JHEP02(2018)161 ) 3 s a 3 at ( − O = Springer ξ January 24, 2018 January 28, 2018 February 26, 2018 only, namely for : : : ξ = 0 and ξ , Received Accepted Published b,c ) level of perturbation theory. It is 4 s a ) QCD generalized Crewther relation MS-scheme this property holds in the Published for SISSA by ( c -function will always be implemented in https://doi.org/10.1007/JHEP02(2018)161 O N β [email protected] , -factorization in the gauge-dependent renormalization β and V.S. Molokoedov ) order of PT it remains valid only for case of the Landau 4 s a b,c ( O -function in the SU( -factorization property? -function factorization in the QCD and QED variants of the β β β . 3 MS-scheme and in two other gauge-independent subtraction schemes, A.L. Kataev = 0. In the 1 a, ξ The Authors. c

The problem of scheme and gauge dependence of the factorization property of , -factorization is valid for three values of the gauge parameter [email protected] 1 and [email protected] β ) approximation. It is demonstrated that if factorization property for the MS-like = 0. The consideration of these two gauge-dependent schemes for the QCD GCR − 3 s , a ξ 3 ( − O Affiliation from 2011 till 2015. 1 = 60th October Anniversary prospect 7a,Moscow Moscow, Institute 117312, of Russia PhysicsInstitusky and per. Technology, 9, Dolgoprudny, MoscowE-mail: region, 141700, Russia viktor Higher School of Economics,Myasnitskaya Math street Department, 20, Moscow, 101000,Institute Russia for Nuclear Research of the Russian Academy of Sciences, b c a Open Access Article funded by SCOAP gauge allows us to conclude that theany factorization MOM-like of renormalization RG schemes with linearthe covariant gauge at GCR. The problem of theschemes possible in QCD is studied. TomMOM investigate and this problem MOMgggg-schemes. we We consider demonstrate thelevel that gauge in the non-invariant the mMOM scheme atξ the QCD GCR at leastall at gauge-invariant this schemes, order. we consider Toof the study the MS-like whether GCR schemes this in in factorizationnamely the QCD property in and is the the true momentum QED-limit firm in MOM the and the existence on-shell of OS the schemes. In these schemes we con- Abstract: the renormalization group (GCR), which connects thelarized flavor non-singlet sum contributions rule to functions, theknown is that Adler investigated and in at Bjorken the the po- gauge-invariant renormalization A.V. Garkusha, Renormalization scheme and gaugethe (in)dependence generalized of Crewther relation:grounds what of are the the real JHEP02(2018)161 1801.06231 Perturbative QCD, Renormalization Group ArXiv ePrint: subject, then the factorizationLandau gauge will in also all occur orders of in perturbation the theory arbitrary asKeywords: MOM-like well. scheme in the schemes is true in all orders of PT, as theoretically indicated in the several works on the JHEP02(2018)161 21 15 26 20 36 14 7 ) level 10 ) approx- 4 s 3 s 16 a a ( ( O O ) approximation 4 s a ) approximation ( 4 s 7 a O ( O ) QCD c 36 3 N MS, MOM and OS schemes 16 − 37 = 1 ξ − (3) QCD GCR in the = 24 c ξ – 1 – 29 MS, MOM and OS schemes in QED MS, MOM and OS schemes in QED 35 MS-scheme 32 23 26 -factorization of the generalized Crewther relation in the β ) 1 4 s a 34 -factorization property manifest itself in the generalized ( -function factorization of the SU β O β mMOM-scheme imation in the MOMgggg-scheme MOM-like schemes mMOM-scheme loop level: the mMOM vs. B.1 The caseB.2 of the Stefanis-Mikhailov The gauge case of the anti-Feynman gauge 4.1 The 4.2 The 4.3 Special features of the Landau and Stefanis-Mikhailov gauges in the QCD 3.5 Asymptotic behavior of the NS Adler and Bjorken functions at the fourth- Crewther relation in the3.1 gauge-dependent schemes in The QCD? reminding3.2 notes on the The miniMOM NS3.3 scheme Adler function in The the NS3.4 mMOM-scheme Bjorken in function the in About the the mMOM-scheme in gauge-dependence the of the generalized Crewther relation in the 2.1 Scheme-dependence of2.2 the PT series The in NS2.3 the Adler SU( function in The the NS2.4 Bjorken function in The the QED generalized Crewther relation in the 1 Introduction The triangle Green function, composed fromone axial-vector-vector of (AVV) fermion the currents, most is interestinginteractions. quantities From in experimental the point modern of theory view of these electromagnetic studies and in strong different kinematic regimes 5 Conclusion A The transformation relations B About partial factorization in the mMOM-scheme at the 4 The generalized Crewther relation in the MOM-like schemes in QCD 3 May the Contents 1 Introduction 2 Scheme-dependence of the GCR in the gauge-invariant schemes JHEP02(2018)161 ]. 3 (1.3) (1.4) (1.5) (1.1) (1.2) decay γγ ) level [ 3 s ] within the → . a 2 ) ( 0 5 s π ) is the singlet O a s ( a ( O , SI ! ) ) + -ratio: s D s a R ( ( 4 s a SI ,  D 0 ) 2 β , − 1 ! µ 2 decay amplitude and measurable β f 137. The Adler function and the ) 1 + / 2 Q ) d µ 1 s γγ . Q 5 6 ( f ≈ → + X R hadrons) → + s γ = 1 ) is the Born approximation for the NS in particular). In the work [ 2 0 0

is the electric charge of the active quark ( → EM β π c f 2 → α 1.1 ds γ N d ) + Q hadrons process, namely the Adler function Born − s , ∞  e ]: → a 0 2 ( 6 + → ) is the Born massless normalization factor with – 2 – Z NS π Bjp − – e s 2 e ( NS C 4 γ − (3 + Q D ) / e s → ( 2 f Born NS Born ( σ 3 s σ ) = − Q is the QCD coupling constant, 2 EM D a 2 e 2 0 s + f Q is identical to the dimension of the quark representation πα β α ( e ) color group, X 1 c c ) = d D s

N N 2 ( c 3 in the l.h.s. of eq. ( π and ) = 4 R N − − µ /π ) Born + ) = s , s s ( µ α a NS ( Bjp D → = C D s − ) = ] the result of application of the conformal symmetry (CS) transformations e a s 1 + ( , e is the Born approximation of the flavor NS contribution to the normalized f ( R Born NS Born σ D ), which has the following general massless renormalized PT expression: The whole Adler function is related to the measurable in the Minkowski region char- s a ( -ratio are related through the following K¨allen-Lehmanndispersion representation The second term contribution to the normalized characteristic of the pure Euclidean process of deep inelastic R and begin to differanalytical in continuation, the studied Minkowski in region refs. from [ three-loop level due to the effects of the where fixed expression of the QED coupling constant of the Lie algebra ofwith the flavor SU( contribution to the Adler function, that begins to manifestacteristic itself of from the the electron-positron annihilation to hadrons process where the number of colors Here Euclidean characteristic of the D currents without the internal gaugeorder particles of perturbation coincides theory with for result,amplitude triangle (and obtained the diagram, at number which the of is the lowest massless proportional quark quark-parton to colors model it wasof proved that the in operator the Born product approximationCS expansion the limit application (OPE) leads to approach the to following the identity: AVV triangle diagram in this formfactors of light mesons.the fundamental From theoretical consequences, point obtainedleads of from to view consideration the the of discovery ofshown detailed the in axial investigation AVV ref. anomaly of Green [ and function, to property of the its triangle non-renormalizability. diagram As was with one flavor non-singlet (NS) axial and two vector fermion allow to obtain the important information on say JHEP02(2018)161 ) ) ). )-  s k 2 a a =1. ( ( 1.6 (1.8) (1.6) (1.7) V ] with /Q g NS NS Bjp 8 2723 . (1 C D 1 O ], where it − 17 = V /g A . g ) level of PT [ ) allowed the authors ) 4 s 2 a ( 1.8 Q , ] by the demonstration of ( can be represented in the O ) s should be considered as the 11 Bjp a a ) corrections to the ( Bjp C 2

C a SI Bjp A ( V ) was proved in ref. [ g g C O

f 1.8 1 6 , Q ) the massive and massless = f X 1.6 ) = 1 dx c a ] that these results are in agreement with ] correspondingly. ] leads to the conclusion that the Crewther ]. Note, that since in this work our main  ( -decay constants with N 9 ) β 14 16 10 2 NS Bjp ) + – 3 – C ) and ( s ], the application of eq. ( ) a x, Q a ( ( 1.2 ( 13 ] and [ ln , 1 NS Bjp g NS 15 12 C D − ) 2 ) = ) in QED-limit in the case when the AVV triangle amplitude s a ) corrections to the AVV triangle Green function in the most ) are the structure functions of these deep-inelastic scattering ( x, Q s ) leading-order (LO) and next-to-leading order (NLO) QCD 2 1.1 ( c α lp 1 ( Bjp ]. The application of CVC hypothesis leads to the result N g 7 C O x, Q 1. Indeed, these effects are the extra sources of the violation of the  ( 1 ln 1 ≥ 0 g Z k is a fixed scale-independent QED coupling constant. Since at this time the ) and 2 ) should be rewritten at the two-loop level as ) functions, which follows from the CS limit and its violation by the procedure is the singlet contribution, which appears first at the are the axial and vector neutron s α/π ] to predict the analytical expression for the a 1.1 V x, Q ( = SI Bjp ( g 10 a C lp 1 NS Bjp ] it was correctly assumed that these insertions do not renormalize the AVV triangle g C In all orders of pqQED the validity of the relation ( At the second stage of theoretical studies it is interesting to learn what will happen and 10 0023 correspondingly [ . A pqQED limit of theresults, obtained SU( previously in refs. [ was clarified that a fixed scale-independentunrenormalized coupling (bare) constant QED coupling in the CS limit, which is realized when the charge where two-loop and three-loop pqQEDwere expressions analytically for calculated the in refs. photon [ of vacuum polarization ref. function [ function in pqQED. It was observed in ref. [ general kinematics. In theof the case theoretically of motivated perturbative surmiserelation of quenched ref. ( QED [ (pqQED) the application with the Crewther relation ( will contain internal propagatorsof of [ photons without internalgraph. fermion At loops. two-loop level this Inthe assumption the was cancellations confirmed work of in the ref. [ aim is to studyand theoretical relations between coefficientsof of renormalizations, the we PT neglect seriescontributions in for with eq. the ( CS in QCD. following form: where the coefficient, analytically evaluated in ref. [ processes, which characterize theg spin distribution of0 quarks and gluons insideIts nucleons, application is sometimesThe coefficient silently function assumed of in the the polarized Bjorken definition sum of rule the r.h.s. of eq. ( which is defined as Here scattering (DIS) of polarized leptons on nucleons, namely to the Bjorken polarized sum rule, JHEP02(2018)161 ] - 2 ), )- β ]. s c a 35 N ( 31 , (1.9) (1.11) (1.10) ]. 34 csb 19 ] by direct ) color group 22 ], turned out to c 14 N MS-scheme [ -function of SU( β ) ], which as demonstrated 23 1.9 ], used in the process of the , 2 , ) level this extra term can be 20 . ) 3 s s a a ( +2 ( ], namely in the i s , ), performed in ref. [ O a ) i 30 csb s 1.8 β a ( 0 ≥ ] in this gauge-independent scheme the K i X ], or to be more precise, of the term, containing  33 − ) 18 s s ], which confirmed the analytical expression a = ) = 1 + ∆ a ( s 2 s 21 β – 4 – a ( ] is fixed by the perturbative expression of the ].  da dµ MS-scheme at the one-loop level in refs. [ 2 32 27 NS Bjp = – µ C ), the Crewther relation, written down in the form of ) 24 3 csb s -contribution to its QED limit, firstly discovered in ref. [ ) = ] it was also explained how to define this pqQED limit, a Z 3 ∆ s ( ζ a 17 ( NS ) QCD it is usually evaluated in the related to dimensional ) function in QCD [ β c s D a ) level: N ( 2 s a NS ( D O ] variant of the MS-like schemes [ ] it was shown that at least at the and can be defined as: ), should be modified. The theoretical reason for this modification is 33 In the work [ 29 , 1.8 /π 1 28 ) 2 µ ( = 1. s -function is responsible for the evolution of the corresponding coupling constant ) or ( ) is the two-loop expression for the defined in eq. ( 3 α β s Z ) contribution to 1.1 a 4 s MS-scheme. As was discovered in ref. [ ( -function, coinciding with anomalous dimension of the trace of energy momentum α ) = β ( 2 β During rather long period it was unclear how the violation of the CS will manifest However, if one will now consider massless limit of the based on SU( Note, that the consideration of the consequences of eq. ( Its more formal definition is given in ref. [ O µ 1 2 ( s theory, analytically calculated in the be rather important forthe the first confirmationunexpected of previously the transcendental validity of the complicated analytical calculation of written down in the following factorized form: here in the Crewther relation receives additional conformalfirst symmetry appearing breaking at contribution ∆ the Moreover, in ref. [ itself in high orders of the PT for the QCD generalization of the Crewther (GCR) relation RG tensor. In theregularization case [ of SU( This RG a either ( related to the appearancewhich inside should AVV be triangle renormalized. graphthe It of CS is the and known leads internal that to divergent the thealmost appearance subgraphs, procedure of at the of conformal renormalizations the anomaly violates [ same time in refs. [ function in thediagram-by-diagram MS-like calculations. and MOM-schemes, previously