<p>We prove that symmetry in the presence of gravity implies a version of the completeness hypothesis. For a broad class of theories, we demonstrate that the existence of finitely many charged particles logically necessitates the existence of infinitely many charged particles populating the entire charge lattice. Our conclusions follow from the consistency of perturbative gravitational scattering and require the following ingredients: (1) a weakly coupled ultraviolet completion of gravity, (2) a non-Abelian symmetry <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>G</mi></math>, gauged or global, whose Cartan subgroup generates the Abelian charge lattice, and (3) a spectrum containing some finite set of charged representations, in the simplest cases taken to be a single particle in the fundamental. Under these conditions, the Abelian charge lattice is completely filled by single-particle states for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>G</mi><mo>=</mo><mi>S</mi><mi>O</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo>≥</mo><mn>5</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>G</mi><mo>=</mo><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo>≥</mo><mn>3</mn></math>, which in turn implies completeness for other symmetry groups such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>p</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>E</mi><mn>8</mn></msub></math>. Curiously, a corollary of our results is that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>O</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></math> grand unified theories have precisely the minimal field content needed to derive completeness using our methodology.</p>

United States

Francesco Calisto

Clifford Cheung

Grant N. Remmen

Spain

Francesco Sciotti

Michele Tarquini

Completeness from gravitational scattering

We perform an exploratory study of a new approach for evaluating Feynman integrals numerically. We apply the recently-proposed framework of physics-informed deep learning to train neural networks to approximate the solution to the differential equations satisfied by the Feynman integrals. This approach relies neither on a canonical form of the differential equations, which is often a bottleneck for the analytical techniques, nor on the availability of a large dataset, and after training yields essentially instantaneous evaluation times. We provide a proof-of-concept implementation within the PyTorch framework, and apply it to a number of one- and two-loop examples, achieving a mean magnitude of relative difference of around 1% at two loops in the physical phase space with network training times on the order of an hour on a laptop GPU.