Nucleon spin structure at low momentum transfers
aa r X i v : . [ h e p - ph ] S e p Nucleon spin structure at low momentum transfers
Roman S. Pasechnik ∗ High Energy Physics, Department of Physics and Astronomy,Uppsala University Box 516, SE-75120 Uppsala, Sweden
Jacques Soffer † Physics Department, Temple UniversityBarton Hall, 1900 N, 13th StreetPhiladelphia, PA 19122-6082, USA
Oleg V. Teryaev ‡ Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna 141980, Russia (Dated: November 19, 2018)
Abstract
The generalized Gerasimov-Drell-Hearn (GDH) sum rule is known to be very sensitive to QCDradiative and power corrections. We improve the previously developed QCD-inspired model forthe Q -dependence of the GDH sum rule. We take into account higher order radiative and highertwist power corrections extracted from precise Jefferson Lab data on the lowest moment of thespin-dependent proton structure function Γ p ( Q ) and on the Bjorken sum rule Γ p − n ( Q ). By usingthe singularity-free analytic perturbation theory we demonstrate that the matching point betweenchiral-like positive- Q expansion and QCD operator product 1 /Q -expansion for the nucleon spinsum rules can be shifted down to rather low Q ≃ Λ QCD leading to a good description of recentproton, neutron, deuteron and Bjorken sum rule data at all accessible Q . PACS numbers: 11.10.Hi, 11.55.Hx, 11.55.Fv, 12.38.Bx, 12.38.Cy ∗ Electronic address: [email protected] † Electronic address: jacques.soff[email protected] ‡ Electronic address: [email protected] . INTRODUCTION The problem of the nucleon spin structure and the peculiarities of its underlying QCDdescription has attracted a lot of attention over the recent years [1, 2]. In particular, this isdue to an enormous progress in experimental studies of the spin sum rules at low momentumtransfer Q , from the very accurate Jefferson Lab data on the lowest moment of the spin-dependent proton structure function Γ p ( Q ) and on the Bjorken sum rule Γ p − n ( Q ) in therange 0 . < Q < [3]. This data provided a good testing ground for combiningboth the perturbative and non-perturbative QCD contributions.Theoretical description of the nucleon spin structure functions g p,n at large Q relies onthe Operator Product Expansion, and at moderate Q their sensitivity to the radiative andhigher twist power corrections becomes significant [4]. Due to such a sensitivity the transi-tion to the entirely non-perturbative Q region is rather cumbersome. This transition wasearlier addressed in the QCD-motivated model [5] for the Q -dependence of the generalizedGerasimov-Drell-Hearn (GDH) sum rule [6] making use of the relation to the Burkhardt-Cottingham sume rule [7] for the structure function g , whose elastic contribution is themain source of a strong Q -dependence, while the contribution of the transverse structurefunction, g T = g + g , is smooth. The successful prediction of this model was the distinct“crossover” point of the proton data for Γ p ( Q ) at low Q ∼ −
250 MeV . Its subsequentmodification [8], including radiative and power QCD corrections, made the description farmore accurate, which was required by the increased accuracy of the data.Now we enter a new level of increasing experimental accuracy, obtained in the recentlypublished proton JLab data [3]. They lie above the model inputs at Q & . (whiledisplaying quite a similar shape) due to a noticeable sensitivity of pQCD part of Γ p,n ( Q )and to poorly known higher twist contributions µ , ,.. , as well as the axial singlet charge a .Our present goal is to improve the model for the generalized GDH sum rule for protonand neutron using the values of the power corrections µ , ,.. and singlet axial charge a ,systematically extracted from the JLab data [9, 10] and by performing a similar programof the smooth interpolation between large Q and Q = 0. As we will see we are able toachieve a rather good description of the data at all Q values.The JLab data were obtained in the low Q region and, therefore, a special attention isneeded to the QCD coupling in this domain. While the 1 /Q term in the OPE works at rel-atively high scales Q & , higher-twist (HT) power corrections 1 /Q , /Q , etc., startto play a significant role at lower scales, where the influence of the ghost singularities in thecoefficient functions within the standard perturbation theory (PT) becomes more noticeable.It affects the results of extraction of the higher twists from the precise experimental dataleading to unstable OPE series and huge error bars [9]. It seems natural that the weakeningor elimination of the unphysical singularities of the QCD coupling would allow shifting theperturbative QCD (pQCD) frontier to a lower energy scale and to get more exact infor-mation about the nonperturbative part of the process described by the higher-twist series[10].In this investigation, in order to avoid the influence of unphysical singularities at Q =Λ QCD ∼
