A Parameter Space Exploration of the Minimal $SU$(5) Unification

We present a phenomenological study of the most minimal realistic <math display="inline"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></math> model that owns its predictivity solely to the gauge symmetry and the representational content. The model is built entirely out of the fields residing in the first five lowest dimensional representations that transform nontrivially under the <math display="inline"><mrow><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow></math> gauge group. It has 18 real parameters and 14 phases, all in all, to address experimental observables of the Standard Model fermions and accomplishes that via simultaneous use of three different mass generation mechanisms. Furthermore, it inextricably links the origin of the neutrino mass to the experimentally observed difference between the down-type quark and charged lepton masses. The main predictions of the model are that (i) the neutrinos are Majorana particles, (ii) one neutrino is massless, (iii) the neutrinos have normal mass ordering, and (iv) there are four new scalar multiplets at or below a 120 TeV mass scale. A one-loop analysis demonstrates that an improvement of the current <math display="inline"><mi>p</mi><mo stretchy="false">→</mo><msup><mi>π</mi><mn>0</mn></msup><msup><mi>e</mi><mo>+</mo></msup></math> partial lifetime limit by a factor of 2, 15, and 96 would require these four scalar multiplets to reside at or below the 100, 10, and 1 TeV mass scales, respectively.