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We study the analytic properties of the ’t Hooft coupling expansion of the beta function at the leading nontrivial

The discovery of four-dimensional asymptotically safe quantum field theories

This quest has revitalized the study of quantum field theories at a large number of flavors

An intriguing property of this limit is that, at each order in the

The first notable study is

In the limit of a large number of matter fields, it is natural to introduce the ’t Hooft coupling

Asymptotic freedom is lost for theories at a finite number of colors and a large number of flavors, and therefore such theories can only be fundamental if they develop an interacting fixed point in the ultraviolet. This cannot occur in perturbation theory without Yukawa interactions

In this paper, we investigate this phenomenon further and make a systematic study of the analytic structure of the ’t Hooft coupling expansion of the leading

The paper is organized as follows: In Sec.

QED is structurally the simplest gauge theory, but it still has a rich perturbative and nonperturbative structure, which we probe here in the

Leading-order

The first few terms in the

Before discussing the analytic structure of the integral representation in Eq.

Our goal in this section is to deduce physical information from a finite number of expansion coefficients

Thus, using Darboux’s theorem (see Appendix

Logarithm of

Thus, the leading behavior of the

Recall that the poles in Eq.

Padé approximation is a commonly used method for studying perturbative expansions in physical systems

Padé approximants construct analytic continuations of truncated Taylor series (i.e., polynomial) approximations to functions, expressing the given polynomial as a ratio of two polynomials of lower order, with coefficients determined purely algorithmically. Padé approximants thus convert a polynomial to a rational function, which can also be expressed as a partial fraction expansion, whose residues and poles are determined by the coefficients of the original truncated Taylor series. This means that Padé approximants tend to be quite good at representing functions with poles, while they are less good at representing functions with branch cuts

The conversion of a truncated Taylor series to a Padé approximant,

We took up to 60 terms of the expansion about

Integrand of the QED beta function

In fact, a full Padé analysis constructs the “Padé table” of all Padé approximants

We have seen that the finite radius of convergence,

This is the case for diagrams containing one resummed gauge chain.

For the QED computation, the value of the

Knowing this, one can devise

The general idea of using Padé approximants to study the behavior of beta functions is significant for analyzing higher orders in the

A new singular structure may emerge closer to the origin,

The current singular structure, and its location, could be further reinforced by higher-order corrections. This possibility is partially supported by the fact that the fermion self-energy amplitude is responsible for the singular structure of the theory. The order of the pole in the integrand might become stronger because

No further singularities emerge, or a new singular structure appears further away from the leading-order one. This would be an indication that the putative fixed point in Eq.

The beta function at order

The simple pole of

Number of poles in each theory, ordered according to Table

Our results across the various theories indicate that the main factor determining the number of coefficients needed to resolve a given pole is the distance of the latter to the origin.

A simple rescaling of the couplings does not change the number of coefficients needed.

As discussed in the end of Sec.In this section, we review and discuss the results obtained in Refs.

In our convention, the coupling

Poles and first residue of the

It is interesting to note that this conclusion holds also in the Novikov-Shifman-Vainshtein-Zakharov (NSVZ) scheme, which can be related to DRED by an order-by-order coupling redefinition

Our analysis of the convergence properties of the leading

We have demonstrated, by direct comparison with the full result, that the analysis of the large-order behavior of the ’t Hooft coupling expansion is able to identify the location and nature of the leading logarithmic singularities, including the overall sign and magnitude of their coefficients. This suggests that a large-order analysis can be used in the near future to tackle the next-to-leading order in the

This work is partially supported by Danish National Research Foundation Grant No. DNRF:90, and is based upon work supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics under Award No. DE-SC0010339.

Darboux’s theorem says that for a convergent series expansion, the behavior of the expansion in the vicinity of a nearby singularity is determined by the large-order growth of the expansion coefficients about another point (say

The

The diagrams that contribute to the beta function at the order

This procedure can be extended to higher orders in the

Another complication at higher order in