High-dimensional inverse problems in Beyond-the-Standard-Model physics

How can we solve the hierarchy problems of the electroweak scale and the cosmological constant? Despite many elaborate attempts of identifying appropriate mathematical structures such as additional fundamental symmetries or extra-dimensions, we have no complete picture of mathematical frameworks to address these big questions.

In the long-term we want to address this question using AI-methods which promise a more efficient exploration than human theoretical physicists can achieve.

The aim of this thesis is to understand mass hierarchies in a well-defined model space, i.e. in the context of flux compactifications of string theory. In this setting mass hierarchies can be studied in the spectrum of moduli fields (describing the dynamics

of extra-dimensions), i.e. hierarchical suppression to the typical mass scale of the problem, the string scale. Such hierarchies are straight-forwardly implemented in phenomenological models addressing for instance the electroweak hierarchy problem (e.g. they set the scale of supersymmetry breaking).

On a methodological level this example is perfectly suited because:

- We can scale the dimensionality from low-dimensional settings to currently numerically intractable solutions in high dimensions.

- Human strategies to generate hierarchies do exist and we want to figure out whether there are different strategies or whether we can prove that there are no other ways to generate hierarchies.

If we find new ways of generating hierarchies, we will understand their structure using numerical tools (e.g. using appropriate methods such as ones we used to automatically discover symmetries). In the other scenario, when we can exclude further scenarios

for hierarchies we understand how we can exclude further solutions to this inverse problem using tools such as automated theorem proving.

Both outcomes promise to impact beyond the Standard Model Physics and theoretical physics model building more generally as they enable for the first time the end-to-end automatic exploration of solutions to inverse problems where one is interested in constructing or excluding models to address a particular observable phenomenon.