Topics:

• Lattice gauge theory
• Composite Higgs
  
• Technicolor
  
• Dense baryonic matter
• Microscopic spectra
  
• Lattice fermions and chiral symmetry
  
• Other
• Hot nuclear matter
• Particle pair production in strong electric fields; the flux tube model
• Charm in the quark–gluon plasma
• Perturbative field theory at high temperature
• Bubbles in the quark–gluon plasma
• Lattice Skyrme model
• Judah

Lattice gauge theory

We have calculated the low-energy constant L₁₀ in a prototype composite Higgs model with dynamical fermions in two different representations of the gauge group. The resulting contribution of the new strong sector to the S parameter is consistent with current bounds on the vacuum misalignment parameter. We end with a brief discussion of future directions.

We calculate the low-energy constant L₁₀ in a two-representation SU(4) lattice gauge theory that is close to a composite-Higgs model. From this we obtain the contribution of the new strong sector to the S parameter. This leads to an upper bound on the vacuum misalignment parameter ξ which is similar to current estimates of this bound. Our result agrees with large-N_c scaling expectations, within large systematic uncertainties.

We have calculated quantities of interest to a theory of compositeness. The lattice model, approximating the candidate theory, is the SU(4) gauge theory coupled to fermions in two color representations. For the composite Higgs, a current correlator gives one of the ingredients of the effective Higgs potential. For the partially composite top quark, we have hyperbaryon matrix elements that govern mixing of the fundamental quark with its heavy composite partner. The matrix elements turn out to be so small that the theory is disfavored as a source of a realistic top mass.

Working in a two-representation lattice gauge theory that is close to a composite Higgs model, we calculate the low-energy constant C_LR which controls the contribution of the electroweak gauge bosons to the Higgs potential. In QCD, the corresponding low-energy constant governs the mass splitting of the pion multiplet. Taking the continuum and chiral limits, we find that C_LR, in units of the pseudoscalar decay constant, is roughly of the same size as its QCD counterpart.

Partial compositeness is a mechanism for the generation of fermion masses which replaces a direct Higgs coupling to the fermions by a linear mixing with heavy composite partners. We present the first calculation of the relevant matrix element in a lattice model which is very close to a candidate theory containing a composite Higgs boson and a partially composite top quark. Specifically, our model is an SU(4) gauge theory coupled to dynamical fermions in the fundamental and two-index antisymmetric (sextet) representations. The matrix element we obtain is small and hence our result disfavors the scenario of obtaining a realistic top mass in this model.

We investigate the phase structure of SU(4) gauge theory with the gauge field simultaneously coupled to two flavors of fermion in the fundamental representation and two flavors of fermion in the two-index antisymmetric representation. We find that the theory has only two phases, a low-temperature phase with both species of fermion confined and chirally broken, and a high-temperature phase with both species of fermion deconfined and chirally restored. The single phase transition in the theory appears to be first order, in agreement with theoretical predictions.

We use lattice simulations to compute the baryon spectrum of SU(4) lattice gauge theory coupled to dynamical fermions in the fundamental and two-index antisymmetric (sextet) representations simultaneously. This model is closely related to a composite Higgs model in which the chimera baryon made up of fermions from both representations plays the role of a composite top-quark partner. The dependence of the baryon masses on each underlying fermion mass is found to be generally consistent with a quark-model description and large-N_c scaling. We combine our numerical results with experimental bounds on the scale of the new strong sector to derive a lower bound on the mass of the top partner.

We have simulated the SU(4) lattice gauge theory coupled to dynamical fermions in the fundamental and two-index antisymmetric (sextet) representations simultaneously.  Such theories arise naturally in the context of composite Higgs models that include a partially composite top quark. We describe the low-lying meson spectrum of the theory and fit the pseudoscalar masses and decay constants to chiral perturbation theory. We infer as well the mass and decay constant of the Goldstone boson corresponding to the non-anomalous U(1) symmetry of the model. Our results are broadly consistent with large-N_c scaling and vector-meson dominance.

We report preliminary results on the finite temperature behavior of SU(4) gauge theory with dynamical quarks in both the fundamental and two-index antisymmetric representations. This system is a candidate to present scale separation behavior, where fermions in different representations condense at different temperature or coupling scales. Our simulations, however, reveal a single finite-temperature phase transition at which both representations deconfine and exhibit chiral restoration. It appears to be strongly first order. We compare our results to previous single-representation simulations. We also describe a Pisarski-Wilczek stability analysis, which suggests that the transition should be first order.

