Euclidean Theory Research Articles

Numerical studies of phase transitions in SU( N)/Z N lattice gauge theories

The average action per plaquette is calculated for the pure SU( N)/Z N , N = 2, 3, 4, 5 and 6 gauge groups using strong coupling expansions up to 13th order for euclidean lattice gauge theory in four space-time dimensions. These expansions are compared with Monte Carlo generated data and agreement is found to be excellent for 0⩽ β⩽ β c , where β c is the critical inverse temperature for the appropriate gauge group considered.

Fluctuations around a meron solution in euclidean Yang-Mills theory

Fluctuations around a meron solution in euclidean Yang-Mills theory

Towards the phase structure of euclidean lattice gauge theories with fermions

We propose the phase structure of abelian and non-abelian lattice gauge theories with fermions. We especially analyse Wilson's lattice action with euclidean discrete space-time. We mainly analyse 〈 ψ nψ n〉 as an order parameter for the fermion-gauge coupled system. The Wilson loop integral and plaquette-plaquette two-point function are also useful in working out abelian phase diagrams. We will discuss physical implications of the phase diagrams, especially for the mass spectrum in the lattice continuum limit and chiral symmetry breaking. The 1/ N expansion and a random walk idea are used in the formulation and play an important role in computing meson and baryon propagators in the strong coupling limit.

Semiclassical functional integrals for self-dual gauge fields

The semiclassical approximation to the functional integral for four-dimensional Euclidean gauge theories is discussed in detail for general stationary points of the action. It is shown how to take the limit from a compact space to flat space, and the zero modes corresponding to global gauge transformations are carefully discussed. The results are specialised to general self-dual multi-instanton gauge fields given by the general construction of Atiyah et al. It is shown how the normalization matrix of the zero modes can be determined and the complete expression for the functional measure is given for the two instanton case. This is shown to factorise for well-separated instantons. Some technical matters are discussed in an appendix and a resume of results for multi-instanton functional determinants is included.

Strong coupling expansions for the mass gap in lattice gauge theories

We derive strong coupling expansions for the mass gap in euclidean lattice gauge theories in any space-time dimension. For gauge groups SU(2), SU(3), Z 2 and Z 3 the series are calculated up to order g −16. They are used to get rough estimates for the lowest glueball mass in continuum SU(2) and SU(3) gauge theories, assuming a sudden crossover from strong to weak coupling behaviour in the lattice theory.

Quark liberation at high temperature: A Monte Carlo study of SU(2) gauge theory

Quark confinement in a finite-temperature $\mathrm{SU}(N)$ gauge theory is formulated as the realization of a global ${Z}_{N}$ symmetry. Spontaneous breakdown corresponds to a transition to a nonconfining, plasma phase. The free energy of a single quark is an order parameter which probes the phase structure, and it may be calculated in the Euclidean theory in terms of a "Wilson line" running the length of the system along the (periodic) time axis. We present results of a Monte Carlo calculation in the SU(2) lattice theory which confirm the transition at a critical temperature computed in terms of the zero-temperature string tension; data for the quark-antiquark potential are presented as well. We discuss the implications of the finite-temperature transition for efforts to calculate zero-temperature quantities on finite-size lattices. Finally, we note that restoration of ${Z}_{N}$ symmetry as the temperature is lowered may be understood as a condensation of instantons and other topological objects.

Phase structure of lattice gauge theories for non-abelian subgroups of SU(3)

We study the phase structure of euclidean lattice gauge theories in four dimensions for certain non-abelian subgroups of SU(3) by using Monte Carlo simulations and strong coupling expansions. As the order of the group increases we observe a splitting of one phase transition into two.

Zero-point oscillations, local stability, and the effective action

We define a gauge-invariant action functional for non-Abelian gauge theories by deriving 't Hooft's generating functional from the vacuum-to-vacuum transition amplitude, and we explicitly demonstrate its gauge-invariance properties. We then examine the effective action in the loop expansion and show that in the presence of a nonvanishing background field the unrenormalized one-loop term can be written as the change in the zero-point energy of the theory. Finally, we consider the interpretation of an imaginary part of the effective action and its relation to local stability by examining the source of such an imaginary part within the Euclidean theory.

Constant Yang-Mills fields

We discuss self-dual solutions with constant potentials for Euclidean and Minkowskian Yang-Mills theories. We show that the action and energy of these solutions must vanish and give two explicit solutions with nonvanishing field-strength tensors. The solutions are not expected to be stable.

Estimate of the relation between scale parameters and the string tension by strong coupling methods

We estimate the relation between the scale parameter ΛL and the string tension α in pure SU(3) gauge theory using strong coupling expansions for euclidean lattice gauge theory up to 12th order. The result is ΛL=3.7 × 10−3 √α. For the more conventional scale parameter ΛMOM this gives ΛMOM = 0.31 √α by use of the proportionality factor of Hasenfratz. Results for the string tension of Z3 lattice gauge theory are also discussed.