evaluated inrenormalizable ref. QCD [ or ofof the the realized coupling in naturephoton constant real renormalization renormalization constant QED by (which the presumes the ultraviolet account divergent expression of the namely which is equivalent to theunrealized language but of theoretically the important investigated finiteanalytical some QED four-loop time program ago [ computations phenomenologically ofof ref. the [ scheme-independent four-loop contribution to the renormalization group QED renormalization constant of QED is fixed to unity in all orders of perturbation theory, JHEP02(2018)161 ] ] ] ) f f ), c n n 50 33 46 N F F MS- ] and T T 1.12 (1.12) ) func- A F 16 s term of C C a ( F csb ) expression NS ,C Bjp of eq. ( 4 s and f ] after getting C a 3 n ( A 48 F K O C T 3 ] F 2 F C 48 , ,C , computed in ref. [ 2 i 2 F A ) functions in the s C C K ]. However, the detailed ] and later on in ref. [ a F ( 54 ) can be represented as the 45 -function. ], the next-to-leading order s ,C NS β Bjp -factorization property in the a )-function, and taking into ac- -function, computed in ref. [ ) level in ref. [ A ( C β s 41 4 s β , C a a K ( 2 )-function (confirmed recently by ( F 40 s ) is in agreement with the performed later O a NS Bjp ,C ( i s ) RG 3 C F ) and c a ) approximation for the 1.11 ], included the information on the ob- s i NS 3 s C N a ) due to the knowledge of the SU( a -factorization property in ∆ ( K D 33 ( β 1 ) was known thanks to the analytical cal- NS O ≥ 1.12 s in eq. ( i X is the number of active quark flavors. These D a 1 ( ]) and next-to-next-to-leading order (NNLO) is expressed through f )–( K – 5 – )-correction [ n 0 s 2 ) = 44 NS ] and in the review [ β ]. The function s a K ( D a 1.10 53 ( 38 ]) and for the O ] and confirmed in ref. [ – 3 K 49 36 ) group structures coefficients MS-scheme. The presented in ref. [ c ] in all orders of PT. At more solid theoretical level this ] (which agrees with the numerical result for this contri- ], was detected at the , while N 52 F 33 43 depends only on the defined in the fundamental representation C is the Casimir operator in the adjoint representation of the , ) expressions for the 1 : c A s 42 K ] in the a N C 51 ) approximations for 3 s a ( O is the Dynkin index and ] LO corrections and the NLO and NNLO-terms, evaluated in ref. [ F 15 . T )-correction [ MS-scheme, which was used in ref. [ 2 f 2 s ] explicit analytical calculations of the three-loop contributions to the AVV correlator, which a n ( 2. Its first term 2 39 F ), discovered in ref. [ , O T -coefficient. ] respectively. 0 F β )-correction, evaluated in ref. [ = 1 47 1.10 C In this work we study this problem at the four-loop level using the method of theoretical Theoretical arguments in favor of the validity of the Seventeen years later the validity of the The corresponding expression for 3 s The appearance of the term proportional to i a 3 ( dependence of the analytical expression of the GCR in QCDon and in QED. ref. [ contain the internal sub-diagramsof responsible the for the running of the coupling constant and the emergence statement was analyzed inunderstanding the and work proof [ ofGCR the is origin still of absent. the existence of this fundamentalexperiment feature and of try the to shed some light on the origin of this feature analyzing scheme- and confirmed in ref.for [ the conformal symmetrywhich breaking is term composed contains of the sixand third color coefficient structures, namely MS-scheme were given in ref. [ eq. ( analytical four-loop SU( the independent calculation of ref.count the [ explicit expression for the three-loop SU( O using a bit different technique.tion The in analytical the tained in ref. [ ref. [ scheme and the two-loop analytical expression for the QCD culations of the(NLO) leading-order (LO) bution, independently obtained in ref. [ quadratic Casimir operator color structures, where Lie algebra, coefficients were obtainedstructure from of eqs. the ( polynomial in powers of with the depending onfor the SU( and at the two-loop level in refs. [ JHEP02(2018)161 ]. Re- renor- (1.13a) (1.13b) ] and in -function 66 renormal- , 62 β – -function in 65 β 57 , of the conformal 33 i ) RG K c -functions and of the N gauge-invariant NS Bjp C , NS gauge non-invariant D . , k s k s a a k k d c 1 1 ], the transformation to the considered ≥ ≥ ), within MS-like schemes up to four-loop 55 k k X X 1.13b -function, which contains two nonzero scheme- condition for factorization of the RG – 6 – β -factorization in the GCR. In view of this the nat- ) = 1 + ) = 1 + s s β a a -functions in various gauge-independent schemes, in- ) in this class of the gauge-invariant schemes. ( ( ) and ( to the Adler and Bjorken polarized sum rule functions, NS Bjp ) expression for the NS flavor contribution to the Adler NS NS Bjp 4 s C 1.12 sufficient NS D Bjp C a MS-results, discussed previously in refs. [ ( 1.13a C O ]. All these schemes are the elements of the considered in and and 68 MS-scheme, but also formulated for simplifying multi-loop cal- NS -Renormalization gauge-independent MS-like schemes. D NS ] and applied at high-order computations in the chiral perturbation δ D 67 -function as well. We clarify the features of the considered analytical β MS-scheme, the momentum subtraction (MOM) scheme and the physical- ], based on the effective charges approach, developed in refs. [ -factorization in the GCR, associated with the manifestation of the conformal 64 , β ] ’t Hooft scheme with its RG ]. We calculate the , determined in eqs. ( 63 k MS + 1)-scheme [ , 56 c 70 ] infinite set of ) QCD and in QED, based on the Abelian U(1) gauge group. In this section we c -function in the generalized Crewther relation in the 48 69 In section 3 we first investigate the question of the possibility of factorization of the In the QED studies to be presented in this work we use three gauge-invariant schemes In section 2 we study this question in two renormalized gauge theories, namely in We start our analysis from the consideration of the PT theoretical QCD expressions for β N and k RG ization schemes. For thisgauge-dependent purpose subtraction we scheme, turn called toin the consideration ref. miniMOM [ of (mMOM) the scheme, popular introduced at present the only, namely the motivated on-shell (OS) scheme. Also wed consider the scheme dependence of thelevel. coefficients After this we examinesymmetry the breaking scheme term dependence in of eq. the ( coefficients mind that the classand of widely gauge-invariant schemes used includes inculations not QCD G-scheme only [ the originaltheory MS-scheme ( ref. [ malization scheme for evaluatingcorresponding PT RG expressions for the PT expressions for the cluding the consideration of the refs. [ symmetry breaking. SU( arrive to the conclusion thatin the the GCR at the fourth order of PT at least is the choice of the independent PT coefficients only (the restof are charge), assumed spoils to the be zero propertyural by of finite question pure renormalizations arises whetherultraviolet there subtraction do schemes, exist whicherty theoretical of provide requirements the the to realization the of choice the of fundamental the prop- It is already known thatthe the GCR property is of scheme-dependent theization even factorization schemes. within of the the Indeed, SU( frameworkin as of ref. was a [ shown gauge-invariant in renormal- ref. [ the NS contributions which are defined as: JHEP02(2018)161 ) ) ) 3 4 s 3 s at − a a ( ( ξ (2.1) = O O 1.13a ξ -function in ) and 3 s β  a ) in this class of 577 is the con- ( 2 . = 0 and 3. Moreover, we 0 µ -schemes) can be ˜ µ O )-factor. Indeed, ( ξ δ − 3 s E ≈ R a γ → = - ] E  π µ ξ γ L 69 2 β ) ) QCD 2 3, we observe at this level c − µ − ( N 2 s 2 L a = 0 and = ), where 1 β ξ E ξ L) 0 γ 1 MS-scheme to another representa- β β − 5 2 − 1 or π + 2 − )-level 3 L 3 s 2 L 0 a = ) approximation in the MOMgggg-scheme 3 0 β ( 3 s β ξ a O ( − exp(log 4 – 7 – O ) + ( 2  2 -function is always possible in an arbitrary MOM- µ µ + 3 are highlighted), we conclude that in the ( β s − = a 5 in comparison with the similar results, obtained in ) at the 2 , L MS-scheme transcendental (log 4 = µ 0 4 ) level the factorization property holds in the mMOM- 1.9 , 4 s β ξ a ) theory. In this extended version of QCD the most effective ( − c = 3 O 1 N f  -factorization is performed in the GCR at the ) n 2 β µ = 0 and ( s ξ a -factorization is valid in MS-like schemes in all orders of PT, then in the ) = β 2 µ 3 are distinguished from an infinite set of values of gauge parameter (˜ − s a , ) change under the transition from one to another scale 1 ) level of PT, because exactly for these values the factorization of 3 s − a ( , -factorization property holds in the 1.13b In section 4 we present an exact recipe for determination of those values of the gauge β O MS-scheme, especially for the Bjorken coefficient function. = 0 the solution of the RG equation ( accomplished by the properMS change to of MS the requires renormalization followingstant scale. of shift Euler-Mascheroni. The ˜ This transition immediately fromand implies Bjorken the that polarized the sum PT rule coefficientsproportional functions for to the in Adler the the MS-scheme absent will in contain the additional terms, regularization the class of theLet MS-like us gauge-invariant ultraviolet find renormalization out how schemes. theand coefficients ( of the physical quantitiesschemes. discussed above It in eqs. is ( knowntive that of the the class transformation of from MS-like the schemes (or following terminology of ref. [ 2 Scheme-dependence of the2.1 GCR in the Scheme-dependence gauge-invariant schemes of theConsider first PT the series case of in SU( method the of SU( performing high-order PT calculations is based on the application of dimensional like renormalization scheme with linearprove that covariant gauges if the MOM-like schemes with the Landauas gauge well. this factorization property persists in all orders between these completely different MOM-typethe schemes. We find that at as well. Based onclear these at why the the first values glanceapproximation the surprising factorization results of (indeed, the it is not immediately ξ the the GCR is fulfilled.scheme At in the the Landauthe gauge partial factorization only. only (see If appendixrenormalization we B). scheme, Then fix namely we consider the another MOMgggg-scheme, gauge-dependent and discover definite similarities PT series for thethe cases of parameter, for which the levels. By explicit calculation we demonstrate that in the mMOM-scheme the gauges and Bjorken polarized sum rule functions and we find the improved convergence of these JHEP02(2018)161 - ) 4 ), β )– 1.9 ) in , in- ) PT i 2.2d (2.3c) (2.4c) (2.2c) 4 s (2.4a) (2.2a) (2.3a) (2.3b) (2.3d) (2.4b) (2.2b) (2.2d) 2.3b K a 1.11 ( )–( . ) coincide O ), into the )–( 3 2.2a L ). The nullifi- 2.4c 1 1.10 d 1.13b 3 0 1.9 )–( β -function in eq. ( , )–( β 3 + 2.4a K 2 0 L β 1.13a  . in the MS-like schemes: 2 − , 2 . The expressions ( d ˜ 2 2 µ 2 L 0 1 csb β K K function in the GCR is true. 0 1 ) function will obey the same → K s 2 β β 0 β µ + 3 a β ( − − 1 , ), we obtain the following relations d at 1 1 1 NS 1 Bjp . k 2.1 K K K k β C d 0 1 2 c 0 )L + 3 β β β β , 2 5 2 . Using the property of the RG-invariance − − − by -function, defined in eq. ( K 2  β 0 ]. It should be emphasized that the scale k ) for = = = = 0 , β d /µ 2 71 2 1 2 2 1 , + c c c L d µ – 8 – 1 2 1 1 L )L + 1 ) RG 1.13b d c d d + 1 3 c )-functions, given in eqs. ( 2 0 K d s 1 K 1 + + + 0 N β d a 0 β β ( 2 2 3 β c c do not affect the expressions for the coefficients of the RG d ), we get the following relations: 1 1 NS ˜ µ d c + 3 + in eq. ( )L + + 2 + 3( , C 2 2 2 1.11 + + 3 1 2 k → d d d c 0 1 3 3 K K K is invariant in the MS-like schemes and the remaining coeffi- µ c c β β )–( 1 1 )-corrections to the CSB term ∆ = = = d ) and + s s , K 3 1 2 + 2 + 2 3 a ) and taking into account that these formulas remain valid for any a 1.10 + ˜ ˜ ˜ ( d L 1 1 ( K K K 4 1 d d O c d 1 2 NS 0 2.3d β β + β D 4 )–( ) is the consequence of the conformal symmetry and it reflects the feature + + ( + ( d , ) in MS-like schemes. 1 2 3 4 are not MS-like schemes invariants. The expressions ( d d d d 2.3a )-function and taking into account eq. ( 1.9 2.3a 3 s = = = = a K ( 1 2 3 4 , ˜ ˜ ˜ ˜ are the coefficients of the SU( d d d d 2 NS i we obtain: β K D 5 ) are valid when the property of factorization of the Now everything is ready to study the scheme-dependence of the coefficients Let us now study the scheme-dependence of the GCR, defined by eqs. ( Note that due to the absence of overallRemind minus that in the transformations definition of the QED RG 4 5 2.3d cients with the results, obtained recently in ref.the [ QED analogues of these formulas contain plusesfunction instead of minuses eq. in ( r.h.s. of these expressions. Note, that the coefficient cluded in the CSB term, withinthe the relations class ( of MS-like schemes.scale, Using equations ( where cation of eq. ( of the absence of( the It is obvious thattransformation coefficients laws after the replacement of the case of application ofapproximations the for gauge-invariant MS-like schemes. SubstitutingGCR, the presented in eqs. ( of the reflecting the transformation law of the coefficients depends on the RG logarithm term L = log ˜ JHEP02(2018)161 ] ]. 