Scheme-Independent Series for Anomalous Dimensions of Higher-Spin Operators at an Infrared Fixed Point in a Gauge Theory
aa r X i v : . [ h e p - ph ] F e b Scheme-Independent Series for Anomalous Dimensions of Higher-Spin Operators atan Infrared Fixed Point in a Gauge Theory
Thomas A. Ryttov a and Robert Shrock b (a) CP -Origins and Danish Institute for Advanced StudySouthern Denmark University, Campusvej 55, Odense, Denmark and(b) C. N. Yang Institute for Theoretical Physics and Department of Physics and Astronomy,Stony Brook University, Stony Brook, New York 11794, USA We consider an asymptotically free vectorial gauge theory, with gauge group G and N f fermionsin a representation R of G , having an infrared fixed point of the renormalization group. We calculatescheme-independent series expansions for the anomalous dimensions of higher-spin bilinear fermionoperators at this infrared fixed point up to O (∆ f ), where ∆ f is an N f -dependent expansion variable.Our general results are evaluated for several special cases, including the case G = SU( N c ) with R equal to the fundamental and adjoint representations. I. INTRODUCTION
An asymptotically free gauge theory with sufficientlymany massless fermions evolves from the deep ultraviolet(UV) to an infrared fixed point (IRFP) of the renormal-ization group at a zero of the beta function. The theoryat this IRFP exhibits scale-invariance due to the vanish-ing of the beta function. The properties of the theoryat this IRFP are of fundamental field-theoretic interest.Among the basic properties are the anomalous dimen-sions γ ( O ) IR of various gauge-invariant operators O .In this paper we consider an asymptotically free vecto-rial gauge theory of this type, with a general gauge group G and N f copies (“flavors”) of massless Dirac fermions ψ i , i = 1 , ..., N f , transforming according to a represen-tation R of G [1]. We present scheme-independent se-ries expansions of the anomalous dimensions of gauge-invariant higher-spin operators that are bilinear in thefermion fields, up to O (∆ f ) inclusive, at the infraredfixed point, where ∆ f is an N f -dependent expansion vari-able defined below, in Eq. (1.8). The operators thatwe consider are of the form (suppressing flavor indices)¯ ψγ µ D µ ...D µ j ψ and ¯ ψσ λµ D µ ...D µ j ψ , where D µ is thecovariant derivative for the gauge theory, and it is under-stood here and below that the operators are symmetrizedover the Lorentz indices µ i , 1 ≤ i ≤ j and have Lorentztraces subtracted, and σ λµ is the commutator of twoDirac matrices (defined in Eq. (2.3)). We consider thecases 1 ≤ j ≤ ψγ µ D µ ...D µ j ψ were considered earlyon in the analysis of approximate Bjorken scaling in deepinelastic lepton scattering and the associated develop-ment of the theory of quantum chromodynamics (QCD).We briefly review this background [3]-[13]. In Euclideanquantum field theory, the short-distance operator prod-uct expansion (OPE) expresses the product of two op-erators A ( x ) and B ( y ) as a sum of local operators O i multiplied by coefficient functions c O i , A ( x ) B ( y ) = X i c O i ( x − y ) O i (( x + y ) / , (1.1)in the limit where x − y →
0. Let us denote the Maxwellian (i.e., free-field) dimension of an operator O in mass units as d O . Then the (free-field) dimension ofthe coefficient function is d c O i = d A + d B − d O i , so c O i ( x − y ) ∼ | x − y | d O i − d A − d B , (1.2)where | x − y | refers to the Euclidean distance. Hence,in the short-distance OPE, the operators with the low-est dimensions dominate, since they are multiplied by thesmallest powers of | x − y | . However, deep inelastic scatter-ing and the associated Bjorken limit probe the light conelimit, ( x − y ) → x − y = 0 in Minkowski space,where x = x µ x µ . With the arguments of two illustrativeLorentz-scalar operators denoted in a symmetric manneras ± x/
2, the light-cone OPE for A ( x/ B ( − x/
2) is A ( x/ B ( − x/
2) = X i,n ¯ c i,n ( x ) x µ · · · x µ n O i,n ; µ ,...,µ n (0)(1.3)in the limit x →
0, where the coefficient functions havebeen written in a form that explicitly indicates the factor x µ · · · x µ n and the operator O i,n ; µ ,...,µ n has spin j = n .Here (suppressing the Lorentz indices on O i,n ; µ ,...,µ n )the dependence of ¯ c i,n on x is¯ c i,n ( x ) ∼ ( x ) ( d O i,n − n − d A − d B ) / (1.4)(with logarithmic corrections in QCD due to anomalousdimensions). Consequently, the operators that have thestrongest singularity in their coefficient function ¯ c i,n ( x )as x → τ [7], where τ is thedimension minus the spin j of the operator, i.e., τ O i,n = d O i,n − j O i,n , (1.5)with j O i,n = n here. Thus, among bilinear fermion oper-ators, in addition to ¯ ψγ µ ψ with dimension 3, spin 1, andhence τ = 2, there are the operators ¯ ψγ µ D µ · · · D µ j ψ ,with dimension 3 + ( j −
1) and spin j , which also have τ = 2. These are the minimum-twist bilinear fermionoperators that contribute to the light-cone OPE (1.3)[14]. In a similar manner, twist-2 operators make thedominant contribution to the right-hand side of the light-cone OPE for the product of two electromagnetic or weakcurrents. The other operators that we consider, namely¯ ψσ λµ D µ ...D µ j ψ , have been relevant for the study oftransversity distributions in QCD [15].Our approach here is complementary to these previ-ous analyses of higher-spin operators, which have fo-cused on applications to QCD. In contrast, we studythe anomalous dimensions of these operators at an in-frared fixed point in a (deconfined) chirally symmetricnon-Abelian Coulomb phase (NACP), where the theoryis scale-invariant and is inferred to be conformally in-variant [16], whence the commonly used term “conformalwindow”. The goal of our calculations is to gain infor-mation about the properties of the conformal field theorythat is defined at this IRFP.Let us recall some further relevant background for ourwork. The evolution of the running gauge coupling g = g ( µ ), as a function of the momentum scale, µ , is describedby the renormalization-group (RG) beta function β = dα/d ln µ , where α ( µ ) = g ( µ ) / (4 π ). From the one-loopterm in the beta function [10, 11], it follows that theproperty of asymptotic freedom restricts N f to be lessthan an upper ( u ) bound, N u , where [17] N u = 11 C A T f . (1.6)Here, C A is the quadratic Casimir invariant for the group G and T f is the trace invariant for the representation R [18]. If N f is slightly less than N u , then this theory hasan infrared zero in the (perturbatively calculated) betafunction, i.e., an IR fixed point of the renormalizationgroup, at a value that we shall denote α IR [19, 20]. Inthe two-loop beta function (with N f < N u as requiredby asymptotic freedom), this IR zero is present if N f islarger than a lower ( ℓ ) value N ℓ , where [19] N ℓ = 17 C A T f (5 C A + 3 C f ) . (1.7)As the scale µ decreases from large values in the UV tosmall values in the IR, α ( µ ) approaches α IR from belowas µ →
