Duality in a Supersymmetric Gauge Theory From a Perturbative Viewpoint

We study duality in $\mathcal{N}=1$ supersymmetric QCD in the non-Abelian Coulomb phase, order-by-order in scheme-independent series expansions. Using exact results, we show how the dimensions of various fundamental and composite chiral superfields, and the quantities $a$, $a/c$, and $b$ at superconformal fixed points of the renormalization group emerge in scheme-independent series expansions in the electric and magnetic theories. We further demonstrate that truncations of these series expansions to modest order yield very accurate approximations to these quantities.


I. INTRODUCTION
Transformations that allow one to deal with a strongly coupled quantum field theory as a weakly coupled field theory in a different form have proved to be very powerful throughout the history of physics. An important example is provided by the lattice formulation of quantum chromodynamics (QCD). Although the property of asymptotic freedom made possible perturbative calculations at large Euclidean energy/momentum scales µ in the deep ultraviolet (UV), the growth of the running gauge coupling g(µ) in the infrared (IR) prevented reliable perturbative calculations at low energies. However, this difficulty was surmounted by Wilson's formulation of the theory on a (Euclidean) lattice [1], in which the plaquette term in the action is multiplied by the coefficient β = 2N c /g 2 0 , where g 0 is the bare gauge coupling. Thus, the strong coupling limit g 0 → ∞ is equivalent to β → 0, allowing analytic strong-coupling Taylor series expansions in powers of β. By means of such an expansion, the area-law behavior of the Wilson loop and hence confinement in QCD were proved for strong g 0 [1]. In a different but related way, a duality transformation links two different regimes of a theory. In statistical mechanics, for a two-dimensional Ising model, a duality transformation maps the high-temperature regime to the lowtemperature regime and led to the calculation of the critical temperature in this model [2]. Another realization of duality occurs in the generalization of electromagnetic theory to include Dirac monopoles.
Here we will consider a theory for which duality relations have been very useful, namely an asymptotically free, vectorial, gauge theory (in d = 4 spacetime dimensions, at zero temperature) with N = 1 supersymmetry, having a gauge group SU(N c ) and N f massless chiral superfields Q i andQ i , i = 1, ..., N f , transforming in the fundamental and conjugate fundamental representations of SU(N c ), respectively. This theory is invariant under a global symmetry group G gb = SU(N f ) ⊗ SU(N f ) ⊗ U(1) B ⊗ U(1) R symmetry, with the representations indicated in Table I. Following common terminology, we call this theory supersymmetric quantum chromodynamics (SQCD). We denote α = g 2 /(4π). This theory has the appeal that many of its properties are well understood at a nonperturbative level [3]- [6]. The property of asymptotic freedom requires that N f < 3N c . In this range, the theory is weakly interacting in the deep UV, so one can self-consistently calculate its properties perturbatively. One can then investigate how it evolves ("flows") from the deep UV to the IR limit as µ → 0. For N f slightly less than 3N c , there is convincing evidence that the theory evolves to an infrared fixed point (IRFP) of the renormalization group (RG) at α IR , at which point it is scale-invariant and is inferred [7] to be (super)conformally invariant. If N f is in the interval I : (3/2)N c < N f < 3N c , the theory flows to an IRFP in a (deconfined, chirally symmetric) non-Abelian Coulomb phase (NACP) [4]. Henceforth, we restrict our consideration to the NACP in this theory.
At this superconformal IRFP, it was conjectured in [4] that the original theory is dual to the IR limit of another N = 1 supersymmetric theory with a gauge group SU(Ñ c ), whereÑ c = N f − N c , with matter content consisting of N f chiral superfields q i andq i , i = 1, ..., N f , in the fundamental and conjugate fundamental representations of SU(Ñ c ), respectively, together with a set of N 2 f gauge-singlet "meson" chiral superfields φ i j , 1 ≤ i, j ≤ N f . The original and dual theories were called "electric" and "magnetic" in [4,5]. The dual theory also allows for a unique superpotential W = λφqq, where λ is the superpotential coupling. Evidence for this conjectured equivalence includes the fact that the dual theory is also invariant under the same global G gb symmetry, and satisfies 't Hooft anomaly matching [4]- [6].
The dual theory is asymptotically free for N f < 3Ñ c , i.e., N f > (3/2)N c and for N f in the intervalĨ : (3/2)Ñ c < N f < 3Ñ c , it flows to a superconformal IRFP in the space of gauge and λ couplings, at which the physics is equivalent to that in the original theory. There is an isomorphism between the intervals I andĨ such that the upper end of I where N f ր 3N c maps to the lower end ofĨ, where N f ց (3/2)Ñ c , and vice versa. The weak-coupling region in the original theory corresponds to strong coupling in the dual theory, and vice versa. This duality is well supported by nonperturbative arguments, but one gains further insight by seeing how the duality relations emerge perturbatively. However, a conventional perturbative calculation, as a series expansion in powers of the gauge coupling, encounters the difficulty that although α IR → 0 as N f ր 3N c at the upper end of the NACP, this theory becomes strongly coupled, and hence not amenable to this type of perturbative approach, as N f ց (3/2)N c at the lower end of the NACP. A similar comment applies to conventional perturbative expansions in the magnetic theory, which has a weak gauge coupling as N f ր 3Ñ c , but is strongly coupled as N f ց (3/2)Ñ c . Furthermore, even in the respective regions of the electric and magnetic theories where they are weakly coupled, perturbative expansions in powers of gauge couplings are scheme-dependent.
