(appears in March 01, 2001, issue of Phys. Rev. D) Scale- and scheme–independent extension of Pad´e approximants; Bjorken polarized sum rule as an example G. Cvetiˇc Asia Pacific Center for Theoretical Physics, Seoul 130-012, Korea and 1 Dept. of Physics, Universidad T´ecnica Federico Santa Mar´ıa, Valpara´ıso, Chile 0 ∗ 0 e-mail: cvetic@fis.utfsm.cl 2 n a R. K¨ogerler J Dept. of Physics, Universit¨at Bielefeld, 33501 Bielefeld, Germany 5 2 e-mail: [email protected] 3 v 8 Abstract 9 0 6 0 0 0 A renormalization–scale–invariant generalization of the diagonal Pad´e / approximants (dPA), developed previously, is extended so that it becomes h p renormalization–scheme–invariant as well. We do this explicitly when two - p terms beyond the leading order (NNLO, α3) are known in the truncated ∼ s e perturbation series (TPS). At first, the scheme dependence shows up as a de- h v: pendence on the first two scheme parameters c2 and c3. Invariance under the i change of the leading parameter c2 is achieved via a variant of the principleof X minimal sensitivity. Thesubleading parameter c is fixed so that a scale– and 3 r a scheme–invariant Borel transform of the resummation approximant gives the correct location of the leading infrared renormalon pole. The leading higher– twist contribution, or a part of it, is thus believed to be contained implicitly in the resummation. We applied the approximant to the Bjorken polarized sum rule (BjPSR) at Q2 =5 and 3 GeV2, for the most recent data and the ph data available until 1997, respectively, and obtained αMS(M2)=0.119+0.003 s Z 0.006 and0.113+0.004,respectively. VerysimilarresultsareobtainedwiththeG−run- 0.019 − berg’s effective charge method and Stevenson’s TPS principle of minimal sen- sitivity, ifwefixc –parameterinthembytheafore-mentionedprocedure. The 3 central values for αMS(M2) increase to 0.120 (0.114) when applying dPA’s, s Z and 0.125 (0.118) when applying NNLO TPS. PACS number(s): 11.10.Hi, 11.80.Fv, 12.38.Bx, 12.38.Cy address after August, 2000 ∗ 1 I. INTRODUCTION The problem of extracting as much information as possible, from an available QCD or QEDtruncatedperturbationseries(TPS) ofanobservable, andincluding thisinformationin a resummed result, was thefocusofseveral works during the last twenty years. Most ofthese resummation methods are based on the available TPS only. Some of these latter methods eliminate the unphysical dependence of the TPS on the renormalization scale (RScl) and scheme (RSch) by fixing them in the TPS itself. Among these methods are the BLM fixing motivated by large–n considerations [1], principle ofminimal sensitivity (PMS) [2], effective f charge method (ECH) [3,4] (cf. Ref. [5] for a related method). Some of the more recent approaches in this direction include approaches related with the method of “commensurate scale relations” [6], an approach using an analytic form of the coupling parameter [7], ECH– related approaches [8], a method using expansions in the two–loop coupling parameter [9] expressed in terms of the Lambert function [10], methods using conformal transformations either for the Borel expansion parameter [11] or for the coupling parameter [12]. A basically different method consists in replacing the TPS by Pad´e approximants (PA’s) which provide a resummation of the TPS such that the resummed results show weakened RScl and RSch dependence [13]. In