Analysis of the screened interaction

To lowest order we may use Eq. (5.52) to eliminate the chemical potential from Eqs. (5.32), (5.33), which define the screened vertex. We find 

This expression strongly depends on the value of IT (A ) = W = uqcN. In the dilute regime W 1, wc clearly can expand out the denominator to recover the normal expansion in powers of ti0 uo(q) = Uq + 0(w ). The regime 

Thus o(q) varies from w0(0) C u0 to u0(q oc) Uq. To extract the typical scale of that variation we determine that momentum (7 where WDp(q 2fN) - 1. For W 1 this momentum is found in the region where Dp can be replaced by its asymptotic behavior (5.51 ii). We thus find 

We may summarize this discussion by noting that uo(q) shows all the aspects of screening. For dilute solutions, W 1. it essentially equals the full interaction uo. For W 1 and q, corresponding to length scales r ( , uo(q) is small compared to n(J, strictly vanishing for q = 0 in the limit W — oo. For q q, i.e. r . the full interdiction u0 is recovered. The essential feature is the suppression of u0(q) for W 1, q q. This effect-enables the loop expansion to deal with solutions of higher concentration, as will be illustrated now. 

To show the loop expansion at work, we evaluate here the relation (5.27) among Hp(n) and cp(w,) to one loop order. The diagrams for R 0,1 (u) are shown in Fig. 5.14. The analytic expression reads 

oc (from above). The bivken line gives uo(q) = w.o (dilute limit) 

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