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The Bootstrap Method for General Amplitudes

Periodic Reporting for period 1 - Amplitudes Bootstrap (The Bootstrap Method for General Amplitudes)

Reporting period: 2018-08-15 to 2020-08-14

If physicists want to discover new physics beyond our Standard Model, we must first understand what the Standard Model predicts. For experiments like the Large Hadron Collider at CERN, this means calculating scattering amplitudes, formulas that let us find the probability that two colliding particles will bounce, or “scatter”, off each other in particular ways.

Current methods to calculate scattering amplitudes are very computationally intensive, but a much more efficient option exists: we can “bootstrap” the amplitudes, starting with a guess in terms of the right kind of mathematical functions then constraining it using what we know about the physics of the problem. This method can be extremely effective, but it does rely on some initial knowledge: both of the right functions, and of the relevant physics.

The objective of this project was to broaden the reach of bootstrap methods, both by investigating new types of mathematical function and by using the bootstrap in new physical contexts. In the course of the project I discovered a new class of functions appearing in scattering amplitudes, related to geometric spaces called Calabi-Yau manifolds, and characterized their properties. I also made progress building knowledge of the physics of new contexts to serve as a foundation for future bootstrap methods.
I worked to investigate functions related to Calabi-Yau manifolds which appear in scattering amplitudes. I characterized their appearance in a wide variety of theories, including proving a bound on their complexity, and found common traits of all known examples.

I pushed bootstrap methods to new heights, bringing them beyond the state of the art in a prior physical context. I also investigated conditions when they fail, and how they compare to alternate methods.
I investigated other theories, in particular the gravitational theories used to make predictions for gravitational wave telescopes such as LIGO. I performed more traditional calculations, with the goal of establishing enough physical knowledge to make bootstrap methods viable in that context.

Finally, I have been working to apply bootstrap methods to a novel context, a so-called “correlation function” that presents new challenges beyond those explored for scattering amplitudes.

I disseminated these results to a technical audience via a current total of six published papers in refereed journals and three preprints, which are either currently under review or will be submitted in the near future, as well as presentations at five conferences and two seminar talks. I also disseminated my work to the public via public talks, regular blog posts, and an article in Scientific American magazine.
I discovered previously unknown facts about the Calabi-Yau manifolds that occur in scattering amplitudes, pushing beyond the state of the art in this respect. I pushed beyond the state of the art in precision in my bootstrap calculations, and in my use of other methods for comparison. Finally, my calculations in gravitational theories were also beyond the current state of the art in precision, as are my expected results bootstrapping a correlation function.

The project’s clearest impact has been to my own career, as well as the career of a PhD student I collaborated with and a Master’s student I mentored: each is in a stronger position to find new positions and advance their career, with the Master’s student already accepted for a PhD position in Padua. More generally, this project advances the state of the art in calculating scattering amplitudes. This lets us understand the consequences of the Standard Model with more precision, allowing us to more effectively identify new physics with the potential to transform the way we live our lives.