Further results about field theory on the world sheet and string formation

The present article is the continuation of the earlier work, which used the world sheet representation and the mean field approximation to sum planar graphs in massless ϕ3 field theory. We improve on the previous work in two respects: a prefactor in the world sheet propagator that had been neglected is now taken into account. In addition, we introduce a non-zero bare mass for the field ϕ. Working with a theory with cutoff and using the mean field approximation, we find that, depending on the range of values of the mass and coupling constant, the model has two phases: a string forming phase and a perturbative field theory phase. We also find the generation of a new degree of freedom, which was not in the model originally. This new degree of freedom can be thought of as the string slope, which is now promoted into a fluctuating dynamical variable. Finally, we show that the introduction of the bare mass makes it possible to renormalize the model.