Generalization of the Brodsky-Lepage-Mackenzie optimization within the {β} -expansion and the Principle of Maximal Conformality
aa r X i v : . [ h e p - ph ] J a n INR-TH-9/2014
Generalization of the Brodsky-Lepage-Mackenzie optimizationwithin the { β } -expansion and the Principle of MaximalConformality A. L. Kataev ∗ Institute for Nuclear Research of the Academy of Sciences of Russia,117312, Moscow, Russia
S. V. Mikhailov † Bogoliubov Laboratory of Theoretical Physics,JINR, 141980 Dubna, Russia (Dated: March 6, 2018)
Abstract
We discuss generalizations of the BLM optimization procedure for renormalization group in-variant quantities. In this respect, we discuss in detail the features and construction of the { β } –expansion representation instead of the standard perturbative series with regards to the Adler D -function and Bjorken polarized sum rules obtained in order of O ( α s ). Based on the { β } –expansionwe analyse different schemes of optimization, including the corrected Principle of Maximal Con-formality, numerically illustrating their results. We suggest our scheme for the series optimizationand apply it to both the above quantities. PACS numbers: 12.38.Bx, 11.10.Hi ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION The problem of scale-scheme dependence ambiguities in the renormalization group (RG)calculations [1] remains important. In the past few years, a new extension of the BLM scale-fixing approach [2], called the Principle of Maximal Conformality (PMC), was started [3]and formulated in more detail in [4–7] with a variety of applications to phenomenologicallyoriented studies.Here we show that the PMC approach is closely related to the sequential BLM (seBLM)method, originally proposed in [8] for the analysis of the N LO QCD prediction for thequantities like e + e − -annihilation R-ratio. This method was based on the renormalizationgroup (RG) inspired representation of the { β } -expansion for perturbative series, the onelater used for other purposes in [9, 10]. The seBLM was constructed as a generalization ofthe works devoted to the extension of the BLM MS–type scale fixing prescription to thelevel of next-to-next-to-leading order (N LO) QCD corrections [11, 12] and beyond [13–16].In this paper, we will use the { β } -expansion representation and the seBLM method tostudy the e + e − -annihilation R-ratio, the related Adler function D EM of the electromagneticquark currents and the Bjorken sum rule S Bjp of the polarized lepton-nucleon deep-inelasticscattering (DIS). We will clarify the concrete theoretical shortcomings of the PMC QCDstudies performed in a number of works on the subject, in particular, in Refs. [4–7], andwill present the results for the corrected PMC approach.Certain problems of the misuse of the PMC approach to the Adler function were alreadyemphasized in [17] but not recognized in the recently published work [7]. We will clarifythese theoretical problems in more detail and consider the existing modification of the N LOPMC analysis, based on application of the seBLM method, which allows one to reproducethe original NLO BLM expression from the considerations performed in [7] and alreadydiscussed in [17]. Note that the necessity of introducing modifications to the analysis of [7]starts to manifest itself from the level of taking into account the second order perturbativecorrections to the R-ratio evaluated analytically in [18] in the minimal subtractions (MS)scheme proposed in [19]. This result was also obtained numerically in [20] and confirmedanalytically in [21] by using the MS-scheme of [22]. At the level of the third order correctionsto D EM , analytically calculated in the MS scheme [23, 24] and confirmed in the independentwork [25], there appear additional differences between the results of the PMC and the seBLMmethods.We present several arguments in favour of theoretical and phenomenological applicationsof the form of the β -expanded expressions for the RG invariant (RGI) quantities proposedin [8] and applied in [9] . In this respect, let us mention the QCD generalization (in MS-scheme) of the Crewther relation [27] based on the { β } -expansion [9]. Using the results ofthese relations we obtain in a self-consistent way the N LO { β } -expansion for S Bjp ( Q ) inQCD with n ˜ g -numbers of gluinos, which can be checked by direct analytical calculations.The article is organized as follows. In Sec. 2, we define single-scale RG invariant quantitiesfor the e + e − -annihilation to hadrons and for the DIS inclusive processes, which will bestudied in this work. The existing theoretical relations between perturbative expressionsfor these characteristics are also summarized. In Sec. 3, the { β } -expansion of the RGI Note, that the { β } -expansion representation is related in part to the considered in [26] expansion of theperturbative terms in the RG invariant (RGI) Green functions through the powers of the first coefficientof the β -function β -function” QCD expression [9] for the MS-schemegeneralization of the Crewther relation [27] we provide the arguments that this expansion isunique. The details of constructing the { β } -expansions for the Adler D EM function and forthe S Bjp sum rule are described at the level of the O ( a s )-corrections, where a s = α s / (4 π ). InSec. 4 1, we consider the relations between certain terms of the { β } -expansion for D EM and S Bjp , which will be obtained from the Crewther relation of [27] and its QCD generalizationof [9], and present the concrete { β } -expanded contributions to the D EM function, R-ratioand the S Bjp sum rule.Using our definition of the { β } -expansion representation we correct the values of the PMCcoefficients and the scales in the related powers of the PMC perturbative expressions for theAdler function D EM and R e + e − –ratio, presented in [4–7], and discuss their correspondenceto the results obtained in [8, 9, 17]. The discussion of the results of the BLM, seBLM, PMCprocedures together with the numerical estimates of the corresponding perturbation theory(PT) coefficients and the couplings at new normalization scales are presented in Sec. 5. Itis demonstrated that in spite of its theoretical prominence following from the conformalsymmetry relations even the corrected PMC procedure does not improve the convergenceof perturbative series for the R -ratio and for the S Bjp sum rule. The methods of furtheroptimizations of these series, which are based on the { β } -expansion, are elaborated in Sec.6. The technical results are presented in the appendices.
2. DEFINITIONS OF THE BASIC QUANTITIES
Consider first the Adler function D EM ( Q ), which is expressed through the two-pointcorrelator of the electromagnetic vector currents j EM µ = P i q i ψ i γ µ ψ i taken at Euclidean − q = Q . Here q i stands for the electric charge of the quark field ψ i . D EM ( Q ) consists ofthe sum of its nonsinglet (NS) and singlet (S) parts D EM (cid:18) Q µ , a s ( µ ) (cid:19) = X i q i ! d R D NS (cid:18) Q µ , a s ( µ ) (cid:19) + X i q i ! d R D S (cid:18) Q µ , a s ( µ ) (cid:19) , (2.1a)where D NS (cid:0) Q /µ , a s ( µ ) (cid:1) = 1 + X l ≥ d NS l (cid:0) Q /µ (cid:1) a ls ( µ ) , (2.1b) D S (cid:0) Q /µ , a s ( µ ) (cid:1) = d abc d abc d R X l ≥ d S l (cid:0) Q /µ (cid:1) a ls ( µ ) . (2.1c)Here d R is the dimension of quark representation of SU ( N c ) color gauge group ( d R = N c )and d abc = 2 T r (cid:0) { λ a , λ b } λ c (cid:1) is the symmetric tensor. Both the NS and S contributions to theAdler-function are the RGI quantities calculable in the Euclidean domain. After applyingthe RG equation they can be represented as D NS ( Q /µ , a s ( µ )) µ = Q −→ D NS ( a s ( Q )) = 1 + X l ≥ d NS l a ls ( Q ) (2.2a) D S ( Q /µ , a s ( µ )) µ = Q −→ D S ( a s ( Q )) = d abc d abc d R X l ≥ d S l a ls ( Q ) . (2.2b)3ue to the cancellation of the logarithms ln k ( Q /µ ) with k ≥ l + 1 in the terms d ai ( Q /µ )(the superscript a defines the contributions to the NS and S parts of the D EM -function) thecoefficients of d al ≡ d al (1) are the numbers in the MS–like schemes.Let us emphasize that in this work we use the perturbative expansion parameter a s ( µ )normalized as a s ( µ ) ≡ α s ( µ ) / π . It obeys the RG equation with the consistently normal-ized SU ( N c )-group β -function µ dd µ a s ( µ ) = β ( a s ) = − a s X i ≥ β i a is , (2.3)where β = (11 / C A − (4 / T R n f ) while other coefficients β i are presented in Appendix A.The quantity related to the observable total cross-section of the e + e → hadrons process R e + e − ( s ) = σ ( e + e − → hadrons) /σ ( e + e − → µ + µ − ) is measured in the Minkowski region( s > D EM -function as R e + e − ( s ) ≡ R ( s, µ = s ) = 12 π i Z − s + iε − s − iε D EM ( σ/µ ; a s ( µ )) σ dσ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ = s = (2.4)= X i q i ! d R X m ≥ r NS m a ms ( s ) ! + X i q i ! d abc d abc d R X n ≥ r S n a ns ( s )The coefficients r am for the a part ( a = NS or S) of R e + e − are associated with the coefficients d al of the D a function by the triangular matrix T a of the relation r am = T aml d al , which will bediscussed in subsection 4 3 and Appendix C.The next observable RGI quantity, we will be interested in, is the Bjorken polarized sumrule S Bjp . It is defined by the integral over the difference of the spin-dependent structurefunctions