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Abstract:

Physics seeks to uncover the laws of Nature and express them through mathematical equations . Despite the vast diversity of natural phenomena, physical equations exhibit structural regularities that set them apart from arbitrary mathematical expressions. While principles such as dimensional analysis have long guided the formulation of physical models, the exploration of more subtle statistical patterns within the equations of physics remains an open question. Here, by analysing four corpora of physics equations and applying advanced implicit-likelihood techniques, we find that the frequency of mathematical operators follows an exponential decay law, in contrast to Zipf's power law for word frequencies in natural languages. This reveals a statistical meta-law of physics, possibly reflecting a combination of communication efficiency and constraints imposed by Nature itself. The meta-law offers practical benefits for symbolic regression by drastically narrowing down the space of physically plausible expressions. More broadly, it may inform the development of language models that can generate coherent mathematical representations, advancing the automation of physical law discovery. This article is part of the discussion meeting issue 'Symbolic regression in the physical sciences'.

Abstract:

Deriving the Yukawa couplings and the resulting fermion masses and mixing angles of the Standard Model (SM) from a more fundamental theory remains one of the central outstanding problems in theoretical high-energy physics. It has long been recognized that string theory provides a framework within which this question can, at least in principle, be addressed. While substantial progress has been made in studying flavor physics in string compactifications over the past few decades, a concrete string construction that reproduces the full set of observed SM flavor parameters remains unknown. Here, we take a significant step in this direction by identifying two explicit $E_8×E_8$ heterotic string models compactified on a Calabi-Yau threefold with Abelian, holomorphic, and polystable vector bundles with minimal supersymmetric (MS) SM spectrum. Subject to reasonable assumptions about the moduli, we show that these models reproduce the correct values of the quark and charged lepton masses, as well as the quark mixing parameters, at some point in their moduli spaces. The resulting four-dimensional theories are $\mathcal{N}=1$ supersymmetric, contain no exotic fields, and realize a $\mathrm{μ}$ -term suppressed to the electroweak scale. While the issues of moduli stabilization and supersymmetry breaking are not addressed here; our main result constitutes a proof of principle: There exist choices of topology and moduli within heterotic string compactifications which allow for an MSSM spectrum with the correct flavor parameters.

Abstract:

We study a class of supersymmetric Froggatt-Nielsen (FN) models with multiple U(1) symmetries and Standard Model (SM) singlets inspired by heterotic string compactifications on Calabi-Yau threefolds. The string-theoretic origin imposes a particular charge pattern on the SM fields and FN singlets, dividing the latter into perturbative and non-perturbative types. Employing systematic and heuristic search strategies, such as genetic algorithms, we identify charge assignments and singlet VEVs that replicate the observed mass and mixing hierarchies in the quark sector, and subsequently refine the Yukawa matrix coefficients to accurately match the observed values for the Higgs VEV, the quark and charged lepton masses and the CKM matrix. This bottom-up approach complements top-down string constructions and our results demonstrate that string FN models possess a sufficiently rich structure to account for flavour physics. On the other hand, the limited number of distinct viable charge patterns identified here indicates that flavour physics imposes tight constraints on string theory models, adding new constraints on particle spectra that are essential for achieving a realistic phenomenology.

Abstract:

The contextual fraction introduced by Abramsky and Brandenburger defines a quantitative measure of contextuality associated with empirical models, i.e. tables of probabilities of measurement outcomes in experimental scenarios. In this paper we define an entanglement monotone relying on the contextual fraction. We first show that any separable state is necessarily non-contextual with respect to any Bell scenario. Then, for 2-qubit states, we associate a state-dependent Bell scenario and show that the corresponding contextual fraction is an entanglement monotone, suggesting contextuality may be regarded as a refinement of entanglement. We call this monotone the quarter-turn contextual fraction, and use it to set an upper bound of approximately 0.601 for the minimum entanglement entropy needed to guarantee contextuality with respect to some Bell scenario.