Scheme-Independent Series for Anomalous Dimensions of Higher-Spin Operators at an Infrared Fixed Point in a Gauge Theory

We consider an asymptotically free vectorial gauge theory, with gauge group $G$ and $N_f$ fermions in a representation $R$ of $G$, having an infrared fixed point of the renormalization group. We calculate scheme-independent series expansions for the anomalous dimensions of higher-spin bilinear fermion operators at this infrared fixed point up to $O(\Delta_f^3)$, where $\Delta_f$ is an $N_f$-dependent expansion variable. Our general results are evaluated for several special cases, including the case $G={\rm SU}(N_c)$ with $R$ equal to the fundamental and adjoint representations.


I. INTRODUCTION
An asymptotically free gauge theory with sufficiently many massless fermions evolves from the deep ultraviolet (UV) to an infrared fixed point (IRFP) of the renormalization group at a zero of the beta function. The theory at this IRFP exhibits scale-invariance due to the vanishing of the beta function. The properties of the theory at this IRFP are of fundamental field-theoretic interest. Among the basic properties are the anomalous dimensions γ (O)

IR of various gauge-invariant operators O.
In this paper we consider an asymptotically free vectorial 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 representation R of G [1]. We present scheme-independent series expansions of the anomalous dimensions of gaugeinvariant higher-spin operators that are bilinear in the fermion fields, up to O(∆ 3 f ) inclusive, at the infrared fixed point, where ∆ f is an N f -dependent expansion variable defined below, in Eq. (1. 8). The operators that we consider are of the form (suppressing flavor indices) ψγ µ1 D µ2 ...D µj ψ andψσ λµ1 D µ2 ...D µj ψ, where D µ is the covariant derivative for the gauge theory, and it is understood here and below that the operators are symmetrized over the Lorentz indices µ i , 1 ≤ i ≤ j and have Lorentz traces subtracted, and σ λµ1 is the commutator of two Dirac matrices (defined in Eq. (2.3)). We consider the cases 1 ≤ j ≤ 3.
The operatorsψγ µ1 D µ2 ...D µj ψ were considered early on in the analysis of approximate Bjorken scaling in deep inelastic lepton scattering and the associated development of the theory of quantum chromodynamics (QCD). We briefly review this background [3]- [13]. In Euclidean quantum field theory, the short-distance operator product expansion (OPE) expresses the product of two op- where |x − y| refers to the Euclidean distance. Hence, in the short-distance OPE, the operators with the lowest dimensions dominate, since they are multiplied by the smallest powers of |x−y|. However, deep inelastic scattering and the associated Bjorken limit probe the light cone limit, (x − y) 2 → 0 with x − y = 0 in Minkowski space, where x 2 = x µ x µ . With the arguments of two illustrative Lorentz-scalar operators denoted in a symmetric manner as ±x/2, the light-cone OPE for A(x/2)B(−x/2) is A(x/2)B(−x/2) = i,nc i,n (x 2 ) x µ1 · · · x µn O i,n;µ1,...,µn (0) (1.3) in the limit x 2 → 0, where the coefficient functions have been written in a form that explicitly indicates the factor x µ1 · · · x µn and the operator O i,n;µ1,...,µn has spin j = n. Here (suppressing the Lorentz indices on O i,n;µ1,...,µn ) the dependence ofc i,n on x 2 is c i,n (x 2 ) ∼ (x 2 ) (dO i,n −n−dA−dB )/2 (1.4) (with logarithmic corrections in QCD due to anomalous dimensions). Consequently, the operators that have the strongest singularity in their coefficient functionc i,n (x 2 ) as x 2 → 0 and hence make the dominant contribution to the right-hand side of the light-cone OPE, Eq. (1. 3) are those with minimal "twist" τ [7], where τ is the dimension minus the spin j of the operator, i.e., [14]. In a similar manner, twist-2 operators make the dominant contribution to the right-hand side of the lightcone OPE for the product of two electromagnetic or weak currents. The other operators that we consider, namelȳ ψσ λµ1 D µ2 ...D µj ψ, have been relevant for the study of transversity distributions in QCD [15]. Our approach here is complementary to these previous analyses of higher-spin operators, which have focused on applications to QCD. In contrast, we study the anomalous dimensions of these operators at an infrared fixed point in a (deconfined) chirally symmetric non-Abelian Coulomb phase (NACP), where the theory is scale-invariant and is inferred to be conformally invariant [16], whence the commonly used term "conformal window". The goal of our calculations is to gain information about the properties of the conformal field theory that is defined at this IRFP.
