Implications of the Principle of Maximum Conformality for the QCD Strong Coupling
Alexandre Deur, Jian-Ming Shen, Xing-Gang Wu, Stanley J. Brodsky, Guy F. de Teramond
SSLAC–PUB–16959JLAB-PHY-17-2394
Implications of the Principle of Maximum Conformality for theQCD Strong Coupling
Alexandre Deur , ∗ Jian-Ming Shen , † Xing-Gang Wu , ‡ Stanley J. Brodsky , § and Guy F. de T´eramond ¶ Thomas Jefferson National Accelerator Facility, Newport News, VA 23606 Department of Physics, Chongqing University, Chongqing 401331, P.R. China SLAC National Accelerator Laboratory,Stanford University, Stanford, California 94309, USA and Universidad de Costa Rica, 11501 San Jos´e, Costa Rica
Abstract
The Principle of Maximum Conformality (PMC) provides scale-fixed perturbative QCD predic-tions which are independent of the choice of the renormalization scheme, as well as the choice ofthe initial renormalization scale. In this article, we will test the PMC by comparing its predic-tions for the strong coupling α sg ( Q ), defined from the Bjorken sum rule, with predictions usingconventional pQCD scale-setting. The two results are found to be compatible with each other andwith the available experimental data. However, the PMC provides a significantly more precisedetermination, although its domain of applicability ( Q (cid:38) . Q (cid:38) PACS numbers: 12.38.Aw, 12.38.Lg ∗ email:[email protected] † email:[email protected] ‡ email:[email protected] § email:[email protected] ¶ email:[email protected] a r X i v : . [ h e p - ph ] M a y . INTRODUCTION The gauge theory of the strong interactions, Quantum Chromodynamics (QCD) is definedto provide objective predictions for physical observables; its predictions should not dependon arbitrary theory conventions, such as the choice of the gauge or the choice of renormal-ization scheme (RS). However, conventional calculations are typically carried out using aperturbative formalism where the truncated high-order predictions are RS-dependent. Fur-thermore, the n ! growth of the n th order coefficient of the resulting series –the renormalonproblem [1]– makes the convergence of the series problematic, even at high momentum trans-fer where the QCD coupling α s becomes small. A methodology to solve these problems hasbeen developed, starting with the BLM procedure [2], extended by Commensurate ScaleRelations [3], and culminating with the Principle of Maximum Conformality (PMC) [4–8].The PMC provides a systematic method to eliminate the renormalization scheme andscale dependences of conventional pQCD predictions for high-momentum transfer processes.It reduces in the Abelian limit ( N c →
0) [9] to the QED Gell-Mann-Low scale-settingmethod [10], and it provides the underlying principle for the BLM procedure, extendingit unambiguously to all orders consistent with renormalization group methods. The PMChas a solid theoretical foundation, satisfying renormalization group invariance [11, 12] andall other self-consistency conditions, such as reflexivity, symmetry, and transitivity derivedfrom the renormalization group [13].The PMC scales in the pQCD series are determined by shifting the arguments of thestrong coupling α s ( Q ) at each order n to eliminate all occurrences of the non-conformal { β i } -terms. The terms involving { β i } are identified at each order using the recursive patterndictated by the renormalization group equation (RGE) [7, 8]. This unambiguous proceduredetermines the scales Q n of the strong coupling at each specific order. As in QED, the PMCscales have a physical meaning in the sense that they are proportional to the virtuality ofthe gluon propagators at each given order, as well as setting the effective number n f ofactive quark flavors. After applying the PMC, the divergent renormalon series disappear,and the pQCD convergence is automatically improved. After normalizing the coupling toexperiment at a single scale, the PMC predictions become scheme-independent. The PMChas been successfully applied to many high-energy processes; see, e.g., Ref. [14].In this paper, we shall test the applicability of the PMC by comparing its prediction for2he evolution of the QCD strong coupling α s ( Q ) to the corresponding prediction based onconventional scale-setting, where the renormalization scale at each order is estimated as atypical momentum