Summing divergent perturbative series in a strong coupling limit. The Gell-Mann-Low function of the ϕ4 theory

Abstract

-An algorithm is proposed for determining asymptotics of the sum of a perturbative series in the strong coupling limit using given values of the expansion coefficients. Application of the algorithm is illustrated, methods for estimating errors are developed, and an optimization procedure is described. Applied to the ϕ 4 theory, the algorithm yields the Gell-Mann-Low function asymptotics of the type β ( g ) ≈ 7.4 g 0.96 for large g . The fact that the exponent is close to unity can be interpreted as a manifestation of the logarithmic branching of the type β ( g ) ~ g (ln g ) -γ (with γ ≈ 0.14), which is confirmed by independent evidence. In any case, the ϕ 4 theory is self-consistent. The procedure of summing perturbative series with arbitrary values of the expansion parameter is discussed.

Strong coupling asymptotics of the β-function in theory and QED

“…(ii) to find out the mechanism leading to this asymptotics; Figure 4: (a) General appearance of the β-function in four-dimensional ϕ 4 theory according to [24] (solid curve), and results obtained by other authors (upper, middle, and lower dashed curves correspond to [11,13,20] respectively). (b) Different estimations of the exponent α according to [24]. Figure 5: Estimations of the exponent α for ϕ 4 theory in two and three dimensions [18,19].…”

“…The described algorithm was successfully tested for a lot of simple examples [24], and now we can apply it to a reconstruction of the Gell-Mann -Low function β(g) of quantum field theories. This function enters the Gell-Mann -Low equation which describes the behavior of the effective charge g as a function of the length scale L:…”

“…In the case of four-dimensional ϕ 4 theory, a realization of this program [24] gives the non-alternating β-function ( Fig. 4, a), with the results for the exponent α shown in Fig.…”

“…In what follows, we identify the function W (g) with the Borel sum of its perturbation series. In the case of ϕ 4 theory, it is possible to test a validity of such identification in one and zero dimensions [24] and to prove the Borel summability in two and three dimensions [16,6]. It is easy to show that the Borel transform B(z) has a singularity at the point z = −1/a ( Fig.…”

“…Estimation of errors was made in a framework of a certain procedure worked out in[24]. Subsequent applications have shown that such estimation is not very reliable.…”