The Chern-Simons Coefficient in Supersymmetric Non-Abelian Chern-Simons Higgs Theories

By taking into account the effect of the would be Chern-Simons term, we calculate the quantum correction to the Chern-Simons coefficient in supersymmetric Chern-Simons Higgs theories with matter fields in the fundamental representation of SU(n). Because of supersymmetry, the corrections in the symmetric and Higgs phases are identical. In particular, the correction is vanishing for N = 3 supersymmetric Chern-Simons Higgs theories. The result should be quite general, and have important implication for the more interesting case when the Higgs is in the adjoint representation. PACS number(s):11.10.Kk, 11.10.Gh, 11.15.Ex, 11.30.Pb Typeset using REVTEX †Email address:[email protected] 1 Chern-Simons theories can give rise to particle excitations with fractional spin and statistics, and thus have been used as effective field theories to study the fractional quantum Hall effect [1–3]. They are also interesting when the Higgs fields with a special sixth order potential are included so that the systems admit a Bogomol’nyi bound in energy [4]. The bound is saturated by solutions satisfying a set of first-order self-duality equations [5]. These solutions have rich structure and have been under extensive study especially when the gauge symmetry is non-abelian with the Higgs in the adjoint representation [6]. It is known that the self-duality in these systems signifies an underlying N = 2 supersymmetry and thus the Bogomol’nyi bound is expected to be preserved in the quantum regime [7]. Furthermore, when these theories are dimensionally reduced, an additional Noether charge appears, which in turns yields a BPS-type of domain wall [8]. The quantum correction to the Chern-Simons coefficient has also attracted a lot of attention. For theories without massless charged particles and the gauge symmetry is not spontaneously broken, Coleman and Hill have shown in the abelian case that only the fermion one-loop diagram can contribute to the correction to the Chern-Simons coefficient and yields 1 4π [9]. The quantization of the correction can be understood with an topological argument in the spinor space by making use of the Ward-Takahashi identity [10]. When there is spontaneous breaking of gauge symmetry, one can show that there exists in the effective action the so-called would be Chern-Simons terms, which induces terms similar to the Chern-Simons one in the Higgs phase [11]. By taking into account the effect of the would be Chern-Simons term, it has been shown that the one-loop correction in the Higgs phase is identical to that in the symmetric phase [12]. On the other hand, if the charged particles, both scalars and spinors can contribute to the correction at two-loop level and it is not quantized [13]. The situation becomes even more intriguing when the gauge symmetry is non-abelian: the Chern-Simons coefficient must be integer multiple of 1 4π for the systems to be invariant under large gauge transformation; otherwise the theories are not quantum-mechanically consistent. Therefore, it is interesting to confirm that the quantization condition is not 2 spoiled by quantum effects. In the symmetric phase, this has been shown to one loop [14]. When there is no bare Chern-Simons term, it is also verified up to two loops considering only the fermionic contribution [15]. In the Higgs phase, it is known for some time that if there is remaining symmetry e.g. SU(n) with n ≥ 3, the quantization condition will still be satisfied [16–18]. However, if the gauge symmetry is completely broken, e.g. SU(2), simple-minded calculation shows that the correction is again complicated and not quantized [11]. Although one may argue that this arises because there is no well-defined symmetry generator in such case, a better way to understand the whole thing is again to note the effect of the would be Chern-Simons terms. They are invariant even under the large gauge transformation, and their coefficients need not to be quantized. Therefore, we must subtract out the their contribution to obtain the correct result. Indeed, more careful calculation shows that for the Higgs being in fundamental SU(n) the quantization condition is always satisfied whether the gauge symmetry is completely broken or not [19]. As a result, a more or less unifying picture of the quantum correction to the Chern-Simons coefficient has emerged. Another interesting aspect of the quantum corrections to the Chern-Simons coefficient is that they depend on whether we introduce the Yang-Mills term as a UV regularizer [20]. Interestingly, it is found that in supersymmetric Chern-Simons theories the corrections become regularization independent [21]. In particular, the corrections are vanishing for N = 2, 3 supersymmetric Chern-Simons theories. Hence, we would like to know what happen if there is also spontaneous breaking of gauge symmetry in the system. In this paper, we calculate the quantum corrections to the Chern-Simons coefficient in supersymmetric Chern-Simons Higgs theories with the Higgs being in the fundamental SU(n). It turns out that the result is partially regularization dependent. If we do not introduce the Yang-Mills term, the quantum corrections are quantized and identical in the symmetric and Higgs phase because of supersymmetry. On the other hand, if we do, the result is more complicated. For n ≥ 3, the quantum corrections are still identical in the two phases. For n = 2, however, the quantum corrections becomes different in the two phases. We conclude with some comments on its implication and possible future direction. 3 With matter fields in the fundamental SU(n), the N = 3 supersymmetric nonabelian Chern-Simons Higgs theories can be simplified to [22]: