Non-renormalization for planar Wess-Zumino model

Abstract

Using a non-perturbative functional method, where the quantum fluctuations are gradually set up, it is shown that the interaction of a N = 1 WessZumino model in 2+1 dimensions does not get renormalized. This result is valid in the framework of the gradient expansion and aims at compensating the lack of non-renormalization theorems. Non-renormalization theorems are based on analyticity properties and are thus not present for N = 1 supersymmetry in 2+1 dimensions, since the odd coordinate of superspace is real. Supersymmetric properties should nevertheless restrict the quantum corrections in this situation, and lead to a control of the renormalization processes. In the framework of relativistic-like effective descriptions of high-temperature superconductivity [1], supersymmetric models in 2+1 dimensions have been introduced [2], where conditions for the elevation to a N = 2 supersymmetry were studied, motivated by the presence of non-renormalization theorems in this case. It is though interesting to look at the possibility to have exact results with N = 1 supersymmetry and a functional method is presented here for this purpose, which gives indications on the renormalized structure of a Wess-Zumino model in 2+1 dimensions. The idea of the method is to control quantum fluctuations with the mass of the bare theory. When this mass is very large, the quantum fluctuations are frozen and the system is classical. As the bare mass decreases, the quantum fluctuations appear and the parameters of the theory get dressed. We can then consider the ”fluctuation flows” of the effective action (the proper graphs generator functional) with the bare mass and, starting from the bare action, follow these flows so as to built the ”full” quantum theory, containing all the quantum effects. There are similarities between this procedure and the blocking procedure [3], since the latter describes the evolution of a theory with a momentum, from the 1 ultraviolet (UV) scales to the infrared (IR) ones. The present method, though, is not based on a splitting the UV from the IR degrees of freedom and thus does not introduce any artificial coarse graining function. It was shown, in previous works [4, 5], that these fluctuation flows recover the usual one-loop results. Beyond one-loop, the results given by these flows do not cöıncide anymore with a loop expansion, since the results are based on the so-called gradient expansion. Note finally that this method does not require any regularization in 2+1 dimensions, which is another advantage to use it. We will find, in the context of the gradient expansion, the exact effective action for a Q Wess-Zumino theory (Q is the scalar superfield and Q leads to the marginal interaction φ, where φ is the scalar component of Q). We will see that only the mass gets renormalized, whereas the interaction does not. We note z = (x, θ) the coordinate of superspace and we take the conventions used in [6]. The bare action is Sλ[Q] = ∫

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