**Principal Investigator**
Anthony G. Williams **Project **g21

Department of Physics and Mathematical Physics, **Machine
**VP

University of Adelaide

**Co-Investigators **Lindsay
R. Dodd and Armin Ardekani

Department of Physics and Mathematical Physics, University of Adelaide

**Research in Computational Physics**

The central aim of this work is to use modern supercomputers to extend our understanding of some of the more complex behaviour exhibited by fundamental theories of matter. This project is concerned with the investigation of the nonperturbative behaviour of quantum field theories with some emphasis on understanding the quark substructure of the atomic nucleus and the associated strong forces which bind the nucleus together. The investigative techniques involve the simulation of field theories on a discrete space-time lattice. We propose to study phase transitions in various model field theories and to study the formation of bound states and extended structures in these theories.

**What are the basic questions addressed?**

As an initial step we are studying the pure boson sector of theories (i.e, without fermions) such as f4 in 2-dimensions. A motivating aspect of this work is the ability to study extended objects (called kinks) on the lattice, which can then be directly compared to semiclassical configurations. In addition these simple models can be used to develop our understanding of spontaneous symmetry breaking, which we know takes place in both quantum chromodynamics (QCD) (the elementary theory of the strong interactions) and in the so-called standard model (the theory which unifies the weak and electromagnetic interactions). Also underway is the extension of these techniques to include fermions, to work in higher dimensions, and to directly study QCD itself. The latter work on QCD is highly computer intensive and is being carried out in collaboration with K.F. Liu and his group at the University of Kentucky in the U.S.A. We are also studying non-lattice approaches to nonperturbative quantum field theory based on the Dyson-Schwinger equations. This includes studies of covariant bound states in terms of the Bethe-Salpeter equation.

**What are the results to date and future of the
work?**

We have obtained some configurations for models including dynamical fermions but some difficulties have become apparent. The generation of configurations is very time-consuming so that direct study of the critical limits which is necessary to obtain physical results may not be the optimal approach. We are therefore examining the possibility of using a combination of analytical and numerical methods to extract information from the simulations. Renormalization group techniques can be applied in certain circumstances. The program has been checked for the Gaussian Model (i.e. for the free field theory of a system of non-interacting bosons). The correlation functions (propagators ) on the lattice were measured and agree within 5% with the analytic forms which are known. Extraction of the particle mass from the correlation length on the lattice was found to have significant systematic finite size effects. An alternative pole definition of the mass was found to be more accurate. The next step in the calculations has been to include the f4 interaction. Currently results from the lattice calculations are being compared to theoretical calculations using lattice perturbation theory in the weak coupling regime. We will then require many hours of CPU time to locate the continuum limit of the theory. (This involves the fine tuning of parameters to identify the phase transitions of the lattice, regarded as a statistical mechanical system). One Ph.D. student (Armin Ardekani) has begun a project to understand how information about the fermion propagators may be derived from correlation functions on the lattice. The model has a rich and complicated spectrum of solutions and we expect that the lifetime of the project will be several years with a need for large amounts (several hundreds of hours of computer time) as the project develops.

The study of nonperturbative renormalization in the presence of spontaneous symmetry breaking in quantum electrodynamics was studied in terms of the Dyson-Schwinger formalism for field theories. This was the first ever consistent treatment of the full subtractive renormalization scheme in this formalism and will allow for the first time a direct comparison of the lattice gauge theory and Dyson-Schwinger based approaches. In addition, the Bethe-Salpeter (BS) equation for scalar-scalar bound states in scalar theories without derivative coupling was formulated and solved in Minkowski space. This was achieved using the perturbation theory integral representation (PTIR), which allows these amplitudes to be expressed as integrals over weight functions and known singularity structures and hence allows us to convert the BS equation into an integral equation involving weight functions. We obtained numerical solutions using this formalism for a number of scattering kernels to illustrate the generality of the approach. It applies even when the naive Wick rotation is invalid. As a check we verified, for example, that this method applied to the special case of the massive ladder exchange kernel reproduces the same results as are obtained by Wick rotation. This was the first numerical solution of the Bethe-Salpeter equation for relativistic bound states using the PTIR approach.

**What computational techniques are used and why
is a supercomputer required?**

The basic ingredients of lattice based studies are degrees of freedom on a lattice representing space-time and Monte-Carlo methods to simulate the dynamics of the physical system. The specific algorithm used here is the Hybrid Monte Carlo method, which combines the best features of the deterministic integration of the equations of motion (leap frog algorithm) and importance sampling (Metropolis method). As the limit of large lattices is taken the computational power needed grows extremely rapidly, meaning that forefront calculations necessarily require the fastest available supercomputers. For the Dyson-Schwinger and Bethe-Salpeter equation studies the system of nonlinear coupled integral equations is solved by iteration from an initial guess on a logarithmic grid in momentum space. We routinely obtain an accuracy of 1 in 10,000 for our solutions over twenty orders of magnitude variation in the momentum

**Publications**

*Chiral symmetry breaking in quenched massive strong
coupling four-dimensional QED* , F.T. Hawes
and A.G. Williams, Phys. Rev. D **51**, 3081-3089 (1995).

*Solving the Bethe-Salpeter equation for scalar
theories in Minkowski space*, K. Kusaka
and A.G. Williams, Phys. Rev. D **51**, 7026-7039 (1995).