Publications of the Department
Balli, FILIPPO MARIA, (2023) - Il Formalismo Worldline incontra il metodo Perturbiner: Calcolo di Ampiezze di Scattering in Teorie di Gauge e Gravitą - , Tesi di dottorato - (, , Universitą degli studi di Modena e Reggio Emilia ) - pagg. -
Abstract: In quantum field theory, arguably the most important class of observables which can be studied are scattering amplitudes, i.e. probability amplitudes for processes of scattering of particles (or strings) off each other. Scattering amplitudes are computed in perturbation theory as a sum of Feynman diagrams, mathematical quantities that depend on the nature of particles involved in the scattering process. In gauge theories, the individual Feynman diagrams can be factorized into a colour part, depending on the structure of the gauge group, and a kinematic part, depending on the momenta and polarizations of the external particles. In the recent years, the color-kinematics duality by Bern, Carrasco, and Johansson (BCJ) has been discovered for gauge theories. It represents a duality where, diagram-by diagram, the kinematic factors are in a representation such that they exhibit the same algebraic structures as their color counterparts. When organised in such a representation, the color factors can be replaced by another copy of kinematic factors. This procedure is known as double-copy and the resulting scattering amplitudes are amplitudes for gravitational theories. The great advantage of the double-copy construction is that, once a suitable representation in gauge theory has been achieved, gravitational amplitudes are computed automatically, and the complexity in the diagrammatic computation directly from Einstein Lagrangian is overcome. In this thesis, we detail recent results addressing BCJ duality and double-copy in the context of the worldline formalism. The worldline formalism represents an equivalent but independent way to study relativistic quantum mechanics with respect to the canonical quantum field theory. Essentially, in the worldline approach the scattering amplitudes are described no more through path integrals over fields but through path integrals over particle coordinates, i.e. integrals over space-time paths (worldline). In the recent years, the worldline approach has had a promising development and it is now used as a powerful tool for the computation of amplitudes at tree- and loop-level. In the first part of this manuscript, we review the way the worldline formalism is used in the computation of dressed propagators. In particular, tree-level scattering amplitudes for a scalar particle coupled to an arbitrary number of photons and a single graviton are computed using worldline techniques. Specifically, we consider the case of a scalar propagator dressed with two photons and one graviton, and, as the amplitude is fully off-shell, we use it to sew together the two external photons and to construct one-loop radiative corrections to the scalar-scalar-graviton vertex. We test our construction by verifying the on-shell gauge and Ward identities. In the second part of the thesis, we develop a novel procedure to construct Berends-Giele (BG) currents using the worldline formalism for one-loop gluon amplitudes (Bern-Kosower formalism). BG currents are fundamental building blocks for on-shell amplitudes in non-abelian gauge theory: applying the so-called pinch procedure of the BK formalism to a suitable special case, the currents are naturally obtained in terms of multi-particle fields in a colour-kinematic-dual representation. Using the same construction from the worldline Bern-Dunbar-Shimada formalism for one-loop gravity amplitudes, we naturally obtain gravity multi-particle polarisation tensors as tensor product of multi-particle fields in the BCJ gauge. This allows us to formulate a new prescription for double-copy gravity BG currents, and to obtain both the colour-dressed Yang-Mills BG currents in the BCJ gauge and the gravitational BG currents explicitly.