Physics of the Non-Abelian Coulomb Phase: Insights from Pad\'e Approximants

We consider a vectorial, asymptotically free SU($N_c$) gauge theory with $N_f$ fermions in a representation $R$ having an infrared (IR) fixed point. We calculate and analyze Pad\'e approximants to scheme-independent series expansions for physical quantities at this IR fixed point, including the anomalous dimension, $\gamma_{\bar\psi\psi,IR}$, to $O(\Delta_f^4)$, and the derivative of the beta function, $\beta'_{IR}$, to $O(\Delta_f^5)$, where $\Delta_f$ is an $N_f$-dependent expansion variable. We consider the fundamental, adjoint, and rank-2 symmetric tensor representations. The results are applied to obtain further estimates of $\gamma_{\bar\psi\psi,IR}$ and $\beta'_{IR}$ for several SU($N_c$) groups and representations $R$, and comparisons are made with lattice measurements. We apply our results to obtain new estimates of the extent of the respective non-Abelian Coulomb phases in several theories. For $R=F$, the limit $N_c \to \infty$ and $N_f \to \infty$ with $N_f/N_c$ fixed is considered. We assess the accuracy of the scheme-independent series expansion of $\gamma_{\bar\psi\psi,IR}$ in comparison with the exactly known expression in an ${\cal N}=1$ supersymmetric gauge theory. It is shown that an expansion of $\gamma_{\bar\psi\psi,IR}$ to $O(\Delta_f^4)$ is quite accurate throughout the entire non-Abelian Coulomb phase of this supersymmetric theory.


I. INTRODUCTION
The properties of a vectorial, asymptotically free gauge theory at an infrared fixed point (IRFP) of the renormalization group (RG) in a conformally invariant regime are of fundamental interest. Of equal interest is the determination of the infrared phase structure of such a theory. Owing to the asymptotic freedom, one can perform reliable perturbative calculations in the deep ultraviolet (UV) where the gauge coupling approaches zero, and then follow the renormalization-group flow to the infrared. This flow is described by the beta function, β = dα/d ln µ, where α = g 2 /(4π), g = g(µ) is the running gauge coupling, and µ is a Euclidean energymomentum scale. For a given gauge group G and fermion representation R of G, the requirement of asymptotic freedom places an upper bound, denoted N u , on the number of fermions, N f , transforming according to this representation. The UV to IR evolution of a theory is determined by G, R, and N f . If N f is slightly less than N u , then the RG flows from the UV to a weakly coupled IRFP in a non-Abelian Coulomb phase (NACP, also called the conformal window) in the infrared. The value of α at the IRFP is denoted α IR . At this IRFP, the theory is scaleinvariant and is deduced to be conformally invariant [1]. Physical quantities at this IRFP can be expressed perturbatively as series expansions in powers of α IR (e.g., [2][3][4][5]). However, beyond the lowest loop orders, the coefficients in these expansions depend on the scheme used for regularization and renormalization of the theory. Consider an asymptotically free vectorial gauge theory with gauge group G and N f massless [6] Dirac fermions ψ i , i = 1, ..., N f , in a representation R of G, such that the RG flow leads to an IRFP, and let N u denote the value of N f at which asymptotic freedom is lost. Since α IR becomes small as N f approaches N u from below, one can reexpress physical quantities as series expansions in the manifestly scheme-independent variable (1.1) Some early work on this was reported in [7]- [8].
Two quantities of considerable interest are anomalous dimensions of (gauge-invariant) fermion bilinear operators and the derivative of the beta function, evaluated at the IRFP, These are physical quantities and hence are schemeindependent [9]. The derivative β ′ IR is equivalent to the anomalous dimension of Tr(F µν F µν ), where F a µν is the field-strength tensor of the theory [10]. In earlier work we had presented values of γψ ψ,IR [2] (see also [3]) and β ′ IR [4] calculated as conventional n-loop series expansions in powers of the n-loop (nℓ) IR coupling α IR,nℓ . In [11]- [17] we calculated values of γψ ψ,IR and β ′ IR via our scheme-independent expansions.
In this paper we use our series expansions of γψ ψ,IR to O(∆ 4 f ) and of β ′ IR to O(∆ 5 f ) at an IR fixed point in vectorial, asymptotically free gauge theories with gauge group SU(N c ) and N f Dirac fermions transforming according to various representations R, to calculate Padé approximants to these quantities. We consider R equal to the fundamental (F ), adjoint (A) and rank-2 symmetric (S 2 ) tensor representations. For technical convenience, as before, we restrict to mass-independent schemes [18] and zero fermion mass [6]. We use these Padé approximants for several purposes, which also constitute motivations for this work. First, as closed-form rational functions of ∆ f , the Padé approximants yield values of γψ ψ,IR and β ′ IR that complement the values obtained from the finite series expansions in powers of ∆ f and can be compared with them. These values are of fundamental importance as properties of conformal field theories and are of value in the resurgent investigation of these theories in four spacetime dimensions. Second, our calculations of these quantities in continuum quantum field theory are complementary to the program of lattice simulations to measure them, and we compare our values with lattice measurements. Third, given an asymptotically free theory with a particular choice of gauge group G and fermion representation R, the question of what the infrared properties are, as a function of N f , is of fundamental field-theoretic importance. This applies, in particular, to the question of the extent of the non-Abelian Coulomb phase as a function of N f . Since the upper end of the NACP at N u (see Eq. (2.1) below) [19] is known exactly, the determination of the extent of the NACP as a function of N f is equivalent to the determination of the value of of N f at the lower end of the NACP, which we denote N f,cr . As in our earlier work, here we use our new calculations of γψ ψ,IR using Padé approximants, together with an upper bound on this anomalous dimension from conformal invariance, to obtain further estimates of N f,cr and hence of the extent of the NACP for several theories. Fourth, we carry out Taylor-series expansions of our Padé approximants in powers of ∆ f to determine their predictions for higher-order coefficients in the scheme-independent expansions of γψ ψ,IR and β ′ IR going beyond the respective orders O(∆ 4 f ) and O(∆ 5 f ) to which we have calculated these. We use these to test our conjecture in earlier work that the coefficients in the series expansion of γψ ψ,IR are positive. (In contrast, we have shown from our calculations for general G and R in [13][14][15] that the coefficients in the series expansions of β ′ IR in powers of ∆ f have mixed signs.) A fifth use of our Padé calculations pertains to phenomenology. In addition to the importance of N f,cr as a basic property describing the UV to IR evolution and infrared phase structure of an asymptotically free gauge theory, this is also important for phenomenological studies, since a knowledge of N f,cr is crucial for the construction of quasi-conformal gauge theories as possible candidates for ultraviolet completions of the Standard Model. This is because, for a given G and R, these constructions of quasi-conformal theories require that one choose N f to be slightly less than N f,cr in order to achieve the quasi-conformal behavior whose spontaneous breaking via chiral symmetry breaking could have the potential to yield a light, dilatonic Higgs-like scalar. As a sixth part of the present work, we present a new analytic result concerning the accuracy of the finite series expansion of γψ ψ,IR in powers of ∆ f as compared with the exactly known result in an N = 1 supersymmetric gauge theory. This analytical result extends our earlier demonstrations [11,[14][15][16] that the truncation of the series for γψ ψ,IR to O(∆ 4 f ) is quite accurate throughout the entire non-Abelian Coulomb phase of this supersymmetric theory. This paper is organized as follows. In Section II we briefly review relevant background and methodology. In Sections III and IV we present our results for γψ ψ,IR and β ′ IR , respectively. We give our conclusions in Section V.

