Symbolic Regression and Differentiable Fits in Beyond the Standard Model Physics

We demonstrate the efficacy of symbolic regression (SR) to probe models of particle physics Beyond the Standard Model (BSM), by considering the so-called Constrained Minimal Supersymmetric Standard Model (CMSSM). Like many incarnations of BSM physics this model has a number (four) of arbitrary parameters, which determine the experimental signals, and cosmological observables such as the dark matter relic density. We show that analysis of the phenomenology can be greatly accelerated by using symbolic expressions derived for the observables in terms of the input parameters. Here we focus on the Higgs mass, the cold dark matter relic density, and the contribution to the anomalous magnetic moment of the muon. We find that SR can produce remarkably accurate expressions. Using them we make global fits to derive the posterior probability densities of the CMSSM input parameters which are in good agreement with those performed using conventional methods. Moreover, we demonstrate a major advantage of SR which is the ability to make fits using differentiable methods rather than sampling methods. We also compare the method with neural network (NN) regression. SR produces more globally robust results, while NNs require data that is focussed on the promising regions in order to be equally performant.

Statistical Patterns in the Equations of Physics and the Emergence of a Meta-Law of Nature

Physics, as a fundamental science, aims to understand the laws of Nature and describe them in mathematical equations. While the physical reality manifests itself in a wide range of phenomena with varying levels of complexity, the equations that describe them display certain statistical regularities and patterns, which we begin to explore here. By drawing inspiration from linguistics, where Zipf's law states that the frequency of any word in a large corpus of text is roughly inversely proportional to its rank in the frequency table, we investigate whether similar patterns for the distribution of operators emerge in the equations of physics. We analyse three corpora of formulae and find, using sophisticated implicit-likelihood methods, that the frequency of operators as a function of their rank in the frequency table is best described by an exponential law with a stable exponent, in contrast with Zipf's inverse power-law. Understanding the underlying reasons behind this statistical pattern may shed light on Nature's modus operandi or reveal recurrent patterns in physicists' attempts to formalise the laws of Nature. It may also provide crucial input for symbolic regression, potentially augmenting language models to generate symbolic models for physical phenomena. By pioneering the study of statistical regularities in the equations of physics, our results open the door for a meta-law of Nature, a (probabilistic) law that all physical laws obey.