0. Here we consider the properties of the theoryat this IRFP in the perturbative beta function. (For adiscussion of an IR zero in a nonperturbatively definedbeta function and its application to QCD, see [21].)Since the anomalous dimensions of gauge-invariant op-erators evaluated at the IRFP are physical, they must beindependent of the scheme used for regularization andrenormalization. In the conventional approach, one firstexpresses these anomalous dimensions as series expan-sions in powers of α or equivalently a = g / (16 π ) = α/ (4 π ), calculated to n -loop order; second, one computesthe IR zero of the beta function, denoted α IR,n , to thesame n -loop order; and third, one sets α = α IR,n inthe series expansion for the given anomalous dimensionto obtain its value at the IR zero of the beta function to this n -loop order. For the operator ¯ ψψ this conven-tional approach to calculate anomalous dimensions at anIR fixed point was carried out to the four-loop level in[22–24] and to the five-loop level in [25]. However, theseconventional series expansions in powers of α , calculatedto a finite order, are scheme-dependent beyond the lead-ing terms. This is a well-known property of higher-orderQCD calculations used to fit actual experimental data,which, in turn, has motivated many studies to reducescheme dependence [26]. These studies dealt with the UVfixed point (UVFP) at α = 0, as is appropriate for QCD.Studies of scheme dependence of quantities calculated ina conventional manner at an IR fixed point at α IR werecarried out in [27]-[31]. In particular, it was shown thatmany scheme transformations that are admissible in thevicinity of the UVFP at α = 0 in an asymptotically freetheory are not admissible away from the origin because ofvarious pathological properties. For example, the schemetransformation ra = tanh( ra ′ ) (depending on a param-eter r ) is an admissible transformation in the neighbor-hood of α = α ′ = 0. However, the inverse of this trans-formation is a ′ = (2 r ) − ln[(1 + ra ) / (1 − ra )], which issingular at an IRFP with a IR ≥ /r , i.e., α IR ≥ π/r , sothat the transformation is not admissible at this IRFP.Refs. [27] derived and studied an explicit scheme trans-formation that removes terms of loop order 3 and higherfrom the beta function in the local vicinity of α = 0, asis relevant to the UVFP in QCD [32], but also showedthat such a scheme transformation cannot, in general beused at an IRFP away from the origin owing to variouspathologies, one of which was illustrated above.It is thus desirable to use a theoretical framework inwhich the series expansions for physical quantities, suchas anomalous dimensions of gauge-invariant operators atthe IRFP, are scheme-independent at any finite order inan expansion variable. Because α IR → N f ap-proaches N u from below (where N f is formally gener-alized here from a non-negative integer to a non-negativereal number [17]), one can reexpress the expansions forphysical quantities at the IRFP as power series in themanifestly scheme-independent quantity [20, 33]∆ f = N u − N f . (1.8)In previous work we have calculated scheme-independentexpansions for anomalous dimensions of several types ofgauge-invariant operators at an IRFP in an asymptoti-cally free gauge theory [34]-[40]. We have compared theresultant values for anomalous dimensions with latticemeasurements where available [35]-[37],[41, 42].In the present paper we extend these calculations tothe case of the higher-spin operators ¯ ψγ µ D µ ...D µ j ψ and ¯ ψσ λµ D µ ...D µ j ψ for 1 ≤ j ≤
3. In addition togeneral formulas, we present results for several differ-ent special cases, including the case where G = SU( N c )and the fermions are in the fundamental ( F ) and ad-joint ( Adj ) representations. We also give results for thelimit N c → ∞ and N f → ∞ with the ratio N f /N c fixedand finite. Our calculations show that these scheme-independent expansions of the anomalous dimensions ofthe operators are reasonably accurate throughout muchof the non-Abelian Coulomb phase. Our results give fur-ther insight into the properties of a theory at an IRFPand should be useful to compare with lattice measure-ments of the anomalous dimensions of these higher-spinoperators when such measurements will be performed[43].This paper is organized as follows. Some relevant back-ground and methods are discussed in Sect. II. Generalstructural forms for the anomalous dimensions of higher-spin bilinear fermion operators are given in Sect. III. InSect. IV we present our scheme-independent calculationsof the anomalous dimensions of these higher-spin Wilsonoperators for a general gauge group G and fermion rep-resentation R . In Sect. V we give results for the casewhere G = SU( N c ) and R is the fundamental representa-tion, and in Sect. VI we present the special case of theseresults for the limit N c → ∞ and N f → ∞ with N f /N c fixed and finite. Anomalous dimension calculations forthe case where G = SU( N c ) and R is the adjoint rep-resentation are presented in Sect. VII. Our conclusionsare given in Sect. VIII and some auxiliary results areincluded in Appendix A. II. CALCULATIONAL METHODS
Let us consider a (gauge-invariant) operator O . Be-cause of the interactions, the full scaling dimension of thisoperator, denoted D O , differs from its free-field value, D O , free ≡ d O : D O = D O , free − γ O , (2.1)where γ O is the anomalous dimension of the operator[44]. Since γ O arises from the gauge interaction, it canbe expressed as the power series γ ( O ) = ∞ X ℓ =1 c ( O ) γ,ℓ a ℓ , (2.2)where c O γ,ℓ is the ℓ -loop coefficient.As stated in the introduction, we shall consider thegauge-invariant operators O µ ...µ j = ¯ ψγ µ D µ ...D µ j ψ and O λµ ...µ j = ¯ ψσ λµ D µ ...D µ j ψ , where σ λµ = i γ λ , γ µ ] . (2.3)We focus on the operators with 1 ≤ j ≤