A. General
Here we briefly review some background and methods relevant for our work. We refer the reader to our previous papers [11]- [17] for details. The requirement of asymptotic freedom implies that N f must be less than an upper (u) bound N u , where [19] Here, C 2 (R) is the quadratic Casimir invariant for the representation R, C A = C 2 (A), where A is the adjoint representation, and T f ≡ T (R) is the trace invariant [20]. At the maximal scheme-independent loop order, namely the two-loop level, the beta function has an IR zero if N f lies in the interval I defined by where N u was given in Eq. (2.1) and Formally generalizing N f from positive integers N + to positive real numbers, R + , one can let N f approach N u from below, thereby making α IR arbitrarily small. Thus, for the UV to IR evolution in this regime of N f , one infers that the theory evolves from weak coupling in the UV to an IRFP in a non-Abelian Coulomb phase. As stated above, we denote the lowest value of N f in this NACP as N f,cr , and correspondingly, we define [21] One of the goals of lattice studies of these types of gauge theories has been to estimate N f,cr for a given G and R [22]. We have also obtained estimates of N f,cr by combining our O(∆ 4 f ) calculations of γψ ψ,IR with our finding of monotonicity and the conformality upper bound [23] γψ ψ,IR ≤ 2 in our earlier work [12][13][14][15]. We will discuss this further below in connection with our new Padé calculations. Our computations assume that the IRFP is exact, as is the case in the non-Abelian Coulomb phase [24]. In the analytic expressions and plots given below, this restriction, that N f lies in the NACP, will be understood implicitly. In Table I we tabulate some relevant values of N ℓ , N u , ∆ f,max , and the intervals with N f ∈ R + and N f ∈ N + .
For the gauge group G = SU(N c ) and the specific fermion representations R considered in this paper, the general formulas above for N ℓ , N u , and ∆ f,max read as follows: , . (2.9) We list these values for the theories under consideration in Table I.

B. Scheme-Independent Expansion for γψ ψ,IR
Since the global chiral symmetry is realized exactly in the non-Abelian Coulomb phase, the bilinear fermion operators can be classified according to their representation properties under this symmetry, including flavor-singlet and flavor-nonsinglet. The anomalous dimensions evaluated at the IRFP, are the same for these flavor-singlet and flavor-nonsinglet fermion bilinears [25]. For R = F , these areψψ ≡ where T b is a generator of SU(N f )), and similarly for other representations. Hence, as in our earlier work, we use the symbol γψ ψ,IR to refer to both. This anomalous dimension at the IRFP has the scheme-independent expansion We denote the truncation of the infinite series in Eq. (2.10) at j = s asγψ ψ,IR,∆ s f . When it is necessary for clarity, we indicate the representation R explicitly in the subscripts as γψ ψ,IR,R , γψ ψ,IR,R,∆ s f , and κ j,R . In general, the calculation of the coefficient κ j requires, as inputs, the ℓ-loop coefficients in the conventional loop expansion of the beta function in powers of α, namely b ℓ , with 1 ≤ ℓ ≤ j + 1, and the ℓ-loop coefficients c ℓ in the corresponding conventional expansion of γψ ψ with 1 ≤ ℓ ≤ j. The coefficients κ j were calculated for general gauge group G and fermion representation R for 1 ≤ j ≤ 3 in [11] and for j = 4 in [15]. The calculation of κ 4 was given for SU (3) and R = F in [12] and for SU(N c ) in [14], Our calculation of κ 4 for general G and R used the b ℓ coefficients up to b 5 from [26] and c ℓ up to c 4 from [27]. Specific expressions for κ j and plots of γψ ψ,IR,∆ s f for G = SU(N c ) and the fundamental, adjoint, and rank-2 symmetric and antisymmetric representations were given in [12]- [15] and for G = SO(N c ) and Sp(N c ) in [17]. In [13] we discussed operational criteria for the accuracy of the ∆ f expansion, and we briefly review some points here.
As with usual perturbative series expansions in powers of interaction couplings in quantum field theories, the ∆ f expansion is generically an asymptotic expansion rather than a Taylor series. This follows from the fact that in order for an expansion in a variable z to be a Taylor series with finite radius of convergence, it is necessary (and sufficient) that the function for which the series is calculated must be analytic at the origin of the complex z plane. In particular, with z = ∆ f , this means that, the properties of the theory should not change qualitatively as one moves from real positive ∆ f through the point ∆ f = 0 to negative ∆ f , i.e., as N f increases through the value N u . However, one knows that the properties of the theory do change qualitatively as N f increases beyond N u , namely it ceases to be asymptotically free. Nevertheless, as with conventional perturbative expansions in powers of the interaction coupling in quantum field theory, one may get a rough estimate of the accuracy of a truncated series by performing the ratio test on the series coefficients that have been calculated. This type of procedure is used, for example, in perturbative quantum electrodynamics and quantum chromodynamics calculations in powers of the respective interaction couplings, and we gave results on the relevant ratios of terms in our series expansions of γψ ψ,IR and β ′ IR in our previous work [11]- [15]. These, in conjunction with plots of curves, gave quantitative evaluations of the accuracy of the ∆ f expansions for these quantities. We will expand upon this earlier work here by comparing O(∆ s f ) expansions for γψ ψ,IR with the exactly known expression for this anomalous dimension in an N = 1 supersymmetric gauge theory below.
Let us denote the full scaling dimension of an operator O as D O and its free-field value as D O,f ree . We define the anomalous dimension of O, denoted γ O , by Given that the theory at an IRFP in the non-Abelian phase is conformally invariant, there is a conformality lower bound on D O from unitarity, namely Dψ ψ ≥ 1 [23]. Since Dψ ψ,f ree = 3, this is equivalent to the upper bound γψ ψ,IR ≤ 2 . (2.12) In [12], for SU(3) and R = F , using our calculation of γψ ψ,IR to O(∆ 4 f ), i.e., γψ ψ,IR,∆ 4 f , we used polynomial extrapolation to obtain estimates of the evaluation of the infinite series (2.10) yielding the value of γψ ψ,IR as a function of N f . We compared our results with lattice measurements for N f = 12 and N f = 10.
In the series expansion (2.10) for γψ ψ,IR , the first two coefficients, κ 1 and κ 2 , are manifestly positive for any gauge group G and fermion representation R [11]. Although κ 3 and κ 4 contain terms with negative as well as positive signs, one of the important results of our explicit calculations of κ 3 and κ 4 for SU(N c ), SO(N c ), and Sp(N c ) gauge groups and a variety of representations, including fundamental, adjoint, and rank-2 symmetric and antisymmetric tensors, was that for all of these theories, κ 3 and κ 4 are also positive [13][14][15]17]. Moreover, as reviewed below, in a gauge theory with N = 1 supersymmetry, an exact expression is known for the anomalous dimension of the (gauge-invariant) fermion bilinear operator, and the Taylor-series expansion of this exact expression in powers of ∆ f yields κ j coefficients that are all positive. These results led to our conjecture in [12], elaborated upon in our later works, that, in addition to the manifestly positive κ 1 and κ 2 , and our findings in [13][14][15]17] that κ 3 and κ 4 are positive for all of the groups and representations for which we calculated them, (i) the higher-order κ j coefficients with j ≥ 5 are also positive in (vectorial, asymptotically free) nonsupersymmetric gauge theories. In turn, this conjecture led to several monotonicity conjectures, namely that (ii) for fixed s, γψ ψ,IR,∆ s f increases monotonically as N f decreases in the non-Abelian Coulomb phase, and (iii) for fixed N f in the NACP, γψ ψ,IR,∆ s f is a monotonically increasing function of s, so that (iv) for fixed N f in the NACP and for finite s, γψ ψ,IR,∆ s f is a lower bound on the actual anomalous dimension γψ ψ,IR , as defined by the infinite series (2.10). By similar reasoning, the analogous conjectures apply for the k'th derivatives of the anomalous dimension as a function of ∆ f . In particular, for the first derivative, one has the analogous conjectures (ii) d for fixed s, Combining our calculations of γψ ψ,IR to O(∆ 4 f ) with these positivity and resultant monotonicity conjectures (used as assumptions), and with the further assumption that γψ ψ,IR saturates its conformality upper bound of 2 in (2.12), as N f decreases to N f,cr at the lower end of the non-Abelian Coulomb phase, we have then derived estimates of N f,cr in various theories [28]. For example, in [12] we inferred that N f,cr = 8 − 9 [12,28]. In [14,15] we extended these O(∆ 4 f ) calculations of γψ ψ,IR from the special case of SU(3) and R = F to general G and R, and, for G = SU(N c ) with various N c and R, we again compared various our values of γψ ψ,IR,∆ 4 f with values from lattice measurements.
Here, we extend this program further via the calculation and evaluation of Padé approximants for G = SU(N c ) with several values of N c and several fermion representations R. There are a number of applications of these calculations: (i) to get further estimates of the value of the anomalous dimension of the fermion bilinear for various N c and fermion representations R; (ii) to estimate N f,cr , as just described; and (iii) via Taylor series expansions of the Padé approximants, to determine their predictions for higher-order coefficients κ j with j ≥ 5. Of course, regarding application (iii), since the Padé approximants that we calculate for γψ ψ,IR are based on the series (2.10) computed only up to O(∆ 4 f ), their predictions for these higher-order κ j with j ≥ 5 only provide a hint as to their actual values.
C. Scheme-Independent Expansion for β ′