6 )– ]. 17 75 ] for 74 ]: (2.5a) (2.5b) 2.5a 73 73 . , -function , , 2 β 1] 2 , ) 1] according to 0 c , i , ) schemes the [0]L -coefficients of N δ [0 2 [0]L k [0 K 3 1 d 4 R d d [0] in eqs. ( d d 1 k 2 d β , , ) within the gauge- β , ) and expanding rela- ) clarify the scheme- 1.10 [0]L [1]L + [0]L [1]L + 3 1] + [2] + 2 2 3 3 [0]L , 3 1 d d d d 2.5b d d [1 2 0 2.2d 4 MS, momentum subtraction -expansion approach [ , namely )–( β d k } ]. As was explained in ref. [ 0 )–( d β β have the following form [ { 20 [1] + [1] + 2 [2] + 2 [1] + 3 [2] + 3 1 2.5a 2 3 3 4 4 k β [1] + d d d d d 2.2a d 3 ], which is the high-order PT QCD d 0 73 1] + β , ) depend on the choice of the concrete ) of the GCR ( [1] = [1] = [2] = [1] = [2] = s 2 3 3 4 4 [0 ˜ ˜ ˜ ˜ ˜ a d d d d d 4 ( d 2.5b [0] + 1 3 csb β d -factorization in the CSB massless perturbative )–( – 9 – β = [0] within the class of MS-like schemes. Actu- 3 2.5a k [3] + , 4 d , d -expanded coefficients , 3 0 } , d [0]L β β ]. The idea of applying the term PMC belongs to authors -expanded formalism ( 1 [0]L { } 2 and obey the same properties as the coefficients of Green [1] [0]L d MS-version of the BLM approach, called the Principle of MS-version of the BLM scale-fixing approach in ref. [ 76 the property of the factorization of the SU( : 2 β 1 d d { d µ 0 [2] + 4 β 1] + d , 2 0 , 1] + , 1] + 2 0 , , condition for the , , , β -expanded representations for } -expansion formalism was used for the construction of the QCD [0] [0 [0] [0 [0 [0] [0] [0] + } β 2 3 3 4 4 4 1 2 ). This gives us the idea that the gauge-invariance of the subtraction { d d d d d d d d β { [1] + ], and analyze the GCR in QED, where the number of widely applicable 4 = ) at an appropriate scale in accordance with this formalism, we obtain 2.4c d the do not violate 72 2 )-series, proposed and used in ref. [ 1] = 1] = 1] = 0 [0] = [0] = [0] = [0] = 2 s sufficient , , , )–( β 2 3 4 1 2.2d a µ 0 ˜ ˜ ˜ ˜ [0 [0 d d d d ( , , d -dependent terms of eqs. ( 3 4 scheme-independent = ˜ ˜ )–( k [0 d d NS 2.4a ], who correctly realized that within the class of MS-like (or 4 2 β [0] + [0] ˜ d D 4 1 76 Q d d 2.2a ), are = = The presented above RG-based expressions ( At Let us now to move aside slightly and consider the For the discussion on the scheme-dependence of the original BLM procedure within QCD see e.g. [ 4 1 6 -independent contributions in d d k 2.5b independent property ofally, the taking into terms accounttions the ( that the MS-like prescription of fixing of ref. [ β ( functions within studied previously finitethis QED property program is [ related to the possibility of defining the CS limit in this Abelian model. In particular, the multiloop generalization of the Maximal Conformality (PMC) [ the PT generalization of the suggested reported in ref. [ gauge-independent schemes is largerwe than in use QCD. three In concrete(MOM) the renormalization and process on-shell schemes, of (OS) namely these gauge-invariant schemes. for considerations representing PT coefficients of the RG invariant quantity, namely for in the PTindependent expression MS-like for schemes, but the onlyformulas CSB modify ( term expressions for ∆ coefficients schemes is the term of the GCR. To study the status of this statement we extend the considerations, transformations JHEP02(2018)161 ], )- s 78 a MS- = 0, ( – ] and A NS Bjp 73 76 , C -function C ) level in -expanded 4 s β -expansion -dependent } a 71 β k , ( β β { O = 1, ) and 69 s , types of charged F a ( 58 C N NS ) color gauge theory, c of the NS Bjorken po- D N k c ]) is on the agenda and will -dependent contributions in , k 61 2 β ) reconsideration of the results 4 s a [0]L ( ] all 1 denotes one of the charged leptons 3 d O l 5 2 76 , [0]L 1 73 d , MS-scheme, which is related through the (here + ) analytical PT approximations for the NS 69 1])L + 4 − , we obtain the following four-loop analytical 2 , l a ( + N [0 l ]. We consider QED with -expanded coefficients – 10 – 3 O MS, MOM and OS schemes in QED [1]L -expanded coefficients of PT series of any RG- } d = 2 } 48 ] it was demonstrated how to define the expressions β → d f β { 60 n { γ ] with taking into account that the QCD anomalous -leptons correspondingly). Fixing ), studied from different points of view in the works [1] + 3 → s 78 2 a – = 1, d ( − [2]L + 3 e R 3 )-function in the + 76 NS d ) expression for the non-singlet QED Adler function in the d Bjp a , e 4 ]. In ref. [ ( e, µ, τ C a coefficients in this model were first obtained in ref. [ ( 78 NS 1] + (2 71 3 , ] the NNLO PMC-type procedure to both , = 0, O D d [1 [3] + 3 69 59 3 for 4 4 d d abcd A , d and 2 ]. 2 ) should be absorbed into the set of scales of the initially defined in the , 1] = d [3] = , 4 = 1, 77 ˜ , [1 d = 1 2.5b 4 ˜ d ] and refs. [ 71 abcd F )–( , N d 59 -expanded coefficients of the Adler and Bjorken functions at the ) QCD without any gluino. The careful QCD } 59 c , 2.5a β We will not discuss anymore these phenomenologically oriented problems, but will Note that in ref. [ In accordance with the proposals of refs. [ N { = 1, 57 F leptons ( T approximation for the K¨allen-Lehmanndispersion representation with thecross-section massless of PT the expression process for the total renormalization schemes we are interested in. 2.2 The NS AdlerIn function order in the to obtainMS the scheme we use the results of ref. [ return to one(in)dependence of of the the CSB theoretical PT contributionform aims to the in of GCR the in this theanalyze different work, factorized by this schemes, namely RG problem including to weAdler the the should function ones study get and commonly of the Bjorken used the polarized in scheme sum QED. rule In function order in to QED in the gauge-invariant of ref. [ dimension of the photon vacuum polarizationstructure function (for has the its own detailed well-defined clarificationbe of considered this elsewhere. point see ref. [ supplemented with multiplet of SUSYexpressions gluino. of The analytical resultsconfirmed for later the on in ref. [ for SU( well and do not respectConformality, the used CS in approximations. the Therefore,should number the be term of used Principle the with of care. related Maximal works on thisfunctions topic was [ applied in the QCD-type model, based on the SU( of [ eqs. ( scheme QCD coupling constant.terms, In the view scales of of the the scheme-dependence resulting of QCD-expansion these parameter become scheme-dependent as The similar relationsinvariant holds physical quantity, for including the larized sum rule function JHEP02(2018)161 . , (2.8) (2.9) (2.7) (2.6c) (2.10) (2.6a) 5 (2.6b) (2.6d) 3 ) N  5 MOM ζ 4 a . ) 5 3 ( ) is determined 5  ) ), were obtained MS + 3 ]. The four-loop a 3 = 1 it agrees with ( ζ N MS 2.7 2.6a -dependence of the 81   a , ( 3 2 N N 2 54 ζ 203  N 1 3 3 N , +  N in eq. ( − ] and [ 11 3 ]. At 144 , ζ 1 2 MS 67 ]. The /π 80 N  − 77 9 972 19 MS 4 6131 3888 12  ) MS + /∂a 7 N − ζ − α − 2 1 /∂a 32 2  N 4 = MOM 54  MOM OS 105 N 151 − a + 5  ( ζ MS 3 2  ∂a ∂a +  -function, which reads  5 3 ζ a ) ) 5 2 N β + + ζ ) to the MOM and OS-schemes we use the − N, 13 36  3 N 3 MS MS ) 2 N 3 3 )   4 a a -functions in the MOM and OS schemes are ζ ζ − 3 3 115 ( ( -function coincides with the Gell-Mann-Low  ζ ζ β 2 3 2.6d 5 MS β 3 1 ζ − a MS MS + 3 MOM + – 11 – 95 5 ( + 864 3 )) or to be more precise of its expression, related )–( β β a 5 − ζ (  ζ N 8 − 11 , a 13 32 1 4 3 N 2 67 32 + 72 23  6 ζ 2.6b ) = ) = 1 4 −  125 + /µ + 4 N + 19 2  2 OS + − + , ) a Q 2 N ( + + 3 MOM k 23 ) 128 N ) ζ 689 384 a MS ( OS 3 a 1 23 ]. 32  29 64 − 32 128 ( MS-scheme was obtained in ref. [ MS β 24 581 a − MOM 22 +  N − ( − − a 3 (1 + Π( MOM 1 3 ( + − =  ζ   β MS k a/ 3 8 N d + + + 1 3 MS 2 ) = + 4 =1 5713 1728 k X 69 128 MS  ) = , d a ( − + 3 4 4157 2048 MS MOM = = = β ), evaluated previously at the same level in ref. [ a ) = 1 + ( ] from the two-loop approximation of the photon vacuum polarization function a , a 3 MS 4 MS 1 MS ( -lepton): 2 79 d d d τ MS, MOM and OS schemes. Note also that in the MOM scheme, determined by NS MS /µ MOM 2 The four-loop expressions for the QED To transform the results of eqs. ( or β D Q subtractions of thenon-zero UV Euclidean divergences momentum, of thefunction, the which QED photon governs the vacuum energy polarization behavior function of the at QEDknown the invariant and charge. have the following form: and take into account theQED properties invariant of charge the RG-invariance andto scheme-independence of the the the results of calculations,term independently was performed obtained in in ref. refs. [ [ following RG-based relations Note, that the first two scheme-independent coefficients,in included ref. in eq. [ ( Π( three-loop coefficient in the The energy behavior of the QEDby coupling constant the four-loop expression of the corresponding e, µ JHEP02(2018)161 , ) in 3 (2.13) (2.12) (2.11) N (2.14c) (2.14a) (2.14b) (2.14d)  5 1.13a ζ 3 5 + N 3 ], and applying  ζ 5 ) were calculated 86 ζ N 4 ) and taking into – )  5 2 54 203 , 2 84 3 2.11 OS 2 + ζ 2.13 + ] correspondingly. The N a 192 3 ( N  ζ 3 83   )–( N ζ 2 3 37 24  972 ζ 6131 7 N 35 96 ) and ( ζ 9 − 2.12 19 and [ ], while the full three-loop cor- log 2 + 7 − 18 − 4 9 7 4 2 ] − 2.10 13  ζ 105 . − . − 2 80 5 + 4 + N, ) log 2 54 N − ) 2 151 5   2 2 ζ 1 OS 3 ζ N  32 ζ OS ζ , a + 4 3 a 5 4  ( 8 of the QED PT series for eq. ( 4 ( + 3 − 2 3 + ) 215 )  + ζ k  + 3  N 2 d 2 8 − 11 + ζ N  N 3 + 5 5 5 6  77 MOM ζ MOM 576  ζ − ζ 3 3 a 3 a 9 8 ) ζ , 5 – 12 – ( − ( ζ (that reflects theoretical freedom in fixing scale  6 k   7 OS − + ) 96 7 125 N 2 1 64 + a 48 3 + 2 OS  ( ζ N  − + 3 3 + 3  m ζ ) 32 N 339 256 2 3 MOM 2 + 3 ζ ζ 1 4 2 ζ + 4 a ζ −  − OS 4 9 = ( 2 3 N a 8 + ] three-loop analytical result coincides with the one, previously com- 3 + ), the obtained expansions ( ) were evaluated in refs. [ = + ( 32 157 − 35 48 2 + 3 80 ) ), one can obtain the following transformation relations: 23 N  ζ 128  − MOM k ]. 3 8 2 MOM 2.11 2.6d d OS + 15 16 + µ − MOM 2 901 2021 2592 695 648 a 1296 84 2.11 + ( 4 =1 scheme-independent contribution to the three-loop corrections of )–( 275 384 +   k = − − X 69 N 128 )–(    + + , d 1 3 N MOM OS ) and ( ) was calculated long time ago in ref. [ 2.6a − + a + a + 2 MS 4157 2048 3 4 2.7 MS and MOM-schemes in QED) and taking into account the scheme- ] and [ µ = = ) = = = = 22 2.10 2.11 = ) = 1 + OS ]. MS MS 2 a ( a a 82 Q MOM 3 MOM 4 MOM 1 MOM d d d OS ) and ( a =1 the obtained in ref. [ β ( N 2.10 Fixing NS MOM For 7 D puted in ref. [ quantity, we get thethe values MOM-scheme: of the coefficients Using expressions ( consideration that the flavor NS Adler function is the renormalization group invariant parameters of the independence of the constantvacuum polarization term function of included the inthe the single results QED of lepton-loop invariant eqs. charge contribution ( [ to the photon The proportional to eqs. ( rections to eqs. ( analytical expressions for thein four-loop the contributions works to of ( [ JHEP02(2018)161 ) 3 . )– )– a d ( are , 2 OS 2 k O d m d 2.12 N (2.15c) (2.15a) (2.15b) (2.15d) 2.15b =  high tran- 7 are scheme- log 2 ζ 2 MOM 2 µ ) and ( ζ NS 3 2 = and D − 5 MS, MOM and OS- 2.14d ζ 2 MS 7 , ζ µ , 2 3 )–( -functions, presented in ζ 4 N β 105  2 3 + ζ MS, MOM and OS-schemes, N 2.14b 5  9 ζ 19 2 -contributions to the QED PT 3 0 ), ( ζ 4 − N 115 terms in the PT coefficients + 3 -function in the 2.6d − 54 N 5 151 ]). corrections to the coefficients of ζ 3 NS N, ζ )–(  17 , D N 6  3 + 125 3 ζ 537 256 N 2.6b N +  + + 3 5  ζ ζ 5 2 8 ζ ζ – 13 – 11 5 3 5 15 16 − + ) QED PT contributions to − 768 2 3 17089 3  + a ζ ζ ( , + − k 4 O 55 32 54 19 ) 203 2 2 3  ζ 32 OS + + a − + + 1 4 ( 3 =  ζ OS k )-corrections in the case, when we choose coefficients in eqs. ( 3 8 11 972 d 144 6131 + a OS 2 are the QED running coupling constants of the MOM and OS ( + 4 -function. − − =1 NS O k X 69 128   D NS OS , d a D − + + 3 4 4157 2048 ). = = = and ) = 1 + 2.11 . This interesting feature is not yet understood. OS 1 OS 3 OS 4 4 OS d d d d a )–( ) terms for the QED expressions of the ( 2 MOM a a NS OS ( ) in three gauge-invariant schemes mentioned above. to the QED scendence contributions to the proportionaland to independent. the consequence of the scheme-independence of the leading renormalon contributions expression for the NSof Adler consideration function of follows the fromquantities pqQED the (for CS, approximation detailed of which explanation is the see valid PT ref. in series [ the for case the RG invariant 2.10 )) relations between QED running coupling constants of D Note also the scheme-independence of the proportional to As explained above, the The scheme-independence of the leading on The scheme-independence of the proportional to O Let us comment the observations, which follow from the comparison of the analytical It is worth emphasizing that the analytical expressions for the coefficients of the 4. 2. 3. 1. 2.13 2.15d ( which do not contain expressions for the ( tion, with the MOMeqs. and ( OS four-loop expressions of the QED and schemes, are the same. This circumstance is a consequence of the presented in eqs. ( where schemes, which at the studied by us level are the solutions of the corresponding RG equa- and in the physically motivated OS-scheme: JHEP02(2018)161 ], 48 N (2.17c) (2.18c) (2.16c) (2.17a) (2.18a) (2.16a) (2.18d) (2.17b) (2.17d) (2.18b) (2.16b) (2.16d)  log 2 2 ζ N 2 3  5 + ) PT QED approxi- , ζ 5 4 2 ζ a ( 24 N 205 24 O . 