Exploring the Truth and Beauty of Theory Landscapes with Machine Learning

TL;DR

We address the problem of selecting Yukawa-sector textures in the SM that are simultaneously true to current measurements and aesthetically pleasing, i.e., compliant with the CKM matrix \(V_{\rm CKM}\) and quark masses \(m_i^{u,d}\). The authors formulate dedicated losses, including \(L_{\rm CKM}\) and \(L_{\rm Jarlskog}\), and three beauty losses for uniform, zero, and symmetric textures, optimizing a reduced 30-parameter space derived from \(M_u=U_u^\dagger M'_u K_u\) and \(M_d=U_d^\dagger M'_d K_d\) with diagonal entries fixed to quark masses. Across ten pseudo-experiments, the method yields mass matrices \(M_u\) and \(M_d\) that exhibit (i) uniform magnitudes, (ii) sparse zero patterns, or (iii) explicit symmetry, while reproducing the CKM data and CP violation quantified by \(J\). This ML-inspired framework offers a transparent, quantitative path to explore theory-building criteria and can be extended to the lepton sector and other observables.

Abstract

Theoretical physicists describe nature by i) building a theory model and ii) determining the model parameters. The latter step involves the dual aspect of both fitting to the existing experimental data and satisfying abstract criteria like beauty, naturalness, etc. We use the Yukawa quark sector as a toy example to demonstrate how both of those tasks can be accomplished with machine learning techniques. We propose loss functions whose minimization results in true models that are also beautiful as measured by three different criteria - uniformity, sparsity, or symmetry.

Figures (9)

• Figure 1: A schematic depiction of the quantitative measure of "beauty" of a theory model as a function of the model parameters $P_1$ and $P_2$. In the absence of any experimental data, model "1" is the most beautiful of all. An experimental measurement of a single observable $O$ enforces a constraint (shown with the dashed line) among the model parameters. Model "2" is the most beautiful model which is consistent with experiment. Model "3" is an example of a true and not-as-beautiful model, while model "4" is an example of a model which is neither true nor beautiful.
• Figure 2: The trained total loss values (see Eq. \ref{['eq:lossuniform']}) along with the detailed breakdown into the four individual components (Eqs. \ref{['eq:loss_CKM']}, \ref{['eq:loss_Jarlskog']}, \ref{['eq:lossuniform_up']}, and \ref{['eq:lossuniform_down']}) are presented for ten representative pseudo-experiments. These experiments are part of the uniform texture analysis discussed in Section \ref{['subsec:democratic']}.
• Figure 3: The learned mass matrices $M_u$ (upper panels) and $M_d$ (lower panels), corresponding to the uniform Yukawa texture case discussed in Sec. \ref{['subsec:democratic']}. Each panel illustrates a specific learned matrix, with the color bar denoting the values of the individual elements of the matrix.
• Figure 4: The zero texture patterns examined in the example of Sec. \ref{['subsec:heterogeneous']}, with circles marking the matrix elements set to zero.
• Figure 5: As in Fig. \ref{['fig:lossuniform']}, it illustrates the trained loss values and their components for the ten $N=3$ zero texture patterns from Fig. \ref{['fig:patterns']}, averaged across 10 different pseudo-experiments.