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The gradient flow and its small flow-time expansion provide a systematic method to represent renormalized composite operators in a regularization-independent manner. This technique has been used to construct Noether currents such as the energy–momentum tensor and the axial-vector current in lattice gauge theory. We apply the same technique to the supercurrent in the four-dimensional supersymmetric theories. In specific, we consider the N=1 super Yang--Mills theory and the N=2 SUSY. Since this approach provides a priori a representation of the properly-normalized conserved supercurrent, our result should be useful for example in lattice numerical simulations of the 4D N = 1 SYM and the N=2 supersymmetric systems; the conservation of the so-constructed supercurrent can be used as a criterion for the supersymmetric point toward which the gluino mass is tuned.

To connect the lattice theory and the continuum theory via gradient flow method, the small flow-time expansion is useful. In this part, we give an explicit example of calculation of some diagrams arised from small flow-time expansion based on [1].