particular, the diagonal Pad´e approximants (dPA’s) were shown to be particularly well motivated since they are RScl–independent in the approximation of the one–loopevolutionofthecouplingα (Q2)[14]. AnadditionaladvantageofPA’sisconnected s withthefactthattheysurmount thepurelypolynomialstructureoftheTPS’s onwhichthey are based, and thus offer a possibility of accounting for at least some of the nonperturbative contributions, via a strong mechanism of quasianalytic continuation implicitly contained in PA’s. Recently, we proposed a generalization of the method of dPA’s which achieves the exact perturbative RScl independence of the resummed result [15]. While this procedure in its original form was restricted to the cases where the number of available TPS terms beyond the leading order (LO: α1) is odd, it was subsequently extended to the remaining cases ∼ where this number is even [16]. This would then apply to those QCD observables where the number of such known terms is two (NNLO, α3).2 In [16] we also speculated on ways ∼ s how to eliminate the leading RSch–dependence from our approximants A, and proposed for the NNLO case a simple way following the principle of minimal sensitivity (PMS). It turns out that the way proposed there does not work properly in practice since no minimum of the PMS equation ∂A/∂c = 0 [cf. Eq. (40) there] can be found. The dependence of 2 our approximants on the RSch–parameters c β /β and c β /β of the original TPS 2 2 0 3 3 0 ≡ ≡ is definitely a problem when the approximants are applied to the low–energy observables like the Bjorken polarized sum rule (BjPSR) at the low momentum transfer of the virtual photon, e.g. Q2 3–5 GeV2 [17]. ph≈ In the present work, we address this problem. For the NNLO TPS case, we construct in Section II an extended version of our approximants, in which the dependence on the A leading RSch-parameter c is successfully eliminated by application of a variant of PMS 2 2 When just one such term is known (NLO), our approximants give the same result as the ECH method. 2 (j) conditions ∂ /∂c =0. This procedure can be extended in a straightforward way to the A 2 cases where moreterms are known inthe TPS, e.g. the NNNLO cases availablenow in QED, but we will not discuss such cases here. In Section III, we apply our approximant to the BjPSRatsuchQ2 wherethreequarkflavorsareassumedactive, e.g. Q2 3–5GeV2. While ph ph≈ the approximant at this stage is an RScl–independent and c –independent generalization of 2 the diagonal Pad´e approximant (dPA) [2/2], it still contains c –dependence comparable to 3 that of the ECH [3] and TPS–PMS [2] methods. Subsequently, we fix the value of c in our, 3 the ECH and the TPS–PMS approximants so that PA’s of a modified (RScl– and RSch– independent) Borel transform of these approximants yield the correct location of the leading infrared (IR) renormalon pole. Thus, in the approximants we implicitly use β–functions which go beyond the last perturbatively calculated order of the observable (NNLO), in order to incorporate the afore–mentioned nonperturbative information. In Section IV we then compare the values of these resummation approximants with the values for the BjPSR extracted from experiments, and obtain predictions for α (M2). We also apply the TPS and s Z various PA methods of resummation to these values of the BjPSR and obtain higher values for α (M2). In Section