Formulation for renormalon-free perturbative predictions beyond large-$\beta_0$ approximation

We propose a general method to obtain an unambiguous (renormalon-free) part from divergent asymptotic series containing renormalons. The renormalon-free part consists of two parts: one is given by series expansion in $\alpha_s$ which does not contain renormalons, and the other is a real part of the Borel integral, where the Borel transform possesses renormalons. In particular, a systematic method to obtain the real part of the Borel integral is proposed. To this end, we adopt an alternative resummation formula to the Borel integral. As an example, we apply our formulation to the fixed-order result of the static QCD potential, whose closest renormalon is already visible.


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 divergent series. 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-β 0 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 part and renormalon uncertainty in the form where each is 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-β 0 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-β 0 approximation is not sufficient to give accurate predictions, because, rigorously speaking, it is accurate only at leading order [O(α s )], 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 propose a general formulation to extract a renormalon-free part from the series containing renormalons, while clearly separating renormalon uncertainties. To this end, we first derive an alternative resummation formula to the Borel integral. This resummation formula is given by an integral with an "ambiguity function." Then, we extract a real part of the resumed quantity, corresponding to an unambiguous part of perturbation theory, systematically by an analytic formulation. This is carried out by introducing a function called preweight, which is given by the dispersive integral of the ambiguity function. In these procedures, we use the techniques in Refs. [3][4][5][6] developed mainly within the large-β 0 approximation.
The formulation advocated in this paper has the following advantages. First, this method can define the perturbative contribution in a renormalization group (RG) invariant way. Secondly, the renormalon uncertainty is given in the form such that it can be cancelled 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 perturbative series is truncated around the order where the term of series gets minimal), these features are not obvious. See Ref. [7] for a recent study on this point.
As a practical application, we apply our method to give a renormalon-free prediction for the static QCD potential starting from the currently known fixed-order result [8][9][10]. Then we can obtain an accurate prediction which is consistent with the fixed-order result and does not suffer from a renormalon uncertainty. 1 In fact, our definition of a renormalonfree part itself reduces to a quite similar one to Ref. [11], which is called Borel resummed quantity therein. The original point in this paper is that we present a systematic method to extract a renormalon-free part and describe how it is related to an ambiguous part of the Borel integral. We also add an insight into a short-distance behavior of the observable. This paper is organized as follows. In Sec. 2, we present a general formulation to extract (or separate) a renormalon-free part from renormalon uncertainties for a given all-order perturbative series. In Sec. 3, we test our formulation by using the all-order perturbative series obtained with certain approximations. We study the Adler function in the large-β 0 approximation, and the static QCD potential with using the RG method in Ref. [6] at leading log (LL) and next-to-LL (NLL). In Sec. 4, we apply our formulation to the static QCD potential starting from the available fixed-order perturbative series. Sec. 5 is devoted to conclusions and discussion. In App. A, we show RG invariance of the Borel integral. In App. B, we present convenient formulae for numerical evaluation of the renormalon-free part.

Formulation
Let us first clarify the notation used in this paper. We consider a general dimensionless observable X(Q) depending on a single scale Q. We denote its perturbative series as where µ is a renormalization scale. Corresponding to this perturbative series, we define the Borel transform as Here, b 0 is the first coefficient of the beta function, which is defined as Explicitly the first two coefficients are given by for QCD. The Λ parameter in the MS scheme is given by The resummation of the perturbative series is given by the Borel integral: In the presence of the IR renormalons (which refer to singularities on the real axis), we can regularize the Borel integral (2.6) by a contour deformation In this case, the Borel sum possesses an imaginary part, whose sign is dependent on which contour is chosen. This imaginary part is regarded as a renormalon uncertainty. The subsequent contents in this Section are as follows. In Sec. 2.1, we derive an alternative resummation formula to the Borel integral. Here, we introduce an "ambiguity function." In Sec. 2.2, we show some formulae and examples of the ambiguity function. In Sec. 2.3, we define a "preweight," which is obtained by the dispersive integral of the ambiguity function. In Secs. 2.4 and 2.5, using the preweight we explain methods to extract an unambiguous perturbative contribution. This is done in two different regularizations: cutoff regularization in Sec. 2.4 and contour regularization in Sec. 2.5. The unambiguous (renormalon-free) part reduces to the same result in both regularizations. As we shall see, regarding the ambiguous part, the method in Sec. 2.5 is superior in the sense that the ambiguous part is compatible with the OPE.

