Estimate of 4-loop Pole-MSbar Mass Relation from Static QCD Potential

We estimate the presently unknown constant in the 4-loop relation between the quark pole mass and the MSbar mass, by requiring stability of the perturbative prediction for E_tot(r)=2m_pole+V_QCD(r) in the intermediate-distance region. The estimate is fairly sharp due to a severe cancellation between 2m_pole and V_QCD(r). This would provide a test, based on general properties of the gauge theory, for the size of ultra-soft contributions to V_QCD(r).

It has become an important theme of today's particle physics to precisely determine the masses of heavy quarks using the frame of perturbative QCD, as their values being indispensable inputs in various fields of modern particle physics. For the purpose of precisely determining heavy quark masses, often the relation between the pole mass and the mass in the modified-minimal-subtraction scheme (MS mass) of a quark becomes necessary. This relation can be expressed in a series expansion in the strong coupling constant as Here, m ≡ m MS (m MS ) denotes the MS mass renormalized at the MS mass scale; α s (µ) = α (n l ) s (µ) represents the strong coupling constant in the MS scheme, where n l is the number of light quark flavors (n l = 3, 4 and 5 for the charm, bottom and top quarks, respectively); the renormalization scale µ is set to m. For the purpose of the analysis in this paper, we use the coupling constant of the theory with n l flavors only as the expansion parameter. The one-loop coefficient is given by d 0 = 4/3. The coefficients d 1 and d 2 are obtained from the two-loop [1] and three-loop [2] * mass relations in the full theory (with n h heavy quarks and n l light quarks), respectively, by rewriting them in terms of the coupling constant of the theory with n l light quarks only. † At present only limited part of d 3 are known [4,5], and there have been increasing demands for its full evaluation recently.
Major estimates of d 3 which have been performed so far rely on the renormalon dominance hypothesis of the pole mass [4,6,7,8]. (This includes the estimate in the so-called "large-β 0 approximation.") In these methods, there is an assumption (with certain grounds, see [9]) on the higher-order behavior of the perturbative expansion: where β i denotes the (i + 1)-loop coefficient of the beta function of α s (µ). Empirically it is known that perturbative series of many observables are approximated well by this form even at relatively low orders. There have also been estimates of d 3 in another method [10]. In this paper we present estimates of d 3 for n l = 0, 3, 4, 5 based on comparatively general assumptions. In particular, our method does not use eq. (2). We consider the total energy of a color-singlet pair of heavy quarks Q andQ, defined by The static QCD potential V QCD (r) represents the potential energy between Q andQ at a distance r, in the static limit. We require stability of the perturbative prediction of E tot (r) at relatively large r, within the range where the perturbative prediction is * The same relation was obtained before in [3] in a certain approximation. † This relation coincides with Eq. (14) of [2]. Note that in the other formulas of [2], the coupling constant of the full theory is used. Figure 1: As a general feature of the gauge theory, a gluon, which couples to static currents j µ a ∝ δ µ0 , couples to the total charge of the system in the IR limit, q → 0. Diagrammatically both self-energy and potential-energy type diagrams are needed for realizing this feature, hence, for a color-singlet system, a cancellation takes place between the two types of diagrams. expected to be valid. Although originally this stability was predicted using the language of the renormalon dominance hypothesis [11], it can be considered to hold as a general property of perturbative QCD beyond the renormalon dominance hypothesis [12]. In fact, a gluon, which couples to static currents j µ a ∝ δ µ0 , couples to the total charge of the system Q tot a = i j 0 a,i (q = 0) (i = Q,Q) in the zero momentum limit q → 0, that is, an infra-red (IR) gluon decouples from the color-singlet system. Diagrammatically an IR gluon observes the total charge when both self-energy diagrams ‡ and potentialenergy diagrams are taken into account. Hence, a cancellation takes place between these two types of diagrams, see Fig. 1. In perturbative QCD, convergence and stability of perturbative series become worse as contributions from IR gluons grow. Oppositely, after cancellation of IR contributions, convergence and stability of perturbative predictions improve. This can be considered as a general property of a gauge theory which is strongly interacting at IR.
