Renormalization group flow

Renormalization group (RG) methods have been developed [ 82 ] to carry out qualitative studies of QFT in regimes, which are not available to perturbation theory in the coupling constant. Hence, they are very much in the spirit of the bootstrap program. For theories in 1+1 space-time dimensions these methods admit particularly powerful realizations in form of explicit constructions of scaling functions. As decribed above such functions may be obtained either from the thermodynamic Bethe ansatz (TBA), from correlations functions involving various components of the energy-momentum tensor [ 122 ,18 ] or from semi-classical studies [126 ]. In general the functions obtained from different approaches differ quantitatively, but nonetheless possess the same qualitative features. As explained in the previous section the renormalization group (RG) flow contains a lot of information and in this context particularly interesting are the staircase patterns related to the formation of unstable particles. In [ 18 ] we adapted the c-theorem of A. Zamolodchikov [ 122 ] and computed these flows for various HSG models [ 18 ,19 ]. The correlation functions of the trace of energy momentum tensor were computed in the way described in (iv). The outcome showed very good agreement with the TBA computations. In addition we carried out such a flow for the analogue of the $\Delta$ -sum rule [40 ] tracing the conformal dimension of particular operators through various RG fixed points. Conceptually there is not very much to be added here and what is left to do is to try to perform such computations for other models. Desirable would be to refine the status of the sum rule, such that degeneracies can be unraveled and also operators which have a vanishing correlation function with the energy momentum tensor can be handled. It appears that this requires an entirely new proposition. Analogues to the $\Delta$ -sum rule in the TBA context are still unknown.