Renormalized expressions

We first consider the grand potential f7[/ip], which according to the Linked Cluster Theorem can be written as 

up to corrections of order 1/n. we may replace the summation over n by integration  

We will mainly be interested in results expressed in terms of concentrations instead of the chemical potential. We thus renormalize Eq. (5.4)  

This derivation is easily extended to all correlation functions of interest. Any grand-canonical cumulant can be expressed by summing appropriate M-chain cumulants. We then use the theorem of renormalizability, and we 

The corresponding results are easily established for endpoint correlations. 

---

The constant a(e) is given in Eq. (A 4.16). Again the renormalized expression is finite in four dimensions and does not refer to a specific microstructure. 

The diagram SESE a), that is called also loop-after-loop , consists of irreducible and reducible parts (see Fig. 3). We consider first of all the irreducible contribution, which can be renormalized and evaluated separately since it does not contain infrared divergencies in the Feynman gauge. The renormalized expression for this contribution can be written as... 

Only the ZP and OP terms are divergent and therefore subject to renormalization. The MP term is finite, though most laborous in numerical evaluations. The renormalized expressions are... 

The renormalized expression for the two-photon self energy A ren (without the irreducible SESE a) term) for the bound-electron state a) reads [30] ... 

As indicated by subscripts the various terms here denote the reducible loop-after-loop (red), the loop-inside-loop (in), the crossed-loops (cr) contributions and an additional counter term (ac) to the SESE b) graph, respectively. The subscript ren always signifies the renormalized expression, though the expres-... 

The renormalized expression for the two-photon self energy (with-... 

Another simple method that allows for the SE calculations in heavy atoms is the partial wave remormalization (PWR) approach first proposed in [69], [70]. Within this approach the renormalized expression for the lowest-order SE is presented in the form of the partial wave expansion ... 

The shown in Fig. 48 excellent conformity of experimental and theoretical MWD curves for DMDAACh, synthesized at c = 1.0 and 2.5 mol/1, confirms the performed renormalization correctness. In its turn, the indicated renormalization expresses one from the main properties of fractals — their automodelity [90],... 

Since this absorbs all the divergences, we do not have to renormalize N. The divergence-free quantity is A2r L,u,N) (R for renormalized expressed in terms of u). Now,. <42 should not depend on L because L is put in by hand. This is ensured by demanding that... 

---