Non - universal corrections to the tension ratios in softly broken N = 2 SU ( N ) gauge theory

Calculation by Douglas and Shenker of the tension ratios for vortices of different N -alities in the softly broken N = 2 supersymmetric SU(N) Yang-Mills theory, is carried to the second order in the adjoint multiplet mass m. Corrections to the ratios violating the well-known sine formula are found, showing that it is not a universal quantity. 1. Recently the tension ratios among the confining vortices corresponding to sources of different N -alities in SU(N) gauge theories have been the subject of some attention, as a quantity characterizing quantitatively the confining phase of these systems. After an interesting suggestion from MQCD that such ratios might have universal values [1], Tk T1 = sin πk N sin π N , (1) a more recent study in string theory based on supergravity duals [2], gave model dependent results for two N = 1 SQCD-like theories. The result of direct measurement in the lattice (non-supersymmetric) SU(N) gauge theories is consistent with Eq.(1) [3, 4]. Derivation of formula such as Eq.(1) in the standard, continuous SU(N) gauge theories still defies us. The first field-theoretic result on this issue was obtained by Douglas and Shenker [5], in the N = 2 supersymmetric SU(N) pure Yang Mills theory, with supersymmetry softly broken toN = 1 by a small adjoint scalar multiplet mass m. They found Eq.(1) for the ratios of the tensions of abelian (AbrikosovNielsen-Olesen) vortices corresponding to different U(1) factors of the low-energy effective (magnetic) U(1) theory. The n-th color component of the quark has charges δn,k − δn,k+1, (k = 1, 2, . . . , N − 1; n = 1, 2, . . . , N) (2) with respect to the various electric Uk(1) gauge groups. The source of the k-th ANO string thus corresponds to the N -ality k multiquark state, |k〉 = |q1q2, . . . qk〉, allowing a re-interpretation of Eq.(1) as referring to the ratio of the tension for different N -ality confining strings [6]. However, physics of the softly broken N = 2 SU(N) pure Yang-Mills theory is quite different from what is expected in QCD. Dynamical SU(N) → U(1) breaking introduces multiple of meson Regge trajectories with different slopes at low masses [6, 7], a feature which is neither seen in Nature nor expected in QCD. For instance, another N -ality k state |k〉′ = |q2q3, . . . qk+1〉 acts as source of the Uk+1(1) vortex and as the sink of the U2(1) vortex, which together bind |k〉′anti |k〉′ states with a tension different from Tk. The Douglas-Shenker prediction is, so to speak, a In fact, the same problem is expected in any confining vacuum in which such a dynamical breaking takes place. ’t Hooft’s original suggestion for QCD ground state [8] is of this type.