We describe our recent lattice study of SU(4) gauge theory with fermions in the fundamental and sextet representations. In this theory, a new type of baryon consists of quarks in both representations. The spectrum of these “chimera baryons” has a straightforward interpretation in terms of a non-relativistic quark model based on SU(4). Our results are particularly relevant to composite Higgs models in which the top quark is partially composite.

The SU(4) gauge theory with two flavors of Dirac fermions in the sextet representation shares features of a candidate for a composite Higgs model. The analogue of the Higgs multiplet of the Standard Model lives in the Goldstone manifold resulting from spontaneous breaking of the global symmetry SU(4) to SO(4). The Higgs potential arises from interaction with the particles of the Standard Model. We have computed the gauge boson contribution to the Higgs potential, using valence overlap fermions on a Wilson-clover sea. The calculation is similar to that of the electromagnetic mass splitting of the pion multiplet in QCD.

We develop methods to calculate the electroweak gauge boson contribution to the effective Higgs potential in the context of composite Higgs models, using lattice gauge theory. The calculation is analogous to that of the electromagnetic mass splitting of the pion multiplet in QCD. We discuss technical details of carrying out this calculation, including modeling of the momentum and fermion-mass dependence of the underlying current-current correlation function; direct integration of the correlation function over momentum; and fits based on the minimal-hadron approximation. We show results of a numerical study using valence overlap fermions, carried out in an SU(4) gauge theory with two flavors of Dirac fermions in the two-index antisymmetric representation.

We present a first look at the spectroscopy of SU(4) gauge theory coupled to two flavors of Dirac fermions in the two-index antisymmetric representation, which is a real representation.  We compute meson and diquark masses, the pseudoscalar and vector meson decay constants, and the masses of six-quark baryons. We make comparisons with large-N_c expectations.

We present a study of spectroscopy of SU(4) lattice gauge theory coupled to two flavors of Dirac fermions in the anti-symmetric two index representation. The fermion representation is real, and the pattern of chiral symmetry breaking is SU(2N_f ) → SO(2N_f ) with N_f  flavors of Dirac fermions. It is an interesting generalization of QCD, for several reasons: it allows direct exploration of an alternate large N_c expansion, it can be simulated at non-zero chemical potential with no sign problem, and several UV completions of composite Higgs systems are built on it. We present preliminary results on the baryon and meson spectra of the theory and compare them with SU(3) results and with expectations for large-N_c scaling.

After some introductory comments on the peculiar features of slowly running theories, I will report results obtained using the Schrödinger functional technique for two gauge theories that are believed to lie near the bottom of the conformal window: the SU(3) theory with two adjoint Dirac fermions, and the SU(4) theory with six Dirac fermions in the two-index antisymmetric representation. In both cases we find a small beta function in strong coupling, but we cannot confirm or rule out an infrared fixed point. In both theories the mass anomalous dimension levels off, staying well below 0.5, much like the theories with fermions in the two-index symmetric representation investigated earlier.

We apply Schrödinger functional methods to two gauge theories with fermions in two-index representations: the SU(3) theory with N_f = 2 adjoint fermions, and the SU(4) theory with N_f = 6 fermions in the two-index antisymmetric representation. Each theory is believed to lie near the bottom of the conformal window for its respective representation. In the SU(3) theory we find a small beta function in strong coupling but we cannot confirm or rule out an infrared fixed point. In the SU(4) theory we find a hint of walking—a beta function that approaches the axis and then turns away from it. In both theories the mass anomalous dimension remains small even at the strongest couplings, much like the theories with fermions in the two-index symmetric representation investigated earlier.

We have calculated the running coupling in SU(2), SU(3), and SU(4) gauge theories to see whether they have infrared fixed points. An infrared fixed point means no confinement: It means that the long-distance physics is conformal, without a mass scale and indeed without a particle spectrum. All these theories run slowly in perturbation theory, so a nonperturbative determination is interesting.

We extend our previous study of the SU(3) gauge theory with N_f = 2 flavors of fermions in the sextet representation of color. Our tool is the Schrödinger functional method. By changing the lattice action, we push the bulk transition of the lattice theory to stronger couplings and thus reveal the beta function and the mass anomalous dimension γ_m over a wider range of coupling, out to g² ~ 11. Our results are consistent with an infrared fixed point, but walking is not ruled out. Our main result is that γ_m never exceeds 0.45, making the model unsuitable for walking technicolor.  We use a novel method of extrapolation to the large-volume/continuum limit, tailored to near-conformal theories.

We complete a program of study of SU(N) gauge theories coupled to two flavors of fermions in the two-index symmetric representation by performing numerical simulations in SU(4). The beta function, defined and calculated via the Schrödinger functional, runs more slowly than the two-loop perturbative result. The mass anomalous dimension levels off in strong coupling at a value of about 0.45, rendering this theory unsuitable for walking technicolor. A large-N comparison of these data with results from SU(2) and SU(3) reveals striking regularities.