205 − 3 N 115 216 3 −  ζ N 3 5 − , ζ ζ , 2 96 N 3 547 605 972 12 N  125 N 768 3 , + + 4373 ζ , MS-scheme results given in ref. [ 2 2 − 115 216 + 5 605 972 2 3 N 12 N N 2 ζ , − , ζ  2711 2304 + − 3 + k 3 2 115 216 )  24 ζ N 16 15 133 N ) QCD  N 21 32 , + − c 29 24 MS 3 133 576 − +  3 MS, MOM and OS schemes in QED ζ a , , 2 N 3 605 972 = ( N ζ N + − k ζ – 14 – 5 3 8 ) 12 2  N + + MS k ) between the QED running coupling constants of  3 576 MS 2 2304 5315 , 3 c 1421 2 − ζ − 865 768 265 576 + k + ) SU(  21 32  ) N 4 4 s MOM =1 + 2.13 + − k X a a , c  21 32 + MS-scheme: + 3 = ( 269 288 OS ( 7 9 2 3 3  2048 3 4 128 4823 2 3 6 a ζ )–( ζ O ζ = ζ ( − − − − − + 3 8 3 8 − − OS k  MOM k  OS 2 MOM 2 = = = 2.12 c c − − ) = 1 + + + 4 4 =1 =1 4 MS 1 MS 3 MS k 2215 1728 X , c k 2951 3456 X , c MS c c c 3 3 ) approximations to the QED analytical expressions for the non- a  3 4 128 4823 2048  3 4 128 4823 2048 4 ( a − − − − − − + + ( MS , O ======) = 1 + ) = 1 + -function in the MOM and OS-schemes: NS Bjp OS 1 OS 3 OS 4 OS c c c C NS Bjp a MOM 1 MOM 3 MOM 4 ( MOM C c c c a ( OS , NS Bjp MOM C , MS, MOM and OS schemes, we get the following analytical NS Bjp C Applying now the relations ( the mations for Consider now the singlet coefficient functioninterested of in. the Using Bjorken theone inverse polarized can sum find the rulepolarized following sum in analytical rule three function expression in schemes for the we the QED are corrections to the Bjorken 2.3 The NS Bjorken function in the JHEP02(2018)161 )- )– a ( (2.19) (2.20) (2.21) (2.22) NS 2.3b D ] pqQED MS-scheme N of the RG )-function, 10 a  1 ( 2 3 coefficients of β ζ 4 NS Bjp c N ) PT expressions C + 9  and 5 5 ζ ζ 0 and 2.11 8 β ) level in the MS-like . 3 2 2 4 s 7 c 125 125 ), ( non-abelian contributions to a ζ N, )-function in the , ( s 2  A -terms: + + a i c 3 O ( C 3 3 ζ 2.10 -factorization in the class of and K ζ ζ NS Bjp 3 β 5 19 ), ( C MS, MOM and OS schemes 8 ζ 16 917 343 , − 3 2.7 ζ − − in the system of equations ( 24 2 163 ), obtained in section 2.1. It is clear β  144 7729 1152 1793 + 2.3d 5 − − ζ   )–( 15 -function by the overall sign. Keeping this in ) QED results of section 2.2 for the + + contributions to the , , – 15 – β 4 − 7 7 2 2 2.3a ) expressions for the , a ζ ζ 3 c N ( ζ 3 MS, MOM and OS-schemes and substituting the N N ), which is valid at the N ζ O 4 4   2 17 315 315 5 5 ζ ζ + 3 1.12 MS, MOM and OS-schemes differ from introduced in + + + MS, MOM and OS schemes. The fourth observed feature 5 5 8 + 5 + 5 )–( 21 ζ ζ 96 3 3 397 − ζ ζ 8 8 )-function are also valid in the case of the 4) in the 715 715 1.10 a = = ( 18 18 )-function we are able to verify whether the conformal symmetry 203 203 ≤ − − a NS OS OS 1 2 3 3 ( k + + ζ ζ MS-scheme to MOM and OS schemes will not spoil the structure D K K NS 8 8 Bjp of the GCR obeys the property of the 61 61 ≤ = = 18 18 C 307 307 + + csb − − MS, MOM and OS schemes and the presented in section 2.3 similar PT -functions in the β MOM MOM   1 2 ), which is one of the CS relations, encoded in the studied in ref. [ 768 768 2471 2471 K K + + (with 1 in MOM and OS-schemes. ) do not contain high-transcendent functions ]) , which is not studied in this section. = = = = 2.3a k 1 c 48 ) determination of the QCD RG -function and scheme-dependent coefficient d MS MS MS 3 1 2 β ), we obtain the following analytical expressions for the 1.9 1.13b MOM Note, that the used in this work and defined in eqs. ( K K K 3 These high-transcendental terms are contained in the proportional to and and 8 K k 1 2.3d the corresponding coefficients of(see the ref. [ PT SU( mind, using the given ind section 2.2 and 2.3 analyticalcorresponding QED expressions expressions for for the twoQED coefficients scheme-independent coefficients ( that eq. ( variant of the originalc Crewther relation, is valid for the scheme-independent coefficients for the QED eq. ( the MS-like schemes only, orschemes, it namely MOM is and also OS fulfilledtransformations schemes. (at from To least understand in thisof QED) we in the assume, that other GCR in gauge-invariant givenschemes QED in the for eqs. sure ( and leads to the eqs. ( Having now atfunction hand in the the analytical expressions for the breaking term ∆ the PT series forconsidered the in the gauge-invariant is violated. Indeed, theeq. proportional ( to 2.4 The QED generalized Crewther relation in the Note, that three from four noticed in section 2.2 properties of the analytical structure of JHEP02(2018)161 ) c N (3.2) (2.23) (2.24) N  2 3 ζ ξ , N. A -function in the schemes, namely  + 9 β Z 3 1 5 ζ − ζ ξ MS-scheme: Z 16 -function in the GCR. 2 135 125 β = − B + ) color gauge group. c 3 of the renormalization sub- ζ term are scheme-independent N ) (3.1) 945 128 renormalization schemes, such ]. Among the number of MOM ,  8  g, ξ csb term only: 391 g gauge-invariant 92 + MS s – Z − N ε MS, MOM and OS-schemes, differ α ( 87 µ OS 3 ]. It is widely used in different QCD- = K MS cg 576 8117 70 Z B = − -function factorization in the GCR. N β ) =  , g gauge invariance  a ,  2 3 gauge-invariant . + c ζ – 16 – c 7 2 to the GCR in QED is valid at least at the fourth in expansion of ∆ ζ Z 2 N MOM s + 9 p csb 4  K α 3 315 ( 5 ζ = ζ + a B 16 and 5 231 condition for factorization of the + 5 ζ 1 3 mMOM cg − K ζ 8 , c Z 715 a µ ]. In this section we will use this mMOM-scheme in the SU( 18 A 735 128 203 − A 100 3  + Z ζ – necessary -contributions, computed in the + 8 p 3 61 93 18 , 307 K . This is the most important requirement which allocates this scheme = + -factorization property manifest itself in the generalized 49 D − MOM 3 β −  K a B, µ 768 2471 A + = = 4 = MS 3  MS (and therefore MS-like), MOM and OS schemes. Moreover, we discover that in OS 3 K K Crewther relation in the gauge-dependent schemes in QCD? -function in the GCR takes place in the where 2 among all the variantsculations. of MOM-like Let schemes us in fix the QCD following and notations greatly for simplifies the concrete QCD cal- renormalization constants: 3.1 The remindingThe notes basic on requirement, which the is miniMOM lyingthe beyond scheme renormalization the constant definition of of the mMOM-like gluon-ghost-antighostto schemes, vertex the is is renormalization that fixed constant by the of its the equality same vertex, computed in the scheme, which was originallyoriented formulated studies in [ ref. [ theory to find outexistence the of constraints the imposed fundamental property by of its the gauge-dependence on the conditions of It is knowngauge-independence. that However, in in theoretical QCD andgauge-dependent the MOM-like phenomenological schemes QCD MS-like are applications used subtractions the subtractions as schemes well schemes [ are one distinguished of by the their most applicable at present is the miniMOM (mMOM) important problem arises, namelytraction whether schemes is a This problem is studied below for the case of3 QCD with SU( May the Completing this section weβ arrive to theas MS-like, conclusion MOM, that OS-schemes really in the QED and factorization MS-like of schemes the in QCD. However, another order of massless PT at leastin in the three widely used inQED QED the first two coefficients and find that the from each other in the linear in the number of leptons These results are convincingconformal symmetry us breaking that term ∆ the factorization property of the JHEP02(2018)161 = . 2 c (3.5) (3.6) (3.7) (3.8) (3.3) (3.4) ) are Z Q 2 2 µ / ( 1 A ξ Z g = Z , , ξ is the coupling =   ) ) g , cg in two considered ,  ,   ) ) Z . c ξ ξ ν ( ( Z 2 q q B B µ , q , ξ , ξ ) ξ ) = 0 ) ) 2 and 2 s s µ − µ a a ( A ( ( ) − Z 2 B s B s MS s q a a = a ( MS-scheme. ( ( 2 A 2 B A B c 1 q  ( ) renormalization constants are related 1 + Π MS 1 + Π 1 + Π c Z  mMOM   c 3.2 mMOM c ) ) ν Z ) and the gauge parameter Π 2 q 2 q µ  µ q , ξ,  , ξ,  ( , s s s ) a a , + – 17 – a 2 in the mMOM scheme the requirements for the ( ( MS A q ( µν 2 Z mMOM = A c MS-scheme but with a replacement of the mMOM g µ enters in the ratios of 1 ), one gets the following relation between coupling Z s ) = 0 − a − 2 mMOM mMOM A c 3.2 µ MS = Z Z ) = is a scale parameter of the dimensional regularization. As ξ 1 + Π  − 2 2 µ 2 2 µ ab ab q = ( MS schemes: q q δ δ ) = ) = ), 2 ξ q , ξ , ξ ) and ( − − ( s s (2 a a / ) are the gluon and ghosts self-energy functions. ( ( 3.1 mMOM s 2 2 ) by their analogies, defined in the ) a ) = ) = q 2 a µ ( q q ” denotes the bare unrenormalized quantities. One should emphasize µ c ( ( mMOM A A ( B µ ab Π µν ab mMOM A mMOM c ∂ , ξ ∆ G ) 2 are the renormalized gluon and ghost fields correspondingly, µ ) and Π ( 1 + Π 1 + Π a s 2 is the gauge parameter, which is included in the QCD Lagrangian in the form q a the residues for the gluon and ghost propagators are equal to unity. The renor- ξ ( , c 2 A a µ µ = 1 in all orders of PT. The defined in eq. ( A ξ = Combining the given above definition of the renormalization constant of gluon-ghost- The similar relations hold in the At the subtraction point In general the MOM-like schemes are determined by the requirement that at Z 2 q where the gauge-dependence on schemes. One should emphasize that thisconstants relation requires of knowledge of the the renormalization gluon and ghost fields only, but not of any vertex structures. Using quantities antighost vertex with ( constants of the mMOM and where the QCD couplingrenormalized in constant the mMOM-scheme. The relation between unrenormalized andpresented in renormalized the two-point Green following form: functions can be where Π residues of these propagators imply the fulfillment of the following conditions: to the renormalization constant of the gluon-ghost-antighost vertex as − malized two-point Green functionsthe for following the form: gluon and ghost fields can be written down in constant, of additive term ( usually the index “ that we work within the theoryply with linear covariant gauge and this fact means that we im- where JHEP02(2018)161 f n (3.9) F 2 A , (3.10) T 2 2 f C (3.11c) (3.12a) (3.11a) (3.11d) (3.11b) ξ 2 F  n 3 3  2 , , C ] with ex- F ξ 3 ξ  4 T 2 ζ ξ  5 )  5 ζ 3  101 ]: ζ 25 81 9 256 5 5 2 )) 128 ζ 2 + − µ + 5 − ( 100 f 2 mMOM s 3 , 3072 (see appendix A): ξ ζ 295 n a 1024 ( MS F  + 93 37 24 3 , T 143 + , ξ 2 f 3 ζ 6912 ) ζ F n 3 − 2 70 ζ 2 C F 3 − µ mMOM , mMOM 3  128 ( T ξ 59 b 3  A  1536 ζ 143 288 + 611 2 MS s = + 1536 C + ) a − +  − 3 ξ ( ξ 2 ) − .  ξ  ξ 55 48 203 ), we obtain the following relation MS c 3 ) 2304  + ζ  2 233 and 3 25 4608 mMOM s  were calculated in ref. [ µ 432 3 ζ A a + ) ( 3.10 4877 mMOM s 11 ( 3 8 2 + C 36864 − 144 . f − a 1 + Π  MS c µ ]. Using results of these computations in ξ MS ( n  ( 3 f 4 ξ )–( +  ξ +  ξ F n ,   mMOM s  3 + + 3 T 100 MS s 3  F 3.9 f ζ f a taken into account. The analogous four-loop )) 5 55 ζ )) mMOM 5 2 a A 2 T 128 n 2 n ζ 2 ζ η ξ 1 9 and Π C 11 64 mMOM 2 µ F 833 F µ MS (  b  − MS and mMOM-schemes reads: ( 13824 T + T – 18 – ξ − 5 2 125 729 − MS A A 5 9 ζ f ξ   + 35 MS MS 7805 n C + 2 12288 12288 5 + 19 + + 72 ) , ξ F F , ξ 216 2 f ) ) 5 5 889 T 175 + A − n C 2 2304 6144 2 ζ ζ − mMOM s 2 − A 2  3 C 3 µ F µ 5 4 5 6 a  ξ 2 ζ ( ζ (  C + T  ξ 2  5 3 mMOM s F − − + 36 ξ 4  MS s ζ MS s a + 5 3 3 3 C ξ 3 a ( a 1 ζ ζ ζ 3 ζ  ( 6144 − 16 ( 7495 ζ mMOM 1 3 18432 47 768 3 3 5 ζ η 16 2 − − MS A 39 ζ MS A 13 36 107 144 223 384 128 + 2304 ξ − ), one can obtain the following relations [ 11 18 + + mMOM 1 8 1 + + − + 1 + + b − 3.8 3  235 − 1 + Π 55 1 + Π + ξ 36864 256 107 648   199 17315 − − 1235 5 110592 − 15552 2592 20736 31104 144 18941 20736 1935757 2985984 1249 25547 169 1152         mMOM ) = ) = ) and ( − − − 2 2 ξ mMOM s + + + + + + − + + µ µ   a  3.7 ( ( = = = = = ) approximation and the expansions ( ), ( coupling constant, expressed through MS 3 s ξ a MS s ( 3.6 mMOM s mMOM a MS s ξ a O a mMOM 3 mMOM 1 mMOM 2 b b b In the necessary orderbetween of gauge accuracy, parameters, required defined for in our the goals, the corresponding PT relation The three-loop results for self-energiesplicit dependence Π on the gauge parameter results were obtained in the recentthe work of ref. [ for eqs. ( JHEP02(2018)161 ] f n 100 ]) the , (3.13) 2 f F (3.14c) (3.12c) (3.14a) (3.14b) (3.12b) n T 93 = 0 and 90 2 F 2 . A , , ξ T C 2 f 2 89 A A  n MS analogue 4 C C 2 F ξ 2 MS   T ξ 4 3 mMOM s 6 ξ 1 F ∂ξ ζ  768 3 C ∂a 1 ζ +  ) 256 − 3 3 ζ 3 , and in refs. [ 13 23 96 MS + 512 ξ f  3 − ) were presented in the n , ξ , evaluated at + ξ 1 + 128 . MS s 72 23 f 7 a f  256 + n ( 3.12a n 725 6144 2 F mMOM + 2 F + ξ T MS ξ β f T ) color gauge group with explicit 2 − 2 γ  F c f n ξ F and coincides with its C n  3 N F C MS 3 ζ ) color group. In our further analysis F 7 T 1  128 ξ ) and ( c ξ + 32 256 T 3 F ζ ξ N F + − C + +  C + ]. The corresponding expression for the A 3 3 ξ A 3.11a 1 4 ζ , C 23 ,  51 C 256  55 48 f 3 MS s  − f 2 ζ  n 4 33 ξ n mMOM s 512 2 − A 1 8 ξ ∂a F F + – 19 – 3 to its analog in the MOM-scheme. C  T 64 ∂a + T + ξ  ) A for the SU( 5 9 55 3 + MS-results for this anomalous dimension are known + C 6144 ξ MS f 37 MS s  ξ − ξ 384 and number of active flavors MS 1 n 2 a 121 + 64 1097 6144 ξ ] in the case of SU( 1536 ( 1 A ξ c F 16 + 3 C T + − ξ N 1 70 3 contains cubic term − 24 MS  + A ζ 2  2 β  ξ  3 3 C ξ + ζ ζ ,  + 5 ξ + 1 137 384 96 3 -function can be computed with the