In this paper, we present a direct perturbative method to solving certain Fokker-Planck equations, which have constant diffusion coefficients and some small parameters in the drift coefficients. The method makes use of the connection between the Fokker-Planck and Schrödinger equations. Two examples are used to illustrate the method. In the first example, the drift coefficient depends only on time but not on space. In the second example, we consider the Uhlenbeck-Ornstein process with a small drift coefficient. These examples show that such perturbative approach can be a useful tool to obtain approximate solutions of Fokker-Planck equations with constant diffusion coefficients.
The Nambu-Jona-Lasinio (NJL) model is a classic theory for the strong dynamics of composite fields and symmetry breaking. Supersymmetric versions of the NJL-type models are certainly of interest too. Particularly, the case with a composite (Higgs) chiral superfield formed by two (quark) chiral superfields has received much attention. Here, we propose a prototype model with a four-chiral-superfield interaction, giving a real superfield composite. It has a spin-one composite vector field with properties being somewhat similar to a massive gauge boson of spontaneously broken gauge symmetry. As such, it is like the first supersymmetric analog to non-supersymmetric models with spin-one composites. The key formulation developed here is the picture of quantum effective action as a superfield functional with parameters like constant superfields, having explicit supersymmetric and Grassmann number dependent supersymmetry breaking parts. Following the standard non-perturbative analysis for NJL-type models, the gap equation analysis shows plausible signature of dynamical supersymmetry breaking which is worth more serious analysis.With an extra superfield model Lagrangian included, comparison between the models and their non-supersymmetric counterparts is discussed, illustrating the notion of supersymmetrization is nontrivial in the setting.
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