Effective Leptonic Weak Mixing Angle

• Effective leptonic weak mixing angle is a parameter that incorporates radiative corrections into Z-boson couplings to charged leptons.
• It is extracted from forward-backward asymmetries in dilepton production near the Z pole using advanced statistical techniques and PDF profiling.
• Its precise measurement underpins Standard Model tests and informs SMEFT renormalisation schemes for potential new physics.

The effective leptonic weak mixing angle, $\sin^2\theta^\ell_{\rm eff}$, is a fundamental parameter of the Standard Model that incorporates radiative corrections into the neutral-current couplings of the $Z$ boson to charged leptons. It governs the vector couplings of the $Z$ boson to leptons, enters precision electroweak observables, and is measured most precisely near the $Z$ pole through asymmetries in fermion production. At hadron colliders, the central observable is the forward-backward asymmetry in Drell-Yan dilepton production; in electroweak theory and SMEFT, the same parameter can also be promoted from a derived quantity to an input parameter in renormalisation schemes (Collaboration, 2014, Biekötter et al., 2023).

1. Definition and relation to neutral-current couplings

The weak mixing angle $\theta_W$ encodes the relative strengths of the weak and electromagnetic interactions through the couplings of fermions to the $Z$ boson,

$$g_V^f = I_3^f - 2 Q_f \sin^2\theta_W,\qquad g_A^f = I_3^f.$$

The effective weak mixing angle incorporates higher-order electroweak corrections and depends on the fermion species: $$\sin^2\theta_{\text{eff}^f} = \frac{1}{4|Q_f|}\left(1-\frac{g_V^f}{g_A^f}\right).$$ The parameter most commonly quoted is the charged-lepton quantity $\sin^2\theta^\ell_{\rm eff}$, which is tightly constrained from precision electroweak measurements and is sensitive to possible new physics (Collaboration, 2014).

A central distinction in the precision literature is that $\sin^2\theta^\ell_{\rm eff}$ is not identical to the on-shell weak mixing angle. In the on-shell scheme,

$Z$0

whereas $Z$1 is extracted from observables near the $Z$2 pole and absorbs radiative corrections. The Tevatron Run II combination made this distinction explicit by quoting both $Z$3 and the inferred on-shell parameter $Z$4, or equivalently $Z$5, through \textsc{zfitter} (Collaboration et al., 2018). A more explicit form-factor definition used in SMEFT analyses is

$Z$6

which makes clear that the effective angle is defined directly from the renormalized $Z$7 amplitude (Biekötter et al., 2023).

2. Extraction from asymmetries near the $Z$8 pole

At hadron colliders, the standard channel is the neutral-current Drell-Yan process

$Z$9

measured as $Z$0 at the Tevatron and as $Z$1 at the LHC. The key observable is the forward-backward asymmetry,

$Z$2

defined with respect to the polar angle of the negatively charged lepton in the Collins-Soper frame. Near the $Z$3 pole, the asymmetry is highly sensitive to $Z$4 because it is generated by interference between vector and axial-vector couplings (Collaboration, 2014, Collaboration, 2018).

The differential angular structure is commonly written as

$Z$5

or, in more complete form near the $Z$6 pole,

$Z$7

This relation motivated two closely related measurement strategies: direct fits to $Z$8 and fits to the unfolded angular coefficient $Z$9 (Collaboration, 2024, Bodek et al., 25 Aug 2025).

The collider environment determines how the quark direction is treated. At a $Z$0 collider, the forward direction is naturally associated with the quark’s original direction in the Collins-Soper frame. In $Z$1 collisions, by contrast, the quark-antiquark assignment is ambiguous and must be resolved statistically using correlations with the dilepton rapidity. The early CMS 7 TeV analysis formalized this through the dilepton rapidity, invariant mass, and decay-angle distributions in a multivariate likelihood method, with a dilution factor that is close to zero at $Z$2 and increases with $Z$3 (Collaboration, 2011).

Modern hadron-collider analyses go well beyond simple forward-minus-backward event counts. CMS introduced angular event weighting, which uses the shape of the angular distributions to reduce statistical and systematic uncertainties and makes the measurement less sensitive to detector acceptance effects (Collaboration, 2018). Later 13 TeV analyses combined the angular-weighted asymmetry method with unfolded $Z$4 measurements, allowing later reinterpretation with updated PDFs and theory inputs (Collaboration, 2024).

