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Large-order asymptotic terms in perturbation theory: The first (4-ε)-expansion correction to renormalization constants in the O(n)-symmetric theory

“…In [3], an analysis of this question for the (4− )-MS scheme showed that the renormalization constants approach their asymptotic values monotonically, but they are nevertheless rather far from them (for example, the exactly calculated value of the renormalization constant Z g , as is seen, is approximately 75 times as great as its asymptotic value for N = 5). This casts some doubt on the accuracy of the Borel resummation in this scheme and requires either continuing the calculation of loop expansions until they approach the asymptotic expressions to the required accuracy or considering the (1/N )-order correction to the asymptotic formula [3].…”

“…Later, a method was suggested in [3] for calculating asymptotic expressions for critical indices in the ϕ 4 -model based on instanton analysis of ultraviolet (UV) divergences, i.e., a method for calculating large-order PT approximations for the renormalization constants was in fact elaborated. There the minimal subtraction (MS) scheme in the (4− )-expansion was used as the most technically advanced method (namely, five PTorder terms were calculated).…”

“…This correction improves the rate of approaching the asymptotic expression for loop expansions in the unit renormalization scheme. But to obtain this correction for the renormalization constants in the (4− )-MS scheme, as in [3], some additional calculations are needed to single out the UV divergences in the N th PT-order asymptotic expressions.…”

“…The renormalization constant Z g (as well as the related renormalization group function β(g)) is one of the main characteristics for determining the asymptotic behavior of critical indices [3]. Precisely this characteristic determines the rate of approaching the loop-calculation asymptotic results, and we therefore confine ourself to only investigating large-order asymptotic terms for the renormalization constant Z g .…”

Large-order asymptotes for dynamic models near equilibrium

“…This behaviour has been thoroughly explored with the aid of instanton analysis and applied to the resummation problem in the prototypical static φ 4 model [2,3,4], which has been widely used as a model of critical behaviour in continuous phase transitions of ferromagnetic type.…”