transfer of the process and where arbitrary range and systematic errorare assigned to estimate the uncertainty of the fixed-order pQCD predictions.The PMC will be applied in this paper in order to determine the behavior of the run-ning coupling α g ( Q ), using the MS-scheme as an auxiliary RS. The coupling α g ( Q ) is an“effective charge” [15] – i.e., an observable – defined from the Bjorken sum rule [16, 17]. Itinvolves the spin-dependent g structure function; hence, its name. The PMC predictionfor α g ( Q ) is RS-independent, whereas the conventional pQCD calculation of α g ( Q ) retainsRS-dependence, typically chosen as the MS scheme.This article is organized as follow: In Sec. II, we recall the formalism which defines the α MS ( Q ) renormalization scheme and the pQCD expansion for the effective charge α g ( Q )using conventional pQCD scale-setting. In Sec. III, we provide the formulae which allowthe computation of α g ( Q ) using the PMC. In Sec. IV, we compare the two calculations. InSec. V, we discuss the possibility of using the PMC in a procedure that employs α s to relatethe fundamental QCD parameter Λ MS to hadron masses or, equivalently, to the confine-ment scale κ emerging from the Light-Front Holographic QCD approach to nonperturbativeQCD [18]. We summarize the results in the final section. II. PQCD COMPUTATION OF THE EFFECTIVE CHARGE α g IN THE MS SCHEME
In the MS-scheme, the effective charge α g has the leading-twist perturbative expansion[19]: α g ( Q ) π = (cid:88) i ≥ a i (cid:18) α MS ( Q ) π (cid:19) i . (1)The perturbative coefficients a i are known up to four loops [20, 21]. (The values are givenexplicitly in Section III, Eq. 10.) The definition of α g stems from the Bjorken sum rule [16,17]. At leading-twist: (cid:90) − g p − n ( x Bj , Q ) dx Bj = g a (cid:34) − (cid:88) i ≥ a i (cid:18) α MS ( Q ) π (cid:19) i (cid:35) ≡ g a (cid:20) − α g ( Q ) π (cid:21) , (2)3here the integration runs over the Bjorken scaling variable x Bj . The nucleon axial chargeis g a and the label p-n indicates the isovector part of the spin structure function g . TheBjorken integral is well measured, including the transition region between perturbative tononperturbative QCD [22]. The Q -evolution of the strong coupling α MS ( Q ) in the MS-scheme is governed by the RGE: Q ∂∂Q (cid:16) α s π (cid:17) = β ( α s ) = − (cid:88) n ≥ (cid:16) α s π (cid:17) n +2 β n , (3)which is known up to 5-loops: β = 11 − n f ,β = 102 − n f ,β = 28572 − n f + 32554 n f ,β = 1497536 + 3564 ξ − (cid:18) ξ (cid:19) n f + (cid:18) ξ (cid:19) n f + 1093729 n f ,β = 815745516 + 6218852 ξ − ξ − ξ + (cid:18) − − ξ + 339356 ξ + 135899527 ξ (cid:19) n f + (cid:18) ξ − ξ − ξ (cid:19) n f + (cid:18) − − ξ + 161827 ξ + 4609 ξ (cid:19) n f + (cid:18) − ξ (cid:19) n f , where ξ n is the Riemann zeta function [23, 24]. The coefficients β i are expressed utilizingthe MS-scheme except for β and β which are scheme independent.Solving Eq. (3) iteratively yields the approximate five-loop expression of α pQCDMS [25], α pQCDMS ( Q ) = 4 πβ t (cid:20) − β β ln( t ) t + β β t (cid:18) ln ( t ) − ln( t ) − β β β (cid:19) + β β t (cid:18) − ln ( t ) + 52 ln ( t )+2ln( t ) − − β β β ln( t ) + β β β (cid:19) + β β t (cid:18) ln ( t ) −
133 ln ( t ) −
32 ln ( t ) + 4ln( t )+ 76 + 3 β β β (cid:0) ( t ) − ln( t ) − (cid:1) − β β β (cid:18) t ) + 16 (cid:19) + 5 β β β + β β β (cid:19)(cid:21) + · · · , (4)where t = ln ( Q / Λ s ) and Λ s is the asymptotic scale. Eqs. (1) to (4) allow us to compute α g ( Q ) in the pQCD domain. Although α g is an observable, the MS RS-dependence remainsin its pQCD approximant due to the truncations of Eqs. (1) to (4).4 II. PMC SCALE-SETTING FOR α g ( Q ) Following the basic PMC procedure, we first identify the conformal and nonconformalpQCD contributions for α g . The corresponding expression (1) is then reorganized as [8, 26] α g ( Q ) π = r , α MS ( Q ) π + ( r , + β r , ) (cid:18) α MS ( Q ) π (cid:19) + ( r , + β r , + 2 β r , + β r , ) (cid:18) α MS ( Q ) π (cid:19) + ( r , + β r , + 2 β r , + 52 β β r , + 3 β r , +3 β r , + β r , ) (cid:18) α MS ( Q ) π (cid:19) + · · · . (5)where the coefficients r i, for i > β = 0, and r i,j for i > , j > { β i } -terms.Here as for Eq. (1), we have implicitly set the initial renormalization scale µ as Q , althoughas a basic