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Does anyone know some theorem or statement about amplitudes involving only Goldstone bosons in theories with spontaneous symmetry breaking in the limit of low energies?

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One very general statement is that if you take one of the external momenta to zero, keeping the other momenta fixed, the amplitude will vanish.

This is simply a consequence of the non-linear (shift) symmetry acting on the Goldstone bosons. This symmetry implies that they can enter the effective Lagrangian only with derivative couplings. Since the derivative becomes multiplication by momentum, when the momentum is zero the Goldstone is decoupled.

If you work a bit more, and take limits in which two momenta vanish, you will find that the amplitudes contains information about the algebra of the spontaneously broken symmetry.

I suggest you to look to Weinberg's book when he discusses pions, or to his original papers (Weinberg's soft pions theorems), or to Chapter 2 of Aspects of Symmetry by Sidney Coleman. A good starting point could even be this blog post: http://golem.ph.utexas.edu/~distler/blog/archives/001787.html

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