We provide algorithms for symbolic integration of hyperlogarithms multiplied by rational functions, which also include multiple polylogarithms when their arguments are rational functions. These algorithms are implemented in Maple and we discuss various applications. In particular, many Feynman integrals can be computed by this method.Keywords: Feynman integrals, hyperlogarithms, polylogarithms, computer algebra, symbolic integration, ε-expansions Solution method: Symbolic integration of rational linear combinations of polylogarithms of rational arguments is obtained using a representation in terms of hyperlogarithms. The algorithms exploit their iterated integral structure. Restrictions: To compute multi-dimensional integrals with this method, the integrand must be linearly reducible, a criterion we state in section 4. As a consequence, only a small subset of all Feynman integrals can be addressed. Unusual features: The complete program works strictly symbolically and the obtained results are exact. Whenever a Feynman graph is linearly reducible, its ε-expansion can be computed to arbitrary order (subject only to time and memory restrictions) in ε, near any even dimension of space-time and for arbitrarily ε-dependent powers of propagators with integer values at ε = 0. Also the method is not restricted to scalar integrals only, but arbitrary tensor integrals can be computed directly. Additional comments: Further applications to parametric integrals, outside the application to Feynman integrals in the Schwinger parametrization, are very likely. Running time: Highly dependent on the particular problem through the number of integrations to be performed (edges of a graph), the number of remaining variables (kinematic invariants), the order in ε and the complexity of the geometry (topology of the graph). Simplest examples finish in seconds, but the time needed increases beyond any bound for sufficiently high orders in ε or graphs with many edges. Program summary References[1] Maple 16 1 . Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario.

We present the perturbative renormalization group functions of O(n)-symmetric φ 4 theory in 4 − 2ε dimensions to the sixth loop order in the minimal subtraction scheme. In addition, we estimate diagrams without subdivergences up to 11 loops and compare these results with the asymptotic behaviour of the beta function. Furthermore we perform a resummation to obtain estimates for critical exponents in three and two dimensions.

We report on calculations of Feynman periods of primitive logdivergent φ 4 graphs up to eleven loops. The structure of φ 4 periods is described by a series of conjectures. In particular, we discuss the possibility that φ 4 periods are a comodule under the Galois coaction. Finally, we compare the results with the periods of primitive log-divergent non-φ 4 graphs up to eight loops and find remarkable differences to φ 4 periods. Explicit results for all periods we could compute are provided in ancillary files. P φ 4 := lin Q P (G) : G primitive log-divergent and φ 4 ⊆ P log := lin Q {P (G) : G primitive log-divergent} ⊂ P denote the Q-vector spaces spanned by primitive log-divergent periods. They are subspaces of the Q-algebra P of periods in the sense of Kontsevich and Zagier [44]. 1 We obtain finite-dimensional subspaces if we restrict the loop order of the graphs, P •,≤n := lin Q {P (G) : h G ≤ n} (1.4)for φ 4 graphs or general primitive log-divergent graphs G, respectively.

This talk summarizes recent developments in the evaluation of Feynman integrals using hyperlogarithms. We discuss extensions of the original method, new results that were obtained with this approach and point out current problems and future directions. Loops and Legs in Quantum Field Theory