property of PMC scale-setting, the determined scales of the coupling Q i at eachorder turn out to be minimally dependent on the initial choice of scale. Any residual initialscale dependence at finite order in pQCD is highly suppressed, especially at the presentlyconsidered four-loop order. (One can test the initial scale dependence by recomputing thePMC predictions for µ (cid:54) = Q ; this can be conveniently done by applying the RGE.)The conformal coefficients r i, are: r , = 34 γ ns1 ,r , = 34 γ ns2 − (cid:0) γ ns1 (cid:1) ,r , = 34 γ ns3 − γ ns2 γ ns1 + 2764 (cid:0) γ ns1 (cid:1) ,r , = 34 γ ns4 − γ ns3 γ ns1 − (cid:0) γ ns2 (cid:1) + 8164 γ ns2 (cid:0) γ ns1 (cid:1) − (cid:0) γ ns1 (cid:1) , and the non-conformal coefficients r i,j read: r , = 34 Π ns1 + K ns1 ,r , = 34 Π ns2 + 12 K ns2 − γ ns1 (cid:18) K ns1 + 94 Π ns1 (cid:19) , r , = 0 ,r , = 34 Π ns3 + 13 K ns3 − γ ns1 ( K ns2 + 3Π ns2 ) − γ ns2 (cid:18) K ns1 + 32 Π ns1 (cid:19) + (cid:0) γ ns1 (cid:1) (cid:18) K ns1 + 274 Π ns1 (cid:19) ,r , = − (cid:0) Π ns1 (cid:1) − K ns1 Π ns1 , r , = 0 , γ ns i , Π ns i and K ns i are given explicitly in Refs. [20, 21].As indicated by Eq. (5), because the running of α MS at each order has its own { β i } -seriesas governed by the RGE, the β -pattern for the pQCD series at each order is a superpositionof all of the { β i } -terms which govern the evolution of the lower-order α s contributions at thisparticular order. All known { β i } -terms should be absorbed into α MS at each order accordingto the RGE [7, 8], thus determining its correct running behavior at each order. Hence, afterapplying PMC scale-setting, only the conformal coefficients remain. The result is: α g ( Q ) π = (cid:88) i ≥ r i, (cid:18) α MS ( Q i ) π (cid:19) i . (6)The elimination of the divergent renormalon terms naturally leads to a pQCD series moreconvergent than the original one in Eq. (5). The PMC scales Q i are functions of Q and read:ln Q Q = − r , r , − β (cid:0) r , r , − r , (cid:1) r , α MS ( Q ) π (7)+ (cid:20) β (cid:18) − r , r , + 2 r , r , r , − r , r , (cid:19) + β (cid:18) r , r , − r , r , (cid:19)(cid:21) (cid:18) α MS ( Q ) π (cid:19) + O (cid:18)(cid:16) α MS π (cid:17) (cid:19) , ln Q Q = − r , r , − β (cid:0) r , r , − r , (cid:1) r , α MS ( Q ) π + O (cid:18)(cid:16) α MS π (cid:17) (cid:19) , (8)ln Q Q = − r , r , + O (cid:16) α MS π (cid:17) . (9)These expressions show that the PMC scales Q i are given as a perturbative series; anyresidual scale dependences in Q i is due to unknown higher-order terms. This is the firstkind of residual scale dependence; the contributions from unknown high-order terms areexponentially suppressed and are thus generally small.A number of PMC applications have been summarized in the review [27]; in each case thePMC works successfully and leads to improved agreement with experiment. Furthermore,this multi-scale PMC approach corresponds to the fact that separate renormalization scalesand effective numbers of quark flavors appear for each skeleton graph. The coefficients ofthe resulting pQCD series match the coefficients of the corresponding conformal theory with β = 0, ensuring the scheme-independence of the PMC predictions at any fixed order.For convenience, we provide the conformal coefficients r i, and PMC scales Q i after sub-stitution of the γ ns i , Π ns i and K ns i into Eq. (5). They are, up to four-loop order:6 , = 1 ,r , = 1 . − . n f ,r , = 5 . − . n f − . n f ,r , = 21 . − . n f + 0 . n f + 0 . n f , ln Q Q = − . . − . n f ) α MS ( Q ) π +(1 . − . n f − . n f ) (cid:18) α MS ( Q ) π (cid:19) + O (cid:18)(cid:16) α MS π (cid:17) (cid:19) , ln Q Q = − . − . n f . − . n f − . n f − . n f + 32 . n f − . . n f − . n f + 2 . α MS ( Q ) π + O (cid:18)(cid:16) α MS π (cid:17) (cid:19) , ln Q Q = − . − . n f + 0 . n f . − . n f − . n f + O (cid:16) α MS π (cid:17) . The PMC scale of the last known order, Q , remains undetermined because the five-loop andhigher order { β i } -terms are unknown. As a test, we can set Q = Q or Q = Q , which leadsto the second kind of residual scale dependence. This scale dependence, however, generatesnegligible uncertainty. For example, we have computed α g ( Q ) using both prescriptions, andthe results are nearly identical because of the fast convergence of the PMC series.We note that the small