Correlation decay for finite lattice gauge theories at weak coupling

We explore a weakly coupled $SU(N_{c})$ gauge theory, examining its fixed-point structure and the transition from infrared conformality to spontaneous symmetry breaking. Following a previous study, we couple the gauge field to $N_s$ scalars and $N_f$ fermions, take the large-$N_c$ limit with $N_s/N_c$ and $N_f/N_c$ fixed, and adjust these ratios to produce weakly coupled infrared fixed points while maintaining asymptotic freedom. We map out renormalization-group trajectories, and describe the transition to symmetry breaking as the ratio $N_s/N_c$ is increased. We examine the breaking by employing the one-loop effective potential. At tree level, a scalar-field vacuum expectation value (VEV) of a certain form is allowed, leading to masses for a subset of the scalar fields. Among the remaining massless scalars, one corresponds to a massless dilaton and the others are Nambu-Goldstone bosons absorbed by the gauge fields. The one-loop potential breaks the scale symmetry explicitly, determining the VEV and generating a dilaton mass suppressed relative to the other masses by a factor of the weak scalar coupling.

Our aim is to give a self-contained review of recent advances in the analytic description of the deconfinement transition and determination of the deconfinement temperature in lattice QCD at large N. We also include some new results, as for instance in the comparison of the analytic results with Monte Carlo simulations. We first review the general set-up of finite temperature lattice gauge theories, using asymmetric lattices, and develop a consistent perturbative expansion in the coupling βs of the spacelike plaquettes. We study in detail the effective models for the Polyakov loop obtained, in the zeroth order approximation in βs, both from the Wilson action (symmetric lattice) and from the heat kernel action (completely asymmetric lattice). The distinctive feature of the heat kernel model is its relation with two-dimensional QCD on a cylinder; the Wilson model, on the other hand, can be exactly reduced to a twisted one-plaquette model via a procedure of the Eguchi–Kawai type. In the weak coupling regime both models can be related to exactly solvable Kazakov–Migdal matrix models. The instability of the weak coupling solution is due in both cases to a condensation of instantons; in the heat kernel case, this is directly related to the Douglas–Kazakov transition of QCD2. A detailed analysis of these results provides rather accurate predictions of the deconfinement temperature. In spite of the zeroth order approximation they are in good agreement with the Monte Carlo simulations in 2 + 1 dimensions, while in 3 + 1 dimensions they only agree with the Monte Carlo results away from the continuum limit.

In this paper we discuss the black hole–string transition of the small Schwarzschild black hole of AdS 5×S5 using the AdS/CFT correspondence at finite temperature. The finite temperature gauge theory effective action, at weak and strong coupling, can be expressed entirely in terms of constant Polyakov lines which are SU(N) matrices. In showing this we have taken into account that there are no Nambu–Goldstone modes associated with the fact that the 10-dimensional black hole solution sits at a point in S5. We show that the phase of the gauge theory in which the eigenvalue spectrum has a gap corresponds to supergravity saddle points in the bulk theory. We identify the third order N=∞ phase transition with the black hole–string transition. This singularity can be resolved using a double scaling limit in the transition region where the large N expansion is organized in terms of powers of N-2/3. The N=∞ transition now becomes a smooth crossover in terms of a renormalized string coupling constant, reflecting the physics of large but finite N. Multiply wound Polyakov lines condense in the crossover region. We also discuss the implications of our results for the resolution of the singularity of the lorenztian section of the small Schwarzschild black hole.

Interpreting the way that the SU(3) bare lattice coupling runs with the lattice spacing is complicated by the fact that there is a smooth cross-over region in which the strong coupling expansion transforms into a weak-coupling one. For N > 4, however, there is a first order bulk transition that cleanly separates the strong and weak coupling regimes. We find that in this case the calculated string tension can be readily fitted throughout the weak coupling region by a standard 3-loop expression modified by lattice spacing corrections of the expected form. While our fits demand the presence of the latter, they do not constrain the perturbative coupling scheme enough to enable us to extract a usefully accurate value of a(beta) in units of Lambda_MSbar. To resolve this ambiguity we turn to SU(3) where we use the Schrodinger Functional coupling scheme to extract a value of r_0 x Lambda_SF as a benchmark. We then find that the Parisi mean-field improved coupling scheme closely reproduces this result. We also develop a comparison between different schemes that does not rely on the calculation of any physical quantity and which can therefore be applied much further into weak coupling. Again the Parisi scheme is favoured over the others that we compare. Using the mean-field scheme we have fitted the values of the string tension that have been calculated for N=2 to N=8, to obtain Lambda_MSbar/sqrt(sigma) = 0.503(2)(40) + 0.33(3)(3)/N^2 for N > 2, where the first error is statistical and the second is our estimate of the systematic error from all sources.

