Towards a Systematic Construction of Realistic D-brane Models on a del Pezzo Singularity

TL;DR

The paper tackles the challenge of constructing a realistic string-derived Standard Model by employing a local, bottom-up framework of D-branes at toric del Pezzo singularities, arguing that the $dP_3$ geometry is the minimal toric setting capable of producing the required Yukawa hierarchies and a Pati-Salam extension. It presents an explicit $dP_3$-based model in which D3-D3 states realize the MSSM spectrum (with a local $B-L$ gauge symmetry or extra Higgs fields depending on breaking patterns), achieve calculable quark and lepton masses with CKM/PMNS mixing, and exhibit suppressed proton-decay operators via anomalous U(1) symmetries and hypercharge embedding from non-abelian factors. Non-perturbative effects through E3-branes generate a mu-term, while right-handed sneutrino vevs induce a seesaw mechanism for neutrino masses; gauge coupling unification is shown to be achievable across a range of scales ($M_s o 10^{12}$–$10^{16}$ GeV) depending on the breaking sequence and threshold effects. The work demonstrates that a concrete, fully stringy construction can reproduce key SM features and predicts observable implications, such as nonzero $ heta_{13}$ in the PMNS matrix and potential TeV-scale remnants, while also outlining crucial open questions about dynamical vev stabilization and global embedding.

Abstract

A systematic approach is followed in order to identify realistic D-brane models at toric del Pezzo singularities. Requiring quark and lepton spectrum and Yukawas from D3 branes and massless hypercharge, we are led to Pati-Salam extensions of the Standard Model. Hierarchies of masses, flavour mixings and control of couplings select higher order del Pezzo singularities, minimising the Higgs sector prefers toric del Pezzos with dP3 providing the most successful compromise. Then a supersymmetric local string model is presented with the following properties at low energies: (i) the MSSM spectrum plus a local B-L gauge field or additional Higgs fields depending on the breaking pattern, (ii) a realistic hierarchy of quark and lepton masses and (iii) realistic flavour mixing between quark and lepton families with computable CKM and PMNS matrices, and CP violation consistent with observations. In this construction, kinetic terms are diagonal and under calculational control suppressing standard FCNC contributions. Proton decay operators of dimension 4, 5, 6 are suppressed, and gauge couplings can unify depending on the breaking scales from string scales at energies in the range 10^{12}-10^{16} GeV, consistent with TeV soft-masses from moduli mediated supersymmetry breaking. The GUT scale model corresponds to D3 branes at dP3 with two copies of the Pati-Salam gauge symmetry SU(4)\timesSU(2)R\timesSU(2)L. D-brane instantons generate a non-vanishing mu-term. Right handed sneutrinos can break the B-L symmetry and induce a see-saw mechanism of neutrino masses and R-parity violating operators with observable low-energy implications.

Figures (6)

• Figure 1: The $dP_3$ quiver with fields labelled in correspondence to the superpotential in Equation \ref{['eq:dp3superpotential']}.
• Figure 2: The quiver diagram of the Pati-Salam realisation on $dP_3.$ The fields $(\Phi_{61},\Psi_{42},\rho_{53})$ will be responsible for gauge group breaking, combining the two copies of every gauge group. The other fields are Standard Model fields including three generation of Higgs fields.
• Figure 3: The $dP_3$ quiver with $D7$ branes included. Indicated in red are the fields that we vev to break the $U(4)\times U(4)$ symmetry, the $D3-D7$ states need vevving to satisfy D-term equation. Similarly the states which break the left and right $U(2)$ symmetries are indicated in purple and orange. The states highlighted in green are $D7$ states that we could vev in accordance with the D-term equations to give large masses to some of the Higgs fields.
• Figure 4: The $D3-D3$ and $D3-D7$ spectrum in the Pati-Salam model. The ranks for the $D7$ branes are chosen to satisfy the anomaly constraints and allow for breaking to the Standard Model gauge group.
• Figure 5: On the left side the breakdown from the $dP_3$ gauge groups to $U(1)_Y$ is shown. The vertices denote the associated breaking scales. At each of these breaking scales we need to apply the matching condition from Equation \ref{['boundarydiagonal']} for the inverse gauge couplings.