An Optimised Perturbation Expansion for a Global O(2) Theory

We use an optimised perturbation expansion called the linear δ-expansion to study the phase transition in a Higgs sector with a continuous symmetry and large couplings. Our results show how to use this non-perturbative method successfully for such problems. We also show how to simplify the method without losing any flexibility. Quantum Field Theory has had many successes over the last fifty years but most are based on the use of small coupling perturbation expansions. However, there are many interesting problems where such an expansion is not appropriate. QCD is the classic example but phase transitions at non-zero temperatures even in models with small coupling constants, e.g. the Electro Weak model, are now known to be in this class. It is this latter example which motivates our work which will focus on the non-perturbative behaviour of the Higgs sector. Alternatives to perturbation theory, non-perturbative methods, have many limitations. For instance it is usually very difficult to go beyond the lowest order in methods such as large N [1] while numerical Monte Carlo codes are relatively expensive [2, 3]. Another method with a long history is the Linked Cluster Expansion [4, 5] which now also exploits large amounts of computing power [6]. In this letter we extend a non-perturbative method called an OPE (Optimised Perturbation Expansion). In this approach one expands around a solvable model which contains several arbitrary parameters, and thus is a variational ansatz. The expansion is not necessarily in terms of some small parameter. One then chooses the variational parameters, invariably using some optimisation criteria which is a highly non-linear procedure. The optimisation criteria has to ensure good results from a few terms of the series but there has been much discussion about what is a good optimisation criteria. As described, the OPE is a very general method so not surprisingly it has been rediscovered several times in different guises, applied to very different applications, and appears under many different names; it is also known as the linear δ-expansion [7], email: [email protected], WWW: http://euclid.tp.ph.ic.ac.uk/~time email: [email protected] email: [email protected], present address: Enrico Fermi Institute, University of Chicago 1 action-variational approach [8], improved gaussian approximation [9], variational perturbation theory [10], method of self-similar approximation [11], or the variational cumulant expansion [12]. For instance the method has been to the evaluation of simple integrals [13, 14, 7], solving non-linear differential equations [15], quantum mechanics [16, 17, 18, 9, 7] to quantum field theory both in the continuum [9, 10] and on a lattice [7, 19, 20, 21, 22, 23, 24, 25, 26, 27, 8, 12]. To motivate this work we first note that the analysis of pure gauge theories on the lattice using OPE used a variety of optimisation schemes when fixing the variational parameters. One can minimise the free energy [22] or demand that the series converges as fast as possible by minimising the high order terms with respect to the lower order terms [8] the principle of fastest convergence. However, we believe the best results for pure gauge models are obtained when using the principle of minimal sensitivity [28], as discussed in [7, 19, 20, 21]. In this case the variational parameters, say {α}, are set separately for each physical quantity, say O, under consideration by demanding that the quantity changes as little as possible if the variational parameters {α} are varied from their optimal values {α̃}, specifically ∂〈O〉 ∂αi ∣