3. Measured values and experimental progression

The empirical program has progressed from first LHC demonstrations to determinations whose quoted precision is comparable to the best LEP and SLD results.

Measurement	Quoted value	Note
CMS 7 TeV dimuon (Collaboration, 2011)	$Z$5	Multivariate likelihood method
D0 $Z$6 (Collaboration, 2014)	$Z$7	Most precise measurement from light quark interactions to date
Tevatron Run II combination (Collaboration et al., 2018)	$Z$8	Most precise result from hadron colliders
CMS 8 TeV combined dilepton (Collaboration, 2018)	$Z$9	Most precise value at the LHC to date
CMS 13 TeV combined dilepton (Khukhunaishvili, 2024)	$\theta_W$0	Most precise measurement at a hadron collider
LHCb 13 TeV dimuon (collaboration et al., 2024)	$\theta_W$1	Forward-region determination

The D0 analysis used $\theta_W$2 of integrated luminosity at $\theta_W$3 and extracted the asymmetry as a function of the dielectron invariant mass around the $\theta_W$4 boson pole. Its value,

$\theta_W$5

or $\theta_W$6 in quadrature, was described as the most precise measurement from light quark interactions to date, with a precision close to the best LEP and SLD results (Collaboration, 2014).

The Tevatron Run II combination of CDF and D0 yielded

$\theta_W$7

and inferred

$\theta_W$8

The combination was stated to be consistent with, and to approach in precision, the best measurements from electron-positron colliders (Collaboration et al., 2018).

At the LHC, CMS first demonstrated the method at 7 TeV, then substantially improved it at 8 TeV using 8.2 million dimuon and 4.9 million dielectron events and Bayesian PDF reweighting, obtaining

$\theta_W$9

The 13 TeV CMS Run 2 analyses pushed the hadron-collider precision to the $Z$0 level. One report quoted

$Z$1

while a complementary CMS measurement based on $Z$2 and unfolded $Z$3 quoted

$Z$4

with CT18Z, and stated that the measured value agrees with the standard model fit result to global experimental data (Khukhunaishvili, 2024, Collaboration, 2024).

LHCb provided a distinct forward-spectrometer determination using $Z$5 of 13 TeV $Z$6 data in the fiducial region $Z$7 GeV, $Z$8, and $Z$9 GeV. The asymmetry was measured in ten intervals of $$g_V^f = I_3^f - 2 Q_f \sin^2\theta_W,\qquad g_A^f = I_3^f.$$0, and the final result was given as an arithmetic average over CT18, MSHT20, and NNPDF31 (collaboration et al., 2024).

A recurrent point in this history is the comparison with LEP and SLD. One 2024 CMS study explicitly noted discrepancies between prior precise measurements at LEP and SLD, differing at $$g_V^f = I_3^f - 2 Q_f \sin^2\theta_W,\qquad g_A^f = I_3^f.$$1, and treated renewed precision studies as motivated by that tension (Khukhunaishvili, 2024). The hadron-collider determinations therefore function both as competitive measurements and as independent cross-checks based on different initial states and different systematic structures.

4. Dominant uncertainties and the central role of PDFs

The dominant limitation in modern hadron-collider determinations is not event yield but proton-structure uncertainty. D0 had already reduced most instrumental effects to a subdominant level through a new, data-driven electron-energy calibration dependent on both $$g_V^f = I_3^f - 2 Q_f \sin^2\theta_W,\qquad g_A^f = I_3^f.$$2 and instantaneous luminosity, by extending electron pseudorapidity acceptance, and by including events previously excluded. In that analysis, the dominant systematics came from electron energy calibration and resolution, while backgrounds were very small, about $$g_V^f = I_3^f - 2 Q_f \sin^2\theta_W,\qquad g_A^f = I_3^f.$$3, and dominated by multijet events faking electrons (Collaboration, 2014).

CMS 8 TeV made the PDF problem explicit and attacked it with Bayesian chi-squared reweighting. The analysis used 100 NNPDF3.0 PDF replicas; replicas that fit the data well got high weights and others low weight. This significantly reduced the PDF-induced uncertainty in the extracted $$g_V^f = I_3^f - 2 Q_f \sin^2\theta_W,\qquad g_A^f = I_3^f.$$4 (Collaboration, 2018).

