Abstract
We employ a recent resummation method to deal with divergent series, based on the Meijer G-function, which gives access to the non-perturbative regime of any QFT from the first few known coefficients in the perturbative expansion. Using this technique, we consider in detail the ϕ4 model where we estimate the non-perturbative β-function and prove that its asymptotic behavior correctly reproduces instantonic effects calculated using semiclassical methods. After reviewing the emergence of the renormalons in this theory, we also speculate on how one can resum them. Finally, we resum the non-perturbative β-function of abelian and non-abelian gauge-fermion theories and analyze the behavior of these theories as a function of the number of fermion flavors. While in the former no fixed points are found, in the latter, a richer phase diagram is uncovered and illustrated by the regions of confinement, large-distance conformality, and asymptotic safety.
Highlights
The perturbative expansion in QFT has zero radius of convergence [1] and the truncated series is strictly valid only for infinitesimal couplings
We focus on a recent method of Borel-hypergeometric resummation proposed in Ref. [5, 6], in which the Padeapproximants are replaced by the more sophisticated hypergeometric functions, and the resummed result admits a representation in terms of Meijer G-functions (MGs)
It is worth noticing that qualitatively this is in full analogy with the 0-dimensional functional discussed above. This confirms the power of Borel-hypergeometric resummation to scan non-perturbative physics from perturbative inputs and shows the natural applicability of the method to reconstruct the non-perturbative β−function from its truncated power series
Summary
The perturbative expansion in QFT has zero radius of convergence [1] and the truncated series is strictly valid only for infinitesimal couplings. A well-known method to perform the analytic continuation is through the Padeapproximants and the whole resummation approach is often called Borel-Paderesummation It should be stressed, that a number of alternatives exist, as for example the large-coupling-expansion which builds a power series expansion in the inverse of the coupling (see [3, 4] for reviews). The resummed series in terms of Meijer G-functions can shed light on another question in QFT, namely, the understanding of the renormalization group (RG) flow in the theory space of couplings This is a fundamental task that is again obscured by a partial knowledge of the β−functions in the form of the divergent truncated series.
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