Abstract
We present a calculation of the weak mixing angle in the overline{mathrm{MS}} renormalization scheme which is relevant for experiments performed at very low energies or momentum transfers. We include higher orders in the perturbative QCD expansion, as well as updated phenomenological and theoretical input, and obtain the result { sin}^2{widehat{theta}}_W(0)=0.23868(5)(2) for the reference values {widehat{alpha}}_sleft({M}_Zright)=0.1182 and {widehat{m}}_cleft({widehat{m}}_cright)=1.272 mathrm{GeV} . The first quoted error is from the current Standard Model evaluation of the mixing angle at the Z boson mass scale. The second error represents the theoretical and parametric uncertainties induced by the evolution to the Thomson limit and is discussed in detail.
Highlights
Measured the weak charge of the proton, QW (p) ∼ 1 − 4 sin2 θW, in polarized electron scattering from a fixed liquid hydrogen target at Q2 ≈ 0.026 GeV2
We present a calculation of the weak mixing angle in the MS renormalization scheme which is relevant for experiments performed at very low energies or momentum transfers
In a very similar setup, the MOLLER Collaboration [5] at JLab will build and improve on the completed E158 experiment [6] at SLAC and measure the analogous weak charge of the electron, QW (e), in polarized Møller scattering at Q2 ≈ 0.0056 GeV2
Summary
In an approximation in which all fermions are either massless and active or infinitely heavy and decoupled, the RGE for the electromagnetic coupling in the MS scheme [24], α, can be written in the form [2], μ2. The RGE for the Z boson vector coupling to fermion f , vf = Tf − 2Qf sin θW , where Tf is the third component of weak isospin of fermion f , is dvf dμ. Represent the singlet contributions to the RGE evolution of the weak mixing angle at four and five loop order. These terms arise from quark-antiquark annihilation (disconnected) diagrams (see figure 1) and are suppressed in perturbative QCD (PQCD). Eq (2.5) together with the solution of the four-loop QCD β-function [26, 27] represents a complete solution, as long as all matching scales μ at which an active particle decouples are known, because there the λi change their values. The matching scales of all bosons [28], charged leptons, and heavy (t, b, and c) quarks [29,30,31] can be calculated as what we call threshold masses mq, where the QCD corrections to the matching relations vanish by definition
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