One - Instanton Tests of the Exact Results in N = 2 Supersymmetric QCD

We use the microscopic instanton calculus to determine the one-instanton contribution to the quantum modulus u3 = 〈Tr(φ3)〉 in N = 2 SU(Nc) supersymmetric QCD with Nf < 2Nc fundamental flavors. This is compared with the corresponding prediction of the hyperelliptic curves which are expected to give exact solutions in this theory. The results agree up to certain regular terms which appear when Nf ≥ 2Nc − 3. The curve prediction for these terms depends upon the curve parameterization which is generically ambiguous when Nf ≥ Nc. In SU(3) theory our instanton calculation of the regular terms is found to disagree with the predictions of all of the suggested curves. For this theory we employ our results as input to improve the curve parameterization for Nf = 3, 4, 5. In their seminal work [1], Seiberg and Witten applied ideas of duality to N = 2 supersymmetric QCD (SQCD) with gauge group SU(2) and Nf = 0, 1, 2, 3, 4 flavors of matter hypermultiplets, and were able to predict exact results, valid at both strong and weak coupling. This was achieved by identifying the quantum moduli spaces of these models with the moduli spaces of certain families of elliptic curves. Exact solutions for the holomorphic prepotential describing the low energy dynamics were then obtained via periods of a meromorphic one-form on the curves. This analysis has subsequently been extended to SU(Nc) models with Nc > 2 and Nf ≤ 2Nc fundamental flavors [2–5]. In the general case, the quantum moduli space is described by a family of genus Nc − 1 hyperelliptic curves. These are parameterized by the gauge invariant quantum moduli un = 〈Tr(φn)〉 (n = 2, 3, . . . , Nc), where φ is the adjoint Higgs. The proposed curves predict exact solutions for these objects as well as for the holomorphic prepotential. The exact solutions can be explicitly expanded in the semiclassical regime [8] to give the expected one-loop perturbative contribution plus predictions for k-instanton corrections. These take the form of rational functions of the vacuum expectation values (VEV’s). Non-trivial tests of the curves can be performed by applying the microscopic instanton calculus to directly evaluate these non-perturbative contributions. In SU(2), instanton calculations have been carried out at the one-instanton [9, 10] and two-instanton [11–15] levels. The results completely agree with the curves for Nf = 0, 1, 2 fundamental flavors. However discrepancies have emerged for Nf = 3 [13] and Nf = 4 [14] fundamental flavors at the two-instanton level (as well as at the one-instanton level for one adjoint flavor in [10]). For general Nc, only the singular part of the one-instanton contribution to u2 has been calculated, in [16, 17]. When Nf < 2Nc − 2 there are no additional regular terms and the result is in full agreement with the curves. It was further claimed in [17] that the results of an evaluation of the regular terms that appear when Nf ≥ 4 in the SU(3) model are in conflict with the predictions of the proposed curves. In [18, 15, 10] it was shown that the SU(2) discrepancies can be resolved in a way consistent with the analysis of Seiberg and Witten, by reinterpreting the parameters of the original curves. In fact it is a generic feature of curve construction that the curve parameterization may not be uniquely fixed when Nf ≥ Nc. In [2–5] possible curve parameterizations are suggested on the basis of various assumed criteria. The results of [17] imply that none of the SU(3) parameterizations are correct and therefore question the validity of these criteria. In this letter, we continue the program of comparing the curve predictions with the results of first-principles instanton calculations, by evaluating the one-instanton contribution to the quantum modulus u3 in N = 2 SQCD with Nc > 2 colors and Nf < 2Nc fundamental flavors. By virtue of the Matone relation [6] (see [7] for instanton based