Resummation formula with ambiguity function
For a given Borel transform B X (u), we decompose it into a singular part and regular part: such that δB X (u) does not possess renormalons, and all the renormalons are involved in B sing X (u). (This decomposition is not unique.) We denote perturbative coefficients involved in δB X (u) by δ n : Since the perturbative series does not contain renormalon divergences, this part shows a more convergent behavior than the original series and is free from the renormalon uncertainties. We refer to this series as the δ part hereafter. On the other hand, we apply the Borel sum to the part including renormalons, (We note thus d n = δ n + d ren n .) Namely, we treat the perturbative contribution as (2.14) In the above decomposition of the perturbative coefficients (or Borel transform), we define the δ part such that it respects the RG invariance. Then, both the δ part and X ren ± (Q) are RG invariant separately, since Eq. (2.13) coincides with the original Borel sum (2.7), which is RG invariant as shown in App. A. Now we derive an alternative resummation formula to the Borel integral for X ren ± . We introduce an ambiguity function Am X , which specifies the imaginary ambiguity in the Borel integral. We define it as Such a function was first introduced in Ref. [3] in the context of the large-β 0 approximation. (In the large-β 0 approximation, x corresponds to the gluon loop momentum. Beyond the large-β 0 approximation we do not have such a diagrammatic correspondence.) This function gives the renormalon uncertainty when we take x = e −1/(b 0 αs(µ)) (≪ 1): as seen from Eq. (2.7). We note that the subtraction of the regular part does not change the renormalon uncertainty. Here, we assume that for small x ≪ 1 the integration contour in the ambiguity function can be deformed as above due to x u = e u log x with log x < 0.
We have an inverted formula of Eq. (2.15), i.e., we obtain B sing X from the ambiguity function as One can show the equality, for instance, for pure imaginary u. Then, if both are analytic functions the equality can be enlarged to the whole complex u-plane. (In practical applications below, we rather use this relation to define B sing X (u) from an ambiguity function.) Using the above inverted formula, we can rewrite the Borel integral in terms of the ambiguity function 2 : . (

2.18)
This is an alternative formula to the Borel integral, which allows us to resum the perturbative series. In this paper, we extensively adopt this resummation formula (with necessary regularization). Let us make comments on the resummation formula (2.18). First, in this expression, the singularity on the integration path (positive real x-axis) is solely given by the simple pole at x = e −1/(b 0 αs(µ)) . This singularity structure is much simpler than the integrand of the Borel integral, which generally has an infinite number of cut singularities on the positive u-axis. This feature makes it easy to handle the all-order resummed series. In this expression, the imaginary ambiguity is correctly obtained from the contribution around this simple pole as 3 where we show only the imaginary part when the symbol ∼ is used. We note that although the pole position is µ dependent, the resulting uncertainty is µ independent, because the Borel integral is RG invariant, and hence so does its imaginary part. The relation between d ren n and the ambiguity function is simply given by expanding the integrand in α s (µ) before integration in Eq. (2.18): That is, we have [cf. Eq. (2.11)] d ren We can obtain this relation also from Eq. (2.17), by taking derivatives with respect to u.

Explicit form of ambiguity function
In this subsection, we present explicit forms of ambiguity functions in some examples. Here and hereafter, we set µ = Q unless otherwise stated and omit the arguments of µ and Q.
Assuming that a Borel transform B sing X (u = Re iθ ) exhibits a good convergence at R → ∞, the ambiguity function is obtained as and where we note that x u = e u log x can be a suppression factor in right or left side of the complex u-plane depending on the sign of log x. Since one expects that the Borel transform is expanded around its singularity at u = u i as the following formulae are convenient to obtain the ambiguity function: for u i > 0 and x < 1 for u i < 0 and x > 1 . (2.26) One can see that the IR renormalons determine the small-x behavior of the ambiguity function, whereas the UV renormalons do the large-x behavior.
As an example, we consider the Borel transform as which possesses renormalons at u = −1 and 2. Then the ambiguity function is given by  As the second example, we consider the Borel transform which possesses a singularity only at u = u i and gives the renormalon uncertainty as Since the ambiguity function can be obtained by the replacement of α s → − 1 b 0 1 log x in the Borel integral (and multiplying b 0 as an overall factor), one sees that the corresponding ambiguity function is given by [cf. Eq. (2.5)] In this way, we can directly obtain the ambiguity function from the renormalon uncertainty and often avoid an explicit calculation of the Borel trasnform. For instance, at the two-loop level (where we set b 2 = b 3 = · · · = 0) we explicitly have In fact, in this case, the explicit form of the Borel transform is inferred as . From this expression, in particular from the factor e −ub 1 /b 2 0 , one sees that the integral to obtain the ambiguity function for small This is a clear exposition why the expression of the above ambiguity function is restricted to the region 0 < x < e b 1 /b 2 0 . We can also confirm (2.33)