of the polarized lepton-proton and lepton-neutron deep-inelastic scattering as S Bjp ( Q ) = Z [ g lp ( x, Q ) − g ln ( x, Q )] dx = g A C Bjp ( Q /µ , a s ( µ )) , (2.5)where g A is the nucleon axial charge as measured in the neutron β -decay, and C Bjp ( a s ) iscoefficient function calculable within perturbation theory and not damped by the inversepowers of Q , i.e., the leading twist term.The application of the operator-product expansion (OPE) method in the MS-like scheme[28] and the knowledge on the perturbative structure of the MS-scheme QCD generalizationof the quark-parton model Crewther relation [27] gained from articles in [29–35] indicatethe existence of the previously undiscussed singlet contribution to C Bjp ( a s ) [36]. Using theresults of this work we define the overall perturbative expression for C Bjp as C Bjp ( Q /µ , a s ( µ )) = C BjpNS ( Q /µ , a s ( µ )) + X i q i ! C BjpS ( Q /µ , a s ( µ )) (2.6)where the NS and S coefficient functions can be written down as C BjpNS ( Q /µ , a s ( µ )) = 1 + X l ≥ c NS l ( Q /µ ) a ls ( µ ) (2.7) C BjpS ( Q /µ , a s ( µ )) = d abc d abc d R X l ≥ c S l ( Q /µ ) a ls ( µ ) , (2.8)4nd have the following RG-improved form C BjpNS ( Q /µ , a s ( µ )) µ = Q −→ C BjpNS ( a s ( Q )) = 1 + X l ≥ c NS l a ls ( Q ) , (2.9a) C BjpS ( Q /µ , a s ( µ )) µ = Q −→ C BjpS ( a s ( Q )) = d abc d abc d R X l ≥ c S l a l +1 s ( Q ) , (2.9b) C Bjp ( a s ( Q )) = C BjpNS ( a s ( Q )) + C BjpS ( a s ( Q )) . (2.9c)The analytical expressions for the NLO and N LO corrections to Eq.(2.9a) in the MS-schemewere evaluated in [37] and [38], respectively, while the corresponding N LO O ( a s )-correctionwas calculated in [33] (its direct analytical form was also presented in [9]). The symbolicexpression for the coefficient c S of the O ( a s )- correction to the singlet contribution C BjpS ( a s )of the Bjorken polarized sum rule was fixed in [36] from the MS-scheme generalization ofthe Crewther relation, which will be presented below.Let us also consider the Gross-Llewellyn Smith (GLS) sum rule of the deep-inelasticneutrino-nucleon scattering. Its leading twist perturbative QCD expression can be definedas S GLS ( Q ) = 12 Z [ F νp ( x, Q ) + F νn ( x, Q )] dx = 3 C GLS ( Q /µ , a s ( µ )) (2.10)where F ( x, Q ) is the structure functions of the deep-inelastic neutrino-nucleon scatteringprocess. The coefficient function in the RHS of Eq.(2.10) also contains both NS and Scontributions, namely C GLS ( Q /µ , a s ( µ )) = C NSGLS ( Q /µ , a s ( µ )) + C SGLS ( Q /µ , a s ( µ )) . (2.11)As a consequence of the chiral invariance, which can be restored in the dimensional regular-ization [39] by means of additional finite renormalizations (for their consequent evaluationin high-loop orders see, e.g., [37, 38, 40, 41]), the NS contributions to the leading twist coef-ficient function of S GLS ( Q ) coincide with a similar NS perturbative contribution S Bjp ( Q ),namely C NSGLS ( Q /µ , a s ( µ )) ≡ C BjpNS ( Q /µ , a s ( µ )) = 1 + X l ≥ c NS l ( Q /µ ) a ls ( µ ) µ = Q −→ C BjpNS ( a s ( Q )) = 1 + X l ≥ c NS l a ls ( Q ) . (2.12)The fulfilment of this identity was explicitly demonstrated in the existing analytical NLOand N LO calculations in [37, 38] and used as the input in the process of determination ofthe analytical expression for the O ( a s ) corrections to S GLS ( Q ) [34].The second (singlet-type) contribution to the coefficient function of Eq.(2.11) has thefollowing form: C SGLS ( Q /µ , a s ( µ )) = n f d abc d abc d R X l ≥ ¯ c l ( Q /µ ) a ls ( µ ) µ = Q −→ C SGLS ( a s ( Q )) = n f d abc d abc d R X l ≥ ¯ c l a ls ( Q ) (2.13)5here ¯ c and ¯ c were evaluated analytically in [38] and [34], respectively.The application of the OPE approach to the three-point functions of axial-vector-vectorcurrents (see [30–32, 35, 36]) leads to the following MS-scheme QCD generalization of theCrewther relation (CR) between the introduced above different coefficient functions of theannihilation and deep-inelastic scattering processes: C Bjp ( a s ) D NS ( a s ) ≡ C GLS ( a s )[ D NS ( a s ) + n f D S ( a s )] (2.14a)= 1l + β ( a s ) a s · K ( a s ) , (2.14b)where 1l was derived in [27] using the conformal symmetry, β ( a s ) is the RG β -funtion, a s = a s ( Q ) and the polynomial K is K ( a s ) = a s K + a s K + a s K + O ( a s ) . (2.15)It contains the coefficients K and K , obtained in [29], while the analytical expression forthe coefficient K = K NS3 + K S3 is the sum of the NS- and S-terms, which are given in[33] and [34], respectively. Note that Eq.(2.14a) was first published in [35] without takinginto account singlet-type contributions to C Bjp . Their more careful analysis of [36] fixes the β -dependent analytical expression of the O ( a s ) contribution to C BjpS 2 .The result of [36] and the general Eq.(2.14a) is not yet confirmed by direct analyticalcalculations. In our further studies we will use the product of their NS parts and related tothis product results of the expansion in Eq.(2.14b), reformulated in [9].
3. GENERAL β -EXPANSION STRUCTURE OF OBSERVABLES1. Formulation of the approach To clarify the main ideas of the { β } -expansion representation proposed in [8] for theperturbative coefficients of the RGI quantities, let us consider the NS part of the Adlerfunction, D NS . Its expression can be rewritten as D NS = 1+ d NS1 · P n ≥ d l a ls , where d NS1 = 3C F is the overall normalization factor. Within the { β } -expansion approach the coefficients d n ,originally fixed in the MS-scheme, are expressed as d = d [0] = 1 , (3.1a) d = β d [1] + d [0] , (3.1b) d = β d [2] + β d [0 ,
1] + β d [1] + d [0] , (3.1c) d = β d [3] + β β d [1 ,
1] + β d [0 , ,
1] + β d [2] + β d [0 ,
1] + β d [1]+ d [0] , (3.1d)... d N = β N − d N [ N −
1] + · · · + d N [0] , (3.1e)where the first argument of the expansion elements d n [ n , n , . . . ] indicates its multiplica-tion to the n -th power of the first coefficient β of the RG β -function, namely to the β n In QED the validity of Eq.(2.14a) follows from the considerations of Ref. [42]. n determines the power of the second multiplication factor,namely β n , and so on. The elements d n [0] ≡ d n [0 , , . . . ,
0] define “refined” β i -independentcorrections with powers n i = 0 of all their β n i i multipliers. These elements coincide withexpressions for the coefficients d n in the imaginary situation of the nullified QCD β –functionin all orders of perturbation theory. This case corresponds to the effective restoration of theconformal symmetry limit of the bare SU ( N c ) model in the case when all normalizationsare not considered. This limit, extensively discussed in [17], will be considered here as atechnical trick. The origins of other elements in Eqs. (3.1) were considered in [8].The first elements d i [ i −
1] of the expansions of Eqs.(3.1b-3.1e) arise from the diagramswith a maximum number of the “fermion 1-loop bubble” insertions and applications of theNaive Non-Abelianization (NNA) approximation [43]). In the case of D NS -function theycan be obtained from the result in [29], which follow from renormalon-type calculations in[44, 45].It would be stressed that the terms β d [1], β d [0 , β d [1] in Eqs.(3.1) were not takeninto account in the variant of the { β } -expansion method used in [4–6]. The omitting ofthese terms leads to the results, which should be corrected by including these terms in theself-consistent variant of the PMC analysis.In high order of perturbation theory one should also consider a similar expression of thesinglet part D S = d S3 · P j ≥ ¯ d j a ( j ) s with the normalization factor d S3 = 11 / − ζ evaluatedfirst in the QED work [46], and the related normalizations of the defined coefficients inEq.(2.13), namely ¯ d j = d S j /d S3 . The { β } -expanded coefficients of this RGI-invariant quantityare expressed as ¯ d = ¯ d [0] = 1 (3.2a)¯ d = β ¯ d [1] + ¯ d [0] , (3.2b)...¯ d j +2 = β j − ¯ d j +2 [ j −
1] + · · · + ¯ d j +2 [0] . (3.2c)The same ordering in the β -function coefficients can be applied to the coefficients c n for the NS coefficient function of the deep-inelastic sum rules C NS of Eq.(2.12) and to thesinglet contribution C S to the GLS sum rule (see Eq. (2.13). Moreover, it is possible to showthat the elements of the corresponding { β } -expansions d n [ n , n , . . . ] and c n [ n , n , . . . ] areclosely related [9]. We will return to a more detailed discussion of this property a bit later.The above { β } –expansion can be interpreted as a “matrix” representation for theRGI quantities: For the quantity D NS expanded up to an order of N , D NS = P Nn =1 a ns P i ≥ D NS ni B ( i ) which is related to the traditional “vector” representation, D NS = P Nn =1 a ns d n with d n = P i D NS ni B ( i ) . Here B ( i ) are the elements that express the struc-ture of { β } -expanded perturbative coefficients and are convolved with the matrix elements D NS ni = d n [ . . . ]. In the case of consideration of the “refined” β i -independent corrections d n [0] ≡ D n and B (0) = 1. The similar matrix representation can be written down for thesinglet part D S with the { β } -expanded coefficient defined in Eqs.