V we redo the calculations by applying PA–type of quasianalytic s Z continuation for the β–functions relevant for our, ECH, and TPS–PMS approximants. We further address the question of higher–twist terms. In Section VI we discuss the obtained numerical results for α (M2), and Section VII contains summary and outlook. s Z A brief version containing a summarized description and application of the method can be found in [18]. In contrast to [18], the numerical analysis of the BjPSR in the present paper (Sections IV, V) uses, in addition, the most recent data of the E155 Collaboration [19]. II. CONSTRUCTION OF C –INDEPENDENT APPROXIMANTS 2 Let us consider a (QCD) observable S, with negligible mass effects, which is normalized so that its perturbative expansion takes the canonical form S = a (1+r a +r a2 +r a3 + ) , (1) 0 1 0 2 0 3 0 ··· where a α(0)/π. We suppose that this expansion is calculated within a specific RSch and 0≡ s using a specific (Euclidean) RScl Q (symbol ‘0’ is generically attached to the RScl and 0 RSch parameters in the TPS) up to NNLO, yielding as the result the TPS S = a (1+r a +r a2) . (2) [2] 0 1 0 2 0 Here, both a and the coefficients r and r are RScl– and RSch–dependent. The coupling 0 1 2 parameter a α /π evolves under the change of the energy scale (RScl) Q, within the given s ≡ RSch, according to the following renormalization group equation (RGE): ∂a(lnQ2;c(0), ) 2 ··· = β a2(1+c a+c(0)a2 +c(0)a3 + ) , (3) ∂ln(Q2) − 0 1 2 3 ··· 3 whereβ andc areuniversal quantities (RScl–andRSch–invariant),3 whereastheremaining 0 1 (0) coefficients c (j 2) are RSch–dependent and their values can – on the other hand – be j ≥ used to characterize the RSch. Consequently, in (2) the coupling parameter a is a function 0 of the RScl and RSch a a(lnQ2;c(0),c(0), ) . (4) 0 ≡ 0 2 3 ··· The NLO and NNLO coefficients in (2) have, due to the RScl and RSch independence of S, the following RScl and RSch dependence: r r (lnQ2) = r (lnQ˜2)+β ln Q2/Q˜2 , 1 ≡ 1 0 1 0 0 r r (lnQ2;c(0)) = r2(lnQ2)+c(cid:16)r (lnQ(cid:17)2) c(0) +ρ , (5) 2 ≡ 2 0 2 1 0 1 1 0 − 2 2 where ρ is RScl– and RSch–invariant. Although the physical quantity S must be indepen- 2 dent of the RScl and RSch, its TPS (2) possesses an unphysical dependence on RScl and RSch which manifests itself in higher order terms ∂S ∂S ∂S [2] a4 [2] [2] . (6) ∂lnQ2 ∼ 0 ∼ ∂c(0) ∼ ∂c(0) 0 2 3 All approximants to S which are based on TPS (2) must fulfill the Minimal Condition: when expanded in powers of a to order a3, they must reproduce TPS (2). Further, since 0 0 the full S is RScl- and RSch–independent, the approximant should preferably share this property with S if it is to bring us closer to the actual value of S. The generalization of the diagonal Pad´e approximants developed in Ref. [15] possesses full RScl independence for massless observables. InitsoriginalformitisaccountableonlytoTPSwithanoddnumber oftermsbeyondthe leading order (LO: a1). Unfortunately, however, QCD observables have been calculated ∼ at most to the NNLO, i.e., at best the TPS (2) is known. Therefore, in Ref. [16] we have extended the method to the cases with even numbers of terms beyond the LO, in particular for the TPS of the type (2). Since within the present paper we are going to apply an extended related procedure to these cases of S , we recapitulate briefly the main steps for [2] treating a TPS of the generic