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Abstract. The appearance of multiple zeta values in anomalous dimensions and β-functions of renormalizable quantum field theories has given evidence towards a motivic interpretation of these renormalization group functions. In this paper we start to hunt the motive, restricting our attention to a subclass of graphs in four dimensional scalar field theory which give scheme independent contributions to the above functions. IntroductionCalculations of Feynman integrals arising in perturbative quantum field theory [4,5] reveal interesting patterns of zeta and multiple zeta values. Clearly, these are motivic in origin, arising from the existence of Tate mixed Hodge structures with periods given by Feynman integrals. We are far from a detailed understanding of this phenomenon. An analysis of the problem leads via the technique of Feynman parameters [12] to the study of motives associated to graph polynomials. By the seminal work of Belkale and Brosnan [3], these motives are known to be quite general, so the question becomes under what conditions on the graph does one find mixed Tate Hodge structures and multiple zeta values.The purpose of this paper is to give an expository account of some general mathematical aspects of these "Feynman motives" and to work out in detail the special case of wheel and spoke graphs. We consider only scalar field theory, and we focus on primitively divergent graphs. (A connected graph Γ is primitively divergent if #Edge(Γ) = 2h 1 (Γ) where h 1 is the Betti number of the graph; and if further for any connected proper subgraph the number of edges is strictly greater than twice the first Betti number.) From a motivic point of view, these play the role of "Calabi-Yau" objects in the sense that they have unique periods. Physically, the corresponding periods are renormalization scheme independent.Date: Sept. 28b, 2005. Graph polynomials are introduced in sections 1 and 2 as special cases of discriminant polynomials associated to configurations. They are homogeneous polynomials written in a preferred coordinate system with variables corresponding to edges of the graph. The corresponding hypersurfaces in projective space are graph hypersurfaces. Section 3 studies coordinate linear spaces contained in the graph hypersurface. The normal cones to these linear spaces are linked to graph polynomials of sub and quotient graphs. Motivically, the chain of integration for our period meets the graph hypersurface along these linear spaces, so the combinatorics of their blowups is important. (It is curious that arithmetically interesting periods seem to arise frequently (cf. multiple zeta values [11] or the study of periods associated to Mahler measure in the non-expansive case [8]) in situations where the polar locus of the integrand meets the chain of integration in combinatorially interesting ways.) Section 4 is not used in the sequel. It exhibits a natural resolution of singularities P(N) → X for a graph hypersurface X. P(N) is a projective bundle over projective space, and the fibres P(N)/X are projecti...

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