The geometry of RG flows in theory space

Renormalisation Group (RG) flows in theory space (the space of couplings) are generated by a vector field – the β 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 β 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 analysed. We illustrate the results for a scalar field theory with a jφ coupling and pointers to other areas are briefly mentioned. ∗Electronic Address : [email protected]//[email protected] 1 In the theory of quantised fields, the notion of the Renormalisation Group (RG) plays a central role in understanding certain non–perturbative issues. It is a well-known fact that an ambiguity arises in the choice of the infinite part of a regularised Feynman amplitude. This necessitates the choice of a renormalisation scheme, which, naturally, implies the existence of a scale or a renormalisation point. The requirement that physical amplitudes, say the one-particle irreducible amplitudes , are independent of a choice of scale leads to the concept of the Renormalisation Group. The RG transformations are finite renormalisations and the RG equation, essentially, follows from the scale invariance of the physical Green functions. The fact that the RG equation can be written as a geometric statement was first noted by Lassig in [1] and later by Dolan [2], [3]. The idea there, was to visualise the β–function as a vector field in the space of couplings (this is the space we refer to as ‘theory space’). Theory space, constructed with the couplings as the coordinates can be thought of as a manifold in its own right. Each point on such a manifold defines a set of values for the various couplings which appear in a given Lagrangian. A curve in theory space, therefore, represents the flow of couplings. Finite renormalisation generates such a flow and thereby, gives rise to RG trajectories which have the β–function as the tangent vector at each point. The RG equation, in its theory space incarnation, can be expressed as a Lie–derivative of the n–point function with respect to this vector field. The change in physical amplitudes due to a diffeomorphism of R (scale transformations of the spatial variables), at fixed couplings, generated by the dilatation generator D is equivalent to the change in amplitudes due to a diffeomorphism of theory space, generated by the β–function, at fixed spatial positions. This fact lies at the heart of the geometric form of the RG equation. As is well known, the definition of the Lie–derivative does not require the concept of a metric or a connection. However, we do need such structures in order to define and analyse a geometry. The natural question which one therefore has to address is – how do we define a metric in theory space ? There have been proposals for such a metric. One of them is related to the spatial integral of a two–point function [4], which, in general, could, as well be, a two-point function of composite operators. We also need to impose the crucial fact that spatial points are well 2 separated. This enables us to avoid the possible divergences of amplitudes which always come up in quantum field theory. The above proposal is akin to that due to Zamolodchikov [5], which he and later authors utilised in order to prove the c-theorem for two–dimensional field theories. Given a definition of a metric we can now investigate the geometry of theory space through the ensuing connection and curvature. This has been worked out for several field theoretic models such as scalar field theories with a jφ term [6], λφ theory, the one dimensional Ising model [6], the O(N) model (for large N) [8] and N = 2 supersymmetric Yang–Mills theory [9]. The location of zeros of the β–function which correspond to fixed points for the flows, (and also imply conformally invariant field theories) coincide, rather surprisingly, with points of diverging Ricci curvature in theory space. The fact that the β function is a vector field in theory space can improve our understanding of the nature of RG flows in a novel way, if we employ the techniques and results primarily used in proving the singularity theorems in General Relativity (GR). This is what we aim to do in this article. We shall also point out that there are some worthwhile directions which may be pursued in greater depth, in future. Firstly, let us recall the analysis from Riemannian geometry and GR which we will use extensively. This is based on the fact that the covariant derivative of the velocity field in a Riemannian/pseudo–Riemannian spacetime is a second rank tensorial object. Therefore vμ;ν can be split into it’s symmetric traceless, trace and antisymmetric parts. These three parts constitute the shear, expansion and rotation of the geodesic flow. Along the families of flow lines, one can therefore write down the differential equations for each of these quantities. It turns out that these equations are coupled and quite difficult to solve completely. However, for simplistic scenarios, there do exist solutions which have been analysed in some amount of detail. In Riemannian geometry and GR one does not quite solve these equations. For general, possibly non-geodesic flows, one has the Raychaudhuri identity given by :