Weinberg AngleEdit
The Weinberg angle, also known as the weak mixing angle, is a fundamental parameter in the electroweak sector of the Standard Model. It encapsulates how the electromagnetic force and the weak nuclear force emerge as two facets of a single gauge theory when the electroweak symmetry is broken. Named for Steven Weinberg, who helped develop the underlying framework, the angle is not just a curiosity of notation; it encodes the relative strengths of the basic interactions and governs how the photon and the Z boson arise from the original gauge fields.
In the electroweak theory, the neutral gauge bosons mix to produce the physical photon and the Z boson. The Weinberg angle is the quantity that describes this mixing. In practical terms, it relates the electromagnetic coupling e to the two electroweak couplings g and g' via e = g sin θW = g' cos θW, and it also links the W boson and Z boson masses through the relation cos θW = mW/mZ in the on-shell (physical) scheme. This angle therefore connects the observable properties of light and weak interactions in a single, predictive framework.
Definition and physical meaning
The electroweak interaction is built from a gauge theory based on the symmetry group SU(2)L × U(1)Y. The neutral components of the gauge fields—the SU(2)L gauge boson W^3 and the U(1)Y gauge boson B—do not correspond directly to a single observed particle. Through the Weinberg angle, these fields mix to form the physical photon (A) and the Z boson (Z): - Aμ is a linear combination of Bμ and W^3μ that becomes the photon. - Zμ is the orthogonal combination, the neutral weak boson responsible for Z-mediated processes.
The angle θW is defined by tan θW = g'/g, where g is the SU(2)L coupling and g' is the U(1)Y coupling. It also fixes the electric charge e through e = g sin θW = g' cos θW. This structure implies that the electromagnetic and weak interactions are two faces of the same underlying gauge symmetry, with θW controlling their relative strengths.
The masses of the W and Z bosons are tied to the same symmetry breaking mechanism (the Higgs mechanism). In the conventional on-shell picture, cos θW = mW/mZ, and sin^2 θW = 1 − (mW^2/mZ^2). Because of quantum corrections, the angle is not a fixed, exact number but a scale-dependent parameter whose precise value depends on how one defines it (the chosen renormalization scheme) and on the energy at which it is probed.
Mathematical framework
In the Standard Model, the electroweak sector is described by gauge fields associated with g (SU(2)L) and g' (U(1)Y). After spontaneous symmetry breaking, the physical fields are obtained by diagonalizing the mass matrix for the neutral gauge bosons. The resulting relations tie the observed photon and Z boson properties to the underlying couplings and the Weinberg angle.
Two common ways of expressing θW are: - On-shell (or pole) scheme: cos θW = mW/mZ, sin^2 θW = 1 − (mW^2/mZ^2). - MSbar (modified minimal subtraction) or other renormalization schemes: sin^2 θW^MSbar(mZ) is defined at the Z-pole with radiative corrections included.
The running of sin^2 θW with energy is a consequence of quantum loops involving all Standard Model particles. Different renormalization schemes yield slightly different numerical values at a given scale, but all are connected by well-defined renormalization group equations. The weak mixing angle acts as a bridge between the gauge structure and the observable couplings of fermions to the photon and Z boson.
Experimental determination and precision tests
Precise measurements at high-energy colliders and deep inelastic scattering experiments pin down the value of the Weinberg angle. Key inputs include: - Z-pole observables from electron-positron colliders, such as asymmetries in Z decays. - W and Z boson production rates and kinematics in hadron colliders. - neutrino-nucleon and electron-nucleon scattering experiments that probe neutral-current interactions.
The result is a highly constrained picture in which sin^2 θW is determined with per-mille precision in some schemes at the Z scale. For example, the MSbar value at the Z mass is commonly quoted as sin^2 θW^MSbar(mZ) ≈ 0.231 with small uncertainties. The effective angle at the Z pole, sin^2 θW^eff, is extracted from asymmetries and can differ slightly depending on the observable considered, reflecting radiative corrections and experimental details.
These measurements provide a stringent test of the electroweak sector of the Standard Model. They also place indirect constraints on possible new physics that would alter the couplings of fermions to the gauge bosons, such as additional heavy particles, extra gauge groups, or modified Higgs-sector dynamics. Historical tensions, such as the NuTeV anomaly in neutrino scattering, sparked discussions about parton distributions and higher-order corrections; subsequent analyses have largely reconciled most of the discrepancy, though they underscore the sensitivity of θW determinations to both theory and experiment.
Running, definitions, and beyond
Because θW is defined within a quantum field theory, its numerical value depends on the energy scale and the renormalization convention used. In the Standard Model, radiative corrections from fermions, bosons, and the Higgs sector shift the apparent mixing angle as one probes different processes or energy scales. This running is not a sign of inconsistency but a prediction of the theory, testable by comparing low-energy precise measurements with high-energy collider data.
Beyond the Standard Model, many theories introduce new particles or interactions that change the way θW appears in observable processes. Extra neutral gauge bosons (often labeled Z′) or modified Higgs sectors can modify neutral-current couplings and alter the relationship between mW and mZ or between e, g, and g′. Precision measurements of sin^2 θW thus remain a critical probe of potential new physics, complementing direct searches for new particles.