IR
Given the property of asymptotic freedom, β is negative in the region 0 < α < α IR , and since β is continuous, it follows that this function passes through zero at α = α IR with positive slope, i.e., β ′ IR > 0. This derivative β ′ IR has the scheme-invariant expansion (2.14) As indicated, β ′ IR has no term linear in ∆ f . In general, the calculation of the scheme-independent coefficient d j requires, as inputs, the ℓ-loop coefficients in the beta function, b ℓ , for 1 ≤ ℓ ≤ j. We denote the truncation of the infinite series in Eq. (2.14) at j = s as β ′ Let the full scaling dimension of Tr(F µν F µν ) be denoted D F 2 µν (with free-field value 4). At an IRFP, D F 2 ,IR = 4 + β ′ IR [10], so β ′ IR = −γ F 2 ,IR . Given that the theory at an IRFP in the non-Abelian phase is conformally invariant, there is a conformality bound from unitarity, namely D F 2 ≥ 1 [23]. Since β ′ IR > 0, this bound is obviously satisfied.
As part of our work, we will calculate Padé approximants to our series expansions to O(∆ 5 f ) for β ′ IR . We will use these for the analogues of the applications (i) and (iii) mentioned above for γψ ψ,IR , namely to obtain additional information about the value of β ′ IR and to get some hints regarding coefficients d j going beyond the order to which we have calculated them, i.e., with j ≥ 6.

D. LNN Limit
For G = SU(N c ) and R = F , it is of interest to consider the limit and ξ(µ) ≡ α(µ)N c is a finite function of µ . (2.15) We will use the symbol lim LN N for this limit, where "LNN" stands for "large N c and N f " with the constraints in Eq. (2.15) imposed. This is also called the 't Hooft-Veneziano limit.
Here we give some background for our calculation of Padé approximants in the LNN limit. We define and The critical value of r such that for r > r cr , the LNN theory is in the NACP and is IR-conformal, while for r < r cr , it exhibits spontaneous chiral symmetry breaking, is denoted r cr and is defined as We define the rescaled scheme-independent expansion parameter for the LNN limit As r decreases from r u to r ℓ in the interval I r , ∆ r increases from 0 to a maximal value ∆ r,max = r u − r ℓ = 75 26 = 2.8846 for r ∈ I r . (2.23) Further, we define the maximum value of ∆ r in the NACP as ∆ r,cr = r u − r cr = 11 2 − r cr .

(2.24)
Since κ j ∝ N −j c , the rescaled coefficientsκ j,F that are finite in the LNN limit arê The anomalous dimension γ IR is also finite in this limit and is given by (2.26) The appropriately rescaled beta function that is finite in the LNN limit is where ξ was defined in Eq. (2.15). Since the derivative dβ ξ /dξ satisfies the relation it follows that β ′ is finite in the LNN limit (2.15). We define the rescaled coefficient which is finite in the LNN limit. Thus, writing We denote the value of β ′ IR,LN N obtained from this series

