Callan-Symanzik

Different from the Lie equation (5.12) is the Callan-Symanzik equation... 

The functional equation formulation of RG theory is due to Bogoliubov and Shirkov [3-5], As we shall see, this approach often is more convenient in practical, numerical calculations than the better known and more widely used formulations based on the Lie and Callan-Symanzik differential equations. 

While the functional equations (5.61) and (5.65), are very well suited to the above mentioned, iterative method for determining the (generalized) time evolution of the dynamic system f(t, g,fo), most presentations of the RG method have instead used an approach based on differential equations. These differential equations can be written directly as equations of evolution for the object function Sf t g) and the related effective coupling function g(t g) [3-5,16] or as a pair of partial differential equations known as the Callan-Symanzik equations [3-5,17]. These three forms of the RG theory are essentially equivalent. However, we personally favor the functional equation approach, not only from a computational point of view but because it provides better insight into the workings of the postulates of the self-similarity based RG technique. 

The Lie and Callan-Symanzik equations for the PFRG are readily obtained from the relationship... 

Also derivable from the PFRG relationship (5.127) are linear partial differential equations of the Callan-Symanzik type. By differentiating this relationship with respect to the scale x and then taking the limit x -> 1, one obtains the equation... 

The Callan-Symanzik equation for the effective coupling function is the same as before, namely, Eq. (5.95). 

The differences between the average squared end-to-end distance a N, g) and the excess partition function 5Q(N, g) also are evident from the associated Callan-Symanzik equations [see Eqs. (5.94) and (5.133)]... 

In both cases, the solution of the second Callan-Symanzik equation [see Eq. (5.95)]... 

In a similar way, one obtains partial differential equations of the Callan-Symanzik type, namely... 

The seeding properties of the system asymptotically close to the critical point eire expressed by the homogeneous Callan-Symanzik equation [4,80] for the renormahzed vertex function ... 

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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