Preweight
As a preparation for extracting an unambiguous part of the perturbative contribution, we introduce a new function, given by the dispersive integral of the ambiguity function, We refer to this function as a preweight. This function is defined in the complex z-plane, and satisfies This function also has a real part. As we shall see below, the real part gives (part of) an unambiguous part of the perturbative prediction. Namely the preweight plays an important role in the resurgence of an unambiguous part, where the ambiguous imaginary part is used as an input.
For later convenience, we also define This function is regular for positive z.
Based on the preweight we define extended Borel transforms as They are in fact related to the Borel transform as Here, we used Eq. (2.17). These formulae are convenient to know asymptotic behaviors of the preweight at z ∼ 0 and z ∼ ∞, since with inverted formulae, and then use the previous result of the z-integration.

Renormalon-free part in cutoff regularization
Now we explain how to extract an unambiguous part of a perturbative contribution. In this subsection, we use cutoff regularization. This formulation is an extension of Ref. [4], performed within the large-β 0 approximation. In this regularization, we consider is free from IR renormalons (singularities on the positive u-axis), which stem from the integral around x ∼ 0. Then, the regularized Borel integral and corresponding expression with the ambiguity function are given by . (2.45) In the last equality, we use log x + 1/(b 0 α s (Q)) > 0 due to the cutoff, which ensures convergence of the u-integral.
We now extract a regularization parameter (μ f ) independent part, which we can identify as an unambiguous part. We can rewrite Eq. (2.45) as . (2.46) Here the contour C a connects the origin z = 0 to z = ∞ in the upper half plane avoiding the pole at z = e −1/(b 0 αs(Q)) , and C b connects the origin z = 0 to z =μ f in the upper half plane. The integral along C a is evaluated as where we rotate the integration path to the negative real axis. [W X+ (z) is defined in Eq. (2.36).] On the other hand, the integral along C b is evaluated as follows. Since we can decompose the preweight as for z > 0, noting the contribution from the pole at z = e −1/(b 0 αs(Q)) , we have . (2.49) As a result, we obtain . (2.50) Thus, we obtain the cutoff-independent part, i.e., unambiguous part as We refer to this part as X RF disp (Q 2 ), which is obtained based on the dispersive integral of the ambiguity function. Interestingly, we have an unambiguous part while starting from the ambiguity function.
As a whole, we have the following perturbative prediction: . (2.52) We obtain a renormalon-free part by the first line: The renormalon-free part consists of the δ part and X RF disp . On the other hand, the second line of Eq. (2.52) remains cutoff dependent, and is regarded as an ambiguity in this regularization.