E. Test of Accuracy of Scheme-Independent Expansion for γψ ψ,IR Using Supersymmetric Gauge Theory
A basic question for the scheme-independent series expansions of physical quantities at an IRFP in powers of ∆ f in the non-Abelian Coulomb phase is how accurate a finite truncation of this series is. We have addressed this question in previous work [11,[13][14][15] by investigating how accurate the truncated, finite-order expansion is, as a function of N f , in a theory where an exact expression for the anomalous dimension of the fermion bilinear is known.
Here we briefly review this analysis and give some new quantitative measures of the accuracy. For this accuracy check, we use a vectorial, asymptotically free gauge theory with N = 1 supersymmetry (ss), gauge group G, and N f pairs of chiral superfields Φ j andΦ j , j = 1, ..., N f , that transform according to the respective representations R andR of G. The requirement of asymptotic freedom in this theory requires that N f must be less than an upper limit, which we will again denote N u , namely (Throughout this subsection, to avoid cumbersome notation, we will use the same notation N u , N ℓ , N f,cr , etc. as in the non-supersymmetric case, but it will be understood implicitly that these quantities refer to this supersymmetric theory.) In this theory, the lower end of the non-Abelian Coulomb phase occurs at N f,cr = N u /2 , so the NACP occupies the range ss : N ACP : Thus, in this theory, ∆ f increases from 0 to a maximum value as N f decreases from N u at the upper end of the NACP to N u /2 at the lower end of the NACP. The anomalous dimension of the quadratic chiral superfield operator productΦΦ, and hence also the fermion bilinear contained in this product, are exactly known in such theories. Defining the mesonic operator M ≡ψψ, with a sum over group indices understood, one has [29]- [31] This anomalous dimension γ M,IR increases monotonically from 0 at N f = N u at the upper end of the NACP to saturate its upper limit of 1 when N f reaches N f,cr = N u /2 at the lower end of the NACP. From this exact expression (2.34), it follows that the coefficient κ j in Eq. (2.10) is (2.35) Thus, κ j is positive for all j, which provided motivation for our conjecture in [12,13] that κ j > 0 ∀ j in the non-supersymmetric theory, in accord with the manifestly positive κ 1 and κ 2 , and the positivity of the κ j with j = 3, 4 that we had calculated for each group and representation that we considered [11][12][13][14][15]17]. From this positivity of the κ j calculated to the highest order, j = 4, the monotonicity property of γψ ψ,IR,∆ 4 f follows. That is, our calculation of γψ ψ,IR,∆ 4 f in the non-supersymmetric theory shares with the exact expression in the N = 1 supersymmetric theory the property that it increases monotonically with decreasing N f in the NACP. Note that this monotonicity does not hold for the scheme-dependent conventional n-loop (nℓ) calculation of γ IR,nℓ ; for example, it was found [2,3] that for various specific theories, such as (non-supersymmetric) SU(N c ) with N c = 2, 3, 4 and R = F , although κ 3 is positive in the NACP, κ 4 (calculated in the widely used MS scheme) is negative, so that, as N f decreases in the NACP, γψ ψ,IR,4ℓ reaches a maximum and then decreases. In the N = 1 supersymmetric theory, we also showed that κ 3 (calculated in the DR scheme) is negative, and, as a consequence, the scheme-dependent three-loop calculation of γ M,IR,3ℓ fails to exhibit the known monotonicity of the exact result [32]. This again demonstrates the advantage of the scheme-independent series expansion at the IRFP, (2.10), in powers of ∆ f , as compared with conventional expansions in powers of the coupling α IR .
The series expansion for γ M,IR is particularly simple in the N = 1 supersymmetric gauge theory, since it is a geometric series. Because the fermions appear together with the Grassmann variable θ in the chiral superfield Φ j = φ j + √ 2 θψ j + θθF j (where F j is an auxiliary field), the conformality lower bound D Φ ≥ 1 on the full scaling dimension of the chiral superfield is equivalent to the conformality upper bound on the (gauge-invariant) fermion bilinear in Φ jΦj . This upper bound is saturated as N f decreases to N f,cr . Thus, as we have observed before [11,13,15,16], both of the assumptions that we make for our estimate of N f,cr in non-supersymmetric theories, namely that (i) κ j > 0 for all j, and (ii) γψ ψ,IR saturates its upper bound from conformal invariance as N f decreases to the lower end of the non-Abelian Coulomb phase, are satisfied in a gauge theory with N = 1 supersymmetry. (As discussed above, the upper bounds themselves are different, namely 2 in the non-supersymmetric theory, Eq. (2.12) and 1 in the supersymmetric theory, Eq. (2.36) [33].) We next determine the accuracy of a finite truncation of the series (2.10). To do this we calculate the fractional difference where we denote the finite series (2.10) for γ M,IR truncated to maximal power j = s as γ M,IR,∆ s f . Using the elementary identity Substituting this into Eq. (2.37), we find Since the maximum value of the ratio ∆ f /N u is 1/2, this fractional difference decreases toward zero exponentially rapidly with s. Quantitatively, if one sets In general, if we require that ǫ ss < ǫ 0 for some ǫ 0 > 0, then this implies that it is necessary to calculate the finite, truncated series in powers of ∆ f up to and including the power to achieve this fractional accuracy, where it is understood that if s is a non-integral real number, then one sets s equal to the closest integer greater than the value in Eq.
(2.40). At the upper end of the non-Abelian Coulomb phase, since ∆ f /N u is small, ln(N u /∆ f ) is large and one can achieve a small fractional difference ǫ 0 with a modest value of s. The most stringent requirement on s to achieve a given fractional accuracy ǫ 0 occurs as N f approaches the lower end of the NACP at N f = N u /2 and is As we have calculated, if one wants to achieve a fractional difference that is less than or equal to 6.25 % for all N f in the NACP, then Eq. (2.40) shows that the expansion to O(∆ 4 f ) is sufficient for this accuracy. These results show quantitatively that finite truncations of the infinite series (2.10), even up to only modest maximal powers such as s = 4, yield very accurate approximations to the exactly known anomalous dimension γ M,IR in this N = 1 supersymmetric gauge theory. In passing, we remark that if an exact expression for β ′ IR were available in this theory, then we could also use it to obtain an additional measure of how accurate a finiteorder truncation of the infinite series (2.14) is to the exact function. However, to our knowledge, an exact expression for β ′ IR is not known for this theory. This N = 1 supersymmetric gauge theory also provides a framework in which to investigate how accurate a [p, q] Padé approximant to a finite-order truncation of the infinite series in Eq. (2.34) would be to the exact result in (2.34) for γ M,IR . In general, if one calculates a [p, q] Padé approximant for a finite truncation of such a simple series as the geometric series in (2.34), then not only does the [0,1] approximant reproduce the exact function (2.34), but so do all of the [p, q] approximants with q = 0. The way that they do this is by inserting factors in the numerator and denominator to yield polynomials of degree p and q, but which cancel precisely, yielding the exact function (2.34) itself.
From the exact expression (2.34), one can also calculate the value of the derivative dγ M,IR /dN f , which is This derivative is always negative in the NACP and increases monotonically in magnitude with decreasing N f . It has the value −1/N u for N f = N u at the upper end of the NACP, and −4/N u for N f = N u /2 at the lower end of the NACP. The curvature is This curvature is positive in the NACP and increases from 2/N 2 u at the upper end, to 16/N 2 u , at the lower end, of the NACP. Given that we have shown that κ 1 and κ 2 are manifestly positive and κ j are positive for all the G and R for which we have evaluated them, it follows that our γψ ψ,IR,∆ 4 f also has positive curvature for N f in the NACP, a property that it shares with the exactly known γ M,IR in the N = 1 supersymmetric theory. We showed in [32] that the three-loop calculation of γ M,IR,3ℓ in the supersymmetric gauge theory, carried out as a conventional scheme-dependent series expansion in powers of α, fails to exhibit the known positive curvature of the exact result, just as it fails to exhibit the known monotonicity of the exact result. This is another advantage of the scheme-independent expansion in powers of ∆ f .
We have noted above that the value of N f at the lower end of the NACP does not, in general, coincide with the value N ℓ at the lower end of the interval I where the two-loop beta function has an IR zero. For this N = 1 supersymmetric theory, one can calculate this difference exactly [32,34]. One has Hence, this difference for this theory is (where here and in the rest of this subsection, N ℓ is given by (2.44) and should not be confused with Eq. (2.3)). This difference can be positive or negative. For example, for G = SU(N c ) and R = F , this difference is the positive quantity .

