Twin Pati-Salam theory of flavour with a TeV scale vector leptoquark

We propose a twin Pati-Salam (PS) theory of flavour broken to the G4321 gauge group at high energies, then to the Standard Model at low energies, yielding a TeV scale vector leptoquark $$ {U}_1^{\mu } $$ U 1 μ (3, 1, 2/3) which has been suggested to address the lepton universality anomalies $$ {R}_{K^{\left(\ast \right)}} $$ R K ∗ and $$ {R}_{D^{\left(\ast \right)}} $$ R D ∗ in B decays. Quark and lepton masses are mediated by vector-like fermions, with personal Higgs doublets for the second and third families, which may be replaced by a two Higgs doublet model (2HDM). The twin PS theory of flavour successfully accounts for all quark and lepton (including neutrino) masses and mixings, and predicts a dominant coupling of $$ {U}_1^{\mu } $$ U 1 μ (3, 1, 2/3) to the third family left-handed doublets. However the predicted mass matrices, assuming natural values of the parameters, are not consistent with the single vector leptoquark solution to the $$ {R}_{D^{\left(\ast \right)}} $$ R D ∗ anomaly, given its current value.