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In field theory textbooks, it is shown that while any gauge invariant Lagrangian must involve massless gauge fields, to obtain massive gauge bosons, we must postulate the existence of a Higgs scalar field, which causes spontaneous symmetry breaking to occur, and thus gives the gauge bosons mass.

While the math behind this is clear to me, I am unable to understand it intuitively - that is, why spontaneous symmetry breaking, of all things? Is it just a wild guess, which works? Or is that a natural, intuitive way to conclude that to obtain massive bosons, we must have spontaneous symmetry breaking?

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  • $\begingroup$ If you break a global symmetry you have massless Goldstone bosons. Breaking a global symmetry in a model that also has local symmetry with a related massless gauge field: the gauge field "eats" the Goldstone boson and becomes massive (but do not fall into the possible misunderstanding physics.stackexchange.com/q/151182/226902 ). Symmetry breaking becomes intuitive in the context of phase transitions (different phases, different symmetries): "massive photons" appear in the phase transition to the superconducting state of a metal... then, the idea worked also for particle physics. $\endgroup$
    – Quillo
    Commented Feb 11, 2022 at 14:35
  • $\begingroup$ @Quillo but why even think about breaking the global (or local) symmetry at all? $\endgroup$
    – Ishan Deo
    Commented Feb 12, 2022 at 5:28
  • $\begingroup$ It may be very subjective, but I personally find the relation "phase transition - spontaneous symm breaking" very intuitive and natural. Example: What is a liquid-crystal transition if not the spontaneous breaking of the "homogenous" liquid state? The crystal is less symmetric. Same in high energy physics: the field acquires a vacuum expectation value (like magnetization in Ising, say) and you break a symmetry/make a phase transition. $\endgroup$
    – Quillo
    Commented Feb 12, 2022 at 18:34
  • $\begingroup$ related: physics.stackexchange.com/q/694461/226902 $\endgroup$
    – Quillo
    Commented Feb 14, 2022 at 12:19

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