We summarize our recent work on gauge theories with two flavors of fermions in the two-index symmetric representation: SU(2) gauge theory with adjoint fermions, SU(3) with sextets, and SU(4) with ten-dimensional-representation fermions. All three systems have beta functions smaller than their perturbative value, approaching a fixed point near the expected two-loop zero. In all cases the mass anomalous dimension is small, under 0.5.

We apply Schrödinger-functional techniques to the SU(2) lattice gauge theory with N_f = 2 flavors of fermions in the adjoint representation. Our use of hypercubic smearing enables us to work at stronger couplings than did previous studies, before encountering a critical point and a bulk phase boundary.  Measurement of the running coupling constant gives evidence of an infrared fixed point g_* where 1/g_*² = 0.20(4)(3). At the fixed point, we find a mass anomalous dimension γ_m(g_*) = 0.31(6).

I report the results of Schrödinger functional calculations in the SU(3) gauge theory with two flavors of color sextet fermions, defined with the Wilson–clover action using nHYP fat links. While we cannot confirm the infrared fixed point seen with thin links, we find very slow evolution of the coupling constant, so slow that extraction of the mass anomalous dimension is straightforward.

We have measured the running coupling constant of SU(3) gauge theory coupled to N_f = 2 flavors of symmetric representation fermions, using the Schrödinger functional scheme. Our lattice action is defined with hypercubic smeared links which, along with the larger lattice sizes, bring us closer to the continuum limit than in our previous study. We observe that the coupling runs more slowly than predicted by asymptotic freedom, but we are unable to observe fixed point behavior before encountering a first order transition to a strong coupling phase. This indicates that the infrared fixed point found with the thin-link action is a lattice artifact. The slow running of the gauge coupling permits an accurate determination of the mass anomalous dimension for this theory, which we observe to be small, γ_m<  0.6, over the range of couplings we can reach.  We also study the bulk and finite-temperature phase transitions in the strong coupling region.

We study the SU(3) lattice gauge theory, with two flavors of sextet Wilson–clover fermions, near its finite-temperature phase transition. We find metastable states that have Wilson line expectation values whose complex phases are near 2π/3 or π. The true equilibrium phase at these couplings and temperatures has its Wilson line oriented only towards the positive real axis, in agreement with perturbation theory.

The methods of lattice gauge theory may be applied to gauge theories besides QCD, in fact to any gauge group and any representation of matter fields (as long as the coupling is not chiral).  Such theories are useful for model building beyond the Standard Model, for instance in technicolor models.  We have carried out Monte Carlo simulations of the SU(3) gauge theory with color sextet fermions.  Our result for its discrete beta function indicate an infrared fixed point that makes the theory conformal rather than confining.  The lattice theory's phase diagram shows no separation of chiral and confinement scales, measured when the quark mass is nonzero.

We have performed numerical simulations of SU(3) gauge theory coupled to N_f = 2 flavors of symmetric representation fermions. The fermions are discretized with the tadpole-improved clover action. Our simulations are  done on lattices of length L = 6, 8, and 12. In all simulation volumes we observe a crossover from  a strongly coupled confined phase to a weak coupling deconfined phase.  Degeneracies in screening masses, plus the behavior of the pseudoscalar decay constant, indicate that the deconfined phase is also a phase in which chiral symmetry is restored.  The movement of the confinement transition as the volume is changed is consistent with avoidance of the basin of attraction of an infrared fixed point of the massless theory.

As a follow up to the previous talk about the beta function of SU(3) gauge theory with N_f=2 symmetric representation (clover) fermions [arXiv:0809.2885 [hep-lat]], we describe our explorations of the β–κ plane, away from the massless limit. Our simulations are mostly done on lattices of length L=8 and 12. We observe a phase transition from a strong coupling confined phase to a deconfined, chirally restored phase. The line of transitions avoids (so far) the location of the infrared fixed point discussed in the last talk.

The SU(3) gauge theory with fermions in the sextet representation is one of several theories of interest for technicolor models. We have carried out a Schrodinger functional (SF) calculation for the lattice theory with two flavors of Wilson fermions. We find that the discrete beta function changes sign when the SF renormalized coupling is in the neighborhood of g²=2.0, showing a breakdown of the perturbative picture even though the coupling is weak. The most straightforward interpretation is an infrared-stable fixed point.