In the perturbative series of E tot (r) up to O(α 3 s ), where the exact terms are known, an improvement of convergence and stability as a result of the cancellation is clearly visible, and the cancellation becomes severer at higher orders. The meaning of the latter statement is as follows. The perturbative series of m pole and V QCD (r), respectively, do not converge well, whereas the perturbative series of E tot (r) converges much more quickly. Let us denote the individual terms of the former as m n and V n , respectively, and of the latter as E n . Then the ratio |E n /m n | or |E n /V n | reduces with n. This means that there is a severer cancellation for larger n: (2|m n | − |V n |)/(2|m n | + |V n |) ≈ E n /(2|V n |). As a consequence, by assuming convergence and stability of E tot (r) up to the next order [O(α 4 s )], we obtain fairly severe constraints on our estimates of d 3 . The leading IR contributions being canceled in the static limit, let us consider the next-to-leading IR contributions which may affect the stability of the perturbative prediction for E tot (r). The interaction of the singlet static QQ pair and IR gluons starts from a dipole interaction S r · E a O a in the multipole expansion in r [13]. The ultra-soft (US) corrections to V QCD (r) originating from this interaction appear first at O(α 4 s ). It has been argued that the US corrections are small at this order [14]. There also exist ‡ In the large mass limit contributions from IR region to the pole mass approximate IR contributions to the self-energy of a static charge.
arguments that these corrections may not be small [7]. In our analysis, we assume that these corrections are small and estimate d 3 by requiring stability of the perturbative prediction for E tot (r) in an IR region. § Subleading IR contributions to m pole , which may also affect the stability of E tot (r), are expected to be suppressed by Λ QCD /m. By increasing m, we render these contributions sufficiently small.
Let us review the behavior of the perturbative series of E tot (r) up to O(α 3 s ) at relatively large r, as analyzed in [15]. Comparing the perturbative series in α s (µ) of E tot (r) and those of m pole and V QCD (r) individually, we observe a drastic improvement in convergence of the series. In the case n l = 4, m = 4.180 GeV and α s (M Z ) = 0.1184, a stable theoretical prediction for E tot (r) is obtained at r < 2.8 GeV −1 . At each r, the scale µ is fixed in two different ways: (1) The scale µ = µ 1 (r) is fixed by demanding stability of E tot (r) against variation of the scale (minimal-sensitivity scale [16]): (2) The scale µ = µ 2 (r) is fixed on the minimum of the absolute value of the O(α 3 s ) term Here and hereafter, we state that a stable theoretical prediction is obtained when both scales exist; in this case, we find that the values of E tot (r) corresponding to both scales agree well, and that the convergence behaviors of both expansions are reasonable. The range of stable prediction extends to larger r as the order of perturbative expansion is raised [up to O(α 3 s )]. E tot (r) is examined also by varying the value of m artificially: whenever stable theoretical predictions for E tot (r) are obtained, the predictions corresponding to different m agree with each other within the estimated theoretical uncertainties, after adding an arbitrary r-independent constant. [Theoretical uncertainties are estimated as order Λ 3 QCD r 2 with Λ QCD ≃ 300 MeV.] As m is increased, the perturbative predictability range of r, where both scales exist, shifts to shorter-distance region. These examinations may be regarded as tests of properties of the SU(3) gauge theory, irrespective of details of the parameters of the theory. ¶ Phenomenologically E tot (r) is compared with typical phenomenological potentials. They are in agreement in the relevant distance range, 0.5 GeV −1 < ∼ r < ∼ 2.8 GeV −1 , within the estimated theoretical uncertainties, after adding an arbitrary r-independent constant to each potential; this is the case independently of the value of m (as long as § In terms of the renormalon language, our standpoint may be phrased as follows. Since there exist uncanceled IR renormalons in E tot (r), starting from the u = 3/2 pole in the Borel plane, they may deteriorate convergence of the perturbative series at higher orders of the perturbative expansion. Ultraviolet (UV) renormalons may also contribute. We assume that both of these contributions are small and negligible in estimating d 3 .
¶ In this analysis, the parameters of the theory can be taken as a dimensionful parameter Λ MS (which sets the unit of mass dimension) and a dimensionless parameter m/Λ MS . Hence, we may vary only m fixing Λ MS , so that we can always consider Λ QCD to be of the order of 300 MeV. a stable prediction is obtained), but only in the cases where realistic values are chosen for Λ MS . There are also similar comparisons with lattice computations of V QCD (r), with good agreements [19,20]. Now we repeat the same analysis including the terms at the next order and varying d 3 in addition. We first set α s (M Z ) = 0.1184, m = 4.180 GeV and n l = 4. (We neglect the masses of the light quarks in the following analysis). We take for V QCD (r) the sum of the perturbative series up to O(α 4 s ) [21,22] and O(α 4 s log α s ) [13,23], as given by eq. (21) of [24]; the O(α 4 s log α s ) term is generated by contributions from the US scale. Roughly speaking, if we choose a value of d 3 close to that of the renormalon estimate [6] or to the large-β 0 value [4] d 3 (large-β 0 ) ≃ 3046.29 − 553.872 n l + 33.568 n 2 l − 0.678141 n 3 l , a cancellation between 2m pole and V QCD (r) takes place and a relatively convergent and stable prediction is obtained. Nevertheless, the level of cancellation depends sensitively on the value of d 3 . For demonstration we show in Fig. 2  These features may be summarized as follows. Whenever a stable prediction is obtained, E tot (r) is consistent with a function of the form Λ MS × f (Λ MS r) up to an additive constant, where f (x) is independent of m; only when a realistic value of Λ MS is chosen it is consistent with phenomenological potentials. In fact, such a function f (Λ MS r) can be explicitly extracted from V QCD (r) as a short-distance dominant (renormalon-free) part, given as a "Coulomb+linear" potential [17,18]. n l Ref. [4] Ref. [6] Ref. [7] Ref. [10] Ref. [  denotes d 3 (large-β 0 ); those of Refs. [6], [7] and [8] are based on the renormalon hypothesis; the estimate of Ref. [10] is derived by an effective charge method.