(3.2a-3.2c).Note that the matrix representation contains new dynamical information about the RGIinvariant quantities, which is not contained in the vector one. Thus, Eqs. (3.1b),(3.2b) can beconsidered as the initial points to apply the standard BLM procedure. The generalization ofthe BLM procedure to higher orders can be constructed using the { β } –expansions of higherorder coefficients of Eqs. (3.1) [8]. However, starting with the N LO the explicit solution ofthis problem is nontrivial. 7 . Explicit determination of the structures of the { β } -expanded series for D NS Let us start the discussion of application of the { β } -expansion procedure in the NLO.Imagine that we deal with the perturbative quenched QCD (pqQCD) approximation forthe D NS -function in the NLO. It is described by the contributions of the three-loop photonvacuum polarization diagrams with closed external loop, formed by quark-antiquark pairand connected by internal gluon propagators, which do not contain any internal quark-loopinsertions. In this theoretical approximation the coefficient d takes the following form: d → d pqQCD = − C F (cid:18) − ζ (cid:19) C A β = 113 C A . (3.3)In this case, it is unclear how to perform the standard BLM scale-fixing prescription in theNLO approximation. Indeed, it is not clear what is the expression for the d [1] coefficientof the β -term of Eq.(3.1b) in the expression for d pqQCD . To obtain explicitly the elementsof the expansion (3.1b) and extract the β -term in (3.3), one should take into account thequark-antiquark one-loop insertion in internal gluon lines of the three-loop approximationfor the hadronic vacuum polarization function. This is equivalent to taking into account inthe pqQCD model of the interacting with gluons of internal quark loops with n f number ofactive quarks. The corresponding parameter n f can be considered as a mark of the chargerenormalization by the quark-antiquark pair. It enters into both d and β expressions andallows one to extract unambiguously the expression for d proportional to the β -term in theMS-scheme. Indeed, fixing T R = T F = we obtain d = − C F (cid:18) ·
11 + 22 − ζ (cid:19) C A − (cid:18) − ζ (cid:19) n f with β = 113 C A − n f . (3.4)To get the appropriate expression of the coefficient d one should take into account the1-loop renormalization of charge. As the result, we immediately obtain from Eq. (3.4) thefollowing expression for the coefficient of Eq.(3.1b): d = C A − C F (cid:18) − ζ (cid:19) (cid:18)
113 C A − n f (cid:19) (3.5a)where d [0] = C A − C F , d [1] = 112 − ζ . (3.5b)This decomposition corresponds to the case of the standard BLM consideration in the MS-scheme [2]. Note that for n f = 0 this decomposition remains valid for the case of pqQCD(QCD at n f = 0) and leads to Eq.(3.5b).Any additional modifications of QCD, say, by means of introducing into considerations n ˜ g multiplets of strong interacting gluino (the element of the MSSM model), will change in theNLO expression for the considered RGI invariant quantity the content of the β -coefficientin the expression for d , calculated in the MS-scheme, but not the “refined” element and thecoefficients at β of Eq.(3.3). Using the result β for the β -function with the n ˜ g multipletof strong interacting gluino (see Eq. (A.1a)) and the D NS -function in the same model (presented in Eq.(A.5b) of the Appendix A), the same result (3.5b) for decomposition can8nambiguously be obtained using the additional marks in Eq.(3.5a), namely the number ofstrong-interacting gluino n ˜ g . Indeed, combining the result d = C A − C F (cid:18) − ζ (cid:19) (cid:18)
113 C A − n f (cid:19) − (11 − ζ ) n ˜ g C A , (3.6a)with β ( n f , n ˜ g ) = 113 C A −
23 ( n f + n ˜ g C A ) (3.6b)we get the expressions for d [0] and d [1], which are identical to the ones presented inEq.(3.5b). Note that these results can be obtained from Eq.(3.6a) and Eq.(3.6b) with gluinodegrees of freedom only ( n ˜ g = 0 , n f = 0), or only with the quark ones ( n f = 0 , n ˜ g = 0), orwith taking into account both of them. The reason of this unambiguity is that the interactionof any new particle accumulated here in the charge renormalization is determined by theuniversal gauge group SU ( N c ).All these possibilities give us a simple tool to restore the β -term in the NLO followingthe BLM precription [2]. Thus, in the NLO we may switch off the gluino degrees of freedom.However, to get the { β } expansion of the N LO term in the form of Eq. (3.1c) we cannotuse the quark degrees of freedom only. Indeed, in this case, we face a problem similar tothat which arises in the process of { β } decomposition of the pqQCD expression for d pqQCD in Eq. (3.3) discussed above.The { β } –expanded form for the d -term was obtained in Ref. [8] by means of a care-ful consideration of the analytical O ( a s ) MS-scheme expression for the Adler function D NS ( a s , n f , n ˜ g ) with the n ˜ g QCD interacting MSSM gluino multiplets obtained in [25] andpresented in Eq. (A.5) together with the corresponding two-loop β -function, β ( n f , n ˜ g ),see Eqs. (A.1a, A.1b).Let us consider this procedure in more detail. The element d [2], which is proportionalto the maximum power β in (3.1c), can be fixed in a straightforward way, using the resultsin [29]. Then one should separate the contributions of β d [0 ,
1] and of β d [1] to the d -term. They both are linear in the number of quark flavours n f , therefore, they could notbe disentangled directly. Their separation is possible if one takes into account additionaldegrees of freedom, e.g. , the gluino contributions mentioned above for both the quantities(the additional mark appears), namely for the D NS -function from Eqs.(A.8c) and for the firsttwo coefficients of the β -function from Eqs.(A.1a,A.1b). In this way, using two equationsone can get the explicit form for the functions n f = n f ( β , β ) and n ˜ g = n ˜ g ( β , β ). Finally,substituting these functions in D = D ( a s , n f ( β , β ) , n ˜ g ( β , β )) its { β } -expanded expressionwas ontained in [8], D NS ( a s , n f , n ˜ g ) = 1 + a s (3 C F ) + a s (3 C F ) · (cid:26) C A − C F (cid:18) − ζ (cid:19) β ( n f , n ˜ g ) (cid:27) + a s (3 C F ) · (cid:26)(cid:18) − ζ (cid:19) β ( n f , n ˜ g ) + (cid:18) − ζ (cid:19) β ( n f , n ˜ g )+ (cid:20) C A (cid:18)
34 + 803 ζ − ζ (cid:19) − C F (18 + 52 ζ − ζ ) (cid:21) β ( n f , n ˜ g )+ (cid:18) − ζ (cid:19) C + 713 C A C F −
232 C (cid:27) , (3.7a) The evaluated in [25] N LO analytical result for the gluino contribution was confirmed in [47]. β ( n f , n ˜ g ) = 343 C A − C A (cid:18) T R n f + n ˜ g C A (cid:19) − (cid:18) T R n f C F + n ˜ g C A C A (cid:19) . (3.7b)Note that in order to write down the O ( a s ) coefficient of D NS , analytically evaluated inthe MS-scheme in [33] for the case of SU ( N c ) in a similar { β } -expanded form of Eq.(3.1d),it is necessary to perform additional calculations, which generalize this result to the caseof SU ( N c ) with n ˜ g multiplets of gluino. Then, one should combine this possible (but notyet existing) generalization with already available analytical expression for the β ( n f , n ˜ g )-coefficient of the β -function in this model, analytically obtained in the MS-scheme in [48].
3. Does { β } -expansion have any ambiguities ? It is instructive to discuss here an attempt [5, 7] to obtain the elements d n [ . . . ] in adifferent way. This is based on the expression for D , rewritten in [49] for the usability ofcurrent 5-loop computation in the form D EM ( a s ) = 12 π (cid:18) γ EMph ( a s ) − β ( a s ) dda s Π EM ( a s ) (cid:19) . (3.8)Here Π EM ( a s ) = Π EM ( L, a s ) ≡ d R / (4 π ) P i ≥ Π i a is is the polarization function of electro-magnetic currents at L ≡ ln( Q /µ ) = 0, and γ EMph ≡ / (4 π ) P j ≥ γ j a js is the anomalousdimension of the photon field. In our notation Eq. (3.8) leads to the expansion for D NS , D NS ( a s ) = 1 + 3 C F a s + (12 γ + 3 β Π ) a s + (48 γ + 3 β Π + 24 β Π ) a s + . . . , (3.9)where the ingredients of the expansion, γ i , Π j , were calculated in [49] up to i = 4 , j = 3,and we take corresponding NS projection in the RHS of Eq.(3.8). The renormalization ofthe charge certainly contributes to the 3-loop anomalous dimension γ ; therefore, it containsa β -term also (one can make sure from the inspection of the explicit formula for γ inEq. (3.12) in [49] and even in Eq.(10) in [20]). Taking into account the explicit form of γ and Π in (3.9) one can recalculate the well-known decomposition for D NS in order O ( a s ) D NS ( a s ) = 1 + 3 C F · a s + 3 C F · ( β d [1] + d [0]) a s + O ( a s ) (3.10)in full accordance with the result in [2] and Eq. (3.1b) (for the related discussions see Ref.[17] as well).Unfortunately the authors of [7] claim, basing on a formal correspondence, that thecoefficient of β is only the term Π /C F in Eq. (3.9) (with the above notation at d normalizedby unity), while the “conformal term” is 4 γ /C F (see Eq. (48a-b) in [2]), which in reality isnot true. The comparison of these terms d [1] = 112 − ζ ≈ . ⇔ Π /C F = 5512 − ζ ≈ − . d [0] = d [0] = C A − C F ⇔ γ /C F = 1112 β − C F /C F with d [1]]. The considerationsof the lead to a shift of the BLM scale Q in the opposite direction, Q ≥ Q , incomparison with the standard value Q = exp ( − d [1]) Q ≈ Q / γ is not “conformal”and depends on β . 10 . PARTIAL β -EXPANSION ELEMENTS FOR D, C , AND R Here we extend our knowledge about the β -expansion elements on NS part of the Bjorken C Bjp basing on CR Eq.(2.14b) for D NS and C BjpNS .
