Abstract
We consider the UV divergences up to sub-subleading order for the four-point on-shell scattering amplitudes in $D=8$ supersymmetric Yang-Mills theory in the planar limit. We trace how the leading, subleading, etc divergences appear in all orders of perturbation theory. The structure of these divergences is typical for any local quantum field theory independently on renormalizability. We show how the generalized renormalization group equations allow one to evaluate the leading, subleading, etc. contributions in all orders of perturbation theory starting from one-, two-, etc. loop diagrams respectively. We focus then on subtraction scheme dependence of the results and show that in full analogy with renormalizable theories the scheme dependence can be absorbed into the redefinition of the couplings. The only difference is that the role of the couplings play dimensionless combinations like ${g}^{2}{s}^{2}$ or ${g}^{2}{t}^{2}$, where $s$ and $t$ are the Mandelstam variables.
Highlights
In recent years maximally supersymmetric gauge theories attracted much attention and served as a theoretical playground promising new insight in to the nature of gauge theories beyond usual perturbation theory
We show how the generalized renormalization group equations allow one to evaluate the leading, subleading, etc. contributions in all orders of perturbation theory starting from one, two, etc. loop diagrams respectively
In this paper we summarize all previous results with addition of the sub-subleading case and focus on the scheme dependence of the counter terms
Summary
In recent years maximally supersymmetric gauge theories attracted much attention and served as a theoretical playground promising new insight in to the nature of gauge theories beyond usual perturbation theory. It is useful to write down the local expression for the KR0 terms (counterterms) equal to KR0 This means that performing the R0-operation one can take care only of the one-, two-, three-loop diagrams surviving after contraction and get the desired leading pole terms via Eq (3) in the leading, subleading and sub-subleading order, respectively. They can be calculated in all loops pure algebraically. As in the ladder case, this recurrence relation takes into account all the diagrams of a given order of PT and allows one to sum all from n o1⁄4rd2ertsooifnPfiTn.itTyh. iDs ceannotbinegacthhieevseudmbybymuΣlðtsip;lty; iznÞg1⁄4boPth∞ns1⁄4id eSsnðosf;EtÞqð.−(1zÞ3n), by ð−zÞn−1, we where z get the g2 ε and summing up following differential equation. One can not just remove the UV regularization and get a finite theory
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