$\beta'_{IR}$ at an Infrared Fixed Point in Chiral Gauge Theories

We present scheme-independent calculations of the derivative of the beta function, denoted $\beta'_{IR}$, at a conformally invariant infrared (IR) fixed point, in several asymptotically free chiral gauge theories, namely SO($4k+2$) with $2 \le k \le 4$ with respective numbers $N_f$ of fermions in the spinor representation, and E$_6$ with fermions in the fundamental representation.


I. INTRODUCTION
The properties of an asymptotically free gauge theory at an infrared fixed point (IRFP) of the renormalization group (RG) in a conformally invariant regime are of fundamental interest. 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 RG 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 momentum scale. The value of α at the IRFP is denoted α IR . For a given gauge group G and fermion representation R of G, the requirement of asymptotic freedom places an upper bound on the number of fermions transforming according to this representation. If the number of these fermions is slightly less than the maximum allowed by asymptotic freedom, then the theory flows from the UV to a weakly coupled IRFP in a non-Abelian Coulomb phase. At this IRFP the theory is scale-invariant and is inferred 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]). However, beyond respective low 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 (VGT) with gauge group G and N f Dirac fermions 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 such that for N f < N u , the theory is asymptotically free. 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 N u − N f [5]. This has the advantage that the coefficients in the expansion are scheme-independent.
Recently, for vectorial gauge theories, we have calculated such scheme-independent expansions in [6]- [10] for anomalous dimensions of (gauge-invariant) fermion bilinears and the derivative of the beta function, evaluated at the IRFP, These are both physical quantities and hence are schemeindependent [11]. 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 [12].
Here we extend our analysis to asymptotically free chiral gauge theories, denoted χGTs (in four spacetime dimensions). We consider several such theories, which can be classified into two general types: (i) theories with special orthogonal gauge groups G = SO(N ), where N = 4k + 2 with k ≥ 2, containing N f chiral fermions transforming according to the spinor representation S of this group; and (ii) theories with the exceptional gauge group G = E 6 , containing N f chiral fermions in the fundamental (27-dimensional) representation. These representations are complex [13]. Without loss of generality, all fermions may be taken as left-handed. We present scheme-independent calculations of β ′ IR to O(∆ 5 f ) in these theories, where The fermions are massless, since fermion mass terms are forbidden by the chiral gauge invariance. For the same reason, the anomalous dimensions γψ ψ,IR are gaugedependent here and hence are not of physical interest, so we focus on β ′ IR . As will be shown in Section II, the constraint of asymptotic freedom limits our consideration of SO(4k + 2) theories to those with k = 2, 3, 4, i.e., SO(10), SO (14), and SO (18), and, for each of these, this constraint yields corresponding upper limits on N f , namely N f ≤ 21 for SO (10), N f ≤ 8 for SO (14), and N f ≤ 2 for SO (18). Similarly, the requirement of asymptotic freedom yields the upper limit N f ≤ 21 in the E 6 theory. Our SO(4k + 2) theories with k ≥ 2 and our E 6 chiral gauge theory have no gauge anomaly [14,15] and no global Witten-type anomaly [16], since the relevant π 4 homotopy groups are trivial [17]. (Theories that have complex representations and vanishing gauge anomaly are commonly called safe.) There is currently renewed interest in four-dimensional conformal field theories (some reviews include [18]). Our results serve as useful input to the further study of conformally invariant gauge theories.