values of Q (around 1 GeV), with n f = 3 lead to an almost zero Q ; this reflects the fact that in the soft Q -region, the intermediate gluons are effectivelynonperturbative, and thus information on the behavior of α s at low momentum is required.We shall adopt a natural extension of the perturbative α s -running behavior as determinedfrom the high Q -region. Then, to avoid having Q enter the nonperturbative region, wewill use as the alternative scale Q = 40 × Q [28]. Although we have also performed thecalculations for values of n f determined by the PMC scale Q i , we will use n f = 3 for theresults in the next sections in order to compare meaningfully with the results reported inRefs. [29–31].The results in this article use α g ( Q ) computed with the scales Q i calculated up to next-to-next leading order. However, for reference, we also provide here their values for n f = 37nd at leading order: Q = 0 . Q,Q = 0 . Q,Q = 40 Q. It is informative to compare the coefficients a i obtained from the conventional pQCDseries, Eq. (1), to the PMC coefficients r i, . The a i values for n f = 3 are [20, 32]: a = 1 ,a = 3 . ,a = 20 . ,a = 175 . ,a ∼ . , (10)which can be compared with the r i, for n f = 3: r , = 1 ,r , = 1 . ,r , = 0 . ,r , = 0 . , (11)The a i values become very large at high orders, a manifestation of the factorial renormalongrowth ( α s /π ) n β n n ! of pQCD series using conventional scale setting. In contrast, the confor-mal coefficients r i, have reasonable values of order 1, as expected from the PMC procedure.This much-improved convergence allows for more precise predictions. IV. COMPARISONS OF THE PMC AND CONVENTIONAL PREDICTIONSFOR THE BJORKEN SUM RULE
The PMC approach can be tested by comparing α g ( Q ) computed using the PMC pre-diction (6) versus the conventional pQCD calculation (1). In each case, the prediction willbe estimated up to fourth order and with n f = 3. For these computations, we will evaluate8 (GeV) ! g1 ( Q ) / " ! g1 / " JLab ! g1( ) / " OPAL ! F3 / "! g1 / " DESY ! g1 / " CERN ! g1 / " SLAC PMCConventionalpQCD in MS -1 FIG. 1: (Color online) the PMC and conventional MS predictions for α g ( Q ) /π computed for n f =3, with Λ ( n f =3) s, MS = 0 . α g ( Q ) or α F ( Q ). α MS up to five loops assuming Λ ( n f =3)MS = 0 . χ minimization.In Fig. 1, we display α g ( Q ) /π calculated using the RS-independent PMC predictionversus the conventional pQCD in the MS-scheme, together with the available experimentaldata [19]. We also show the experimental data for α F ( Q ), since the two effective charges α F and α g are in practice nearly identical [19]. We compute α g ( Q ) for values of the argumentof α MS ( µ ) greater than 1 GeV, µ > α g ( Q )the renormalization scale is directly set to Q and α g ( Q ) is computed for Q > µ > α g ( Q ) is computed for Q > . • The uncertainty of the perturbative approximant for α MS , which we estimate by takingthe difference between the expressions of α MS at order β and at order β . • The 17 MeV uncertainty on the value of Λ ( n f =3)MS [33];9 The truncation uncertainty in the PMC series (6) or in the conventional MS series(1). For the PMC series, it is estimated by taking the difference between the fourthorder and third order terms: (cid:0) α MS /π (cid:1) (cid:0) r , α MS /π − r , (cid:1) . For the conventional MSpQCD series, it is taken as the difference between the estimated fifth order term andthe calculated fourth order term: (cid:0) α MS /π (cid:1) (cid:0) a α MS /π − a (cid:1) .Fig. 1 shows that the four-loop PMC and the conventional pQCD calculations of α g ( Q )are consistent with each other, although only marginally for Q below a few GeV.We have also performed the same calculations by computing the value of the quark flavorvariable n f , according to the quark mass threshold as determined by the value of Q , in thecase of the conventional pQCD calculation of α g ( Q ), or the values of the Q i PMC scalesfor the PMC calculation. The results are similar to that shown in Fig. 1.A notable feature in Fig. 1 is that the theoretical uncertainty of the PMC prediction issignificantly smaller than that of the conventional pQCD prediction. As seen from Eqs. (10)and (11), this is due to the fact that the pQCD series using PMC scale-setting convergesmuch faster than the conventional pQCD series.
