The diagrammatic coaction beyond one loop

The diagrammatic coaction maps any given Feynman graph into pairs of graphs and cut graphs such that, conjecturally, when these graphs are replaced by the corresponding Feynman integrals one obtains a coaction on the respective functions. The coaction on the functions is constructed by pairing a basis of differential forms, corresponding to master integrals, with a basis of integration contours, corresponding to independent cut integrals. At one loop, a general diagrammatic coaction was established using dimensional regularisation, which may be realised in terms of a global coaction on hypergeometric functions, or equivalently, order by order in the ϵ expansion, via a local coaction on multiple polylogarithms. The present paper takes the first steps in generalising the diagrammatic coaction beyond one loop. We first establish general properties that govern the diagrammatic coaction at any loop order. We then focus on examples of two-loop topologies for which all integrals expand into polylogarithms. In each case we determine bases of master integrals and cuts in terms of hypergeometric functions, and then use the global coaction to establish the diagrammatic coaction of all master integrals in the topology. The diagrammatic coaction encodes the complete set of discontinuities of Feynman integrals, as well as the differential equations they satisfy, providing a general tool to understand their physical and mathematical properties.

Double Box Motive

The motive associated to the second Symanzik polynomial of the double-box two-loop Feynman graph with generic masses and momenta is shown to be an elliptic curve.

Random tensor models—the multi-orientable (MO) model

In its first section, this chapter presents the definition of the multi-orientable tensor model. The 1/N expansion and the large N limit of this model are exposed in the second section of the chapter. In the third section, a thorough enumerative combinatorial analysis of the general term of the 1/N expansion is presented. The implementation of the double scaling mechanism is then exhibited in the fourth section. This chapter presents the multi-orientable (MO) tensor model and it follows the review article. This rank three model, having O(N) U(N) O(N) symetry, can be seen as an intermediate step between the U(N) invariant model presented in the previous chapter, and the O(N) invariant model presented in the following chapter. The class of Feynman graph generated by perturbative expansion of MO model is strictly larger than the class of Feynman graphs of the U(N) invariant model and strictly smaller than the one of the O(N) invariant model.

FROM DIMENSIONAL REDUCTION OF 4d SPIN FOAM MODEL TO ADDING NON-GRAVITATIONAL FIELDS TO 3d SPIN FOAM MODEL

A Kaluza–Klein-like approach for a 4d spin foam model is considered. By applying this approach to a model based on group field theory in 4d (TOCY model), and using the Peter–Weyl expansion of the gravitational field, reconstruction of new non-gravitational fields and interactions in the action are found. The perturbative expansion of the partition function produces graphs colored with SU(2) algebraic data, from which one can reconstruct a 3d simplicial complex representing space-time and its geometry (like in the Ponzano–Regge formulation of pure 3d quantum gravity), as well as the Feynman graph for typical matter fields. Thus a mechanism for generation of matter and construction of new dimensions are found from pure gravity.

GRAVITATIONAL RENORMALIZATION OF QUANTUM FIELD THEORY

We propose to include gravity in quantum field theory nonperturbatively, by modifying the propagators so that each virtual particle in a Feynman graph move in the space–time determined by the four-momenta of the other particles in the same graph. By making additional working assumptions, we are able to put this idea at work in a simplified context, and obtain a modified Feynman propagator for the massless neutral scalar field. Our expression shows a suppression at high momentum, strong enough to entail finite results, to all loop orders, for processes involving at least two virtual particles.