Crossover scaling functions and an extended minimal subtraction scheme ⋆

A field theoretic renormalization group method is presented which is capable of dealing with crossover problems associated with a change in the upper critical dimension. The method leads to flow functions for the parameters and coupling constants of the model which depend on the set of parameters which characterize the fixed point landscape of the underlying problem. Similar to Nelson’s trajectory integral method any vertex function can be expressed as a line integral along a renormalization group trajectory, which in the field theoretic formulation are given by the characteristics of the corresponding Callan-Symanzik equation. The field theoretic renormalization automatically leads to a separation of the regular and singular parts of all crossover scaling functions. The method is exemplified for the crossover problem in magnetic phase transitions, percolation problems and quantum phase transitions. The broad applicability of the method is emphasized.