Parametrization of the system

In this section, we consider systems at temperatures for which the effective two-body interaction is very small. In other words, we assume that we have z 4, 1, y 0 and, for d = 3 - % no other parameter. However, as the three-body interaction is marginal at d = 3, it will be necessary to calculate logarithmic effects depending on (S/s0) and not on y, which remains finite at d 3 when S - oo. In what follows, we consider the system for a space dimension d - 3 — s in order to calculate various exponents, and subsequently we go to the limit e - 0. Here, we use a method very similar to the method which we applied, in the purely repulsive case (good solvent), in order to calculate logarithmic corrections for d = 4 and S/s0 1 (see Chapter 12, Section 3.3.4). 

For tricritical chains, the important physical parameter is the osmotic parameter h, which is proportional to the third virial coefficient) and not g as in the purely repulsive case. In the following, we keep only linear terms in z. The values of h, g, and 3E0 are 

The factors (y, S, s0) and i0(y, S/s0) renormalize the two-body interaction. When S/s0 - oo, these renormalization factors must be equivalent to each other and also to the renormalization factor, 3 4(y, S/s0) associated with the centre of a polymer star consisting of four (very long) branches with three-body interactions (see Fig. 14.20(b)). In fact, when the space dimension is d = 3 — % 

For d = 3, y is a number which remains constant when /s0 - ao on the contrary, as will be shown later, the osmotic parameter h goes to zero when S/s0 - oo. Thus, h is the good parameter and, in the following, we re-express all physical quantities in terms of h. But, first, let us examine the behaviour of k when S/s0 - 00. 

---