V. MATCHING TO THE NONPERTURBATIVE DOMAIN
In Refs. [29, 30], a method has been proposed to relate the perturbative QCD asymptoticscale Λ s to the hadron mass scale such as the proton mass. The scale Q which signifies thetransition between the perturbative and nonperturbative domains of QCD is also determinedby this method. Both Λ s and Q are obtained in any renormalization scheme in the pQCDdomain. This method uses the analytic form of α g [34] predicted in the nonperturbativedomain by Light Front Holographic QCD (LFHQCD) [18]: α g ( Q ) = π exp (cid:18) − Q κ (cid:19) , (12)where κ is a universal nonperturbative scale derived from hadron masses, for example, κ = M ρ / √ .
548 GeV, where M ρ is the mass of the ρ –meson. Alternatively, κ can be obtainedfrom fits to hadron form-factors, the Regge slopes, or the Bjorken sum rule Eq. (2). Althoughthe value of κ is universal, in practice, the approximations used in LFHQCD induce a (cid:39) κ = 0 . α s ( Q ) [30, 36], including the recent result based on Schwinger-Dyson Equations [37].The basis for the matching procedure to determine Λ s is the overlap of the domains ofapplicability of LFHQCD ( Q (cid:46) . Q (cid:38) . α g ( Q ) and its first derivative implies that Eq. (1) and Eq. (12), as well as their corresponding β –functions, can be equated in the overlap region. The simultaneous solution to these twoequations provides an analytical relation between Λ s and κ , as well as the transition scale Q . This leads to a determination of Λ ( n f =3)MS = 0 . on par with that of the averaged world data of 0.332(17) GeV [33].Since the PMC provides a more precise determination of α g ( Q ) than conventional renor-malization scale-setting, it is interesting to investigate if the procedure is also applicableusing Eq. (6) rather than Eq. (1) to improve the determination of Λ s . Q (GeV) ! g1 ( Q ) / " ! g1 / " JLab ! g1( ) / " OPAL ! F3 / "! g1 / " DESY ! g1 / " CERN ! g1 / " SLAC LFHQCD+pQCD (2016)LFHQCD+pQCD (PMC)LFHQCD PMCConventionalpQCD in MS -1 FIG. 2: (Color online) Matching procedure applied to PMC calculation (blue line). It is matchedto the LFHQCD results (magenta line) by requiring the continuity of both α g and its β -function.The blue band is the PMC prediction evaluated down to Q = 1 . κ = 0 .
523 GeV for LFHQCD.
Following the same matching procedure, we have computed the PMC prediction using κ = 0 .
523 GeV. To reach the matching point Q , it is necessary to extrapolate the PMC11rediction down to Q = 1 GeV, which implies that the α MS ( µ ) must be extrapolated down to µ = 0 .
68 GeV. The result is shown in Fig. 2. As a comparison, we also show the conventionalMS prediction [31] in the figure.The matching of the PMC prediction to LFHQCD yields a large value for Λ ( n f =3) s, MS =0.406(17) GeV. This explains why, compared to Fig. 1, a better agreement between thematched PMC curve (blue line) and the conventional MS pQCD calculations (red band) isobserved in Fig. 2.The determined transition scale, Q = 1 .
