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D  =  4   ,    N  =  4    super‐Yang–Mills theory has an off‐shell superspace formulation in terms of pure spinor superfields, which is directly inherited from the    D  =  10    theory. That superspace, in particular the choice of pure spinor variables, is less suitable for dealing with fields that are inherently 4‐dimensional, such as the superfields based on the scalars, which are gauge‐covariant, and traces of powers of scalars, which are gauge‐invariant. We give a reformulation of    D  =  4   ,    N  =  4    super‐Yang–Mills theory in    N  =  4    superspace, using inherently 4‐dimensional pure spinors. All local degrees of freedom reside in a superfield based on the physical scalars. The formalism should be suited for calculations of correlators of traces of scalar superfields.

An Off‐Shell Superspace Reformulation of D = 4, N = 4 Super‐Yang–Mills Theory