The origin of the mass of the Nambu–Goldstone bosons

We explain the origin of the mass for the Nambu–Goldstone bosons when there is a chemical potential in the action which explicitly breaks the symmetry. The method is based on the number of independent histories for the interaction of the pair of Nambu–Goldstone bosons with the degenerate vacuum (triangle relations). The analysis suggests that under some circumstances, pairs of massive Nambu–Goldstone bosons can become a single degree of freedom with an effective mass defined by the superposition of the individual masses of each boson. Possible mass oscillations for the Nambu–Goldstone bosons are discussed.