1. What constraints Crewther relation gives
In the case of the β -function has identically zero coefficients β i = 0 for i ≥ D NS0 · C Bjp0 = 1l , (4.1)where the expansions for the functions D NS0 and C Bjp0 , analogous to the ones of Eq. (3.1b-3.1e), contain the coefficients of genuine content only, namely, d n ( c n ) ≡ d n [0] ( c n [0]). Equa-tion (4.1) provides an evident relation between the genuine elements in any loops, namely, c NS n [0] + d NS n [0] + n − X l =1 d NS l [0] c NS n − l [0] = 0 , (4.2)where d NS n [0] = d NS1 · d n [0] and c NS n [0] = c NS1 · c n [0] in virtue of the normalization condition.From Eq. (4.2) at n = 1 immediately follows that c NS1 = − d NS1 . The relation (4.2) can beused to obtain the unknown genuine parts of the 4-loop term c NS3 [0], through the 4-loopresults already known from the analysis in [8]: c NS3 [0] = − d NS3 [0] + 2 d NS1 d NS2 [0] − ( d NS1 ) , (4.3a)or, in the other normalized terms, c [0] = d [0] − d NS1 d [0] + ( d NS1 ) , (4.3b)It is useful to relate the unknown elements c NS4 [0] , d
NS4 [0] in a 5-loop calculation with theknown elements of the 4-loop results, viz, c NS4 [0] + d NS4 [0] = 2 d NS1 d NS3 [0] − d NS1 ) d NS2 [0] + ( d NS2 [0]) + ( d NS1 ) . (4.4)Let us consider now the generalized CR in Eq.(2.14b), which includes the terms propor-tional to the conformal anomaly, β ( a s ) /a s , appearing due to violation of the the conformalsymmetry in the renormalized SU(N c ) interaction (in the MS-scheme). As was shown in [9],this relation can be rewritten in the following multiple power representation: D NS · C BjpNS = 1l + β ( a s ) a s · K ( a s ) = 1l + β ( a s ) a s · X n ≥ (cid:18) β ( a s ) a s (cid:19) n − P n ( a s ) , (4.5)where P n ( a s ) are the polynomials in a s that can be expressed only in terms of the elements d k [ . . . ] , c k [ . . . ]. In this sense P n do not depend on the β -function, all the charge renormaliza-tions being accumulated by ( β ( a s ) /a s ) n . Below we present the first two polynomials (factor11ut − c NS1 = d NS1 = 3 C F ) P ( a s ) = − a s d NS1 (cid:26) d [1] − c [1] + a s (cid:2) d [1] − c [1] − d NS1 ( d [1] + c [1]) (cid:3) (4.6a)+ a s h d [1] − c [1] − d NS1 ( d [1] + c [1] + d [0] c [1] + d [1] c [0]) i(cid:27) , (4.6b) P ( a s ) = a s d NS1 (cid:26) d [2] − c [2] + a s h d [2] − c [2] + d NS1 ( c [2] + d [2]) i(cid:27) , (4.6c)which were obtained and verified in N LO in [9, 10] in another normalization. The con-struction of the β -term in the RHS of (4.5) also creates constraints for combinations of the β -expansion elements. A few chains of these constraints were obtained in [9]. Further weshall use the relation d [1] − c [1] = d [0 , − c [0 ,
1] = . . . = d n [0 , , . . . , | {z } n − ] − c n [0 , , . . . , | {z } n − ] = (cid:18) − ζ (cid:19) , (4.7)that corresponds to Eq.(30) in [9].If the terms c [1] , d [1] and c [2] , d [2] are missed in the { β } -expansion of D NS and C BjpNS ,as in the variant of the expansion in [4–7], the structure of the generalized CR in (4.6) iscorrupted. That structure certainly contradicts the explicit results of analytical calculationsof D NS ( a s ) and C BjpNS ( a s ), preformed in the N LO in [23–25] and in N LO in [33].
2. Nonsinglet parts of D and C Bjp
Following the approach discussed in Sec. 3 and taking into account a certain definitionof the β -function coefficients in Eq. (2.3) we can obtain the β -expansion for D – and C –functions. For the Adler function D NS it reads d NS1 = 3C F ; d = 1; (4.8a) d [1] = 112 − ζ ; d [0] = C A − C F d [2] = 3029 − ζ ≈ . d [0 ,
1] = 10112 − ζ ≈ − . d [1] = C A (cid:18)
34 + 803 ζ − ζ (cid:19) − C F (18 + 52 ζ − ζ ) ≈ . d [0] = (cid:18) − ζ (cid:19) C + 713 C A C F −
232 C ≈ − . , (4.8e)which differs from the ones presented in Ref. [9] (see its “natural form” in the AppendixB), by the normalization factor only. It looks more convenient for a certain BLM task (thepresentation corresponds to one in [8]) due to setting of the first PT coefficient, d ( c ), equalto 1. Let us emphasize that gluinos are used here as a pure technical device to reconstructthe β -function expansion of the perturbative coefficients.In this connection, we mention the relation d [0 ,
1] = d n [0 , . . . ,
1] = d [1] proposed in [5]and based on a “special degeneracy of the coefficients” suggested there (see Eq. (6) in [5])12n an analogy with the perturbative series rearrangement, d i → d ′ i under the change of thecoupling renormalization scale, a ( µ ) → a ′ ( µ ′ ) (see below the discussions around Eq. (5.3)).This rearrangement has an outside reason with respect to d i and “does not know” about the intrinsic structure of the initial coefficient d i under consideration. This relation is artificialand by this reason it is not supported by the direct calculations. The explicit result ofthis rearrangement is presented in Eq. (5.3), it is the initial step for any BLM optimizationprocedure that will be discussed in Sec.5 in detail.Let us compare now Eqs. (4.8) with the results presented in [7] and based on the inter-pretation of the term (cid:0) γ + 3 β Π + 24 β Π (cid:1) a s in the presentation of (3.9). The first andthe third terms of the sum form the term proportional to β γ + 24 β Π −→ β (cid:18) − ζ = d [2] (cid:19) (4.9)that can be unambiguously obtained by extracting the n f terms in γ and the β -term in Π ,see the corresponding explicit expressions in [49]. The second term there, β Π , certainlycontributes to the value of the element d [0 , β ,in both the γ and β Π terms that also contribute to d [0 , β .The final explicit expressions given in [49] are not sufficient for this separation, as it wasalready discussed in Subsec.3 2.Let us consider the β -expansion of the Bjorken coefficient function C BjpNS of the DIS sumrules. Based on CR (4.2) for n = 2 and n = 3 and the already fixed d [0] and d [0]-terms we get expression (4.3b) for the c [0] and c [0] elements of C BjpNS , namely, c [0] = d [0] − d NS1 d [0] + ( d NS1 ) , see the explicit expression in (4.10e). The knowledge of c [0]allowed us to fix all other elements c [ . . . ] of the PT coefficient c without involving additionaldegrees of freedom [9]. It is instructive to consider this in detail. Indeed, the terms c [0] aswell as the coefficient c [2] of the β (maximum power of n f ) can be found independently.Therefore, the Casimir structure of the rest of c , c − c [0] − β c [2], contains basis elements(we factor out c NS1 = − C F ): c − c [0] − β c [2] : (cid:26) C , C , C A C F , T R n f C F , T R n f C A β , β This Casimir expansion of the rest should be equated to the β -expansion of the one (seedecomposition (3.1)), c [0 , · β + ( x · C F + y · C A ) · β that contains 3 unknown coefficients c [0 , , x, y .The C -terms in the explicit result for c [38] (see App.A, Eq.(A.7)) and in the expressionfor c [0] in Eq.(4.10e) coincide to one another; therefore, the term C is canceled in therest. This confirms the fact that its β -expansion does not contain C . So we have constraints (not ) for the coefficients c [0 , , x, y . This overdetermined system is a systemof simultaneous equations; the fact provides us with an independent confirmation of this β -expansion. The explicit form of the elements were first obtained in [9]; below we presentthem at the same normalization as Eq. (4.8) (cf. (4.8e) with (4.10e)): c NS1 = − F ; c = 1; (4.10a) c [1] = 2; c [0] = (cid:18)
13 C A −
72 C F (cid:19) = −
113 = − . [2] = 11518 = 6 . c [0 ,
1] = (cid:18) − ζ (cid:19) ≈ . c [1] = − (cid:18) − ζ (cid:19) C F − (cid:18) − ζ + 403 ζ (cid:19) C A ≈ . c [0] = (cid:18) − ζ (cid:19) C + 653 C F C A + C ≈ − . . (4.10e)The same results can be obtained if one fixes first the element c [0 ,
1] = d [0 , − d [1] + c [1]from relation (4.7), the latter originates from another source – the symmetry breaking termproportional to β ( a s ) in the generalized CR. Therefore, the results (4.10) are in mutualagreement with both the terms in the RHS of CR and can be obtained independently fromeach of them.These elements of decomposition in (4.10) allows one to make a new prediction for thelight gluino contribution to C BjpNS . Indeed, for the considered here RGI quantities the effectsof charge renormalization appear in two ways: the elements c [ ... ] – the coefficients of the β -function products (named B j in 3 1) – are formed following gauge interaction; the effect ofvarious degrees of freedom, say, gluino, which reveals itself only in intrinsic loops, changes thecontent of the β -coefficients β i with the corresponding mark, say, n ˜ g . Therefore, to obtain C Bjp → C Bjp ( a s , n f , n ˜ g ) with the light MSSM gluino one should replace the β -coefficients β i → β i ( n f , n ˜ g ) and compose them with the elements from Eq.(4.10), C BjpNS ( a s , n f , n ˜ g ) = 1 + a s ( − C F ) (4.11a)+ a s ( − C F ) · (cid:26)
13 C A −
72 C F + 2 β ( n f , n ˜ g ) (cid:27) (4.11b)+ a s ( − C F ) · (cid:26) β ( n f , n ˜ g ) + (cid:18) − ζ (cid:19) β ( n f , n ˜ g ) − (cid:20)(cid:18) − ζ + 403 ζ (cid:19) C A + (cid:18) − ζ (cid:19) C F (cid:21) β ( n f , n ˜ g )+ (cid:18) − ζ (cid:19) C + 653 C F C A + C (cid:27) . (4.11c)This logic can be reverted: the values of c [0] , c [0 ,
1] and then the CR can be checked fromthe direct calculation of C BjpNS ( a s , n f , n ˜ g ) with the MSSM massless gluino.
