We show how to use the Hopf algebra structure of quantum field theory to derive nonperturbative results for the short-distance singular sector of a renormalizable quantum field theory in a simple but generic example. We discuss renormalized Green functions GR(α, L) in such circumstances which depend on a single scale L = ln q 2 /µ 2 and start from an expansion in the scale GR(α,We derive recursion relations between the γ k which make full use of the renormalization group. We then show how to determine the Green function by the use of a Mellin transform on suitable integral kernels. We exhibit our approach in an example for which we find a functional equation relating weak and strong coupling expansions.

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We give an expression for the solution to propagator-type DysonSchwinger equations with one primitive at 1 loop as an expansion over rooted connected chord diagrams. Along the way we give a refinement of a classical recurrence of rooted connected chord diagrams, and a representation in terms of binary trees.

We study the splitting functions for the evolution of fragmentation distributions and the coefficient functions for single-hadron production in semi-inclusive e + e − annihilation in massless perturbative QCD for small values of the momentum fraction and scaling variable x, where their fixed-order approximations are completely destabilized by huge double logarithms of the form α n s x −1 ln 2n−a x. Complete analytic all-order expressions in Mellin-N space are presented for the resummation of these terms at the next-to-next-to-leading logarithmic accuracy. The poles for the first moments, related to the evolution of hadron multiplicities, and the small-x instabilities of the next-to-leading order splitting and coefficient functions are removed by this resummation, which leads to an oscillatory small-x behaviour and functions that can be used at N = 1 and down to extremely small values of x. First steps are presented towards extending these results to the higher accuracy required for an all-x combination with the state-of-the-art next-to-next-to-leading order large-x results.If the constants up to F (m) n,ℓ are known for all n and ℓ, then the splitting functions and coefficient functions can be determined at N m LL accuracy at all orders of the strong coupling. As observed in Ref. [19], the n th order small-x contributions to F T, φ are built up from n terms of the form(2.4)Since the terms with ε −2n+1 , . . . , ε −n−1 have to cancel in sum (2.1), there are n−1 relations between the LL coefficients A n,k which lead to the constants F (0) n,ℓ in Eq. (2.3), n−2 relations between the NLL coefficients B n,k etc. As discussed above, a N m LO calculation fixes the (nonvanishing) coefficients of ε −n , . . . , ε −n+m at all orders n, adding m + 1 more relations between the coefficients in Eq. (2.4). Consequently the highest m+1 double logarithms, i.e., the N m LL approximation, can be determined order by order from the N m LO results. Finally the resulting series, here calculated to order α 18 s using FORM and TFORM [24], can be employed to find their generating functions via over-constrained systems of linear equations. The whole procedure is analogous to, if computationally more involved than, the large-x resummation in Ref. [25].

Abstract. The c2 invariant of a Feynman graph is an arithmetic invariant which detects many properties of the corresponding Feynman integral. In this paper, we define the c2 invariant in momentum space and prove that it equals the c2 invariant in parametric space for overall log-divergent graphs. Then we show that the c2 invariant of a graph vanishes whenever it contains subdivergences. Finally, we investigate how the c2 invariant relates to identities such as the four-term relation in knot theory. IntroductionLet G be a connected graph. The graph polynomial of G is defined by associating a variable x e to every edge e of G and settingwhere the sum is over all spanning trees T of G. These polynomials first appeared in Kirchhoff's work on currents in electrical networks [16]. Let N G denote the number of edges of G, and let h G denote the number of independent cycles in G (the first Betti number). Of particular interest is the case when G is primitive and overall logarithmically divergent:N γ > 2h γ for all strict non-trivial subgraphs γ G .For such graphs, the corresponding Feynman integral (or residue) is independent of the choice of renormalization scheme and can be defined by the following convergent integral in parametric spaceThe numbers I G are notoriously difficult to calculate, and have been investigated intensively from the numerical [5,20] and algebro-geometric points of view [4,9]. For graphs in φ 4 theory with subdivergences, the renormalised Karen Yeats is supported by an NSERC discovery grant and would like to thank Samson Black for explaining knots. Francis Brown is partially supported by ERC grant 257638. Francis Brown and Oliver Schnetz thank Humboldt University, Berlin, for support as visiting guest scientists. All three authors thank Humboldt University for hospitality.

We give an expression for the solution to propagator-type Dyson-Schwinger equations with one primitive at 1 loop as an expansion over rooted connected chord diagrams. Along the way we give a refinement of a classical recurrence of rooted connected chord diagrams, and a representation in terms of binary trees.

We investigate combinatorial properties of a graph polynomial indexed by half-edges of a graph which was introduced recently to understand the connection between Feynman rules for scalar field theory and Feynman rules for gauge theory. We investigate the new graph polynomial as a stand-alone object.arXiv:1207.5460v1 [math.CO]

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