Scheme-Independent Calculations of Properties at a Conformal Infrared Fixed Point in Gauge Theories with Multiple Fermion Representations

In previous work we have presented scheme-independent calculations of physical properties of operators at a conformally invariant infrared fixed point in an asymptotically free gauge theory with gauge group $G$ and $N_f$ fermions in a representation $R$ of $G$. Here we generalize this analysis to the case of fermions in multiple representations, focusing on the case of two different representations. Our results include the calculation of the anomalous dimensions of gauge-invariant fermion bilinear operators, and the derivative of the beta function, evaluated at the infrared fixed point. We illustrate our results in an SU($N_c$) gauge theory with $N_F$ fermions in the fundamental representation and $N_{Adj}$ fermions in the adjoint representation.


I. INTRODUCTION
In this paper we shall consider a vectorial, asymptotically free gauge theory (in four spacetime dimensions, at zero temperature) with gauge group G with massless fermions transforming according to multiple different representations of G, which has an exact infrared (IR) fixed point (IRFP) of the renormalization group [1]. For technical simplicity, we will restrict ourselves to two different representations. We thus take the theory to contain N f copies (flavors) of Dirac fermions, denoted f , in the representation R of G, and N f ′ copies of fermions, denoted f ′ , in a different representation R ′ of G. In the case in which f ′ transforms according to a self-conjugate representation, the number N f ′ refers equivalently to a theory with N f ′ Dirac fermions or 2N f ′ Majorana fermions and hence in this case N f ′ may take on half-integral as well as integral values. One motivation for such theories is a possible direction for ultraviolet completions of the Standard Model (e.g., [2,3]). In [3] we studied the infrared evolution and phase structure of this type of theory. Here we go beyond Refs. [2,3] in presenting (scheme-independent) calculations of anomalous dimensions of gauge-invariant operators.
We denote the running gauge coupling as g = g(µ), where µ is the Euclidean energy/momentum scale at which this coupling is measured. We define α(µ) = g(µ) 2 /(4π). Since the theory is asymptotically free, its properties can be computed reliably in the deep ultraviolet (UV) region at large µ, where the coupling approaches zero. The dependence of α(µ) on µ is described by the renormalization-group (RG) beta function, β = dα(µ)/dt, where dt = d ln µ (the argument µ will often be suppressed in the notation). We will consider a theory in which the fermion content is such that the RG flow from the UV to the IR ends in an exact IR fixed point, as determined by the zero in the beta function nearest to the origin for physical coupling, denoted α IR . Since β = 0 at α = α IR , the resultant theory in this IR limit is scale-invariant, and is deduced also to be conformally invariant [4].
The properties of the resultant conformal field theory at this IRFP are of considerable importance. Physical quantities defined at the IRFP obviously cannot depend on the scheme used for the regularization and renormalization of the theory. In conventional computations of these quantities, one first writes them as series expansions in powers of the coupling, and then evaluates these series expansions with α set equal to α IR , calculated to a given loop order. These calculations have been performed for anomalous dimensions of gauge-invariant fermion bilinears in a theory with a single fermion representation up to four-loop level [5]- [7] and to five-loop level [8]. However, as is well known, these conventional (finiteorder) series expansions are scheme-dependent beyond the leading terms. Indeed, this is a generic property of higher-order calculations in quantum field theory, such as computations in quantum chromodynamics (QCD) used to compare with data from the Fermilab Tevatron and CERN Large Hadron Collider (LHC).
