chapter  8
18 Pages

Renormalization Group

To obtain a more explicit form of equation (8.1) we assume Wto be the unique point at which all of parameters are defined: the mass squared m2 according to (7.64), the wave function renormalization constant Zcp according to (7.75) and the coupling constant A according to (7.68) and (7.77). The renormalization procedure makes it necessary to clarify our notation. The bare field 'Po does not depend on J1 and the renormalised field 'Pphys = 'Po does (here, we extend definition (7.77) to the total field including the background as well as fluctuations). In this chapter 'Pphys is called the field and the quantities r~~~s are defined as the coefficients of the functional expansion in this field (7.78). Below we shall omit the subscript phys for the sake of compactness. Equation (8.1) takes the form

where In J1 is considered as the variable. The dependence of theory parameters on the renormalization point

is most transparently manifested in the case of the coupling constant A. Here, we use the calculation results and the definition of A (7.68) from the previous chapter. First, let us make the definition (7.68) more specific. For this purpose let us fix the renormalization point of r(4)(kl ... k4 ) by choosing a set of momenta kl = kf, ... k4 = k!! (see expression (7.68) and the discussion thereafter). We shall call the change of the renormalization point the change of the overall scale of momenta. That is, the renormalization point at an arbitrary J1 is defined as kl = J1kf ... k4 = J1k!!, where AN N N AN N N . kl = kl Ilkl I··· k4 = k4 IIk4 I are umt vectors. Because of (7.68) and (7.78) the renormalised expression for the vertex function is

The dependence of the mass and the constant Zcp on J1 is not as straightforward. It depends on the context of the problem. If the subject of interest is the propagation and the interaction of quanta of the 'P field, then the most reasonable definition of m2 is as the pole of propagator (7.64) and Zcp is defined as the constant normalising the residue at this

Let us give an example in which the definition of m2 and Z'" is not so obvious. Let the field rp describes the local magnetisation or another order parameter of a sample near a second-order phase transition. The effective action in this case is the free energy of the system. Strictly speaking, the system is three-dimensional and one has to reproduce the treatment of the previous chapter for this case. One should have four dimensions only in the case of very low temperature when quantum effects are essential but we can hardly imagine the system which exhibits a second-order phase transition at such a temperature. Thus let us return to equation (8.2) and discuss the meaning of m2 and Z'" depending on J1 despite the incorrect dimension.