400 MeV, we deal with the ghost-free analytic perturbation theory (APT) [11] (fora review on APT concepts and algorithms, see also Ref. [12]), which was recently proven tobe an intriguing candidate for a quantitative description of light quarkonia spectra withinthe Bethe-Salpeter approach [13], as well as in the recent higher-twist analysis of the deepinelastic scattering data on the F structure function [14]. For completeness, we compare our2esults obtained with conventional PT and APT couplings and, finally, discuss the relateduncertainties and stability issues. II. FORMALISMA. OPE regime Q > Λ QCD
To recall the basic ideas of the approach let us consider the lowest moments of spin-dependent proton and neutron structure functions g p,n defined asΓ p,n ( Q ) = Z dx g p,n ( x, Q ) , (2.1)From now on, it is understood that the elastic contribution at x = 1 is excluded from themoments, since it is the “inelastic” contribution which can be matched with GDH sum rule.At large Q the moments Γ p,n ( Q ) are given by the OPE series in powers of 1 /Q withthe expansion coefficients (see, e.g., Ref. [15]). In the limit Q ≫ M the moments aredominated by the leading twist contribution, µ p,n ( Q ), which can be decomposed into flavorsinglet and nonsinglet contributions:Γ p,n ( Q ) = 112 (cid:20)(cid:18) ± a + 13 a (cid:19) E NS ( Q ) + 43 a E S ( Q ) (cid:21) + ∞ X i =2 µ p,n i ( Q ) Q i − , (2.2)where E S and E NS are the singlet and nonsinglet Wilson coefficients, respectively, calculatedas series in powers of α s [16]. These coefficient functions for n f = 3 active flavors in the MSscheme are E NS ( Q ) = 1 − α s π − . (cid:16) α s π (cid:17) − . (cid:16) α s π (cid:17) − O ( α s ) , (2.3) E S ( Q ) = 1 − α s π − . (cid:16) α s π (cid:17) − O ( α s ) . (2.4)The triplet and octet axial charges a ≡ g A = 1 . ± .
004 [17] and a = 0 . ± .
025 [18],respectively, are extracted from weak decay matrix elements. As for the singlet axial charge a , it is convenient to work with its renormalization group (RG) invariant definition in theMS scheme a = a ( Q = ∞ ), in which all the Q dependence is factorized into the definitionof the Wilson coefficient E S ( Q ). For detailed discussion of the higher-loop stability of thecoefficient functions and prescriptions used in actual calculations, see Ref. [10].We address both proton and neutron spin sum rules (SSRs), and the singlet and octetcontributions are canceled out in their difference Γ p − Γ n resulting in the Bjorken sum rule[19] Γ p − n ( Q ) = g A E NS ( Q ) + ∞ X i =2 µ p − n i ( Q ) Q i − . (2.5)The unphysical singularities at Q ∼ Λ QCD in the PT series for the coefficient functions E S ( Q ) (2.4) and E NS ( Q ) (2.3) strongly affect the analysis of the spin sum rules at low Q [10]. Their influence becomes essential at Q < α s models are free of such a problem, thus providinga more reliable tool of investigating the behavior of the spin sum rules in the low-energydomain.The moments of the structure functions are analytic functions in the complex Q planewith a cut along the negative real axis, as demonstrated in Refs. [20, 21]. On the other hand,the standard PT approach does not support these analytic properties. The APT method[11] gives the possibility of combining the RG resummation with correct analytic propertiesof the QCD corrections. The consequence of requiring these properties to hold in the DISdescription was studied previously in Refs. [22, 23].Let us recall that the expression for Γ p,n ( Q ) in the framework of the analytic approachis completely similar to the one in the standard PT (2.2):Γ p,n ,AP T ( Q ) = 112 (cid:20)(cid:18) ± a + 13 a (cid:19) E AP TNS ( Q ) + 43 a inv E AP TS ( Q ) (cid:21) + ∞ X i =2 µ AP T i ; p,n ( Q ) Q i − . (2.6)The corresponding NNLO APT modification of the singlet and nonsinglet coefficient func-tions is E AP TNS ( Q ) = 1 − . A (3)1 ( Q ) − . A (3)2 ( Q ) − ... ,E AP TS ( Q ) = 1 − . A (3)1 ( Q ) − . A (3)2 ( Q ) − ... , (2.7)where A (3) k is the analyticized k -th power of three-loop PT coupling in the Euclidean domainand defined as A ( n ) k ( Q ) = 1 π Z + ∞ Im([ α ( n ) s ( − σ, n f )] k ) dσσ + Q , n = 3 . (2.8)In the one-loop case, the APT Euclidean functions are simple enough [11]: A (1)1 ( Q ) = 1 β (cid:20) L + Λ Λ − Q (cid:21) , L = ln (cid:18) Q Λ (cid:19) , (2.9) A (1)2 ( l ) = 1 β (cid:20) L − Q Λ ( Q − Λ ) (cid:21) , A (1) k +1 = − k β d A (1) k dL . Analogous two- and three-loop level expressions are more involved. However, according tothe “effective log” approach [24] in the region
Q < effective logarithm L ∗ : A (3)1 , , ( L ) → A mod , , = A (1)1 , , ( L ∗ ) , L ∗ ≃ Q/ Λ (1)eff ) , Λ (1)eff ≃ .
