On the relation between Green functions of the SUSY theory with and without soft terms

We study possible relations between the full Green functions of softly broken supersymmetric theories and the full Green functions of rigid supersymmetric theories on the example of the supersymmetric quantum mechanics and find that algebraic relations exist and can be written in a simple form. These algebraic relations between the Green functions have been derived by transforming the path integral of the rigid theory. In this approach soft terms appear as the Berezinians of general changes of coordinates in the superspace. E-mail: [email protected] On leave for absence of the Laboratory of Nuclear Research, JINR, Dubna, Russia The renormalization of softly broken supersymmetric Yang-Millls theories in four spacetime dimensions into the framework of the superfield technique and spurions method was studied from different points of view in Refs. ([1]-[6]). The soft terms were introduced as the finite series in terms of Grassmanian coordinates of the superspace. It results in dependence of renormalization constants of the theory on the Grassmanian coordinates. As it was understood, there are direct relations between the renormalization constants of the rigid theory – theory without soft terms – and the renormalization constants of its softly broken counterpart. The first investigation in this direction was performed by Yamada [1] and it was developed in [4, 5], where simple rules to calculate these constants starting from well-known results for the rigid theory have been derived. These rules are realized in terms of differential operators acting in the coupling space of the rigid theory. In the Ref.[5] it was found that this approach is in good agreement with nonperturbative results [2]. The reason why these rules exist is that the soft breaking terms can be considered as background x-independent superfields [5, 6] in which our theory is embedded. These external superfields must be substituted into infinite parts of the Green functions instead of the couplings of the rigid theory. The algebraic relations between the renormalizations constants are consequences of these substitutions [5]. However, it would be nice to have relations for the complete Green functions of the theory that includes soft terms and for its counterpart without them. Here we find these relations for the case of the supersymmetric quantum mechanics as the easiest example where the method proposed below can work. The supersymmetric quantum mechanics was developed in ([7]-[9]) and its action is 3 S = ∫