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In this paper we discuss in detail computational methods and new results forone-loop virtual corrections to N = 4 super Yang-Mills scattering amplitudescalculated to all orders in epsilon, the dimensional regularization parameter.It is often the case that one-loop gauge theory computations are carried out toorder epsilon^0, since higher order in epsilon contributions vanish in thesmall epsilon limit. We will show, however, that the higher order contributionsare actually quite useful. In the context of maximally supersymmetricYang-Mills, we consider two examples in detail to illustrate our point. Firstwe will concentrate on computations with gluonic external states and argue thatN = 4 supersymmetry implies a simple relation between all-orders-in-epsilonone-loop N = 4 super Yang-Mills amplitudes and the first and second stringycorrections to analogous tree-level superstring amplitudes. For our secondexample we will derive a new result for the all-orders-in-epsilon one-loopsuperamplitude for planar six-particle NMHV scattering, an object which allowsone to easily obtain six-point NMHV amplitudes with arbitrary external states.We will then discuss the relevance of this computation to the evaluation of theratio of the planar two-loop six-point NMHV superamplitude to the planartwo-loop six-point MHV superamplitude, a quantity which is expected to haveremarkable properties and has been the subject of much recent investigation. Tomake the presentation as self-contained as possible, we extensively review theprerequisites necessary to understand the main results of this work.

One-Loop N = 4 Super Yang-Mills Scattering Amplitudes to All Orders in the Dimensional Regularization Parameter