(2.46)
This decreases to zero as N c → ∞. The resultant fractional difference between these values is This fractional difference (2.47) has the values 0.143 and 0.0588 for N c = 2 and N c = 3, respectively, and also decreases toward zero with increasing N c . In contrast, for both the adjoint and symmetric rank-2 tensor representations, the difference N ℓ − N f,cr is negative and does not vanish as N c → ∞ (with N f fixed).
In the LNN limit of the N = 1 supersymmetric gauge theory with G = SU(N c ) and R = F , In this section we report our calculation of Padé approximants to our scheme-independent O(∆ 4 f ) series for γψ ψ,IR , in SU(N c ) theories with various fermion representations R. (For a review of Padé approximants, see, e.g., [35].) It may be recalled that resummation methods such as Padé approximants have been useful in in the analysis of series expansions in both quantum field theories and in statistical mechanics [36] (e.g. [37]- [42]), and our current work extends to higher order our earlier calculations of Padé approximants for these types of gauge theories [2,5,12,13].
In general, given the series calculated to maximal order s, denoted γψ ψ,IR,∆ s f as above (with N c and R implicit in the notation), we write this as and calculate the [p, q] Padé approximant to the expression in square brackets, with p + q = s − 1. This takes the form The [p, q] Padé approximant is thus a rational function whose p + q coefficients are determined uniquely by the condition that the Taylor series expansion of this approximant must match the s − 1 coefficients in the series in square brackets. With the prefactor κ 1 ∆ f thus extracted, the Padé approximant in the square brackets is normalized to be equal to 1 at ∆ f = 0. By construction, the series expansion of each closed-form approximant γψ ψ,IR,[p,q] exactly reproduces the series expansion of γψ ψ,IR up to the maximal order to which we have calculated it, s = 4. In addition to providing a closedform rational-function approximation to the finite series, a Padé approximant also can be used in another way, namely to yield a hint of higher-order terms. This information is obtained by carrying this Taylor series expansion of [p, q] Padé approximants with q = 0 to higher order. We will use the Padé approximants for both of these applications. Note that the [s − 1, 0] Padé approximant is just the series itself, i.e., As a special case of Eq. (3.4) for j max = 4, γψ ψ,IR, [3,0] is just the original polynomial γψ ψ,IR,∆ 4 f itself, and hence we do not consider it, since we have already obtained evaluations of this truncated series in previous work.
By construction, the [p, q] Padé approximant in (3.2) is analytic at ∆ f = 0, and if it has q = 0, then it is a meromorphic function with q poles. A necessary condition that must be satisfied for a Padé approximant to be useful for our analysis here is that it must not have a pole for any ∆ f in the interval 0 < ∆ f < ∆ cr , or equivalently, for any N f in the non-Abelian Coulomb phase. Since N f,cr is not precisely known for all values of N c and all fermion representations R under consideration here, we will also use another condition, namely that the Padé approximant should not have any poles for N f in the interval I where the two-loop beta function has an IR zero, or equivalently, for ∆ f in the interval 0 < ∆ f < ∆ f,max . This second condition can be applied in a straightforward manner for each N c and R, since the upper and lower ends of this interval I, namely N u and N ℓ , and thus ∆ f,max , are known (listed above in Eqs. (2.1), (2.3), and (2.4)). Since the [p, q] Padé approximant in Eq. (3.2) is an analytic function at ∆ f = 0, and, for q = 0, is meromorphic, the radius of convergence of its Taylor series expansion is set by the magnitude of the pole closest to the origin in the complex ∆ f plane. Let us denote this radius of convergence as ∆ f,conv. . We shall also require that ∆ f,conv. be greater than ∆ f,max and ∆ f,cr , since we would like the Taylor series expansion of (3.2) to accurately reproduce the series (2.10) in this disk. We will do this to be as careful as possible, even though the actual expansion (2.10) is, in general, only expected to be an asymptotic expansion.
B. G = SU(Nc), R = F With G = SU(N c ) and R = F , the explicit expression for κ 1,F is .
The explicit numerical expressions for the schemeindependent series expansions of γψ ψ,IR to order ∆ 4 f for R = F and N c = 2, 3, 4 are as follows: In these equations, For G = SU(2) with R = F , the general formulas (2.1) and (2.3) give N u = 11 and N ℓ = 5.551, so the interval I with N f ∈ R + is 5.551 < N f < 11, with ∆ f,max = N u −N ℓ = 5.449, and the physical interval with N f ∈ N + is 6 ≤ N f ≤ 10. This information is summarized in Table  I. For this SU(2) theory with R = F , we calculate the following [p, q] Padé approximants with q = 0: γψ ψ,F, [1,2] (3.12) The [2,1] Padé approximant in γψ ψ,F, [2,1] has a pole at ∆ f = 3.721 i.e., at N f = 7.279. This is in the interval I and in the estimated NACP, so we cannot use this [2,1] Padé approximant for our analysis. The [1,2] Padé approximant in γψ ψ,F, [1,2] has poles at ∆ f = −1.6405, i.e., N f = 12.6405, and at ∆ f = 9.544, i.e., N f = 1.456. The first of these occurs at a value of N f greater than N u , while the second occurs at a value of N f well below N ℓ and N f,cr , so neither is in the interval I or in the NACP. Finally, the [0,3] Padé approximant has a pole at ∆ f = 6.289, i.e., at N f = 4.731, which is below N ℓ and slightly below the estimated N f,cr . In addition, this [0,3] approximant has a complex-conjugate pair of poles at ∆ f = −2.3195 ± 9.273i, whose magnitude is 9.558, considerably greater than ∆ f,max = 5.449 and ∆ cr ≃ 5.5. Hence, we can use the [1,2] and [0,3] Padé approximants for our analysis. In Table II we list values of γψ ψ,IR,F, [1,2] and γψ ψ,IR,F,[0,3] for this SU(2) theory with R = F as a function of N f and, for comparison, values of γψ ψ,IR,F,∆ s f for 1 ≤ s ≤ 4 from [13,15]. In Fig. 1 we plot these values. We next make some general comments about these SU(2), R = F calculations, which also will apply to our calculations for SU(3) and SU(4) with R = F . In earlier work [11]- [17], we have noted that, in addition to the manifestly positive κ 1 and κ 2 , the higher-order coefficients κ j with j = 3 and j = 4 are positive for all of the groups, SU(N c ), SO(N c ), and Sp(N c ) and for all of the representations, namely F , A, S 2 , and A 2 , for which we have performed these calculations. Here (at an IR fixed point in the non-Abelian Coulomb phase), this means that, at least with s in the range 1 ≤ s ≤ 4, (i) for fixed s, γψ ψ,IR,F,∆ s f monotonically increases with decreasing N f ; and (ii) for fixed N f and hence ∆ f , γψ ψ,IR,F,∆ s f is a monotonically increasing function of s. As is evident from Table II and Fig. 1, the analogue of the monotonicity property (i) is also true for the Padé approximants, namely that both the [1,2] and [0,3] Padé approximants increase monotonically with decreasing N f values listed in the table and shown in the figure. The value of the anomalous dimension obtained via the [1,2] Padé approximant, γψ ψ,IR,F, [1,2] , is quite close to γψ ψ,IR,F,∆ 4 f , increasing slightly above it as N f decreases toward the lower part of the non-Abelian Coulomb phase. The curve for γψ ψ,IR,F,[0,3] lies above that for γψ ψ,IR,F, [1,2] and increases more rapidly with decreasing N f .
In [14,15] we compared our results for γψ ψ,IR,F,∆ s f with s up to 4 in this SU(2) theory with our earlier conventional n-loop calculations in [2] and with lattice measurements for N f = 8 [43,44]. These lattice measurements are consistent with the SU(2), R = F , N f = 8 theory being IR-conformal, so our calculations in the non-Abelian Coulomb phase are applicable. As listed in Table II,  We now go further to combine our scheme-independent series calculation of γψ ψ,IR,∆ 4 f with our Padé approximant computation to estimate N f,cr for this theory. We require that both of the two Padé-based values, namely γψ ψ,IR,F, [1,2] and γψ ψ,IR,F,[0, 3] , should obey the conformality upper bound (2.12). We also assume that the larger of these values saturates this upper bound at the lower end of the NACP, as the exactly known γ M,IR saturates its upper bound in the supersymmetric gauge theory discussed above. Then the N f value at which the larger of these two values exceeds the conformality upper bound yields the new Padé-based estimate of N f,cr . For this SU(2) theory (and for the SU(3) and SU(4) theories to be discussed below) the larger Padé-based value is γψ ψ,IR,F,[0,3] . From Fig. 1, we therefore infer that This result is consistent with a recent lattice study [45] of the SU(2) theory with R = F , N f = 6, which finds this theory is IR-conformal. (Earlier lattice studies of this theory include [46].) With this estimate that N f,cr < ∼ 6, it follows that the non-Abelian Coulomb phase occupies the physical interval 6 ≤ N f ≤ 10, the same as the interval I with N f ∈ N + .
As another application, we can calculate Taylor series expansions of these Padé approximants to see what they predict for higher-order coefficients, namely the κ j,F with j ≥ 5. (Recall that the κ j,R were given for general G and R in [13][14][15] and were listed numerically for G = SU(2), R = F in Eq. (3.6) above.) We find that the Padé approximant γψ ψ,IR,F,[0,3] that we used to estimate N f,cr yields coefficients κ j,F with j ≥ 5 that are all positive to the highest order to which we have calculated them, namely j = 200. This is an important result, because it shows that our use of this approximant, γψ ψ,IR,F,[0, 3] , to estimate N f,cr is self-consistent. That is, our use assumed that, in addition to the known positive κ j,F with 1 ≤ j ≤ 4, the higher-order κ j with j ≥ 5 are positive, and our Taylor series expansion of γψ ψ,IR,F,[0,3] is consistent with this. In Table IV we list the higher-order coefficients κ j,F,[0,3] with 5 ≤ j ≤ 10 for this theory obtained from the Taylor series expansions of the Padé approximant γψ ψ,IR,F,[0,3] . Since we did not use the γψ ψ,IR,F, [2,1] for our estimate of N f,cr , it is not of direct relevance what the signs of the κ j,F with j ≥ 5 from the Taylor series expansion of this [2,1] approximant. However, for completeness, we mention that they include both positive and negative ones in an alternating manner. The first few are κ 5,F, [1,2] = −(0.4344482 × 10 −4 ), κ 6,F, [1,2] = 0.323207 × 10 −4 , κ 7,F, [1,2] = −(1.90897 × 10 −5 ), etc. This difference in the signs of the κ j,F,[0,3] and κ j,F, [1,2] for j ≥ 5 accounts for the fact that γψ ψ,IR,F,[0,3] > γψ ψ,IR,F, [1,2] , as observed in Table II and Fig. 1. Similar comments apply for the SU(3) and SU(4) theories with R = F to be discussed next.