As already stated, at NNLO both scales µ 1 (r) and µ 2 (r) [eqs. (4) and (5)] exist up to r ≃ 2.8 GeV −1 . We require that at NNNLO both scales also exist at least up to the same r, such that the perturbative stability is not deteriorated at this order. This requirement leads to an upper bound for d 3 : d 3 < 0.96 × d 3 (large-β 0 ). We also vary m artificially to 8 GeV and 16 GeV and require that at NNNLO the two scales exist at least up to the same r as at NNLO. (The corresponding values of r are 1.4 GeV −1 and 0.7 GeV −1 , respectively.) In these cases, a common value 0.97 × d 3 (large-β 0 ) is obtained as upper bounds for d 3 . All the upper bounds are fairly solid, in the sense that as soon as we assign a larger value to d 3 in each case, we observe a strong instability of the perturbative prediction; see If we assign a value much smaller than d 3 (large-β 0 ) to d 3 , qualitatively the perturbative series of E tot (r) at NNNLO tends to become unstable and exhibit a poorer convergence behavior. For instance, the scales fixed at NNLO and NNNLO [eq. (4) or eq. (5)] tend to be separated farther; the crossing points of the solid lines in Fig. 2, which are centered to a small region, tend to be separated apart. We quantify this feature by further demanding that the difference between the NNLO and NNNLO predictions be smaller than the perturbative uncertainty * Λ 3 QCD r 2 (Λ QCD = 300 MeV) for each of the scale choices µ = µ 1 (r) and µ = µ 2 (r); since the estimate Λ 3 QCD r 2 is meaningful only in an IR region, we apply this requirement in the range r > 1 GeV −1 . This requirement sets a lower bound for d 3 corresponding to each value of m. † We obtain d 3 > ∼ 0.90 × d 3 (large-β 0 ). This value, however, depends on our choice Λ QCD = 300 MeV. * The estimate of order Λ 3 QCD r 2 is among the predictions of the renormalon dominance hypothesis. It can, however, be derived also in a more general framework, without assuming the renormalon dominance, eq. (2). Namely, within the effective field theory "potential non-relativistic QCD" (pNRQCD), the leading IR contribution to E tot (r) can be absorbed into a matrix element of a non-local gluon condensate, whose size is of order Λ 3 QCD r 2 by dimensional analysis [13]. In this case only UV contributions remain in the perturbative expansion of the Wilson coefficient, which should be more convergent than the perturbative series of E tot (r). Thus, IR contributions to E tot (r) generates (at most) order Λ 3 QCD r 2 uncertainties. (See also [18].) † Since we require consistency with the NNLO prediction, the perturbative prediction of E tot (r) at NNNLO also agrees with typical phenomenological potentials if we take a realistic value for Λ MS .
Thus, in comparison to the upper bound, the lower bound is to some extent obscure.
After choosing a value of d 3 within the range determined by the above two requirements, qualitatively the perturbative prediction for E tot (r) becomes stable and the perturbative series exhibits an optimally convergent behavior. This effect is enhanced especially at larger r. An optimal estimate is d 3 ≈ 0.95 × d 3 (large-β 0 ); see Fig. 2. We repeat the same analyses for n l = 0, 3 and 5 and find qualitatively similar results. Our estimates of d 3 are summarized in Table 1. Other estimates of d 3 are also listed in the same table for comparison. We note that the large-β 0 values and some of the renormalon estimates for n l = 4, 5 lie above the upper bounds of our estimates.
Once d 3 is computed exactly in the future, a comparison with our estimates will test our understanding of the perturbative series of E tot (r). In particular, it will test our assumption that the US contributions (the leading residual IR contributions in the multipole expansion that may affect the stability of the perturbative series) are small and do not deteriorate the perturbative convergence observed up to NNLO. Other assumptions are based on general properties of a gauge theory strongly interacting at IR and are independent of the renormalon dominance eq. (2). We obtained fairly constrained estimates for d 3 reflecting a severe cancellation between 2m pole and V QCD (r).