We have carried out a Schrodinger functional (SF) calculation for the SU(3) lattice gauge theory with two flavors of Wilson fermions in the sextet representation of the gauge group. We find that the discrete beta function, which governs the change in the running coupling under a discrete change of spatial scale, changes sign when the SF renormalized coupling is in the neighborhood of g²=2.0. The simplest explanation is that the theory has an infrared-attractive fixed point, but more complicated possibilities are allowed by the data. While we compare rescalings by factors of 2 and 4/3, we work at a single lattice spacing.

Because the staggered fermion determinant is complex at nonzero μ, taking its fourth root leads to phase ambiguities. These unphysical effects cause the measure to become discontinuous; the problem becomes acute when Re μ ≳ m_π/2 (when T > 0 this rough bound probably moves towards larger μ). We show how to overcome the problem, but only very close to the continuum limit. This regime may be beyond reach with current resources.

The staggered fermion determinant is complex when the quark chemical potential μ is nonzero. Its fourth root, used in simulations with dynamical fermions, will have phase ambiguities that become acute when Re μ is sufficiently large. We show how to resolve these ambiguities, but our prescription only works very close to the continuum limit. We argue that this regime is very far from current capabilities. Other procedures require being even closer to the continuum limit, or fail altogether, because of unphysical discontinuities in the measure. At zero temperature the breakdown is expected when Re μ ≳ m_π/2. Estimates of the location of the breakdown at nonzero temperature are less certain.

We calculate the spectrum of excitations in strong-coupling lattice QCD in a background of fixed baryon density, at a substantial fraction of the saturation density. We employ a next-nearest-neighbor fermion formulation that possesses the SU(N_f)xSU(N_f) chiral symmetry of the continuum theory. We find two types of massless excitations: type I Goldstone bosons with linear dispersion relations and type II Goldstone bosons with quadratic dispersion relations.  Some of the type I bosons originate as type II bosons of the nearest-neighbor theory. Bosons of either type can develop anisotropic dispersion relations, depending on the value of N_f and the baryon density.

We determine the patterns of spontaneous symmetry breaking in strong-coupling lattice QCD in a fixed background baryon density. We employ a next-nearest-neighbor fermion formulation that possesses the SU(N_f)xSU(N_f) chiral symmetry of the continuum theory. We find that the global symmetry of the ground state varies with N_f and with the background baryon density. In all cases the condensate breaks the discrete rotational symmetry of the lattice as well as part of the chiral symmetry group.

We study the effective Hamiltonian for strong-coupling lattice QCD in the case of non-zero baryon density. In leading order the effective Hamiltonian is a generalized antiferromagnet. For naive fermions, the symmetry is U(4N_f) and the spins belong to a representation that depends on the local baryon number. Next-nearest neighbor (nnn) terms in the Hamiltonian break the symmetry to U(N_f)xU(N_f). We transform the quantum problem to a Euclidean sigma model which we analyze in a 1/N_c expansion. In the vacuum sector we recover spontaneous breaking of chiral symmetry for the nearest-neighbor and nnn theories. For non-zero baryon density we study the nearest-neighbor theory only, and show that the pattern of spontaneous symmetry breaking depends on the baryon density.

We apply strong-coupling perturbation theory to the QCD lattice Hamiltonian. We begin with naive, nearest-neighbor fermions and subsequently break the doubling symmetry with next-nearest-neighbor terms. The effective Hamiltonian is that of an antiferromagnet with an added kinetic term for baryonic "impurities," reminiscent of the t–J model of high-T_c superconductivity. As a first step, we fix the locations of the baryons and make them static. Following analyses of the t–J model, we apply large-N methods to obtain a phase diagram in the (N_c,N_f) plane at zero temperature and baryon density. Next we study a simplified U(3) toy model, in which we add baryons to the vacuum. We use a coherent state formalism to write a path integral which we analyze with mean field theory, obtaining a phase diagram in the (n_B,T) plane.

We calculate an analogue of the average phase factor of the staggered fermion determinant at imaginary chemical potential. Our results from the lattice agree well with the analytical predictions in the microscopic regime for both quenched and phase-quenched QCD. We demonstrate that the average phase factor in the microscopic domain is dominated by the lowest-lying Dirac eigenvalues.

We consider the chiral limit of QCD subjected to an imaginary isospin chemical potential. In the ε-regime of the theory we can perform precise analytical calculations based on the zero-momentum Goldstone modes in the low-energy effective theory. We present results for the spectral correlation functions of the associated Dirac operators.

We calculate the response of the microscopic Dirac spectrum to an imaginary isospin chemical potential for QCD with two dynamical flavors in the chiral limit. This extends our previous calculation from the quenched to the unquenched theory. The resulting spectral correlation function in the ε-regime provides here, too, a new and efficient way to measure F_π on the lattice. We test the method in a hybrid Monte Carlo simulation of the theory with two staggered quarks.