D. SU(3)
For G = SU(3) with R = F , the general formulas (2.1) and (2.3) yield the values N u = 16.5, N ℓ = 8.053. Thus, for this theory, the interval I for N f ∈ R + is 8.053 < N f < 16.5 with ∆ f,max = 8.447, and the physical interval I with N f ∈ N + is 9 ≤ N f ≤ 16 (see Table I). From our calculation of γψ ψ,IR,F,∆ 4 f in [12], we presented polynomial extrapolations to infinite order to obtain estimates of lim s→∞ γψ ψ,IR,∆ s f . Combining these with the conformality upper bound (2.12) and the assumption that, as in the supersymmetric case, γψ ψ,IR saturates this upper bound at the lower end of the NACP, we estimated that [12] SU(3), R = F : N f,cr ≃ 8 − 9 , (3.14) in agreement with several lattice estimates [22], [47]- [53]. As we will discuss, our new calculations presented here are consistent, to within the intrinsic theoretical uncertainties involved, with our estimate of N f,cr given in [12].
The general features that we remarked on for the SU (2) theory with R = F are also evident here. For N f in the upper part of the non-Abelian Coulomb phase, the values of γψ ψ,IR,F, [1,2] and γψ ψ,IR,F,[0,3] are quite close to γψ ψ,IR,R,∆ 4 f . As N f decreases, γψ ψ,IR,F, [1,2] continues to be close to γψ ψ,IR,F,∆ 4 f , while γψ ψ,IR,R,[0,3] becomes progressively larger than γψ ψ,IR,F,∆ 4 f and γψ ψ,IR,F, [1,2] . For small N f near to the lower end of the non-Abelian Coulomb phase, γψ ψ,IR,F,[0,3] rises up and eventually exceeds the conformality upper bound (2.12) for N f between 8 and 9. Using the value of N f where γψ ψ,IR,F,[0,3] exceeds this upper bound as an estimate of N f,cr , we derive the result N f,cr ∼ 8 − 9, in agreement with Eq. (3.14) from [12] and with most lattice estimates.
It is also worthwhile to compare our results with other lattice studies, bearing in mind that (i) our calculations assume an exact IR fixed point, as is true in the non-Abelian Coulomb phase, and (ii) as N f decreases toward the lower end of the NACP and ∆ f increases, one generally needs more terms in a series expansion in powers of ∆ f to achieve a given accuracy. For SU (3), R = F , and N f = 10 (see Table II), again rounding off to two significant figures, we obtain the Padé approximant values γψ ψ,IR,F, [1,2] = 0.63 and γψ ψ,IR,F,[0,3] = 0.76. The first of these Padé-based values is close to our highest-order scheme-independent series calculation [12], γψ ψ,IR,F,∆ 4 f = 0.62, while the second is slightly higher. A study of the SU(3) theory with R = F and N f = 10 was reported in [56], with the result γψ ψ,IR,F ∼ O(1). To within the estimated uncertainties, our values for this theory from [12], as augmented by our new results from Padé approximants, are in reasonable agreement with this estimate of γψ ψ,IR,F from [56]. For SU(3) with R = F and N f = 8, as is evident in Table II, there is a significant difference between the values of our two Padé approximants, indicating that a calculation of the series to higher order in ∆ f than O(∆ 4 f ) would be desirable.
This theory with N f = 8 been the subject of a number of lattice studies, including [57,58], which have observed quasi-conformal behavior. However, there has not yet been a decisive conclusion on whether or not this SU(3) theory with R = F and N f = 8 is in the (chirally symmetric) non-Abelian Coulomb phase, where the IRFP is exact and our calculations apply, or in the chirally broken phase [24]. The Taylor series expansion of the Padé approximant γψ ψ,IR,F,[0,3] to calculate higher-order coefficients κ j,F with j ≥ 5 is again of interest for this SU(3) theory. We list the higher-order coefficients κ j,F with 5 ≤ j ≤ 10 from the Taylor series expansion of γψ ψ,IR,F,[0,3] in Table  IV. As was the case with the SU(2) theory, these coefficients are all positive. We find the same positivity for the highest order, j = 200, to which we have calculated the series expansion of this Padé approximant in powers of ∆ f . As with SU(2), this shows the self-consistency of our use of γψ ψ,IR,F, [0,3] here to estimate N f,cr , which assumed this positivity of higher-order coefficients. Similarly to the SU(2) theory, the higher-order coefficients κ j,F from the Taylor series expansion of the other approximant, γψ ψ,IR,F, [1,2] , which we did not use to estimate N f,cr (since it is smaller than use of γψ ψ,IR,F,[0,3] ) have alternating signs, starting with a negative κ 5,F .
In Table II we list the values of the Padé approximants γψ ψ,IR,F, [1,2] and γψ ψ,IR,F,[0,3] for this SU(4) theory, as a function of N f . For comparison, we also include the values of γψ ψ,IR,F,∆ s f with 1 ≤ s ≤ 4 from [13,15]. In Fig. 3 we plot all of these values. The general features of these results for SU(4) are similar to the features that we have already discussed for SU(2) and SU (3). Again using the larger of the two Padé-based values, γψ ψ,IR,F,[0, 3] , and inferring N f,cr as the value of N f where this exceeds the conformality upper bound, we derive the estimate SU(4) : N f,cr ∼ 11 . (3.21) The self-consistency of our procedure is again shown by the fact that the higher-order coefficients κ j,F with j ≥ 5 from the Taylor series expansion of γψ ψ,IR,F,[0,3] are positive. We list these κ j,F in Table IV. (black). The curves for γψ ψ,IR,F, [1,2] and γψ ψ,IR,F,[0,3] are dashed magenta and dotted magenta. Over most of the range of N f , the dashed curve for the extrapolated value γψ ψ,IR,F lies highest with cyan color online.