The 13 TeV CMS analyses sharpened this program further. One study summarized the total uncertainty as

$$g_V^f = I_3^f - 2 Q_f \sin^2\theta_W,\qquad g_A^f = I_3^f.$$5

showing that the PDF term remained dominant even after in-situ profiling. The extraction was performed through a simultaneous $$g_V^f = I_3^f - 2 Q_f \sin^2\theta_W,\qquad g_A^f = I_3^f.$$6 fit to $$g_V^f = I_3^f - 2 Q_f \sin^2\theta_W,\qquad g_A^f = I_3^f.$$7 distributions across all channels and years, with over 14,000 bins and more than 3,300 nuisance parameters (Khukhunaishvili, 2024).

A CMS-based reanalysis then used the dilepton mass dependence of $$g_V^f = I_3^f - 2 Q_f \sin^2\theta_W,\qquad g_A^f = I_3^f.$$8 to profile PDFs and added new CMS measurements of the $$g_V^f = I_3^f - 2 Q_f \sin^2\theta_W,\qquad g_A^f = I_3^f.$$9-boson decay lepton asymmetry and the $$\sin^2\theta_{\text{eff}^f} = \frac{1}{4|Q_f|}\left(1-\frac{g_V^f}{g_A^f}\right).$$0 cross section ratio at 13 TeV. In this framework, the uncertainty published by CMS was described as dominated by uncertainties in Parton Distribution Functions, which are reduced by PDF profiling using the dilepton mass dependence of $$\sin^2\theta_{\text{eff}^f} = \frac{1}{4|Q_f|}\left(1-\frac{g_V^f}{g_A^f}\right).$$1. The reanalysis obtained

$$\sin^2\theta_{\text{eff}^f} = \frac{1}{4|Q_f|}\left(1-\frac{g_V^f}{g_A^f}\right).$$2

described as the most precise single measurement to date, and a later summary quoted

$$\sin^2\theta_{\text{eff}^f} = \frac{1}{4|Q_f|}\left(1-\frac{g_V^f}{g_A^f}\right).$$3

after incorporating complementary CMS observables (Bodek et al., 25 Aug 2025, Bodek et al., 28 Jan 2026).

The uncertainty reductions quoted in the CMS profiling summary make the methodological point concrete: for the nominal PDF set, the uncertainty was stated to move from $$\sin^2\theta_{\text{eff}^f} = \frac{1}{4|Q_f|}\left(1-\frac{g_V^f}{g_A^f}\right).$$4 before profiling to $$\sin^2\theta_{\text{eff}^f} = \frac{1}{4|Q_f|}\left(1-\frac{g_V^f}{g_A^f}\right).$$5 after $$\sin^2\theta_{\text{eff}^f} = \frac{1}{4|Q_f|}\left(1-\frac{g_V^f}{g_A^f}\right).$$6 profiling and to $$\sin^2\theta_{\text{eff}^f} = \frac{1}{4|Q_f|}\left(1-\frac{g_V^f}{g_A^f}\right).$$7 after including $$\sin^2\theta_{\text{eff}^f} = \frac{1}{4|Q_f|}\left(1-\frac{g_V^f}{g_A^f}\right).$$8 asymmetry and $$\sin^2\theta_{\text{eff}^f} = \frac{1}{4|Q_f|}\left(1-\frac{g_V^f}{g_A^f}\right).$$9 ratios (Bodek et al., 28 Jan 2026). This suggests that the next increments in precision are tied less to raw luminosity than to correlated PDF control and to the joint use of observables that probe distinct parton-density combinations.

5. Renormalisation schemes, electroweak fits, and SMEFT use

Precision measurements of $\sin^2\theta^\ell_{\rm eff}$0 are not only outputs of collider analyses; they also define renormalisation schemes. A one-loop Standard Model study proposed using $\sin^2\theta^\ell_{\rm eff}$1 as a direct input parameter, alongside $\sin^2\theta^\ell_{\rm eff}$2 or $\sin^2\theta^\ell_{\rm eff}$3 and $\sin^2\theta^\ell_{\rm eff}$4, for the prediction of the forward-backward asymmetry in neutral-current Drell-Yan production. In that framework, the proposed input scheme was described as suitable for a direct determination of the effective leptonic weak mixing angle from the experimental data, with reduced sensitivity to poorly known quantities such as $\sin^2\theta^\ell_{\rm eff}$5 or the top mass (chiesa et al., 2019).

