Mixed Hodge structures and renormalization in physics

1.1. This paper is a collaboration between a mathematician and a physicist. It is based on the observation that renormalization of Feynman amplitudes in physics is closely related to the theory of limiting mixed Hodge structures in mathematics. Whereas classical physical renormalization methods involve manipulations with the integrand of a divergent integral, limiting Hodge theory involves moving the chain of integration so the integral becomes convergent and studying the monodromy as the chain varies. Even methods like minimal subtraction in the context of dimensional or analytic regularization implicitly modify the integrand through the definition of a measure ∫ dk via analytic continuation. Still, as a regulator dimensional regularization is close to our approach in so far as it leaves the rational integrand assigned to a graph unchanged. Minimal subtraction as a renormalization scheme differs though from the renormalization schemes which we consider -momentum subtractions essentiallyby a finite renormalization. Many of the nice algebro-geometric structures developed below are not transparent in that scheme. The advantages of the limiting Hodge method are firstly that it is linked to a very central and powerful program in mathematics: the study of Hodge structures and their variations. As a consequence, one gains a number of tools, like weight, Hodge, and monodromy filtrations to study and classify the Feynman amplitudes. Secondly, the method depends on the integration chain, and hence on the graph, but it is in some sense independent of the integrand. For this reason it should adapt naturally e.g. to gauge theories where the numerator of the integrand is complicated. An important point is to analyse the nature of the poles. Limiting mixed Hodge structures demand that the divergent subintegrals have at worst log poles. This does not imply that we can not apply our approach to perturbative amplitudes which have worse than logarithmic degree of divergence. It only means that we have to correctly isolate the polynomials in masses and external momenta which accompany those divergences such that the corresponding integrands have singularities provided by log-poles. This is essentially automatic from the notion of a residue available by our very methods. As a very pleasant byproduct, we learn that physical renormalization schemes -on-shell subtractions, momentum subtractions, Weinberg’s scheme,belong to a class of schemes for which this is indeed automatic. Moreover, for technical reasons, it is convenient to work with projective rather than affine integrals. One of the central physics results in this paper is that the renormalization problem can be reduced to the study of logarithmically divergent, projective integrals. This is again familiar from analytic regulators. The fact that it can be achieved here by leaving the integrand completely intact will hopefully some fine day allow to understand the nature of the periods assigned to renormalized values in quantum field theory. A remark for Mathematicians: our focus in this paper has been renormalization, which is a problem arising in physics. We suspect, however, that similar methods will apply more generally for example to period integrals whenever the domain of integration is contained in R and the integrand is a rational function with polar locus defined by a polynomial with non-negative real coefficients. The toric methods and the monodromy computations should go through in that generality.