We derive the two-point spectral correlation function of the Dirac operator with a specific external source in the ε-regime of QCD. This correlation function has a unique and strong dependence on F_π, and thus provides an novel way to extract F_π from lattice simulations. We test the method in a quenched lattice simulation with staggered fermions.

It has been conjectured that spontaneous chiral symmetry breaking in strongly coupled vectorlike gauge theories falls into only three different depending on the gauge group and the representations carried by the fermions. We test this proposal by studying SU(2), SU(3) and SU(4) lattice gauge theories with staggered fermions in different irreducible representations. Staggered fermions away from the continuum limit should, for all complex representations, still belong to the continuum class of spontaneous symmetry breaking. But for all real and pseudo-real representations we show that staggered fermions should belong to incorrect symmetry breaking classes away from the continuum, thus generalizing previous results. As an unambiguous signal for whether chiral symmetry breaks, and which breaking pattern it follows, we look at the smallest Dirac eigenvalue distributions. We find that the patterns of symmetry breaking are precisely those conjectured.

I review how the phenomenology of localization applies to fermions in lattice gauge theory, present measurements of the localization length and other quantities, and discuss the consequences for things like the overlap kernel.

Lattice fermions in a fluctuating gauge field can show localization, much like electrons in a disordered potential. We study the spectrum of localized and extended states of supercritical Wilson fermions in gauge ensembles generated with plaquette and improved actions. When the Wilson fermion operator is used to construct the overlap kernel, the mobility edge, that is the boundary between the localized and extended states, determines the range of the kernel.

We determine the location λ_c of the mobility edge in the spectrum of the hermitian Wilson operator in pure-gauge ensembles with plaquette, Iwasaki, and DBW2 gauge actions. The results allow mapping a portion of the (quenched) Aoki phase diagram. We use Green function techniques to study the localized and extended modes. Where λ_c>0 we characterize the localized modes in terms of an average support length and an average localization length, the latter determined from the asymptotic decay rate of the mode density. We argue that, since the overlap operator is commonly constructed from the Wilson operator, its range is set by the value of λ_c⁻¹ for the Wilson operator. It follows from our numerical results that overlap simulations carried out with a cutoff of 1 GeV, even with improved gauge actions, could be afflicted by unphysical degrees of freedom as light as 250 MeV.

We determine the location λ_c of the mobility edge in the spectrum of the hermitian Wilson operator on quenched ensembles. We confirm a theoretical picture of localization proposed for the Aoki phase diagram. When λ_c>0 we also determine some key properties of the localized eigenmodes with eigenvalues |λ|<λ_c. Our results lead to simple tests for the validity of simulations with overlap and domain-wall fermions.

We derive the effective Lagrangian for mesons in lattice gauge theory with domain-wall fermions in the strong-coupling and large-N_c limits. We use the formalism of supergroups to deal with the Pauli–Villars fields, needed to regulate the contributions of the heavy fermions. We calculate the spectrum of pseudo-Goldstone bosons and show that domain wall fermions are doubled and massive in this regime. Since we take the extent and lattice spacing of the fifth dimension to infinity and zero respectively, our conclusions apply also to overlap fermions.

We apply strong-coupling perturbation theory to gauge theories containing domain-wall fermions in Shamir's surface version. We construct the effective Hamiltonian for the color-singlet degrees of freedom that constitute the low-lying spectrum at strong coupling. We show that the effective theory is identical to that derived from naive, doubled fermions with a mass term, and hence that domain-wall fermions at strong coupling suffer both doubling and explicit breaking of chiral symmetry. Since we employ a continuous fifth dimension whose extent tends to infinity, our result applies to overlap fermions as well.

We have carried out a Schrödinger-functional calculation for the Abelian gauge theory with N_f = 2 four-component fermions in three dimensions. We find no fixed point in the beta function, meaning that the theory is confining rather than conformal.

Normalized hypercubic smearing improves the behavior of dynamical Wilson-clover fermions, but has the unwanted side effect that it can occasionally produce spikes in the fermion force. These spikes originate in the chain rule connecting the derivative with respect to the smeared links to the derivative with respect to the dynamical links, and are associated with the presence of dislocations in the dynamical gauge field. We propose and study an action designed to suppress these dislocations. We present evidence for improved performance of the hybrid Monte Carlo algorithm. A side benefit is improvement in the properties of valence chiral fermions.

We apply the Schrödinger functional method to the Abelian gauge theory in three dimensions with N_f = 2 four-component fermions. We find that the calculated beta function does not cross zero in the range of coupling we study. This implies that the theory exhibits confinement and mass generation, rather than a conformal infared regime.

