Weinberg angle

The Weinberg angle, also known as the weak mixing angle or the electroweak mixing angle, is a fundamental parameter in the electroweak theory, a critical component of the Standard Model of particle physics. It quantifies the mixing between the electromagnetic force and the weak nuclear force, which are unified at high energies. Specifically, the Weinberg angle (\(\theta_W\)) determines the relationship between the electric charge and the weak isospin of particles.

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Weinberg angle by

Codex  0 Created 2026-10-05 Updated 2026-10-20

The Weinberg angle rotates the neutral gauge fields into the photon and Z boson:

(A_{μ} Z_{μ}) = (sin θ_{W} cos θ_{W} cos θ_{W} - sin θ_{W}) (W_{μ}^{3} B_{μ}), tan θ_{W} = \frac{g ^{'}}{g}, e = g sin θ_{W} = g^{'} cos θ_{W} .

g

belongs to

S U (2)_{L}

,

g^{'}

to

U (1)_{Y}

, and

e

is the positive magnitude of the electron electric charge. These definitions use

Q = T_{3} + Y

; doubling the definition of hypercharge requires halving its coefficient in the covariant derivative. The electromagnetic coupling of a weak multiplet follows by substituting this rotation.

For a fermion component, the corresponding neutral coefficient is

g cos θ_{W} T_{3} - g^{'} sin θ_{W} Y = \frac{g}{c o s θ _{W}} (T_{3} - Q sin^{2} θ_{W}) .

For a left-handed charged lepton,

T_{3} = - 1/2

and

Q = - 1

g_{L} = - 1/2 + sin^{2} θ_{W}

. Its right-handed component has

T_{3} = 0

g_{R} = sin^{2} θ_{W}

. This algebra links the neutral current to the electromagnetic charge assignment. Together with the Higgs doublet, it also gives the tree-level electroweak gauge-boson masses with

m_{W} / m_{Z} = cos θ_{W}

; precise comparisons require a defined renormalization convention and radiative corrections.

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