F. LNN Limit
Here, in the LNN limit, we calculate Padé approximants for our series γψ ψ,IR,∆ 4 r from s = 4 in [13,15]. The values ofκ j,F were given (analytically) in [13] for 1 ≤ j ≤ 3 and in [15] for j = 4, and arê and calculate the [p, q] Padé approximant to the expression in square brackets, with p + q = s − 1. We have calculated analytic results for Padé approximants to γψ ψ,IR,F,∆ 4 r . It is again simplest to present these in numerical form.
We find The [2,1] Padé approximant has a pole at ∆ r = 2.28032, or equivalently, r = 3.21968, which lies in the interval I r and also in the inferred NACP (see Eq. (2.21 below)), and hence we cannot use this approximant for our analysis. The [1,2] Padé approximant has poles at ∆ r = −0.291997 and ∆ r = 4.26813, i.e., at r = 5.79200 and r = 1.23187. The first pole lies above r u , where the theory is not asymptotically free, and the second pole lies below r ℓ . Considering γψ ψ,IR,F,LN N, [1,2] as an analytic function, the pole at ∆ r = −0.291997 lies much closer to the origin ∆ r = 0 than the radii of both of the disks |∆ r | < ∆ r,max = 2.8846 and |∆ r | < ∆ r,cr = 2.6 (where ∆ r,cr is given below in Eq. (2.24)). Consequently, we cannot use this [1,2] Padé fully reliably for our analysis. However, it turns out that because the pole lies on the opposite side of the origin in the ∆ r plane, at negative ∆ f , relative to the interval I : 0 < ∆ r < 2.8846 and the NACP, where 0 < ∆ r < ∼ 2.9, the approximant γψ ψ,IR,F,LN N, [1,2] is actually rather close to γψ ψ,IR,F,∆ 4 r . The [0,3] Padé approximant has a zero at ∆ r = 3.03365, which is larger than ∆ r,max and ∆ r,cr . In terms of r, this pole is at r = 2.46635, which lies below r ℓ . This [0,3] approximant also has complex poles at ∆ r = −1.370865 ± 5.210899i with magnitude |∆ r | = 5.38820, which is larger than the values ∆ r,max = 2.8846 and ∆ r,cr = 2.6. Therefore, we can use the [0,3] Padé ap-proximant reliably.
In Table III we list values of γψ ψ,IR,F, [1,2] and γψ ψ,IR,F,[0,3] for this LNN limit and, for comparison, values of γψ ψ,IR,F,∆ s r for 1 ≤ s ≤ 4 from [13,15]. We have remarked above that although one of the poles in γψ ψ,IR,F, [1,2] lies within the disk |∆ r | < ∆ r,max in the complex ∆ r plane, this pole does not occur in the region of positive ∆ r of interest here and hence does not strongly affect the values of γψ ψ,IR,F, [1,2] in this region. As was the case with the specific SU(N c ) theories with R = F discussed above, for each of the given r values in Table III, γψ ψ,IR,F,[0,3] is larger than γψ ψ,IR,F, [1,2] . At the upper end of the NACP, these are very close to each other, as they are to γψ ψ,F,∆ 4 r . As r decreases sufficiently, γψ ψ,IR,F,[0,3] first exceeds the conformality upper bound at r ≃ 2.9. Thus, if we use this value as the estimate of the lower end of the NACP, we infer that LN N : r cr ≃ 2.9 (3.31) and hence LN N : ∆ r,cr ≃ 2.6 . (3.32) Our inferred value of r cr defining the lower end of the NACP is slightly larger than the value of r defining the lower end of the interval I, namely, r ℓ = 2.615, and correspondingly, the maximal value of ∆ r in the NACP, ∆ r,cr ≃ 2.6, is slightly smaller than the maximal value of ∆ r in the interval I, namely ∆ r,max = 2.8846, as given in (2.23). It is of interest to investigate how close the N f,cr values inferred for SU(N c ) theories with R = F and finite N c and N f are to our result (3.31) in the LNN limit. To make this comparison, we compute LNN-based reference (LN N r) values, defined as N f,cr,LN N r ≡ r cr N c . above. This shows that the approach to the LNN limit appears to be rather rapid, even for relatively small values of N c . In Table V we list the higher-order κ j,F,LN N,[0,3] with 5 ≤ j ≤ 10 calculated via a Taylor series expansion of γψ ψ,F,LN N, [0,3] . We focus on this approximant, since it is the only one free from poles in the disks |∆ r | < ∆ r,max and |∆ r | < ∆ r,cr . As was the case with the specific SU(N c ) theories with R = F that we have studied above, all of these higher-order coefficients are positive. We have further verified this positivity up to a much higher order, j = 200.

G. Adjoint Representation
For SU(N c ) and R = A, the adjoint representation, the values of N ℓ , N u , and ∆ f,max were given above in Eq. (2.8). As indicated in Table I, there is only a single integral value of N f in the interval I for these theories, namely N f = 2. The SU(2) theory with R = A and N f = 2 has been of some previous theoretical interest [61,62]. As we have discussed before [2], since this is a real representation, one could also take N f to be halfintegral, corresponding to Majorana fermions, but the N f = 2 value will be sufficient for our study here. In [2] we carried out n-loop calculations of γψ ψ,IR,nℓ for 2 ≤ n ≤ 4. Our two highest-order values for SU (2)  We gave explicit analytic results for the schemeindependent expansion coefficients κ j,A with 1 ≤ j ≤ 3 in [13] and for j = 4 in [14]. Both κ 1,A = 4/9 and κ 2,A = 341/1458 are independent of N c , while the k j,A for j ≥ 3 depend on N c . With the prefactor κ 1,A ∆ f extracted and the Padé approximants normalized to unity at ∆ f = 0 as in Eq. (3.2), we calculate the following approximants to γψ ψ,IR,A,∆ 4 f (in addition to γψ ψ,IR,A, [3,0] = γψ ψ,IR,A,∆ 4 f .) For SU(2) we find ter (the coefficient β of the plaquette term in the lattice action) and hence more work was needed to determine the actual γψ ψ,IR .

H. Symmetric Rank-2 Tensor Representation
For our SU(N c ) theories with R = S 2 , the symmetric, rank-2 tensor representation, the values of N ℓ , N u , and ∆ f,max were given above in Eq. (2.9) and are listed numerically in Table I. In the case of SU(2), the S 2 representation is the same as the adjoint representation, which we have already analyzed above. Thus, as in our earlier work, we focus on the two illustrative theories, SU(3) and SU (4). With both of these theories, the interval I contains two integral values of N f , namely N f = 2 and N f = 3. Explicit analytic results for κ j,S2 in SU(N c ) theories with R = S 2 were given for 1 ≤ j ≤ 3 in [13] and for j = 4 in [15]. The lowest-order coefficient is . (3.45) This has the values κ 1,S2 = 200/519 = 0.385356 for SU(3) and κ 1,S2 = 54/155 = 0.348387 for SU (4).
With the prefactor κ 1,S2 ∆ f extracted and the Padé approximant normalized as in Eq. (3.2), we calculate the following Padé approximants, in addition to γψ ψ,IR,S2, [3,0] All of these three Padé approximants with q = 0 for the SU(3) theory with R = S 2 have poles in the interval I. For the corresponding SU(4) theory, we find that the Padé approximants share the property with our SU(3) results of all having poles in the interval I. Consequently, for these theories, our Padé analysis does not add to our previous study of the γψ ψ,IR,S2 to O(∆ 3 f ) in [13] and to O(∆ 4 f ) in [14,15]. There have been several lattice studies of the SU(3) theory with N f = 2 fermions in the symmetric rank-2 tensor (sextet) representation. These include Ref. [70], which concluded that it is IR-conformal and obtained γ IR < 0.45 and Ref. [71], which concluded that it is not IR-conformal, but instead exhibits spontaneous chiral symmetry breaking and obtained an effective γ IR ≃ 1.