We study non-perturbative improvement in SU(3) lattice gauge theory coupled to fermions in the fundamental and two-index symmetric representations. Our lattice action is defined with hypercubic smeared links incorporated into the Wilson–clover fermion kernel. Using standard Schrödinger-functional techniques we estimate the clover coefficient c_SW and find that discretization errors are much smaller than in thin-link theories.

We use a variational wave function to calculate the energy of the interaction between external charges in the compact Abelian gauge theory in 2+1 dimensions with mixed action. Our variational wave functions preserve the compact gauge invariance of the theory both in the vacuum and in the charged sectors. We find that a good estimate of the interaction energy is obtained only when we allow more variational parameters in the charged sector than in the vacuum sector. These extra parameters are the profile of an induced electric field. We find that the theory has a two-phase structure: When the charge-2 coupling is large and negative there is no mass gap in the theory and no confinement, while otherwise a mass gap is generated dynamically and the theory confines charges. The pure Wilson theory is in the confining phase.

Spatial correlations—bubbles, domain walls, etc.—can best be studied by concentrating on the degrees of freedom most relevant to the problem. For the finite temperature confinement transition, I integrate out all gauge degrees of freedom, leaving only spins—Ising or Potts—related to the Wilson line. I present problems that arise in the course of this transformation and some results for the effective spin action.

I review evidence for the stability of bubbles in the quark–gluon plasma near the confinement phase transition. In analogy with the much-studied oil–water emulsions, this raises the possibility that there are many phases between the pure plasma and the pure hadron gas, characterized by spontaneous inhomogeneity and modulation. In studying emulsions, statistical physicists have reproduced many of their phases with microscopic models based on Ising-like theories with competing interactions. Hence we seek an effective Ising Hamiltonian for the SU(3) gauge theory near its transition.

We attempt the numerical construction of an effective action in three dimensions for Ising spins which represent the Wilson lines in the four-dimensional SU(3) gauge theory at finite temperature. For each configuration of the gauge theory, each spin is determined by averaging the Wilson lines over a small neighborhood and then projecting the average to ±1 according to whether the neighborhood is ordered or disordered. The effective Ising action, determined via the lattice Schwinger–Dyson equations, contains even (two-spin) and odd (one- and three-spin) terms with short range. We find that the truncation to Ising degrees of freedom produces an effective action which is discontinuous across the gauge theory's phase transition. This discontinuity may disappear if the effective action is made more elaborate.

We show that the method of Fourier acceleration can be used to speed up tunnelling between degenerate and near-degenerate minima of the effective action in Langevin simulations of statistical systems. Acceleration factors of two orders of magnitude are attained. Such improvement is important for the study of weak first-order phase transitions, for simulations of spin glasses, and for the method of simulated annealing in optimization problems.

I review recent results in lattice gauge theory which pertain to the physics of high temperature QCD and quark matter. Topics covered include: the transition temperature in the pure gauge theory, scaling and finite temperature behavior in QCD with quarks, problems with non-zero chemical potential, the Debye length in the high temperature plasma, and quantities relevant to thermodynamics and hydrodynamics in relativistic heavy ion collisions.

Lattice Skyrme model

We propose using the Skyrme model on a lattice as an effective field theory of meson–baryon interactions. To this end we construct a local topological density that involves the volumes of tetrahedra in the target space S³ and we make use of Coxeter's formula for the Schläfli function to implement it. We calculate the mean-square radius of a skyrmion in the three-dimensional Skyrme model, and find some surprises.

We offer the Skyrme model on a lattice as an effective field theory—fully quantized—of baryon–meson interactions at temperatures below the chiral phase transition. We define a local topological density that involves the volumes of tetrahedra in the target space S³ and we make use of Coxeter's formula for the Schläfli function to implement it. This permits us to calculate the mean-square radius of a skyrmion in the three-dimensional lattice Skyrme model, which may be viewed as a Ginzburg–Landau effective theory for the full quantum theory at finite temperature. We find that, contrary to expectations, the skyrmion shrinks as quantum and thermal fluctuations are enhanced. We ascribe this to a large number of metastable states that become accessible as the temperature is raised.

Hot nuclear matter

We study hot nuclear matter in a model based on nucleon interactions deriving from the exchange of scalar and vector mesons. The main new feature of our work is the treatment of the scale breaking of quantum chromodynamics through the introduction of a dilaton field. Although the dilaton effects are quite small quantitatively, they affect the high-temperature phase transition appreciably. We find that inclusion of the dilaton leads to a metastable high-density state at zero pressure, similar to that found by Glendenning who considered instead the admixture of higher baryon resonances.

