Fast-convergent resummation algorithm and critical exponents of \(\phi^4\)-theory in three dimensions. (English) Zbl 1013.81039
Summary: 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 information: the knowledge of the large-order behavior proportional to \((-\alpha)^k k!k^\beta g^k_B\), with a known growth parameter \(\alpha\), and the knowledge of the approach to scaling of the type \(c+c'/g^\omega_B\), 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 a recent satellite experiment.
MSC:
| 81T15 | Perturbative methods of renormalization applied to problems in quantum field theory |
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 65D15 | Algorithms for approximation of functions |
| 82B27 | Critical phenomena in equilibrium statistical mechanics |
Keywords:
divergent perturbation expansions; bare coupling constant; large-order behavior; approach to scalingReferences:
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