RG/Pade Estimate of the Three-Loop Contribution to the QCD Static Potential Function

The three renormalization-group-accessible three-loop coefficients of powers of logarithms within the \bar{MS} series momentum-space for the QCD static potential are calculated and compared to values obtained via asymptotic Pad\'e-approximant methods. The leading and next-to-leading logarithmic coefficients are both found to be in exact agreement with their asymptotic Pad\'e-predictions. The predicted value for the third RG-accessible coefficient is found to be within 7% relative |error| of its true value for n_f leq 6, and is shown to be in exact agreement with its true value in the n_f \to \infty limit. Asymptotic Pad\'e estimates are also obtained for the remaining (RG-inaccessible) three-loop coefficient. Comparison is also made with recent estimates of the three-loop contribution to the configuration-space static-potential function.

The perturbative portion of the QCD static potential is presently known to two subleading orders of perturbation theory [1,2,3]. This potential may be expressed as an integral over an MS perturbative-QCD series in momentum space, where x(µ) ≡ α s (µ)/π, L(µ, q 2 ) ≡ log(µ 2 / q 2 ), (2) and where the momentum-space series within the integrand of (1) is of the form with the following known coefficients [1]: with [5] β To leading and next-to-leading orders in perturbation theory, Eq. (6) is manifestly satisfied by the known coefficients (4) for the perturbative series (3): The coefficients of x 2 , x 3 and x 3 L in (9) are all seen to vanish for known series coefficients (4) and β-function coefficients (8): i.e., the known values of b 1 and b 2 are seen to uphold the RG-equation (6) by satisfying its perturbative formulation (9). However, it is important to note that (9) may also be utilized to extract all but one of the three-loop coefficients c k in the series (3). The coefficients of x 4 , x 4 L and x 4 L 2 in (9) respectively vanish provided Consequently, the only RG-inaccessible three-loop-order term in the series (3) is c 0 . This coefficient can be obtained only via a direct perturbative calculation, which has not yet been performed. In the absence of such a three-loop calculation, we employ the anticipated error of Padé approximants in predicting next-order terms of a field theoretical series in order to obtain an estimate of all four three-loop coefficients c k within (3). The predictions for {c 1 , c 2 , c 3 } can then be compared to their true values (10)(11)(12) to check the validity of the estimation procedure. The procedure we describe below has already been employed in a large number of applications: QCD β-and γ-functions [6,7,8], the SQCD β-function [7,9], QCD current-correlation functions [8,10], the renormalization-group functions of O(N)-symmetric massive scalar field theory [6,8,11], Higgs decays [8,10,12], Higgs-mediated scattering processes [13], and QCD corrections to inclusive semileptonic B-decays [14]. The general method we employ is described in refs. [7], [8] and [14]; we restate its development here for convenience.

Consider a perturbative series
For many such series, the series sum can be approximated by an [N|M] Padéapproximant, where N and M are respectively the degrees of numerator and denominator polynomials within the approximant. If only the next-to-leading term R 1 is known, for example, the Padé approximant would predict a value of R 2 1 for the coefficient R 2 in (13). Somewhat more realistically, if R 1 and R 2 in (13) are both known, the Padé approximant (15) leads to the predicted value R 2 2 /R 1 for the coefficient R 3 in (13). Generally, one finds that the higher the degree of the approximant, the more accurate the prediction of the next unknown coefficient of the series will be. Suppose one now utilises an [N − 1|1] approximant to estimate the coefficient R N +1 within (13), based upon knowledge of all previous series coefficients {R 1 , R 2 , ...., R N }. For perturbative field-theoretical series, it is often found that the relative error in such an estimate is inversely proportional to N [6,7,15]: The constant A in (16) can be estimated by comparing the [0|1]-approximant estimate for R 2 (i.e., R pred. In the series (13), let us suppose we only know the subleading and NNLO coefficients R 1 and R 2 , as is the case for the series (3). If the [1|1] approximant prediction R pred.

3
= R 2 2 /R 1 has a relative error described by (16), we can substitute (17) into (16) to obtain the "true" value for R 3 algebraically [8,14]: Of course, the validity of this result can be ascertained only by seeing how well it predicts coefficients that can be extracted by other means. 2 For the case of the series (3), we identify the known coefficients R 1 and R 2 as polynomials in the logarithm L [1]: Substituting (19) and (20) into (18), we obtain the following "large-L" series expansion for R 3 : 3 where As is evident from (3), R 3 should be a degree-3 polynomial in the logarithm L. A direct comparison of equivalent powers of L in (3) and in (21) leads to the following predictions: in addition to the predicted equivalence of the unknown coefficient c 0 with the lengthy square-bracketed term in (21).
