Schwarzschild deformed supergravity background: possible geometry origin of fermion generations and mass hierarchy

TL;DR

This paper explores a 10-dimensional supergravity model with a Schwarzschild deformation to explain fermion mass hierarchies and generations, matching observed mass ratios through geometric and boundary condition analysis.

Contribution

It introduces a novel geometric background that naturally produces three fermion generations and accurately predicts their mass ratios.

Findings

Three fermion generations arise from angular modes on the deformed background.

The model reproduces the observed ratio of down and strange quark masses as e^{-3}.

Non-chiral modes follow Schrödinger equation solutions with hypergeometric functions.

Abstract

TThe problem of fermion masses hierarchy in the Standard Model is considered on a toy model of a 10-dimensional space-time with a IIA supergravity type background. Dirac equation on this background, after compactification of extra 4- and 1-dimensional subspaces, gives the spectrum of Fermi fields which profiles in 5 dimensions and corresponding Higgs generated masses in 4 dimensions depend on the eigenvalues of Dirac operator on the named compact subspaces. Schwarzschild Euclidean deformation of the supergravity throat with the "apple-shaped" conical singularity permits to leave only three non-divergent angular modes interpreted as three generations of the down-type quarks. Calculated ratio $m_{d} / m_{s} = e^{-3}$ exactly coincides with its experimentally observed value for integer values of two free parameters of the 10-dimensional background. Equations for non-chiral modes coincide with the non-relativistic Schr\"odinger equation for an electron moving in a Coulomb field; the corresponding small fermion masses generated by the twisted boundary conditions are expressed through the degenerate hypergeometric functions.