explicit dependence on f 16 2 3 ) = is the anomalous dimension of gauge, which due to the Slavnov- 11 12 3 ζ 3 1 β n ζ mMOM 24 512 1 ζ − − 2 + F β + − ) color group in ref. [ µ ξ 3 T ξ + − 16 c ] in terms of 143 512 1 3 1 8 N mMOM 5 13 − log 2009 1152 641 576 12 192 371 576 − − + 100 21 , ξ 1 only, while the three-loop coefficient -function starts to depend on the gauge from the two-loop level: 2048 − − − − − A ξ/d β −     C  97 144 5591 4608 , 17 24 9655 4608   + + + + + log 11 12   ] and [ s mMOM d = = a = 0 = = = ( ) group when the corresponding Casimir operators were kept non-expanded on 93 , c = ), was presented in ref. [ using the following relation: ξ N 70 γ -function with the arbitrary gauge in mMOM-scheme, that can be obtained from 3.13 β mMOM mMOM 2 1 The mMOM-scheme mMOM mMOM mMOM mMOM mMOM 2 0 1 η η ξ β β β β mMOM The two-loop coefficient at dependence on number of colors for SU( number of colors. AsmMOM-scheme in all previously used MOM-like QCD schemes (see e.g. [ Taylor identities coincides withwith the the expression opposite sign. for The thefor three-loop gluon an anomalous arbitrary dimension SU( RG taken eq. ( ξ where One should noteworks that [ expressions similarwe to will use ( the results,dependence totally on related to the the gauge mMOM-scheme, parameter transforming everywhere the JHEP02(2018)161 ], 5 ζ 2 4 3 F A 48 ξ ξ C C 55 24 ]. We    F (3.15c) (3.15e) 2 (3.15a) 5 3 (3.15d) (3.15b) 3 f − ζ ζ C ζ -function n 51 3  3 8 , ζ β 4 F 32 33 , ξ A 165 256 T 50 + f 828805 147456 2  C A A n 5 − − 3 F C C − ζ F 1536 3 5 18929 C F F T ζ ζ  A 4157 2048   C C F ) approximation 2 −  C 5   35 4 s + ξ C ζ 2 3 2 F a  2 16384 + 3 ξ ξ 5 2 3465 1024 8192 3695 ( 3 C ζ  2 ζ A  − + + 207 + O 3 2 3 2048 9 3072 C 8 3 + ζ 50575 ξ 3 ζ ) level we need to analyze ζ 2 1024 F ζ 256 4 s  + 4411 24 23 a − 3 C ξ ( − 2639 4096 − 256 + 93  + 4 − 2 8192 O  2 f 2048 5667 7 ξ ξ 23 + 13 16 process in the mMOM-scheme:  192 n ζ − + 207 + the values of the gauge parameter  ξ 2 1024 F 7 +  MS-scheme, presented in ref. [ 3 − 3 2 ζ T ζ 64 A 3 385 µ +  8192 + F 128 C 503 7 − ξ C − − 32 F ζ 49152 185 512  3 + 1155  49152 abcd F 5 C = 171455 3  c ξ 3 ζ hadrons d 5 ζ +  4 + 2 ζ ζ 16 2 N + − 5 q 1155 ξ 3 − 3 ζ → 4 − abcd F 55 ξ  4096 3 3 2 MS analogue, presented in refs. [ d γ − ) expressions for the non-singlet contributions 64  1273 3072 c +  f + 5 23 96 5 + 64 49152 – 20 – ξ ζ ζ n − 3 2 N 5575 → . 2907785 − +  ζ ξ ξ 2  + 2 + f − 3 µ  )-function in the  5 + − 3 3 e ζ ), we obtain the following analytical expression for 32 n 32 s , 5  16 ζ 5 4096 3 2255 143 ζ + 5 a + F k ζ ζ ( ζ e ) − T ξ + 5 6 121 128 + − 5 4 3 2 3 64 F NS  ζ 165 3.12c 3 ζ + 3 + C ζ − D 5 6144 4 ζ −  11 3 ζ 509815 )–( 3 3 5 ζ 768 4 mMOM s ζ 1355 24576 ζ ζ 103 139 128 − a 2048 1504181 5 − 143 256 ( 89 12 − − + − 4405 4096 − − + 3.11a  + 569 192 3 3423 2048 3 16 + ζ  + 3  + ζ + mMOM k 835 3 F 73728 d 2048 192 12288 2063 3072 2 + 4 F 12305 1850345 2033 C abcd A C c 49152 697 4 , − − =1 d 2048 − − − 5674729 29 96 k X 6144 3 69 N ) result for the F 15845 32 128         4 s C abcd F a + + + + + + + + − − + ( d 3 4 O = = 1 + = = = 1, differs from its gauge-independent − In order to study what will happen with the property of the factorization of NS mMOM 4 mMOM 1 mMOM 2 mMOM 3 mMOM = d d d d D Using the and the results ofthis eqs. Eucledian characteristic ( of the the product of the analyticalto general the SU( Adler functionThese expressions and will Bjorken be polarized obtained below. sum rule in3.2 this gauge-dependent The scheme. NS Adler function in the mMOM-scheme in the remind here that at fixed renormalizationand point the coupling constantfrom are their fixed initial normalization and values start to run whenin the the GCR energy in scales the move gauge-dependent away mMOM-scheme at the ξ JHEP02(2018)161 - . 5 R ζ 2 f 3 f ξ n 5 n 2 36. F ζ 12 ]. The  695 3 (3.16a) / F 5 T , f 5 ζ T 70 32 2 F n + F 8 = 5 C F 15 − 3 C  ζ 3 c T 2  3 ζ + 2 A 5 + 18) ζ ) approxima- 3 ζ /N C 2 c ζ 4 s )-approximation + 5 3 and other values F 197 128 256 4 s 3 2 a N 5 12977 MS-scheme to the C a ( abcd F f 6 ζ + − (  d 3 − n c 3 + 4 O 2 − O 25 ζ ξ N 4 7 6 c abcd F  29 6409 3072 + 256 96 3 d N ]. + ζ 3 ,  ζ − ξ , k 2, 96 1)( + ) 1 from the / 18432 67 24  1 2 , defined in this renormaliza-  ξ 256 =3 the numerical expressions 2027833 3 24 ξ − 281 f ζ  − + = 2 + = 5 c 2 n MS s 3 − 7 2  − 3 ζ a 24 c F N  ] and [ ζ mMOM s ( + a 11 32 + 4 23 + 94 /N ( 11 f 125 384 6144 ,T f − = n 13 24 c 5 n − )-function in the − F − abcd A ζ s F 7 N d T = 0 with −  a ζ 5 T  ( abcd F 3 16 F mMOM k c ξ = A d  c + C 8 + abcd F − N C NS Bjp A 105 2 f  d 3 3 + MS-scheme results and the presented in sec- 4 =1 2 7 F n ζ C ξ abcd F k X ζ 2 ] 3 2 − C F d 2 – 21 –  ζ = 0 in refs. [  5 ,C 3 T 4 48 2 ζ 493 256 ξ ζ = 3, A , 105 1 + ξ c − C A 5 ) version of QCD. Its analytical expression can be 1  − + = 1 + c ζ 16 F 5 128 C N 2 5 ζ c 1085 2 N C 2 + 6) ζ 35 − 3,  ) are consistent with the ones, obtained in ref. [ N 2 15 16 c − 2 / 8 68479 18432 − ξ 3 N 235 + ζ = 3 mMOM   23 ζ , = 4 48 3 3 3.15e 3072 − F 1)( + ζ ζ 2 F 3 3 32  C 3 4 NS Bjp 32 3 − ζ ζ )–( 429 2109 16 C ) in Minkowski region and related to Euclidean Adler function, C 2 + c − 8 17 − + + 2 64 1.3 N 489 ξ ( + 29 9 3.15b ) theory we have:  − 512 32 c 2 7 3 = ζ ζ N − − 2304 4608 (3) case 73339 287 256 48451 c 3 35 16 64      abcd A c + + + + + + + d N abcd F d tion The corresponding expression for the PT coefficient function of the polarized Bjorken In the particular case of the Landau gauge sum rule, calculated in the mMOM-scheme, reads in the mMOM-scheme inobtained the from SU( the summarizedtion in 3.1 transformations ref. of [ gauge-dependent the mMOM-scheme with QCD the coupling gaugetion constant parameter scheme. was analyzed in the mMOM-scheme at 3.3 The NS Bjorken function in the mMOM-scheme inLet the us now turn to the calculation of the for the coefficients ( detailed study of theof numerical behavior gauges of will these beratio, PT presented series defined below. for in other eq. The ( scheme-dependence of the PT series for the while in the SU Note that in SU( JHEP02(2018)161 ], f 3 A n 33 ζ F C ξ f T 3 F n (3.16c) (3.16e)  (3.16b) (3.16d) 2 A 5 C F ξ f 2 f ζ C  T n  49763 73728 2 n 2 F 3 5 3 F F ξ ξ 2 16 F ζ C + C T T  3   + A 3 9 9 2 A ξ f 5 ζ 3 ξ 512 ξ C ζ n 2048 C  ζ F 5   F F − ζ 5 95 12  + 3 C T 49 96 ζ C 121 5 ζ 16 1024  2 A ξ −  ζ 442368 2 3927799 5 2 3 − C 5 2 − ξ + ζ ξ 349 4096 )-approximations of the F 9 3072 − 4405 4096 1 4 s − 96 , C 5 1024 13  a  − 24 192 3  109 128 ( f − ζ 3 4 − + + 8257 + n  18432 ζ 3 O ξ 1339 5 + + 2 ξ 12288 A ζ F ξ + ζ A + ξ T − C 1 ξ  C 48 1 − 59 2 3 F F  317 144 13 16 2  F  2048 ζ 48 3072 3 C   C 3  − 1623 2048 C 1045 8 ζ 12 ζ  + 5  − + 1 + 4 + 33 12 + ζ 2 + 2 256 3 − 3 ξ A 3 + 7 3 ξ abcd F ζ 5 6 c ξ ζ ζ + d C + . 13 96 N 151 F 11 12 − 21 A 21 3072 11 256 192 1 3 f  16 96 3 8192 C 385 971 abcd F C  n ζ +  1536  d + − F 3 215 + 107569 147456 4 3 8 F 7 − 3072 7 − f + 3 ξ ζ ξ C T + ζ −  5 n ξ  2 + ,  F  2 ζ 2 3 4 ξ 35 16 – 22 – 5 6 2 35 ξ + 2 f 21 + ζ C + ζ 128 ξ 9  n 13 2 21  3  5 + 5 − 13307 18432 5 2 5 ζ − 4096 3 F 5 72 ζ  ζ 5 144 − 64 5 ζ 1375 ζ ζ T 5 ζ − 3 35 − ζ + 4 5 + 55 24 5 F ζ 96 + 2 32 − 121  16384 f MS-scheme structure of the GCR, discovered in ref. [ 2 f ξ 15 16 72 C ξ 3 36 205 35 8192 3695 + + n + 11 12 n + + ζ  − 5 3 3 3 2 + 4 F 3 3 24 F + F 32 4 3 − − 7 ζ ζ ζ ζ − 3 T 3 ζ T ζ C 3 3 ζ ζ 2 − F 2 +  F ζ ζ + 3 5 6144 2 C 117 512 33 A C 64 90169 ζ 512 415 385 72 85  3  8192 2048 11 C 3 8 16 1415 2304 288 512 512 3 − 192 107 1739 3 1237 3 + − F + ζ  + + + ζ 36 − − − − C 5 − − + ζ +   − 3  3 F 41 869 , ξ 864 9216 3072 1117 6144 1283 4823 2048 13 36 20585 864 C have the following form + 49152 6229 abcd A F c − − d 1728 221184 k 9 49152 28303 2 − − − − C F 1891 3456 12265 2543485 3 − − c N 345755 147456 128 1024 3 4            C  abcd F + + + + + + + + + + − + + + − 32 d 21 + = = = = mMOM 4 mMOM 1 mMOM 2 mMOM 3 c c c c We will use the analyticalThe first results, problem presented we in areples section interested of 3.1, in gauge is 3.2 quantum field whether and theory thewill 3.3, unexplained be for essentially yet two modified from purposes. in the theis first mMOM-scheme. related princi- to The the second comparison problem of is the more asymptotic landed behavior and of the Its coefficients JHEP02(2018)161 1 ], K 106 (3.17) (3.18) mMOM k , c -function 105 ) = 0. 3 was first can not be q ( − csb µν ab = mMOM G β ξ µ q ), which follows from -function is factorized can be factorized at any ) imposes certain restric- 2.3a ) we conclude that in the MS-scheme. Note also that csb mMOM 2.3c β 2.3b . F C ]. That is why the renormalization  MS-scheme and the mMOM-scheme , 3 102 ζ = 0 3. The extra more detailed theoretical = 0 is often used in multiloop calculations + 3 ξ 1 (more precise of the class of gauges with − ξ 8 − 21 , MS expression, namely + 3 1 from section 3.2 and for the coefficients 3 are less studied. However, some attractive = 4. The anti-Yennie gauge = − 2 ) mMOM-scheme coefficient in ∆ − − – 23 – 3 s ξ ξ f and in several fixed gauges.  a . Indeed, equation ( n , ( f ξ = = + 4 n O mMOM k ξ 3 ], where it was shown that for these values of gauge the = 0 d ξ ] to clarify the special features of renormalizations of )-functions in the 88 s ξ a ( thanks to the fulfillment of eq. ( mMOM 3 we find that the two-loop 1 104 , ξ 1 and K ) for any NS Bjp ≤ − C 103 k ]). Its most vivid feature is the validity of the property of the non- = 1.11 ≤ ξ and coincides with its 100 , ) and s ξ ) we find that the a 96 ). Taking first into account the relation ( ( , NS 2.3c 94 , D 3.16e in the mMOM-scheme is chosen the same as in the 93 , )–( cg Z 70 mMOM-scheme 3.14a 1) was noticed in the work of [ ) approximation the conformal symmetry breaking term ∆ Other two gauges It should be stressed that the Landau gauge 2 s a ( | ≤ ξ used in QCD bygauge-invariant Stefanis definition [ of QCDline. quark correlator, This formulatedwhere gauge with it the was was help independently demonstratedrenormalization of that applied constant when Wilson of later this the gauge on gluon is field by chosen is Mikhailov proportional the to one-loop in the correction refs. first to scheme-independent [ the features of using in theMOM-schemes PT with QCD anti-Feynman gauge expansions of| measurable physical quantities theone-loop class of QCD corrections toof MOM-scheme Green effective functions, charges, are defined rather as small the at combination (see e.g. [ renormalization of the gluon-ghost-antighost vertex [ constant in the Landau gaugeand the therefore longitudinal its part PT of approximation the has renormalized a gluon transverse propagator structure, vanishes namely namely for threeclarification specific of values these foundations will be presented below in the separate section. represented in the formtions ( on the factorization conditionsanalytical results in for mMOM-scheme. the coefficients Taking intofrom account section the 3.3 concrete within 1 the GCR only forthe certain following equation: values of gauge parameter, which are fixed by the solution of does not depend on Using now eq. ( is factorized in theeqs. ( GCR. This problemO will be analyzedvalue of using the the gauge analyticalthe parameter results property of of scheme-independence of this relation. This implies that the coefficient for different numbers of quark flavors 3.4 About the gauge-dependence of the generalizedConsider Crewther now relation the in question the whether (or when) the RG gauge-dependent PT series for JHEP02(2018)161 ) ) 3 ), ), c s − a N ( = 0, = 1.12 ξ NS 3.16e ξ D at )–( NS Bjp -function also C 3.14a β ]. In the mMOM- ) and the one is not 48 of three-loop mMOM of the renormalization 2.3d )-approximation and are 4 s a 3 of the gauge parameter. -contribution (see appendix ( f − ) approximation of the GCR. O n 3 s , a F 1 ( 5, while the results, summarized T coefficient, included in eq. ( , − O A 4 , , C only. ) theory in the conformal symmetry F c = 0 C = 3 ) mMOM expressions for )-approximation for two basic functions gauge invariance N mMOM MS-scheme in ref. [ 2 4 s ξ 4 s f a a ) in the K n ( = 0 will be also presented in section 4. ( partial factorization also contains these six color structures and O property for the factorization of the SU( ξ O 3 1.11 K MS-results in the – 24 – ] for the case of the Landau gauge. It was also 3 only 70 ) coefficient of the GCR is more complicated. We − 4 s MS-scheme ) and the analytical mMOM results of ( a -term contains six color structures, first revealed in -term at ( 3 = O K partial factorization -function of SU( 2.3c ξ ]). From phenomenological point of view one may realize β mMOM 3 108 K , of the 1 and ξ -function. In what follows we call this anti-Yennie gauge − 107 β )-expression of the generalized Crewther relation in QCD, presented = 4 s ), which enter the GCR and both were extracted from the concrete of the mMOM-variant of the GCR takes place. The corresponding s a ξ ( a ( O csb NS Bjp C -function in the CSB term of eq. ( is the sufficient but not a necessary β ) and s loop level: the mMOM vs. a ( These results are compared with the Thus from the study of the structure of the GCR in the mMOM-scheme in the Landau In the case of The dependence on Taking now into account eq. ( NS -function in the -function obeys the property of -function in the analytical expression for the CSB term of the GCR is observed. Indeed, within mMOM scheme with three specific values presented in the tableagree 1. with the Note, ones,recently that checked obtained that the in the mMOM-scheme given ref. in numerical [ the results table 1 for Let us study theD asymptotic behavior of the experimental data (see [ that these studies are ofabove may interest attract for definite interest the to values the behavior of the corresponding PT series, obtained β in the concrete renormalizationsubtractions schemes, of which renormalizations are of fixed the by Green kinematics functions,3.5 related conditions to of the the QCD Asymptotic vertexes. behavior of the NS Adler and Bjorken functions at the fourth- B) Therefore for theseβ two gauges the three-loop approximation ofgauge the we come mMOM-scheme to conclusionschemes that the property of scheme when the Landau gaugewe is conclude chosen, that inpersists in this the case GCR. total However,only in factorization five the from of case these of thedetermined, six anti-Feynman three-loop namely and color the Stefanis-Mikhailov QCD structures coefficient, gauges may proportional to be found from eq. ( analytical expression of the β the analytical expressionthe of case the of application of the gauge-independent mMOM The concrete results and the analysis of their structurefound will that be for presented the inthe case section of three-loop 4. the mMOM-scheme Landaubreaking gauge term the fundamental ∆ property of factorization of as the Stefanis-Mikhailov gauge. specified to theobtain cases the of explicit analytical the expressionsfor Landau, of these three the anti-Feynman gauges, which and do not Stefanis-Mikhailov violate gauges, the property of we the factorization of the two-loop coefficient of the QCD JHEP02(2018)161 s a s 4 4 s 4 s 4 s 4 s 4 s 4 s 4 s 4 s 4 s a a 4 s 4 s a a a a a a a a a a ) at the NS Bjp s C 74950¯ 40202¯ a . . 