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Amplitudes and form factors via integrability

Periodic Reporting for period 1 - AFFINITY (Amplitudes and form factors via integrability)

Reporting period: 2018-02-01 to 2020-01-31

The idea that our understanding of physical reality should be phrased in a mathematical language goes back to G. Galilei and it is still the common thread throughout several research directions in theoretical physics. The quantum theory of fields is the framework we currently use to encode a variety of physical phenomena, ranging from the interactions among fundamental particles to the description of statistical systems, such as a gas or a magnet. The canonical approach to the study of quantum field theories consists in computing physical observables in the approximation of weak interactions. In this context physical observables are expressed as power series expansions in the coupling constant, the small parameter governing the strength of the interaction.

A concrete example of physical observable that can be measured at colliders, such as the Large Hadron Collider (LHC), is the differential cross section, whose computation involves different ingredients. This project focuses on two crucial ones, i.e. scattering amplitudes, describing the evolution between the initial and the final state of a scattering process, and form factors, describing the production or annihilation of fundamental particles by the action of some external energetic probe – as in Drell-Yan (DY) or Deep Inelastic Scattering (DIS) processes. The standard technique for computing such quantities has a graphic interpretation in terms of Feynman diagrams, for which the number of loops is related to the order in the perturbative expansion. However one important drawback of the method is that the complexity of the computation grows quickly (more precisely factorially) with the number of particles and standard techniques soon cease to be a viable path.

Improving the currently known computational methods and understanding quantum field theories at finite value of the coupling are two of the main challenges of current research in theoretical physics. This proposal aims to give a contribution in both of these directions through the study of amplitudes and form factors in the context of the AdS/CFT correspondence. This duality established a remarkable relation between very different theories and it allows to access the strong coupling regime of quantum field theories by studying the weak coupling regime of a string moving in a particular geometry called Anti de Sitter (AdS) space. Furthermore, the emergent integrability of the model allows to hope for an exact solution which would yield an unprecedented example of quantum field theory which can be solved exactly.
The collaboration with the amplitudes group at QMUL led to the derivation of a novel recursive relation for loop-level form factors and to the understanding of the mechanism implementing dual conformal symmetry in weakly coupled form factors for N=4 Super Yang-Mills (SYM) theory. In particular, the former result provides a recipe to obtain the loop level integrand for the N=4 SYM form factor using results for lower loops or lower number of legs. This is an important progress in the exploration of the perturbative structure of form factors in the weakly coupled regime and it is likely to lead to new powerful methods for the computation of high loop results. The latter achievement, on the other hand, is conceptually relevant as it shows the realization of a hidden symmetry, which had been discovered long ago in the study of scattering amplitudes in N=4 SYM, but it was believed to be absent for the case of form factors.

In a parallel research line, with the aim of identifying the Wilson loop dual of form factors, the researcher carried out some independent work focused on the insertion of local operators on Wilson lines. As it often happens in scientific research, this work led to important connections with the very active research area of conformal defects, leading the researcher to make important progress in that field. In particular, he studied four-dimensional theories with N=2 superconformal symmetry where he managed to prove an exact formula for the energy emitted by an accelerated charged particle, first by developing a new way of imposing superconformal invariance in a defect setting and then by using defect methods to study a small deformation of the background geometry. The former paper, in particular, was accepted for publication in the prestigious journal Physical Review Letters.

The Researcher also undertook an intense activity of communication and dissemination of his results. He presented his work in five international conferences and was invited for eleven seminars in high-level international institutions during the duration of the fellowship. The invitations he received provide a further confirmation of the excellent quality of the research carried out during the fellowship. He also took part in several outreach activities, including the publication of an article on a non-scientific journal and a public talk during his secondment. The latter was carried out from March 1st 2018 to August 31st 2018 at the University of Torino, where he had an extremely fruitful collaboration resulted in the publication of a paper with members of the Torino string theory group as well as in a long-lasting collaboration with the Italian Association of Physics Students, which involved the researcher in a variety of outreach activities.
The scientific results that have been obtained in the project made significant progress in our understanding of form factors and defect field theory. The presence of dual conformal invariance in the perturbative structure of form factors in N=4 SYM was the object of an active debate and sceptical comments were available in the literature. This project has shown the explicit mechanism for the realization of this symmetry. Furthermore, the discovery of a loop recursion relation for form factors allows for the development of new powerful techniques for the computation of loop-level results. In the area of defect field theories, the defect interpretation of Wilson lines has led to the achievement of exact results for the radiation emitted by an accelerating particle for supersymmetric theories in three and four dimensions. Such a radiation, often called Bremsstrahlung, is one of the simplest and most fundamental processes of physical interest and the achievement of exact results, despite in the context of unphysical theories, is of primary importance to improve our understanding of quantum field theories in their non-perturbative regime.

From a technical point of view, the researcher has had useful interactions with internationally leading figures in different fields, thus enlarging his expertise, allowing him to make relevant contribution in a wide range of topics. The ability to compute scattering amplitudes and form factors in various regimes of the AdS/CFT correspondence has been supplemented with a renewed interest in the field of conformal field theories in the presence of extended excitations. Being purely theoretical in nature, this project does not have an immediate and measurable impact on our societal needs. Nevertheless, the non-perturbative understanding of quantum field theories will give us a deeper and more precise picture of the way fundamental particles interact and, in a long term perspective, it will give us a better understanding of the fundamental laws of nature.