Physics and Astronomy

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This is the collection for the University of Waterloo's Department of Physics and Astronomy.

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Browsing Physics and Astronomy by Author "Cachazo, Freddy"

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Now showing 1 - 4 of 4
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    Aspects of Scattering Amplitudes and Moduli Space Localization
    (University of Waterloo, 2019-08-19) Mizera, Sebastian; Cachazo, Freddy; Dittrich, Bianca
    We propose that intersection numbers of certain cohomology classes on the moduli space of genus-zero Riemann surfaces with $n$ punctures, $\mathcal{M}_{0,n}$, compute tree-level scattering amplitudes in quantum field theories with a finite spectrum of particles. The relevant cohomology groups are twisted by representations of the fundamental group $\pi_1(\mathcal{M}_{0,n})$ that describes how punctures braid around each other on the Riemann surface. Such a structure can be used to link the space of Riemann surfaces with the space of kinematic invariants. Intersection numbers of said cohomology classes—whose representatives we call twisted forms—can be shown to fully localize on the boundaries of $\mathcal{M}_{0,n}$, which are in a one-to-one correspondence with trivalent trees that have an interpretation as Feynman diagrams. In this work we develop systematic approaches towards accessing such boundary information. We prove that when twisted forms are logarithmic, their intersection numbers have a simple expansion in terms of trivalent Feynman diagrams weighted by residues, allowing only for massless propagators on the internal and external lines. It is also known that in the massless limit intersection numbers have a different localization formula on the support of so-called scattering equations. Nevertheless, for physical applications one also needs to study non-logarithmic forms as they are responsible for propagation of massive states. We utilize the natural fibre bundle structure of $\mathcal{M}_{0,n}$—which allows for a direct access to the boundaries—to introduce recursion relations for intersection numbers that "integrate out" puncture-by-puncture. The resulting recursion involves only linear algebra of certain matrices describing braiding properties of $\mathcal{M}_{0,n}$ and evaluating one-dimensional residues, thus paving a way for explicit analytic computations of scattering amplitudes. Together with a reformulation of the tree-level S-matrix of string theory in terms of twisted forms, the results of this work complete a unified geometric framework for studying scattering amplitudes from genus-zero Riemann surfaces. We show that a web of dualities between different homology and cohomology groups allows for deriving a host of identities among various types of amplitudes computed from the moduli space, which in this setup become a consequence of linear algebra. Throughout this work we emphasize that algebraic computations can be supplemented—or indeed replaced—by combinatorial, geometric, and topological ones.
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    Bootstrapping Quantum Field Theories
    (University of Waterloo, 2024-01-23) Homrich, Alexandre; Vieira, Pedro; Cachazo, Freddy
    This thesis compiles a few developments on the S-matrix bootstrap, conformal field theory, and integrability in N = 4 SYM. After an introduction contextualizing the various works that compose this thesis, we present a number of results in independent chapters, followed by a conclusion discussing some future directions.
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    Explorations in the Conformal Bootstrap
    (University of Waterloo, 2017-07-19) Mazac, Dalimil; Gaiotto, Davide; Cachazo, Freddy
    We investigate properties of various conformally invariant quantum systems, especially from the point of view of the conformal bootstrap. First, we study twist line defects in three-dimensional conformal field theories. Numerical results from lattice simulations point to the existence of such conformal defect in the critical 3D Ising model. We show that this fact is supported by both epsilon expansion and the conformal bootstrap calculations. We find that our results are in a good agreement with the numerical data. We also make new predictions for operator dimensions and OPE coefficients from the bootstrap approach. In the process we derive universal bounds on one-dimensional conformal field theories and conformal line defects. Second, we analyze the constraints imposed by the conformal bootstrap for theories with four supercharges in spacetime dimension between 2 and 4. We show how superconformal algebras with four Poincaré supercharges can be treated in a formalism applicable to any, in principle continuous, value of d and use this to construct the superconformal blocks for any dimension between 2 and 4. We then use numerical bootstrap techniques to derive upper bounds on the conformal dimension of the first unprotected operator appearing in the OPE of a chiral and an anti-chiral superconformal primary. We obtain an intriguing structure of three distinct kinks. We argue that one of the kinks smoothly interpolates between the d=2, N=(2, 2) minimal model with central charge c=1 and the theory of a free chiral multiplet in d=4, passing through the critical Wess-Zumino model with cubic superpotential in intermediate dimensions. Finally, we turn to the question of the analytic origin of the conformal bootstrap bounds. To this end, we introduce a new class of linear functionals acting on the conformal bootstrap equation. In 1D, we use the new basis to construct extremal functionals leading to the optimal upper bound on the gap above identity in the OPE of two identical primary operators of integer or half-integer scaling dimension. We also prove an upper bound on the twist gap in 2D theories with global conformal symmetry. When the external scaling dimensions are large, our functionals provide a direct point of contact between crossing in a 1D CFT and scattering of massive particles in large AdS. In particular, CFT crossing can be shown to imply that appropriate OPE coefficients exhibit an exponential suppression characteristic of massive bound states, and that the 2D flat-space S-matrix should be analytic away from the real axis.
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    The S-Matrix of Gauge and Gravity Theories and The Two-Black Hole Problem
    (University of Waterloo, 2020-10-01) Guevara, Alfredo; Cachazo, Freddy
    This thesis is devoted to diverse aspects of scattering amplitudes in gauge theory and gravity including interactions with matter particles. In Part I we focus on the applications of massive scattering amplitudes in gravity to the Black Hole two-body problem. For this we construct a classical limit putting especial emphasis on the multipole expansion of certain massive amplitudes, which we will use to model spinning black holes in a large distance effective regime or particle approximation. In Part II we study scattering amplitudes in six dimensions, and construct a compact formula analogous to the four-dimensional Witten-RSV/rational maps formulation. This provides a supersymmetric extension of moduli space localization formulae such as the CHY integral. We explore the cases of Super Yang-Mills and Maximal Supergravity theories, among others.