There is thus strong motivation to calculate and analyze series expansions for physical properties at the IRFP which are scheme-independent at each finite order. The fact that makes this possible is simple but powerful. To review this, we first specialize to a theory with N f fermions in a single representation, R, of the gauge group G. The constraint of asymptotic freedom means that N f must be less than a certain upper (u) bound, denoted N f,u . Here and below, we will often formally generalize the number(s) of fermions in one or multiple representations from non-negative integers to non-negative real numbers, with the understanding that for a physical quantity one restricts to integral values. Furthermore, as noted above, if an f ′ fermion transforms according to a self-conjugate representation, then the number N f ′ refers equivalently to a theory with N f ′ Dirac fermions or 2N f ′ Majorana fermions, so that in this case, N f ′ may take on half-integral physical values. As N f approaches N f,u from below, the value of the IRFP, α IR , approaches zero. This means that one can re-express series expansions for physical quantities at this IRFP in powers of the manifestly scheme-independent variable [9,10] ∆ f = N f,u − N f . (1.1) In recent work, for theories with N f fermions in a single representation of the gauge group G, we have calculated scheme-independent series expansions for the anomalous dimensions of gauge-invariant fermion bilinears and the derivative dβ/dα, both evaluated at the IRFP, to the respective orders O(∆ 4 f ) and O(∆ 5 f ) [11]- [19]. These are the highest orders to which these quantities have been calculated. We gave explicit expressions for the case G = SU(N c ) and R equal to the fundamental, adjoint, and rank-2 symmetric and antisymmetric tensor representations, and for other Lie groups, including orthogonal, symplectic, and exceptional groups.
In this paper we shall generalize our previous schemeindependent series calculations of physical quantities at an IRFP from the case of an asymptotically free gauge theory with N f fermions in a single representation of the gauge group G to the case of fermions in multiple different representations. Specifically, we consider a theory with N f fermions in a representation R of G and N f ′ fermions in a different representation, R ′ , of G. We present scheme-independent calculations of the anomalous dimensions of gauge-invariant fermion bilinear operators to cubic order in the respective expansion variable and to quartic order in ∆ f and ∆ f ′ for the derivative of the beta function, evaluated at the infrared fixed point.
The condition of asymptotic freedom requires that the value of a certain linear combination of N f ′ and N f must be less than an upper bound given below by Eq. (2.3). For a fixed N f ′ , this implies an upper bound denoted as N f < N f,u , and for a fixed N f , this implies the upper bound N f ′ < N f ′ ,u given respectively in Eqs. (2.4) and (2.5) below. For fixed N f ′ , as N f approaches N f,u from below, α IR approaches zero. Therefore, one can rewrite the series expansions for physical quantities as power series in the variable ∆ f . The coefficients in these series expansions depend on N f ′ . If ∆ f is small, the value of α IR is also small, so that the resultant IR theory may be inferred to be in a (deconfined) non-Abelian Coulomb phase (NACP), often called the conformal window. Strong evidence for this in the single-representation case comes from fully nonperturbative lattice simulations [20][21][22]. In the same way, for fixed N f , one can rewrite the series expansions for physical quantities as power series in the variable For a general operator O, we denote the full scaling dimension as D O and its free-field value as D O,f ree . The anomalous dimension of this operator, denoted γ O , is defined via the relation [23] Let us denote the fermions of type f as ψ i , i = 1, ..., N f and the fermions of type f ′ as χ j , j = 1, ..., N f ′ . We shall calculate scheme-independent series expansions for the anomalous dimensions, denoted γψ ψ,IR and γχ χ,IR of the respective (gauge-invariant) fermion bilinears The anomalous dimension ofψψ is the same as that of the (gauge-invariant) bilinear where T a is a generator of the Lie algebra of SU(N f ) [24], and we shall use the symbol γψ ψ,IR to refer to both. An analogous comment applies to γχ χ,IR . We write the schemeindependent series expansions of γf f,IR as We shall illustrate our general results in an SU(N c ) gauge theory with N F fermions of type f in the fundamental (F ) representation and N Adj fermions of type f ′ in the adjoint (Adj) representation. For this theory we will also use an explicit notation with coefficients κ (f ) = κ (F ) and κ (f ′ ) = κ (Adj) .
We shall calculate two equivalent scheme-independent series expansions of the derivative β ′ IR . With N f ′ fixed, and N f variable, one may write the series as an expansion in powers of ∆ f : Alternately, one may take N f to be fixed and write β ′ IR as a series expansion in powers of ∆ f ′ , as Note that d 1 =d 1 = 0 for all G and fermion representations. This paper is organized as follows. In Section II we discuss the methodology for our calculations. In Sections III and V we present our new results for schemeindependent expansions of the anomalous dimensions of fermion bilinears and dβ/dα, both evaluated at the infrared fixed point. We discuss the special cases of the anomalous dimension and β ′ IR results for an illustrative theory with gauge group SU(N c ) containing fermions in the fundamental and adjoint representations in Sections IV and VI, respectively. Our conclusions are given in Section VII, and some relevant group-theoretic results are reviewed in Appendix A.