50 Λ (3) . (2.10)Thus, instead of the exact three-loop expressions for the APT functions, in Eq. (2.7) onecan use the one-loop expressions (2.9) with the effective Λ parameter Λ mod = Λ (1)eff whosevalue is given by the last relation (2.10). This model was successfully applied for higher-twist analysis of low-energy JLab data in Refs. [9, 10], and also in the Υ decay analysis inRef. [25]. Note also that the APT couplings are stable with respect to different loop ordersat low-energy scales Q . [12], contrary to the standard PT approach.4he APT functions A k contain the ( Q ) − k power contributions, which effectively changethe values of the µ -terms, when going from the PT to the APT framework. In particular,by subtracting an extra ( Q ) − term induced by the APT series for the Bjorken sum ruleΓ p − n ,AP T ( Q ) ≃ g A f (cid:18) Q / Λ (1)eff 2 ) (cid:19) + κ Λ (1)eff 2 Q + O (cid:18) Q (cid:19) where κ ≃ .
43 and Λ (1)eff ∼ .
18 GeV is the effective one-loop Λ
QCD parameter, we get therelation between µ p − n ,AP T coming into the APT expression (2.6) and the conventional µ p − n from Eq. (2.2): µ p − n (1 GeV ) M ≃ µ p − n ,AP T + κ Λ (1)eff 2 M . (2.11)Along with the conventional PT scheme, we will also apply the APT approach basedon Eqs. (2.6) and (2.7) to construct the improved model for smooth continuation of per-turbative expressions for Γ p,n ( Q ) and its non-singlet combination Γ p − n ( Q ) down to thenon-perturbative region Q → B. “Chiral” regime Q . Λ QCD
For the purpose of a smooth continuation of Γ p,n ( Q ) to the non-perturbative region0 ≤ Q . Λ QCD [5], we consider firstly the Q -evolution of the integral I ( Q ) ≡ M Q Γ ( Q ) = 2 M Q Z dx g ( x, Q ) , (2.12)which is equivalent to the integral over all energies of the spin-dependent photon-nucleoncross-section, whose value at Q = 0 is defined by the GDH sum rule [6] I (0) = − µ A , (2.13)where µ A is the nucleon anomalous magnetic moment. Then, the function I ( Q ) can bewritten as a difference I ( Q ) = I T ( Q ) − I ( Q ) , (2.14)where I T ( Q ) = 2 M Q Z dx g T ( x, Q ) , I ( Q ) = 2 M Q Z dx g ( x, Q ) . (2.15)The well-known Burkhardt-Cottingham (BC) sum rule [7] provides us with an exactexpression for I ( Q ), in terms of familiar electric G E and magnetic G M Sachs form factorsas I ( Q ) = 14 µG M ( Q ) µG M ( Q ) − G E ( Q )1 + Q / M , (2.16)5here µ is the nucleon magnetic moment. As a consequence of the strong Q behavior ofthe r.h.s. of Eq. (2.16), we get for large Q Z g ( x, Q ) dx (cid:12)(cid:12) Q →∞ = 0 , (2.17)so I is much smaller than I for large Q . Now from the BC sum rule (2.16), it follows that I (0) = µ A + µ A e e is the nucleon charge. Then the GDH value (2.13) is reproduced with I T (0) = µ A e . (2.19)To summarize, from the above equalities (2.16), (2.17) and (2.19), we can conclude thatthe BC and GDH sum rules together, lead to positivity of I T ( Q ) for all Q in the protoncase and a