Let us recall some further relevant background for our work. The evolution of the running gauge coupling g = g(µ), as a function of the momentum scale, µ, is described by the renormalization-group (RG) beta function β = dα/d ln µ, where α(µ) = g(µ) 2 /(4π). From the one-loop term in the beta function [10,11], it follows that the property of asymptotic freedom restricts N f to be less than an upper (u) bound, N u , where [17] 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 has an infrared zero in the (perturbatively calculated) beta function, i.e., an IR fixed point of the renormalization group, at a value that we shall denote α IR [19,20]. In the two-loop beta function (with N f < N u as required by asymptotic freedom), this IR zero is present if N f is larger than a lower (ℓ) value N ℓ , where [19] As the scale µ decreases from large values in the UV to small values in the IR, α(µ) approaches α IR from below as µ → 0. Here we consider the properties of the theory at this IRFP in the perturbative beta function. (For a discussion of an IR zero in a nonperturbatively defined beta function and its application to QCD, see [21].) Since the anomalous dimensions of gauge-invariant operators evaluated at the IRFP are physical, they must be independent of the scheme used for regularization and renormalization. In the conventional approach, one first expresses these anomalous dimensions as series expansions in powers of α or equivalently a = g 2 /(16π 2 ) = α/(4π), calculated to n-loop order; second, one computes the IR zero of the beta function, denoted α IR,n , to the same n-loop order; and third, one sets α = α IR,n in the series expansion for the given anomalous dimension to obtain its value at the IR zero of the beta function to this n-loop order. For the operatorψψ this conventional approach to calculate anomalous dimensions at an IR fixed point was carried out to the four-loop level in [22][23][24] and to the five-loop level in [25]. However, these conventional series expansions in powers of α, calculated to a finite order, are scheme-dependent beyond the leading terms. This is a well-known property of higher-order QCD calculations used to fit actual experimental data, which, in turn, has motivated many studies to reduce scheme dependence [26]. These studies dealt with the UV fixed point (UVFP) at α = 0, as is appropriate for QCD. Studies of scheme dependence of quantities calculated in a conventional manner at an IR fixed point at α IR were carried out in [27]- [31]. In particular, it was shown that many scheme transformations that are admissible in the vicinity of the UVFP at α = 0 in an asymptotically free theory are not admissible away from the origin because of various pathological properties. For example, the scheme transformation ra = tanh(ra ′ ) (depending on a parameter r) is an admissible transformation in the neighborhood of α = α ′ = 0. However, the inverse of this transformation is a ′ = (2r) −1 ln[(1 + ra)/(1 − ra)], which is singular at an IRFP with a IR ≥ 1/r, i.e., α IR ≥ 4π/r, so that the transformation is not admissible at this IRFP. Refs. [27] derived and studied an explicit scheme transformation that removes terms of loop order 3 and higher from the beta function in the local vicinity of α = 0, as is relevant to the UVFP in QCD [32], but also showed that such a scheme transformation cannot, in general be used at an IRFP away from the origin owing to various pathologies, one of which was illustrated above.
It is thus desirable to use a theoretical framework in which the series expansions for physical quantities, such as anomalous dimensions of gauge-invariant operators at the IRFP, are scheme-independent at any finite order in an expansion variable. Because α IR → 0 as N f approaches N u from below (where N f is formally generalized here from a non-negative integer to a non-negative real number [17]), one can reexpress the expansions for physical quantities at the IRFP as power series in the manifestly scheme-independent quantity [20,33] (1. 8) In previous work we have calculated scheme-independent expansions for anomalous dimensions of several types of gauge-invariant operators at an IRFP in an asymptotically free gauge theory [34]- [40]. We have compared the resultant values for anomalous dimensions with lattice measurements where available [35]- [37], [41,42].
In the present paper we extend these calculations to the case of the higher-spin operatorsψγ µ1 D µ2 ...D µj ψ andψσ λµ1 D µ2 ...D µj ψ for 1 ≤ j ≤ 3. In addition to general formulas, we present results for several different special cases, including the case where G = SU(N c ) and the fermions are in the fundamental (F ) and adjoint (Adj) representations. We also give results for the limit N c → ∞ and N f → ∞ with the ratio N f /N c fixed and finite. Our calculations show that these scheme-independent expansions of the anomalous dimensions of the operators are reasonably accurate throughout much of the non-Abelian Coulomb phase. Our results give further insight into the properties of a theory at an IRFP and should be useful to compare with lattice measurements of the anomalous dimensions of these higher-spin operators when such measurements will be performed [43].
