# Goldstone boson

In particle physics, Goldstone bosons are bosons that appear in models with spontaneously broken symmetry.

In certain supersymmetric models, "Goldstone fermions," or "Goldstinos" also appear.

The simplest model (almost trivial) with a Goldstone boson is as follows:

We have a complex scalar field φ (phi), with the constraint that φ*φ=k2. One way to get a constraint of that sort is by including a potential

[itex]\lambda^2(\phi^*\phi - k^2)^2 \,[itex]

and taking the limit as λ goes to infinity. The field can be redefined to give a real scalar, θ, without a constraint by using

[itex]\phi = k e^{i\theta} \,[itex]

where θ is the Goldstone boson (actually kθ is) with the Lagrangian density given by:

[itex]L=-\frac{1}{2}(\partial^\mu \phi^*)\partial_\mu \phi +m^2 \phi^* \phi = -\frac{1}{2}(-ik e^{-i\theta} \partial^\mu \theta)(ik e^{i\theta} \partial_\mu \theta) + m^2 k^2=-\frac{k^2}{2}(\partial^\mu \theta)(\partial_\mu \theta) + m^2 k^2.[itex]

Note that the constant term m2k2 has no physical significance and the other term is simply the kinetic term for a massless scalar. In general the Goldstone boson is always massless, and parametrises the curve of possible vacuum states.

A somewhat simpler but more detailed presentation of the same concepts is presented in the article on the Yukawa interaction.

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