Renormalon-free part in contour regularization
In this subsection, we extract an unambiguous part by starting from the contour regularization as Eq. (2.7). This is an extension of Ref. [3], performed within the large-β 0 approximation. Let us consider X ren + (Q 2 ): In this case it is convenient to use the following relation to give the Borel transform from the ambiguity function [cf. Eq. (2.17)], This slight deformation of the integration contour does not change the result of the integral (and thus correctly gives the Borel transform) as long as the ambiguity function has a good convergence property at infinity. Then, we deform the integration contour in the u-plane and make use of this relation: . (2.56) In the final step, we use Im[log x + 1/(b 0 α s (Q))] < 0, which ensures convergence of the v-integral. In a parallel manner, one sees that X − corresponds to the x-integration along C + . Thus, we obtain 5 . (2.57) Now we rewrite Eq. (2.57) in a form where its real (unambiguous) part and imaginary (ambiguous) part are clearly separated. By using a preweight in Eq. (2.34) and its property (2.35), we obtain (2.58) by noting the existence of a pole at z = e −1/(b 0 αs(Q)) . By using this equation, we have By rotating the integration paths C ± to the negative real axis without hitting the pole, we obtain This is one of the main results in this paper. The first line is a real part and corresponds to an unambiguous part. We denote it by X RF disp (Q 2 ), which is completely the same as Eq. (2.51). In the last equality, we decompose X RF disp into two parts. The first line and second line give qaulitatively different behaviors and such a decomposition is useful to understand the short-distance behavior of an observable, as we shall see. In particular, the second line of Eq. (2.61) gives a non-trivial power-like behavior to an observable.
We have a pure imaginary part in the second line of Eq. (2.60), which indeed coincides with the renormalon uncertainty appearing in the Borel integral. In this way, we can obtain an explicit result where the unambiguous part and ambiguous part are clearly separated.
As a whole, we have the following perturbative prediction: The first line is real, which we call a renormalon-free part: This is again the same as Eq. (2.53). The renormalon-free part gives a net and reliable part of the originally divergent asymptotic series. We note that the last term in Eq. (2.52) and that in Eq. (2.62) (which are regarded as ambiguous parts) differ. This is because we adopt different regularizations. We consider that the method in the present subsection is superior in the sense that the ambiguity coincides with the renormalon uncertainty of the Borel integral, which can potentially be cancelled against an uncertainty of a nonperturbative matrix element in the OPE. Thus, this method to calculate the perturbative contribution would be optimal to be systematically combined with the OPE framework.

Test of formulation
In this section, we test our formulation by using the all-order perturbative series obtained in certain methods. In particular, we check behaviors of δ n explicitly, and check validity of our renormalon-free predictions. In Sec. 3.1, we consider the Adler function in the large-β 0 approximation and briefly explain how the previous result in Refs. [3,4,12] is reproduced with the method in Sec. 2. In Secs. 3.2 and 3.3, we study the static QCD potential with the RG method in Ref. [6], which allows us to obtain all-order perturbative series (in principle). We note that although a method to extract a renormalon-free prediction was developed in Ref.
[6], we do not adopt it here. We only use the perturbative series obtained with the method of Ref. [6], and apply the formulation in Sec. 2 to extract a renormalon-free part.

Adler function in the large-β 0 approximation
The Adler function D(Q 2 ) is defined from the correlator of the electro-magnetic quark with Q 2 = −q 2 > 0. The Borel transform in the large-β 0 approximation for µ = Q is given by [13,14] B D (u) = 32N C C F 3π with N C = 3 and C F = 4/3. We identify this Borel transform as B sing D (u) and set δB D (u) = 0. Thus, we do not have a δ part in this case. The Borel transform has both UV and IR renormalons; the singularities are located at u = · · · , −2, −1, 2, 3, · · · . Then, from Eq. (2.15), one can obtain the ambiguity function as [5] Am The behavior of Am D (x) for x < e −5/3 is determined from the IR renormalons while that for x > e −5/3 from the UV renormalons. Corresponding to the first IR renormalon at u = 2 and first UV renormalon at u = −1, the ambiguity function behaves as Am D (x) ∼ x 2 for small x and Am D (x) ∼ x −1 for large x. A preweight W D can be analytically calculated and the result is presented in Eq. (38) of [4]. Then, one can extract a renormalon-free part according to Eq. (2.61). This is the same result as that in Refs. [3,4,12].

