Left-right symmetry, orbifold $S^1/Z_2$, and radiative breaking of $U(1)_{\rm R} \times U(1)_{\rm B-L}$

We study the origin of electroweak symmetry under the assumption that $SU(4)_{\rm C} \times SU(2)_{\rm L} \times SU(2)_{\rm R}$ is realized on a five-dimensional space-time. The Pati-Salam type gauge symmetry is reduced to $SU(3)_{\rm C} \times SU(2)_{\rm L} \times U(1)_{\rm R} \times U(1)_{\rm B-L}$ by orbifold breaking mechanism on the orbifold $S^1/Z_2$. The breakdown of residual gauge symmetries occurs radiatively via the Coleman-Weinberg mechanism, such that the $U(1)_{\rm R} \times U(1)_{\rm B-L}$ symmetry is broken down to $U(1)_{\rm Y}$ by the vacuum expectation value of an $SU(2)_{\rm L}$ singlet scalar field and the $SU(2)_{\rm L} \times U(1)_{\rm Y}$ symmetry is broken down to the electric one $U(1)_{\rm EM}$ by the vacuum expectation value of an $SU(2)_{\rm L}$ doublet scalar field regarded as the Higgs doublet. The negative Higgs squared mass term is originated from an interaction between the Higgs doublet and an $SU(2)_{\rm L}$ singlet scalar field as a Higgs portal. The vacuum stability is recovered due to the contributions from Kaluza-Klein modes of gauge bosons.