Why isn't there two Higgs bosons? As I understand, in the SM Lagrangian the Higgs field, $\phi$ is actually a column vector of two complex scalar fields: $\phi_1+i\phi_2$ and $\phi_3+i\phi_4$. Shouldn't there be a particle corresponding to each of these fields? Are they indistinguishable? Or is there something I missed in the Higgs mechanism?
 A: Let me define $H^\pm~=~\phi_1~\pm~\phi_2$. These two are absorbed into $W^\pm$. The massless fields $W^\pm$ have no longitudinal component, and as such are not subject to the divergence and causality problems a longitudinal field would have at high energy. However, at the transition energy the $H^\pm$ are absorbed into $W^\pm$ and the longitudinal degree of freedom for the now massive $W^\pm$. We can make a similar argument for the other complex scalar, but now one of them is absorbed into the neutral current or $Z$. The now massive $Z$ has a longitudinal degree of freedom gained by absorbing one of the Higgs in the doublet or complex scalar. The other boson left is the photon $\gamma$, that does not absorb the remaining Higgs boson $h_0$. The $h_0$ is what was detected in 2012.
The parts that are absorbed as the Goldstone bosons, and the remaining $h_0$ is the Higgs particle.  This is the reason there is only one Higgs particle. With the minimal supersymmetric theory there are four additional Higgs particles, two charged and two pseudoscalar Higgs. However, the LHC is not finding any hint of low energy supersymmetric theory or supersymmetric standard model.  The ICHEP meeting has concluded so far that nothing has been found. About four decades of low mass/energy supersymmetric phenomenology is on the threshold of being sent to the shredder.
