Fractional Charge and Confinement of Quarks
Philipp Scior, Sam R. Edwards, Lorenz von Smekal
Tue, 15:20, Seminar Room E -- Parallels 3E (Slides)

In quantum chromodynamics with static quarks the confinement-deconfinement phase transition is connected to the spontaneous breaking of a global $$Z_3$$ center symmetry. This symmetry is lost when one considers dynamical quarks. Owing to the fractional electric charge of quarks, we recover a global $$Z_6$$ center symmetry when QCD is regarded as a part of the Standard Model. We will present results from QCD-like theories extended by electromagnetic interactions and show that the weak coupling limit of the QED part of the model results in a center-like symmetry with disorder in the vacuum. This can be seen explicitly in a character expansion of the fermion determinant. Further, we will show that corresponding center averages project the fermion determinant on N-ality zero. We will also discuss whether the additional center symmetry can be used to eliminate the fermion sign problem in QCD with fundamental quarks.

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Vortex liquid in superconducting vacuum of QCD induced by strong magnetic field.
Andrey Kotov, Victor Braguta, Pavel Buividovich, Maxim Chernodub, Mikhail Polikarpov
Tue, 15:40, Seminar Room E -- Parallels 3E (Slides)

In the background of the strong magnetic field the vacuum is suggested to possess an electro- magnetically superconducting phase characterised by the charged $$\rho$$ mesons condensate. The $$\rho$$- meson condensates are inhomogeneous due to the presence of the stringlike defects which are parallel to the magnetic field. In agreement with these expectations, we have observed the presence of the $$\rho$$ vortices in numerical simulations of the vacuum of the quenched two-color lattice QCD in strong magnetic field background. We have found that in the quenched QCD the $$\rho$$ vortices form a liquid.

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Correlation functions and confinement in scalar QCD
Tajdar Mufti, Axel Maas
Poster Session

One of the choices to understand a theory, and hence the physics involved, is understanding the correlation functions. QCD in confinement regime serves as one of the ideal areas for non-perturbative calculations since they are not accessible in perturbative sense. A number of scenarios have been proposed about QCD confinement, thus necessitating non-perturbative calculations to be performed in order to check which one(s) of them is(are) acceptable. Since the physics of QCD confinement does not depend upon the number of colors and whether the matter fields are scalars or not, cheaper computation can be carried out using lattice methods with scalar matter fields and 2 colors in order to be able to address the problem mentioned above. Propagators and interaction vertices of matter and gauge fields, calculated in Landau gauge, are presented.

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Measuring the ground-state wave functional of SU(2) Yang-Mills theory in 3+1 dimensions: Abelian plane waves
Stefan Olejnik, Jeff Greensite
Poster Session

A method of measuring relative probabilities of various gauge-field configurations in the Yang-Mills vacuum was proposed long ago [Greensite and Iwasaki, Phys. Lett. B 223 (1989) 207]. We apply this method to compute the square of the YM vacuum wave functional (VWF) in numerical simulations of SU(2) lattice gauge theory in 3+1 dimensions for sets of Abelian plane waves. The results are compared to predictions based on various VWF proposals in the literature. None of them describes the data satisfactorily at large plane-wave momenta.

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Flux tubes and coherence length in the SU(3) vacuum
Francesca Cuteri, Paolo Cea, Leonardo Cosmai, Alessandro Papa
Poster Session

An estimate of the London penetration and coherence lengths in the vacuum of the SU(3) pure gauge theory is given downstream an analysis of the transverse profile of the chromoelectric flux tubes. Within ordinary superconductivity, a simple variational model for the magnitude of the normalized order parameter of an isolated vortex produces an analytic expression for magnetic field and supercurrent density. In the picture of SU(3) vacuum as dual superconductor, this expression provides us with the function that fits the chromoelectric field data. The smearing procedure is used in order to reduce noise.

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The static potential from the selected intermediate states of gluons
Yoshiaki Koma, Miho Koma
Poster Session

By employing the multilevel algorithm, we investigate the static inter-quark potential from the Polyakov loop correlation function (PLCF) constructed from the selected intermediate states of gluons. While the use of partial intermediate states cannot guarantee gauge invariance of the PLCF, we find that the functional forms of the PLCF and then the static potential are unchanged from the gauge invariant ones if certain intermediate states are selected. We discuss its implication and possible applications.

