Resums

Resums, short for resummations, are mathematical and numerical techniques used to convert divergent or asymptotic series into finite, meaningful expressions. They arise when perturbation theory yields series that do not converge, yet the partial sums hint at a well-defined value. The aim of resummation is to extract reliable predictions from these series by reorganizing or extending the sum over terms to all orders.

Several common methods are used in physics and applied mathematics. Borel resummation transforms the original series into a Borel transform, which can be integrated to define a finite value under suitable analyticity conditions. Padé approximants replace a power series by a ratio of polynomials to improve convergence or extrapolate beyond the known terms, and Padé–Borel combines these ideas. Conformal mapping of the Borel plane reshapes the domain of analyticity to enhance convergence properties. Renormalization-group–improved resummation uses renormalization group equations to sum leading logarithms and related contributions, often in quantum field theory. In high-energy processes, Sudakov resummation and related techniques systematically sum large logarithms associated with soft and collinear emissions.

Applications span quantum chromodynamics, electroweak theory, and critical phenomena in statistical mechanics, where resummation yields more accurate predictions from perturbative expansions. Limitations include non-uniqueness and scheme dependence, as different resummation prescriptions may yield different finite results. Moreover, not all series are resummable, and the success of a method depends on the analytic structure of the underlying function. Resums remain a central tool for bridging perturbation theory and finite, physically meaningful outcomes.