A. General
In this section we report our computation and analysis of Padé approximants for β ′ IR , using our calculation of β ′ IR to O(∆ 4 f ) in [13] and to O(∆ 5 f ) in [14,15]. As noted, β ′ IR is equivalent to the anomalous dimension of Tr(F µν F µν ) [10]. We also discuss the behavior of β ′ IR toward the lower end of the non-Abelian Coulomb phase at N f = N f,cr . This behavior is relevant to the change in the properties of the theory as N f increases through N f,cr [72].
For a given truncation of the series (2.14) to maximal order s we write β ′ IR as . We do not consider this, since we have already obtained evaluations of this series truncation in previous work [14,15].
We focus here on SU(N c ) theories with fermions in the fundamental, R = F and consider two illustrative values of N c , namely 2 and 3. For SU(N c ), with R = F , . Although the lowest two nonzero coefficients d 2 and d 3 are manifestly positive for any gauge group G and fermion representation R, our calculation of d 4 in [13] and d 5 in [14,15] showed that these are both negative for SU(N c ) and R = F . Explicitly, for SU(2) and SU(3) [14,15], For SU (2) The The  [52] for this SU(3) theory with R = F and N f = 12, finding agreement. Here, we extend this comparison with the new input from our Padé calculation. We recall that the conventional higher-order nloop calculations in powers of α are β ′ IR,3ℓ,F = 0.2955 and β ′ IR,4ℓ,F = 0.282 [4], which agree with this lattice measurement. As we noted in [15], our higher-order scheme-independent values, namely, β ′ f ,F = 0.228, are also in agreement with this lattice value from [52]. Our new Padé value, γψ ψ,IR,F, [1,2] = 0.231, again in reasonable agreement with both our earlier values from our higherorder n-loop and scheme-independent series expansions and with the lattice value from [52].

D. Discussion
The behavior of β ′ IR in the middle and upper part of the non-Abelian Coulomb phase can be accurately described by the series expansion (2.14), since ∆ f approaches zero as N f approaches N u from below. The behavior of β ′ IR toward the lower end of the NACP is also of considerable interest. As with γψ ψ,IR , one can gain a useful perspective concerning this behavior from known results for a vectorial, asymptotically free N = 1 supersymmetric theory with a gauge group G and N f chiral superfields Φ j andΦ j transforming according to respective representations R andR of G [29]- [31]. For this supersymmetric gauge theory, the NACP occupies the range (2.32), and β ′ IR → 0 at the lower end, as well as the upper end, of the NACP [73]. In [16] we calculated Padé approximants for β ′ IR from finite series expansions in ∆ f and found consistency with the vanishing of β ′ IR at the lower end of the NACP (as well as the obvious zero of β ′ IR at the upper end, where ∆ f → 0). Returning to the non-supersymmetric theories under consideration here, although d 2 and d 3 are manifestly positive for any G and R, we found that d 4,F and d 5,F are negative for SU(N c ) and R = F [13]. Consequently, as N f decreases below N u , i.e., ∆ f increases from zero, β ′ IR is initially positive, and has positive slope. As N f decreases further, i.e., ∆ f increases further, the negative d 4,F ∆ 4 f + d 5,F ∆ 5 f terms become progressively more important. From Table VII, one sees that, for the range of N f included, β ′ IR,F,∆ 5 f and the resultant Padé approximants are continuing to increase with decreasing N f . Evidently, the negative d 4,F ∆ 4 f + d 5,F ∆ 5 f terms are not sufficiently large in magnitude to cause β ′ IR to vanish at the lower end of the NACP. Hence, one needs to calculate the scheme-independent series expansion for β ′ IR , Eq. (2.14), to higher order in ∆ f to see this turnover.
Insofar as the behavior of β ′ IR in the N = 1 supersymmetric gauge theory is at least a qualitative guide to the non-supersymmetric gauge theory, then it suggests that the exact β ′ IR would reach a maximum in the NACP and then would decrease and vanish as N f decreased to the lower end of this NACP. We suggest that this is a plausible behavior for the non-supersymmetric theory.

V. CONCLUSIONS
In this paper, we have presented several new results on the anomalous dimension, γψ ψ,IR , and the derivative of the beta function, β ′ IR , at an infrared fixed point of the renormalization group in vectorial, asymptotically free SU(N c ) gauge theories with N f fermions transforming according to several representations R, including the fundamental, adjoint, and rank-2 symmetric tensor. We have used our series for γψ ψ,IR to O(∆ 4 f ) to calculate Padé approximants and have evaluated these to obtain further estimates of γψ ψ,IR . Our new results using these Padé approximants are consistent with our earlier results using the series themselves calculated to O(∆ 4 f ). We have compared the values of γψ ψ,IR with lattice measurements for various theories. Taylor-series expansions of the Padé approximants have been calculated to determine their predictions for higher-order coefficients. We have found that all of the Padé approximants that we have calculated that satisfy the requisite constraints (absence of poles in the disks |∆ f | < ∆ f,max and |∆ f | < ∆ f,cr in the complex ∆ f plane) yield Taylor-series expansions with positive coefficients κ j , providing further support for our earlier conjecture that the κ j are positive. We have also used our Padé results to obtain new estimates of the value of N f,cr at the lower end of the non-Abelian Coulomb phase for various N c and R. Since, for a given SU(N c ) gauge group and fermion representation R, the upper end of the NACP, namely N u , is known exactly, these estimates of N f,cr are equivalently estimates of the extent of the non-Abelian Coulomb phase, as a function of N f , for each of the theories that we have considered. In a different but related application, our values of N f,cr are useful for the phenomenological program of constructing and studying quasi-conformal gauge theories to explore ideas for possible ultraviolet completions of the Standard Model. This is because, for a given gauge group G and fermion representation R, one must choose N f to be slightly below N f,cr (requiring that one know N f,cr ) in order to achieve the quasi-conformal behavior whose spontaneous breaking via formation of fermion condensates could have the potential to yield a light, dilatonic Higgs-like scalar. We have carried out calculations of Padé approximants for β ′ IR , using our series to O(∆ 5 f ) for these theories. Again, the results for β ′ IR obtained from these Padé approximants are consistent with, and extend, our earlier analyses using the series themselves. Our values for γψ ψ,IR and β ′ IR obtained with Padé approximants provide further information about fundamental properties of conformal field theories. Finally, we have presented new analytic and numerical results assessing the accuracy of a series expansion of γψ ψ,IR to finite order in powers of ∆ f by comparison with the exactly known expression in an N = 1 supersymmetric gauge theory, showing that an expansion to O(∆ 4 f ) is quite accurate throughout the entire non-Abelian Coulomb phase of this supersymmetric theory. I: N ℓ , Nu, ∆ f,max , and interval I in terms of N f , for G = SU(Nc) with fermions in the representation R equal to fundamental (F), adjoint (A), and rank-2 symmetric (S 2 ) tensor. The interval I is listed for N f formally generalized to real numbers, R + and for physical, integral values of N f ∈ N + . Note that for R = A, N ℓ and Nu are independent of Nc.       [2,1] and β ′ IR,F, [1,2] for SU(Nc) with Nc = 2, 3 and R = F (fundamental) as a function of N f . For comparison, we also include the values of β ′ IR,F,∆ s f from [13,15]. The columns list β ′ IR,F,∆ s f for 2 ≤ j ≤ 5, and then β ′ IR,F, [2,1] and β ′ IR,F, [1,2] . As discussed in the text, for SU (3)