The prediction (23) is in exact agreement with (12), the RG-determination of c 3 , as is evident by substituting (8a) into (23). Surprisingly, the predicted value (24) for c 2 is also in exact agreement with the RG value (11), as is evident from direct substitution of (8a), (8b) and (4a) into (24). Note that this agreement for both coefficients is true for all values of n f , indicating that the asymptotic error formula (16) replicates the RG-invariance of the series (3) to leading and next-to-leading order in the logarithm L, a most surprising result.
The formula (16) cannot, of course, replicate RG invariance to all orders in L, since the infinite series (21) which follows from it is not a degree-3 polynomial in L. Nevertheless, the coefficient of L in (21) is strikingly close to the corresponding coefficient c 1 within (3), as obtained via RG-methods in (10). In Table 1, such RG determinations of c 1 are compared to the prediction (25). As is evident from the Table, the predicted values for c 1 underestimate corresponding RG values by less than 7% for n f ≤ 6, with the best agreement seen curiously to occur at n f = 6. This feature may be understood by noting that the large-n f behaviour of the estimate (25), is in exact agreement with that of (12), the RG-determination of c 1 . Table 1 also presents estimates of the coefficient c 0 , as obtained from the (square-bracketed) L 0 term in (21). This coefficient, as noted earlier, cannot be extracted from lower-order terms via RG-methods. It is nevertheless encouraging to note that corresponding predictions for c 3 and c 2 are exact, and that predictions of c 1 are nearly so. Thus, we have obtained in Table 1 asymptotic Padé-approximant estimates for the three-loop coefficient c 0 , which, in conjunction with explicit RG-determinations (10-12) of the other three-loop coefficients {c 1 , c 2 , c 3 } occuring within the perturbative series W [x, L] (3), constitute a prediction for the full three-loop contribution to the static-potential integrand (1).
The coordinate-space potential corresponding to the series (3) can be obtained via (1) through use of the following identities [3]: where γ E = 0.577216 and ζ(3) = 1.202057. Following ref [16], we set µ = 1/r and find that For arbitrary n f , values of {a 0 , a 1 , b 0 , b 1 , b 2 } are given by (4), and values of {c 1 , c 2 , c 3 } are given by (10,11,12). In Table 2 we display values of the coefficients V 1−3 obtained via (29) for n f = {3, 4, 5}. The estimate for V 3 is obtained through use of (10, 11, 12) and the estimated values for c 0 in the final column of Table 1. In Table 2 we also list values of V 3 estimated via renormalon-matching (RM) considerations [16], as well as corresponding large-β 0 estimates of V 3 [17]. Striking agreement of all three estimation procedures is clearly evident in Table 2. However, it must be noted that V 3 is not very sensitive to c 0 [the only RG-inaccessible coefficient in (29d)] when µ = 1/r. If one uses (29d) to extract c 0 from V 3 , for example, one finds that the V 3 values -38.4, -37.34 (the RM value), and -34.06 (large β 0 ) tabulated in Table 2 for n f = 3 respectively correspond to c 0 values of 142 (our Table  1 RG/Padé estimate), 116, and 40. An alternative approach to estimating c 0 follows from a least-squares fit of the asymptotic Padé-approximant prediction (18) to the three-loop momentum-space contribution's explicit dependence on L, over the entire ultraviolet (µ 2 > q 2 ) region, a procedure which has been employed previously in a number of different applications [12,13,14]. If we define w ≡ q 2 /µ 2 [i.e., log(w) = −L], such a procedure entails optimization of and where the set of known coefficients {a 0 , a 1 , b 0 , b 1 , b 2 , c 1 , c 2 , c 3 } is given by (4) and (10)(11)(12). Unlike previous applications in which large-L expansions of (18) are quite consistent with least-squares fits, 4 such a fit is seen to lead to values of c 0 that are ∼50% larger than those of Table 1: In assessing the accuracy of (33-36), it should be noted that such leastsquares fitting could also be employed to fit simultaneously all four three-loop coefficients {c 0 , c 1 , c 2 , c 3 }, as has been done before in a number of applications [12,13,14] in which the fitted values for {c 1 , c 2 , c 3 } closely approximated their known RG values. However, such a procedure completely fails for the series (3): when n f = 3, optimization of (31) with respect to c 0−3 yields values [c 0 = 258, c 1 = 54.7, c 2 = 66.3, c 3 = 10.2] that differ substantially from true values [c 1 = 137.46, c 2 = 49.078, c 3 = 11.391] obtained from eqs. (10,11,12). A similarly large estimate of c 0 for the n f = 3 case is obtained directly via (18) in the L = 0 (small-log) limit, in which R 2 = b 0 (4b) and R 1 = a 0 (4a). Such an approach yields c 0 = 273, a value quite comparable to that obtained above (c 0 = 258) by simultaneous least-squares fitting of all four three loop coefficients c 0−3 . If one inputs this small-log estimate for c 0 (= 273) into (31) and then minimizes with respect to the RG-accessible coefficients c 1−3 , the estimated values for these coefficients will be even worse than those characterising the full least squares fit (c 1 ≃ 33, c 2 ≃ 74, c 3 ≃ 9.6) -values which are inconsistent (particularly c 1 ) with the RG determinations of these same parameters.