vanishing difference between I T ( Q ) and I ( Q ) for large Q . Thus, I pT ( Q ) isa smooth and monotonous function, and it is possible to obtain its smooth interpolationbetween large Q and Q = 0 [5]. III. IMPROVED MODEL FOR SMOOTH INTERPOLATION OF I T ( Q ) To improve the agreement between the model predictions and the experimental data, weconsider the general asymptotic expression I p,n ,pert ( Q ) = 2 M Q " (cid:18) ± a + 13 a (cid:19) E NS ( Q ) + 19 a E S ( Q ) + ∞ X i =2 µ p,n i ( Q ) Q i − , (3.1)where the nonsinglet E NS and singlet E S coefficient functions are defined in Eqs. (2.3) and(2.4), respectively. Then, the perturbative expression for I T , defined above the matchingpoint Q , is I T,pert ( Q ) = Θ( Q − Q ) [ I ,pert ( Q ) + I ( Q )] , (3.2)where I ,pert ( Q ) is calculated from Eq. (3.1), while I ( Q ) in known from the BC sum rule(2.16). The smooth interpolation to the GDH value at Q = 0 (2.19) is difficult and cannotbe performed analytically. Following the procedure developed in Ref. [5], we instead makeuse of the smooth extrapolation of the perturbative expression (3.2), to the nonperturbativedomain Q < Q defining the polynomial in positive powers of Q as I T,nonpert ( Q ) = Θ( Q − Q ) N X n =0 n ! ∂ n I T,pert ∂ ( Q ) n (cid:12)(cid:12)(cid:12) Q = Q ( Q − Q ) n , (3.3)where N is the number of derivatives, which is a free parameter of the model, together withthe matching point Q = Q , which have to be chosen to satisfy I T,nonpert (0) = µ A e . (3.4)6n practice, the easiest way to solve the problem is to fix the number of derivatives N andthen to vary the Q value until the relation (3.4) is satisfied. It is interesting to note thattaking N = 1 does not allow for such a solution.Such a procedure can be considered as a matching of the “twist-like” expansion in negativepowers of Q and the “chiral-like” expansion in positive powers of Q [5], which is similarto the matching of the expansions in direct and inverse coupling constants.Once we have obtained the parameters N and Q , then the all- Q expressions for themoments Γ p,n can be restored from I T ( Q ) defined by Eqs. (3.1) – (3.3), by using Eqs. (2.12)and (2.14). As we will see below, this can be done within both the standard PT andsingularity-free APT in the same way, leading to rather similar curves for Q -evolution,except that in the APT case the matching point Q playing a role of the “pQCD frontier” inthis interpolation scheme is noticeably shifted down to lower Q scales (see the next Section). IV. HIGHER-TWIST ANALYSIS
A detailed higher twist analysis of the recent Jefferson Lab data [3] on the lowest momentsof the spin-dependent proton and neutron structure functions Γ p,n ( Q ) and Γ p − n ( Q ) in therange 0 . < Q < was performed in Refs. [9, 10]. In particular, including only threeterms of the OPE expansion µ , , in Eq. (2.2), a satisfactory description of the data hasbeen achieved down to Q ≃ .
