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

We present a new method for the decomposition of multi-loop Euclidean Feynman integrals into quasi-finite Feynman integrals. These are defined in shifted dimensions with higher powers of the propagators, make explicit both infrared and ultraviolet divergences, and allow for an immediate and trivial expansion in the parameter of dimensional regularization. Our approach avoids the introduction of spurious structures and thereby leaves integrals particularly accessible to direct analytical integration techniques. Alternatively, the resulting convergent Feynman parameter integrals may be evaluated numerically. Our approach is guided by previous work by the second author but overcomes practical limitations of the original procedure by employing integration by parts reduction.

We calculate the complete quark and gluon cusp anomalous dimensions in four-loop massless QCD analytically from first principles. In addition, we determine the complete matter dependence of the quark and gluon collinear anomalous dimensions. Our approach is to Laurent expand four-loop quark and gluon form factors in the parameter of dimensional regularization. We employ finite field and syzygy techniques to reduce the relevant Feynman integrals to a basis of finite integrals, and subsequently evaluate the basis integrals directly from their standard parametric representations.

Hyperlogarithms provide a tool to carry out Feynman integrals in Schwinger parameters. So far, this method has been applied successfully mostly to finite single-scale processes. However, it can be employed in more general situations.We give examples of integrations of three-and four-point integrals in Schwinger parameters with non-trivial kinematic dependence, including setups with off-shell external momenta and differently massive internal propagators. The full set of Feynman graphs admissible to parametric integration is not yet understood and we discuss some counterexamples to the crucial property of linear reducibility. In special cases we observe how a change of variables can restore this prerequisite for direct integration and thereby enlarge the set of accessible graphs.Working in dimensional regularization, we furthermore clarify how a simple application of partial integration can be used to convert divergent parametric integrands to convergent ones. In contrast to the subtraction of counterterms, this scheme is ideally suited for our method of integration.

We study shift relations between Feynman integrals via the Mellin transform through parametric annihilation operators. These contain the momentum space integration by parts relations, which are well-known in the physics literature. Applying a result of Loeser and Sabbah, we conclude that the number of master integrals is computed by the Euler characteristic of the Lee-Pomeransky polynomial. We illustrate techniques to compute this Euler characteristic in various examples and compare it with numbers of master integrals obtained in previous works. for interesting discussions and helpful comments on integration by parts for Feynman integrals, Jørgen Rennemo for suggesting literature relevant for section 3.1 and Viktor Levandovskyy for communication and explanations around D-modules and in particular their implementation in SINGULAR.Thomas Bitoun thanks Claude Sabbah for feedback and bringing the work [58-60] to our attention. Thomas acknowledges funding through EPSRC grant EP/L005190/1. Christian Bogner thanks Deutsche Forschungsgemeinschaft for support under the project BO 4500/1-1.This research was supported by the Munich Institute for Astro-and Particle Physics (MIAPP) of the DFG cluster of excellence "Origin and Structure of the Universe". Furthermore we are grateful for support from the Kolleg Mathematik Physik Berlin (KMPB) and hospitality at Humboldt-Universität Berlin.Images in this paper were created with JaxoDraw [11] (based on Axodraw [93]) and FeynArts [36].