Such discrepancies suggest that the large-L (i.e. large µ or short distance) c 0 estimates of Table 1, which reproduce exact RG values for c 2 and c 3 and closely approximate RG values for c 1 , be taken more seriously than either the c 0 estimates (33-36) obtained via least-squares fitting over a broad range of µ, or other (e.g. small-log) approaches to estimating c 0 within an asymptotic Padé-approximant context. It is evident that such alternative asymptotic Padé-approximant estimation procedures are of little value if not tied to some way of successfully estimating c 1−3 , the RG-accessible three loop coefficients. By this criterion, the large-L estimates of Table 1 have the most substantial credibility.
However, it is important to remain cognisant of the relative insensitivity (noted earlier) of the configuration-space static potential to the parameter c 0 at its benchmark µ = 1/r length scale. At this scale, RG-accessible coefficients alone dictate that the n f = 3 three-loop contribution V 3 is given by a result which follows from substitution of (10-12) into (29d give a factor-of-two-accuracy estimate for the magnitude of c 0 , RG-accessible coefficients alone are sufficient to extract a surprisingly concise range for the three-loop contribution V 3 . In other words, the µ = 1/r estimate for V 3 , the three-loop quantity ultimately of phenomenological interest to us, is subject to substantially less theoretical uncertainty than the parameter c 0 . A final and necessary caveat, however, is the possibility that new diagrammatic topologies (and their corresponding group theoretical factors) known to enter the QCD static potential at three-loop order [4] may further circumscribe the applicability of using lower-order terms to predict the threeloop contribution c 0 , as in a Padé-approximant approach. This situation is entirely analogous to the theoretically uncertain light-by-light scattering contributions known to enter the muon's anomalous magnetic moment at sufficiently high order, as well as the quartic Casimir terms first appearing in the QCD β-function series at four-loop order. Padé-approximant based techniques cannot be expected to predict terms characterised by new higher-order group-theoretical factors [7]. Nevertheless, such contributions do not necessarily dominate the first order in which they appear, nor do they necessarily devalidate Padé estimates for that order. For example, the asymptotic Padéapproximant estimate of the (N c = 3) four-loop contribution to the QCD β-function [7] β 3 = 23600 -6400n f + 350n 2 f + 1.49931n 3 f (estimated numbers are italicised) is in quite reasonable agreement with the exact result [19] β 3 = 29243.0 -6946.30n f + 405.089n 2 f + 1.49931n 3 f , though in much closer agreement with the calculated result with quartic Casimir terms excised [7]: β 3 = 24633 -6375n f + 398.5n 2 f + 1.49931n 3 f . Thus, based on the limited information available, the remarkable success of the asymptotic Padé-approximate large-L expansion in predicting those three-loop momentum-space static potential terms that are also extractable by RG methods encourages some confidence in the corresponding large-L prediction of the RG-inaccessible parameter c 0 (modulo the above-mentioned uncertainties characterising Padé approaches), at least for purposes of predicting V 3 , the three-loop contribution to the configuration-space static potential.  Table 2: Configuration-space coefficients (28) of the configuration-space static potential. The column labeled V 3 is obtained using RG-determinations of c 1 , c 2 , c 3 and the Table 1 estimate of c 0 within eq. (29d). The column labeled V RM 3 is obtained from eq. (22) of ref. [16]. The column labeled V Lβ 0 3 lists large-β 0 estimates [17] that are also tabulated in ref. [16].