Generalized lattice gauge theory, spin foams and state sum invariants

Higher gauge theory is a higher order version of gauge theory that makes possible the definition of 2-dimensional holonomy along surfaces embedded in a manifold where a gauge 2-connection is present. In this paper, we study Hamiltonian models for discrete higher gauge theory on a lattice decomposition of a manifold. We show that a construction for higher lattice gauge theory is well-defined, including in particular a Hamiltonian for topological phases of matter in [Formula: see text] dimensions. Our construction builds upon the Kitaev quantum double model, replacing the finite gauge connection with a finite gauge 2-group 2-connection. Our Hamiltonian higher lattice gauge theory model is defined on spatial manifolds of arbitrary dimension presented by slightly combinatorialized CW-decompositions (2-lattice decompositions), whose 1-cells and 2-cells carry discrete 1-dimensional and 2-dimensional holonomy data. We prove that the ground-state degeneracy of Hamiltonian higher lattice gauge theory is a topological invariant of manifolds, coinciding with the number of homotopy classes of maps from the manifold to the classifying space of the underlying gauge 2-group. The operators of our Hamiltonian model are closely related to discrete 2-dimensional holonomy operators for discretized 2-connections on manifolds with a 2-lattice decomposition. We therefore address the definition of discrete 2-dimensional holonomy for surfaces embedded in 2-lattices. Several results concerning the well-definedness of discrete 2-dimensional holonomy, and its construction in a combinatorial and algebraic topological setting are presented.

A technique for analytical calculation of observables in lattice gauge theories

The action of SU( N) lattice gauge theory in terms of the heat kernel on the group manifold

Using weak coupling methods McLerran and Venugopalan~\\cite{LV1} expressed the\nparton distributions in large nuclei as correlation functions of a two\ndimensional Euclidean field theory. The theory has the dimensionful coupling\n$g^2 \\mu $, where $\\mu^2\\sim A^{1/3}$ is the valence quark color charge squared\nper unit area. We use a lattice regularization to investigate these correlation\nfunctions both analytically and numerically for the simplified case of $SU(2)$\ngauge theory. In weak coupling ($g^2\\mu L<< 5$), where $L$ is the transverse\nsize of the nucleus, the numerical results agree with the {\\it analytic}\nlattice weak coupling results. For $g^2\\mu L>> 5$, no solutions exist at\nO($a^4$) where $a$ is the lattice spacing. This suggests an ill-defined\ninfrared behavior for the two dimensional theory. A recent proposal of McLerran\net al.~\\cite{Kovner} for an {\\it analytic} solution of the classical problem is\ndiscussed briefly.\n

