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One of the main challenges in obtaining predictions for collider experiments from perturbative quantum field theory, is the direct evaluation of the Feynman integrals it gives rise to. In this chapter, we review an alternative bootstrap…
Inspired by the recent work of Nima Arkani Hamed and collaborators who introduced the notion of positive geometry to account for the structure of tree-level scattering amplitudes in bi-adjoint $\phi^3$ theory, which led to one-loop…
Within the non-perturbative 1/N expansion, we discuss numerical methods for calculating multi-loop Feynman graph needed to derive physical scattering amplitudes. We apply higher order 1/N methods to the scalar sector of the standard model,…
We introduce a new, probability-level approach to calculations in scalar field particle scattering. The approach involves the implicit summation over final states, which makes causality manifest since retarded propagators emerge naturally.…
We present a numerically efficient and accurate Multiple Scattering formalism, which is a generalization of the Multiple Scattering method with a truncated basis set [X. -G. Zhang and W. H. Butler, Phys. Rev. B 46,7433 (1992)]. Compared to…
We propose a new framework to represent the perturbative S-matrix which is well-defined for all quantum field theories of massless particles, constructed from tree-level amplitudes and integrable term-by-term. This representation is derived…
A geometrical approach to the calculation of N-point Feynman diagrams is reviewed. It is shown that the geometrical splitting yields useful connections between Feynman integrals with different momenta and masses. It is demonstrated how…
I analyze the algebraic patterns underlying the structure of scattering amplitudes in quantum field theory. I focus on the decomposition of amplitudes in terms of independent functions and the systems of differential equations the latter…
Ab initio predictions of two-loop electroweak contributions to observables are increasingly essential for precision collider experiments, yet their evaluation remains very challenging. We connect recurrence techniques and dispersive method…
A framework to represent and compute two-loop $N$-point Feynman diagrams as double-integrals is discussed. The integrands are 'generalised one-loop type" multi-point functions multiplied by simple weighting factors. The final integrations…
We present analytic results for all planar two-loop Feynman integrals contributing to five-particle scattering amplitudes with one external massive leg. We express the integrals in terms of a basis of algebraically-independent…
On the basis of the explicit formulae for the action of the unitary group of exponentials corresponding to almost solvable extensions of a given closed symmetric operator with equal deficiency indices, we derive a new representation for the…
Current all-orders proofs of factorization of hard processes are made by extracting the leading power behavior of Feynman graphs, i.e., by extracting asymptotics strictly order-by-order in perturbation theory. The resulting parton densities…
The free energy of a field theory can be considered as a functional of the free correlation function. As such it obeys a nonlinear functional differential equation which can be turned into a recursion relation. This is solved order by order…
We outline ideas to connect the analytic structure of Feynman amplitudes to the structure of Karen Vogtmann's {\em Outer Space}. We focus on the role of cubical chain complexes in this context, and also investigate the bordification problem…
In the wake of the growing popularity of machine learning in particle physics, this work finds a new application of geometric deep learning on Feynman diagrams to make accurate and fast matrix element predictions with the potential to be…
One problem which plagues the numerical evaluation of one-loop Feynman diagrams using recursive integration by part relations is a numerical instability near exceptional momentum configurations. In this contribution we will discuss a…
A T-dualized selfdual inspired formulation of massive vector fields coupled to arbitrary matter is generated; subsequently its perturbative series modeling a spontaneously broken gauge theory is analyzed. The new Feynman rules and external…
We use the homological perturbation lemma to produce explicit formulas computing the class in the twisted de Rham complex represented by an arbitrary polynomial. This is a non-asymptotic version of the method of Feynman diagrams. In…
We present a formalism for the scattering of an arbitrary linear or acyclic branched structure build by joining mutually non-interacting arbitrary functional sub-units. The formalism consists of three equations expressing the structural…