A. Beta Function and Series Expansions for Physical Quantities
In this section we discuss some background and the calculational methods that are relevant for our present work. The series expansion of β in powers of the squared gauge coupling is where a = g 2 /(16π 2 ) = α/(4π) and b ℓ is the ℓ-loop coefficient. With an overall minus sign extracted, as in Eq. (2.1), the condition of asymptotic freedom is that b 1 > 0. The one-loop coefficient, b 1 , is independent of the scheme used for regularization and renormalization. Mass-independent schemes include minimal subtraction [25] and modified minimal subtraction, denoted MS [26]. For mass-independent schemes, the two-loop coefficient, b 2 , is also independent of the specific scheme used [27]. For a theory with a general gauge group G and N f fermions in a single representation, R, the coefficients b 1 and b 2 were calculated in [28] and [29], while b 3 , b 4 , and b 5 were calculated in the commonly used MS scheme in [30], [31], and [32], respectively (see also [33]). For the analysis of a theory with fermions in multiple different representations, one needs generalizations of these results. These are straightforward to derive in the case of b 1 and b 2 , but new calculations are required for higherloop coefficients. These have recently been performed in [34] (again in the MS scheme) up to four-loop order, and we use the results of Ref. [34] here. The expansion of the anomalous dimension of the fermion bilinear γψ ψ in powers of the squared gauge coupling is where c (f ) ℓ is the ℓ-loop coefficient. The analogous expansion applies for γχ χ with the replacement c is scheme-independent, while the c  [35] and ℓ = 5 in [36]. For the case of multiple fermion representations, the anomalous dimension coefficients for the fermion bilinears have been calculated up to four-loop order in [37]. We use the results of [37] up to three-loop order here.
Concerning scheme-independent series expansions, the calculation of the coefficient κ (f ) j in Eq. (1.6) requires, as inputs, the values of the b ℓ for 1 ≤ ℓ ≤ j + 1 and the c (f ) ℓ for 1 ≤ ℓ ≤ j, and similarly for κ . The calculation of the coefficients d j andd j in Eqs. (1.8) and (1.9) requires, as inputs, the values of the b ℓ for 1 ≤ ℓ ≤ j.
Thus, using the calculation of the beta function for multiple fermion representation to four-loop order in [34], together with the calculation of the anomalous dimensions of the fermion bilinears in [37] up to three-loop order, we can calculate γψ ψ,IR to order O(∆ 3 f ) and γχ χ,IR to O(∆ 3 f ′ ) for the case of multiple fermion representations. (Note that we cannot make use of the four-loop calculation of the anomalous dimensions of fermion bilinears in [37] to compute γψ ψ,IR to order O(∆ 4 f ) and γχ χ,IR to O(∆ 4 f ′ ), because this would require, as an input, the five-loop coefficient b 5 in the beta function for this case of multiple fermion representations, and, to our knowledge, this has not been calculated.) Similarly, using the four-loop beta function from [34], we can calculate the d j andd j for β ′ IR to order j = 4. We denote the truncation of these series to maximal power j = p as γψ ψ,IR,∆ p f , γχ χ,IR,∆ p f , β ′ IR,∆ p f , and β ′ IR,∆ p f ′ , respectively. Although we use these coefficients as calculated in the MS scheme below, we emphasize that our results are scheme-independent, so the specific scheme used for their calculation does not matter. An explicit illustration of this using several schemes is given in [38]. We refer the reader to our previous work for detailed discussions of the procedure for calculating the coefficients κ j and d j in the case of a theory with N f fermions in a single representation of G.