17 GeV in conventional PT and down to Q ≃ .
10 GeV inthe APT.The lower Q involved, the higher twist contribution is needed to describe the data. Aswas shown in Ref. [10], there is some sensitivity of fitted values of µ to the minimal scale Q min variations; namely, it increases in magnitude when one incorporates into the fit thedata points at lower energies. This property of the fit was treated as the slow (logarithmic)evolution µ ( Q ) (and a ( Q ) in the singlet case) with Q which becomes more noticeable forbroader fitting ranges in Q , as discussed above. Indeed, fit results for µ with taking intoaccount the RG evolution with Q = 1 GeV, as a normalization point become more stablewith respect to Q min variations.However, there is still a problem how to treat the evolution of higher-twist terms µ , ,.. ( Q )which again may turn out to be important when one goes to lower Q , since the fit becomesmore sensitive to very small variations of µ , ,.. with Q . Since the evolution of the highertwists µ , is still theoretically unknown, they can be taken as free parameters [26]. Thisprocedure leads to rather small χ D.o.f ∼ . Q . On the other hand, by including µ , ,.. into the fit, one observes only a smallchange in µ (1 GeV ) [10], which demonstrates its stability down to lower Q . Taking thisinto account, in order to reduce the number of free parameters, in the current work we applyanother fitting procedure and determine, first, µ and a at higher scale Q = 1 GeV. Thenwe extract µ applying the known QCD evolution for µ ( Q ) and a ( Q ) and fixing themat 1 GeV from the previous fit. The number of free parameters does not grow in this case.The fitting domain is restricted from below by Q min defined by the condition χ /D.o.f ≤ µ ( Q ), whichtends to be quite noticeable at lower Q , we do not go below Q min and do not take intoaccount µ -term here.The advantage of the APT analysis is the infrared and higher loop stability of the radiativecorrections, as well as the stability w.r.t. Λ QCD variations, leading to the stability andconvergence of the higher twist series extracted from the data. Indeed, as we see from7able I, in the APT case the applicability of the perturbative expansion (2.6) is somewhatshifted down to lower Q , due to the absence of Landau singularities (see also Ref. [10] andreferences therein). V. ALL- Q SPIN SUM RULES
In the perturbative expression (3.1) we take into account the two-loop perturbative cor-rection in the singlet E S and non-singlet E NS coefficient functions, as well as the twist-4,6contributions discussed in the previous section. To explore the infrared sensitivity of themodel of the smooth continuation to Q = 0, we used two different sets of higher twist terms(with µ and µ , , respectively) and the corresponding singlet axial charge extracted fromthe data above a certain minimal scale Q min . Q -0.020.020.040.060.08 Q =1.0 GeV (PT&APT)
Γ ( ) p Q Q -0.020.020.040.060.08 Γ ( ) p Q Q =0.3 GeV (APT)
Q =0.8 GeV (PT)
FIG. 1:
Proton spin sum rule function Γ p ( Q ) with respect to the combined set of JLab and SLACdata. Results are shown with an account of the twist-4 term (left panel) and the twist-4,6 terms(right panel). Corresponding perturbative parts are calculated in the framework of conventional PT(dotted lines) and APT (dashed lines). All- Q model function obtained by the smooth interpolationof I pT ( Q ) is also presented in PT (dash-dotted lines) and APT (solid lines). In Fig. 1 we present the proton spin sum rule function Γ p ( Q ) obtained by the smoothinterpolation of the perturbative part I pT,perp ( Q ) to the non-perturbative region Q → p − n ( Q ) calculated at any Q inthe similar way as Γ p ( Q ).The all- Q model functions Γ p ( Q ) and Γ p − n ( Q ) in both versions of the perturbation the-ory (dash-dotted and solid lines) are rather close to each other demonstrating the agreementbetween the singularity-free APT analysis at lower Q and the usual PT one at relativelyhigher Q . Also, as one can see from the comparison of the left and right panels the resultsof the interpolation do not strongly depend on the number of higher twists included and,hence, on the border Q between perturbative and non-perturbative regimes. This exhibitsa sort of duality between them implying that the experimental data in the wide intermediateregion Λ QCD . Q ∼ can be described equally well either by OPE 1 /Q -series or by“chiral-like” Q -series.We studied the sensitivity of above results w.r.t. variations of the number of derivatives N in Eq. (3.3) being the number of positive ∼ Q i power terms. As mentioned above, at8 ABLE I:
Combined fit results of JLab and SLAC data on the Bjorken SR and proton SSR for the singletaxial charge a , and the higher-twist terms µ and µ defined at the normalization point Q = 1 GeV in the APT and the standard PT approaches, along with the matching value Q . Corresponding curvesfor Γ p ( Q ) and Γ p − n ( Q ) are shown in Figs. 1 and 2, respectively. Typical values of χ /D.o.f are closeto unity. Method Target Q min , GeV a µ /M µ /M Q , GeV p-n 1.0 – − . − .
060 0.010(2) 0.8(2)proton 1.0 0.34(3) − . − .
056 0.010(2) 0.8(2)p-n 1.0 – − . − .
058 0.010(1) 0.3(1)proton 1.0 0.37(2) − . − .
