Divergent Perturbation Series

Various perturbation series are factorially divergent. The behavior of their high-order terms can be found by Lipatov’s method, according to which they are determined by instanton configurations of appropriate functional integrals. When the Lipatov asymptotics is known and several lowest order terms of the perturbation series are found by direct calculation of diagrams, one can gain insight into the behavior of the remaining terms of the series. Summing it, one can solve (in a certain approximation) various strong-coupling problems. This approach is demonstrated by determining the Gell-Mann–Low functions in φ4 theory, QED, and QCD for arbitrary coupling constants. An overview of the mathematical theory of divergent series is presented, and interpretation of perturbation series is discussed. Explicit derivations of the Lipatov asymptotics are presented for some basic problems in theoretical physics. A solution is proposed to the problem of renormalon contributions, which hampered progress in this field in the late 1970s. Practical perturbation-series summation schemes are described for a coupling constant of order unity and in the strong-coupling limit. An interpretation of the Borel integral is given for “non-Borel-summable” series. High-order corrections to the Lipatov asymptotics are discussed. DIVERGENT PERTURBATION SERIES 1189 1. DYSON’S ARGUMENT: IMPORTANT PERTURBATIVE SERIES HAVE ZERO RADIUS OF CONVERGENCE Classical books on diagrammatic techniques [2–4] describe the construction of diagram series as if they were well defined. However, almost all important perturbation series are hopelessly divergent since they have zero radii of convergence. The first argument to this effect was given by Dyson [5] with regard to quantum electrodynamics. Here, it is reiterated by using simpler examples. Consider a Fermi gas with a delta-function interaction g δ ( r – r ') and the corresponding perturbation series in terms of the coupling constant g. Its radius of convergence is determined by the distance from the origin to the nearest singular point in the complex plane and can be found as follows. In the case of a repulsive interaction ( g > 0), the ground state of the system is a Fermi liquid. When the interaction is attractive ( g < 0), the Cooper instability leads to superconductivity (see Fig. 1a). As g is varied, the ground state qualitatively changes at g = 0. Thus, the nearest singular point is located at the origin, and the convergence radius of the series is zero. An even simpler example is the energy spectrum of a quantum particle in the one-dimensional anharmonic potential (1.1) Whereas the system has well-defined energy levels U x ( ) x gx. + = when g > 0, these levels are metastable when g < 0 since the particle can escape to infinity (see Fig. 1b). Therefore, the perturbation series in terms of g is divergent for any finite g as it can be tested by direct calculation of its coefficients. The calculation of the first 150 coefficients of this series in [6] was the first demonstration of its divergence and gave possibility of its detailed study. Zero radius of convergence looks “accidental” in quantum-mechanical problems: it takes a place when a potential of special form is taken and a “bad” definition of coupling constant is chosen. However, zero radius of convergence is encountered in all fundamental quantum field theories with a single coupling constant. Even though Dyson’s argument is unquestionable, it was hushed up or decried for many years: the scientific community was not ready to face the problem of the hopeless divergency of perturbation series. 2. LIPATOV’S METHOD: QUANTITATIVE ESTIMATION OF DIVERGENCY OF SERIES A further step was made in 1977, when Lipatov’s method was proposed [7] as a tool for calculating high-order terms in perturbation series and making quantitative estimates for its divergence. The idea of the method is as follows. If a function F ( g ) can be