Our procedure for calculating scheme-independent series expansions requires that the IRFP be exact, and hence we restrict our consideration to the non-Abelian Coulomb phase, where this condition is satisfied. For sufficiently smaller values of N f and/or N f ′ , there is spontaneous chiral symmetry breaking (SχSB), giving rise to dynamical masses for the f and/or f ′ fermions [39]. Most-attractive channel arguments suggest that as N f and/or N f ′ decrease(s) and α IR increases, the fermion with the largest value of C f would be the first to form bilinear fermion condensates and hence obtain dynamical masses and be integrated out of the low-energy effective field theory (EFT). Assuming that this happens and, say, the f ′ fermions condense out, then one would proceed to examine the resultant EFT with the remaining massless f fermions to determine the further evolution of this theory into the infrared. The details of the construction of the EFT will not be relevant here, since we restrict our analysis to the (chirally symmetric) non-Abelian Coulomb phase.
Since we require that the theory should be asymptotically free and since our scheme-independent calculational method requires an exact IR fixed point, which is satisfied in the non-Abelian Coulomb phase, a first step is to discuss the corresponding values of the pair (N f , N f ′ ) that satisfy these conditions. We denote this set of values, or more generally, the region in the first quadrant of the R 2 plane defined by the generalization of (N f , N f ′ ) from non-negative integers (or half-integers in the case of a Majorana fermion in a self-conjugate representation) to non-negative real numbers, where the theory has an IRFP in the non-Abelian Coulomb phase as the region R N ACP . We next discuss the boundaries of this region.
For a specified gauge group G and fermion representations R and R ′ , the numbers N f and N f ′ are bounded above by the asymptotic freedom (AF) condition that b 1 > 0. This condition is expressed as the inequality on the linear combination and similarly, for fixed N f , the AF condition implies that The upper boundary of this asymptotically free region, which is also the upper boundary of the region R N ACP , in N f and N f ′ is the locus of solutions to the condition b 1 = 0. This is a finite segment of the line We may picture the first quadrant in the R 2 space defined by non-negative (N f , N f ′ ) to be such that N f is the horizontal axis and N f ′ is the vertical axis. Then the line segment bounding the asymptotically free region is an oblique line segment running from the upper left to the lower right, with slope This line segment intersects the horizontal axis at the point (N f , N f ′ ) = (11C A /(4T f ), 0) and the vertical axis ). Without loss of generality, we take f to be the (nonsinglet) fermion representation of smaller dimension. The respective scheme-independent expansions in powers of ∆ f and ∆ f ′ amount to moving into the interior of the non-Abelian Coulomb phase from the upper boundary line horizontally (moving leftward) and vertically (moving downward). In our earlier work on theories with N f fermions in a single fermion representation of the gauge group, we denoted the lower boundary of the NACP as N f,cr . In that case, we assumed that N f was in the NACP interval I N ACP : N f,cr < N f < N f,u . Here the generalization of this is the set of physical values of N f and N f ′ in the region R N ACP . Even in the case of a single fermion representation, the value of N f,cr is not known precisely. This question of the value of N f,cr for various specific theories has been investigated in a number of lattice studies [20,21], which continue at present. As noted above, we have previously presented approximate analytic results relevant for this study in [2,3] Corresponding lattice studies could be carried out for theories with multiple different fermion representations to study properties of the respective theories. An example is a recent lattice study of an SU(4) gauge theory with N f = 2 Dirac fermions in the fundamental representation and N f ′ = 2 Dirac fermions in the (selfconjugate) antisymmetric rank-2 tensor representation [40,41], which finds that the (zero-temperature) theory is in the phase with chiral symmetry breaking for both types of fermions. Since our results are restricted to an exact infrared fixed point in the (conformally invariant) non-Abelian Coulomb phase, they are not directly applicable to this theory.