3. Singlet parts and the R -ratio Here we present the singlet part of the Adler function, d , that can be obtained based onthe result for c of C BjpS and CR [34], d S = β ( n f ) · d S [1] + d S [0] , (4.12) d S [0] = (cid:18) − ζ − ζ + 2051536 (cid:19) C A + (cid:18) − ζ + 58 ζ − (cid:19) C F , (4.13) d S [1] = − ζ − ζ + 516 ζ + 149576 , (4.14)14 S = c S [0] + β ( n f ) · c S [1] , (4.15) c S [0] = (cid:18) ζ + 532 ζ − (cid:19) C A + (cid:18) ζ − ζ + 37128 (cid:19) C F , (4.16) c S [1] = − ζ (3) + 18 ζ (3) − ζ (5) (4.17)The integral transform D → R e + e − , R e + e − ( s ) ≡ R ( s, µ = s ) = 12 π i Z − s + iε − s − iε D EM ( σ/µ ; a s ( µ )) σ dσ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ = s = (4.18)= X i q i ! d R X m ≥ r NS m a ms ( s ) ! + X i q i ! d abc d abc d R X n ≥ r S n a ns ( s ) , can be realized as a linear relation by means of the matrix T , r j = T ji d i , or for the vectorrepresentation R = T D = P a js T ji d i . The triangular matrix T of the relation can beobtained at any fixed order of perturbative theory [50]. The elements of this matrix belowthe units on the diagonal contain so-called kinematic “ π -terms” multiplied by the β -functioncoefficients , see an example of T ji in Appendix C. Taking into account that the β -structureof the normalized coefficients r i = r NS i /r NS1 is like that for the coefficients d i , Eqs.(3.1), onecan rewrite the results from the matrix in Table 1, r = d ; r NS1 = d NS1 ; r = 1 ; r = d ; (4.19a) r [2] = d [2] − π r S3 [2] = d S3 [2]; (4.19b) r [3] = d [3] − π d [1]; r [2] = d [2] − π d [0]; r [1 ,
1] = d [1 , − π ; (4.19c) r [4] = d [4]+ π − π d [2]; r [0 ,
2] = d [0 , − π r [2 ,
1] = d [2 , − π (cid:18) d [1] + d [0 , (cid:19) ; r [1 , ,
1] = d [1 , , − π ; r [1 ,
1] = d [1 , − π d [0]; r [3] = d [3] − π d [1]; r [2] = d [2] − π d [0] , (4.19d)while the other elements in r i coincide with ones in d i ( i ≤ T Snl whichrelates the coefficients r S n and d S l , can be constructed as well. However, in this work we willnot consider the π -dependent effects of analytical continuation, which in the singlet caseappear first at the O ( a s )-level. Further, we shall use Eqs.(4.19) to construct PT optimizedseries for R . These terms can be obtained for any order of perturbative theory (constrained mainly by the value ofRAM) with “Mathematica” routine constructed by V. L. Khandramai and S. V. Mikhailov. . BLM AND PMC PROCEDURES AND THE RESULTS1. General basis The re-expansion of the running coupling ¯ a ( t ) = a (∆ , a ′ ) and its powers in terms of t − t ′ = ∆ = ln ( µ /µ ′ ) and new coupling a ′ reads,¯ a ( t ) = a (∆ , a ′ ) = a ′ − β ( a ′ ) ∆1! + β ( a ′ ) ∂ a ′ β ( a ′ ) ∆
2! + β ( a ′ ) ∂ a ′ ( β ( a ′ ) ∂ a ′ β ( a ′ )) ∆
3! + . . . = exp ( − ∆ β (¯ a ) ∂ ¯ a ) ¯ a | ¯ a = a ′ , (5.1)which is the way to write the corresponding RG solution for a ( t ) through the operatorexp ( − ∆ β ( a ) ∂ a ) [ . . . ] | a = a ′ (see [8] and refs therein). The shift of the logarithmic scale ∆ inits turn can be expanded in perturbative series in powers of a ′ β t ′ ≡ t − ∆ , ∆ ≡ ∆( a ′ ) = ∆ + a ′ β ∆ + ( a ′ β ) ∆ + . . . , (5.2)where the argument of the new coupling a ′ depends on t ′ = t − ∆. It is sufficient to take thisrenormalization scale for the a ′ argument, which corresponds to the solution on the previousstep, rather than to solve the exact equation a ( t − ∆( a ′ )) = a ′ . Re-expansion a in termsof a ′ leads to rearrangement of the series of perturbative expansion for the RGI quantity D a ( C Bjp ), a i d i → a ′ i d ′ i , where the r.h.s. are expressed in a rather long but evident formulae.n the square brackets below, we write them explicitly: a · d → a ′ · [ d ′ = 1]; a · d → a ′ · (cid:2) d ′ = β d [1] + d [0] − β ∆ (cid:3) ; (5.3a) a · d → a ′ · h d ′ = β (cid:0) d [2] − d [1] ∆ + ∆ (cid:1) + β ( d [0 , − ∆ ) + (5.3b) β ( d [1] − d [0] ∆ ) + d [0] − β ∆ i ; (5.3c) a · d → a ′ · h d ′ = β (cid:0) d [3] − d [2] ∆ + 3 d [1] ∆ − ∆ − ∆ − d [1]) ∆ (cid:1) + β β (cid:18) d [1 , − (3 d [0 ,
1] + 2 d [1]) ∆ + 52 ∆ − ∆ (cid:19) + β ( d [0 , , − ∆ ) + (5.3d) β (cid:0) d [2] − d [1] ∆ + 3 d [0] ∆ − d [0] ∆ (cid:1) + β ( d [0 , − d [0] ∆ ) + (5.3e) β ( d [1] − d [0] ∆ ) + d [0] − β ∆ i ; (5.3f) . . . . . .a n · d n → a ′ n · h d ′ n = β n − d n [ n −
1] + . . . i . (5.3g)The standard BLM fixes the scale ∆ by the requirement ∆ = d [1], accumulating 1-loop renormalization of charge just in this new scale [2], at the same time the coefficient d → d [0] – its “conformal part”. Numerically, ∆ = d [1] = 112 − ζ = 0 . . . . ≈ ln(2) = 0 . . . . , (5.4)16herefore, Q BLM = exp( − ∆ ) Q ≈ Q / { ∆ i } . The system of Eqs. (5.3a-5.3g) for d ′ i is the basisto construct different BLM generalizations. It is instructive to consider these coefficients { d ′ i } after the first BLM step; taking ∆ = d [1] one obtains d ′ = d [0]; (5.5a) d ′ = β ( d [2] − d [1] ) + β ( d [0 , − d [1]) + β ( d [1] − d [0] d [1]) + d [0] − β ∆ ; (5.5b) d ′ = β (cid:0) d [3] − d [2] d [1] + 2 d [1] (cid:1) + β ( d [0 , , − d [1]) β β (cid:0) d [1 , − d [0 , d [1] + d [1] / − ∆ (cid:1) + (5.5c) β (cid:0) d [2] − d [1] d [1] + 3 d [0] d [1] − d [0] ∆ (cid:1) + (5.5d) β ( d [0 , − d [0] d [1]) + (5.5e) β ( d [1] − d [0] d [1]) + d [0] − β ∆ ; (5.5f) . . . . . .d ′ n = β n − d n [ n −
1] + . . . (5.5g)The detailed analysis of the d ′ i structure was made in [8] in Sec.5. Here we mention acommon property of this transform – to obtain the rearrangement of the coefficient at anorder n + 1, d n +1 → d ′ n +1 , one should know its β -structure up to the previous order n . Forthe partial case of relation d n +1 [ n ] = ( d [1]) n the β n -terms are canceled (underlined terms)in all the orders even due to the first BLM step. Correspondingly, the special conditions d i [0 , . . . ,
1] = d [1] will remove the next terms with the leading coefficient β i − in everyorder, see double underlined terms in Eqs. (5.5b,5.5c). The latter conditions were proposedin [5] (see the discussion in Sec.4 2 after Eq. (4.8)), though both of the above hypotheses arefar from the results of the direct calculations at O ( a s ) in (4.8), really d [2] − d [1] ≈ . − . d [0 , − d [1] ≈ . − . . (5.6)Even more, in QCD the elements d n +1 [ n ] grow as n ! due to renormalon contributions [29]and the role of these terms becomes more and more important. To construct the next stepsof the PT-optimization with ∆ , ∆ , . . . , one should get more detailed knowledge or providea hypothesis about the different contributions to d ′ n .
