The difference between a Goldstone boson and a pseudo-Goldstone boson lies in their origin and properties within the context of symmetry breaking in particle physics.
Goldstone bosons: Goldstone bosons are associated with spontaneous symmetry breaking. When a continuous symmetry is spontaneously broken, Goldstone's theorem states that there will be massless particles called Goldstone bosons in the theory. These bosons arise due to the presence of "unstable" directions in the potential energy landscape of the system, resulting in a degenerate ground state.
A well-known example is the Higgs mechanism in the Standard Model of particle physics. The spontaneous breaking of the electroweak symmetry gives rise to three Goldstone bosons: two are absorbed by the W and Z bosons, acquiring mass and becoming the longitudinal components of the weak force carriers, while the remaining one, the Higgs boson, remains as an observable particle.
Pseudo-Goldstone bosons: Pseudo-Goldstone bosons, on the other hand, are associated with approximate or partial symmetry breaking. In certain situations, a symmetry may not be perfectly conserved, but only approximately so. This can happen when there are small explicit symmetry-breaking terms present in the theory.
In such cases, the resulting bosons are not exactly massless but have small masses due to the explicit symmetry-breaking effects. These bosons are known as pseudo-Goldstone bosons. They still retain some of the properties of Goldstone bosons, such as being associated with broken symmetries, but their mass is non-zero.
A well-known example of pseudo-Goldstone bosons is found in the theory of Quantum Chromodynamics (QCD), which describes the strong interaction. QCD exhibits a chiral symmetry, which is approximately conserved in the limit of massless quarks. However, quarks in the real world have small masses, which explicitly break the chiral symmetry. As a result, the pions, which are the pseudo-Goldstone bosons associated with chiral symmetry breaking, acquire small but non-zero masses.
In summary, the distinction between Goldstone bosons and pseudo-Goldstone bosons lies in the nature of symmetry breaking: Goldstone bosons arise from spontaneous symmetry breaking and are exactly massless, while pseudo-Goldstone bosons arise from approximate symmetry breaking and have small non-zero masses due to explicit symmetry-breaking effects.