the last statement, that all counter-terms will be consistent with world sheet locality, is in
fact a corii.--ture put forward by Thorn in many of his papers on the subject. If Thorn's
(, ,ii. hi e were not true, it would make the formalism considerably less interesting.
We consider in this chapter renormalization to one loop in the context of the
Lightcone World Sheet, for a general class of Supersymmetric gauge theories formulated
earlier. We will see that with the rather unusual regulators, with discrete and p+ are
(equivalently, cutoffs in p- and ix-), we still obtain the correct well known result for
gauge coupling renormalization. In the calculations that follow we shall first revisit older
work done on the subject where close contact with the Feynman diagram picture has been
kept, because, even though the world sheet diagrams are completely equivalent, organizing
diagram according to loop order is only natural and apparent in the Feynman diagrams.
Following this rather thorough treatment we turn to a world sheet organization which
allows for some insights the Feynman picture does not. We shall also later see that the
world sheet diagrams organize themselves in a way much better suited for Monte Carlo
methods which are the second subject of this thesis.
We present here with permission, the work which was published before in a paper
titled "One loop calculations in gauge theories regulated on an x+p+ lattice" [6]. This
work constitutes in part the graduate work which was done towards the completion of
this thesis and is therefore presented here without additions or major modification. It also
relates very strongly to the second theme of the thesis, namely the Monte Carlo approach
of the Lightcone World Sheet formalism.
3.1 Gluon Self Energy
In addition to setting up the basic formulation of the x+, p+ lattice, Bering et. al.
[10] also calculated the one-loop gluon self-energy diagram as a check of the faithfulness of
the lattice as a regulator of divergences. We recap their results but with the world sheet
conventions used here. In addition we consider a general particle content with Nf number
of fermions and N, number of scalars. The gluon self energy to one loop can be extracted