Path integral for gauge theories with fermions

The Atiyah-Singer index theorem indicates that a naive unitary transformation of basis vectors for fermions interacting with gauge fields is not allowed in general. On the basis of this observation, it was previously shown that the path-integral measure of a gauge-invariant fermion theory is transformed nontrivially under the chiral transformation, and thus leads to a simple derivation of "anomalous" chiral Ward-Takahashi identities. We here clarify some of the technical aspects associated with the discussion. It is shown that the Jacobian factor in the path-integral measure, which corresponds to the Adler-Bell-Jackiw anomaly, is independent of any smooth regularization procedure of large eigenvalues of $D$ in Euclidean theory; this property holds in any even-dimensional space-time and also for the gravitational anomaly. The appearance of the anomaly and its connection with the index theorem are thus related to the fact that the primary importance is attached to the Lorentz-covariant "energy" operator $D$ and that $D$ and ${\ensuremath{\gamma}}_{5}$ do not commute. The abnormal behavior of the path-integral measure at the zero-frequency sector in the presence of instantons and its connection with spontaneous symmetry breaking is also clarified. We comment on several other problems associated with the anomaly and on the Pauli-Villars regularization method.

Vanishing of the energy-momentum tensor in euclidean Yang-Mills theories

Certain global properties of the energy-momentum tensor in euclidean field theories are established. These properties combined with the inversion invariance are used to show that the energy-momentum tensor corresponding to finite-action solutions of euclidean YM theories must vanish everywhere.

Spezielle Geradenkongruenzen im einfach isotropen Raum

In this paper we study three kinds of congruences (pseudospherical, W — andV — congruences) in an isotropic space of degree oneJ3(1), i.e. a three dimensional real affine space with the metricds2=dx2+dy2. It is proved that many results from the euclidean theory are applicable inJ3(1).

Monte Carlo study of Abelian lattice gauge theories

Using Monte Carlo techniques, we study the thermodynamics of four-dimensional Euclidean lattice gauge theories, with gauge groups ${Z}_{N}$ and U(1). For $N\ensuremath{\le}4$ the models exhibit a single first-order phase transition, while for $N\ensuremath{\ge}5$ we observe two transitions of higher order. As $N$ increases, one of these transitions moves toward zero temperature, whereas the other remains at finite temperature and survives in the U(1) limit. The behavior of the Wilson loop factor is also analyzed for the ${Z}_{2}$ and ${Z}_{6}$ models.

Vacuum polarization effects in lattice gauge theories

In a 4-dimensional euclidean lattice gauge theory, vacuum polarization is investigated within a strong coupling expansion. In leading order, analogue potentials are calculated for quark-antiquark and three-quark systems. They show a characteristic plateau behaviour for large distances. There the values of the potentials are determined by hadronic masses.

A pathology of the Landau gauge in euclidean Yang-Mills theories

It is argued that the Landau gauge in euclidean Yang-Mills theories exhibits the following pathological features similar to those of the Coulomb gauge: (1) As far as asymptotically O(3) symmetric potentials are concerned, the potential for the field configuration with | q|1 is necessarily singular, where q is the topological quantum number of the configuration. (2) If furthermore the boundary condition r A→0 as r→∞ is imposed, then even the | q|=1 configuration necesstitates singularities of the potential.

Cluster expansion for lattice gauge theories with fermions

A cluster expansion is constructed for Euclidean lattice gauge theories including fermions. For sufficiently small values of the bare qauge coupling constant g−2 and the fermion propagation constant k, we prove the existence of gauge invariant infinite volume vacuum expectation values, convergence of Wilson’s strong coupling expansion, and an inequality related to quark confinement.

Complex solutions of Yang-Mills

Complex solutions to classical euclidean Yang-Mills theory are found with zero action. There are both configurations which are neither self-dual nor anti-self-dual, and self-dual solutions. The quantity PeφAμdxμ is computed. The working of a Bäcklund transformation between these finite (zero) action solutions and the multi-instantons is illustrated.

Singular pseudoparticles in Higgs theories

We present a new class of pseudoparticles in four-dimensional Euclidean non-Abelian gauge theories with Higgs fields. These pseudoparticles, in the classical limit, have nonzero Lagrangian density at only one single point. We discuss the topological and analytic construction of these pseudoparticles and the related fractional-topological-charge configurations. The action of a unit-topological-charge pseudoparticle is the usual $\frac{8{\ensuremath{\pi}}^{2}}{{g}^{2}}$. These pseudoparticles are not field configurations, but are limiting cases of field configurations. Owing to quantum fluctuations, their size is proportional to $\sqrt{\ensuremath{\hbar}}$ if $\ensuremath{\hbar}$ is small.

Construction of a selfadjoint, strictly positive transfer matrix for Euclidean lattice gauge theories

It is shown that physical positivity holds in Wilson's lattice gauge theories, i.e. transition probabilities between gauge invariant states are non-negative and the quantum mechanical Hamiltonian has real eigenvalues only.