$W$ Boson Mass and Grand Unification via the Type-$\mathrm{I}\mathrm{I}$ Seesaw-like Mechanism

We propose an SU(5) GUT model extended with two additional pairs of $\mathbf{10}$ representation vector-like fermions. The CDF collaboration $W$ boson mass anomaly is explained by using the VEV of a real $\mathrm{S}\mathrm{U}(2{)}_{\mathrm{L}}$ triplet scalar coming from the $\mathbf{24}$ representation Higgs. The vector-like fermions are decomposed partly into vector-like quark doublets. Those vector-like quark doublets acquire mass from two sources; through the Yukawa interaction with the real $\mathrm{S}\mathrm{U}(2{)}_{\mathrm{L}}$ triplet via a type-$\mathrm{I}\mathrm{I}$ seesaw-like mechanism. And, they acquire mass from the $\mathbf{24}$ representation Higgs. We assume that the mass for the vector-like quark doublets is expressed in terms of the real triplets mass. By combining the constraints on the vector-like quark masses with those on the heavy Higgs boson masses, we can obtain the narrow allowed mass ranges for the vector-like quark doublet and the real triplet. Therefore, our model can be tested in searches for these particles in the near future. In addition, the two additional pairs of vector-like fermions allow the SM gauge couplings to unify successfully at $M_{\mathrm{G}\mathrm{U}\mathrm{T}}\approx 5.1\times {10}^{15}$~GeV. Our model is also testable by the future Hyper-Kamiokande experiment via the proton decay lifetime ${\tau }_p(p\to {\pi }^0{e}^{+})<1.0\times {10}^{35}$~years.