According to present day knowledge all fundamental interactions in nature are described by gauge theories. In the first part we summarized concepts and properties of continuum gauge theories which are needed in the remaining chapters of this book. These include local gauge transformations, gauge potential and field strength, covariant derivative, parallel transport and gauge invariant Lagrangians for Euclidean Higgs models. Following Wilson we discretize gauge theories on a space-time lattice by replacing the Lie algebra valued continuum gauge field by Lie group valued variables on the links. Thereby the functional integral becomes a finite-dimensional integral which can be estimated by stochastic simulation techniques. We study the weak and strong coupling limits of lattice Higgs models, calculate the mean field approximations and discuss the expected phase diagram at zero temperature. In the second part we prove Elitzur’s theorem, sketch the strong coupling expansion, introduce the string tension and show in detail how to extract glueball masses. Towards the end we comment on gauge theories at finite temperature and in particular the breaking of center symmetry in pure lattice gauge theories.KeywordsGauge TheoryPartition FunctionGauge GroupGauge TransformationWilson LoopThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A general field-theoretic formulation of the Anderson model for the localization of wave functions in a random potential is given in terms of $n$-component replicated fields in the limit $n\ensuremath{\rightarrow}0$, and is analyzed primarily for spatial dimension $d\ensuremath{\ge}4$. Lengths ${\ensuremath{\xi}}_{1}$ and ${\ensuremath{\xi}}_{2}$ associated with the spatial decay of correlations in the single-particle and two-particle Green's functions, respectively, are introduced. Two different regimes, the weak coupling and strong coupling, are distinguished depending on whether ${\ensuremath{\xi}}_{1}^{\ensuremath{-}1}$ or ${\ensuremath{\xi}}_{2}^{\ensuremath{-}1}$, respectively, vanishes as the mobility energy, ${E}_{c}$, is approached. The weak-coupling regime vanishes as $d\ensuremath{\rightarrow}4+$. Mean-field theory is developed from the uniform minimum of the Lagrangian for both the strong- and weak-coupling cases. For the strong-coupling case it gives the exponents ${v}_{a}=\frac{1}{4}$, ${\ensuremath{\gamma}}_{a}={\ensuremath{\beta}}_{a}=\frac{1}{2}$, $\ensuremath{\eta}=0$, and $\ensuremath{\mu}=1$, where ${\ensuremath{\beta}}_{a}$ is the exponent associated with the density of extended states and $\ensuremath{\mu}$ is that associated with the conductivity. Simple heuristic arguments are used to verify the correctness of these unusual mean-field values. Infrared divergences in perturbation theory for the strong-coupling case occur for $d<8$, and an $\ensuremath{\epsilon}$ expansion ($\ensuremath{\epsilon}=8\ensuremath{-}d$) is developed which is found to be identical to that previously analyzed for the statistics of lattice animals and which gives ${\ensuremath{\beta}}_{a}=\frac{1}{2}\ensuremath{-}\frac{\ensuremath{\epsilon}}{12}$, $\ensuremath{\eta}=\ensuremath{-}\frac{\ensuremath{\epsilon}}{9}$, ${v}_{a}=\frac{1}{4}+\frac{\ensuremath{\epsilon}}{36}$, and $\ensuremath{\mu}=1\ensuremath{-}\frac{5\ensuremath{\epsilon}}{36}$. The results are consistent with the Ward identity, which in combination with scaling arguments requires that ${\ensuremath{\beta}}_{a}+{\ensuremath{\gamma}}_{a}=1$. The treatment takes account of the fact that the average of the on-site Green's function ${[G(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}},\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}};E)]}_{\mathrm{av}}$ is nonzero and is predicated on this quantity being real, i.e., on the density of states vanishing at the mobility edge. We also show that localized states emerge naturally from local minima of finite action in the Lagrangian. These instanton solutions are analyzed on a lattice where the cutoff produced by the lattice constant leads to lattice instantons which exist for all $d$, in contrast to the case for the continuum model where instanton solutions seem not to occur for $d>4$. This analysis leads to a density of localized states ${\ensuremath{\rho}}_{\mathrm{loc}}$ satisfying $\mathrm{ln}{\ensuremath{\rho}}_{\mathrm{loc}}\ensuremath{\sim}\ensuremath{-}{E}^{2}$ at large $E$ and $\mathrm{ln}{\ensuremath{\rho}}_{\mathrm{loc}}\ensuremath{\sim}\ensuremath{-}{|E\ensuremath{-}{E}_{c}|}^{\ensuremath{-}\ensuremath{\zeta}}$ at the mobility edge, where for the weak-coupling case $\ensuremath{\zeta}=(\frac{1}{2})(d\ensuremath{-}4)$ and for the strong-coupling case $\ensuremath{\zeta}=(d\ensuremath{-}2+\ensuremath{\eta}){v}_{a}\ensuremath{-}{2\ensuremath{\beta}}_{a}=\frac{1}{2}+\frac{\ensuremath{\epsilon}}{18}$ for $d<8$ and $\ensuremath{\zeta}=(\frac{1}{4})(d\ensuremath{-}6)$ for $d>8$. A brief discussion of the relationship between this work and the theories of localization below four dimensions is presented.

Coupled maps and analytic function spaces

Singular phases of Seiberg–Witten integrable systems: weak and strong coupling

In axial gauge, the (2+1)-dimensional SU($N$) Yang-Mills theory is equivalent to a set of (1+1)-dimensional integrable models with a non-local coupling between charge densities. This fact makes it possible to determine the static potential between charges at weak coupling in an anisotropic version of the theory, and understand features of the spectrum.

We outline a partial-fractions decomposition method for determining the one-particle spectral function and single-particle density of states of a correlated electronic system on a finite lattice in the non self-consistent T-matrix approximation to arbitrary numerical accuracy, and demonstrate the application of these ideas to the attractive Hubbard model. We then demonstrate the effectiveness of a finite-size scaling ansatz which allows for the extraction of quantities of interest in the thermodynamic limit from this method. In this approximation, in one or two dimensions, for any finite lattice or in the thermodynamic limit, a pseudogap is present and its energy diverges as T c is approached from above; this is an unphysical manifestation of using an approximation that predicts a spurious phase transition in one or two dimensions. However, in three dimensions one expects the transition predicted by the approximation to represent a true continuous phase transition, and whether or not a pseudo gap exists in the thermodynamic limit in three dimensions remains an open question. We have applied our method to the attractive Hubbard model on a three-dimensional simple cubic lattice, and find, similar to previous work, that for intermediate coupling a prominent pseudogap is found in the single-particle density of states, and this gap persists over a large temperature range. In addition, we also show that for weak coupling (an on-site Hubbard energy equal to one quarter the bandwidth) a pseudogap is also present. The pseudogap energy at the transition temperature is almost a factor of three larger than the T = 0 BCS gap for intermediate coupling, whereas for weak coupling the pseudogap and T = 0 BCS gap energies are essentially equal. These results show that a pseudogap due to superconducting fluctuations occurs in three dimensions even in the weak-coupling limit.