form S . The trick consisted in introducing – in addition to [2] S – the auxiliary observable S˜ S S, which then gets the following formal canonical form: ≡ ∗ S˜ = (S)2 = a (0+a +R a2 +R a3 + ), (7) 0 0 2 0 3 0 ··· where : R = 2r , R = r2 +2r ,... (8) 2 1 3 1 2 S˜ is then known formally to NNNLO ( a4) and the method can thus be applied, yielding ∼ [2/2] ˜ [2/2] an approximant to S. The corresponding approximant to S is A which has the AS2 S2 form [16] q A[2/2] = α˜ a(lnQ˜2;c(0),c(0),...) a(lnQ˜2;c(0),c(0),...) 1/2 = S + (a4) , (9) S˜ 0 1 2 3 − 2 2 3 [2] O 0 r n h io (cid:16) (cid:17) 3 β =(11 2n /3)/4, c =(102 38n /3)/(16β ), where n is the number of active quark flavors. 0 f 1 f 0 f − − 4 and it is again exactly RScl–invariant. Here, the two scales Q˜ (j=1,2) and the factor α˜ j 0 are independent of the RScl Q and determined by the identities 0 ln(Q˜2/Q2) 1 1 2 0 = ˜b ˜b2 4˜b , α˜ = , (10) ln(Q˜21/Q20)! 2β0 (cid:20) 1 ±q 1 − 2(cid:21) 0 ˜b21 −4˜b2 3 q ˜b = c 2r , ˜b = c2 +c(0) +c r +3r2 2r . (11) 1 1 − 1 2 −2 1 2 1 1 1 − 2 (0) If we ignore all higher than one–loop evolution effects, i.e., if we set c =0=c in (10)– 1 2 (11) and replace the two coupling parameters in (9) by their one–loop evolved (from RScl Q2 to Q˜2) counterparts, then the approximant (9) becomes the square root of the [2/2] 0 j Pad´e approximant of S˜. This follows from general considerations in [15,16], but can also be 1/2 verified directly in this special case. The approximant [2/2] preserves the RScl–invariance S˜ only approximately (in the one–loop RGE approximation). Although the RScl dependence is eliminated completely by using the approximant (9), (0) there remains a RSch–dependence, i.e., dependence on c (j 2). It manifests itself to j ≥ ˜ (0) ˜ (0) a large degree due to ∂b /∂c = 0 (∂b /∂c = 3). In Ref. [16] we speculated that the 2 2 6 2 2 (0) dependence on the leading RSch–parameter c could be eliminated by imposing the PMS 2 condition of local independence (cf. Eq. (40) in [16]) dA[2/2] lnQ˜2(c(0)) ;c(0),c(0),... S˜ j 2 j 2 3 (cid:18)n o (cid:19) = 0 , (12) (0) dc2 (cid:12)(cid:12)c(0),... (cid:12) 3 (cid:12) where implicitly “=0” should be understood as “ a6” sin(cid:12)ce in general this derivative is a5. ∼ 0 ∼ 0 However, expansion of this expression in powers of the coupling a (or: any a) yields 0 [2/2] dA S˜ = 10c a5 + (a6) . (13) dc(20) (cid:12)(cid:12)c(0),... − 1 0 O 0 (cid:12) 3 (cid:12) This implies that the approximan(cid:12)t (9) to S has no stationary (PMS) point with respect to (0) the RSch–parameter c , since the coefficient of the leading term in the expansion of the 2 derivative is constant and cannot be made equal zero by a change of the RSch. Also actual numerical calculations for various observables S confirm this. Therefore, wewillmodifytheapproximant(9)sothatthenewonewillallowustoremove, (0) by a PMS condition, the dependence on the leading RSch–parameter c . This modification 2 must, of course, be such that the afore–mentioned Minimal Condition is satisfied and that the RScl–invariance is preserved. We do this in the following way. We keep the overall (0) functional structure of (9). However, we replace the single set of RSch–parameters c j (1) (j 2), which we inherited from the TPS, by two sets of apriori arbitrary parameters c ≥ j (2) and c (j 2) in the two coupling parameters, respectively, and we also admit new values j ≥ of the reference momenta Q2 and Q2 1 2 [2/2] = α˜ a(lnQ2;c(1),c(1),...) a(lnQ2;c(2),c(2),...) 