There is an aspect of the Higgs mechanism I am troubled about. This applies to any theory with Higgs, but for sake of definiteness I will restrict the discussion to the Standard Model. The (unbroken) theory starts with a complex Higgs doublet, which corresponds to four degrees of freedom. Then Higgs develops a vacuum expectation value, and through the usual well known machinery three of these four degrees of freedom are absorbed in the longitudinal components of the gauge bosons. The two (oppositely) charged Higgs get absorbed in W+ and W-, and one combination of the neutral Higgs gets absorbed in Z. Before the symmetry breakdown, the gauge bosons were massless, and they had only two physical (transverse) components. After the symmetry breakdown, they acquire a new degree of freedom from the Higgs field, which shows up as the longitudinal component.(adsbygoogle = window.adsbygoogle || []).push({});

Now, why does that not break the conservation of Angular momentum J? To be sure, the third component of the angular momentum, J_{3}is OK in this case: J_{3}= 0, and it is conserved. But J^{2}appears not to be conserved. How did a scalar degree of freedom in Higgs became a part of the three degrees of freedom in a spin-1 particle without violating J^{2}conservation? I had believed all these years that the longitudinal component of a spin-1 boson is not equivalent to a scalar. Is this belief wrong? Is it not possible to distinguish it from a scalar? And yet in Higgs mechanism it appears to have come from a scalar.

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# Higgs mechanism and Angular momentum

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