Here we surmount this difficulty and present, for the first time, a scheme-independent perturbative understanding of the duality in the non-Abelian Coulomb phase of SQCD. An important property that we utilize is the fact that α IR → 0 as N f ր 3N c , so that, as observed in [8] (see also [9]), one can alternatively express physical quantities at an IRFP in the NACP as series in powers of the manifestly scheme-independent quantity ∆ f = 3N c − N f . As N f decreases from 3N c to (3/2)N c , ∆ f increases from 0 to its maximal value of (3/2)N c in the NACP. We have calculated schemeindependent expansions of physical quantities such as anomalous dimensions at an IRFP in various theories [10,11]. By comparisons of finite truncations of such expansions with exactly known expressions for operator dimensions in supersymmetric gauge theories, we have shown that these series truncated to a modest order, such as O(∆ 4 f ), can provide quite accurate approximations to these anomalous dimensions throughout the entire NACP [11]. The expansion parameter in the dual We will study dimensions of various (gauge-invariant) chiral superfield operators and of certain quantities a, a/c, and b characterizing RG flows, in both the original and dual theories, as expansions in both ∆ f and∆ f , and will show how various relations emerge order-by-order in these expansions. From this analysis, we will show that a combination of finite-order expansions in these two dual expansion parameters ∆ f and∆ f can yield quite accurate approxima-tions to physical quantities throughout the entire NACP. Our work thus demonstrates how perturbative calculations in the well-chosen scheme-independent expansion parameters ∆ f and∆ f can provide insight into results based on abstract nonperturbative methods.
For a (gauge-invariant) quantity P at the superconformal IRFP, we write the expansions of P in powers of ∆ f and∆ f as In the first term of Eq. (1.1), one takes N c as fixed and computes P as a function of the variable ∆ f , or equivalently, N f . In the second term, for the dual theory, one takesÑ c = N f − N c as fixed; while varying N f , one can keepÑ c fixed by formally varying N c oppositely to N f . As noted, these expansions in (1.1) have the advantage, relative to conventional perturbative expansions in powers of couplings, of being scheme-independent at each order. Furthermore, the n'th-order coefficients have a definite relation to terms in conventional perturbative expansions. One important example is where P is the (full) scaling dimension D O of a physical operator O, which, in general, differs from its free-field value, due to interactions. We write For n ≥ 1, the coefficient D O,n depends on the terms in the beta function of the theory up to loop order ℓ = n+1 and on the terms in the conventional expansion of D O in powers of α IR up to loop order ℓ = n, inclusive, but does not receive contributions from higher-loop terms. This is a very powerful observation which can be used in two different ways. First, it allows for a calculation of any physical quantity in a manifestly scheme-independent way, orderby-order, even if one only knows the gauge coupling beta function and D O to some finite loop order. Second, if, on the other hand, one knows an exact expression for a quantity P at the IRFP, then one can calculate its expansions in powers of ∆ f and∆ f in Eq. (1.1) exactly, to all orders, without doing any explicit loop computations.
In the original (electric) theory, gauge-singlet composite chiral superfields include the meson-type operators M i j = Q iQ j , and baryon-and antibaryon-type op- Similarly, in the dual (magnetic) theory, in addition to φ i j , one has the dual baryon and antibaryon operators b i1···iÑ c = ǫ i1···iÑ c q i1 · · · q iÑ c andb i1···iÑ c = ǫ i1···iÑ cq i1 · · ·q iÑ c . Duality dictates that the meson operators are matched in the electric and magnetic theories, and similarly for the baryon operators and the antibaryon operators. Hence, the meson operators in the electric and magnetic theories must have the same dimensions, and similarly for the baryon and antibaryon operators.
In a superconformal theory, the dimension D O of a gauge-invariant chiral superfield operator O is related to the R-charge, R O , of the operator via D O = (3/2)R O . This implies, in particular, that the scaling dimension of the (composite) electric and (fundamental) magnetic mesons is Note that D M,f ree = 2 while D φ,f ree = 1. We want to understand how Eq. (1.2) emerges order-by-order in the scheme-independent expansions in the original and dual theories. Calculating the series expansion of D in powers of ∆ f , we find Equivalently, in the dual theory, we find for fixed N f , D s , as calculated in the electric theory, decreases monotonically with s; and (iii) D s , as calculated in the magnetic theory, increases monotonically with s. As is evident from Fig. 1, the respective fractional accuracies of the ∆ f and∆ f series expansions in (1.3) and (1.4) are highest near the upper and lower ends of the NACP, respectively. Thus, by combining these two perturbative calculations, we achieve an excellent approximation to the exact expression (1.2) throughout all of the NACP, even with a modest value of the truncation order, s such as s = 4. This makes use of the full power of the duality, since it allows one to treat the strong-coupling regime in the original theory via a perturbative calculation in the weak-coupling regime of the dual theory, and vice versa.
