We study how to compute the operator product expansion coefficients in the exact renormalization group formalism. After discussing possible strategies, we consider some examples explicitly, within the $\ensuremath{\epsilon}$ expansions, for the Wilson-Fisher fixed points of the real scalar theory in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and the Lee-Yang model in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions. Finally we discuss how our formalism may be extended beyond perturbation theory.
In dealing with critical phenomena of complex systems that simulate realistic materials, the full structure of the renormalization group is often unnecessarily cumbersome. For approximate calculations and for systems with special properties, specialized generators are simpler to apply. We derive several such exact and approximate differential generators and solve a number of interesting practical problems to illustrate this approach: (i) We derive a new approximate differential generator based on the Wilson incomplete-integration generator. Using this generator we calculate for an
n-component spin system the eigenvalues (critical-point exponents) associated with a critical point of arbitrary order
O and "propagator exponent" $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}$ to first order in the expansion parameter ${\ensuremath{\epsilon}}_{\mathcal{O}}(\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}})=d+\mathcal{O}(\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}\ensuremath{-}d)$; this extends previous work for $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}=2$,
O arbitrary; $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}\ensuremath{\le}2$,
O=2 (long-rang forces); and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}=4$,
O=2 (the "Lifshitz point"). Our results agree with those obtainable using an approximate generator based on the Wegner-Houghton equation. The cases $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}=2L$ (
L a positive integer \ensuremath{\ge}2) describe the onset of helical ordering for which $|\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}|\ensuremath{\sim}{(\ensuremath{-}p)}^{{\ensuremath{\beta}}_{k}}$, where ${\ensuremath{\beta}}_{k}=\frac{1}{2(L\ensuremath{-}1)}+o({\ensuremath{\epsilon}}_{\mathcal{O}}^{2}(2L))$ and
p parametrizes the hypersurface of critical points. For
p>0, the ordered phase is uniform; for
p<0 there is spiral order. The point
p=0, at which such nonuniform ordering commences, we term a generalized Lifshitz point of Lifschitz character
L. (ii) We consider the full Wilson and Wegner-Houghton generators in the paired spin-momenta limit and the $n\ensuremath{\rightarrow}\ensuremath{\infty}$ limit for even-order critical points. These limiting generators are identical for both full generators. This demonstrates that at least in these cases the Wilson and Wegner-Houghton generators agree exactly, without recourse to perturbation theory. These simple exact generators should provide "anchors" for calculations of exponents for higher-order critical points. (iii) We derive approximate generators which are suitable for compressible magnetic systems and more general systems with constraints for which the spin momenta are grouped in any arbitrary manner. We apply this to the case of a simple compressible magnet model and obtain the exact renormalization-group trajectories to order $\ensuremath{\epsilon}$ with $\ensuremath{\epsilon}\ensuremath{\equiv}{\ensuremath{\epsilon}}_{2}(2)=4\ensuremath{-}d$.
The large distance behaviour, near the critical temperature, of continuous phase transitions with short range interactions can be reproduced by an effective local field theory. Within this framework, the universal properties of the large distance can be determined by finding fixed points of general, functional renormalization group (RG) equations (Wilson, Wegner). For dimension D > 4, the large scale behavior is governed by the Gaussian fixed point (related to mean field theory), while for D < 4 the Gaussian fixed point is unstable. Near D = 4, the RG equations can be solved in the form of an ǫ = 4 − D expansion, a new stable fixed point is found and this leads to the famous Wilson-Fisher expansion for critical exponents and other universal quantities. Moreover, one realizes that, in this framework, the effective field theory can be reduced to a renormalizable field theory and, at leading order in the critical domain, the much simpler quantum field theory renormalization group can be used. A number of physicists, including the Saclay group (Brezin, Le Guillou, Zinn-Justin) have developed sophisticated field theory techniques, to prove general scaling laws and to calculate exponents to higher orders [1].
Abstract These introductory notes are about functional renormalization group equations and some of their applications. It is emphasised that the applicability of this method extends well beyond critical systems, it actually provides us a general purpose algorithm to solve strongly coupled quantum field theories. The renormalization group equation of F. Wegner and A. Houghton is shown to resum the loop-expansion. Another version, due to J. Polchinski, is obtained by the method of collective coordinates and can be used for the resummation of the perturbation series. The genuinely non-perturbative evolution equation is obtained by a manner reminiscent of the Schwinger-Dyson equations. Two variants of this scheme are presented where the scale which determines the order of the successive elimination of the modes is extracted from external and internal spaces. The renormalization of composite operators is discussed briefly as an alternative way to arrive at the renormalization group equation. The scaling laws and fixed points are considered from local and global points of view. Instability induced renormalization and new scaling laws are shown to occur in the symmetry broken phase of the scaler theory. The flattening of the effective potential of a compact variable is demonstrated in case of the sine-Gordon model. Finally, a manifestly gauge invariant evolution equation is given for QED.