3. We introducethe following compact notation for these operators: O ( γD ) µ µ ≡ ¯ ψγ µ D µ ψ (2.4) O ( γDD ) µ µ µ ≡ ¯ ψγ µ D µ D µ ψ (2.5) O ( γDDD ) µ µ µ µ ≡ ¯ ψγ µ D µ D µ D µ ψ (2.6) O ( σD ) λµ µ ≡ ¯ ψσ λµ D µ ψ (2.7) O ( σDD ) λµ µ µ ≡ ¯ ψσ λµ D µ D µ ψ (2.8)and O ( σDDD ) λµ µ µ ≡ ¯ ψσ λµ D µ D µ D µ ψ . (2.9)For brevity of notation, we suppress the flavor indices onthe fields ψ .For a given operator O , we write the scheme-independent expansion of its anomalous dimension γ ( O ) evaluated at the IRFP, denoted γ ( O ) IR , as γ ( O ) IR = ∞ X n =1 κ ( O ) n ∆ nf . (2.10)The truncation of right-hand side of Eq. (2.10) to maxi-mal power p is denoted γ ( O ) IR, ∆ pf = p X n =1 κ ( O ) n ∆ nf . (2.11)We use a further shorthand notation for the anomalousdimensions in which the superscript in γ ( O ) IR is replacedby a symbol for the quantity standing between ¯ ψ and ψ in the operator O . These shorthand symbols are asfollows: γ ( γD ) IR for the anomalous dimension of the oper-ator O ( γD ) µ µ = ¯ ψγ µ D µ ψ at the IRFP, and so forth forthe other operators. In comparing with our previous cal-culations in [34]-[39], we also use the notation γ (1) IR and γ ( σ ) IR for the anomalous dimensions of ¯ ψψ and ¯ ψσ λµ ψ atthe IRFP. (The anomalous dimension γ ( σ ) IR was denoted γ T,IR in [36], where the subscript T referred to the Diractensor σ µν .)As discussed in [34, 36], the calculation of the coef-ficient κ ( O ) n in Eq. (2.10) requires, as inputs, the betafunction coefficients at loop order 1 ≤ ℓ ≤ n + 1 andthe anomalous dimension coefficients c ( O ) γ,ℓ at loop order1 ≤ ℓ ≤ n . The method of calculation requires that theIR fixed point must be exact, which is the case in the non-Abelian Coulomb phase. As in our earlier work [34]-[39],we thus restrict our consideration to the non-AbelianCoulomb phase (conformal window) [45]. For a givengauge group G and fermion representation R , the con-formal window extends from an upper end at N f = N u to a lower end at a value that is commonly denoted N f,cr .In contrast to the exactly known value of N u (given in Eq.(1.6)), the value of N f,cr is not precisely known and hasbeen investigated extensively for several groups G andfermion representations R [41, 42, 45]. For values of N f in the non-Abelian Coulomb phase such that ∆ f is nottoo large, one may expect the expansion (2.10) of γ ( O ) IR in a series in powers of ∆ f to yield reasonably accurateperturbative calculations of the anomalous dimension. Inour earlier works, using our explicit calculations, we haveshown that this is, in fact, the case.We recall some relevant properties of the theory re-garding global flavor symmetries. Because the N f fermions are massless, the Lagrangian is invariant un-der the classical global flavor ( f l ) symmetry G fl,cl =U( N f ) L ⊗ U( N f ) R , or equivalently, G fl,cl = SU( N f ) L ⊗ SU( N f ) R ⊗ U(1) V ⊗ U(1) A (2.12)(where V and A denote vector and axial-vector). TheU(1) V represents fermion number, which is conserved bythe bilinear operators that we consider. The U(1) A sym-metry is broken by instantons, so the actual nonanoma-lous global flavor symmetry is G fl = SU( N f ) L ⊗ SU( N f ) R ⊗ U(1) V . (2.13)This G fl symmetry is respected in the non-AbelianCoulomb phase, since there is no spontaneous chiral sym-metry breaking in this phase [41, 42]. For our operators,the flavor matrix between ¯ ψ and ψ is either the iden-tity or the operator T a , a generator of SU( N f ), whichcan be viewed as acting either to the right on ψ or tothe left on ¯ ψ . These yield the same anomalous dimen-sions [46]. As a consequence of the unbroken global flavorsymmetry, our operators transform as representations ofthe global flavor group G fl . The invariance under thefull G fl in the non-Abelian Coulomb phase is differentfrom the situation in the QCD-like phase at smaller N f ,where the chiral part of G fl is spontaneously broken bythe QCD bilinear quark condensate to the vectorial sub-group SU( N f ) V and operators are classified according towhether they are singlet or nonsinglet (adjoint) underthis vectorial SU( N f ) symmetry. In particular, in theconsideration of flavor-singlet operators, in QCD, onemust take into account mixing with gluonic operators.Here, in contrast, there is no mixing between any of ourbilinear fermion operators and gluonic operators, sincethe latter are singlets under G fl .The operators O with an even number of Dirac γ ma-trices, symbolically denoted Γ e , link left with right chiralcomponents of ψ , while the operators O with an oddnumber of Dirac γ matrices, Γ o , link left with left andright with right components:¯ ψ Γ e ψ = ¯ ψ L Γ e ψ R + ¯ ψ R Γ e ψ L (2.14)¯ ψ Γ o ψ = ¯ ψ L Γ o ψ L + ¯ ψ R Γ o ψ R , (2.15)where ¯ ψ = ψ † γ . In the non-Abelian Coulomb phasewhere the flavor symmetry is (2.13), one may regard the T b in the term ¯ ψ L T b ψ R acting to the right as an ele-ment of SU( N f ) R and acting to the left as an element ofSU( N f ) L .Given that the theory at the IR fixed point is confor-mally invariant [16], there is an important lower boundon the full dimension of an operator O and hence, with our definition (2.1), an upper bound on the anomalous di-mension γ ( O ) that follows from the conformal invariance.To state this, we first recall that a (finite-dimensional)representation of the Lorentz group is specified by theset ( j , j ), where j and j take on nonnegative integralor half-integral values [47]. A Lorentz scalar operatorthus transforms as (0 , / , / F aµν as (1 , ⊕ (0 , D O ≥ j + j + 1 , (2.16)i.e., the upper bound γ O ≤ D O , free − ( j + j + 1) . (2.17)We have studied the constraints from the upper bound(2.17) in our previous calculations of anomalous dimen-sions in [22, 25], [36]-[39]. Anticipating the results givenbelow, since our calculations yield negative values for theanomalous dimensions of higher-spin Wilson operators,they obviously satisfy these conformality upper bounds. III. SOME GENERAL STRUCTURALPROPERTIES OF γ ( O ) IR From our previous calculations [34]-[39] for the anoma-lous dimensions of ¯ ψψ and ¯ ψσ µν ψ , in conjunction withour new results on the anomalous dimensions γ ( O ) IR ofhigher-spin twist-2 bilinear fermion operators O , we findsome general structural properties of the coefficients κ ( O ) n in the scheme-independent series expansions of theanomalous dimensions γ ( O ) IR . These involve various groupinvariants, including the quadratic Casimir invariants C A ≡ C ( G ), C f ≡ C ( R ), the trace invariant T ( R ), andthe quartic trace invariants d abcdR d abcdR ′ /d A , where d A de-notes the dimension of the adjoint representation [18, 49].For compact notation, it is convenient to define a factorthat occurs in the denominators of these κ ( O ) n coefficients,namely. D = 7 C A + 11 C f . (3.1)(not to be confused with covariant derivative). We ex-hibit this general form here, using a ( O ) j,k for various (con-stant) numerical coefficients: κ ( O )1 = c ( O )1 C f T f C A D , (3.2) κ ( O )2 = C f T f ( a ( O )2 , C A + a ( O )2 , C A C f + a ( O )2 , C f ) C A D , (3.3)and κ ( O )3 = C f T f C A D (cid:20) a ( O )3 , C A T f + a ( O )3 , C A C f T f + a ( O )3 , C A C f T f + a ( O )3 , C A C f T f + a ( O )3 , C A C f T f + a ( O )3 , C A T f d abcdA d abcdA d A + a ( O )3 , C f T f d abcdA d abcdA d A + a ( O )3 , C A T f d abcdR d abcdA d A + a ( O )3 , C A C f T f d abcdR d abcdA d A + a ( O )3 , C A d abcdR d abcdR d A + a ( O )3 , C A C f d abcdR d abcdR d A (cid:21) . (3.4) IV. ANOMALOUS DIMENSIONS γ ( O ) IR OFHIGHER-SPIN OPERATORSA. General
In this section we present the results of our calculationsof the coefficients in the scheme-independent series ex-pansions up to O (∆ f ) for the various higher-spin opera-tors considered here. As was noted above, the calculationof the O (∆ nf ) coefficient, κ ( O ) n , for the anomalous dimen-sion of an operator O at the IRFP requires, as inputs, thebeta function coefficients at loop order 1 ≤ ℓ ≤ n + 1 andthe anomalous dimension coefficients c ( O ) ℓ at loop order1 ≤ ℓ ≤ n . Hence, we use the beta function coefficientsfrom one-loop up to the four-loop level [10, 19], [50, 51],together with the anomalous dimension coefficients cal-culated in the conventional series expansion in powersof a up to the three-loop level [11], [46], [52]–[57]. Thehigher-order terms in the beta function and anomalousdimensions that we use have been calculated in the MSscheme [58], but our results are independent of this since they are scheme-independent. (The beta function has ac-tually been calculated up to five-loop order [59, 60], butthese results will not be needed here.) B. γ ( γD ) IR For the anomalous dimension γ ( γD ) IR of the operator¯ ψγ µ D µ ψ at the IRFP, we calculate κ ( γD )1 = − C f T f C A D , (4.1) κ ( γD )2 = 2 C f T f (cid:16) C A − C A C f − C f (cid:17) C A D , (4.2)and κ ( γD )3 = − C f T f C A D (cid:20) C A T f ( − ζ ) + C A C f T f (2764440 + 145152 ζ )+ C A C f T f (8940028 − ζ ) + C A C f T f ( − − ζ ) + C A C f T f (3841024 + 5018112 ζ )+ C A T f d abcdA d abcdA d A ( − ζ ) + C f T f d abcdA d abcdA d A ( − ζ )+ C A T f d abcdR d abcdA d A (2838528 − ζ ) + C A C f T f d abcdR d abcdA d A (4460544 − ζ )+ C A d abcdR d abcdR d A ( − ζ ) + C A C f d abcdR d abcdR d A ( − ζ ) (cid:21) . (4.3)In these expressions and the following ones, we have in-dicated the factorizations of the numbers in the denom-inators, since they are rather simple. In general, thenumbers in the numerators do not have such simple fac-torizations. With these coefficients, the anomalous dimension γ ( γD ) IR calculated to order O (∆ pf ), denoted γ ( γD ) IR,F, ∆ pf , isgiven by Eq. (2.11) with O = ¯ ψγ µ D µ ψ . Our resultshere yield γ ( γD ) IR,F, ∆ pf with p = 1 , ,
3. Analogous state-ments apply to the anomalous dimensions of the otheroperators for which we have performed calculations, andwe proceed to present the coefficients for these next. C. γ ( γDD ) IR For the anomalous dimension γ ( γDD ) IR of the operator¯ ψγ µ D µ D µ ψ at the IRFP, we calculate κ ( γDD )1 = − C f T f C A D , (4.4) κ ( γDD )2 = 10 C f T f (cid:16) C A − C A C f − C f (cid:17) C A D , (4.5)and κ ( γDD )3 = − C f T f C A D (cid:20) C A T f (1538649 + 2794176 ζ ) + C A C f T f (14860881 + 399168 ζ )+ C A C f T f (40821518 − ζ ) + C A C f T f ( − − ζ ) + C A C f T f (19308575 + 13799808 ζ )+ C A T f d abcdA d abcdA d A ( − ζ ) + C f T f d abcdA d abcdA d A ( − ζ )+ C A T f d abcdR d abcdA d A (14192640 − ζ ) + C A C f T f d abcdR d abcdA d A (22302720 − ζ )+ C A d abcdR d abcdR d A ( − ζ ) + C A C f d abcdR d abcdR d A ( − ζ ) (cid:21) . (4.6) D. γ ( γDDD ) IR Proceeding to the anomalous dimension γ ( γDDD ) IR ofthe operator ¯ ψγ µ D µ D µ D µ ψ at the IRFP, we find κ ( γDDD )1 = − C f T f · C A D , (4.7) κ ( γDDD )2 = 2 C f T f (cid:16) C A − C A C f − C f (cid:17) · C A D , (4.8)and κ ( γDDD )3 = 2 C f T f · C A D (cid:20) C A T f ( − − ζ ) + C A C f T f ( − − ζ )+ C A C f T f ( − ζ ) + C A C f T f (709569531572 + 51285960000 ζ )+ C A C f T f ( − − ζ )+ C A T f d abcdA d abcdA d A (15825600000 − ζ ) + C f T f d abcdA d abcdA d A (24868800000 − ζ )+ C A T f d abcdR d abcdA d A ( − ζ ) + C A C f T f d abcdR d abcdA d A ( − ζ )+ C A d abcdR d abcdR d A (1053193680000 − ζ ) + C A C f d abcdR d abcdR d A (1655018640000 − ζ ) (cid:21) . (4.9) E. γ ( σD ) IR For the anomalous dimension γ ( σD ) IR of the operator¯ ψσ λµ D µ ψ at the IRFP, we calculate κ ( σD )1 = − C f T f C A D , (4.10) κ ( σD )2 = 4 C f T f (cid:16) C A − C A C f − C f (cid:17) C A D , (4.11) κ ( σD )3 = 4 C f T f C A D (cid:20) C A T f − C A C f T f − C A C f T f + 738144 C A C f T f − C A C f T f + C A T f d abcdA d abcdA d A (17920 − ζ ) + C f T f d abcdA d abcdA d A (28160 − ζ )+ C A T f d abcdR d abcdA d A ( − ζ ) + C A C f T f d abcdR d abcdA d A ( − ζ )+ C A d abcdR d abcdR d A (1192576 − ζ ) + C A C f d abcdR d abcdR d A (1874048 − ζ ) (cid:21) . (4.12) F. γ ( σDD ) IR For the anomalous dimension γ ( σDD ) IR we calculate κ ( σDD )1 = − C f T f C A D , (4.13) κ ( σDD )2 = 4 C f T f (cid:16) C A − C A C f − C f (cid:17) C A D , (4.14)and κ ( σDD )3 = − C f T f C A D (cid:20) C A T f (2935737 + 4064256 ζ ) + C A C f T f (39906468 + 580608 ζ )+ C A C f T f (107242456 − ζ ) + C A C f T f ( − − ζ )+ C A C f T f (43045024 + 20072448 ζ )+ 3 C A T f d abcdA d abcdA d A ( − ζ ) + 3 C f T f d abcdA d abcdA d A ( − ζ )+ 3 C A T f d abcdR d abcdA d A (12300288 − ζ ) + 3 C A C f T f d abcdR d abcdA d A (19329024 − ζ )+ 3 C A d abcdR d abcdR d A ( − ζ ) + 3 C A C f d abcdR d abcdR d A ( − ζ ) (cid:21) . (4.15) G. γ ( σDDD ) IR Finally, for the anomalous dimension γ σDDD ) IR we ob-tain κ ( σDDD )1 = − C f T f C A D , (4.16) κ ( σDDD )2 = 2 C f T f (cid:16) C A − C A C f − C f (cid:17) C A D , (4.17)and κ ( σDDD )3 = − C f T f C A D (cid:20) C A T f (5213502 + 2667168 ζ ) + C A C f T f (25185069 + 381024 ζ )+ C A C f T f (58268711 − ζ ) + C A C f T f ( − − ζ )+ C A C f T f (36476660 + 13172544 ζ )+ C A T f d abcdA d abcdA d A ( − ζ ) + C f T f d abcdA d abcdA d A ( − ζ )+ C A T f d abcdR d abcdA d A (22708224 − ζ ) + C A C f T f d abcdR d abcdA d A (35684352 − ζ )+ C A d abcdR d abcdR d A ( − ζ ) + C A C f d abcdR d abcdR d A ( − ζ ) (cid:21) . (4.18) V. EVALUATION OF κ ( O ) n FOR G = SU( N c ) AND R = F In this section we evaluate our general results for theseanomalous dimensions γ ( O ) IR in the important special casewhere the gauge group is G = SU( N c ) and the N f fermions are in the fundamental representation of thisgroup, R = F . A. γ ( γD ) IR,
SU( N c ) ,F Substituting G = SU( N c ) and R = F in our generalresults (4.1)-(4.3), we obtain κ ( γD )1 , SU( N c ) ,F = − ( N c − N c (25 N c − , (5.1) κ ( γD )2 , SU( N c ) ,F = − ( N c − N c − N c + 385)3 N c (25 N c − , (5.2)and κ ( γD )3 , SU( N c ) ,F = − ( N c − N c (25 N c − (cid:20) N c + (1831104 − ζ ) N c + ( − ζ ) N c + (6282342 − ζ ) N c + 240064 + 313632 ζ (cid:21) . (5.3)Then, for this case G = SU(3), R = F , the anoma-lous dimension γ ( γD ) IR calculated to order O (∆ pf ), denoted γ ( γD ) IR,F, ∆ pf , is given by Eq. (2.11) with O = ¯ ψγ µ D µ ψ . B. γ ( γDD ) IR,
SU( N c ) ,F Substituting G = SU( N c ) and R = F in our generalresults (4.4)-(4.6), we obtain κ ( γDD )1 , SU( N c ) ,F = − N c − N c (25 N c − , (5.4) κ ( γDD )2 , SU( N c ) ,F = − N c − N c − N c + 9383)2 · N c (25 N c − , (5.5)and κ ( γDD )3 , SU( N c ) ,F = − N c − · N c (25 N c − (cid:20) N c + (160969860 − ζ ) N c + ( − ζ ) N c + (499565484 − ζ ) N c + 19308575 + 13799808 ζ (cid:21) . (5.6) C. γ ( γDDD ) IR,
SU( N c ) ,F In a similar manner, from our general formulas (4.7)-(4.9), we find κ ( γDDD )1 , SU( N c ) ,F = − N c − · N c (25 N c − , (5.7) κ ( γDDD )2 , SU( N c ) ,F = − ( N c − N c − N c + 7719767)2 · · N c (25 N c − , (5.8)and κ ( γDDD )3 , SU( N c ) ,F = − ( N c − · · N c (25 N c − (cid:20) N c + (3455659520100 − ζ ) N c + ( − ζ ) N c + (9616576686156 − ζ ) N c + 433168554247 + 225658224000 ζ ) (cid:21) . (5.9) D. γ ( σD ) IR,
SU( N c ) ,F From our general results (4.10)-(4.12), we obtain κ ( σD )1 , SU( N c ) ,F = − N c − N c (25 N c − , (5.10) κ ( σD )2 , SU( N c ) ,F = − N c − N c − N c + 44)3 N c (25 N c − , (5.11)and κ ( σD )3 , SU( N c ) ,F = − ( N c − N c (25 N c − (cid:20) N c + (78610 − ζ ) N c + ( − ζ ) N c + (209836 − ζ ) N c + 13794 (cid:21) . (5.12) E. γ ( σDD ) IR,
SU( N c ) ,F From our general results (4.13)-(4.15), we obtain κ ( σDD )1 , SU( N c ) ,F = − N c − N c (25 N c − , (5.13)0 κ ( σDD )2 , SU( N c ) ,F = − N c − N c − N c + 5588)3 N c (25 N c − , (5.14)and κ ( σDD )3 , SU( N c ) ,F = − ( N c − N c (25 N c − (cid:20) N c + (30539268 − ζ ) N c + ( − ζ ) N c + (84228606 − ζ ) N c + 2690314 + 1254528 ζ (cid:21) . (5.15) F. γ ( σDDD ) IR,
SU( N c ) ,F For this case we have κ ( σDDD )1 , SU( N c ) ,F = − ( N c − N c (25 N c − , (5.16) κ ( σDDD )2 , SU( N c ) ,F = − ( N c − N c − N c + 7832)3 N c (25 N c − , (5.17)and κ ( σDDD )3 , SU( N c ) ,F = − ( N c − N c (25 N c − (cid:20) (90949802 N c + (70557192 − ζ ) N c + ( − ζ ) N c + (194401944 − ζ ) N c + 9119165 + 3293136 ζ (cid:21) . (5.18)Below, where the meaning is clear, we will often omit theSU(3) in the subscript.We remark on the signs of these coefficients. It is ev-ident from Eqs. (4.1), (4.4), (4.7), (4.10), (4.13), and(4.16) that κ ( γD )1 , κ ( γDD )1 , κ ( γDDD )1 , κ ( σD )1 , κ ( σDD )1 , and κ ( σDDD )1 are all negative for any G and R . We findthat the O (∆ f ) and O (∆ f ) coefficients, κ ( O )2 and κ ( O )3 ,for these operators are also negative for the theory with G = SU( N c ) and fermions in the fundamental represen-tation, R = F , in the full range N c ≥ κ ( O ) n for the operators in this theory. For comparison, we alsoinclude the signs of κ (1) n for ¯ ψψ and κ ( σ ) n for ¯ ψσ µν ψ thatwe obtained in our earlier calculations (which hold for all N c ).It is interesting to note that for all of the higher-spinoperators O that we consider, the anomalous dimensions γ ( O ) IR that we calculate are negative (with our sign conven-tion in (2.1) [44])). They thus have the same sign as thesign of the anomalous dimension of the operator ¯ ψσ µν ψ and are opposite in sign relative to the anomalous dimen-sions that we calculated for ¯ ψψ in our previous work [22],[34]-[39].In Tables II-VIII we list values of the anomalous di-mensions γ ( γD ) IR , γ ( γDD ) IR , γ ( γDDD ) IR , γ ( σ ) IR , γ ( σD ) IR , γ ( σDD ) IR ,and γ ( σDDD ) IR for the theory with G = SU(3) and fermionsin the fundamental representation, R = F , calculated to O (∆ pf ), denoted γ ( γD ) IR,F, ∆ pf , etc., with p = 1 , ,
3, as func-1tions of N f for a relevant range of N f values extendingdownward from the upper end of the conformal regimeat N f = N u (i.e., ∆ f = 0) within this conformal window[61]. The numbers in Table V) are evaluations of ouranalytic results given in [36] and are included for com-parison.In Figs. 1-7 we show plots of these anomalous dimen-sions for the SU(3) theory with R = F . The plot of theanomalous dimension for ¯ ψσ λµ ψ is based on the analyticresults of our earlier paper [36] but was not given thereand is new here. As can be seen from these tables and fig-ures, the higher-order terms in the ∆ f expansion are suf-ficiently small that it is expected to be reliable through-out much of the non-Abelian Coulomb phase (i.e., con-formal window). As is obvious, since our calculations arefinite series expansions in powers of ∆ f , they are mostaccurate in the upper part of the NACP, where this ex-pansion parameter ∆ f is small. This is similar to whatwe found in our earlier scheme-independent calculationsof anomalous dimensions [34]-[39]. In the figures, this isevident from the fact that the curves for the anomalousdimensions calculated to O (∆ f ) are reasonably close tothe corresponding curves for these anomalous dimensionscalculated to order O (∆ f ). VI. LNN LIMIT FOR γ ( O ) IR,
