Progress and bottlenecks in simulating lattice gauge theory with fermions

We present a progress report in lattice gauge theory computer simulations which includes the effects of light, dynamical fermions. Microcanonical and hybrid microcanonical-Langevin alogrithms are presented and discussed. A method for “accelerating” stochastic differential equations and defeating critical slowing down is reviewed. Physics applications such as the thermodynamics of quantum chromodynamics, hierarchal energy scales in unified gauge theories, and the phase diagram of theories with many fermion species are discussed. Prospects for future research are assessed.     

Scalar lattice gauge theory

Scalar lattice gauge theories are models for scalar fields with local gauge symmetries. No fundamental gauge fields, or link variables in a lattice regularization, are introduced. The latter rather emerge as collective excitations composed from scalars. For suitable parameters scalar lattice gauge theories lead to confinement, with all continuum observables identical to usual lattice gauge theories. These models or their fermionic counterpart may be helpful for a realization of gauge theories by ultracold atoms. We conclude that the gauge bosons of the standard model of particle physics can arise as collective fields within models formulated for other “fundamental” degrees of freedom.     

Adjoint sources in lattice gauge theory

Variance reduction techniques for the evaluation of Wilson loops in lattice gauge theory are analysed. The method is extended to Wilson loops in the adjoint representation. Variational methods are also applied to adjoint sources. The combination of these techniques allows the potential V(R) between two static adjoint sources to be determined in SU(2) gauge theory. One isolated static adjoint source is also studied and the energy and distribution of the gluon field of this “glue-lump” is obtained. This is relevant to the saturation of the adjoint potential V(R) at large R.     

Some calculations in lattice gauge theory

We look at the action proposed by Wilson on a lattice and calculate static constants like f_π and two-body decay amplitudes in a certain approximation. Results are good to factors of four to six. There is good agreement for some of the predicted meson masses.     

Loop states in lattice gauge theory

We reformulate d-dimensional SU(2) lattice gauge theory in terms of gauge invariant loop state variables by solving the SU(2) Gauss law as well as the corresponding Mandelstam constraints. The loop states satisfying the Gauss law and the Mandelstam constraints in d dimension are explicitly constructed in terms of the SU(2) harmonic oscillator prepotential operators. We show that these mutually independent (orthonormal) loop states carry certain non-negative integer Abelian fluxes over the lattice links and are characterized by 3(d−1) gauge invariant angular momentum quantum numbers per lattice site. Thus, they provide a complete orthonormal loop basis in the physical Hilbert space of the gauge theory. Further, we derive the loop Hamiltonian and show that it counts, creates and annihilates the Abelian fluxes over the plaquettes. The generalization to SU(N) gauge group is discussed.     

Ising lattice gauge theory in three dimensions

By raising the transfer matrix to a finite power the partition function for a finite lattice Z(2) gauge model is obtained exactly. The zeros of the resultant polynomial are found and some plaquette-plaquette expectation values are extracted. An exponential fit for the inverse correlation length matches onto both strong- and weak-coupling results but breaks down close to the second-order phase transition point.Similar calculations for the three-dimensional Ising model are also discussed.     

Bosonic lattice gauge theory with noise

We apply the recently suggested linear updating algorithm of Kennedy and Kuti to four-dimensional SU(3) bosonic gauge theory with the Wilson action. The change in the action for each link update is estimated stochastically, and we find that the algorithm gives the mean plaquette correctly for reasonable parameter values. Our results indicate that this method should be efficient for Monte Carlo computations with complicated “improved” actions, and they also show the feasibility of using such “noisy” methods to include the dynamical effects of fermions.     

Three-dimensional lattice gauge theory and strings

We consider SU(2) lattice gauge theory in three dimensions. The Wilson loops are found to be well described by a simple string model in the approximate scaling region.     

Generalized actions in Zp lattice gauge theory

In Z_p lattice gauge theory we generalize the Wilson action to include all group representations. We review the implications of duality for these models. With Monte Carlo methods, we find a rich phase structure for the cases p = 4 and 5.     

Schwinger-Dyson equations and currents in lattice gauge theory

After introducing appropriate derivatives, the structure of Schwinger-Dyson equations, currents and Ward-Takahashi identities (including the anomalous ones) on a finite lattice is completely clarified. A general relation between correlation functions without and with gauge fixing is given.     

Finite-temperature gauge theory from the transverse lattice

Numerical computations are performed and analytic bounds are obtained on the excited spectrum of glueballs in SU(inifinity) gauge theory, by transverse lattice Hamiltonian methods. We find an exponential growth of the density of states, implying a finite critical (Hagedorn) temperature. It is argued that the Nambu-Goto string model lies in a different universality class.     

Chiral and conformal anomalies in lattice gauge theory

The continuum limit of the chiral and conformal (Weyl) Ward-Takahashi identities in the lattice Wilson action is studied. The Wilson term works for the chiral anomaly, but it gives rise to-15 times the conventional conformal anomaly for a smallr-parameter and a very sensitiver-dependence of the Λ-parameter. This shows that the strong symmetry breaking by the Wilson term by itself does not necessarily generate correct anomalies. In the lattice regularization the functional Jacobian factors becomec-numbers and do not contribute to anomalies, corresponding to the cut-off of short distance components; the naive continuum limit of lattice WT identities can thus behave differently from continuum ones. To reconstruct conventional identities from lattice relations, the lattice composite operators should be rewritten in terms of relevant continuum operators. In general, this identification of relevant operators is facilitated either by the procedure corresponding to Zimmermann's normal product algorithm or simply by the use of auxiliary regulators such as the dimensional regulator.     

Non-perturbative decoupling theorems in lattice gauge theory simulations

We illustrate that massless fermions which condense at high energy decouple from low energy sectors of a gauge theory by considering a lattice theory with (almost) massless quarks in the fundamental and adjoint representations of the gauge group. We confirm that the low energy chiral condensate of the fundamental fermions satisfies the asymptotic freedom scaling law with the adjoint fermions decoupled.     

Implicit schemes for real-time lattice gauge theory

We develop new gauge-covariant implicit numerical schemes for classical real-time lattice gauge theory. A new semi-implicit scheme is used to cure a numerical instability encountered in three-dimensional classical Yang-Mills simulations of heavy-ion collisions by allowing for wave propagation along one lattice direction free of numerical dispersion. We show that the scheme is gauge covariant and that the Gauss constraint is conserved even for large time steps.