Particle pair production in strong electric fields; the flux tube model

I review some of the history of the flux-tube model, concentrating on work done by the Los Alamos–Tel Aviv collaboration on particle creation and back-reaction in uniform fields and in the central rapidity region. I discuss the incorporation of more realistic geometry and structure of the flux tube via application of the dual superconductor model of confinement.

We have studied plasma oscillations in a flux tube created in a dual superconductor. Starting from a static flux tube configuration, with electric charges at either end, we release a fluid of electric charges in the system that accelerate and screen the electric field. The weakening of the electric field allows the flux tube to collapse, and the inertia of the charges forces it open again. We find strong radiation of electric flux into the superconductor in all regimes of couplings.

We study plasma oscillations in a flux tube of the dual superconductor model of 't Hooft and Mandelstam. A magnetic condensate is coupled to an electromagnetic field by its dual vector potential, and fixed electric charges set up a flux tube. An electrically charged fluid (a quark plasma) flows in the tube and screens the fixed charges via plasma oscillations. We investigate both Type I and Type II superconductors, with plasma frequencies both above and below the threshold for radiation into the Higgs vacuum. We find strong radiation of electric flux into the superconductor in all regimes, and argue that this invalidates the use of the simplest dual superconductor model for dynamical problems.

We study pair production from a strong electric field in boost-invariant coordinates as a simple model for the central rapidity region of a heavy-ion collision. We derive and solve the renormalized equations for the time evolution of the mean electric field and current of the produced particles, when the field is taken to be a function only of the fluid proper time. We find that a relativistic transport theory with a Schwinger source term modified to take Pauli blocking (or Bose enhancement) into account gives a good description of the numerical solution to the field equations. We also compute the renormalized energy–momentum tensor of the produced particles and compare the effective pressure, energy and entropy density to that expected from hydrodynamic models of energy and momentum flow of the plasma.

We review recent achievements in the solution of the initial-value problem for quantum back-reaction in scalar and spinor QED. The problem is formulated and solved in the semiclassical mean-field approximation for a homogeneous, time-dependent electric field. Our primary motivation in examining back-reaction has to do with applications to theoretical models of production of the quark–gluon plasma, though we here address practicable solutions for back-reaction in general. We review the application of the method of adiabatic regularization to the Klein–Gordon and Dirac fields in order to renormalize the expectation value of the current and derive a finite coupled set of ordinary differential equations for the time evolution of the system. Three time scales are involved in the problem and therefore caution is needed to achieve numerical stability for this system. Several physical features, like plasma oscillations and plateaus in the current, appear in the solution. From the plateau of the electric current one can estimate the number of pairs before the onset of plasma oscillations, while the plasma oscillations themselves yield the number of particles from the plasma frequency.

We compare the field-theory solution to a simple model based on a relativistic Boltzmann–Vlasov equation, with a particle production source term inferred from the Schwinger particle creation rate and a Pauli-blocking (or Bose-enhancement) factor. This model reproduces very well the time behavior of the electric field and the creation rate of charged pairs of the semiclassical calculation. It therefore provides a simple intuitive understanding of the nature of the solution since nearly all the physical features can be expressed in terms of the classical distribution function.

The initial-value problem for the quantum back-reaction in spinor QED is formulated and solved in the semiclassical mean field approximation, for a homogeneous but time-dependent electric field E(t). We apply the method of adiabatic regularization to the Dirac equation in order to renormalize the expectation value of the current and derive a finite coupled set of ordinary differential equations for the time evolution of the system. We solve this system in (1+1) dimensions numerically and compare the solution to a simple model based on a relativistic Boltzmann–Vlasov equation, with a particle production source term inferred from the Schwinger particle creation rate and a Pauli-blocking factor. This model reproduces very well the time behavior of the electric field and the creation rate of electron–positron pairs of the semiclassical calculation.

Dedicated to Wiesław Czyż on the occasion of his sixty-fifth birthday

We present a summary of the present status of efforts to solve the problem in which pairs are produced in a strong electric field, are accelerated by it, and then react back on it through the counter-field produced by their current. This picture has been used by Białas and Czyż and others as a model for effects that may possibly arise in the study of the quark–gluon plasma. We here give a didactic review of recent developments in this back-reaction problem. We first present a simple version of the theory of pair tunneling from a fixed electric field, and then sketch how this has been applied to the quark–gluon plasma. Then we turn to a field formulation of the problem for charged bosons, which leads to the need to carry out a renormalization program, outlined again in simple terms. Numerical results for this program are presented for one spatial dimension, the corresponding physical behavior of the system is discussed, and the implications for three spatial dimensions are considered. We exhibit a phenomenological transport equation embodying physics that is essentially identical to that of the field formulation, thus helping to tie the model of Białas and Czyż for the quark–gluon plasma to a field-theory formulation. Last, we note the status of extensions to (i) the problem with three space dimensions; (ii) the fermion case; (iii) the formulation in terms of boost-invariant variables (as desirable for the quark–gluon plasma); and (iv) transport equations derived in a fundamental and consistent fashion.

