Numerical Study of Periodic Instanton Configurations in Two-dimensional Abelian Higgs Theory

Numerical minimization of the Euclidean action of the two-dimensional Abelian Higgs model is used to construct periodic instantons, the euclidean field configurations with two turning points describing transitions between the vicinities of topologically distinct vacua. Periodic instantons are found at any energy ( up to the sphaleron energy $E_{sph}$ ) and for wide range of parameters of the theory. We obtain the dependence of the action and the energy of periodic instanton on its period; these quantities directly determine the probability of certain multiparticle scattering events.Comment: 8 pages, 6 figures available upon request, LaTeX, ITP-SB-92-7

Skewed Sudakov Regime, Harmonic Numbers, and Multiple Polylogarithms

On the example of massless QED we study an asymptotic of the vertex when only one of the two virtualities of the external fermions is sent to zero. We call this regime the skewed Sudakov regime. First, we show that the asymptotic is described with a single form factor, for which we derive a linear evolution equation. The linear operator involved in this equation has a discrete spectrum. Its eigenfunctions and eigenvalues are found. The spectrum is a shifted sequence of harmonic numbers. With the spectrum found, we represent the expansion of the asymptotic in the fine structure constant in terms of multiple polylogarithms. Using this representation, the exponentiation of the doubly logarithmic corrections of the Sudakov form factor is recovered. It is pointed out that the form factor of the skewed Sudakov regime is growing with the virtuality of a fermion decreasing at a fixed virtuality of another fermion.Comment: 6 page

The Challenge of Light-Front Quantisation: Recent Results

We explain what is the challenge of light-front quantisation, and how we can now answer it because of recent progress in solving the problem of zero modes in the case of non-Abelian gauge theories. We also give a description of the light-front Hamiltonian for SU(2) finite volume gluodynamics resulting from this recent solution to the problem of light-front zero modes.Comment: 17 pages, lecture delivered by GBP at the XXXIV PNPI Winter School, Repino, St.Petersburg, Russia, February 14-20, 2000, version to appear in the Proceeding

Complex-Temperature Phase Diagram of the 1D Z6Z_6 Clock Model and its Connection with Higher-Dimensional Models

We determine the exact complex-temperature (CT) phase diagram of the 1D $Z_6$ clock model. This is of interest because it is the first exactly solved system with a CT phase boundary exhibiting a finite-$K$ intersection point where an odd number of curves (namely, three) meet, and yields a deeper insight into this phenomenon. Such intersection points occur in the 3D spin 1/2 Ising model and appear to occur in the 2D spin 1 Ising model. Further, extending our earlier work on the higher-spin Ising model, we point out an intriguing connection between the CT phase diagrams for the 1D and 2D $Z_6$ clock models.Comment: 10 pages, latex, with two epsf figure

A Connection Between Complex-Temperature Properties of the 1D and 2D Spin ss Ising Model

Although the physical properties of the 2D and 1D Ising models are quite different, we point out an interesting connection between their complex-temperature phase diagrams. We carry out an exact determination of the complex-temperature phase diagram for the 1D Ising model for arbitrary spin $s$ and show that in the $u_s=e^{-K/s^2}$ plane (i) it consists of $N_{c,1D}=4s^2$ infinite regions separated by an equal number of boundary curves where the free energy is non-analytic; (ii) these curves extend from the origin to complex infinity, and in both limits are oriented along the angles $\theta_n = (1+2n)\pi/(4s^2)$, for $n=0,..., 4s^2-1$; (iii) of these curves, there are $N_{c,NE,1D}=N_{c,NW,1D}=[s^2]$ in the first and second (NE and NW) quadrants; and (iv) there is a boundary curve (line) along the negative real $u_s$ axis if and only if $s$ is half-integral. We note a close relation between these results and the number of arcs of zeros protruding into the FM phase in our recent calculation of partition function zeros for the 2D spin $s$ Ising model.Comment: 8 pages, latex, 2 uuencoded figure