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?
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$\begingroup$ Have a look at my answer here: physics.stackexchange.com/a/267503/84488. There is no reason why there should be more than one Higgs field. $\endgroup$– gentedCommented Aug 7, 2016 at 22:11
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$\begingroup$ Thanks for asking the question, I tried a few versions of the same question as this but received no answers. Delighted to get somewhere at last on this. $\endgroup$– user108787Commented Aug 8, 2016 at 0:45
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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.
answered Aug 7, 2016 at 22:52
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$\begingroup$ What do your $H, \phi_{1,2}$ and $W^{\pm},Z$ represent? Why does the question have anything to do with absorption, radiation and longitudinal components? :o $\endgroup$– gentedCommented Aug 7, 2016 at 22:59
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1$\begingroup$ I think this is beginning to make more sense. I gathered from Grifiths inteoduction to elementary particles that, after symmetry breaking of the higgs field, the lagrangian is rewritten so that the gauge bosons have a mass term and we must pick a gauge to eliminate the existence of a goldstone boson. you're saying that if $\phi=\langle \phi_1+i\phi_2,\phi_3+i\phi_4\rangle$ then a linear combination of the first two components is absorbed by the W fields, taking away those two degrees of freedom, while one of the 3rd/4th components is absorbed by the Z field leaving only one degree of freedom $\endgroup$ Commented Aug 7, 2016 at 23:36
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1$\begingroup$ And the remaining degree of freedom gives rise to the observable higgs boson at high energies? If I'm properly understanding this so far I would love to see a more mathematically rigorous explanation. $\endgroup$ Commented Aug 7, 2016 at 23:38
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$\begingroup$ Basically that is it. I don't know why I got down voted on this. Yes, Gennaro the Goldstone boson degree of freedom goes into a longitudinal mode, which is due to mass. $\endgroup$ Commented Aug 8, 2016 at 0:19
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1$\begingroup$ There is the old book by Taylor "Gauge Theory of Weak Interactions." Any QFT book will do. Essentially if you have a spinning particle, say $\ell~=~1$, then if it is massive there is no $m~=~0$ state, because there must be a rest mass for the particle. A photon has no such frame. If there is an $m~=~0$ state the particle has mass and its wave has a longitudinal component. This is important because in old physics such gauge bosons at high energy have problems. In effect the longitudinal field has causal interaction greater than the speed of light. $\endgroup$ Commented Aug 8, 2016 at 2:21