This paper is organized as follows. Some relevant background and methods are discussed in Sect. II. General structural forms for the anomalous dimensions of higherspin bilinear fermion operators are given in Sect. III. In Sect. IV we present our scheme-independent calculations of the anomalous dimensions of these higher-spin Wilson operators for a general gauge group G and fermion representation R. In Sect. V we give results for the case where G = SU(N c ) and R is the fundamental representation, and in Sect. VI we present the special case of these results for the limit N c → ∞ and N f → ∞ with N f /N c fixed and finite. Anomalous dimension calculations for the case where G = SU(N c ) and R is the adjoint representation are presented in Sect. VII. Our conclusions are given in Sect. VIII and some auxiliary results are included in Appendix A.

Let us consider a (gauge-invariant) operator O.
Because of the interactions, the full scaling dimension of this operator, denoted D O , differs from its free-field value, where γ O is the anomalous dimension of the operator [44]. Since γ O arises from the gauge interaction, it can be expressed as the power series We focus on the operators with 1 ≤ j ≤ 3. We introduce the following compact notation for these operators: (2.9) For brevity of notation, we suppress the flavor indices on the fields ψ. For a given operator O, we write the schemeindependent expansion of its anomalous dimension γ (O) evaluated at the IRFP, denoted γ The truncation of right-hand side of Eq. (2.10) to maximal power p is denoted We use a further shorthand notation for the anomalous dimensions in which the superscript in γ µ1µ2 =ψγ µ1 D µ2 ψ at the IRFP, and so forth for the other operators. In comparing with our previous calculations in [34]- [39], we also use the notation γ IR was denoted γ T,IR in [36], where the subscript T referred to the Dirac tensor σ µν .) As discussed in [34,36], the calculation of the coefficient κ (O) n in Eq. (2.10) requires, as inputs, the beta function coefficients at loop order 1 ≤ ℓ ≤ n + 1 and the anomalous dimension coefficients c (O) γ,ℓ at loop order 1 ≤ ℓ ≤ n. The method of calculation requires that the IR 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-Abelian Coulomb phase (conformal window) [45]. For a given gauge group G and fermion representation R, the conformal 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 has been investigated extensively for several groups G and fermion representations R [41,42,45]. For values of N f in the non-Abelian Coulomb phase such that ∆ f is not too large, one may expect the expansion (2.10) of γ in a series in powers of ∆ f to yield reasonably accurate perturbative calculations of the anomalous dimension. In our earlier works, using our explicit calculations, we have shown that this is, in fact, the case.
We recall some relevant properties of the theory regarding global flavor symmetries. Because the N f fermions are massless, the Lagrangian is invariant under the classical global flavor (f l) symmetry (where V and A denote vector and axial-vector). The U(1) V represents fermion number, which is conserved by the bilinear operators that we consider. The U(1) A symmetry is broken by instantons, so the actual nonanomalous global flavor symmetry is (2.13) This G f l symmetry is respected in the non-Abelian Coulomb phase, since there is no spontaneous chiral symmetry breaking in this phase [41,42]. For our operators, the flavor matrix betweenψ and ψ is either the identity or the operator T a , a generator of SU(N f ), which can be viewed as acting either to the right on ψ or to the left onψ. These yield the same anomalous dimensions [46]. As a consequence of the unbroken global flavor symmetry, our operators transform as representations of the global flavor group G f l . The invariance under the full G f l in the non-Abelian Coulomb phase is different from the situation in the QCD-like phase at smaller N f , where the chiral part of G f l is spontaneously broken by the QCD bilinear quark condensate to the vectorial subgroup SU(N f ) V and operators are classified according to whether they are singlet or nonsinglet (adjoint) under this vectorial SU(N f ) symmetry. In particular, in the consideration of flavor-singlet operators, in QCD, one must take into account mixing with gluonic operators.
