Geometry of renormalization group flows in theory space

Renormalization group (RG) flows in theory space (the space of couplings) are generated by a vector field-the beta function. Using a specific metric ansatz in theory space and certain methods employed largely in the context of general relativity, we examine the nature of the expansion, shear and rotation of geodesic RG flows. The expansion turns out to be a negative quantity inversely related to the norm of the beta function. This implies the focusing of the flows towards the fixed points of a given field theory. The evolution equation for the expansion along geodesic RG flows is written down and analyzed. We illustrate the results for a scalar field theory with a j phi coupling and pointers to other areas are briefly mentioned.