Linked Cluster Expansions on non-trivial topologies

Linked cluster expansions provide a useful tool both for analytical and numerical investigations of lattice field theories. The expansion parameter is the interaction strength fields at neighboured lattice sites are coupled. They result into convergent series for free energies, correlation functions and susceptibilities. The expansions have been generalized to field theories at finite temperature and to a finite volume. Detailed information on critical behaviour can be extracted from the high order behaviour of the susceptibility series. We outline some of the steps by which the 20th order is achieved.Comment: 3 pages, Talk presented at LATTICE96(Theoretical Developments

Chiral symmetry restoration of QCD and the Gross-Neveu model

Two flavour massless QCD has a second order chiral transition which has been argued to belong to the universality class of the $3d$ O(4) spin model. The arguments have been questioned recently, and the transition was claimed to be mean field behaved. We discuss this issue at the example of the $3d$ Gross-Neveu model. A solution is obtained by applying various well established analytical methods.Comment: LATTICE98(hightemp

Dynamical linke cluster expansions: Algorithmic aspects and applications

Dynamical linked cluster expansions are linked cluster expansions with hopping parameter terms endowed with their own dynamics. They amount to a generalization of series expansions from 2-point to point-link-point interactions. We outline an associated multiple-line graph theory involving extended notions of connectivity and indicate an algorithmic implementation of graphs. Fields of applications are SU(N) gauge Higgs systems within variational estimates, spin glasses and partially annealed neural networks. We present results for the critical line in an SU(2) gauge Higgs model for the electroweak phase transition. The results agree well with corresponding high precision Monte Carlo results.Comment: LATTICE98(algorithms

Renormalization of lattice gauge theories with massless Ginsparg Wilson fermions

Using functional techniques, we prove, to all orders of perturbation theory, that lattice vector gauge theories with Ginsparg Wilson fermions are renormalizable. For two or more massless fermions, they satisfy a flavour mixing axial vector Ward identity. It involves a lattice specific part that is quadratic in the vertex functional and classically irrelevant. We show that it stays irrelevant under renormalization. This means that in the continuum limit the (standard) chiral symmetry becomes restored. In particular, the flavour mixing current does not require renormalization.Comment: 13 pages, Latex2

Background field technique and renormalization in lattice gauge theory

Lattice gauge theory with a background gauge field is shown to be renormalizable to all orders of perturbation theory. No additional counterterms are required besides those already needed in the absence of the background field. The argument closely follows the treatment given earlier for the case of dimensional regularization by Kluberg-Stern and Zuber. It is based on the BRS, background gauge and shift symmetries of the lattice functional integral.Comment: 26 pages, uuencoded compressed postscript fil

Finite Size Scaling Analysis with Linked Cluster Expansions

Linked cluster expansions are generalized from an infinite to a finite volume on a $d$-dimensional hypercubic lattice. They are performed to 20th order in the expansion parameter to investigate the phase structure of scalar $O(N)$ models for the cases of $N=1$ and $N=4$ in 3 dimensions. In particular we propose a new criterion to distinguish first from second order transitions via the volume dependence of response functions for couplings close to but not at the critical value. The criterion is applicable to Monte Carlo simulations as well. Here it is used to localize the tricritical line in a $\Phi^4 + \Phi^6$ theory. We indicate further applications to the electroweak transition.Comment: 3 pages, 1 figure, Talk presented at LATTICE96(Theoretical Developments

Coordinate space methods for the evaluation of Feynman diagrams in lattice field theory

We describe an efficient position space technique to calculate lattice Feynman integrals in infinite volume. The method applies to diagrams with massless propagators. For illustration a set of two-loop integrals is worked out explicitly. An important ingredient is an observation of Vohwinkel that the free lattice propagator can be evaluated recursively and is expressible as a linear function of its values near the origin.Comment: 27 pages, postscript file, now compressed & uuencode

High temperature critical O(N) field models by LCE series

The critical properties of renormalizable O(N) field models are determined by means of the high order ($\geq 18$) behaviour of convergent linked cluster series on finite temperature lattices. It is shown that those models become weakly coupled at the phase transition. The critical exponents agree to those of the corresponding superrenormalizable 3-dimensional models. Concerning critical amplitudes and subcritical behaviour, corrections induced by renormalizable couplings are measurable.Comment: 10 pages, late

Hopping Parameter Series Construction for Models with Nontrivial Vacuum

Hopping parameter expansions are convergent power series. Under general conditions they allow for the quantitative investigation of phase transition and critical behaviour. The critical information is encoded in the high order coefficients. Recently, 20th order computations have become feasible and used for a large class of lattice field models both in finite and infinite volume. They have been applied to quantum spin models and field theories at finite temperature. The models considered are subject to a global ${\bf Z}_2$ symmetry or to an even larger symmetry group such as O(N) with $N\geq 2$. In this paper we are concerned with the technical details of series computations to allow for a nontrivial vacuum expectation value $\not=0$, which is typical for models that break a global ${\bf Z}_2$ symmetry. Examples are scalar fields coupled to an external field, or manifestly gauge invariant effective models of Higgs field condensates in the electroweak theory, even in the high temperature phase. A nonvanishing tadpole implies an enormous proliferation of graphs and limits the graphical series computation to the 10th order. To achieve the hopping parameter series to comparable order as in the ${\bf Z}_2$ symmetric case, the graphical expansion is replaced by an expansion into new algebraic objects called vertex structures. In this way the 18th order becomes feasible.Comment: 19 pages, latex2