2. seBLM and PMC procedures
One of the hypotheses mentioned above is based on the empirical relation between theQCD β –function coefficients β i , β i ∼ β i +10 . This can be easily verified for perturbativequenced QCD ( n f = 0) numerically and this works in the range of n f = 0 ÷ β -coefficients; compare the expressions in Eqs. (A.1,A.2), β i ∼ β i +10 , c i = β i /β i +10 = O (1) . (5.7)This relation allows one to set a hierarchy of contributions in order of the “large value of β ” ( β = 11(9) at n f = 0(3)) [8]. Of course, relation (5.7) should be broken at some17arge enough order of expansion i in virtue of expected Lipatov like asymptotics for the β -function β i ∼ ( i !) β i +10 . Therefore, this hierarchy has a restricted field of application thatdescribes the term “practical approach” in the title of [8].For this hierarchy the most important terms are of an order of ( β a s ) n /β in order n – underlined below in Eq. (5.8). For illustration we shall use the R NS ( s )-ratio taking intoaccount the result (5.5) for D and relations in Eq. (4.19) r ′ = d [0] , (5.8a) r ′ = β ( d [2] − d [1] − π /
3) + β ( d [0 , − d [1]) (5.8b)+ β ( d [1] − d [0] d [1]) − β ∆ + d [0]; (5.8c) r ′ = β (cid:0) d [3] − d [2] d [1] + 2 d [1] − π d [1] (cid:1) + β ( d [0 , , − d [1]) β β (cid:16) d [1 , − d [0 , d [1] + d [1] / − / π − ∆ (cid:17) + (5.8d) β (cid:16) d [2] − d [1] d [1] + 3 d [0] d [1] − π d [0] − d [0] ∆ (cid:17) + (5.8e) β (cid:16) d [0 , − d [0] d [1] (cid:17) + (5.8f) β (cid:18) d [1] − d [0] d [1] (cid:19) − β ∆ + d [0]; (5.8g) r ′ n = β n − d n [ n −
1] + . . . + . . . (5.8h)The less important terms are suppressed by β − in this order; ( β a s ) n /β , they are double-underlined in (5.8c,5.8e,5.8f), and so on. Following the hierarchy one fixes the values of ∆ , ∆ , . . . consequently nullifying at first the most important (1-underlined) β -terms inevery order. After that the procedure repeated with the less important terms (double un-derlined) in all orders, etc. This procedure was called sequential BLM (seBLM) and its resultwas presented in detail in Sec. 6 of [8] (see Eqs. (6.7,6.8) there). The discussed hierarchycan also be used for generalization of the NNA approximation, see Appendix C in [50].The above invented hierarchy ignores a possible difference of values of the elements d n [ . . . ]tacitly suggesting that they are of the same order of magnitude. Of course, one can abandonthe suggestion of the hierarchy, and to remove all the β -terms in one mold consequentlyorder by order. This approach leads to other values of ∆ i = ¯∆ i : ¯∆ = d [1]; (5.9a) ¯∆ = 1 β h β (cid:0) d [2] − d [1] − π / (cid:1) + β ( d [0 , − d [1]) + β ( d [1] − d [1] d [0]) i (5.9b) ¯∆ = 1 β (cid:2) β (cid:0) d [3] − d [1] d [2] + 2( d [1]) − π d [1] (cid:1) + β ( d [0 , , − d [1])+ β β (cid:18) d [1 , − d [0 , d [1] + 32 ( d [1]) − d [2] − π / (cid:19) + β /β ( d [1] − d [0 , β (cid:0) . . . (cid:1) + . . . (cid:3) , (5.9c)which differ by the underlined “suppressed in the 1 /β ” terms from the previous ones in[8]. The complete form for ∆ looks cumbersome and it is outlined in Appendix D. The18rocedure like this was called PMC later on [3], though for both the cases, seBLM andcorrected PMC, the final PT series has the same “conformal terms” d n [0] as the coefficientsof new expansion. The new normalization scale s ′ follows from Eq. (5.2), taking into accountcertain expressions for ∆ i in (5.9), R e + e − ( s ) = X i q i ! · d A R NS + X i q i ! · d A R S R NS ( s ) = 1 + 3 C F (cid:8) a ( s ′ ) + d [0] · a ( s ′ ) + d [0] · a ( s ′ ) + d [0] · a ( s ′ ) + . . . (cid:9) (5.10a)ln( s/s ′ ) = ¯∆ + a ′ β ¯∆ + ( a ′ β ) ¯∆ + . . . . (5.10b)The formulae Eqs.(5.9) and Eqs.(5.10) are the main results of these subsections.
3. Numerical estimates, discussion of PMC/seBLM results
Here we apply the results of the procedure accumulated in Eqs. (5.8,5.9,5.10) for thenumerical estimates of the expansion coefficients for a few processes starting with the non-singlet part R NS of the R e + e − ( s )-ratio. The corresponding singlet part R S can be optimizedindependently; moreover it is not very important numerically. For the sake of illustration,we put the value n f = 3 for all estimates below. At the very beginning we have the followingnumerical structure of r i , r = β · .
69 + 13 ≈ .
56; (5.11a) r = − β · . − β · . β · . − . ≈ − . − . − . . − . r ≈ − .
29 (5.11c)At the first BLM setting ¯∆ ≈ . a s ( s ) → a ′ s = a s ( se − . ≈ s/
2) we obtain for thecoefficients r ′ , r ′ — Eqs. (5.12a,5.12b) — the explicit result of the BLM procedure. Thevalue of the second coefficient r ′ diminishes by an order of magnitude, while r ′ becomesmoderately larger, compare (5.11b) with (5.12b,5.13a), r ′ = 13 ; (5.12a) r ′ = − β · . − β · .
892 + β · . − . ≈ − .
7; (5.12b) − . − . . − . r ′ ≈ − . . (5.12c)At the second step (PMC), we obtain ¯∆ ≈ .
98 following Eq.(5.9b) and Eq.(5.2), r ′′ ≈ − . , ¯∆ ≈ . , (5.13a) r ′′ ≈ − . , (5.13b) a ′ s → a ′′ s = a s (cid:16) s · e − . − . β a ′ s ( s ) (cid:17) , (5.13c)19hile r ′′ = d [0] following the main aim of PMC, see Eq. (5.10a). Due to the strong suppres-sion of the normalization scale by a factor of exp [ − . − . β a ′ s ( s )] the applicability ofPT is shifted to the region of very large s ; simultaneously the coefficient r ′′ increases 3 times(cf. (5.11b)). So this procedure makes the convergence of PT worse.Within the same framework we obtain for the coefficient of the Bjorken function C BjpNS c = β · −
113 = 14 . c = β · .
39 + β · . β · . − . ≈ .
44; (5.14b)517 . . . − . .c ≈ . . (5.14c)At the first BLM step, we do not obtain a significant profit in the first coefficient c → c ′ ,as it was in the previous case of r ′ . But the next order coefficient c ≈ .
05 in (5.14b)diminished by two orders (!) of magnitude, c → c ′ ≈ .
444 at a s ( Q ) → a ′ s = a s ( Q e − ≈ Q · . c ′ = −
113 ; (5.15a) c ′ = β · . − β · .
892 + β · . − . ≈ . . − .
06 + 491 . − . c ′ ≈ . . (5.15c)It is interesting that the far fourth coefficient c , (5.14c), reduces twice, c → c ′ , (5.15c). Atthe second step (PMC) ¯∆ ≈ .
32 and a ′ s → a ′′ s = a s ( Q · exp [ − − . β a ′ ( Q )]); so theregion of applicability of PT is shifted far from the scale of a few GeV . While the value of | c ′′ | goes up to the previous order of magnitude, compare (5.15b) with (5.16), c ′′ = −
113 ; c ′′ ≈ − . . (5.16)It is instructive to compare this result with one from seBLM (Sec. 5 2), where we removethe first 2 terms in (5.15b) converting them into the normalization scale and holding thelast two terms in c ′′ . For this prescription we obtain ∆ ≈ . a ′ s → a ′′ s = a s (cid:0) Q exp (cid:2) − − . β a ′ ( Q ) (cid:3)(cid:1) and c ′′ ≈ − r n ( c n ), see the discussion in Sec.6-7 in [8]. It is clear that one should notremove and absorb all the β -terms for the PT-optimization but leave a part of them forcomplete cancellation with the d n [0]-term. We shall treat the circumstances in this way inthe next section.
6. OPTIMIZATION OF THE GENERALIZED BLM PROCEDURE
Indeed, it is not mandatory to absorb all the β -terms as a whole into the new scale∆ (∆ i ) following BLM/PMC, but take instead only those parts of it that are appropriate20or optimization (nullification) of the current order coefficient r ( r i +2 ). At the same time,one should care for the size of the ∆ i – PT coefficients for the shift of scale ∆ in (5.2) – notto violate just this expansion.Let us consider the optimization of R NS at the second BLM step starting with the first stepexpressions in Eqs.(5.12) and using the general results in (5.8c,5.8e,5.8d). This expressionfor r ′′ can be rewritten as r ′′ = r ′ − β ∆ = r − β d [1] − β d [1] − β d [0] d [1] − β ∆ . (6.1)The optimization requirement, e.g., r ′′ = 0 leads to the expressions for ∆ and r ′′ r ′′ = 0 ⇒ ∆ = r ′ /β = r /β − d [1] − β /β d [1] − /β d [0] d [1] , (6.2a) r ′′ = r ′ − r ′ ( β /β + 2 d [0]) . (6.2b)Numerical calculation at n f = 4 gives the estimates for the values of the quantities inEqs.(6.2), r ′′ = 0 , ∆ ≈ − . , (6.3a) r ′′ ≈ − . , (6.3b) a ′ s → a ′′ s = a s (cid:16) s · e − . . β a ′ s ( s ) (cid:17) . (6.3c)One may conclude that the PT expansion R NS = 1 + 3 C F (cid:26) a ′′ s + 13 · ( a ′′ s ) + 0 · ( a ′′ s ) + r ′′ · ( a ′′ s ) + . . . (cid:27) (6.4)significantly improves:(i) r ′ = 0, while the value of r ′′ in (6.5e) is less than in (5.11c, 5.12c) and reduces twice incomparison with the PMC estimate in (5.13b) (taken for n f = 4).(ii) Domain of applicability of the approach extends to a wider region due to the oppositesign at ∆ , compare with one for PMC in (5.13a). This makes the NLO “shift” ∆ less,which tends numerically to 0 at the boundary of applicability, ∆ = d [1] + ∆ β a ′ s ( s ) ≈− .
692 + 3 . β a ′ s ( s ).Indeed, following the usual PT condition | d [1] | & | ∆ β a ′ s ( s ) | or ∆ . s &
10 GeV , as it is illustrated in Fig. 1(Left). The factor exp [ − ∆], enteringin the argument of a ′′ s in Eq.(6.3c), see solid (red) upper line in Fig. 1(Left), satisfies theconditions 1 & exp[ − .
692 + 3 . β a ′ s ( s )] > /
2, this factor slowly decreases with s from thevalue 1. It looks tempting to get and use the exact solution for the coupling a ⋆s , followingfrom Eq.(5.2), a ⋆s ( s ) = a s ( s exp[ − ∆ − ∆ β a ⋆s ( s )]) , rather than its iteration a ′′ s ( s ). It is easy to obtain the useful inequality a ′ s ( s ) > a ⋆s ( s ) > a ′′ s ( s );moreover, the numerical calculation gives that the difference between a ⋆s and a ′′ s becomesnoticeable below s = 1 GeV for this optimized quantity and for the next one discussedbelow.Similar optimization can be performed for C BjpNS ( Q ) ( n f = 4). We apply the generalcombined equations, analogous to the ones of Eqs.(5.3). In these C BjpNS -oriented expressionswe fix the conditions c ′′ = 0 and c ′′ = 0. This leads to the following equations: c ′ = 0 , ∆ = c /β ≈ . , , (6.5a) a s → a ′ s = a s (cid:0) Q · e − . (cid:1) (6.5b)21
10 15 20 25 300.40.50.60.70.80.91.01.11.2 s [GeV ]exp[ − ∆] Q [GeV ]exp[ − ∆] Figure 1:
Factors exp [ − ∆] at coupling scale: (Left) for R NS ( s ). Solid (red) upper line – the NLO factorexp [ − .