For the present study, with the axes of the firstquadrant quarter plane in (N f , N f ′ ) ∈ R 2 as defined above, the upper boundary of the NACP is the line segment resulting from the b 1 = 0 condition. The analogue of the lower boundary of the NACP at N f,cr for the present study with two fermion representations is a line segment or nonlinear curve displaced in the direction to the lower left relative to the oblique b 1 = 0 line, so that the resultant NACP forms a region in which physical values of N f and N f ′ define possible IR theories. This lower boundary of the NACP intersects the horizontal axis at the point (N f , N f ′ ) = (N f,cr , 0) and intersects the vertical axis at the point (N f , N f ′ ) = (0, N f ′ ,cr ). Although this lower boundary of the NACP is not known, one can get a rough idea of where it lies by generalizing the analysis that we gave in our previous work for theories with a single fermion representation [12,13,15]. This analysis was based on the observation that the two-loop beta function has an IR zero if N f is sufficiently large that b 2 is negative (with b 1 > 0). In this case of a single fermion representation, for small N f , b 2 is positive, and turns negative when N f exceeds a certain lower (ℓ) value Thus, in this single-representation case, if and only if N f lies in an interval that we have denoted previously as I IRZ , the two-loop beta function has an IR zero (IRZ). This interval I IRZ is Although N f,ℓ is not, in general, equal to N f,cr , it is moderately close to the latter in theories that have been studied. As an example, in the case of an SU(N c ) gauge theory with N f fermions in the fundamental (F ) representation, In the intensively studied case N c = 3 theory, N ℓ = 153/19 ≃ 8.05. This is close to the estimates of N f,cr for this theory from our previous studies and from a number of lattice simulations [12,15,20,21].
In our present asymptotically free theory with two fermion representations, the two-loop beta function has an IR zero if and only if b 2 < 0, which is the inequality (2.10) This IR zero of the two-loop (2ℓ) beta function occurs at α = α IR,2ℓ , where We thus define the two-dimensional region in the first quadrant of the R 2 plane defined by non-negative real values of (N f , N f ′ ) where the theory is asymptotically free and the two-loop beta function has an IR zero as the region R IRZ , given by the conditions (2.3) and (2.10). The upper boundary of R IRZ is the same as the upper boundary of R N ACP , while the lower boundary of R IRZ can provide a rough guide to the lower boundary of R N ACP and has the advantage that it is exactly calculable. This lower boundary of the region R IRZ is given by the solution of the condition that b 2 = 0 in the first quadrant of the R 2 plane. This condition is obtained from Eq. (2.10) by replacing the inequality by an equality. The corresponding line defining the lower boundary of R IRZ has the slope This lower boundary of the region R IRZ crosses the horizontal axis in the ( , where N f,ℓ was given above in Eq. (2.7), and it crosses the vertical axis at the corresponding value (0, As noted, the lower boundary of this R IRZ region provides a rough guide to the actual lower boundary of the NACP region R N ACP . The determination of the true lower boundary of R N ACP would require a fully nonperturbative analysis, e.g., via lattice simulations. Although our calculational methods require the IRFP to be exact and hence, strictly speaking, apply only in the non-Abelian Coulomb phase, they could also be useful for the investigation of quasi-conformal gauge theories. In turn, the latter have been of interest as possible ultraviolet completions of the Standard Model. Specifically, (a) if the transition from the lower part of the non-Abelian Coulomb phase to the quasi-conformal regime in the variables (N f , N f ′ ) is continuous, and (b) if our series calculations are sufficiently accurate in this region, our results for γψ ψ,IR , γχ χ,IR , and β ′ IR could provide approximate estimates for the values of these quantities in the quasi-conformal regime just below the lower boundary with the NACP.

C. Example with Fermions in the Fundamental and Adjoint Representations
As an illustrative example, we consider a theory with the gauge group SU(N c ) that contains N f ≡ N F fermions in the fundamental (F ) representation and N f ′ ≡ N Adj fermions in the adjoint representation, Adj. We denote this as the FA theory. Here the upper boundary of the NACP region R N ACP , which is also the upper boundary of the region R IRZ , is given by the line while the line b 2 = 0 has slope FA theory : In Table I  In order for our perturbative analysis to be selfconsistent, it is necessary that α IR should not be excessively large, and so one may require, say, that α IR,2ℓ < 1. Our perturbative analysis is expected to be most accurate for the (N F , N Adj ) FA theories with small d u and hence small α IR,2ℓ in the upper part of the NACP. We will discuss this illustrative two-representation FA theory further below.

