Counter terms

We could inquire in a manner similar to the electron case as to what changes must be made in the hamiltonian so as to guarantee that Eq. (11-144) be satisfied. Since the change should affect the one-photon states, we would expect that the counter terms to be added are of the form... 

Asymptotic Condition.—In Section 11.1, we exhibited the equivalence of the formulation of quantum electrodynamics in the Coulomb and Lorentz gauges in so far as observable quantities were concerned (t.e., scattering amplitudes). We also noted that both of these formulations, when based on a hamiltonian not containing mass renormalization counter terms, suffered from the difficulty that the... 

However, one readily verifies, using invariance techniques similar to those outlined in the last section, that if/( ) is given by Eq. (11-484), it doss not satisfy the criterion that <0 /( ) p, > = 0. It is easy to show that Eq. (11-476) can be satisfied if we add the (mass renormalization) counter term Anufi(x) to the right side of Eq. (11-484). In fact... 

However, we have no guarantee that the current operator defined by Eq. (11-490) will leave Eq. (11-483) satisfied. Stated differently, we have no guarantee that the charge -e, which occurs in the definition (11-490) in the current operator, is the same as the total charge of the one-electron state. In order to be able to satisfy this requirement we shall add another counter term of the form L"Z2A(x) to the expression (11-490) of the current operator. The addition of such a term does not affect Eq. (11-491). Hence, finally, the current operator is defined as... 

We see that the modes near the Fermi surface contributes only some parts of the axial anomaly. As in the vector current, the rest should come from modes in the deep Fermi sea and from anti-particles. To recover the full axial anomaly we add a counter term (See Fig. 7), which is two thirds of the axial anomaly plus a Chern-Simons term ... 

Then, given the bare mass of the electron as mo, we have the mass of the electron as m = mo — bm. By the Dirac equation this also contributes a counter term into the Lagrangian 8m j/ /. 

The propagator F 2 for the free electron is approximated by loops that are given by the function —/S(fe2) connected by electron propagators of the form i/( 2 — m2). So the propagator that computes the free electron with the mass counter term is then given by... 

This is a matter of replacing all of the correction terms with a finite number, in this case one, counter terms that may be evaluated. 

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... 

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 energy shift of a bound-electron state contracted terms of the second-order S matrix element taking into account the mass-renormalization counter term provided by S l according to... 

As is well known, the mass counter term in equation (6) accounts for the use of the observable instead of the bare mass of the electron. The latter is not observable due to the interaction of an electrical charge with its own radiation field which is always present. The magnitude of 5m is fixed by... 

The renormalized first-order bound-electron self energy shift for a state a( ) = ( a) can formally be written as the real part of the bound self energy with the mass counter term subtracted... 

We now turn to the second part of Eq. (1), i.e. the bound-state mass counter term. This term is defined as the free-electron self energy, in Feynman gauge, calculated for a momentum distribution determined by the bound state (French and Weisskopf [17]) and can be expressed in the following symmetric partial-wave form which is analogous to Eq. (9)... 

It can be shown from power counting arguments that only the first two terms in this expansion, the zero-potential and the one-potential terms, lead to divergent contributions and hence only these terms need to be considered in the A dependent part. In addition also the mass counter term has to be considered. 

Consider now the denominator difference between the zero-potential term and the mass counter term, which we can express as... 

The second term corresponds to a counter-term, and now the integral is always... 

The effect of these counter-terms in the contributions of the diagram is the elimination all the self energy terms at k = 0 indeed, no such term should appear, since the value aH is exact therefore the process consists only in performing the necessary subtractions, order by order. Thus, when this prescription is followed, the contributions of the diagrams are obtained in a simple and convergent manner. 

In a very general way, divergences in diagrams can be eliminated with the help of counter-terms these counter-terms can be interpreted as resulting from additional interaction terms and the latter are absorbed by renormalization of the partition functions and of the bare interactions. This is exactly what we did in Chapter 10, Section 4.2.6, for the model with purely repulsive two-body interactions, and, in this case, it is easy to see that the process amounts to dimensional regularization. 

When the operator 91 is applied to diagram contributions, the essentially divergent terms, 4( s e S must be interpreted as renormalization terms or counter-terms they are functions of s0, which become infinite when s0 0, and... 

The counter term m(x) represents some sort of average effect of the interelec-tronic interaction which has been added to the operator so that... 

For the moment we shall confine our discussion to the two leading terms. The first order term reduces to the expectation of the counter-term. 

As a result, we may view the orbital invariant SS-MRCEPA, termed by us as SS-MRCEPA(I) (I, for invariant), as the optimal approximation to the parent SS-MRCC method, which includes all the EPV terms exactly and which utilizes only those counter terms of the equations which eliminate the lack of extensivity of the attendant non-EPV terms in an orbital invariant manner [59]. In this article, we will present a couple of invariant SS-MRCEPA methods, viz. SS-MRCEPA(O) and SS-MRCEPA(I), for general open-shell systems using spin-free unitary generator adapted cluster operators starting from explicitly spin-free full-blown parent SS-MRCC formalism. Eor a detailed discussion of the allied issues pertaining to all the SS-MRCEPA-like methods, we refer to our recent SS-MRCEPA papers [58,59] and an earlier expose by Szalay [66]. 

If this excitation /r 1 on is also an allowed process on by Pauli Exclusion Principle, then the corresponding excitation involves orbitals which are not among those distinguishing and Such terms are then disconnected. However, for every such (fij, where this is true, there is a counter term in which together leads... 

For cases when one does not desire the interface to have any curvature-driven motion, the term on the right-hand side is still maintained in order to keep the sharp, hyperbolic tangent profile near the interface however, a so-called counter term is subtracted from the right-hand side in order to cancel the main curvamre-driven flow. The resulting equation then becomes [4]... 

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