Three-body interactions and counter terms

The dimensional regularization method enables one to eliminate a large number of divergences automatically. However, these divergences have a physical meaning and this is why, in this section, and in the two following ones, their nature is briefly analysed. 

In a very general way, divergences in diagrams can be eliminated with the help of counter-terms these counter-terms can be interpreted as resulting from additional interaction terms and the latter are absorbed by renormalization of the partition functions and of the bare interactions. This is exactly what we did in Chapter 10, Section 4.2.6, for the model with purely repulsive two-body interactions, and, in this case, it is easy to see that the process amounts to dimensional regularization. 

When three-body interactions are also present, the question becomes more complicated, but again we can determine the nature of the divergent terms by using the following dimension equations which can be easily deduced from (14.6.1) and (14.6.2) 

Let us first consider the renormalization of the partition functions such a renormalization, for a polymer, can be written in the form 

The dimension of A is thus given by A L 2 and if such a term results from diagrams with p two-body interactions and q three-body interactions, we then have 

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