96225 34185 54404 49192 09174 54819 14145 95977¯ ( 61660 57175 ...... 3 7 22 12 31 18 44 31 21 175 102 41 NS MS-scheme (¯ − − − − − − − − − − − − D 3 s 3 s 3 s 3 s 3 s 3 s 3 s 3 s 3 s s 3 3 s 3 s schemes a a a a a a a a a a a a ] demonstrate the functions, evaluated 3 and MS 33 5099 3566 4262 0375 2312 8633 0317 5170 9269 2153¯ 8503¯ 8402¯ . . − ...... NS Bjp , 1 7 C 1 20 13 + 4 + 3 + 1 + 3 + 0 + 1 − + 0 + 1 − − 2 s 2 s − − 2 s 2 s 2 s 2 s 2 s 2 s 2 s 2 s , a a s 2 2 s a a a a a a a a ], based on the QCD calcu- and a a 785 972 = 0 896 840 083 028 333 278 222 917¯ ...... 110 NS 583¯ 250¯ 0 0 ξ 0 0 1 1 0 0 0 . . 2 )-function [ 5 are more pronounced, than in 3 3 D s , − − − − − − − − − − in mMOM and a 4 s s − − s s s s s s s s ( , a a s s ¯ a a a a a a a a ¯ ¯ a a − − The flavor NS Bjorken function NS − − − − − − − − Bjp 1 1 = 3 − − 1 1 1 1 1 1 1 1 C ) even at more high level of the related 1 1 f s n a ( 4 s 4 s 4 s 4 s 4 s 4 s 4 s s 4 4 s 4 s 4 s 4 s = 5 in the latter case these features are even a a a a a a a a a a a a MS-scheme calculations of the Borel image NS f Bjp NS n C D – 25 – 71854 25793 28190 39217 90258 10284 39054 07570¯ 38880¯ 12575 38863 21018¯ ...... + 3 + 6 + 9 + 11 + 18 + 13 + 18 + 15 + 23 + 28 + 49 + 27 3 s 3 s 3 s 3 s 3 s 3 s 3 s 3 s 3 s 3 s 3 s 3 s schemes a a a ) PT expansions for the a a a a a a a a a MS-results, presented in table 1, give extra argument 4 s MS-scheme). a ( MS 8241 3579 6814¯ 3). However, without any estimates of the numerical values 8133 6039 2862 0159 1797 3517 3251 3710¯ 7586¯ . . . O ...... 4 4 0 − 5 6 5 6 0 1 2 ] and the one for the − − − − − − − − − − + 6 + 2 = ]). The concrete 2 s 2 s 2 s 2 s 2 s 2 s 2 s 2 s 2 s 2 s 2 s 2 s 114 a a a ξ a a a a a a a a a 048 860 409¯ 113 885 723 698 535 610 448 285 640¯ 525¯ . . . , ...... 5 active flavors in mMOM-scheme with 1 0 , 0 0 0 0 1 1 1 ], are consistent with the results of ref. [ 4 ) approximation. The inconstancy of signs of PT series in the mMOM- − − + 1 ]. − − − − − − − + 1 + 1 , in mMOM and 3 s 112 s s s )-function. Moreover, for s s s s s s s s s a a a a s = 3 and 109 a a a a a a a a a ( ) contribution and for the The flavor NS Adler function 112 a = 3 4 s f ( O )-function [ a 1 + 1 + 1 + ¯ n f The existing at present point of view on the structure of the asymptotic PT s 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + ¯ 1 + ¯ ( n NS a 9 ( O D ]. f 3 4 5 3 4 5 3 4 5 3 4 5 NS n . The comparison of the ) with D 111 s a )-approximation for all considered values of ( The comparison of numerical ξ 0 4 s -1 -3 — We are grateful to G. Cvetiˇcfor the confirmation of this agreement. a 9 ( NS Bjp the case of violated at the scheme is observed for bothC physical quantities (apart of the case of mMOM-PT series for PT expansions [ in flavor of this renormalon-motivatedsum guess. rule the Indeed, sign-constant in patternO the and the case related of asymptotic the growth Bjorken of polarized the coefficients of series by the correspondingsions ultraviolet renormalon see (UVR) reviews effects [ for (for the the detailed discus- interplay of the IRR andlevel above UVR effects are manifesting itself in the case of in ref. [ QCD series for the physical quantities,MS-scheme defined the in coefficients the of Euclidean these region, PT(IRR) indicates expansions that generated in obey sign-constant the the factorial patterncompete of growth infrared up with renormalon to sign-alternating the factorial level, contributions, when generated they in are starting the to related PT QCD in QCD with corresponds to the calculation in the presented in ref. [ lation within the variant of Analytical Perturbation Theory (APT), developed previously Table 1 JHEP02(2018)161 : ξ in NS Bjp C (4.2) (4.1) ] and csb ) and 115 . ) 3.14a ]. , determined in i mMOM s ), eq. ( is increasing from a 117 K ( f ) for any value of ) allow us to find all n 3.11a 4.1 ) approximation of the 3.17 ]), which should include 4 s mMOM . a ( K ) 110 MS PT approximations for s ) O a ( K coefficients are much smaller than MS-scheme. Naturally, the equal-  mMOM s ) 4 s ) coefficients are decreasing modulo a s 4 s a ( s a a . a ( mMOM s ( -factorization of the CSB term ∆ a β β O and MS 1  . Indeed, the eq. ( 3 s mMOM ξ K a β , = ), lead to the conclusion that in the GCR the 2 s in the arbitrary gauge ( a – 26 – )) = ) = 1 + 2.3d s -factorization in gauge-dependent mMOM scheme. a ( β mMOM 1 )–( mMOM s K NS a Bjp mMOM s a 2.3c C the values of the ] and in the most detailed work of ref. [ ( ) ) corrections we can not make definite conclusion whether s f 5 s -function factorization property at definite orders of PT: a n a MS s β ( 116 ( a ( O NS -function in the conformal symmetry breaking term is possible through MS D β K MS s MS-scheme. This difference may be essential in the process of study -factorization of the generalized Crewther relation in the a )) β ) ) 4 s , for which the factorization is possible. ) we reproduce the result, presented above in eq. ( ) QCD in the case of application of the gauge-dependent mMOM-scheme, ξ a mMOM s c ( = 5. The situation for the Bjorken polarized sum rule function is somewhat a 4.1 N mMOM s ( f O ) level of PT QCD the GCR relation is defined as a 4 s ( n MS s a ( ) and is not satisfied for any values of a MS s ( )-function of their mMOM-scheme analogs are growing when ) is valid only under the assumption of the O a s mMOM-scheme a MS 1.11 ( 4.1 Now we find out the criteria for β = 3 to NS f these values of Using the expansion the relation ( This equation permits usthe mMOM-scheme, to and obtain their the analogs,ity evaluated relations ( in between the coefficients eq. ( only for the certain specificdetail values we of write the down gauge the parameter.gauge following parameter To equality, study respect which this allows the property us in to more find out what values of the As already mentioned above in sectionmMOM-scheme 3.4, with the using direct eqs. computations, performed ( factorization by of us in the the RG Let us return toGCR in analysis SU( of thedescribed analytical in section structure 3.4. ofat the We the remind that in the class of gauge-invariant MS-like schemes next-to-leading order in ref. [ 4 The generalized Crewther4.1 relation in the The MOM-like schemes in QCD in both schemes. Onecatches the more eyes: interesting the feature absolutethe of values of ones, the the obtained mMOM-scheme in PTof series the for scheme-dependence uncertaintiesexperimental of data for the the possible Bjorkenvirtual polarized new heavy-quark sum more massive rule effects, careful (see calculated ref. analysis at [ of the leading the order of PT in ref. [ in the mMOM-scheme theor considered sign-nonregular asymptotic structure.D PT Next, QCD on series then have contrary sign-alternating to the different: here with growth of of the unknown at present JHEP02(2018)161 (4.4) (4.5) (4.6) (4.3) coeffi- , if the 0 -function, /β β f , namely for mMOM 2 -factorization ξ n β K A A F A C C T C F F A . F C C C to the GCR in the ) C   of the RG 2 3 3  MS 1 ξ ζ ζ 0 3 can be found from ζ csb β K 8 4 + MS and mMOM schemes, 91 47 2  25 ξ ( ) and has the following solu- + + MS + 2 1 24 , K 3.18 identically and therefore the difference 96 mMOM 1 3 − 192 48 b 2591 1327 695 ξ 2 A MS 1 1 − − − β 64 C + 2     . − 3 + + + , ξ 2 f ξ 2 -function in the n 2 2 2 F F F 1 ξ 64 β F 5 f f f C C C mMOM 1 96 T 1 0 n n n    24 β − 5 5 5 A β ) analytical expressions of the F F F + 2 ζ ζ ζ c − T T T + ξ – 27 – ξ coincides with θC N F F F 15 15 15 ξ is some real number. Using the explicit expressions 5 does not contain terms proportional to 1 1 3 96 C C C MS 1 13 1 θ − − − 24 192    β 4). Thus, we demonstrate that the − 3 3 3 3 3 3 + / ζ ζ ζ ζ ζ ζ  mMOM 1 2 = = the relation with ξ A 3 3 3 3 β , 2 2 2 ) and looks like as: θ θ + 17 17 17 3 mMOM 2 − − − θC 1 3 13 192 − 11 12 , where + + + K 4.1 ( MS 2 8 8 8 12 11  A 31 31 31      , K    C ), ( 96 96 96 = = 0 397 397 397 0) mMOM 2 ) in the mMOM-scheme, have the following form: = + + + ,    is proportional to the leading coefficient 1 θβ K 1 this conclusion also follows from the fact that for these two values of the = = = − − 3.14b 1.12 = ( , ) is nullified. 3 1 , mMOM 1 − − 4.3 ), ( ) approximation of the CSB contribution ∆ mMOM 2 β =0 = = mMOM = 0 1 0) 3 s The corresponding SU( , β K a , ξ , ξ , ξ ξ − mMOM mMOM mMOM ( 2 2 2 mMOM 1 − 10 O 3.14a K K K β MS in eq. ( 1 3. − β MS 1 ), ( − β ) = (0 , MS 1 1 β mMOM 1 ξ, θ − β , 3.11a − In the case of = 0 10 MS 1 gauge parameter the two-loop coefficient β tions ( property for mMOM-scheme is possibleξ only for threecients, values defined of in eq. the ( gauge parameter This system leads to the presented above single equation ( Hence, we obtain the following system of equations: difference namely for the one and two-loopwe coefficients get of the the RG following equations eqs. ( In this equation the coefficient For the coefficient JHEP02(2018)161 , - ) 1 2 0 f )– − n 4.6 /β (4.8) (4.7) F 4.4 = )) T -term, )–( 3 ξ F MS 1 K C ) level at mMOM 1 3. There- 4.4 β 3 s b ) − a 3 only, since − ( 3. Moreover, -factorization = -contributions − O − β coefficients: f we mean either , ξ 0 n 1 = β mMOM 1 F X MS 3 − mMOM 1 β ξ T 2 β , -expression and the K F ( 1 and ). Indeed, due to the − C ). The reason for this = 0 − and is equal to zero in case of 4.3 ξ MS 1 = 2 mMOM 0 2 -factor, in eqs. ( β -function in the QED-limit mMOM 1 2.20 ξ 2 F K β β /β (3 ) level in the mMOM-scheme C A 4 s )) + a C mMOM 3 ( 2 ), can always be chosen to be zero F MS to mMOM-scheme in the next MS 1 K and ( f O C β n 0 MS 2  − 7 /β K ζ mMOM 2 0 β  ) from 3 F β 8 105 C − -term to the 4.1 )-approximation of the GCR reduces to in- mMOM 1  A mMOM 1 4 s MS 1 − 7 β b MS a 2 C ζ ( 5 ) mMOM 1 ( , is nullified (here under the K β ζ b F -factorization property holds at – 28 – X  1 O 4 β C ) + 315 β . However, in Landau gauge the sum of these two + 3 ξ 48 2 − ) 3685 + mMOM MS 1 1 mMOM 1 β -structure. However, as one can see these β 5 3. β − ζ f 2 MS 1 − − n 3 β ). Thus, it is clear that the question of factorization of ζ − 8 F mMOM 1 = mMOM 1 715 0 T b β 4.3 8 ξ β MS 1 2 ) level. We study this problem for F 0 (and as a consequence to 451 − 4 s − β β ) is quite understandable. Other typical features of eqs. ( C + ( 3 a mMOM 1 . Nothing like this is observed for + ζ ( ) we notice that this relation contains three new additional frac- 0 β MS 1 -function in the ( , (3 8 β O 61 2.20 0 β β 4.3 -factorization property holds at mMOM 1  β b /β + mMOM 2 ) 4608 b + = 0 only and it is not performed for the anti-Feynman gauge 132421 2 mMOM 1 ξ β   1, and differs from zero for Stefanis-Mikhailov gauge 768 MS 3 2471 − + + + mMOM K 2  β = = = − ξ 3 this fraction is irreducible. The remaining two fractions can not be divided − MS 2 =0 β , ξ = mMOM mMOM 3 3 Summarizing the foregoing, we get the following analytical expression for Considering now the basic transformation ( ) are the gauge-dependence of the ξ K K = 0 and 4.6 computed in the Landau gauge: at individually at all considered valuesfractions of are divided by fore we find thatin the Landau gauge and Stefanis-Mikhailov gauge in sum or separately,property in then the we GCR can at at confidently these state values the ofleast. existence gauge The of extra parameter appearing the the fractionξ ( Comparing it with eq. ( tions ( which did not enterthe into three-loop eq. mMOM ( vestigating the divisibility of these fractions. Indeed, if all these fractions are contractible are different. order of PT we arrive to the following relation between difference becomes clear uponscheme-independence careful of consideration the first ofthe two the contribution, coefficients proportional eq. of to ( theMOM or RG OS renormalizationterm, schemes, proportional defined to in QED).in Unlike QED the due QCD to case the corresponding the normalizations. second That is why these in the QED-limitschemes with in QED the in eq. results,( ( obtained ingauge-independence three of gauge-invariant the renormalizations contributions differ from its QED analogs, included in eq. ( The coincidence of the analytical terms, proportional to JHEP02(2018)161 , ) → AS 1 . MS 1 β ) (4.9) β 2 f ) ap- (4.10) in the − 3 s n − 1 2 a F b ( MS AS 1 1 T β β mMOM i F O ( C , b AS 1  ) in QCD. To . 