Static QCD potential with RG method at LL
We consider the static QCD potential in this and next subsections. The static QCD potential is extracted from an expectation value of a rectangular Wilson loop. It can be written as with the V -scheme coupling α V (q). In Ref. [6], all-order perturbative series is obtained by RG estimate. In this method, at N k LL one considers log k (µ/q)α s (µ) n+k+1 terms for arbitrary n in α V (q) , and then performing the q-integral, one obtains all-order perturbative series for V QCD (r), which contains renormalon divergences. There are only IR renormalons in this observable (with this treatment) and this is a distinct difference from the Adler function.
In this subsection, we work at LL. The renormalon uncertainty in this method has been revealed in Ref. [15] at general order of the RG improvement. The renormalon uncertainty for the dimensionless potential v = rV QCD (r) at LL is given by where C is shown in Fig. 2 Am v (x) = −C F sin(x 1/2 ) . (3.7) Note that Λ 2 MS r 2 = e −1/(b 0 αs(r −1 )) at LL. We adopt this form for x < 1: (3.9) Figure 2. Contour C. q * represents a singularity of the running coupling. At LL, it is a simple pole. Beyond LL, the singularity is given as the cut singularity.
In this case, we define B sing v (u) from the ambiguity function of Eq. (3.8) through Eq. (2.17). The Borel transform itself is given by and one can confirm that Eq. (3.9) gives the singular parts correctly (with µ = 1/r): We now examine the δ part. Namely, we evaluate We can obtain d n at an arbitrary order by performing the q-integral of the LL result of α V (q)| LL = α s (q) = α s (µ) + α s (µ) 2 b 0 log (µ 2 /q 2 ) + · · · . The results for d n /b n 0 and δ n /b n 0 are given in Table 1. We can confirm that the perturbative coefficients δ n are significantly smaller than d n as a consequence of renormalon subtraction. In Fig. 3, we show the δ part,  Table 1. Original perturbative coefficient d n and renormalon subtracted perturbative coefficient δ n . We take n f = 3.  where α s (1/r) is obtained at LL. One sees that the δ part exhibits much better convergence than the original series. Now we study the renormalon-free part obtained via a preweight. The preweight is given by from the ambiguity function of Eq. (3.8). It is possible to give an analytic expression for the preweight. Using this function, the renormalon-free part corresponding to X RF disp is given by The total renormalon-free prediction, which is the sum of the δ part and v RF disp , is shown in Fig. 5. This is compared with a result obtained with the method in Ref. [6] to subtract renormalons (right panel in Fig. 5). They show quite similar behaviors. In fact, the result in the right panel is obtained by adopting the ambiguity function of Eq. (3.7) for whole x, i.e., 0 < x < ∞. (In this case, we do not have a δ part.) In this sense, we observe consistency among the two different schemes.

