CHAPTER 3
PERTURBATION THEORY ON THE WORLD SHEET
As we have seen the Lightcone World Sheet formulation is set up in the framework of
the Lightcone. The = ix+ and p+ lattice as constructed by Bering, Rozowsky and Thorn
[10] is of course the starting point for the world sheet and the perturbative issues faced in
that work, of course remain present in the world sheet picture.
When using the discretized world sheet to calculate processes to a given order in
perturbation theory the insertions have been designed to exactly reproduce the cubic
vertices of the Lightcone Feynman rules in the continuum limit. The precise meaning of
this limit is that every solid line in the diagram is many lattice steps long and also is many
lattice steps away from every other solid line. Clearly a diagram in which one of these
criteria is not met is sensitive to the details of our discretization choice. In tree diagrams
one can always avoid these dangerous situations by restricting the external legs so that
, carry p+ so that the differences |p+ p+ are several units of m for any pair i,j, and
so that the time of evolution, T, between initial and final states are also several units of
a. However, a diagram containing one or more loops will involve sums over intermediate
states that violate these inequalities, and because of field theoretic divergences the
dangerous regions of these sums can produce significant effects in the continuum limit.
In particular we should expect these effects to include a violation of Lorentz invariance,
in addition to the usual harmless effects that are absorbed into renormalized couplings.
Indeed, when a solid line is of order a few lattice steps in length, it produces a gap in
the gluon energy spectrum that is forbidden by Lorentz invariance. This effect can be
cancelled by a counter-term that represents a local modification of the world sheet action.
The hope is that all counter-terms needed for a consistent renormalization program can
be implemented by local modifications of the world sheet dynamics. A slight weakening of