Here, in contrast, there is no mixing between any of our bilinear fermion operators and gluonic operators, since the latter are singlets under G f l . The operators O with an even number of Dirac γ matrices, symbolically denoted Γ e , link left with right chiral components of ψ, while the operators O with an odd number of Dirac γ matrices, Γ o , link left with left and right with right components: In the non-Abelian Coulomb phase where the flavor symmetry is (2.13), one may regard the T b in the termψ L T b ψ R acting to the right as an element of SU(N f ) R and acting to the left as an element of SU(N f ) L . Given that the theory at the IR fixed point is conformally invariant [16], there is an important lower bound on the full dimension of an operator O and hence, with our definition (2.1), an upper bound on the anomalous dimension 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 the set (j 1 , j 2 ), where j 1 and j 2 take on nonnegative integral or half-integral values [47]. A Lorentz scalar operator thus transforms as (0, 0), a Lorentz vector as (1/2, 1/2), an antisymmetric tensor like the field-strength tensor F a µν as (1, 0) ⊕ (0, 1), etc. Then the requirement of unitarity in a conformally invariant theory implies the lower bound [48] 16) i.e., the upper bound We have studied the constraints from the upper bound (2.17) in our previous calculations of anomalous dimensions in [22,25], [36]- [39]. Anticipating the results given below, since our calculations yield negative values for the anomalous dimensions of higher-spin Wilson operators, they obviously satisfy these conformality upper bounds.
III. SOME GENERAL STRUCTURAL PROPERTIES OF γ IR . These involve various group invariants, including the quadratic Casimir invariants C A ≡ C 2 (G), C f ≡ C 2 (R), the trace invariant T (R), and the quartic trace invariants d abcd R d abcd R ′ /d A , where d A denotes the dimension of the adjoint representation [18,49]. For compact notation, it is convenient to define a factor that occurs in the denominators of these κ (O) n coefficients, namely.
(not to be confused with covariant derivative). We exhibit this general form here, using a (O) j,k for various (constant) numerical coefficients: and In this section we present the results of our calculations of the coefficients in the scheme-independent series expansions up to O(∆ 3 f ) for the various higher-spin operators considered here. As was noted above, the calculation n , for the anomalous dimension of an operator O at the IRFP requires, as inputs, the beta function coefficients at loop order 1 ≤ ℓ ≤ n + 1 and the anomalous dimension coefficients c (O) ℓ at loop order 1 ≤ ℓ ≤ n. Hence, we use the beta function coefficients from one-loop up to the four-loop level [10,19], [50,51], together with the anomalous dimension coefficients calculated in the conventional series expansion in powers of a up to the three-loop level [11], [46], [52]- [57]. The higher-order terms in the beta function and anomalous dimensions that we use have been calculated in the MS scheme [58], but our results are independent of this since they are scheme-independent. (The beta function has actually been calculated up to five-loop order [59,60], but these results will not be needed here.) For the anomalous dimension γ (γD) IR of the operator ψγ µ1 D µ2 ψ at the IRFP, we calculate and κ (γD) 3 In these expressions and the following ones, we have indicated the factorizations of the numbers in the denominators, since they are rather simple. In general, the numbers in the numerators do not have such simple factorizations.
With these coefficients, the anomalous dimension γ with p = 1, 2, 3. Analogous state-ments apply to the anomalous dimensions of the other operators for which we have performed calculations, and we proceed to present the coefficients for these next.

C. γ (γDD) IR
For the anomalous dimension γ (γDD) IR of the operator ψγ µ1 D µ2 D µ3 ψ at the IRFP, we calculate Proceeding to the anomalous dimension γ (γDDD) IR of the operatorψγ µ1 D µ2 D µ3 D µ 4 ψ at the IRFP, we find and of the operator ψσ λµ1 D µ2 ψ at the IRFP, we calculate , (4.14) and Finally, for the anomalous dimension γ σDDD) IR we obtain and , In a similar manner, from our general formulas (4.7)-(4.9), we find and For this case we have We remark on the signs of these coefficients. It is evident from Eqs. 3 , for these operators are also negative for the theory with G = SU(N c ) and fermions in the fundamental representation, R = F , in the full range N c ≥ 2 of relevance here. In Table IV we list the signs of these coefficients κ It is interesting to note that for all of the higher-spin operators O that we consider, the anomalous dimensions γ IR that we calculate are negative (with our sign convention in (2.1) [44])). They thus have the same sign as the sign of the anomalous dimension of the operatorψσ µν ψ and are opposite in sign relative to the anomalous dimensions that we calculated forψψ in our previous work [22], [34]- [39].