1/2 = S + (a4) . (14) AS˜ 1 2 3 − 2 2 3 [2] O 0 r n h io (cid:16) (cid:17) 5 (1) (2) Theparametersc andc willbeappropriatelyfixed. Theywillturnouttobeindependent j j of the RSch–parameters c(0) and of the RScl Q2 of the original TPS, just like the scales Q2 j 0 1 and Q2 and the parameter α˜ will be.4 We will now require c(1) = c(2), in contrast to (9) 2 2 6 2 which led us to the problem (13). This requirement is not unnatural, since the forms (9) and (14) have Q˜2=Q˜2 and Q2=Q2, respectively. The two new momentum scales Q and the 16 2 16 2 j (j) parameter α˜ in (14) will be determined, in terms of c ’s (k=2,3; j=1,2), by expanding the k two coupling parameters in power series of the original coupling a (4) and requiring that 0 [2/2] the Minimal Condition be fulfilled, i.e., that the power series for coincides with that of AS2 S˜ (7)-(8) up to (and including) a4. For this purpose we use the expansion for the general ∼ 0 a a(lnQ2;c ,c ,...) in powers of a a(lnQ2;c(0),c(0),...) as obtained in Appendix A ≡ 2 3 0 ≡ 0 2 3 [Eqs. (A.7)–(A.9)], and apply it to as yet unspecified parameters Q2, Q2 and c(j) (j=1,2). 1 2 k The resulting expressions, when introduced into the square of the right–hand side of (14), yield an expansion in powers of a . According to the Minimal Condition, it should coincide 0 with (7) up to a4. Comparison of the coefficients of an (n=2,3,4) leads to the following ∼ 0 0 relations: ( 1) ( 1) at a2 : 1 = α˜(x x ) , = α˜ = − = − . (15) 0 − 1− 2 ⇒ (x x ) β ln(Q2/Q2) 1− 2 0 1 2 at a3 : 2r = (x2 x2) c (x x )+δc /(x x ) , (16) 0 1 − 1− 2 − 1 1− 2 2 1− 2 h 5 i at a4 : 2r +r2 = (x3 x3)+ c (x2 x2) c(0)(x x ) 0 2 1 − − 1− 2 2 1 1− 2 − 2 1− 2 h 1 (1) (2) 3(x δc x δc )+ δc /(x x ) , (17) − 1 2 − 2 2 2 3 1− 2 i where we have used the notations x β ln(Q2/Q2) , δc(j) c(j) c(0) (j=1,2) , (18) j ≡ 0 j 0 2 ≡ 2 − 2 (1) (2) (1) (2) δc c c , δc c c . (19) 2 ≡ 2 − 2 3 ≡ 3 − 3 Eqs. (16) and (17) are the two equations which determine the two scales Q and Q ( 1 2 ⇔ (j) parameters x and x ) as functions of c ’s (k=2,3; j=1,2). In order to see that these two 1 2 k (0) scales are independent of the original RScl (Q ) and of the original RSch (c , k 2), we 0 k ≥ introduce x˜ β ln(Q2/Λ˜2) (j=1,2) , (20) j ≡ 0 j where Λ˜ is the universal QCD scale appearing in the Stevenson equation (A.1), so it is RScl– and RSch–invariant. After some algebra, we can rewrite Eqs. (16) and (17) as a system of equations for x˜ j 4 Parameters c(1) and c(2) will be chosen later in the Section, by following a variant of the PMS; 2 2 (1) (2) c and c will be set equal to each other and fixed in the next Sections. 3 3 6 δc 2 2ρ +c = (x˜ +x˜ )+ , (21) 1 1 1 2 (x˜ x˜ ) 1 2 − (1) (2) 5 (x˜ c x˜ c ) δc 2ρ +3ρ2 2c ρ = (x˜2+x˜ x˜ +x˜2) c (x˜ +x˜ )+3 1 2 − 2 2 3 , (22) 2 1 − 1 1 1 1 2 2 − 2 1 1 2 (x˜ x˜ ) − 2(x˜ x˜ ) 1 2 1 2 − − where ρ and ρ are the usual RScl– and RSch–invariants as defined, e.g., in [2]5 [cf. also 1 2 (5)] ρ = β ln(Q2/Λ˜2) r , (23) 1 0 0 − 1 ρ = r r2 c r +c(0) . (24) 2 2 − 1 − 1 1 2 (1) (2) (1) (2) Therefore, Eqs. (21)–(22) show the following: If c and c and δc c c are chosen 2 2 3≡ 3 − 3 and fixed, then the solutions x˜ and thus the scales Q (j=1,2) are independent of the RScl j j (0) (0) (Q ) and of the RSch (c ,c ,...). Thus, we have 0 2 3 ( 1) Q2 = Q2(c(1),c(2);δc ) (j=1,2) , α˜ = − = α˜(c(1),c(2);δc ) . (25) j j 2 2 3 β ln(Q2/Q2) 2 2 3 0 1 2 (j) Therefore, our approximant (14) will be regarded from now on as a function of only c k [2/2] (1) (2) (1) (2) parameters (k 2; j=1,2): (c ,c ;c ,c ;...). For actually solving the equations ≥ AS2 2 2 3 3 for the scales Q and Q , it is more convenient to use Eqs. (16)–(17). For the subsequent 1 2 use, we rewrite them in the following form: 1 3 y4 y2z2(c(s))+y (5c δc δc ) (δc )2 = 0 , (26) − − − 0 2 −4 1 2 − 3 − 16 2 1 1δc 2 r + c = y , (27) 1 1 + − 2 − 4 y − where we use the notations 1 Q2 Q2 y β ln 1 ln 2 , (28) ± ≡ 2 0" Q20 ± Q20# 1 (1) (2) (s) (1) (2) δc c c , c (c +c ) (k=2,3) , (29) k ≡ k − k k ≡ 2 k k 7 z2 2ρ + c2 3c(s) z2(c(s)) , (30) 0 ≡ 2 4 1 − 2 ≡ 0 2 (cid:18) (cid:19) where ρ is given by (24). Incidentally, it can be explicitly checked that in the special case 2 (1) (2) (0) (1) (2) (0) of c =c =c and c =c =c Eqs. (26)–(30) and (15) recover the old approximant 2 2 2 3 3 3 (9)–(11) of Ref. [16]. (j) (j) The next question is how to fix parameters c and c (j= 1,2). Above all, we have to 2 3 (j) fix the leading parameters c ’s since otherwise their arbitrariness would reflect the fact that 2 5 Ra¸czka [20] used the sum of the absolute values of terms in ρ for a formulation of criteria 2 for acceptable RScl’s and RSch’s in NNLO TPS. He concluded that the strong RScl and RSch dependence of the NNLO TPS of the BjPSR (with n =3) presents a serious practical problem. f 7 (0) the leading RSch–dependence (i.e., the dependence on c ) has not been eliminated from 2 the approximant. We do this by requiring the local independence of the approximant with (1) (2) respect to variation of c and of c separately. This condition is a variant of the principle 2 2 of minimal sensitivity (PMS), or a PMS–type ansatz [2/2] [2/2] [2/2] [2/2] ∂ ∂ ∂ ∂ AS˜ = 0 = AS˜ AS˜ = 0 = AS˜ (31) ∂c(21) (cid:12)(cid:12)(cid:12)c(22) ∂c(22) (cid:12)(cid:12)(cid:12)c(21) ⇐⇒ ∂c(2s) (cid:12)(cid:12)(cid:12)δc2 ∂(δc2)(cid:12)(cid:12)(cid:12)c(2s) (cid:12) (cid:12) (cid:12) (cid:12) Here, “=0” should(cid:12) be understood as(cid:12) “ a6” since in gener(cid:12)al these derivativ(cid:12)es are a5. These ∼ 0 ∼ 0 (1) (2) two equations then give us solutions for the leading parameters c and c , once the values 2 2 of the subleading parameters c(s) (c(1)+c(2))/2 and δc c(1) c(2) have been chosen.6 3 ≡ 3 3 3 ≡ 3 − 3 However, using Eq. (A.5) and the fact that Q2 are independent of c(s) [cf. (25)], we can show j 3 (s) the following dependence of the approximant on c (at constant δc ): 3 3 1 dln [2/2] = d(c(s)) (a3+a2a +a a2+a3)+ (a4) < d(c(s)) a 3 , (32) rAS˜ ! 3 4 1 1 2 1 2 2 O j ≈ 3 | 1| where a a(lnQ2;c(j),c(j),...) (j = 1,2) and we took the index convention a a . j ≡ j 2 3 | 1| ≥ | 2| (s) This means that the dependence on c cannot be eliminated in the considered case, not 3 even by a PMS variant. In this respect, the situation is analogous to the usual TPS–PMS [2] and the ECH [3] methods. These two methods (cf. Appendix C), while fixing RScl (Q Q =Q ) and c RSch–parameter (c(0) cPMS or cECH) in the original TPS 0 7→ ECH PMS 2 2 7→ 2 2 (2), leave the value of the subleading parameter c there unspecified, with the residual 3 c –dependence of the (TPS–)approximant 3 dln S(X) d(c )a3 /2 , (33) [2] ≈ 3 X (cid:16) (cid:17) where label ‘X’ stands either for ‘ECH’ of ‘TPS–PMS’. Comparing (32) and (33), we see (s) that the c –dependence of our approximant could be up to twice as strong as that of the 3 TPS–PMS and ECH methods. (1) (2) Hence, varyingc andc parametersinourapproximant atthispointwouldapparently 3 3 not lead to any new insight. For the sake of simplicity, we choose from now on these two subleading parameters to be equal to each other (1) (2) c = c c (δc = 0) , (34) 3 3 ≡ 3 3 but we will adjust the common parameter c later to a physically motivated value. 