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I'm reading about the Higgs mechanism considering the Lagrangian: $$ \mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu} + \left(D_\mu\phi\right)\left(D^\mu\phi\right)^\dagger-\mu^2\left(\phi\phi\right)^\dagger-\lambda\left(\phi\phi^\dagger\right)^2 $$

Breaking the symmetry one arrives to the existence of Goldsone bosons: $$ \mathcal{L} = \frac{1}{2}\left(\partial^\mu\eta\right)\left(\partial_\mu\eta\right)-\lambda v^2\eta^2+\frac{1}{2}\left(\partial_\mu\xi\right)\left(\partial^\mu\xi\right)-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\frac{1}{2}q^2v^2A_\mu A^\mu+qvA_\mu\partial^\mu\xi-V_{int}\left(\eta,\xi,A^\mu\right) $$ where $\phi=\frac{1}{\sqrt{2}}(v+\eta(x)+i\xi(x))$

Are these bosons measurable?

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    $\begingroup$ Search phrase: "the pion is the Goldstone boson of QCD." $\endgroup$
    – rob ♦
    Commented Mar 27, 2021 at 15:19
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    $\begingroup$ If the context is the Higgs mechanism, then the would-be Goldstone bosons are "eaten" by the gauge fields, making them massive. So yes, we observe the effect of the would-be Goldstone bosons by observing that the $W$ and $Z$ bosons are massive. Is that what you're asking? $\endgroup$
    – Chiral Anomaly
    Commented Mar 27, 2021 at 16:12
  • $\begingroup$ @ChiralAnomaly you have reason, but I remake the question ¿can you measure a Goldstone boson and a non massive W and Z or only massive W and Z? ¿Are Goldstone bosons a ``mathematical object'' or are present in nature? $\endgroup$
    – Abel Gutiérrez
    Commented Mar 28, 2021 at 18:10
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    $\begingroup$ In the Higgs mechanism, which the body of the question describes, there are no Goldstone bosons (the Goldstones that would have existed get mathematically "eaten" by the fact that the symmetry group is gauged). @rob's comment about pions in QCD is a better example of Goldstone bosons being present in nature as independent objects. They're not exactly massless in that case, but they would be if quark masses were absent. Other examples of Goldstone bosons in nature include phonons. Are you asking about the Higgs mechanism, or true Goldstone bosons? $\endgroup$
    – Chiral Anomaly
    Commented Mar 28, 2021 at 18:48
  • $\begingroup$ I understand, so depends on the situation. My question is related more about the Higgs mechanism, but I'm studying only the basic breaking simmetry U(1), i.e., photon and Higgs boson, no W and Z bosons. Since at some point in the calculations you get a term that is the Goldstone boson, I was wondering if you can measure it before you make a gauge where it desapears. $\endgroup$
    – Abel Gutiérrez
    Commented Mar 30, 2021 at 9:49

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