III. SCHEME-INDEPENDENT CALCULATION OF ANOMALOUS DIMENSIONS OF FERMION BILINEAR OPERATORS
In this section, for a theory with a general gauge group G containing N f fermions in a representation R and N f ′ fermions in a representation R ′ , we present our new calculations of the coefficients κ in the scheme-independent expansions of the anomalous dimensions γψ ψ,IR and γχ χ,IR in Eqs. (1.6) and the analogue for γχ χ,IR with 1 ≤ j ≤ 3. It will be useful to define a factor that occurs repeatedly in the denominators of various expressions, namely In the previously studied theory with a single fermion representation, i.e., N f ′ = 0, this factor D reduces as where as defined in Eq. (2.13) of our earlier work [13,15]. For the first two coefficients we calculate and For the third coefficient, we write It follows that the A (f ) 0 term is independent of N f ′ and hence, taking into account the difference in the prefactor, it is equal to C A times the terms in the square bracket of Eq. (6.7) in our earlier Ref. [13] or equivalently Eq. (3.4) in our Ref. [15]. We have where ζ s = ∞ n=1 n −s is the Riemann zeta function. Here, the group invariants

For the other
and (3.10) The coefficients κ (f ′ ) j are obtained from these κ and so forth for the other expressions. An important result that we found in our previous work [12]- [16] was that for a theory with a single representation, κ are also positive. This property implied several monotonicity relations for our calculation of γψ ψ to maximal power ∆ p f , denoted γψ ψ,∆ p f , namely that (i) for fixed p, γψ ψ,∆ p f is a monotonically increasing function of ∆ f , i.e., a monotonically increasing function of decreasing N f , and (ii) for fixed N f , γψ ψ,∆ p f is a monotonically increasing function of the maximal power p.
A basic question that we may ask concerning these results is how a coefficient κ (f ) changes as one goes from the single-representation theory with N ′ f = 0 to theories with an increasing number N f ′ of fermions in a different representation, and vice versa for the dependence of κ (f ′ ) on N f . For the purpose of this discussion, we recall that, by convention, we take f to be the fermion in the representation with a smaller dimension. In the cases with which we deal, this also means that C f < C f ′ . The question is readily answered in the case of κ (f ) 1 and κ on N f for indices j = 2, 3 will be analyzed below for particular theories.
Concerning the question of the positivity of κ  are positive for all of the orders that we have calculated, namely j = 1, 2, 3.
In our earlier work [11]- [17] on scheme-independent series calculations for theories with a N f fermions transforming according to a single type of representation, we carried out detailed studies of the reliability of these expansions using a variety of methods. One of the simplest procedures is to analyze the fractional change in a quantity, calculated to a given order O(∆ p f ), as one increases the maximal power p of the expansion. Here we shall apply this method in our illustrative theory discussed in the next section.

IV. ANOMALOUS DIMENSIONS IN A THEORY WITH FERMIONS IN THE FUNDAMENTAL AND ADJOINT REPRESENTATIONS OF SU(Nc)
In this section we discuss our scheme-independent calculations of γψ ψ,IR and γχ χ,IR for the illustrative case of a theory with gauge group SU(N c ) containing N f ≡ N F fermions in the fundamental representation and N f ′ ≡ N Adj fermions in the adjoint representation. As before, we call this the FA theory. In this case, the denominator factor D f takes the form FA theory : (4.1) We have given the values of (N F , N Adj ) in Table I for the region R IRZ . For the first-order coefficients we calculate .
Our results for the third-order coefficients are as follows: and κ (F )  If N Adj = 0, then the coefficient κ reduces to the ex-pression in Eq. (6.10) of our earlier work [13], while if N F = 0, then κ (Adj) 3 reduces to Eq. (6.20) of [13]. The agreement of these reductions of κ (F ) j for N Adj = 0 and of κ (Adj) j for N F = 0 with our earlier calculations in [13] for j = 1, 2, 3 serves as a check on our present results. As was discussed in [13,15], these coefficients have the leading large-N c dependence As specific examples of these FA theories, we consider the following sets of SU(3) gauge theories in R IRZ with the indicated fermion content: The respective positions of these theories in the regions R IRZ and R N ACP can be ascertained by referring to Table I. The corresponding values of the coefficients κ (F ) j with j = 1, 2, 3, as functions of N Adj , are listed in Table  II, and the values of κ (Adj) j with j = 1, 2, 3, as functions of N F , are listed in Table III.
