Instanton-induced Effective Vertex in the Seiberg-Witten Theory with Matter

The instanton-induced effective vertex is derived for N = 2 supersymmetric QCD (SQCD) with arbitrary mass matter hypermultiplets for the case of SU(2). The leading term of the low energy effective lagrangian obtained from this vertex agrees with oneinstanton effective term of the Seiberg-Witten result. [email protected] [email protected] During the past year there has been a lot of progress in N = 2 supersymmetric YangMills theories in four dimensions. Using the idea of duality and holomorphy, Seiberg and Witten determined the exact low energy effective lagrangian for the gauge group SU(2) without [1] and with matter hypermultiplets [2]. This low energy effective lagrangian is determined by a single holomorphic function: the prepotential F . By carefully studying of its singular structures in moduli space, they determined not only the form but the numerical cefficients of the prepotential exactly. The moduli spaces of these theories are described by hyper elliptic curves and can be related to integrable models [3]. One of the most interesting feature of this prepotential is that it contains an infinite series of instanton contributions [4]. Noticing this important feature several authors tried to check the Seiberg -Witten result with semi-classical instanton calculation at weak coupling limit without using any duality conjecture [5]. This direct microscopic instanton calculation for N = 2 SU(2) SUSY Yang-Mills theory provides a nontrivial check of the idea of duality and has been carried out at the oneinstanton [5], and two-instanton levels using the ADHM construction [6, 7]. Shortly thereafter multi-instanton calculation for SUSY Yang-Mills theory coupling to matter has been performed by the two independent groups [7, 9]. They found that there is an excellent agreement between the Seiberg-Witten result and the semi-classical instanton calculation except for the Nf = 3, 4 cases. At the one instanton level this microscopic calculation has been extented to the group SU(N), again with and without matter [11, 12] and also to the semi-simple Lie groups [13]. Another direction of studying the instanton effects in the low energy effective lagrangian has been suggested by Yung [14]. In that approach the nonperturbative instanton effect was represented, according to the perturbation theory language, as a four fermion vertex attached to the tree level lagrangian and one can derive one instantoninduced effective vertex and find that in the low energy limit the leading term coincides with the Seiberg-Witten effective action. This provides get another nontrivial check on the exact results. In this letter we consider the one instanton-induced effective vertex for N = 2 SU(2) 1 SUSY gauge theory with matter hypermultiplets which have arbitrary masses. Experience from the Seiberg-Witten theory tells us that simple addition of matter can lead to quite different structure compare to the pure Yang-Mills case [2]. The model we are considering is the N = 2 SQCD which has an N = 1 chiral multiplet Φ = (φ, ψ) in the adjoint representation of the group SU(2) and N = 1 vector multiplet Wα = (λ, vμ), which form an N = 2 vector multiplet. There are also N = 1 chiral multiplets Qk = (qk, ψmk) and Q̃k = (q̃k, ψ̃mk) (k = 1, · · · , Nf), which form the N = 2 matter hypermultiplets in the fundamental representation of the group. There exist global SU(2)R under which the superfields transform as follows: λ↔ ψ, q → q̃, q̃ → −q, (1) while gauge and scalar fields are singlets under the transformation. The lagrangian of the model is given by LSQCD = LSYM + Lmatter + LY ukawa + Lmass, (2) where each term is given as follows: LSYM = 1 4g ∫ dθW a αW αa + 1 4g ∫ dθ̄W̄ W̄ a α̇ , (3) Lmatter = ∫