5 β renormalization ζ X 2 AS X β mMOM i + 5 β 0 in the corresponding 2 − 3 A β ζ -factorization property , C f 7 2 β MS F 2 n X 3 β C F + β MS-scheme with arbitrary 1 T 0  ) can be rewritten in more 6 − 2 f 3 β A K 49 ) are valid not only for the ζ 2 n C 4.7 ]). F − MS 3 F 4.7 + 3 T β 72 C )) and the special feature, that in ) contributions to the GCR obey  128 A ), ( 2113 4 s (the more detailed clarification and ), ( = C a + ( QED − 3 4.3 F (3) QCD GCR in the f , . Therefore, the differences , calculated in the c MS 4.3 5 − O 3 3.11a C n ζ N = X 3 QED K F  -function in any renormalization schemes , 2 3 , ξ -factor, and the expression T ), ( K X 3 mMOM 3 β ζ 0 = 2 48 F β 5 2 2975 K C 4.2 , b +  and + ) monomials QED 5 – 29 – c 5 3 , ζ X 2 and ζ X ζ 3 X 2 N β 2 1 75 K 3 0 − − 125 β 3 the mMOM = + in these expressions can be represented through values 3 i 3072 − − , ξ MS 2 ζ mMOM b ,K 3 152329 -function, namely 3 β ζ β = K + 24 599 = ξ -factorization only. It is violated by the extra non-factorized QED 24 539 , β MS − 2 − -function factorization in the GCR for the concrete choice of the K -function of QED, the discussed above possibility of fixing the QED β , are always divided by β . This means that observed in section 2.4 X 2 18432 = 1 and 144 2 0 b 1273 1840145 ) are in full agreement with the results, obtained in section 2.4 (see AS β 1 − 1152 β − − 71251 2 = QED 4.10    , − X 2 ξ + + + K MS 1 -function factorization of the SU β β (the QED analog of the QCD relation ( , 3 ). X AS 2 AS i β proximation in the MOMgggg-scheme , b Moreover, in the case of QED, the relations ( Note, for the QED case all coefficients of One should emphasize that relations ( − AS i MS 2 β 4.2 The It is important to findthe out property whether of there the RG are other MOM-schemes in QCD, which respect scheme, have the following form: The expressions ( also the unpublished work, presented in the talk of ref. [ QED the second and thirdof terms corresponding coefficients of we get that in QED the coefficients compact form. Actually,coefficients taking of into the accountvalues RG the of scale scheme-independence parameters, ofexpression which the are which first leading relates to two thescheme nullification the QED first coupling coefficient constants in the are proportional to the number ofβ charged leptons is always divided by of the QED version of the GCR will be valid in all renormalization schemes in QED. mMOM-scheme, but and forachieve this any goal MOM-like it renormalization is onlyby schemes necessary ( to the replace all corresponding quantities,( calculated quantities in mMOM in scheme, any other MOM-like scheme ( For the gauges the property of thecontributions, partial proportional toexpressions the for SU( the coefficients derivation of these statements are given in the appendix B below). JHEP02(2018)161 ]. , f 118 3, but n (4.12) (4.13) (4.14) (4.15) (4.11) √ . /  = 3 and ) c iz  9  16 -function in N 9 9 832045 and 16 , 16 β . ] 2 , 9 , , 16 -function in this , 9 118 9 16  β 16 1 -function has more ≈  (3) group when the   β Φ 1 c 1 1 3  Φ Φ -function factorization 1 9  4) β  / 3 11331 12800 , 6400 33993 -ratio in the MOMgggg-scheme, 4 + 25600 / 373923 R + ) level  (3 2 arccos 3 s − 3 4 1  2 arccos a  ( 2 3 4  ,  2 4 3 3 4 O , 3 4  , 1 + Cl 4 3 3 the factorization of the  16) are expressed through the Clausen + Cl Φ 1 / −  3 8 1 9 Φ  , = 9 4 Φ 3  . + ) level. However, considering the MOMgggg 1 ξ 16 2 3 3 s

√ + 16 99  / a 2 x  ) approximation for the GCR in the MOMgggg (  4 3  – 30 – 3 s (3) case, we obtain the following relation: (9 − ] at the two-loop approximation in terms of pow- c O a 1 4 3  ) approximation for the fixed anti-Feynman gauge (  3 s  2 sin O 3 4 118 a

log ]. (  4), Φ log 2 arccos O 2 arccos / log  3  118 log 400 2 , , 1179 2 dx 4 200 3537 / 94 ) in the SU − Θ 2Cl 2Cl 0 (3 −  800  Z 1 4.3 , we are convinced that the 38907 5 2 30 c 4 107 11 − √ √ 20 + N 333 − = = −  ] as: 80 403614.   = . + -function in the 4173 ) and taking into account the explicit form of the RG 3 4 3 (Θ) = 9 118 β )-functions. 16 ], were expressed through more familiar functions such as the derivative of the logarithm , 2 = , 3 ≈ 3 4 4.3 94 Cl − x, y 9  ( 16 =3 = 1 (Θ) [ 1 c 16) 2  Φ , ξ / 1 0 MOMgggg 1 ,N 9 β 3 Φ β , − − 3, we get the following analytical expression: 16 = / Initially there was no indication that at In general the two-loop coefficient for the MOMgggg-scheme QCD Using eq. ( − , ξ One should note that the analytical three-loop QCD results for MOMgggg 1 and contains additional transcendental logarithmic and Clausen-function terms [ MS 1 (9 11 β = 1 β 4 computed in ref. [ of the Euler Γ-functionnot and through the Φ imaginary part of polylogarithm function with argument exp( Φ the GCR will beanalog valid in of this the scheme r.h.s. at of eq. ( The numerical values of these extra terms are Φ where the contributions Φ function Cl ξ complicated analytical expression thancoefficient, its calculated mMOM-scheme in analog.ξ the Indeed, MOMgggg-scheme, the depends two-loop The on complicated the structure remains fourthStefanis-Mikhailov even power gauge in of the is case gauge chosen. of Indeed, QCD fixing with in SU the results of ref. [ ers of number of colors property is also valid inviolated for this anti-Feynman scheme gauge. for Therefore thetion we Landau conclude of that and the the Stefanis-Mikhailov QCD property gaugesis of and the the factoriza- is peculiarity of the mMOM-scheme. scheme, defined by renormalizationultraviolet divergences of in the the quarticconcrete symmetric calculations gluon subtraction in vertex refs. point, through [ investigated subtractions and of used inMOMgggg-scheme, calculated the in ref. [ gauge parameter. Let us consider the JHEP02(2018)161 , 3 2 − K . . = f (4.17) (4.18) (4.16) n  ξ f     n 9 9 16 9  9 16 9 16 16 16 -coefficient  , 9 2 , 16 , , ,  9 9 K 9 16 16 9 , 9 16 9 16 16 9 16 = 0 and 16  , 9   16  1  ξ 1 , 1 1 9 1 16 Φ  Φ Φ Φ 9 1 Φ 16  MS-scheme, but and Φ 1 7 ) into the MOMgggg-  96 1 Φ 7 ) level. Therefore, it is 313 96 1600 3 s 7219 6400 50533 51200 2191 + Φ 1 12800 a 12 4.15 ] for relating the coupling + ( + 1 +  − -coefficient. Following the 12 − + O 1 3 4     β 118 + 3 4  4 3 ,  3 4 3 4 3 4 3 4 3 4 ,  , , , , 3 4 3 4 , 3 4  f , 3 4 -function factorization in the GCR 3 4 1 3 4 n 3 4   β ,   -factor without any residue. As was 1 Φ 1  3 4 1 1  0 19 24 1 Φ Φ 1 Φ β Φ  49 96 Φ 5 4 1 Φ − 49 96 85 32 − Φ 7 35 32 + 12 595 256 8 − 51 7 +  12  − -function are nullified and we arrive to the + − 4 3  4 3  = β + 4 3     4 3  – 31 – 4 3 4 3 4 3   =3  c  4 3 log   log log  ,N = 0 all transcendental functions included into the log 49 24 MS-expression for the log log log =0 ξ 49 24 50 log 7 3 − 25 1967 , ξ 862 MOMgggg MS-schemes, taking into account equations from the ap- 1 7 3 − -term, substituting it and eq. ( ) is divided by the + β 100 400 − 36 65 3017 − + 13769 -function holds in the GCR at -function coincides with the one of the  65 36 9 10 β β +  9 + + 10 45 − 2108 5 + 360 ζ − 15973 5  MOMgggg 1  MOMgggg 1 ζ b 3 3 3 270 ), we find the following analytical expression of the coefficient   9337 80 β ζ ζ 8640 3 3 3 280073 80 ζ ζ − + − + 4.3 − − + − + = 3: MS 1 = β − =3 c =3 ), namely = c ) level. Using now the two-loop results of ref. [ ,N 3 s ξ 3 a ,N − ( = =0 3.11a O , ξ , ξ MOMgggg 2 MOMgggg 2 K K namely the mMOM and MOMgggg schemes,the allows factorization us to of discover that theimportant at RG to understand why thisloop happens coefficient not of only the forfor the QCD Landau the gauge, Stefanis-Mikhailov where gauge the as two- well. Next, in section 4.1 we have shown that in the Thus the consideration of completely different gauge-dependent renormalization schemes, which as expecteddescribed is above equal considerations, to wein also the the obtain Landau the gauge: following expression for In the casetwo-loop of coefficient the of Landau thefollowing gauge result MOMgggg RG of eq. ( version of relation ( defined at the difference ( already discussed above, this feature isat essential for the the constants of the MOMggggpendix and A, getting the analytical expression for one-loop coefficient in the MOMgggg-analog Thus, like in the mMOM-scheme, in the MOMgggg-scheme in the Stefanis-Mikhailov gauge JHEP02(2018)161 - 3 3 12 β − − 1 as = (4.20) (4.19) = − ξ ξ = -function -factor. ) that at 0 MOM-like β ξ β . Indeed, as ) level in the ξ ) that in the 4.20 all 3 s a ( = 0 and AS 4.20 ξ O → MOM-like schemes MS all , which respect the

ξ ) level in 3 s -term. AS s MS a ( ∂a ∂ξ /∂ξ O ) 3 is the one-loop coefficient 1 / , f MS ∂b n ) -function in the GCR is always , ξ scheme. Next, since at ξ F β ( T 1 ∂ξ MS s a + ∂b AS ( ) will always be divided by , and taking into account the formulas ) A ξ ( AS 1 MS ξ +1 C k β γ ) 8) MS 0 -factor. It is clear from eq. ( − / MS 0 , it is obviously from eq. ( AS s ξγ ξ 0 β a ) level, depend on the concrete renormalization MS MS 1 β ( 3 s – 32 – − + ξ k β a − b ( )-level we should consider the issue whether the 3 s MS 1 AS s MS s O =1 a β k ( 24 + ∂a 3 = ∂a / MS-scheme, through the coupling constant of another O = / ) P ]. f 13 n + MS s AS 1 − -function factorization will take place in MOM-like schemes a F β 106 ): ( , T β AS s a ) is divided by = ( MS -factorization property is always valid at the 105 3.13 β β = -function factorization property is valid in the mMOM-scheme at AS 1 12 + β β 3 the one-loop expression for renormalization constant of gluon field is propor- MS 0 / MS s γ − ) = -function factorization may also hold at other values of ) and the arbitrary formal PT series, containing the expansion of the A − a and therefore the factorization of β = C 3 the AS ξ − MS 1 11 4.19 scheme with linear covariant gauge. Note, that these statements do not , ξ and MS 1 β − β = AS s AS a ξ AS = ( = ξ ) approximation at least. Moreover, we may make the guess that in the higher denotes any MOM-like renormalization scheme with linear covariant gauge. 4 s was noted in refs. [ 0 AS a = γ ) level. These foundations give us the hint that for the gauges 0 AS β ( s 4 β β AS ξ a O MOM-like schemes ( O The fact that at -scheme, namely = 0 12 well. Now we can makefunction more factorization property strict at statement the thatscheme other and values are of determined by the special behavior of the tional to Stefanis-Mikhailov gauge the expression ( Therefore at GCR in the contradict that the was already demonstrated above this feature holds in the mMOM-scheme at sign. As follows fromfactorization section in 4.1, the toexpression GCR study for the at validity ( the ofξ the property ofpossible the in thewe Landau have gauge at this level in the where of the anomalous dimensioncorresponding of coefficient gauge. of Due the to gluon its field linear anomalous covariance dimension it but coincides with with the the opposite where Considering eq. ( coupling constant, defined inAS the of appendix A, we get: The summarized above assumptionssimilar can to be the proved one using of eq. the ( following relation, which is in the case ofin Landau more gauge detail. as well. In the next4.3 section we will Special study features these of assumptions the Landau and Stefanis-Mikhailov gauges in the QCD the the fundamental property ofschemes, factorization while will for be the Landau true gaugein at this the the property will beorders fulfilled in of PT the property of Landau gauge only the JHEP02(2018)161 ]. ), . ): MS .  