at Bristol University. Possible applications The use of pions in medical radiation therapy, such as for cancer, was explored at a number of research institutions, including the Los Alamos National Laboratory's Meson Physics Facility, which treated 228 patients between 1974 and 1981 in New Mexico, and the TRIUMF laboratory in Vancouver, British Columbia. Theoretical overview In the standard understanding of the strong force interaction as defined by quantum chromodynamics, pions are loosely portrayed as Goldstone bosons of spontaneously broken chiral symmetry. That explains why the masses of the three kinds of pions are considerably less than that of the other mesons, such as the scalar or vector mesons. If their current quarks were massless particles, it could make the chiral symmetry exact and thus the Goldstone theorem would dictate that all pions have a zero mass. In fact, it was shown by Gell-Mann, Oakes and Renner (GMOR) that the square of the pion mass is proportional to the sum of the quark masses times the quark condensate: , with the quark condensate. This is often known as the GMOR relation and it explicitly shows that in the massless quark limit. The same result also follows from Light-front holography. Empirically, since the light quarks actually have minuscule nonzero masses, the pions also have nonzero rest masses. However, those masses are almost an order of magnitude smaller than that of the nucleons, roughly ≈ ≈ 45 MeV, where are the relevant current-quark masses in MeV, around 5−10 MeV. The pion can be thought of as one of the particles that mediate the residual strong interaction between a pair of nucleons. This interaction is attractive: it pulls the nucleons together. Written in a non-relativistic form, it is called the Yukawa potential. The pion, being spinless, has kinematics described by the Klein–Gordon equation. In the terms of quantum field theory, the effective field theory Lagrangian describing the pion-nucleon interaction is called the Yukawa interaction. The nearly identical masses of and indicate that there must be a symmetry at play: this symmetry is called the SU(2) flavour symmetry or isospin. The reason that there are three pions, , and , is that these are understood to belong to the triplet representation or the adjoint representation 3 of SU(2). By contrast, the up and down quarks transform according to the fundamental representation 2 of SU(2), whereas the anti-quarks transform according to the conjugate representation 2*. With