Computation of Quark Masses from String Theory

TL;DR

This work demonstrates a proof-of-concept for computing physical Yukawa couplings in a heterotic string model by numerically constructing the required geometric data with neural networks. The authors develop a reference-quantity framework (Fubini–Study metric, ambient bundle data) and train networks to obtain the Ricci-flat Calabi–Yau metric, Hermitian-Yang-Mills bundle metrics, and harmonic bundle-valued forms, enabling the calculation of up-type Yukawas along a complex-structure locus. They find Yukawa predictions within about 10% of analytic expectations and show that matter-field normalizations significantly affect hierarchical structure, while degeneracies due to extra U(1) symmetries can be lifted away from symmetric loci. The study establishes a scalable methodology for evaluating fermion masses in broad classes of string compactifications and points to future work extending to larger moduli spaces, non-Abelian bundles, and F-theory contexts, with implications for connecting string geometry to Standard Model flavour. Overall, this work provides a concrete computational bridge between intricate string geometry and phenomenological Yukawa structures, paving the way for systematic searches for realistic flavor patterns.

Abstract

We present a numerical computation, based on neural network techniques, of the physical Yukawa couplings in a heterotic string theory compactification on a smooth Calabi-Yau threefold with non-standard embedding. The model belongs to a large class of heterotic line bundle models that have previously been identified and whose low-energy spectrum precisely matches that of the MSSM plus fields uncharged under the Standard Model group. The relevant quantities for the calculation, that is, the Ricci-flat Calabi-Yau metric, the Hermitian Yang-Mills bundle metrics and the harmonic bundle-valued forms, are all computed by training suitable neural networks. For illustration, we consider a one-parameter family in complex structure moduli space. The computation at each point along this locus takes about half a day on a single twelve-core CPU. Our results for the Yukawa couplings are estimated to be within 10% of the expected analytic result. We find that the effect of the matter field normalisation can be significant and can contribute towards generating hierarchical couplings. We also demonstrate that a zeroth order, semi-analytic calculation, based on the Fubini-Study metric and its counterparts for the bundle metric and the bundle-valued forms, leads to roughly correct results, about 25% away from the numerical ones. The method can be applied to other heterotic line bundle models and generalised to other constructions, including to F-theory models.

Figures (7)

• Figure 1: Monge-Ampère loss for the neural network approximating the function $\phi$ defined in Eq. \ref{['phidef']}. For this example, we have chosen $\psi_0=2$ and $\psi=1$.
• Figure 2: Transition loss for the neural network approximating the function $\phi$ defined in Eq. \ref{['phidef']}. For this example, we have chosen $\psi_0=2$ and $\psi=1$.
• Figure 3: A typical HYM loss, $\mathscr{L}_{\rm{HYM}}$, as given in Eq. \ref{['eqBetaLoss']}, for the neural network approximating the function $\beta$ defined in Eq. \ref{['betaeq']}, and the line bundle $\mathcal{L}_2$. For this example, we have chosen $\psi_0=2$ and $\psi=1$.
• Figure 4: A typical transition loss, $\mathscr{L}_{\rm{transition}}$, as given in Eq. \ref{['eqBetaLoss']}, for the neural network approximating the function $\beta$ defined in Eq. \ref{['betaeq']}, for the line bundle $\mathcal{L}_2$. For this example, we have chosen $\psi_0=2$ and $\psi=1$.
• Figure 5: A typical Laplacian loss, $\mathscr{L}_{\Delta}$, as in Eq. \ref{['eqSigmaLoss']}, for the neural network approximating the section $\sigma$ defined in Eq. \ref{['sigmaeq']}, for the case of the up-Higgs. For this example, we have chosen $\psi_0=2$ and $\psi=1$.