Isospin - Modern Understanding of Isospin

Modern Understanding of Isospin

Observation of the light baryons (those made of up, down and strange quarks) lead us to believe that some of these particles are so similar in terms of their strong interactions that they can be treated as different states of the same particle. In the modern understanding of quantum chromodynamics, this is because up and down quarks are very similar in mass, and have the same strong interactions. Particles made of the same numbers of up and down quarks have similar masses and are grouped together. For examples, the particles known as the Delta baryons—baryons of spin 3⁄₂ made of a mix of three up and down quarks—are grouped together because they all have nearly the same mass (approximately 1,232 MeV/c2), and interact in nearly the same way.

However, because the up and down quarks have different charges (2⁄₃ e and −1⁄₃ e respectively), the four Deltas also have different charges (Δ++ (uuu), Δ+ (uud), Δ0 (udd), Δ− (ddd)). These Deltas could be treated as the same particle and the difference in charge being due to the particle being in different states. Isospin was devised as a parallel to spin to associate an isospin projection (denoted I₃) to each charged state. Since there were four Deltas, four projections were needed. Because isospin was modeled on spin, the isospin projections were made to vary in increments of 1 and to have four increments of 1, you needed an isospin value of 3⁄₂ (giving the projections I₃ = 3⁄₂, 1⁄₂, −1⁄₂, −3⁄₂). Thus, all the Deltas were said to have isospin I = 3⁄₂ and each individual charge had different I₃ (e.g. the Δ++ was associated with I₃ = +3⁄₂). In the isospin picture, the four Deltas and the two nucleons were thought to be the different states of two particles. In the quark model, the Deltas can be thought of as the excited states of the nucleons.

After the quark model was elaborated, it was noted that the isospin projection was related to the up and down quark content of particles. The relation is

where n_u and n_d are the numbers of up and down quarks respectively, and n_u and n_d are the numbers of up and down antiquarks respectively.

By this, the value of I₃ of the nucleons proton (symbol p) and neutron (symbol n) is determined by their quark composition, uud for the proton and udd for the neutron.