SU( N c ) ,F In a theory with gauge group SU( N c ) and fermions inthe fundamental representation, R = F , it is of interestto consider the limit N c → ∞ , N F → ∞ with r ≡ N F N c fixed and finiteand ξ ( µ ) ≡ α ( µ ) N c is a finite function of µ . (6.1)This limit is denoted as lim LNN (where “LNN” connotes“large N c and N F ” with the constraints in Eq. (6.1)imposed). It is also often called the ’t Hooft-Venezianolimit. It has the simplifying feature that rather thandepending on N c and N f , the properties of the theoryonly depend on their ratio, r . The scheme-independentexpansion parameter in this LNN limit is∆ r ≡ lim LNN ∆ f N c = 112 − r . (6.2) r u = lim LNN N u N c , (6.3)and r ℓ = lim LNN N ℓ N c , (6.4)with values r u = 112 = 5 . r ℓ = 3413 = 2 . . (6.6)With I IRZ : N ℓ < N f < N u , it follows that the corre-sponding interval in the ratio r is I IRZ,r : 3413 < r < , i.e., . < r < . G = SU( N c ) and R = F , in the LNN limit. The rescaled coefficients thatare finite in the LNN limit areˆ κ ( O ) n = lim N c →∞ N nc κ ( O ) n (6.8)The anomalous dimension γ ( O ) IR is also finite in this limitand is given bylim LNN γ ( O ) IR,
SU( N c ) ,F = ∞ X n =1 κ ( O ) n ∆ nf = ∞ X n =1 ˆ κ ( O ) n ∆ nr . (6.9)As r decreases from its upper limit, r u , to r ℓ , the expan-sion variable ∆ r increases from 0 to(∆ r ) max = 7526 = 2 . r ∈ I IRZ,r . (6.10)In this LNN limit, the values of ˆ κ ( O ) n with 1 ≤ n ≤ O considered here are listed in TableIX. For comparison, we also include the correspondingvalues of ˆ κ ( O ) n for the operators ¯ ψψ and ¯ ψσ µν ψ that wehad calculated in [36]. VII. EVALUATION OF ANOMALOUSDIMENSIONS γ ( O ) IR FOR G = SU( N c ) AND R = Adj
For the case where G = SU( N c ) and the fermions are inthe adjoint representation, R = Adj , our general resultsfor the scheme-independent expansion coefficients for theanomalous dimensions of the operators under considera-tion are as follows: : κ ( γD )1 , SU( N c ) ,Adj = − = − . , (7.1) κ ( γD )2 , SU( N c ) ,Adj = − = − . , (7.2) κ ( γD )3 , SU( N c ) ,Adj = − + 47363 N c = − . . N c , (7.3) κ ( γDD )1 , SU( N c ) ,Adj = − = − . , (7.4)2 κ ( γDD )2 , SU( N c ) ,Adj = − · = − . , (7.5) κ ( γDD )3 , SU( N c ) ,Adj = − · + 74003 N c = − . . N c , (7.6) κ ( γDDD )1 , SU( N c ) ,Adj = − · − . , (7.7) κ ( γDDD )2 , SU( N c ) ,Adj = − · · = − . , (7.8)and κ ( γDDD )3 , SU( N c ) ,Adj = − · · + 464723 · N c = − . . N c , (7.9) κ ( σD )1 , SU( N c ) ,Adj = − = − . , (7.10) κ ( σD )2 , SU( N c ) ,Adj = − · = − . , (7.11)and κ ( σD )3 , SU( N c ) ,Adj = − · + 5923 N c = − . . N c , (7.12) κ ( σDD )1 , SU( N c ) ,Adj = − = − . , (7.13) κ ( σDD )2 , SU( N c ) ,Adj = − · = − . , (7.14) κ ( σDD )3 , SU( N c ) ,Adj = − · + 76963 N c = − . . N c , (7.15) κ ( σDDD )1 , SU( N c ) ,Adj = − = − . , (7.16) κ ( σDDD )2 , SU( N c ) ,Adj = − = − . , (7.17)and κ ( σDDD )3 , SU( N c ) ,Adj = − · + 94723 N c = − . . N c . (7.18)For all of these operators O , the coefficients κ ( O ) n, SU( N c ) ,Adj are negative for n = 1 and n = 2 and for all N c . The coefficient κ ( σD )3 , SU( N c ) ,Adj is negative for all N c ,while the coefficients κ ( O )3 , SU( N c ) ,Adj for the other operatorsare positive for N c = 2, i.e., G = SU(2), and are negativefor N c ≥ VIII. CONCLUSIONS
In conclusion, in this paper we have calculatedscheme-independent expansions up to O (∆ f ) inclu-sive for the anomalous dimensions of the higher-spin,twist-2 bilinear fermion operators ¯ ψγ µ D µ ...D µ j ψ and¯ ψσ λµ D µ ...D µ j ψ with j up to 3, evaluated at an IR fixedpoint in the non-Abelian Coulomb phase of an asymp-totically free gauge theory with gauge group G and N f fermions transforming according to a representation R of G . Our general results are evaluated for several specialcases, including the case G = SU( N c ) with R equal tothe fundamental and adjoint representations. We havepresented our results in convenient tabular and graphi-cal formats. For fermions in the fundamental represen-tation, we also analyze the limit N c → ∞ and N f → ∞ with N f /N c fixed and finite. A comparison with ourprevious scheme-independent calculations of the corre-sponding anomalous dimensions of ¯ ψψ and ¯ ψσ µν ψ hasalso been given. Our new results further elucidate theproperties of conformal field theories. With the requi-site inputs, one could extend these scheme-independentcalculations to higher-spin operators and to higher orderin powers of ∆ f . It is hoped that lattice measurementsof these anomalous dimensions of higher-spin operatorsin the conformal window will be performed in the future,and it will be of interest to compare our calculations withlattice results when they will become available. Acknowledgments
This research was supported in part by the DanishNational Research Foundation grant DNRF90 to CP -Origins at SDU (T.A.R.) and by the U.S. NSF GrantsNSF-PHY-1620628 and NSF-PHY-1915093 (R.S.). Appendix A: Previous Results on γ (1) and γ ( σ ) for G = SU(3) and R = F In this appendix, for comparison with our new results,we list our previous results from [36] (see also [37]) for the3scheme-independent series expansions of the anomalousdimensions γ ( O ) IR for O = ¯ ψψ and O = ¯ ψσ µν ψ . Followingthe same shorthand notation as in the text, we denote thecoefficients at order O (∆ nf ) in the scheme-independentseries expansions (2.10) for these anomalous dimensionsas κ (1) n and κ ( σ ) n . We calculated κ (1)1 = 8 T f C f C A D , (A1) κ (1)2 = 4 T f C f (5 C A + 88 C f )(7 C A + 4 C f )3 C A D , (A2)and κ (1)3 = 4 T f C f C A D (cid:20) − T f C A + 432012 T f C A C f + 5632 T f C f d abcdA d abcdA d A ( − ζ )+ 16 C A (cid:18) T f C f + 6776 d abcdR d abcdR d A ( −
11 + 24 ζ ) (cid:19) + 704 C A (cid:18) T f C f + 112 T f d abcdR d abcdA d A (4 − ζ ) + 242 C f d abcdR d abcdR d A ( −