We investigate the mechanism of pair creation in scalar QED from spatially homogeneous strong electric fields in 1+1 dimensions. Solution of the semiclassical field equations shows particle creation followed by plasma oscillations. We compare our results with a model based on a relativistic Boltzmann–Vlasov equation with a pair-creation source term related to the Schwinger mechanism. The time evolution of the electric field and the current obtained from the Boltzmann–Vlasov model is surprisingly similar to that found in the semiclassical calculation.

Charm in the quark–gluon plasma

A charmed quark experiences drag and diffusion in the quark–gluon plasma, as well as strong interaction with the plasma surface. Our simulations indicate that charmed quarks created in heavy ion collisions will be trapped in the mixed phase and will come to equilibrium in it. Their momentum distribution will thus reflect the temperature at the confinement phase transition.

Also given (under the title Charm transverse momentum as a thermometer of the quark–gluon plasma) at the International Europhysics Conference on High Energy Physics, 19–26 August 1997, Jerusalem, Israel, published in proceedings, ed. by D. Lellouch, G. Mikenberg, and E. Rabinovici (Springer-Verlag, Berlin, 1999).

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We employ a modified cascade hydrodynamics code to simulate the phase transition of an expanding quark–gluon plasma and the passage of a charmed particle through it. When inside the plasma droplets, the charmed quark experiences drag and diffusion forces. When outside the plasma, the quark travels as a D meson and experiences collisions with pions. Additional energy transfer takes place when the quark enters or leaves a droplet. We find that the transverse momentum of D mesons provides a rough thermometer of the phase transition.

We present a model of charmonium as two heavy quarks propagating classically in a weakly coupled quark–gluon plasma. The quarks interact via a static, color-dependent potential and also suffer collisions with the plasma particles. We calculate the radiation width of the color octet state (for fixed, classical q–q separation) and find that it is long-lived provided a finite gluon mass is used to provide a threshold energy.

The transverse momentum of charmed particles should be degraded by their passage through the mixed-phase fluid existing during the course of the quark–hadron phase transition. The mechanism is multiple hadronization, the repeated conversion of quark to meson and back as phase boundaries are crossed. The effect can be reduced by thermalization via ordinary collisions with particles of the plasma or the hadron gas, but passage through a homogeneous dense medium cannot duplicate the momentum degradation. The effect, if observed, thus constitutes a simple signal of the confinement phase transition in ultrarelativistic heavy ion collisions.

I review some features of proposed signals of the quark–gluon plasma in order to motivate a new one: measuring the transverse momentum of charmed particles. A sharp reduction in the mean transverse momentum will show that the matter produced in a collision has cooled through a confinement phase transition. I argue that other mechanisms will not duplicate the effect.

Perturbative field theory at high temperature

We study the spectrum of fermionic excitations in a hot relativistic electron plasma. Numerical analysis of the one-loop electron propagator shows the appearance of new quasiparticle modes, as already studied in the massless fermion limit, as the temperature is raised to exceed the electron mass m. We calculate the relevant spectral densities and show that one mode, whose splitting in energy from the original electron is of the order of m, moves as the temperature is raised towards higher frequency along with the electron while retaining a narrow width. We compare results derived in Feynman and Coulomb gauge. The role of the zero temperature counterterms is discussed.

Bubbles in the quark–gluon plasma

The small surface tension of vacuum bubbles in the quark–gluon plasma, along with a negative curvature tension, implies that the uniform plasma is unstable against the formation of such bubbles. We show that spherical bubbles are in turn unstable against the growth of irregularities.

I argue that the quark–gluon plasma is unstable against the formation of bubbles of confining vacuum.

We calculate the free energy of finite droplets of quark–gluon plasma, and of finite hadronic bubbles in the bulk plasma, near the confinement phase transition. We sum over free quark and gluon energy levels in the presence of MIT bag boundary conditions. We find that the curvature term in the free energy, proportional to the radius of the droplet/bubble, is far more important than the contribution of the surface tension, proportional to the radius squared. This affects the critical radius for nucleation of plasma droplets in the superheated hadron gas, and seems to lead to instability of the plasma (even when not supercooled) against nucleation of hadron gas bubbles.

Judah

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