4.1 101 )  (4.23) (4.24) (4.25) (4.26) (4.21) (4.22) ξ ( 4.21 (0) . ∂ξ 4 MS 0 b ) level. , and using -function in 3 4 s ∂γ k MS 1 , a β ) ) level in the ) ( − 4 s ξ  K  ( a AS s O 1  ( a (0) (0) ( (0) O 2 2 3 ξη k 3 b b (0) η b 1 − 0 b − . ), we find that β − ) =1 MS-scheme in all orders k ξ  MS 1 ]. Since we have already ( . (0) 1 β (0) P ∂ξ 3 1 (0) + 4 4.23 3 1 (0) b 1 93 b 2 ∂η ) , b + MS 1 AS b ) ξ − ξ MOM-like scheme with linear − ( ξ 70 K (0) 2 ( − AS 2 ∂ξ 3 + β (0) b (0) ∂b (0) 2 MS 0 AS 0 2 2 ) (0) b − AS b β b 2 1 ξ ξγ ξ ( b ) into eq. ( (0)  MS 2 (0) 1 + 3 0 = )+ MS 1 0 b β (0) + 6 β ξ b 2 2 1 4.22 ( ) has the following form: 3 ξγ b MS MOM-type scheme at the 2  ) and taking into account the relation ( MS ] arguments in favor of the validity of the  0 ξ MS − 1 K 4.7 β (0) 54 γ MS (0) + 1 2 (0) + AS  4.26 – 2 1 b 1 β (0) ) − b – 33 – b 1 ) ξ 52 b ( 11 )–( ξ + 2 ( MS 1 from eq. ( b 1  (0) + 2 − -function does not occur at the β b MS-schemes: 2 1 + 3 ) allows to confirm the results of calculations of the ) β 4.24 b − (0) + ξ (0) =0 + 2 ) (0) ( 2 1 b MS 4 1 ξ 3 , ξ MS b 1 ), ( 2 ( 4.21 and AS b 2 β MS MS 2 2 1 K 0 4  β b β γ  = 4.22 0  + AS 0 = β β ) + (0) + 2 (0) + ), ( AS 3 MS 3, at this PT level the Landau gauge will be only applied. At ξ 2 2 -function in the mMOM-scheme [ 1 =0 ( -function for the GCR in gauge-invariant -scheme we obtain the following simplified version of eq. ( b b K β − ) proves the validity of the factorization of the RG 1 K β 4.3 , ξ  ) + η AS 2 (0) + ξ 0 MS 2 AS = 1 ( β (0) MS β ), ( 2 β b 1 ) is the two-loop coefficient of the anomalous dimension of gauge [ 1 4.24 -scheme when the Landau gauge is chosen: ξ b β ξ b  MOM-like scheme, namely ( ) MS 4.2 3 + 2 + ξ MS 1 AS β ( + 3 MS 1 β 1 ∂ξ AS γ = 0 the analog of the relation ( MS MS 3 4 ), we obtain the following expressions for the four- and five-loop coefficients + , + 3 ∂b MS β β ξ 3 ξ AS K = = MS 2 ξ 4.19 + β = = =0 =0 = ξ , ξ , ξ AS 3 AS AS This fact leads us to the natural assumption of the realization of the property of the Taking into account the PT relation between gauge parameters, defined in the 3 4 -function in β β AS 2 K β = 0 in an arbitrary β -function factorization in the GCR, evaluated in the Keeping in mind thefactorization of presented the in RG refs.of [ PT, using eqs. ( relation ( of Thus, the relation ( the Landau gauge in the GCR in arbitrary β covariant gauge in any order of perturbation theory in the Landau gauge. Indeed, using the Substituting the expression for In the limit three-loop coefficient of found out that themMOM-scheme at factorization of ξ Here The application of the relation ( and arbitrary the transformation equations fromthree-loop appendix coefficients A, of we the obtain the following relation between JHEP02(2018)161 ) 3 s a ( 3 and (4.27) (4.28) O − ), which condition = MS 2 ξ 1.12 K  1. However, in -function in the − ) approximations β (0) 4 s 2 = a b ( ) do not contain the ξ ) of eq. ( , O (0) s 1 a b 4.28 ( -function factorization is MS 1 β K K )–( (0) + (0) 3 3 4.27 b b  -function holds in the GCR at the ) analytical -function factorization property of β c + 4 + 5 N sufficient but not necessary MS MS β 2 MS 3 K K   ) expressions for the GCR with (0) c 2 1 (0) -function factorization only, whereas in Landau N b 2 1 MS-scheme persists in all orders of PT, then it – 34 – β b -factor. Therefore we conclude that in the MOM- 3 gauges. We also find out that in the mMOM- 0 − ] variant of the GCR, with its conformal symmetry /β 57 = (0) + (0) + 2 2 2 = 0: ξ b b ξ 4  -factorization property of the GCR is valid at the -factorization property remains valid in all orders of PT in  )-coefficients of the polynomial β β ) the obtained expressions ( 6 s -function factorization in the conformal symmetry breaking + 5 + a β ( 4.7 O MS MS 4 3 . ), ( K K -scheme at 4.3 MS 1 (0) (0) ) and 1 1 K 5 s b b AS a ( ) level. There are no obstacles to obtain similar expressions for any value (0) + 5 + 4 6 s 4 O b a ( -function in the GCR in the MS MS 5 4 O β + 5 K K ) approximation the mMOM SU( 4 s = = a 1 satisfy the property of the partial ( ) and AS AS 4 5 − O s 5 a K K = 0 and the Stefanis-Mikhailov ( = We also show that in QEDany the ultraviolet subtraction scheme. Wewill expect, be that true the for similar the QCDbreaking proposed and term, QED in expressed properties ref. through [ and the the two-fold coupling series constant in of powers these of fundamental the gauge conformal theories. anomaly of the RG is also true ingauge-invariance all of MOM-like the scheme renormalizationfor with schemes the linear is manifestation covariant of Landauterm the gauge. of the Therefore QCDfor the GCR. Adler We and obtain Bjorken functions the in explicit the mMOM-scheme SU( with the arbitrary covariant gauge. scheme in this orderalso of valid PT in approximation one more thethe extra property gauge, of namely the anti-Feynman gaugeξ with gauge this problem do not manifest itself. Moreover, we conclude that if the factorization schemes in QCD andwell. QED, Considering but two gauge in non-invariant QCD theschemes, schemes, we gauge-dependent namely mMOM- discover subtraction and that schemes MOMgggg- level the in in QCD all as MOM-like renormalizationξ schemes with linear covariant gauge at the Landau 5 Conclusion In this work wegeneralized find Crewther relation out really that holds true the not property only in of gauge-invariant factorization renormalization of the RG of order of PT.GCR Thus is we most probably concludegauge true that in in since all all the MOM-type ordersfrom schemes of the in PT, first any this principles order of property of perturbative also PT. QCD takes The is place explanation the in of important Landau this opened new problem. feature Unlike the relations ( terms, proportional to the powerstype of schemes 1 in the LandauO gauge the factorization of are defined in the we obtain the JHEP02(2018)161 (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) (A.1) (A.2) defined ) ξ ξ ( 2 1 υ dξ 2 d . ) ξ ) correspondingly,  ( ξ ) 2 1 , ( , ξ i ω (  2  ω 1 ) dξ ) ξ 4 s 4 s 1 2 a dω a ( ( ) − ξ O O ( ) , ) and 1 ξ + ξ ) + . . ( ( ξ 3 s 2 i ξω 3 s dξ (  a  υ a 1 ) ) ) dξ ) − dυ 3 s 3 s ξ ξ ) ) ( a dω a ( ξ ξ 3 ( ( 3 ) ( ( υ b ξ 1 1 O O ( υ 1 + + ) + ξω + , 2 s ξ 2 s 2 s 2 s ξω ( a ) a a 1 ), depending on gauge parameter a ) ) ξ ) ) + ) ξ ξ ( ξ ω ξ ( ξ ξ ( ( 1 5 ) + i dξ ( ( ( 2 2 ξ 2 3 1 2 , ω b υ – 35 – ( dυ − η υ ω ) 1 ) ) of coefficients ) 5 ξ + + υ ξ ξ ξ ( + + ) ( s ( s ( 1 − i s ξ 1 s a 2 a 1 AS, in the following expressions: ( ω η ) a ) a ) ) and 1 ω ) ) ξ ξ ξ ξω ξ − ) solve the stated problem of finding the required coeffi- ( ξ ( ξ ω ( ( ), ( − ( 1 1 i 1 ξ 1 1 b υ υ υ ) ( η ) + ) A.8 i ω ) = ξ ) + ξ b ξ ( ξ ξ ( ( 2 ( ( 1 + 2 )–( 2 1 1 + 1 MS-scheme). 2 ω 1 1 +  υ υ 1 +  s ω   s  a ξ A.5 ) a ξ ξ , η ( ) + = = ) + 5 ) ) + 2 = = 1 dξ ξ ξ ξ ξ s ( ξ ( ( ( s ξ ) in the AS-scheme (note, that in the context of our studies, presented 2 a dυ 3 1 2 a ξ υ ξ υ υ ω ( i − + − − − η ) = ) = ) = ) = ξ ξ ξ ξ AS we imply the ) and ( ( ( ( ξ 2 3 1 2 ( b b b η i b The presented relations ( cients in this work, under theunder AS the scheme we mean either the mMOM or MOMgggg-schemes, and tion equations: Using the Taylor expansion it is straightforward to obtain the following set of transforma- Our aim is todetermined in find another analogies scheme AS at the same order of perturbation theory: us explain this inexplicit details, form considering of the thein following coefficients some example. bar-renormalization scheme Suppose that we know the A The transformation relations In this part we present transformation relations,in which one allow us particular to renormalization obtain scheme, the PT if coefficients coefficients in another scheme are known. Let The authors are gratefulsions. to The G.Cvetiˇc,S.V.Mikhailov work and ofNo. N.G.Stefanis A.K 14-22-00161. for and useful V.M discus- is supported by the Russian Science Foundation Grant Acknowledgments JHEP02(2018)161 ) (B.2) (B.1) 3.14a term at ) level f 4 s 3 , n a 2 f F K ( 3 f 3 A 0 n T f n 2 β 2 C A O 2 F A n f 3 f F F T − C F C n . n ) we obtain the T F C T F 2 F F F F 2 f  2 C A T C C T 4.6 2 C 3 n 6  3  C F 6 ζ A A θ 2 F 2 3 7 θ F C C C ζ ζ T f ), ( 1 3 1 + C 2 3 F F A n θ f 384  C C 4499 96 32 1 3 F C + 629 385 n 5   3.17 T F 2 f θ F 7 7 + + 3 F n − + ζ ζ 2 f C 3 T 5 2 12 11 3  C n A F 5 5 ), ( A ζ 8 − 2 3 ζ ζ 2 35  T F C C ζ 7 16 − -function factorization property 2 T F F ζ 5 6 1155 F = − β 3 2 F 3 f C C 48 3 θ C 5 576 3.16a 4 ξ n 4 3 C 2825 144 + − 5 ζ − 3 1 32725 θ θ 3 105 10505  θ F 5 5 = 1 3 5  + ζ ζ mMOM ξ T ), ( − ζ

11 12 ) must be equal to the + + + 72 3 F 1 3 + + ) level in the Stefanis-Mikhailov gauge: 5 ζ 4385 2 3 f ζ 36 C 25 4 s − ζ ζ 665 96 f 2 f K n B.1  a 2 7865 n n − 2 ( 5 A F 3.15a + β − 2 ζ F F 24 3 T C 715 O 3 768 + ζ 3 3 5 T T 2 ζ + F 2 27673 384 F ζ 3 9216 ), ( A A 20405 − 2 ζ – 36 – C C 376445 + 4 C C + 2 3 72 must contain the same group structures as in the K 3 144 599 θ − ζ θ 2 6077 F F − 1 ζ 144 96 3 1567 671 β C C 3.14c 6 7 + − 61 24 11 12 + K   2 − − A + + 4 6 + − ), ( 2 θ θ C 27648 c A 432 27648 F 1273 49 18 2499749 747967 2 11 12 11 12 C d C 432 110592 6912 3 9216 3 F − − − − 12245 10031531 124009 27181 2471 2304 3.14b − − θ + C         2 5 3 1 − + θ θ d θ ), ( + + + + + + + +  1 3 1 3 1 A c 11 12   ) in the case of existence of the C = + 2 F 2.3d − + + 3 C c 2 2.3d = ), namely 1 θ d 3 4.8 + K + are unknown analytical coefficients. Taking into account expression ( 0 3 F 4 6 β c ) level of the GCR the expression ( C θ 4 s 1 − + θ a − ( 4 1 = d O θ 3. In this case the coefficient 3 − K = where one can find the following expression: According to eq. ( at the ξ Landau gauge ( B.1 The case ofUsing the the Stefanis-Mikhailov gauge relations ( following mMOM-expression, defined at the B About partial factorization in the mMOM-scheme at the JHEP02(2018)161 , ) f c 2 f n n N F 2 F (B.3) T T 2 A A C f C F n F can not be C F 5 T 2 A ,C θ f 2 A f 3 A C . n C n 2 F C F 2 f ) level. However, F F F C T 4 s n T C  , a A A C 2 F  A 7 ( 2  3 C C 2 ζ T 3 f term. 2 ζ C 3 O ζ 3 2 F F A ζ n 2 F f C C 32 F C n 385 32   96 409 T F 24 713 F 7 7 ,C 335 3 3 at F + ζ ζ 2 f C f T − , −  C n − 5 n + A 8 7 5 2 3 ζ 2 35  5 F ζ F ζ 5 C ζ 7 ζ 16 , = ζ T T ζ F 1155 5 6 − 7 2 8 F 3 3 f F ξ 1 1 C ζ 5 105 48 n 4 C 144 48 − + 5 ζ − 2975 3 105 − 10505 2825 θ  F 5 5 576 ,C 4 = − 5 ζ ζ ξ mMOM T 32725 315 2 A + ζ

5 + + + , = 72 F 1 ζ 3 C 5 5 4385 2 3 3 − + ζ 36 C 25 ξ ζ 2 ζ ζ ζ F 96 665 K 3 5  2 7865 12 − ζ ζ 5 2 955 + β 75 − ζ 24 3 ,C . 715 3 + ζ 3 5 3 8 256 − ζ A + 384 9671 term we obtain for Stefanis-Mikhailov gauge: + ζ 5 contribution at the group weights 3 715 8448 3 20405 − ζ 2 ζ C 3 576 376445 + 3 – 37 – ζ 5 + ζ 3 3 26255 72 F ) except the − θ K 3 144 599 − ζ K 6077 1 + ζ 96 144 C 3 + 5 factorization for 0 671 1567 − β ζ 7 6 24 32 − β 3 61 24 599 B.2 + 1855 β ζ 8 − − + − 61 + 7 2 + − 2 + 27648 c + 27648 432 + 2523437 747967 1273 49 18 2 d 432 13824 6912 101376 9216 144 6 − − − − 12245 2304 1287481 124009 27181 2471 10031531 1273 49 ) with + 2304 768         2471 67997 term is incompatible. Therefore the coefficient − − − 3 − d + + + + + + + + 3 B.1  1 = = = = = c K 3 3 3 3 3 = 0 + − − − − − β 3 = = = = = − c ξ ξ ξ ξ ξ 2 3 4 6 1 1 θ θ θ θ θ d in + 2 f in group structures in ( 4 n c i 2 F θ T + A 4 d C F C It is interesting togroup note structures that the in seven this coincidecontributions expression with which among the is the results proportional nine for to Stefanis-Mikhailov possible gauge, terms namely in SU( Similarly one can obtain: and found indicating the absenceone of should the emphasize thatcoefficients there holds a partial factorizationB.2 for five of The the case six of possible the anti-Feynman gauge The system of equations containing the Equating the expression ( JHEP02(2018)161 ] , 3 ). , f ) − n s ( Eur. F B.2 C 40 = R , T Phys. A ξ , Phys. , C , 2 3 F (1991) 144 ]. − (1971) 980 = ,C ξ 6 2 A θ Chin. Phys. and multiloop C D 3 arXiv:1004.4125 SPIRE = ) mMOM-scheme B 259 , 2 F ) [ 4 s in expansion ( IN s 1 C a ]. ( f − (1995) 235 ( ][ R n = O ξ 6 F T SPIRE A A 10 Phys. Rev. IN C (2010) 085 Phys. Lett. , , θ F ][ , 3 C 06 − 5 ]. θ = -corrections to Constraints on anomalies . coefficients: ξ 4 ) 3 s θ i 2 3 α θ ζ in QCD ( = JHEP SPIRE ) arXiv:1501.06739 O , Higher-order QCD perturbation theory in 1 22 [ IN 335 − Review of particle physics = ][ Massless propagators, Mod. Phys. Lett. ξ 4 − The , 5 ζ arXiv:1303.4021 hadrons [ (2015) 3 – 38 – + , θ 48 τ 3 2975 ν − Estimates of the higher order QCD corrections to ]. + = → ξ 2 1. 3 θ ζ − ]. − 261-262 τ (2013) 348 = SPIRE ), which permits any use, distribution and reproduction in = Γ( 1 ]. arXiv:1707.00668 IN ξ 528 − [ [ 26255 SPIRE , C. Patrignani et al., = ξ 2 and IN B 723 + [ ) SPIRE 3 and IN Electron–positron annihilation into hadrons at the higher-loop levels − ][ , θ CC-BY 4.0 3 Nonperturbative evaluation of the anomalies in low-energy theorems = (2017) 844 in anti-Feynman gauge are identically equal to the results for − Conformal symmetry and three-point functions 12672 . (1972) 1421 hadrons This article is distributed under the terms of the Creative Commons ξ ). Separately it should be explained that of these seven contributions, 1287481 = 3 f ξ The singlet contribution to the Bjorken sum rule for polarized deep inelastic 1 Phys. Lett. n θ − 28 → (1972) 2982 3 , F 1 we obtain the following values of the B.1 C 77 ]. ]. ]. − T = = − e group weights. Nevertheless it is quite surprising that these gauge-dependent Nucl. Part. Phys. Proc. F 1 1 + and deep inelastic scattering sum rules D 6 , 2 f = e − − ) ( n ,C ξ = = τ SPIRE SPIRE SPIRE 2 F ξ ξ ( 1 3 2 f tot factorization property is again violated by the term hep-ph/9502348 IN IN IN θ θ T n scattering QCD Rev. Phys. J. Particle Data Group (2016) 100001 R [ different schemes: from FOPT[ to CIPT to FAPT Rev. Lett. σ [ [ 2 A S.L. Adler, C.G. Callan Jr., D.J. Gross and R. Jackiw, P.A. Baikov, K.G. Chetyrkin and J.H. K¨uhn, A.V. Nesterenko, S.A. Larin, A.L. Kataev and V.V. Starshenko, A.P. Bakulev, S.V. Mikhailov and N.G. Stefanis, R.J. Crewther, S.G. Gorishnii, A.L. Kataev and S.A. Larin, E.J. Schreier, β F For C T [9] [6] [7] [8] [4] [5] [2] [3] [1] 2 F F [10] References Therefore in this caseapproximation of we the observe GCR the as partial well. factorizationOpen of Access. the Attribution License ( any medium, provided the original author(s) and source are credited. 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