We next consider the baryon-type operators. In both the electric and magnetic theories, their scaling dimensions have to agree, and are As with the mesons, we want to understand how this expression for D ′ emerges order-by-order in perturbation theory, as calculated in both the original and dual theories. In the original (electric) theory, we find while in the dual (magnetic) theory we find (1.7) As before, these series converge throughout the entire NACP. The truncations of these series to order n = s are denoted D ′ s . In Fig. 2 Fig. 1 hold here, so that again, by combining these two perturbative calculations, we obtain an excellent approximation to the exact expression (1.5) throughout all of the NACP even with a modest truncation order, s, such as s = 4.
Corresponding to the global symmetry groups in G gb , there are conserved currents. We will focus on the conserved current J for this current (in flat space) is Here b is a function of the couplings of the theory and changes from b UV to b IR along the RG flow. For SQCD, b UV is given by its respective free-field values, b UV = 2N f /N c and b UV = 2N f /Ñ c in the electric and magnetic theories, while b IR = 6 at the IRFP in the non-Abelian Coulomb phase in both of these theories [13]. Hence, calculating b UV − b IR , we have, in the electric theory, and, in the magnetic theory, (1.10) These results show that higher-order contributions to b IR in powers of ∆ f , and equivalently in powers of ∆ f , vanish. This again shows the value of the schemeindependent series expansion method, since the zero coefficients of the respective ∆ n f and∆ n f terms with n ≥ 2 in Eqs. (1.9) and (1.10) involve complicated cancellations when computed via conventional (scheme-dependent) series expansions in powers of couplings.
The trace of the energy-momentum tensor in four spacetime dimensions, in the presence of a curved background metric g µν , is [14] T where W µνρσ is the Weyl tensor (so W µνρσ W µνρσ = R µνρσ R µνρσ − 2R µν R µν + (1/3)R 2 ) and E 4 = R µνρσ R µνρσ −4R µν R µν +R 2 is the Euler density, satisfying d 4 x |det(g µν )| E 4 = χ E , the Euler-Poincaré characteristic. Here, R µνρσ and R µν are the Riemann and Ricci tensors, and R is the scalar curvature of the manifold defined by the background metric g µν . In d = 2, it was proved that c decreases monotonically along an RG flow [15], but this monotonicity does not hold in d = 4. The quantity a encodes important information about the flow of a quantum field theory between two RG fixed points and satisfies the inequality that a UV − a IR > 0 (called the a theorem) [16]- [21], [13]. This is in accord with the Wilsonian notion of thinning of degrees of freedom along an RG flow [22,23]. For an asymptotically free theory, a UV is given by the (massless) free-field content of the theory [14]: where here N v and N χ denote the numbers of vector and chiral superfields, respectively [14,24]. In the original (electric) theory, f . The duality at the superconformal IRFP dictates that the value of a IR must be identical in the electric and magnetic theories; it is [13] Calculating a series expansion for a IR in the electric theory, in powers of ∆ f , we obtain (1.12) Eq. (1.12) shows how the a theorem is satisfied at each order in powers of ∆ f . In the magnetic theory, we find Here again, the a theorem is satisfied at each order in powers of∆ n f ; in this case, the result follows because the leading-order term in the square brackets, 21∆ 2 f , dominates over higher-order terms ∝∆ n f with 3 ≤ n ≤ 10 with opposite sign.
In Fig. 3 we plot a IR as computed to first through fourth order in ∆ f in the electric theory and in∆ f in the magnetic theory, together with the exact result, for the illustrative case N c = 3.
Although c does not obey a monotonicity relation along RG flows, there is a bound 1/2 ≤ a/c ≤ 3/2 in a superconformal theory from the positivity of the energy flux [25]- [27]. As with the other quantities analyzed here, one gains insight by investigating how finite-order expansions of this ratio at the superconformal IRFP in the original and dual theories approach the exactly known expression. In Fig. 4 we show results from these O(∆ s f ) and O(∆ s f ) expansions [28], together with the exact result.
In conclusion, we have shown, for the first time, how exact relations for dimensions of chiral superfields and for the quantities a, a/c, and b in the non-Abelian Coulomb phase of SQCD emerge order-by-order in scheme-independent perturbative series expansions, as calculated in the original (electric) theory in powers of ∆ f and in the dual magnetic theory in powers of∆ f . We have demonstrated that truncated series expansions of modest order yield quite accurate approximations to exact results for these quantities.