Static QCD potential with RG method at NLL
As an analysis beyond the large-β 0 or LL approximation, we extend the analysis in Sec. 3.2 to the NLL approximation. The renormalon uncertainty at NLL is obtained as [15] Im where we change the integration variable asq = q/Λ MS . (Note that α s (q) in α V (q) is actually a function of q/Λ MS .) Then, we can adopt the ambiguity function as (3.18) and 0 for the other region [cf. Sec. 2.2]. The behavior of the ambiguity function is shown in Fig. 6. Here, we perform theq-integral along C numerically. We can compare this result with an asymptotic behavior of the ambiguity function. The asymptotic behavior at x ∼ 0 is obtained as  3.05 × 10 2 9 −3.97236 × 10 8 5.98 × 10 2 10 −8.24602 × 10 9 −5.93 × 10 3 Table 2. Original perturbative coefficient d n and renormalon subtracted perturbative coefficient δ n . We take n f = 3.
from the u = 1/2, 3/2, · · · renormalons [cf. (2.31)], where N 1/2 and N 3/2 are defined such that the renormalon uncertainty is given as (3.20) One should note the relation in the convention where one defines parameters in an expansion of the Borel transform around u = u i as In Ref. [15], the normalization constants K u i Γ(1 + ν i ) for u i = 1/2 and 3/2 are explicitly obtained (which are denoted as N i therein), and should be converted via Eq. (3.21). We have for n f = 3. In Fig. 6, we also show the asymptotic form of the ambiguity function (3.19) with the above normalization constants. If the ambiguity function up to the u = 3/2 renormalon is included, it coincides well with the whole ambiguity function. From the defined ambiguity function, 7 we can obtain d ren n and then δ n . (d n is calculated by the q-integral of the NLL result for α V (q).) The results for d n /b n 0 and δ n /b n 0 are given in Table 2. The δ part [Eq. (2.10)] is shown as a function of rΛ MS in Fig. 7. It is compared with the original series containing renormalon divergences, and one can see that the δ part exhibits a good convergence property also at NLL. We also study the δ part when we define  it with subtracting only first few renormalons. In Fig. 8, we can see that the subtraction up to the u = 3/2 renormalon is sufficient at the order we work [O(α 11 s )]. In contrast, only the u = 1/2 renormalon subtraction seems not satisfatory around this order. Now we calculate the renormalon-free part corresponding to X RF disp [cf. (2.61)]. Here, we approximate the ambiguity function by the first two terms of Eq. (3.19) (corresponding to the first two renormalons). (In this case, strictly speaking, we need to modify the δ part accordingly but its modification is small and not significant.) Then, we obtain a preweight and V RF disp (r) with this ambiguity function. At NLL, it is difficult to calculate the preweight [Eq. (2.34)] analytically and its integral in Eq. (2.61), and then we perform all the integrals numerically. We use convenient formulae collected in Appendix B. V RF disp (r) is shown in Fig. 9.
As a result, we obtain the renormalon-free prediction, which is the sum of the δ part and V RF disp (r). The result is shown in Fig. 10 (left panel). In the right panel, as a consistency check, we compare the renormalon-free prediction with fixed-order results. In plotting the fixed-order results, we adjust the height of the potential at rΛ MS = 0.05. This adjustment corresponds to subtracting the u = 1/2 renormalon (whose uncertainty is an r-independent constant) and the perturbative series exhibits convergent behavior. See the right panel of   4 Renormalon-free prediction for static QCD potential at NNNLO We apply our formulation to the static QCD potential starting from its fixed-order result. Thus, the analysis here does not rely on particluar methods or assumptions. We have explicit perturbative series to O(α 4 s ) (NNNLO). Let us state the current understanding of the renormalons for this quantity. The structure of the first IR renormalon at u = 1/2 was investigated [15,17,18], and its uncertainty is exactly proportional to Λ MS . This determines the form of the ambiguity function for the u = 1/2 renormalon. The overall constant was investigated in Refs. [11,19] and the latest result at NNNLO has been obtained in Ref. [15] Table 3. Original perturbative coefficient d n and renormalon subtracted perturbative coefficient δ n . We take n f = 3.
by using the technique developed in Ref. [20]. It was confirmed that the estimate of the normalization constant at u = 1/2 is stable against including higher order result and varying the renormalization scale. This indicates that the normalization constant can be obtained reliably with reasonably small error. The second IR renormalon at u = 3/2 has been investigated recently [15], and its uncertainty takes a form of ∼ Λ 3 MS r 2 [1 + O(α s (1/r))]. In Ref. [15], however, it was shown that the normalization constant for the u = 3/2 renormalon cannot be estimated reliably from the currently available perturbative series. It may indicate that the u = 3/2 renormalon does not have a significant effect to the currently available series, and in this analysis, we only take into account the u = 1/2 renormalon.
From the above reasoning we consider the ambiguity function as [cf. Eq. (2.30)] Here we use the four-loop beta function β(α s ) = − 3 i=0 b i α i+2 s . We choose the above range x < e −1 so that 1/ log (1/x) < 1, which can be regarded as an expansion parameter of the ambiguity function [Eq. (2.25)]. The normalization constant for the u = 1/2 renormalon has been determined from the NNNLO result as [15] We take n f = 3 here and hereafter. We note that the NNNLO perturbative coefficient contains IR divergence [21,22], and we remove the pole in 1/ǫ (where the dimension is set as d = 4 − 2ǫ in dimensional regularization) and associated logarithm. (This scheme is called the scheme A in Ref. [15].) We obtain δ n after calculating d ren n according to Eq. (2.21) with the above ambiguity function. The results for d n and δ n are presented in Table 3. One can confirm that a large part of d n is cancelled in δ n . We show a behavior of the δ part [Eq. (2.10)], which is compared to that of the original series in Fig. 11.