derivations) between the prepotential and u2, it actually suffices to consider only the solutions for the quantum moduli as independent predictions of the curves when Nf < 2Nc. 1 We determine the most singular part of the answer, which for Nf < 2Nc − 3 is the complete answer and agrees exactly with the prediction which we extract from the curves. Our analysis also gives the coefficients of the regular terms which arise in the SU(3) theory when Nf ≥ 3, and we find disagreement with the numbers obtained from the curves, for all of the suggested curve parameterizations. We then employ the set of microscopic instanton calculations in the SU(3) theory to improve the parameterizations of the Nf = 3, 4, 5 curves. For Nf = 3, 4 flavors the curves should be completely fixed, so that no discrepancies can appear at higher order instanton levels. The field content of SU(Nc) N = 2 SQCD is as follows. There is an N = 2 hypermultiplet comprised of two N = 1 superfields, Φ and Wα, which transforms under the adjoint representation of the gauge group. Components of the chiral superfield Φ are a complex scalar Higgs φ and its fermionic superpartner, the Higgsino ψ. The vector superfield Wα contains the gauge boson Aμ and its superpartner, the gaugino λ. The additional Nf matter hypermultiplets consist of chiral superfields Qf and Q̃f (f = 1, 2, . . . , Nf ), which transform under the fundamental and its conjugate representation respectively. The associated component fields are the squarks qf and quarks χf along with their conjugate representation counterparts, q̃f and χ̃f . In this paper we adopt the leading-order short-distance constrained instanton [19] approach to semiclassical analysis as reviewed in Sections 3 and 4 of [11]. The constrained Euclideanized Euler-Lagrange equations in the short-distance region |x| < 1/M , where M represents a typical W-boson mass, can be solved perturbatively in the coupling constant g. Only the leadingorder terms contribute to the holomorphic prepotential and quantum moduli un. The defining equations of the instanton configuration are consequently [11–15] Fμν = F̃μν , (1) /̄ Dλ = 0, /̄ Dψ = 0, /̄ Dχ = 0, /̄ Dχ̃ = 0, (2) Dφ = √ 2ig[λ, ψ], D2φ†a = √ 2igχ̃T χ, (3) Dq = √ 2igλχ, Dq̃ = − √ 2igχ̃λ, D2q† = √ 2igχ̃ψ, D2q̃† = √ 2igψχ. (4) We use the convention /̄ D α̇α = ē μ Dμ where ēμ is the Hermitian conjugate of eμ = (i~σ,1). For notational clarity we have dropped the flavor indices on the quark and squark fields. In the Coulomb branch of the theory, the moduli space of vacua results from a potential term V (φ) ∼ Tr([φ, φ†]2) in the Lagrangian. Up to gauge transformations, the Higgs field acquires the matrix of vacuum expectation values 〈φ〉 = diag(a1, a2, . . . , aNc). (5) The ai are complex parameters satisfying the constraint ∑Nc i=1 ai = 0 which ensures that 〈φ〉 lives in the Lie algebra of the group SU(Nc). This imposes a boundary condition on the instanton solution for the Higgs field, since it must approach its matrix of VEV’s at large distances. 2 The required self-dual solution to Eq. (1) of unit topological charge is given by the standard SU(2) pure gauge field (BPST) instanton [20] ‘minimally embedded’ in the SU(Nc) Lie algebra [21]. In singular gauge this is Aμ = 2ρ g yν η̄ a μν y2(y2 + ρ2) T , (6) where yμ = (x − x0)μ, and x0 and ρ give the location and size of the instanton respectively. We make use of the usual ’t Hooft η-symbol [22] and choose a basis of generators such that T 1,2,3 ij = 1 2 σ ij (ie. these are normalized Pauli matrices in the ‘upper left corner’). The above configuration is subject to global gauge transformations which rotate it into the space of the SU(Nc) Lie algebra. However for the purposes of the instanton calculation we can choose to preserve the upper left embedding of the BPST instanton, and perform global gauge transformations of the matrix of VEV’s (5) instead [23]. In this case the boundary condition on the Higgs becomes lim |y|→∞ φ = Ω†〈φ〉Ω = (