Direct evaluation in three dimensions

Avoiding the e-expansion, we can work directly with renormalized perturbation theory in powers of u, evaluated in d = 3 dimensions. We still have to decide what quantities to expand, and it clearly will make some difference whether we for instance use the result (12.25) for Pr(p, ft/, /) or rather expand In Pr(P- nR, /). Having chosen some basic set of quantities, however, we consistently evaluate observables of interest without further expansion. We for instance calculate ii2, perturbatively, constructing ijj from Eq. (12.33) without further manipulation. Clearly then some trivial change in the definition of %fj has no deeper consequences. 

We must however add a word of caution. The method as sketched here has been established to deal with the small momentum behavior of the scaling functions. For large momenta we may run into problems in that the subleading momentum dependence is not treated correctly. On the quantitative level however this does not seriously invalidate our results. We will discuss this aspect further in Sect. 15.2. See also Sect. 13.2.3, 

We should not close this section without touching upon some delicate technical point. A priori even for a discrete chain the critical chemical potential g s, defined as chemical potential per segment of an infinitely long chain, does not exist outside the -expansion. We already encountered this problem in Sect. 7.2, where we found that rC proportional to p, for d 4 in naive perturbation theory suffers from infrared singularities. That problem has been considered in the field theoretic framework. The results, expressed in polymer language, show that the u-expansion is not invalidated, as long as we consider quantities which do not involve explicitly. Almost all the quantities of interest to us are of this type. Only in the equation of state relating gp(n) and cp(n) does an explicit contribution g n occur. But even 

In field theoretic context the method of dimensional regularization and minimal subtraction has been proposed, in [tHV72]. To the level considered here it is discussed in standard textbooks [ZJ89, Ami84], There the explicit calculation of the -factors can also be found. The method has been applied directly to the polymer system in the continuous chain limit in [Dup86a], where different versions of the approach are compared, 

The evaluation of the scaling functions directly in d = 3 has been advocated in [SD89, SD90], where the method has been shown to work for small momenta. Also the treatment of the additive mass renormalization (corresponding to th introduction of p. J has been carefully considered there. 

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