We observe that all of these coefficients are positive, and so the generalizations of the monotonicity relations that we found in our earlier work for the theory with fermions in a single representation also hold for this FA theory, namely (i) for fixed N Adj , γψ ψ,IR is a monotonically increasing function of ∆ F , i.e., a monotonically increasing function of decreasing N F ; (ii) for fixed N F , γχ χ,IR is a monotonically increasing function of ∆ Adj , i.e., a monotonically increasing function of decreasing N Adj ; (iii) for fixed N f ′ , γψ ψ,IR,∆ p F is a monotonically increasing function of p; and (iv) for fixed N f , γχ χ,IR,∆ p Adj is a monotonically increasing function of p.
Separately, we also note a generalization of the monotonicity relation that we proved for κ Having calculated these coefficients κ (F ) j and κ (Adj) j with j = 1, 2, 3 for this FA theory, we next proceed to substitute them in the general scheme-independent expansions (1.6) for f = F and the analogue for f ′ = Adj. Explicitly, with f = ψ and f ′ = χ, and where with For reference, we list the values of N F,u and N Adj,u from Eqs. (4.22) and (4.24) for these (N F , N Adj ) FA SU(3) theories in Table IV. In Table V we list the values of γψ ψ,IR calculated to O(∆ p F ) for p = 1, 2, 3, denoted as γψ ψ,IR,∆ p F . Similarly, in Table VI we list the values of γχ χ,IR calculated to O(∆ p Adj ) for p = 1, 2, 3, denoted as γχ χ,IR,∆ p Adj . The monotonicity relations noted above are evident in these tables. From an examination of the fractional changes in the anomalous dimensions as one increases the order of calculation, one may infer that these scheme-independent expansions should be reasonably reliable. For example, in the SU(3) FA theory with (N F , N Adj ) = (12, 1/2) theory, the fractional change in the γψ ψ,IR anomalous dimension is yielding identical entries listed to three significant figures in Table V. Similar comments apply to the calculations of γχ χ,IR,∆ p Adj .

IR
In this section we return to the general asymptotically free gauge theory with gauge group G containing N f and N f ′ fermions in the respective representations R and R ′ and present our calculations of the coefficients d j andd j in the scheme-independent expansions of the derivative of the beta function evaluated at the IR fixed point, β ′ IR , in powers of ∆ f in Eqs. (1.8) and in powers of ∆ f ′ in Eq. (1.9), respectively. As before in this paper, this IR fixed point is taken to be in the non-Abelian Coulomb phase. Part of the physical interest in the quantity β ′ IR stems from the fact that, owing to the trace anomaly relation [42], it is equivalent to the anomalous dimension of the field-strength tensor term Tr(F a µν F aµν ) in the Lagrangian [13,43]. As noted above, generalizing our result for the single-representation case, d 1 =d 1 = 0 for arbitrary G, R, and R ′ .