11 + 24 ζ ) (cid:19) + 32 T f C A (cid:18) T f C f − C f d abcdR d abcdA d A ( − ζ ) + 112 T f d abcdA d abcdA d A ( − ζ ) (cid:19)(cid:21) (A3)(where the denominator factor D was defined in Eq.(3.1)). In [37, 39] we presented results for the next-higherorder coefficient, κ (1)4 , but these are not needed here.For the κ ( σ ) n we found κ ( σ )1 = − C f T f C A D (A4) κ ( σ )2 = − C f T f (259 C A + 428 C A C f − C f )9 C A D (A5)and κ ( σ )3 = 4 C f T f C A D (cid:20) C A T f (cid:26) C A ( − ζ ) + C A C f ( − ζ ) + C A C f (83616 − ζ )+ C A C f (1385472 − ζ ) + C f ( − ζ ) (cid:27) − T f D ( − ζ ) d abcdA d abcdA d A − C A D ( −
11 + 24 ζ ) d abcdR d abcdR d A + 11264 C A T f D ( − ζ ) d abcdR d abcdA d A (cid:21) . (A6)For G = SU( N c ) and R = F , in the LNN limit, theseyield the rescaled coefficientsˆ κ (1)1 = 45 = 0 . , (A7)ˆ κ (1)2 = 5885 = 0 . , (A8)ˆ κ (1)3 = 21939443 · = 0 . × − , (A9) ˆ κ ( σ )1 = − · = − . , (A10)ˆ κ ( σ )2 = − · = − (0 . × − ) , (A11)and ˆ κ ( σ )3 = 1844563 · = 2 . × − . (A12)4 [1] This assumption of massless fermions does not entail anyloss of generality, since a fermion with nonzero mass m would be integrated out of the low-energy effective fieldtheory that describes the physics at Euclidean momen-tum scales µ < m and hence would not affect the infraredlimit µ → , 1499 (1969); K. G. Wil-son and W. Zimmermann, Commun. Math. Phys. , 87(1972).[3] There is an extensive literature on operator product ex-pansions, the renormalization group, and applications todeep inelastic scattering and QCD. 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We use standard normalizationsfor SU( N c ) so that, e.g., for the fundamental representa-tion F , T ( F ) = 1 / C ( F ) = ( N c − / (2 N c ).[19] W. E. Caswell, Phys. Rev. Lett. , 244 (1974); D. R. T.Jones, Nucl. Phys. B , 531 (1974).[20] T. Banks and A. Zaks, Nucl. Phys. B , 189 (1982).[21] A. Deur, S. J. Brodsky, and G. F. de T´eramond, Prog.Part. Nucl. Phys. , 1 (2016) and references therein.[22] T. A. Ryttov and R. Shrock, Phys. Rev. D , 056011(2011).[23] C. Pica and F. Sannino, Phys. Rev. D , 035013 (2011).[24] R. Shrock, Phys. Rev. D , 105005 (2013); R. Shrock,Phys. Rev. D , 116007 (2013).[25] T. A. Ryttov and R. Shrock, Phys. Rev. D , 105015(2016).[26] See, e.g., S. J. Brodsky and X.-G. Wu, Phys. Rev. Lett. , 042002 (2012); M. Mojaza, S. J. Brodsky, and X.-G.Wu, Phys. Rev. Lett. , 192001 (2013); X.-G. Wu, Y.Ma, S.-Q. Wang, H.-B. Fu, H.-H. Ma, S. J. Brodsky, andM. Mojaza, Rept. Prog. Phys. , 126201 (2015), andreferences therein.[27] T. A. Ryttov and R. Shrock, Phys. Rev. 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D , 105004 (2017).[38] T. A. Ryttov and R. Shrock, Phys. Rev. D , 076009(2012); Phys. Rev. D , 105018 (2017); Phys. Rev. D , 065020 (2018).[39] T. A. Ryttov and R. Shrock, Phys. Rev. D , 105015(2017); Phys. Rev. D , 016020 (2018); Phys. Rev. D , 025004 (2018); Phys. Rev. D , 096003 (2018); G.Girmohanta, T. A. Ryttov, and R. Shrock, Phys. Rev. D , 116022 (2019). [40] J. A. Gracey, T. A. Ryttov, and R. Shrock, Phys. Rev.D ψψ , one could take the illustrative case of G = SU(3)and R = F , for which N u = 16 . N ℓ = 8 .
05, sothat the value N f = 12 lies roughly midway between N ℓ and N u . Some lattice measurements of the anomalousdimension of ¯ ψψ in the IR for this SU(3) theory with R = F and N f = 12 include T. Appelquist et al. (LSDCollab.), Phys. Rev. D , 054501 (2011); T. DeGrand,Phys. Rev. D , 116901 (2011); Y. Aoki et al. (LatKMICollab.), Phys. Rev. D , 054506 (2012); A. Hasenfratz,A. Cheng, G. Petropoulos, and D. Schaich, JHEP ,061 (2013); A. Hasenfratz and D. Schaich, JHEP ,132 (2018) 132; M. P. Lombardo, K. Miura, T. J. Nunesda Silva, and E. Pallante, JHEP , 183 (2014); Z. Fodoret al., Phys. Rev. D , 091501 (2016); Z. Fodor et al.,Phys. Lett. B , 230 (2018).[44] Some authors use the opposite sign convention for theanomalous dimension, writing D O = D O , free + γ O . Oursign convention is the same as the one used in the latticegauge theory literature [41, 42]. The equivalent notation γ ( O ) ≡ γ O will often be used.[45] In contrast, in the confined QCD-type phase at lower N f , although the UV beta function may exhibit a for-mal IR zero, as the theory flows from the UV to theIR, the running coupling becomes large enough so thatspontaneous chiral symmetry breaking occurs, giving thefermions dynamical masses, so that they are integratedout of the low-energy effective field theory that is appli-cable deeper in the IR, and hence the resultant beta func-tion is that of a pure gauge theory, which does not havea (perturbative) IR zero. If N f is only slightly less than N f,cr , these theories thus exhibit quasi-conformal behav-ior. If the boundary in N f between the NACP and theQCD-type regime involves a continuous, although non-analytic, change in the values of anomalous dimensions,then our previous scheme-independent results in [34]-[39]and our results here may give some approximate guide tothe values of the corresponding anomalous dimensions inquasi-conformal theories with N f slightly less than N f,cr .[46] J. A. Gracey, Phys. Lett. B , 175 (2000).[47] See, e.g., I. M. Gelfand, R. A. Minlos, and Z. Ya Shapiro, Representations of the Rotation and Lorentz Groups andTheir Applications (Pergamon Press, New York, 1963);W.-K. Tung,
Group Theory in Physics (World Scientific,Singapore, 1985). [48] G. Mack, Commun. Math. Phys. , 1 (1977); B. Grin-stein, K. Intriligator, and I. Rothstein, Phys. Lett. B ,367 (2008); Y. Nakayama, Phys. Repts. , 1 (2015).[49] For a given representation R of G , d abcdR =(1 / R [ T a ( T b T c T d + T b T d T c + T c T b T d + T c T d T b + T d T b T c + T d T c T b )]; see, e.g., T. van Ritbergen, A. N.Schellekens, and J. A. M. Vermaseren, Int. J. Mod. Phys.A , 41 (1999).[50] O. V. Tarasov, A. A. Vladimirov, and A. Yu. Zharkov,Phys. Lett. B , 429 (1980); S. A. Larin and J. A. M.Vermaseren, Phys. Lett. B , 334 (1993).[51] T. van Ritbergen, J. A. M. Vermaseren, and S. A. Larin,Phys. Lett. B , 379 (1997).[52] K. G. Chetyrkin, Phys. Lett. B , 161 (1997); J. A.M. Vermaseren, S. A. Larin, and T. van Ritbergen, Phys.Lett. B , 327 (1997).[53] E. G. Floratos, D. A. Ross, and C. T. Sachrajda, Nucl.Phys. B , 66 (1977); erratum Nucl. Phys. B , 545(1978).[54] J. A. Gracey, Nucl. Phys. B , 242 (2003).[55] J. A. Gracey, JHEP (2006) 040.[56] See also S. Kumano and M. Miyama, Phys. Rev. D ,R2504 (1997); V. N. Velizhanin, Nucl. Phys. B , 113(2012).[57] With our definition (2.1) [44], the inputs for our calcu-lations of anomalous dimensions of higher-spin operatorsfrom Refs. [54] and [55] involve multiplication by a factorof ( − , 3998 (1978).[59] F. Herzog, B. Ruijl, T. Ueda, J. A. M. Vermaseren, andA. Vogt, JHEP 02 (2017) 090.[60] P. A. Baikov, K. G. Chetyrkin, and J. H. K¨uhn, Phys.Rev. Lett. , 082002 (2017).[61] Some of the lowest values of N f in these tables and figuresmay lie below N f,cr . However, since at present there isnot a complete consensus among lattice groups concern-ing the value of N f,cr for this SU(3) theory with fermionsin the fundamental representation, R = F [41, 42], we listthese values for completeness. TABLE I:
