Vertex functions

The obvious disadvantage of this simple LG model is the necessity to cut off the infinite expansion (26) at some order, while no rigorous justification of doing that can be found. In addition, evaluation of the vertex function for all possible zero combinations of the reciprocal wave vectors becomes very awkward for low symmetries. Instead of evaluating the partition function in the saddle point, the minimization of the free energy can be done within the self-consistent field theory (SCFT) [38 -1]. Using the integral representation of the delta functionals, the total partition function, Z [Eq. (22)], can be written as... 

The construction of the phase diagram of a heteropolymer liquid in the framework of the WSL theory is based on the procedure of minimization of the Landau free energy T presented as a truncated functional series in powers of the order parameter with components i a(r) proportional to Apa(r). The coefficients of this series, known as vertex functions, are governed by the chemical structure of heteropolymer molecules. More precisely, the values of these coefficients are entirely specified by the generating functions of the chemical correlators. Hence, before constructing the phase diagram of the specimen of a heteropolymer liquid, one is supposed to preliminarily find these statistical characteristics of the chemical structure of this specimen. Here a pronounced interplay of the statistical physics and statistical chemistry of polymers is explicitly manifested. 

Here q is a wavevector (eqn 1.6), ip(q) is the Fourier transform of />(r), and S(q) is the structure factor (Fourier transform of the two-point correlation function). The cubic term, ft, is zero for a symmetric system and otherwise may be chosen to be positive. The quartic term, y, is then positive to ensure stability. For block copolymers, these coefficients may be expressed in terms of vertex functions calculated in the random phase approximation (RPA) by Leibler (1980). The structure factor is given by... 

From the above expressions it is evident that for the calculations of the vertex function and the propagator the knowledge of four-particle correlation functions are required. For simplification, Gaussian approximation has been assumed and the four-particle correlation function is written as the product of two two-particle correlation functions. 

Next the calculation of the vertex function is described. The projection operator Q in Eq. (184) will introduce products of the form (Av Lft(q)). Considering time inversion symmetry, only (A4 Z-/ (q)) will survive. 

The matrix inversion in the above equation can be performed easily because both S p) and pxd/dpz are even in p. Thus the final expression for the vertex function is written as... 

Next the expression for the vertex function [given by Eq. (195)] and that of the propagator [given by Eq. (186)] are substituted in Eq. (183), the summation is replaced as integral, angular average is performed, and variable p is renamed as q. Thus the expression for the rjspp can be written as... 

Following the same procedure, the vertex function can be calculated, and the final expression of the transverse current contribution to the viscosity is written as... 

As discussed by Kirkpatrick [30], T( ) can be replaced by T( m) since there is a marked softening near this wavenumber. The maximum contribution from Eq. (225) in the long time comes when both the dynamic structure factors are evaluated near qm. The Gaussian part of the dynamic structure factor can also be neglected in the asymptotic limit. Thus the long-time part of the memory function now contains the vertex function and a bilinear product of the dynamic structure factors, all evaluated at or near qm. To make the analysis simpler, the wavenumber dependence of all the quantities are not written explicitly. 

The above equation is an integrodifferential equation that has an unusual structure. Here (co2)1 2 is the frequency of the free oscillator and y is the damping constant. The fourth term on the left-hand side of Eq. (226) has the form of the memory kernel, and its strength is controlled by the dimensionless coupling constant X which contains the contribution from the vertex function. 

Next, for simplicity, an approximation for the vertex function is considered. It has been shown by Balucani that y dn q) can approximately be given... 

Note that even without the approximation of the vertex function (given by Eq. (278)], it is possible to demonstrate that Rtt z — 0) oc crjj, but the analysis becomes cumbersome. Thus the approximation for the vertex function is not necessary, but it is made to simplify the analysis. 

We now want to extract the leading microstructure dependence of the theory, i.e. the leading dependence on the cut off o. We therefore analyze the ultraviolet properties of the theory for d = 4, noting that our approach is based on an expansion in e = 4 — d. As has been pointed out in Sect. 7.2 it is the virtue of the field theoretic formulation that we only have to consider the one-line-irreducible (1 - -%) vertex functions defined by the sum of all... 

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... 

Three types of processes are divergent as a result of this coupling to virtual quanta the self-energy of the electron, vacuum polarizations, and vertex functions. 

In the perturbative approach the first order (or higher order) expressions for the self-energy and the polarization operator are used. The other possibility is to summarize further the diagrams and obtain the self-consistent approximations (Figs. 18,19), which include, however, a new unknown function, called vertex function. We shall write these expressions analytically, including the Hartree-Fock part into unperturbed Green function Gq(1, 2). 

The equation for the vertex function can not be closed diagrammatically (Fig. 20). Nevertheless, it is possible to write close set of equations Redin s equations), which are exact equations for full Green functions written through a functional derivative. Hedin s equations are equations (373)-(376) and the equation for the vertex function... 

In the subsequent discussion we shall need the irreducible 2- and 3-point functions i.e. the electron self energy Sy(p), the vacuum polarisation /7k,pv( ) and the full vertex function rv, PuP2% as knowledge of I vip) and determines the corresponding propagators Gy(p) and Dk,mv( ) completely and fy. iPuPz) represents the perturbative corrections to the free vertex The connection between these quantities is established by the Dyson equations (see e.g. [26])... 

One encounters ultraviolet (UV) divergent contributions to the electron and photon propagators as well as the vertex function already in the first order of the perturbation expansion. These UV-divergent first order contributions (1-loop contributions) correspond to the following diagrams for the irreducible 2- and 3-point functions ... 

This effc )t is most simply handled by considering the inverse of the Greens-function. We introduce the so-called vertex function defined as... 

---