Fast-Convergent Resummation Algorithm and Critical Exponents of phi⁴-Theory in Three Dimensions

We develop an efficient algorithm for evaluating divergent perturbation expansions of field theories in the bare coupling constant g_B for which we possess a finite number L of expansion coefficients plus two more informations: The knowledge of the large-order behavior proportional to (- alpha)^kk!k^beta g_B^k, with a known growth parameter alpha, and the knowledge of the approach to scaling being of the type c+c'/g_B^ omega, with constants c,c' and a critical exponent of approach omega . The latter information leads to an increase in the speed of convergence and a high accuracy of the results. The algorithm is applied to the six- and seven-loop expansions for the critical exponents of O(N)-symmetric phi^4-theories, and the result for the critical exponent alpha is compared with the recent satellite experiment.