3 With the chosen restriction (34), the problem of finding our approximant (14) to the TPS (2) basically reduces to the problem of solving the system of three coupled equations (s) (1) (2) (26) and (31) for the three unknowns y [=β ln(Q /Q )] and δc and c ( c and c ). − 0 1 2 2 2 ⇔ 2 2 For completeness, the PMS–like equations (31), when δc =0=δc , are written explicitly 3 4 6 Also a value of δc c(1) c(2) has to be chosen – see later. 4≡ 4 − 4 8 in Appendix B, to the relevant order a5 at which we solve them – Eqs. (B.1)–(B.2). ∼ 0 From there and from (26) we explicitly see that these three equations contain only the (s) three unknowns (y , c and δc ) and the (known) RScl– and RSch–invariants ρ (24) and 2 2 2 − (1) (2) c =β /β . Interestingly enough, these three equations do not depend on c (=c =c ). 1 1 0 3 3 3 (j) In addition, they do not depend on any other higher order parameters c (k 4; j=1,2) k ≥ appearing in a a(lnQ2;c(j),c ,c(j),...), except on δc c(1) c(2) which was taken to be j ≡ j 2 3 4 4≡ 4 − 4 (j) zero in Eqs. (B.1)–(B.2). Hence, Q and c (j=1,2) will be functions of ρ and c only, j 2 2 1 thus explicitly RScl– and RSch–invariant. For simplicity, we want the solutions Q2 and j c(j) (j =1,2) to be independent of any higher order parameter c(j) (k 3) that possibly 2 k ≥ (1) (2) appears in our approximant, therefore we choose from now on also δc ( c c )= 0. 4 ≡ 4 − 4 The solution of the mentioned three coupled equations in any specific case can be found numerically, e.g. by using Mathematica or some other comparable software for numerical iteration. Certainly we have to ensure that the program scans through a sufficiently wide range of the initial trial values y(in.), (c(s))(in.) and (δc )(in.) for iterations, in order not to miss 2 2 − any solution. The solutions which result in either α˜ 1 or α˜ 1 should be discarded | |≫ | |≪ since they signal numerical instabilities of the approximant [ α˜ 1 Q2 Q2 – cf. (15)] | |≫ ⇒ 1≈ 2 or are in addition physically unacceptable ( α˜ 1 Q2 Q2 or Q2 Q2). We have | | ≪ ⇒ 1 ≪ 2 2 ≪ 1 apparently two possibilities: (s) y , c and δc are all real numbers (and thus the intial trial values as well); • − 2 2 (s) c and its initial values are real; y and δc and their initial values are imaginary • n2umbers (c(1) and c(2) are complex co−njugate 2to each other, as are Q2 and Q2). 2 2 1 2 In both cases, the approximant itself turns out to be real, as long as c is real. 3 If we encounter several solutions which give different values for the approximant, we should choose, again within the PMS–logic, among them the solution with the smallest (1) (2) curvature with respect to c and c . For such cases, we define two almost equivalent 2 2 expressions for such curvature in Appendix B – cf. Eqs. (B.4)–(B.5). III. BJORKEN POLARIZED SUM RULE (BPSR): C -FIXING 3 We will now apply the described method to the case of the Bjorken polarized sum rule (BjPSR) [21]. It