Signs of scheme-independent expansion coefficients κ ( O ) n for gauge group G = SU( N c ) with N c ≥ R = F (fundamental). O κ ( O )1 , SU( N c ) ,F κ ( O )2 , SU( N c ) ,F κ ( O )3 , SU( N c ) ,F ¯ ψψ + + +¯ ψσ λµ ψ − − +¯ ψγ µ D µ ψ − − − ¯ ψγ µ D µ D µ ψ − − − ¯ ψγ µ D µ D µ D µ ψ − − − ¯ ψσ λµ D µ ψ − − − ¯ ψσ λµ D µ D µ ψ − − − ¯ ψσ λµ D µ D µ D µ ψ − − − TABLE II:
Values of the anomalous dimension γ ( γD ) IR,F calculatedto O (∆ pf ), denoted γ ( γD ) IR,F, ∆ pf , with 1 ≤ p ≤
3, for G = SU(3), as afunction of N f . N f γ ( γD ) IR,F, ∆ f γ ( γD ) IR,F, ∆ f γ ( γD ) IR,F, ∆ f − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . TABLE III:
Values of the anomalous dimension γ ( γDD ) IR,F calcu-lated to O (∆ pf ), denoted γ ( γDD ) IR,F, ∆ pf , with 1 ≤ p ≤
3, for G = SU(3),as a function of N f . N f γ ( γDD ) IR,F, ∆ f γ ( γDD ) IR,F, ∆ f γ ( γDD ) IR,F, ∆ f − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . Values of the anomalous dimension γ ( γDDD ) IR,F calcu-lated to O (∆ pf ), denoted γ ( γDDD ) IR,F, ∆ pf , with 1 ≤ p ≤
3, for G = SU(3),as a function of N f . N f γ ( γDDD ) IR,F, ∆ f γ ( γDDD ) IR,F, ∆ f γ ( γDDD ) IR,F, ∆ f − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . Values of the anomalous dimension γ ( σ ) IR,F calculatedto O (∆ pf ), denoted γ ( σ ) IR,F, ∆ pf , with 1 ≤ p ≤
3, for G = SU(3), as afunction of N f . N f γ ( σ ) IR,F, ∆ f γ ( σ ) IR,F, ∆ f γ ( σ ) IR,F, ∆ f − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . TABLE VI:
Values of the anomalous dimension γ ( σD ) IR,F calculatedto O (∆ pf ), denoted γ ( σD ) IR,F, ∆ pf , with 1 ≤ p ≤
3, for G = SU(3), as afunction of N f . N f γ ( σD ) IR,F, ∆ f γ ( σD ) IR,F, ∆ f γ ( σD ) IR,F, ∆ f − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . Values of the anomalous dimension γ ( σDD ) IR,F calcu-lated to O (∆ pf ), denoted γ ( σDD ) IR,F, ∆ pf , with 1 ≤ p ≤
3, for G = SU(3),as a function of N f . N f γ ( σDD ) IR,F, ∆ f γ ( σDD ) IR,F, ∆ f γ ( σDD ) IR,F, ∆ f − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . Values of the anomalous dimension γ ( σDDD ) IR,F calcu-lated to O (∆ pf ), denoted γ ( σDDD ) IR,F, ∆ pf , with 1 ≤ p ≤
3, for G = SU(3),as a function of N f . N f γ ( σDDD ) IR,F, ∆ f γ ( σDDD ) IR,F, ∆ f γ ( σDDD ) IR,F, ∆ f − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . TABLE IX:
Values of the ˆ κ ( O ) n coefficients for G = SU( N c ) and R = F in the LNN limit. The operators are indicated by theirshorthand symbols, so 1 refers to ¯ ψψ ; σ refers to ¯ ψσ λµ ψ ; γD to¯ ψγ µ D µ ψ , etc. The notation a e- n means a × − n . O ˆ κ ( O )1 ˆ κ ( O )2 ˆ κ ( O )3 σ − . − . γD − . − . − . γDD − . − . − . γDDD − . − . − . σD − . − . − . σDD − . − . − . σDDD − . − . − . γ ( γD ) IR,F of the oper-ator ¯ ψγ µ D µ ψ at the IRFP for the theory with G = SU(3),and N f fermions in the fundamental representation, calcu-lated to order O (∆ pf ), where p = 1 , ,
3. Denoting theanomalous dimension calculated to order O (∆ pf ) as γ ( γD ) IR,F, ∆ pf ,the curves, from top to bottom (with colors online), refer to γ ( γD ) IR,F, ∆ f (red), γ ( γD ) IR,F, ∆ f (green), and γ ( γD ) IR,F, ∆ f (blue). FIG. 2: Plot of the anomalous dimension γ ( γDD ) IR,F of the op-erator ¯ ψγ µ D µ D µ ψ at the IRFP for G = SU(3), and N f fermions in the fundamental representation, calculated to or-der O (∆ pf ), where p = 1 , ,
3. Denoting the anomalousdimension calculated to order O (∆ pf ) as γ ( γDD ) IR,F, ∆ pf , the curves,from top to bottom (with colors online), refer to γ ( γDD ) IR,F, ∆ f (red), γ ( γDD ) IR,F, ∆ f (green), and γ ( γDD ) IR,F, ∆ f (blue).FIG. 3: Plot of the anomalous dimension γ ( γDDD ) IR,F of theoperator ¯ ψγ µ D µ D µ D µ ψ at the IRFP for G = SU(3), and N f fermions in the fundamental representation, calculated toorder O (∆ pf ), where p = 1 , ,
3. Denoting the calculationto order O (∆ pf ) as γ ( γDDD ) IR,F, ∆ pf , from top to bottom (with colorsonline), the colors refer to γ ( γDDD ) IR,F, ∆ f (red), γ ( γDDD ) IR,F, ∆ f (green),and γ ( γDDD ) IR,F, ∆ f (blue). FIG. 4: Plot of the anomalous dimension γ ( σ ) IR,F of the opera-tor ¯ ψσ λµ ψ at the IRFP for G = SU(3), and N f fermions inthe fundamental representation, calculated to order O (∆ pf ),where p = 1 , ,
3. Denoting the calculation to order O (∆ pf )as γ ( σ ) IR,F, ∆ pf , from top to botton (with colors online), the col-ors refer to γ ( σ ) IR,F, ∆ f (red), γ ( σ ) IR,F, ∆ f (green), and γ ( σ ) IR,F, ∆ f (blue).FIG. 5: Plot of the anomalous dimension γ ( σD ) IR,F of the opera-tor ¯ ψσ λµ D µ ψ at the IRFP for G = SU(3), and N f fermionsin the fundamental representation, calculated to order O (∆ pf ),where p = 1 , ,
3. Denoting the calculation to order O (∆ pf )as γ ( σD ) IR,F, ∆ pf , from top to botton (with colors online), the col-ors refer to γ ( σD ) IR,F, ∆ f (red), γ ( σD ) IR,F, ∆ f (green), and γ ( σD ) IR,F, ∆ f (blue). FIG. 6: Plot of the anomalous dimension γ ( σDD ) IR,F of the op-erator ¯ ψσ λµ D µ D µ ψ at the IRFP for G = SU(3), and N f fermions in the fundamental representation, calculated to or-der O (∆ pf ), where p = 1 , ,
3. Denoting the calculation toorder O (∆ pf ) as γ ( σDD ) IR,F, ∆ pf , from top to botton (with colors on-line), the colors refer to γ ( σDD ) IR,F, ∆ f (red), γ ( σDD ) IR,F, ∆ f (green), and γ ( σDD ) IR,F, ∆ f (blue).FIG. 7: Plot of the anomalous dimension γ ( σDDD ) IR,F of theoperator ¯ ψσ λµ D µ D µ D µ ψ at the IRFP for G = SU(3), and N f fermions in the fundamental representation, calculated toorder O (∆ pf ), where p = 1 , ,
3. Denoting the calculationto order O (∆ pf ) as γ ( σDDD ) IR,F, ∆ pf , from top to bottom (with colorsonline), the colors refer to γ ( σDDD ) IR,F, ∆ f (red), γ ( σDDD ) IR,F, ∆ f (green),and γ ( σDDD ) IR,F, ∆ ff