063 0.011(1) 0.3(1) Q Γ ( ) p-n Q Q =1.1 GeV (PT)
Q =1.0 GeV (APT) Q Γ ( ) p-n Q Q =0.8 GeV (PT)
Q =0.3 GeV (APT)
FIG. 2:
Bjorken sum rule function Γ p − n ( Q ) with respect to the combined set of JLab and SLACdata. The meaning of curves here is the same as in Fig. 1. lower Q we need more higher 1 /Q -power twist terms. In the same way, going up from verylow Q we observe analogously that to describe the data at higher Q we need more ∼ Q i power terms, i.e. a higher value N .The minimal number of derivatives N min , which is necessary to perform the smoothextrapolation according to Eq. (3.3) in the conventional PT case and with one µ term only,is N min = 4. Corresponding matching value between perturbative and non-perturbativedomains in this case is found to be Q = 1 . ± . for the proton SSR and Q =1 . ± . for the Bjorken SR (see Table I). However, if one increases the numberof Q -power term up to N = 6, the applicability of the “chiral-like” expansion raises upto Q ≃ . for the proton SSR and Q ≃ . for the Bjorken SR. Similarobservation was made earlier in Ref. [5].In the framework of APT the minimal number of derivatives N min = 3 is even smallerthan in the conventional PT. In this case, if only one µ term is included then the matching9alue turned out to be same as in PT: Q ≃ . for both the proton SSR and p − n demonstrating the similarity of the APT and PT predictions at Q & Q including two µ , terms leads to quite different Q valuesfor PT and APT. In this case, for the proton SSR and Bjorken SR we have Q ≃ . ± . (PT, N min = 3) and Q ≃ . ± . (APT, N min = 2). Such a shiftof the border between perturbative and non-perturbative domains in the APT is a directconsequence of the disappearance of the unphysical singularities in the radiative corrections,and confirms the similar conclusion made in Refs. [9, 10]. Q -0.10-0.08-0.06-0.04-0.02 Q =0.5 GeV (PT) Q =0.3 GeV (APT)
Γ ( ) n Q - Q Γ ( ) d Q - Q =0.3 GeV (APT)
20 2
Q =0.8 GeV (PT)
FIG. 3:
Neutron (left) and deuteron (right) spin sum rule functions, Γ n ( Q ) and Γ d ( Q ) , withrespect to the combined set of JLab and SLAC data. Results are shown with an account of twist-4,6terms. The meaning of curves here is the same as in Fig. 1. Finally, in Fig. 3 we show the neutron spin sum rule function Γ n ( Q ), which is simplyobtained from the difference Γ p ( Q ) − Γ p − n ( Q ), and the deuteron spin sum rule Γ d ( Q ).We also present its perturbative PT and APT parts together with less precise data. Bothversions of the perturbation theory predict monotonous curves for Γ n,d ( Q ) at any Q . Com-parison between them and the results of Ref. [10] demonstrates the consistence of the directfits to the data and the predictions of the generalized GDH sum rule. VI. CONCLUSION
In the current paper we have considered the all- Q model for the generalized GDH sumrule, constructed by the smooth interpolation of I T ( Q ) between large Q and Q = 0, in theframework of both the conventional PT and the ghost-free Analytic Perturbation Theory.We used the values of the power corrections µ , ,.. and singlet axial charge a , systematicallyextracted from the precise JLab data. We achieve a rather good description of the protondata on Γ p ( Q ) at any Q values. We also present an improved description of the neutrondata, as well as the Bjorken sum rule data at all experimentally accessed Q .The results of the smooth interpolation Γ p ( Q ) and Γ p − n ( Q ) do not strongly dependon the number of higher-twist terms, and on the border Q between perturbative and non-perturbative regimes. This exhibits a sort of duality between them implying that the exper-imental data in the wide intermediate region Λ QCD . Q ∼ can be described equallywell either by OPE 1 /Q -series or by non-perturbative “chiral-like” Q -series. Within theanalytic PT the “pQCD frontier” being the matching value between Q - and 1 /Q -power10eries naturally decreases from 1 . with single µ down to 0 . with extra µ -term included, which is significantly lower than the corresponding value in conventional PT Q ≃ . . Such a shift of the border between perturbative and non-perturbative do-mains in the APT is a direct consequence of the disappearance of the unphysical singularitiesin the radiative corrections. ACKNOWLEDGMENTS
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