hypercharge

Hypercharge, denoted Y, is a quantum number associated with the U(1)_Y gauge symmetry of the Standard Model of particle physics. It combines with the SU(2)_L weak isospin to mediate electroweak interactions. The electric charge Q of a particle is related to its weak isospin T3 and hypercharge by Q = T3 + Y/2. Hypercharge interactions are carried by the U(1)_Y gauge field, commonly labeled B_mu, with coupling constant g'.

In the electroweak theory, the SU(2)_L × U(1)_Y symmetry is spontaneously broken to U(1)_em, and the gauge fields B_mu and W^3_mu mix to produce the photon A_mu and the Z boson. The photon remains massless and mediates electromagnetic interactions, while the Z boson acquires mass through the Higgs mechanism.

Hypercharge assignments in the Standard Model (one generation) are as follows: the left-handed lepton doublet L_L = (ν_L, e_L) has Y = -1; the right-handed electron e_R has Y = -2. The left-handed quark doublet Q_L = (u_L, d_L) has Y = 1/3; the right-handed up quark u_R has Y = 4/3; the right-handed down quark d_R has Y = -2/3. The Higgs doublet φ has Y = 1. Right-handed neutrinos are not included in the minimal Standard Model, so they do not carry hypercharge.

Hypercharge values are chosen to cancel gauge anomalies across a full generation, ensuring quantum consistency. In grand unified theories, hypercharge emerges as a generator of a larger gauge group, with different normalizations used in various contexts.