Exact scheme independence at two loops

Abstract

We further develop an algorithmic and diagrammatic computational framework for very general exact renormalization groups, where the embedded regularisation scheme, parametrised by a general cutoff function and infinitely many higher point vertices, is left unspecified. Calculations proceed iteratively, by integrating by parts with respect to the effective cutoff, thus introducing effective propagators, and differentials of vertices that can be expanded using the flow equations; many cancellations occur on using the fact that the effective propagator is the inverse of the classical Wilsonian two-point vertex. We demonstrate the power of these methods by computing the beta function up to two loops in massless four dimensional scalar field theory, obtaining the expected universal coefficients, independent of the details of the regularisation scheme.

On the renormalization of theories of a scalar chiral superfield

“…As an aside, it is well worth mentioning that, in the past, cancellations of the seed action were demonstrated using elaborate (though increasing sophisticated) diagrammatics [45,47,50,51,56,60,61]. However, as recognized in [8], by employing the dual action, these cancellations can instead be done with a few lines of algebra, as has been done here.…”

“…Given the choice (2.2), and a choice of cutoff function, the seed action encodes the residual blocking freedom. The only restrictions on the seed action are that it is infinitely differentiable and leads to convergent loop integrals [44,47]. The first requirement is that of 'quasi-locality' (mentioned in the introduction), which must apply to all ingredients of the flow equation.…”

“…The rule for determining how many legs each of these vertices hasequivalently, the rule for decorating the diagrams on the right-hand side-is that the n f available legs are distributed in all possible, independent ways. For much greater detail on the diagrammatics, see [47,55,56].…”

“…In the case of scalar field theory, such a flow equation (withŜ I = 0) was first considered in [52]; the version with more general seed action has been considered in [47,53].…”

“…whereŜ is the 'seed action' [44][45][46][47], a nonuniversal input which controls the flow but of which all physical quantities should be independent. Given the choice (2.2), and a choice of cutoff function, the seed action encodes the residual blocking freedom.…”