Basic Resummation Theory

The power series in the coupling constant g or in ε derived so far are unsuitable for numeric calculations since they are divergent for any coupling strength. They are called asymptotic series. Resummation procedures are necessary to extract physical results. The crudest method to approximate a divergent function f (g) whose first L expansion coefficients are known employs Padé approximants [1]. These are rational functions with the same power series expansions as f (g). A better approximation can be found by taking into account the knowledge of the large-order behavior of the expansion coefficients. This is provided by a semiclassical analysis of the functional integral (2.6) of the φ 4-theory, from which it follows that the coefficients grow factorially with the order. Such a factorial growth can be taken into account by means of Borel transformations. The Padé-Borel method applies the Padé approximation to the Borel transform. We shall see below that the Borel transform has a left-hand cut in the complex g-plane. The Padé approximation replaces the cut by a string of poles. This approximation can be improved further by a conformal mapping technique in which the complex g-plane is mapped into a unit circle which contains the original left-hand cut on its circumference. More efficient resummation techniques are based on re-expansions of the asymptotic truncated series for f (g) in terms of special basis functions I n (g). These can be chosen in different ways to possess precisely the analytic behavior responsible for the divergence of the original series. Thus the basis functions contain from the outset the necessary information on the large-order behavior of the expansion coefficients, thus containing information beyond the expansion coefficients of the perturbation series. All these methods will be explained in this chapter. The most efficient technique will be applied in Chapters 17 and 18 to calculate critical exponents. A completely different approach to the critical exponents and the resummation problem will be developed in Chapters 19 and 20. There we shall consider the problem of calculating the critical behavior of φ 4-theory as a strong-coupling problem in the bare coupling constant g B. At large orders, the perturbation coefficients of the renormalization group functions grow like a factorial of the order, with alternating signs. This behavior has a simple origin. If we plot the potential part of the energy density in (2.1) for small negative values of the coupling constant g as …