Formulation for renormalon-free perturbative predictions beyond large- β 0 approximation

Published for SISSA by Springer Received: February 24, 2020 Revised: August 20, 2020 Accepted: September 10, 2020 Published: October 7, 2020 Hiromasa Takaura Department of Physics, Kyushu University, Fukuoka, 819-0395, Japan E-mail: Abstract: We present a formulation to give renormalon-free predictions consistently with fixed order perturbative results. The formulation has a similarity to Lee’s method in that the renormalon-free part consists of two parts: one is given by a series expansion which does not contain renormalons and the other is the real part of the Borel integral of a singular Borel transform. The main novel aspect is to evaluate the latter object using a dispersion relation technique, which was possible only in the large-β0 approximation. Here, we introduce an “ ambiguity function,” which is defined by inverse Mellin transform of the singular Borel transform. With the ambiguity function, we can rewrite the Borel integral in an alternative formula, which allows us to obtain the real part using analytic techniques similarly to the case of the large-β0 approximation. We also present detailed studies of renormalization group properties of the formulation. As an example, we apply our formulation to the fixed-order result of the static QCD potential, whose closest renormalon is already visible. Keywords: Renormalization Regularization and Renormalons, Resummation, Perturbative QCD ArXiv ePrint: 2002.00428 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP10(2020)039 JHEP10(2020)039 Formulation for renormalon-free perturbative predictions beyond large-β0 approximation Contents 1 2 Formulation 2.1 Resummation formula with ambiguity function 2.2 Explicit form of ambiguity function 2.3 Preweight 2.4 Renormalon-free part in cutoff regularization 2.5 Renormalon-free part in contour regularization 2.6 Renormalization group properties 2.7 Practical use of the formulation and discussion on error 3 5 7 10 11 13 15 21 3 Test of formulation 3.1 Adler function in the large-β0 approximation 3.2 Static QCD potential with RG method at LL 3.3 Static QCD potential with RG method at NLL 23 24 24 28 4 Renormalon-free prediction for static QCD potential at NNNLO 32 5 Conclusions and discussion 37 A RG invariance of Borel integral 39 B Convenient formulae for numerical evaluation 40 1 Introduction Perturbation theory is a very basic tool in quantum field theory, yet perturbative series are expected to be divergent asymptotic series. In QCD, due to this property, perturbative predictions have inevitable uncertainties, and in particular renormalon uncertainties can practically limit accuracies of predictions. (See ref. [1] for a review on renormalon.) It is generally a non-trivial task to extract an unambiguous part or meaningful prediction from such a divergent series, particularly when the number of known perturbative coefficients is limited. Nevertheless, it is necessary to systematically assign a definite value to perturbation theory in order to go beyond perturbation theory with using the operator product expansion (OPE); one should systematically add a nonperturbative matrix element to the perturbative contribution for this purpose. Within the large-β0 approximation [2], methods to extract an unambiguous part from the series containing renormalons were developed [3, 4]. In these methods, one can give a renormalon-free (unambiguous) part and renormalon uncertainty in the form where each is –1– JHEP10(2020)039 1 Introduction –2– JHEP10(2020)039 clearly separated. The renormalon-free part is given in a semi-analytic form and is useful to gain insight into short-distance behaviors of observables [4]. However, the methods are applicable within the large-β0 approximation, because they rely on the feature that the series is given by the one-loop integral with respect to the momentum of a dressed gluon propagator. The large-β0 approximation is not sufficient to give accurate predictions, because, rigorously speaking, it is accurate only at leading order [O(αs )], and a systematic way to improve this approximation is unclear. In particular, it is not possible to incorporate exact results of fixed order perturbation theory, which have been computed currently up to a few to several orders. In this paper, we devise a general formulation beyond the large-β0 approximation to extract a renormalon-free part from the series containing renormalons, while clearly separating renormalon uncertainties. Our formulation has similarities to Lee’s method [5, 6] in the following points. We consider the Borel transform which is consistent with fixed order perturbative results and with the structure of renormalons. Then the Borel transform is given by the sum of a regular part [δB(u)] and singular part containing the renormalons [B sing (u)], i.e., B(u) = δB(u) + B sing (u). For this Borel transform, the Borel integral is considered. This is the same procedure as refs. [5, 6]. We evaluate the Borel integral of the regular Borel transform by a series expansion in αs , as it does not contain renormalons. A novel point of the present paper is to devise a procedure to evaluate the Borel integral of the singular Borel transform. We introduce an “ambiguity function”, which is defined by inverse Mellin transform of the singular Borel transform. With the use of the ambiguity function, we obtain a resummation formula alternative to the Borel integral. This resummation formula is given by the one-dimensional integral which has similar features to the resummation formula in the large-β0 approximation. Then, it is possible to use a dispersion relation technique to obtain the real part of the quantity (an unambiguous part of the Borel integral) in a parallel manner to the case of the large-β0 approximation [3, 4, 15]. This work can be regarded as an extension of the preceding studies [3, 4, 8, 9], developed mainly within the large-β0 approximation. As a result, we obtain the unambiguous prediction in a closed form from the resummation formula. The final renormalon-free result is consistent with fixed order perturbation theory and does not suffer from renormalon uncertainties similarly to refs. [5, 6]. We also study renormalization group (RG) properties of the formulation in detail. The method using the Borel resummation, as done in refs. [5, 6] and in the present paper, has the following advantages. First, one can (in principle) define the perturbative contribution in a renormalization group (RG) invariant way. This feature is assumed in the OPE argument to discuss renormalon structure and the Borel resummation respects this property. Secondly, the renormalon uncertainty is given in the form such that it can be canceled against a nonperturbative matrix element in the OPE. These features are obvious in our construction and quite useful to go beyond perturbation theory with using the OPE. On the other hand, in minimal term truncation methods (where pertur (...truncated)