For the higher coefficients we find and 3) to maintain the same notation as in our earlier works [13,15], where we found that in the case of fermions in a single representation R = F , d 4 is negative.) As was the case with A term in d 4 is independent of N f ′ and hence, taking into account the difference in the prefactor, it is equal to C A times the terms in the square bracket of Eq. (5.11) in our earlier Ref. [13] or equivalently, Eq. (4.8) of our Ref. [15]. We have In this section we discuss the special case of our general calculation of β ′ IR for an SU(N c ) theory with N f fermions in the fundamental representation and N Adj fermions in the adjoint representation (i.e., the FA theory). We write Eqs. (1.8) and (1.9) as where ∆ F and ∆ Adj were given explicitly in Eqs. (4.21)-(4.24). We calculate [13] for j = 1, 2, 3 serves as a check on our present calculations. As was discussed in [13,15], these coefficients have the leading large-N c dependence In Table IX we present our scheme-independent calculations of β ′ IR to order O(∆ p F ) via the expansion (6.1) and to O(∆ p Adj ) via the expansion (6.2), with p = 1, 2, 3, where ∆ F and ∆ Adj were defined in Eqs. tively. Graphically, in the first quadrant of R 2 defined by (N F , N adj ) (formally generalized to non-negative real numbers), the series (6.1) is an expansion in a leftward horizontal direction from the b 1 = 0 line toward a given point (N F , N Adj ) in the NACP, while the series (6.1) is an expansion inward in a downward vertical direction from the b 1 = 0 line toward this point (N F , N Adj ). Since these are two alternate expansions for the same quantity, one expects that as the maximal power p in the series increases, they should yield similar values, and we see that this expectation is satisfied by our results at the highest order, p = 3, as listed in Table IX. The agreement between the two series is best when the (N F , N Adj ) theory is near to the upper end of the non-Abelian Coulomb phase, since in this case the expansion parameters ∆ F and ∆ Adj are the smallest. Some explicit examples that demonstrate this accuracy are provided by the following fractional differences:

VII. CONCLUSIONS
In this paper, generalizing our previous work, we have considered an asymptotically free gauge theory with gauge group G and two different fermion representations, with the property that it exhibits an infrared fixed point such that the infrared theory is in a non-Abelian Coulomb phase. Specifically, we have considered a theory with N f fermions transforming according to a representation R of G and N f ′ fermions transforming according to a different representation, R ′ . We have calculated scheme-independent series expansions of the anomalous dimensions of gauge invariant fermion bilinears and the derivative β ′ IR evaluated at the IR fixed point in the respective expansion parameters ∆ f and ∆ f ′ . As an explicit application, we have presented calculations for an SU(N c ) theory with N F fermions in the fundamental representation and N Adj fermions in the adjoint representation. Our results for scheme-independent expansions of gauge-invariant fermion bilinears extend up to O(∆ 3 F ) and O(∆ 3 Adj ), while our results for β ′ IR extend up to O(∆ 4 F ) and O(∆ 4 Adj ). These results provide further information about the properties of these conformal field theories. To the extent that the transition from the lower part of the non-Abelian Coulomb phase to the quasi-conformal regime in the variables (N f , N f ′ ) is con-tinuous and our finite-order perturbative calculations in the lower part of the non-Abelian Coulomb phase are sufficiently accurate, our present results can also be useful for the investigation of quasi-conformal theories with possible relevance to ultraviolet completions of the Standard Model.
where the f abc are the associated structure constants of this Lie algebra. Here and elsewhere, a sum over repeated indices is understood. We denote the dimension of a given representation R as d R = dim(R). In particular, we denote the adjoint representation by A, with the dimension d A equal to the number of generators of the group, i.e., the order of the group. The trace invariant is given by The quadratic Casimir invariant C 2 (R) is defined by where I is the d R × d R identity matrix. For a fermion f transforming according to a representation R, we often use the equivalent compact notation T f ≡ T (R) and C f ≡ C 2 (R). We also use the notation C A ≡ C 2 (A) ≡ C 2 (G). The invariants T (R) and C 2 (R) satisfy the relation C A = N c and for R equal to the fundamental representation, T (R) = 1/2 and C 2 (R) = (N 2 c − 1)/(2N c ). At the four-loop and five-loop level, one encounters traces of quartic products of the Lie algebra generators. For a given representation R of G, As with the quadratic invariants, for a fermion f in the represenation R of G, we often use the notation d abcd   j , j = 1, 2, 3, for the scheme-independent expansion of the anomalous dimension γψ ψ,IR in an SU(3) gauge theory with fermions in the fundamental and adjoint representations, as functions of N Adj . Half-integral values of N Adj correspond to 2N Adj copies of Majorana fermions in the adjoint representation. The notation ae-n means a × 10 −n . 4.56e-2 3.39e-3 1.835e-4 1 4.20e-2 3.03e-3 1.51e-4 3 2 3.89e-2 2.71e-3 1.31e-4 2 3.63e-2 2.44e-3 1.16e-4      N Adj d2 d3 d4 0 0.831e-2 0.983e-3 −0.463e-4 1/2 0.760e-2 0.8225e-3 −2.44e-5 1 0.700e-2 0.698e-3 −1.24e-5 3/2 0.649e-2 0.600e-3 −0.578e-5 2 0.605e-2 0.521e-3 −2.12e-6