SU2×U1Y

SU(2) × U(1)Y is the gauge group of the electroweak sector of the Standard Model before electroweak symmetry breaking. It is the direct product of the non-abelian SU(2) group with the abelian U(1) hypercharge group, with corresponding gauge fields W^aμ (a = 1,2,3) and Bμ, and couplings g and g′. The Lie algebra is su(2) ⊕ u(1).

In the Standard Model, fermions transform under representations of SU(2) × U(1)Y according to their chirality. Left-handed fermions occur in SU(2) doublets with weak hypercharge Y, while right-handed fermions are SU(2) singlets with their own Y. Electric charge is given by Q = T3 + Y/2, where T3 is the third component of weak isospin. This assignment yields the observed charges of leptons and quarks, such as L_L = (νL,eL) with Y = −1, eR with Y = −2, Q_L = (uL,dL) with Y = 1/3, uR with Y = 4/3, and dR with Y = −2/3.

Gauge interactions are described by the covariant derivative Dμ = ∂μ − i g T^a W^aμ − i g′(Y/2) Bμ. The SU(2) × U(1)Y symmetry is spontaneously broken to U(1)em by the Higgs field, a complex SU(2) doublet with Y = 1/2. This breaking yields the massive W± and Z bosons and the massless photon A. The mixing is governed by the Weinberg angle θW, with tan θW = g′/g and sin²θW = g′²/(g² + g′²). Mass relations at tree level include MW = (1/2) g v and MZ = MW / cos θW, where v is the Higgs vacuum expectation value.

The SU(2) × U(1)Y structure is central to electroweak interactions, parity violation, and the unification of electromagnetic and weak forces in the Standard Model. The group is a direct product, not a simple group, and its hypercharge assignments ensure anomaly cancellation across fermion generations.