  The formalisms of Wyld [2] and Martin, Siggia, and Rose (MSR) [3] address the
closure problem of a statistical treatment of homogeneous isotropic turbulence
(HIT) based on techniques primarily developed for quantum field theory. In the
Wyld formalism, there is a well-known double-counting problem, for which an
{\it ad hoc} solution was suggested by Lee [44]. We show how to implement this
correction in a more natural way from the basic equations of the formalism.
This leads to what we call the "Improved Wyld-Lee Renormalized Perturbation
Theory". MSR had noted that their formalism had more vertex functions than
Wyld's formalism and based on this felt Wyld's formalism was incorrect. However
a careful comparison of both formalisms here shows that the Wyld formalism
follows a different procedure to that of the MSR formalism and so the treatment
of vertex corrections appears in different ways in the two formalisms. Taking
that into account, along with clarifications made to both formalisms, we find
that they are equivalent and we demonstrate this up to fourth order.