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Classification of quark-antiquark sources in Yang-Mills Theories
Andrea Guerrieri, Silvano Petrarca, Argia Rubeo, Massimo Testa
Poster Session

We present preliminary results of a lattice numerical investigation of the structure of physical states contributing to the sector of a pure Yang-Mills Theory, in the presence of infinitely heavy quark-antiquark sources, based on G.C.Rossi, M.Testa Phys.Rev.D87, 085014 (2013).

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Confinement From The Gauge Invariant Abelian Decomposition
Nigel Cundy, Yongmin Cho, Weonjong Lee
Poster Session

A common approach while considering confinement is to study the dominance of an Abelian subgroup of the SU(3) gauge Links. A good way to find the Abelian component of the field is through the Cho-Guan-De gauge invariant Abelian Decomposition, which uses a carefully chosen direction vector $$n$$ to split the gauge field into an Abelian restricted field and a remnant coloured field. The restricted field can be further subdivided into topological and non-topological terms. We show that there is a choice of $$n$$ which allows us to exactly represent the Wilson Loop of full QCD as a function of only the restricted Abelian field without requiring any path ordering or additional path integrals. We present numerical evidence showing that the topological part of the restricted field dominates the string tension. We also show that $$n$$ contains certain topological objects, which, if they exist, will be at least partially responsible for confinement. These leave distinctive patterns in the restricted field strength, and we search for these structures in quenched lattice QCD.

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Towards the continuum limit of SU(2) Landau gauge gluodynamics
Igor Bogolubsky, Ernst-Michael Ilgenfritz, Andre Sternbeck
Poster Session

We give an update on our long-term project to provide properly continuum-limit extrapolated data for the Landau-gauge gluon and ghost propagators of SU(2) lattice gauge theory. In our simulations we keep the physical volume fixed at (9.6 fm)$$^4$$ and vary the lattice coupling parameter such that lattice spacings from 0.17 down to 0.09 fermi are reached. We will discuss the difficulties one encounters with the renormalization caused by lattice discretization artefacts.

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't Hooft loop and the phases of SU(2) lattice gauge theory
Giuseppe Burgio
Poster Session

We analyze the vacuum structure of SU(2) lattice gauge theories in different dimensions, concentrating on the stability of 't Hooft loops. We recognize three, well separated phases as the couplings vary; high precision calculations to identify their boundaries have been performed in d=2+1, but similar results hold also for d=3+1 and d=1+1. We discuss the impact of our findings on the continuum limit of Yang-Mills theories.

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Stable and Quasi-Stable confining SU(N) strings in D=2+1
Andreas Athenodorou, Michael Teper
Poster Session

We investigate the low-lying spectrum of closed confining flux tubes that wind around a spatial torus in D=2+1 and carry flux in different representations of SU(N). We focus on our most recent calculations on N=6 and beta = 171, where the calculated low-energy physics is very close to the continuum and large-N limit. We investigate the adjoint, 84, 120, k=2A, 2S and k=3A, 3M, 3S representations and show that the corresponding flux tubes do exist. Similarly to the results for the fundamental representation, the ground state of a flux tube with momentum along its axis appears to be well described by Nambu-Goto all the way down to very short flux tubes. In contrast, excited states have much larger deviations from Nambu-Goto. We discuss whether these states are non-string-like associated to excitations of massive flux-tube modes.

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SU(3) quark-antiquark QCD flux tube
Pedro Bicudo, Nuno Cardoso, Marco Cardoso
Poster Session

We compute the quark-antiquark flux tube for pure gauge SU(3) in space-time 3 + 1 dimentions. To increase the signal over noise ratio, we apply the improved multihit and extended smearing techniques. We fit the field densities with an appropriate ansatz and we observe both the screening of the colour fields and the quantum widening of the flux tube in the mediator plane.

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Form factor and width of a quantum string
David Weir, Kari Rummukainen, Arttu Rajantie
Poster Session

We show how the form factor for a quantum string can be obtained from field correlation functions calculated in lattice Monte Carlo simulations. As an example, we apply this technique for simulations of the Ising model. We demonstrate that the form factor shows the same logarithmic broadening as observed by other quantities. Various difficulties in finding the intrinsic width of a string are discussed.

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