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$SU(5)$ grand unification and $W$-boson mass

A realistic extension of the minimal $SU(5)$ theory consisting of the addition of an adjoint fermion is known to predict light real fermion and scalar weak triplets, potentially accessible at the LHC. These particles, in addition to playing a key role in gauge coupling unification, have profound phenomenological implications. The fermion triplet, that through the seesaw mechanism offers a testable origin of neutrino mass, has been already extensively discussed. The scalar triplet develops a vacuum expectation value that modifies the $W$-boson mass. We show that its low-energy effective theory is remarkably predictive: in the leading approximation, all the relevant physical processes involving the scalar triplet depend only on its mass and the deviation from the Standard Model $W$-mass value.