In Tables II-VIII we  ., ∆ f = 0) within this conformal window [61]. The numbers in Table V) are evaluations of our analytic results given in [36] and are included for comparison.
In Figs. 1-7 we show plots of these anomalous dimensions for the SU(3) theory with R = F . The plot of the anomalous dimension forψσ λµ1 ψ is based on the analytic results of our earlier paper [36] but was not given there and is new here. As can be seen from these tables and figures, the higher-order terms in the ∆ f expansion are sufficiently small that it is expected to be reliable throughout much of the non-Abelian Coulomb phase (i.e., conformal window). As is obvious, since our calculations are finite series expansions in powers of ∆ f , they are most accurate in the upper part of the NACP, where this expansion parameter ∆ f is small. This is similar to what we found in our earlier scheme-independent calculations of anomalous dimensions [34]- [39]. In the figures, this is evident from the fact that the curves for the anomalous dimensions calculated to O(∆ 3 f ) are reasonably close to the corresponding curves for these anomalous dimensions calculated to order O(∆ 2 f ).
In a theory with gauge group SU(N c ) and fermions in the fundamental representation, R = F , it is of interest to consider the limit N c → ∞ , N F → ∞ with r ≡ N F N c fixed and finite and 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-Veneziano limit. It has the simplifying feature that rather than depending on N c and N f , the properties of the theory only depend on their ratio, r. The scheme-independent expansion parameter in this LNN limit is and Here we evaluate these scheme-independent anomalous dimension coefficients in a theory with G = SU(N c ) and R = F , in the LNN limit. The rescaled coefficients that are finite in the LNN limit arê The anomalous dimension γ IR is also finite in this limit and is given by n ∆ n r . (6.9) As r decreases from its upper limit, r u , to r ℓ , the expansion variable ∆ r increases from 0 to (∆ r ) max = 75 26 = 2.8846 for r ∈ I IRZ,r . (6.10) In this LNN limit, the values ofκ with 1 ≤ n ≤ 3 for the operators O considered here are listed in Table  IX. For comparison, we also include the corresponding values ofκ (O) n for the operatorsψψ andψσ µν ψ that we had calculated in [36].

VIII. CONCLUSIONS
In conclusion, in this paper we have calculated scheme-independent expansions up to O(∆ 3 f ) inclusive for the anomalous dimensions of the higher-spin, twist-2 bilinear fermion operatorsψγ µ1 D µ2 ...D µj ψ and ψσ λµ1 D µ2 ...D µj ψ with j up to 3, evaluated at an IR fixed point in the non-Abelian Coulomb phase of an asymptotically 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 special cases, including the case G = SU(N c ) with R equal to the fundamental and adjoint representations. We have presented our results in convenient tabular and graphical formats. For fermions in the fundamental representation, we also analyze the limit N c → ∞ and N f → ∞ with N f /N c fixed and finite. A comparison with our previous scheme-independent calculations of the corresponding anomalous dimensions ofψψ andψσ µν ψ has also been given. Our new results further elucidate the properties of conformal field theories. With the requisite inputs, one could extend these scheme-independent calculations to higher-spin operators and to higher order in powers of ∆ f . It is hoped that lattice measurements of these anomalous dimensions of higher-spin operators in the conformal window will be performed in the future, and it will be of interest to compare our calculations with lattice results when they will become available. scheme-independent series expansions of the anomalous dimensions γ (O) IR for O =ψψ and O =ψσ µν ψ. Following the same shorthand notation as in the text, we denote the coefficients at order O(∆ n f ) in the scheme-independent series expansions (2.10) for these anomalous dimensions as κ (1) n and κ (σ) n . We calculated and κ (1) 3 (where the denominator factor D was defined in Eq. (3.1)). In [37,39] we presented results for the next-higher order coefficient, κ 4 , but these are not needed here. For the κ (σ) n we found κ (σ) For G = SU(N c ) and R = F , in the LNN limit, these yield the rescaled coefficientŝ      n coefficients for G = SU(Nc) and R = F in the LNN limit. The operators are indicated by their shorthand symbols, so 1 refers toψψ; σ refers toψσ λµ 1 ψ; γD tō ψγµ 1 Dµ 2 ψ, etc. The notation ae-n means a × 10 −n .