A. General
Here we briefly review some background and methods relevant for our work. We refer the reader to our previous papers [6]- [10] for details. As noted, we focus on chiral gauge theories that have complex representations but for which the gauge anomaly vanishes identically. These include theories with the gauge groups G = SO(N ), where N = 4k + 2 with k ≥ 2 and G = E 6 [14,15]. The requirement of asymptotic freedom implies that N f must be less than an upper (u) bound N u , where Here, C 2 (R) is the quadratic Casimir invariant for the representation R, For a chiral gauge theory, the two-loop (2ℓ) beta function (which is scheme-independent) has an IR zero if N f lies in the interval I defined by This IR zero occurs at 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,2ℓ 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 (NACP, also called conformal window). We denote the lowest value of N f in this NACP as N f,cr . Our calculations assume that the IRFP is exact, as is the case in the non-Abelian Coulomb In the analytic expressions and plots given below, this assumption will be understood implicitly. 1 Here and elsewhere, when an expression is given for N f that formally evaluates to a non-integral real value, it is understood implicitly that one infers an appropriate integral value from it. 2 For sufficiently small N f , nonperturbative phenomena involving strong coupling can occur, including possible spontaneous chiral symmetry breaking with generic breaking of the chiral gauge symmetry, or, if 't Hooft anomaly-matching conditions are satisfied, confinement without global or gauge symmetry breaking. For a recent discussion and references to the literature, see, e.g., [19]. We do not consider such phenomena at smaller N f here, instead restricting to the non-Abelian Coulomb phase, where the gauge and chiral symmetries are exact.
B. Interval I for SO(4k + 2) Theories For our SO(N ) theories with N = 4k + 2, k ≥ 2, and chiral fermions in the spinor representation S, one has T S = 2 (N/2)−4 and C 2 (S) = N (N − 1)/16, so Here N u takes on the values (i) 22 for k = 2, i.e., SO (10) N ℓ takes on the value (i) 4352/455 = 9.564835 for SO(10); (ii) 816/251 = 3.250996 for SO (14); and (iii) 1088/1099 = 0.9899909 for SO (18). In Table I we list the resultant intervals I in N f for which the asymptotically free chiral gauge theories of SO(4k + 2) type have a two-loop beta function with an IR zero. For each case, we give two ranges, namely one for N f formally generalized to R + , and the second for physical, integral N f ∈ N + .

C. Interval I for E6 Theory
For the E 6 chiral gauge theory with N f fermions in the fundamental (27-dimensional) representation, F , C A ≡ C 2 (G) = 12, T F = 3, and C 2 (F ) = 26/3, so N u = 22. Hence, to maintain asymptotic freedom in this E 6 theory, we require that N f < 22. Furthermore, we calculate that N ℓ = 408/43 = 9.488372. Therefore, the interval I for this E 6 theory is D. 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 dervative β ′ IR has the scheme-invariant expansion 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. For our calculation of β ′ IR to O(∆ 5 f ) in [9], we thus made use of the five-loop beta function from [21]. In the literature, the beta function coefficients have usually been given for a vectorial gauge theory with N f Dirac fermions in a representation R of the gauge group G. In the case of a chiral gauge theory with fermions in a single representation of the gauge group, one can take over these results with the replacement N f → N f /2, reflecting the replacement of Dirac with chiral fermions. In particular, we can use our previous calculations of the d j with 2 ≤ j ≤ 4 in [8] and d 5 in [9] in a VGT for the χGTs under consideration, with the correspondence 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 Let the full scaling dimension of Tr(F µν F µν ) be denoted D F 2 (with free-field value 4). At an IRFP, D F 2 ,IR = 4 + β ′ IR [12], 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 [20]. Since β ′ IR > 0, this bound is obviously satisfied.
For the SO(N ) theories with N = 4k + 2 considered here, namely SO(10), SO (14), and SO (18) with N f fermions in the spinor representation, and N f in the respective intervals in Table I, we calculate 3 Some authors use the opposite sign convention for the anomalous dimension, writing and where ζ s = ∞ n=1 n −s is the Riemann zeta function. Concerning the signs of these coefficients, d 2 and d 3 are manifestly positive (for a general G and R) [8], while the signs of d 4 and d 5 depend on the theory.
Evaluating these for the SO(N ) theories under consideration, we obtain the following results for β ′ IR calculated up to O(∆ 5 f ) order (in floating-point format): SO (14) :