is the isotriplet combination of the first moments over x of proton and Bj neutron polarized structure functions 1 1 dx g(p)(x ;Q2 ) g(n)(x ;Q2 ) = g 1 S(Q2 ) , (35) Bj 1 Bj ph − 1 Bj ph 6| A| − ph Z0 h i h i where p2= Q2 < 0 is the momentum transfer carried by the virtual photon. The quantity − ph S(Q2 ) has the canonical form (1). It has been calculated to the NNLO [22,23], in the MS ph RSch and with the RScl Q2 = Q2 . The pertaining values of r and r , for those Q2 where 0 ph 1 2 ph three quark flavors are assumed active (n =3), e.g. at Q2 =3 or 5 GeV2, are r = 3.5833 f ph 1 [22] and r = 20.2153 [23], so that 2 9 S (Q2 ;Q2 = Q2 ;cMS,cMS) = a (1+3.5833a +20.2153a2) , (36) [2] ph 0 ph 2 3 0 0 0 with : a = a(lnQ2;cMS,cMS,...) , n = 3 , cMS = 4.471, cMS = 20.99 . (37) 0 0 2 3 f 2 3 The constant g appearing in (35) is known from β–decay measurements [24] (it is denoted A | | there as g /g ) A V | | g = 1.2670 0.0035 . (38) A | | ± (s) Solving the coupled system of (26) and (B.1)–(B.2) for the three unknowns y , c and δc , 2 2 as discussed in the previous Section, results in this case in one physical solut−ion only7 1 Q2 y β ln 1 = 1.514 ( α˜ = 0.3301) , (39) − ≡ 2 0 Q2 − ⇒ (cid:18) 2(cid:19) (s) (1) (2) c = 3.301 , δc = 3.672 c = 1.465 , c = 5.137 . (40) 2 2 − ⇒ 2 2 Parameter y , defined in (28), is then obtained from (27). The resulting scales Q , Q are + 1 2 then 0.767 GeV, 1.504 GeV (Q2 =5 GeV2) and 0.594 GeV, 1.165 GeV (Q2 =3 GeV2). ph ph (j) We stress that these results are independent of the value of c (34) and of c and other c 3 4 k (k 5; j=1,2) in the approximant √ (14), and are independent of the choice of RScl S2 ≥ A (0) Q and RSch (c , k 2) in the original TPS S . In TPS (36), the choice was Q =Q and 0 k ≥ [2] 0 ph c(0)=cMS (=4.471). Knowing Q and c(j) (j=1,2), for the actual evaluation of approximant 2 2 j 2 (14) we need to assume a certain value for a (37) (at RScl Q ). The value of α˜ is obtained 0 0 from (15) (α˜=0.3303); the value of the coupling parameter a a(lnQ2;c(j),c ,c ,c(j),...) j≡ j 2 3 4 5 (j=1,2) can be obtained, for example, by solving the subtracted Stevenson equation (A.2) β ln Q2j = 1 +c ln c1aj + ajdx (c(2j)+c3x) 0 Q20 aj 1 1+c1aj! Z0 (1+c1x)(1+c1x+c(2j)x2+c3x3) 1 c ln c1a0 a0dx (cM2 S+cM3 Sx) (j=1,2) . (41) 1 −a0 − (cid:18)1+c1a0(cid:19)−Z0 (1+c1x)(1+c1x+cM2 Sx2+cM3 Sx3) In (41) we ignored terms c(j) and higher since they are not known (cMS is not known, ∝ 4 4 either). Stated otherwise, we set here and in the rest of this Section: c(1)=c(2)=cMS=0 for k k k k 4, i.e. the β–functions pertaining to the approximant are taken in the TPS form to the ≥ four–loop order. Hence, the only free parameter in the approximant √ (14) is now c S2 3 A [cf. condition (34)], all the other nonzero parameters (Q2, c(j), α˜) have been determined and j 2 are c – and RScl– and RSch–independent. Further, any effects due to the mass thresholds 3 7 Formally, we get two solutions, butthey give the same approximant, since the second solution is (1) (2) obtained from the first by Q Q and c c . Further, if ignoring in PMS conditions (B.1)– 1 ↔ 2 2 ↔ 2 (B.2) the denominators, one arrives at two additional solutions, both having c(s) = (6ρ 7c2/4)/7; 2 2− 1 however, one can check that also the denominators are then zero and the derivative (B.1) reduces to 2(2δc 15c y )a¯5/(3y ) which turns out to be finite and nozero. 2− 1 − 0 − 10