692 + 3 . β a ′ s ( s )]; long dashed (blue) line – the LO BLM one exp[ − . C BjpNS ( Q ).Solid (red) upper line – the NLO factor exp (cid:2) − .
56 + 0 . β a ′ s ( Q ) (cid:3) , long dashed (blue) line – the LOexp[ − . c ′′ = c ′ = 0 , (6.5c) c ′′ = 0 , ∆ ≈ − . , (6.5d) c ′′ ≈ . , (6.5e) a ′ s → a ′′ s = a s (cid:16) Q · e − . . β a ′ s ( Q ) (cid:17) . (6.5f)The new “optimized scale” behaviour of factor exp [ − ∆] is illustrated in Fig. 1(Right) bysolid (red) line, while the broken (blue) line there corresponds to the condition c ′ = 0 , ∆ = c /β that is not the BLM one. This transformation significantly improves the perturbativeseries for C BjpNS , C BjpNS ( Q ) = 1 − C F (cid:8) a ′′ s + 0 · ( a ′′ s ) + 0 · ( a ′′ s ) + c ′′ · ( a ′′ s ) + . . . (cid:9) (6.6)in comparison with Eqs.(5.14, 5.15, 5.16). We conclude that for both of the consideredquantities the PT series are improved, the corresponding Eqs.(6.3c,6.4) and Eqs.(6.5f,6.6)consist of the main results of this Sec. We did not perform the next step of optimizationwith the coefficients r ′′ , c ′′ because in this case we lose control under accuracy.It is clear that Eqs.(6.5,6.4) and Eqs.(6.5,6.6)) are not unique optimal solutions becausedifferent efficiency functions may be called “optimal”. Therefore, one can satisfy his ownefficiency function with the coefficients { c i } basing on the combined Eqs.(5.3) in the plane(∆ , ∆ ) or space (∆ , ∆ , ∆ , . . . ) of fitting free parameters ∆ j .
7. CONCLUSION
We have considered the general structure of perturbation expansion of renormalizationgroup invariant quantities in MS-schemes to clarify the effects of charge renormalization andthe conformal symmetry breakdown. Following the line started in [8] we arrived at the matrixrepresentation for this expansion, named the { β } -expansion [9], instead of the standardperturbation series. We discussed in great detail the unambiguity of this representation forAdler D NS -function (or related R e + e − –ratio) and for the Bjorken polarized sum rule S Bjp (with the coefficient function C BjpNS ) for DIS in order O ( α s ). The expansion for S Bjp was22btained by using different parts of the Crewther relation [9] for D NS and the coefficientfunction C BjpNS . Others attempts of this presentation [3–6] were discussed too. We providednew prediction for C BjpNS ( a s , n f , n ˜ g ) with the MSSM massless gluino n ˜ g in order O ( α s ) inEqs.(4.11), Sec.4 2, as a byproduct of our consideration.Based on the { β } -expansion we constructed renormalization group transformation for theperturbation series of the considered quantities, Eqs.(5.3) in Sec.5. The initial expansionwas split into two parts: A new series for the expansion coefficients, while the other one – forthe shift of the normalization scale of the coupling α s . The contributions from each ordercan be balanced between these two series. Different procedures of the PT optimization,including PMC [4, 5], and seBLM [8], were discussed and illustrated by numerical estimates.We conclude that the corrected PMC does not provide better PT series convergence andsuggest our own scheme of the series optimization in order of O ( α s ); the working formulaefor R NS of the R e + e − –ratio and C BjpNS are presented in Sec.6.
Acknowledgments
We would like to thank A. V. Bednyakov, K. G. Chetyrkin, A. G. Grozin, V. L. Khan-dramai, and N. G. Stefanis for the fruitful discussion. The work is done within the scientificprogram of the Russian Foundation for Basic Research, Grant No. 14-01-00647. The workby A.K. was supported in part by the Russian Science Foundation, Grant N 14-22-00161.The work of M.S. was supported in part by the BelRFFR–JINR, grant F14D-007.
Appendix A: Explicit formulas for β ( n f , n ˜ g ) and D ( n f , n ˜ g ) The required β -function coefficients with the Minimal Supersymmetric Model (MSSM)light gluinos [48] calculated in the MS scheme are, β ( n f , n ˜ g ) = 113 C A − (cid:18) T R n f + n ˜ g C A (cid:19) ; (A.1a) β ( n f , n ˜ g ) = 343 C A − C A (cid:18) T R n f + n ˜ g C A (cid:19) − (cid:18) T R n f C F + n ˜ g C A C A (cid:19) ; (A.1b) β ( n f , n ˜ g ) = 285754 C A − n f T R (cid:18) C A + 2059 C A C F − C F (cid:19) + ( n f T R ) (cid:18) C F + 15827 C A (cid:19) − n ˜ g C A ( C A ) + n ˜ g C A n f T R (cid:18) C A C F + 22427 C A (cid:19) + ( n ˜ g C A ) C A . (A.1c)23he β coefficient, which includes the MSSM light gluinos, is not yet known, so we presentit here in the standard [52, 53] simplest form β ( n f ) = C A (cid:18) − ζ (cid:19) + C A T R n f (cid:18) − ζ (cid:19) + C F T R n f (cid:18) − ζ (cid:19) + C A C F T R n f (cid:18) ζ (cid:19) + C A C F T R n f (cid:18) − ζ (cid:19) + 424243 C A T R n f + C A C F T R n f (cid:18) − ζ (cid:19) + C A T R n f (cid:18) ζ (cid:19) + 1232243 C F T R n f +46 C F T R n f + n f d abcdF d abcdA N A (cid:18) − ζ (cid:19) + n f d abcdF d abcdF N A (cid:18) − ζ (cid:19) + d abcdA d abcdA N A (cid:18) −
809 + 7043 ζ (cid:19) . (A.2)For the SU c ( N )-group with fundamental fermions the invariants read: T R = 12 , C F = N − N , C A = N, d abc d abc = ( N − N A N ; N A = 2 C F C A ≡ N − . (A.3) d abcdF d abcdA N A = N ( N + 6)48 , d abcdA d abcdA N A = N ( N + 36)24 , d abcdF d abcdF N A = N − N + 1896 N . (A.4)The D NS -function evaluated in [25] in the same model in the case when the masses ofgluion are neglected reads D NS ( a s , n f , n ˜ g ) = 1 + a s · (3 C F ) (A.5a)+ a s (cid:26) − C F + 2 C F (cid:20) − ζ − (11 − ζ ) n ˜ g (cid:21) C A − C F (11 − ζ ) n f T R (cid:27) (A.5b)+ a s (cid:26) − C F − C F C A [127 + 572 ζ − ζ − (36 + 104 ζ − ζ ) n ˜ g ]+ C F C A (cid:20) − ζ − ζ − (cid:18) − ζ − ζ (cid:19) n ˜ g + (cid:18) − ζ (cid:19) n g (cid:21) − n f T R C F [29 − ζ + 320 ζ ] − n f T R C F C A (cid:20) − ζ − ζ − (cid:18) − ζ (cid:19) n ˜ g (cid:21) +3 C F (cid:20) − ζ (cid:21) (cid:18) T R n f (cid:19) ) = (A.5c) In the numerical case this expression from [25] coincides with the result of the related numerical calculationof [54].
