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Abstract
Maximally supersymmetric field theories in various dimensions are believed to possess special properties due to extended supersymmetry. In four dimensions, they are free from UV divergences but are IR divergent on shell; in higher dimensions, on the contrary, they are IR finite but UV divergent. In what follows, we consider the four-point on-shell scattering amplitudes in D = 6 , 8 , 10 supersymmetric Yang–Mills theory in the planar limit within the spinor-helicity and on-shell supersymmetric formalism. We study the UV divergences and demonstrate how one can sum them over all orders of PT. Analyzing the R -operation, we obtain the recursive relations and derive differential equations that sum all leading, subleading, etc., divergences in all loops generalizing the standard RG formalism for the case of nonrenormalizable interactions. We then perform the renormalization procedure, which differs from the ordinary one in that the renormalization constant becomes the operator depending on kinematics. Solving the obtained RG equations for particular sets of diagrams analytically and for the general case numerically, we analyze their high energy behavior and find that, while each term of PT increases as a power of energy, the total sum behaves differently: in D = 6 two partial amplitudes decrease with energy and the third one increases exponentially, while in D = 8 and 10 the amplitudes possess an infinite number of periodic poles at finite energy.
Highlights
In the last decade, we witnessed serious progress in understanding the structure of the amplitudes in gauge theories in various dimensions
The polarization vectors for massless fermions can be chosen as Weyl spinors while the polarization vectors for scalars are trivial. Using this representation of momenta and polarization vectors in terms of Weyl spinors, one can always write down the scattering amplitude in the gauge theory in arbitrary even dimension, which is the function of the Lorentz invariant products of momenta and polarization vectors in terms of the spinor products corresponding to the momenta of external particles only
The discussed procedure makes it possible to derive solutions for a fixed number of loops and to obtain the recurrence relations in any loop order. We demonstrate this derivation by the example of the horizontal ladder-type diagrams in D = 8 [17]
Summary
We witnessed serious progress in understanding the structure of the amplitudes (the S-matrix) in gauge theories in various dimensions (for review see, for example, [1,2,3,4,5,6,7]). In the sequence of papers [14,15,16,17], we considered the leading and subleading UV divergences of the on-shell four-point scattering amplitudes for all three cases of maximally supersymmetric SYM theories, D = 6 (N = 2 SUSY), D = 8 (N = 1 SUSY) and D = 10 (N = 1 SUSY). We explain how the R-operation works and derive the recursive relations that allow one to get the all-loop expressions for the leading, subleading, etc., divergences We convert these relations into differential equations, which are the generalization of the familiar RG equations for the case of non-renormalizable interactions.
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