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Exact SU(5) Yukawa matrix unification in the General Flavour Violating MSSM

We investigate the possibility of satisfying the $SU(5)$ boundary condition $\mathbf{Y}^d=\mathbf{Y}^{e\,T}$ at the GUT scale within the renormalizable $R$-parity conserving Minimal Supersymmetric Standard Model (MSSM). Working in the super-CKM basis, we consider non-zero flavour off-diagonal entries in the soft SUSY-breaking mass matrices and the $A$-terms. At the same time, the diagonal $A$-terms are assumed to be suppressed by the respective Yukawa couplings. We show that a non-trivial flavour structure of the soft SUSY-breaking sector can contribute to achieving precise Yukawa coupling unification for all the three families. However, large non-zero values of the flavour-violating parameter $(m^2_{dl})_{12}$ lead to a strong tension with the Lepton Flavour Violating (LFV) observables. Nevertheless, the LFV problem does not arise when the Yukawa coupling unification requirement is restricted to the second and third families only. We demonstrate that such a scenario is consistent with a wide set of experimental constraints, including flavour and electroweak observables, Higgs physics and the LHC bounds. We also point out that in order to provide a proper value for the relic density of dark matter, the lightest neutralino needs to be almost purely bino-like and with mass in the range of 200$÷$600 GeV. Such a clear experimental prediction makes the flavour-violating $SU(5)$ Yukawa unification scenario fully testable at the LHC 14 TeV.