Short-range divergences in diagrams and dimensional regularization

3 Short-range divergences in diagrams and dimensional regularization 

We want to calculate the swelling of a chain and the osmotic pressure of a set of chains in the vicinity of the Flory point. For this purpose, we can calculate, on one side + (k, — H S) to first-order in k2, on the other side the quantities (iVxS). 

Of course, these various quantities depend on the cut-off s0 and diverge if s0 - 0. However, these divergences cannot be eliminated by a simple renormalization of the partition functions as could be done in the purely repulsive case (see Chapter 10). The bare interaction has also to be renormalized. In particular, as will be shown later in more detail, the three-body interactions act as if they were generating cut-off dependent two-body interactions. The elimination of the cut-off dependent terms then results from a re-interpretation of the various terms of the diagram expansion nevertheless, the process is complex. 

However, a simple method, the dimensional regularization, enables us to eliminate the cut-off in a rational way. The polymer is embedded in a space of dimension d and in principle we can calculate diagrams for any (non-integer) value of d. The contribution of a diagram associated with a partition function +y(...) is a function 2(d, s0). It is easy to show [see (10.4.57)] that if the real part of d is small enough (Red d0,d0 0), then 

The importance of dimensional regularization comes from the fact that it is equivalent to a renormalization. Actually, the fact has not been rigorously proved in polymer theory, but it seems very likely to be true, and in Appendix 

---