Renormalization schemes subtraction

The results of the previous chapter are completely general. They are valid for any field theoretic renormalization scheme, i.e. independent of the specific choice of the renormalization factors, For quantitative calculations we of course have to specify the Z-factors, and as pointed out in Sect. 11,1, we have some freedom there. We will use the scheme of dimensional regularization and minimal subtraction . This scheme is most efficient for actual calcular tions, but its underlying basis is a little bit delicate, It needs some careful explanation. 

For low order calculations of the scaling functions a variety of implementations of the RG have been used. The present formulation has grown out of the work [Sch84]. The basic philosophy is the same, but in this earlier work the renormalization scheme was based on field theoretic renormalization conditions1. This amounts to using a non-minimally subtracted theory, where the Z-factors are determined by imposing specific values to certain renormalized field theoretic vertex functions. The renormalized coupling, for instance, is defined as the value of (qi, qa, qg. qi) at some special momenta of order... 

In this paper we adopt the PWE renormalization scheme. Accordingly, all bound terms in Eqs. (18)-(21) appear as double sums over partial waves. These double sums arise from the product of two matrix elements containing exp(i w r). Each individual partial wave contribution is finite and only the sum over partial waves diverges. In our calculation this divergency is removed by term-by-term subtraction of the corresponding counterterms. The PWE for the sum of all terms in Eq. (17) thus ensures a correct cancellation. 

There are more problems. The result (12.34), for instance, yields no information on chain-length or temperature dependence, which is hidden in or ip. To extract it, we have to write down RG flow equations for these variables. This results in the so-caUed direct renormalization scheme [Clo81], which however has not been pushed to the same high order as the flow equations derived for- n, Nfi in minimal subtraction. 

The decomposition of the irreducible part of the self-energy wave-function correction term is depicted in Fig. 2. The divergent terms are these with zero and one interaction in the binding potential present, below referred to as zero-potential term and one-potential term , respectively. The charge divergences cancel between both terms. In addition, a mass counter term dm has to be subtracted to obtain proper mass renormalization similar to the case of the free self energy [47] (for our schemes see also [44]). The zero- and one-potential... 

The problem, however, does not ruin our construction of the renormalized theory, and it does not keep us from using the results in some region u = fu > u. Concerning the RG flow we note that we will use the special scheme of minimal subtraction , where the flow equations depend on d only trivially... 

A description of the partial-wave renormalization (PWR), used for calculating the first-order self energy and certain higher-order effects for the energy levels in highly cllarged ions, is presented. We put special emphasis on correction terms which have to be considered due to the use of the non-covariant subtraction scheme used in PWR. 

Here, r is a rescaling parameter, which defines the scale of the external momenta in the minimal subtraction scheme. In the same way as in the Callan-Symanzik equation (77) the coefficients in (86) define the renormalization group functions ... 

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