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  1. Mexican Hat Potential

    Linked via "gauge degree of freedom"

    The Massless Mode ($\phi_2$): This corresponds to the excitation along the tangential direction (moving along the trough). Its mass squared is zero:
    $$m^2{\phi2} = \left. \frac{\partial^2 V}{\partial \phi2^2} \right|{\phi=v} = 0$$
    This mode manifests as the Goldstone boson, which "eats" a gauge degree of freedom during the symmetry-breaking process.