We now give V RF disp (r) [Eq. the result in Fig. 12. We calculate the preweight and its integral in Eq. (2.61) numerically, where we use the formulae in App. B. The first line of Eq. (2.61) gives a Coulomb-like potential and the second line of Eq. (2.61) gives a linear-like potential. We note that such a behavior is obtained as an unambiguous part of the perturbative contribution. Such a behavior in perturbation theory was first clarified in Ref. [16]. Using a different formulation, we arrive at a similar conclusion. In particular, we emphasize that this behavior is obtained originally from the ambiguity function corresponding to the u = 1/2 renormalon.
We finally obtain a renormalon-free prediction, which is the sum of the δ part and V RF disp (r). We show it in Fig. 13. For the δ part, we use the highest order O(α 4 s ) result. As a consistency check, we compare V RF (r) = 3 k=0 δ k α s (1/r) k+1 + V RF disp (r) with fixedorder results in Fig. 14. For the renormalization scale Λ MS /µ = 0.0026, we can confirm an agreement at short distances. On the other hand, for Λ MS /µ = 0.173, we observe an agreement around the region Λ MS r ∼ 0.173. These sound plausible taking into account the fact that fixed order perturbation theory is reliable around µ ∼ 1/r. We note that V RF (r) contains all-order perturbative series in the sense that the expansion of V RF disp (r) in α s gives the infinite series ∞ n=0 d ren n α n+1 s [cf. Eq. (2.21)]. However, we also note that the uncertainty coming from this divergent series is removed from V RF disp (r).
Before closing this section, we make comments on relation with Ref. [11]. In the present analysis, we gave the prediction which is consistent with fixed order perturbation theory but does not suffer from the u = 1/2 renormalon uncertainty. In Ref. [11], this was carried out by considering the "bilocal expansion" and then "Borel resummed" quantity (the real part of the Borel integral). At least numerically, our result should produce a quite similar result to the one in Ref. [11]. The novel point in the present paper is that we used the systematic method explained in Sec. 2 to evaluate the real part (unambiguous part) of the Borel integral, and described how the ambiguous part relates to the unambiguous part of the perturbative contribution.
-25 -In this paper, we presented a formulation to extract an unambiguous perturbative prediction from divergent asymptotic series for a general observable X(Q 2 ). We refer to such an unambiguous part as a renormalon-free part. The renormalon-free part consists of two parts. One is given by series expansion in α s which does not contain renormalons (δ part), and the other is a real part of the Borel integral (X RF disp (Q 2 )), where the Borel transform possesses renormalons. In particular, we proposed a systematic method to obtain X RF disp (Q 2 ). To obtain the real part of the Borel integral, we first rewrote the Borel integral by an alternative resummation formula. This formula is written as the x-integral of an "ambiguity function," as shown in Eq. (2.18). The ambiguity function is deeply connected with renormalon uncertainties and specified by singularities of the Borel transform. The advantages to adopt this resummation formula can be stated as follows. First, the integrand of the x-integral has only a simple pole as the singularity structure. This structure is much simpler than that of the Borel integral, whose integrand has an infinite number of cut singularities. This feature makes it easier to handle the resummation formula. Secondly, one may avoid an explicit calculation of the Borel transform since the ambiguity function can often be calculated directly from the renormalon uncertainty. The renormalon uncertainty (in particular its form) can be detected by an OPE argument, and this argument itself does not need an explicit form of the Borel transform.
In particular, this resummation formula allows us to use the techniques developed in the large-β 0 approximation. We introduced a "preweight," which is given by the dispersive integral of the ambiguity function, and plays an important role in the resurgence of an unambiguous part. Then using the preweight we presented an analytic formulation to extract an unambiguous part. The main result is given in Eq. (2.61). In this method, the ambiguous part, identified as the renormalon uncertainty, is simultaneously obtained explicitly.
We applied this formulation to the Adler function and static QCD potential. For the Adler function, as a test of the formulation, we considered the large-β 0 approximation, where the all-order perturbative series is known. In this case, we do not have a δ part and the result completely reduces to the one studied in Refs. [3,12]. We also studied the static QCD potential with the RG method [6], where the all-order perturbative series containing renormalon divergences can be obtained. We confirmed that the δ parts exhibit much better convergence than the original perturbative series. 9 We also confirmed that our renormalon-free predictions are reasonable by comparison with other calculations. Then we applied the formulation to the fixed-order result for the static QCD potential at NNNLO. The first IR renormalon at u = 1/2 already has a significant effect to this series, and we removed this uncertainty and gave a stable result.
There are several useful features of this method. First, the renormalon uncertainty is compatible with the OPE structure, and it can be cancelled against an uncertainty of a nonperturbative matrix element in the OPE. In addition to this, the perturbative con-tribution, identified as the renormalon-free part, is constructed as a clearly RG invariant quantity, and this property is also compatible with the OPE. Secondly, our formulation can remove subleading renormalons, as done in Sec. 3.2 and Sec. 3.3, without difficulties (although generally it is not an easy task to investigate renormalon structures of the subleading ones 10 ).
It would be possible to apply the present formulation to other observables such as the Adler function (beyond the large-β 0 approximation). The formulation would also be useful to give a clear definition of the gluon condensate (see Ref. [23] for discussion on this issue within the large-β 0 approximation) and its precise determination. We would like to discuss these issues in near future.

A RG invariance of Borel integral
We show that the Borel integral is independent of the renormalization scale µ. Here, it is convenient to adopt the following definitions B X (t; Q, µ) = rather than the definition adopted in the main part of this paper (although they are certainly equivalent). The derivative of the Borel integral with respect to µ is given by We note that the perturbative coefficients satisfy the RG equation, Then, we obtain