14 GeV, is below the scale at which the presentPMC calculation is applicable ( Q ≈ .
48 GeV). The failure of this self-consistency checkindicates that the matching procedure cannot be used with the PMC calculation, at leastwhen MS is used as an auxiliary RS. This explains why the matching procedure yieldsΛ ( n f =3)MS = 0 . µ is fixed at its initialvalue Q . In contrast, as shown by Eqs. (8) and (9), the determined PMC scale Q i for eachorder is a function of Q which can result in Q i scales that are larger or smaller than Q . Thishas consequences for the matching procedure proposed in Ref. [29], which requires that thetransition between nonperturbative and perturbative QCD occurs at a point Q rather thanover a non-zero Q range.In the case of conventional scale-setting, the meaning of the inflection point Q is un-ambiguous: α s ( Q ) has perturbative behavior for Q > Q and nonperturbative behavior for Q < Q . These are determined by pQCD and LFHQCD, respectively. On the other hand,in the case of the PMC scale-setting, some PMC scales are smaller than the determined Q , thus leading to an apparent incompatibility; i.e., if the determined PMC scale Q i isless than Q , the meaning of Q is questionable since Q i is now within the nonperturbativeregion. This is indeed the case for the present procedure. Thus, due to the fast convergenceof PMC series, we have α g ( Q ) ∼ α MS ( Q ), where the PMC scale Q = 0 .
45 GeV is signif-icantly smaller than the transition scale Q = 1 .
14 GeV. This conflict could be due to thefact that some of nonperturbative effects which are not accounted for in the (perturbative)derivation of the PMC scales Q i , such as those from the high-twist terms [39], may havealready come into the higher-order calculations. For example, the renormalization scale forthe heavy-quark loop which appears in the three-gluon coupling depends nontrivially on the12irtualities of the three gluons entering the three-gluon vertex [40]. −2 −1 Q (GeV) α s ( Q ) MSMOM( ξ = 0)V g FIG. 3: (Color online) The strong coupling α s ( Q ) for various renormalization schemes [30]. Thelines in the perturbative region are the perturbative calculations done at order β . The dashedcurves are their matched LFHQCD continuations into the nonperturbative domain. This problem may be solved by transforming to a different MS-like scheme; e.g., the R δ scheme [7, 8] where the subtraction ln 4 π − Γ E − δ is used within the minimal subtractionprocedure. (The conventional MS-scheme is the R δ -scheme corresponding to δ = 0.) Thescheme transformation between different R δ -schemes corresponds simply to a displacementof their corresponding scales; µ δ = µ exp( δ ); thus a proper choice of δ may avoid thesmall scale problem found for the MS-scheme. This problem may also be solved by usinga different auxiliary RS, such as the MOM scheme with ξ = 0 (Landau gauge) [41], or the V scheme [42]. This possibility is motivated by comparing the running behaviors of α s fordifferent schemes; examples are presented in Fig. 3. It shows that to ensure the scheme-independence of the couplings, e.g. α MS ( µ MS ) = α V ( µ V ) = α MOM ( µ MOM ), we must have µ MS < µ V < µ MOM . This fact has been observed by the LO commensurate scale relationsamong different effective couplings [3]. Thus a larger PMC scale can be achieved whenthe V -scheme or MOM-scheme is adopted as the auxiliary RS. For example, in the case ofthe V -scheme, the PMC prediction is applicable down to Q = 1 GeV if the perturbativebehavior of α MS ( µ ) can be extrapolated down to µ = 0 .
85 GeV [3], which is larger than thecorresponding value of µ = 0 .
68 GeV required for the MS-scheme.13nother avenue to address the problem could be to use the single-scale approach for thePMC [43], where a single effective scale replaces the individual PMC scales in the sense ofa mean value theorem; this can avoid the small scale problem which can appear at specificorders in the multi-scale PMC approach. These investigations will be reported in a futurepublication.
VI. SUMMARY AND CONCLUSION
In this paper, we have tested the PMC scale-setting procedure by comparing its predictionin the MS scheme for the effective charge α g ( Q ) defined from the Bjorken sum rule withthe prediction obtained using conventional renormalization scale-setting. To this end, wehave calculated the necessary PMC coefficients and renormalization scales. We have verifiedthat the PMC series converges much faster than the conventional MS pQCD series, whichresults in a significantly smaller uncertainty for the PMC pQCD prediction. Thus the centralobjective of the PMC is realized: it provides a determination of α g ( Q ) compatible with thedata and the conventional pQCD calculation, but without scheme-dependence and withsignificantly improved precision.As an important application, we have investigated the possibility of determining Λ MS fromhadronic scales by matching the PMC calculation for pQCD to the nonperturbative light-front holographic QCD prediction for α g ( Q ). This had been done previously using the con-ventional scale-setting pQCD prediction; this worked well, giving Λ ( n f =3)MS = 0 . Q . A detailed investigation for solving thisproblem, using alternative renormalization schemes and/or the single-scale PMC procedurefor pQCD, is in preparation. Acknowledgments
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