24 1 + a s (3 C F ) + a s (3 C F ) · (cid:26) C A − C F (cid:18) − ζ (cid:19) β ( n f , n ˜ g ) (cid:27) (A.6a)+ a s (3 C F ) · (cid:26)(cid:18) − ζ (cid:19) β ( n f , n ˜ g ) + (cid:18) − ζ (cid:19) β ( n f , n ˜ g )+ (cid:20) C A (cid:18)
34 + 803 ζ − ζ (cid:19) − C F (18 + 52 ζ − ζ ) (cid:21) β ( n f , n ˜ g )+ (cid:18) − ζ (cid:19) C + 713 C A C F −
232 C (cid:27) (A.6b)The Bjorken coefficient function C BjpNS of the DIS sum rules calculated first in [38] C BjpNS ( a s , n f ) = 1+ a s ( − C F ) (A.7a)+ a s ( − C F ) (cid:20) − C F + 233 C A − T R n f (cid:21) (A.7b)+ a s ( − C F ) (cid:26) C F C F C A (cid:20) ζ − (cid:21) + C A (cid:20) − ζ (cid:21) (A.7c)+ n f T R C F (cid:20) − ζ (cid:21) − n f T R C A (cid:20) ζ − ζ (cid:21) + 11518 (cid:18) n f T R (cid:19) ) . (A.7d)The prediction for C Bjp obtained in Sec.4 2 of this article under the same conditions asEq.(A.5) reads C BjpNS ( a s , n f , n ˜ g ) = 1 + a s ( − C F ) (A.8a)+ a s ( − C F ) · (cid:26)
13 C A −
72 C F + 2 β ( n f , n ˜ g ) (cid:27) (A.8b)+ a s ( − C F ) · (cid:26) β ( n f , n ˜ g ) + (cid:18) − ζ (cid:19) β ( n f , n ˜ g ) − (cid:20)(cid:18) − ζ + 403 ζ (cid:19) C A + (cid:18) − ζ (cid:19) C F (cid:21) β ( n f , n ˜ g )+ (cid:18) − ζ (cid:19) C + 653 C F C A + C (cid:27) . (A.8c) Appendix B: Natural forms for β -expansion of D NS and C NS Here we present for completeness the results of (4.8,4.10) in their “natural form”changing only the normalization factors [9], which correspond to the coupling α s π with β = 14 (cid:18) C A − (cid:18) T R n f + n ˜ g C A (cid:19)(cid:19) , . . . ,25 NS1 = 34 C F , (B.1a) d NS2 [1] = (cid:18) − ζ (cid:19) C F , d NS2 [0] = −
332 C + 116 C F C A , (B.1b) d NS3 [2] = (cid:18) − ζ (cid:19) C F , d NS3 [0 ,
1] = (cid:18) − ζ (cid:19) C F , (B.1c) d NS3 [1] = (cid:18) − − ζ + 15 ζ (cid:19) C − (cid:18) − ζ + 52 ζ (cid:19) C F C A , (B.1d) d [0] = − + 7164 C C A + (cid:18) − ζ (cid:19) C F C . (B.1e) c NS1 = −
34 C F , (B.2a) c NS2 [1] = −
32 C F , c NS2 [0] = 2132 C −
116 C F C A , (B.2b) c NS3 [2] = − F , c NS3 [1] = (cid:18) − ζ (cid:19) C + (cid:18) − ζ + 52 ζ (cid:19) C F C A , (B.2c) c NS3 [0 ,
1] = (cid:18) − ζ (cid:19) C F , (B.2d) c NS3 [0] = − − C A − (cid:18) − ζ (cid:19) C F C . (B.2e) Appendix C: R-ratio
Table 1 exemplifies the structure of a few first coefficients r m of the conventional expansionof the R -ratio. Every coefficient r m contains a number of d k ( k ≤ m ) terms in its expansion,which are shown in the corresponding row. In other words, r m = T mk d k (summation in k = 1 , . . . , m is assumed), where T mk are the Table entries. Note that the content of thistable is limited here only by the restricted place.26 able 1: The table of the T mk -matrix. The one-loop contributions are marked by black, two-loop contributions are marked by red, three-loop contribution — by blue, while the four-loopcontribution is colored by green. d d d d d d r r r − ( π β ) r − π β β − ( πβ ) r a ( π β ) − π β − π β β − π β β − ( πβ ) r − π β β − π β β − π β β − ( πβ ) − π β − π β β − π β β − ( π β ) a This expression for r was presented first in [51] ppendix D: Explicit formulae for ∆ i The explicit expressions for the elements of the proper scales ∆ and ∆ are given by∆ = d [1]; (D.1)∆ = d [2] − d [1] − π / β β ( d [0 , − d [1]) + 1 β (cid:16) d [1] − d [1] d [0] (cid:17) ; (D.2)∆ = (cid:0) d [3] − d [1] d [2] + 2( d [1]) − π d [1] (cid:1) + β /β ( d [0 , , − d [1]) + β /β (cid:20) d [1 , − d [0 , d [1] + 32 ( d [1]) − d [2] − π / (cid:21) + β /β ( d [1] − d [0 , β /β (cid:16) d [0 , − d [1] − d [0] ( d [0 , − d [1]) (cid:17) + (D.3)1 /β (cid:20) d [2] − d [1] d [1] + d [0] (cid:0) d [1] − d [2] − π / (cid:1)(cid:21) + (D.4)1 /β (cid:20) d [1] − d [0] d [1] + 2 d [0](2 d [1] d [0] − d [1]) (cid:21) . (D.5) [1] A. A. Vladimirov and D. V. Shirkov, Sov. Phys. Usp. , 860 (1979) [Usp. Fiz. Nauk ,407 (1979)].[2] S. J. Brodsky, G. P. Lepage and P. B. Mackenzie, Phys. Rev. D (1983) 228.[3] S. J. Brodsky and L. Di Giustino, Phys. Rev. D , 085026 (2012) [arXiv:1107.0338 [hep-ph]].[4] S. J. Brodsky and X. -G. Wu, Phys. Rev. D , 034038 (2012) [Erratum-ibid. D , 079903(2012)] [arXiv:1111.6175 [hep-ph]].[5] M. Mojaza, S. J. Brodsky and X. -G. Wu, Phys. Rev. Lett. , 192001 (2013),[arXiv:1212.0049 [hep-ph]].[6] X. -G. Wu, S. J. Brodsky and M. Mojaza, Prog. Part. Nucl. Phys. , 44 (2013)[arXiv:1302.0599 [hep-ph]].[7] S. J. Brodsky, M. Mojaza and X. -G. Wu, Phys. Rev. D , 014027 (2014) [arXiv:1304.4631[hep-ph]].[8] S. V. Mikhailov, JHEP (2007) 009 [arXiv:hep-ph/0411397].[9] A. L. Kataev and S. V. Mikhailov, Theor. Math. Phys. , 139 (2012) [arXiv:1011.5248].[10] A. L. Kataev and S. V. Mikhailov, Proc. Sci. , QFTHEP , 014 (2010), [arXiv:1104.5598[hep-ph]].[11] G. Grunberg and A. L. Kataev, Phys. Lett. B (1992) 352.[12] A. L. Kataev, Preprint CERN-TH.6485/92; “QCD scale scheme fixing prescriptions at thenext next-to-leading level,” In *Les Arcs 1992, Perturbative QCD and hadronic interactions*123-129.[13] M. Beneke and V. M. Braun, Phys. Lett. B , 513 (1995) [hep-ph/9411229].[14] M. Neubert, Phys. Rev. D , 5924 (1995) [hep-ph/9412265].[15] P. Ball, M. Beneke and V. M. Braun, Nucl. Phys. B 452 , 563 (1995) [hep-ph/9502300].[16] K. Hornbostel, G. P. Lepage and C. Morningstar, Phys. Rev. D , 034023 (2003)[hep-ph/0208224].[17] A. L. Kataev, JHEP , 092 (2014) [arXiv:1305.4605 [hep-th]].
18] K. G. Chetyrkin, A. L. Kataev and F. V. Tkachov, Phys. Lett. B (1979) 277.[19] G. ’t Hooft, Nucl. Phys. B 61 , 455 (1973).[20] M. Dine and J. R. Sapirstein, Phys. Rev. Lett. (1979) 668.[21] W. Celmaster and R. J. Gonsalves, Phys. Rev. Lett. (1980) 560.[22] W. A. Bardeen, A. J. Buras, D. W. Duke and T. Muta, Phys. Rev. D , 3998 (1978).[23] S. G. Gorishny, A. L. Kataev and S. A. Larin, Phys. Lett. B (1991) 144.[24] L. R. Surguladze and M. A. Samuel, Phys. Rev. Lett. (1991) 560 [Erratum-ibid. (1991)2416].[25] K. G. Chetyrkin, Phys. Lett. B (1997) 402.[26] C. N. Lovett-Turner and C. J. Maxwell, Nucl. Phys. B 452 (1995) 188.[27] R. J. Crewther, Phys. Rev. Lett. (1972) 1421.[28] S. G. Gorishny and S. A. Larin, Nucl. Phys. B 283 , 452 (1987).[29] D. J. Broadhurst and A. L. Kataev, Phys. Lett. B (1993) 179.[30] G. T. Gabadadze and A. L. Kataev, JETP Lett. (1995) 448 [Pisma Zh. Eksp. Teor. Fiz. (1995) 439].[31] R. J. Crewther, Phys. Lett. B (1997) 137.[32] V. M. Braun, G. P. Korchemsky and D. Mueller, Prog. Part. Nucl. Phys. (2003) 311.[33] P. A. Baikov, K. G. Chetyrkin and J. H. Kuhn, Phys. Rev. Lett. (2010) 132004.[34] P.A. Baikov, K.G. Chetyrkin, J.H. Kuhn, and J. Rittinger, Phys.Lett. B , (2012) 62;arXiv:1206.1288[hep-ph].[35] A. L. Kataev, JETP Lett. , 789 (2011) [Pisma Zh. Eksp. Teor. Fiz. , 867 (2011)][arXiv:1108.2898 [hep-ph]].[36] S. A. Larin, Phys. Lett. B (2013) 348.[37] S. G. Gorishny and S. A. Larin, Phys. Lett. B , 109 (1986).[38] S. A. Larin and J. A. M. Vermaseren, Phys. Lett. B , 345 (1991).[39] G. ’t Hooft and M. J. G. Veltman, Nucl. Phys. B 44 , 189 (1972).[40] I. Antoniadis, Phys. Lett. B , 223 (1979).[41] T. L. Trueman, Phys. Lett. B , 331 (1979).[42] S. L. Adler, C. G. Callan, D. J. Gross and R. Jackiw, Phys. Rev. D (1972) 2982.[43] D. J. Broadhurst and A. G. Grozin, Phys. Rev. D , 4082 (1995).[44] D. J. Broadhurst, Z. Phys. C , 339 (1993).[45] M. Beneke, Nucl. Phys. B 405 , 424 (1993).[46] S. G. Gorishnii, A. L. Kataev, S. A. Larin and L. R. Surguladze, Phys. Lett. B , 81 (1991).[47] L. J. Clavelli and L. R. Surguladze, Phys. Rev. Lett. (1997) 1632.[48] L. Clavelli, P. W. Coulter and L. R. Surguladze, Phys. Rev. D , 4268 (1997).[49] P. A. Baikov, K.G. Chetyrkin, J.H. Kuhn, and J. Rittinger, JHEP (2012) 017;[arXiv:1206.1284[hep-ph]][50] A.P. Bakulev, S.V. Mikhailov, N.G. Stefanis JHEP (2010) 085 [arXiv: 1004.4125].[51] A. L. Kataev and V. V. Starshenko, Mod. Phys. Lett. A , 235 (1995) [hep-ph/9502348].[52] T. van Ritbergen, J.A.M. Vermaseren, S.A. Larin, Phys. Lett. B , 379–384 (1997).[53] M. Czakon, Nucl. Phys. B 710 , 485 (2005), [arXiv: 0411261v2].[54] A. L. Kataev and A. A. Pivovarov, JETP Lett. (1983) 369 [Pisma Zh. Eksp. Teor. Fiz. (1983) 309].(1983) 309].