The SMEFT extension of this idea was developed to NLO through the $\sin^2\theta^\ell_{\rm eff}$6 and $\sin^2\theta^\ell_{\rm eff}$7 schemes, with inputs $\sin^2\theta^\ell_{\rm eff}$8 and $\sin^2\theta^\ell_{\rm eff}$9. An attractive feature is that large corrections from top-quark loops appearing in other schemes are absorbed into the definition of the effective weak mixing angle. The renormalisation condition is imposed by matching the renormalized SMEFT parameter $\sin^2\theta^\ell_{\rm eff}$0 to the experimentally measured effective weak mixing angle,

$\sin^2\theta^\ell_{\rm eff}$1

and in the large-$\sin^2\theta^\ell_{\rm eff}$2 limit the top-induced one-loop correction to the weak-angle counterterm vanishes,

$\sin^2\theta^\ell_{\rm eff}$3

The same work emphasized a practical complication: the renormalisation condition involves a large number of flavour-specific SMEFT couplings between the $\sin^2\theta^\ell_{\rm eff}$4 boson and charged leptons, motivating simple flavour assumptions such as minimal flavour violation for practical applications (Biekötter et al., 2023).

The flavor issue is numerically substantial. Under general flavour assumptions, up to about 93 Wilson coefficients can contribute at NLO to leptonic observables, whereas under minimal flavour violation that number is reduced to about 34. The purpose of the scheme is not to replace conventional inputs universally, but to provide a valuable new component for estimating systematic uncertainties in SMEFT fits by performing analyses in multiple input schemes (Biekötter et al., 2023).

In standard electroweak fits, $\sin^2\theta^\ell_{\rm eff}$5 remains tightly linked to the broader precision program. The Tevatron combination’s use of \textsc{zfitter} to infer $\sin^2\theta^\ell_{\rm eff}$6 and $\sin^2\theta^\ell_{\rm eff}$7 from $\sin^2\theta^\ell_{\rm eff}$8 is one explicit example (Collaboration et al., 2018). More broadly, comparisons between direct measurements of $\sin^2\theta^\ell_{\rm eff}$9 and the Standard Model fit value are repeatedly used as electroweak consistency tests.

6. Future precision program and extensions beyond present hadron-collider benchmarks

Several future facilities aim to move the uncertainty on $Z$00 below the present hadron-collider level. A CEPC study proposed a two-year running period around the $Z$01 boson mass pole with $Z$02 $Z$03 candidates in total. It stated that the uncertainty on $Z$04 could be one order of magnitude lower than any previous measurement at LEP, SLC, Tevatron and LHC, with an overall projected precision of $Z$05 in both lepton and $Z$06 final states. The same study also examined off-pole running and quoted a precision of $Z$07 for $Z$08 measured at $Z$09 GeV from the $Z$10 quark final state with one month of data (Zhao et al., 2022).

A complementary proposal for a super $Z$11-factory focused on determining flavor-dependent effective angles, especially $Z$12 for $Z$13, through forward-backward, left-right, and left-right-forward-backward asymmetries of doubly heavy-flavored hadrons such as $Z$14, $Z$15, $Z$16, $Z$17, and $Z$18. The claimed advantage is that the doubly heavy flavor(s) and the out-going direction of the produced doubly-heavy hadron can be experimentally determined precisely, avoiding errors from missing identification of the heavy flavor(s) and from determining the thrust axis of produced jets (Zheng et al., 2018).

For the LHC itself, the outlook papers emphasize further improvement through PDF control rather than a purely statistical strategy. One review projected that combining 13 and 13.6 TeV data should yield an uncertainty of $Z$19 to $Z$20, and also discussed a measurement of $Z$21 for $Z$22-quarks in the initial state and a measurement of the running of $Z$23 up to $Z$24 TeV (Bodek et al., 25 Aug 2025).

Beyond the $Z$25 pole, low-energy and wide-scale determinations probe the running of the weak mixing angle in complementary ways. The MOLLER experiment at Jefferson Lab targets a low-energy effective weak mixing angle measurement with a precision of $Z$26, using parity-violating M{\o}ller scattering and an expected Standard Model asymmetry of about $Z$27 parts per billion (Collaboration et al., 2014). The $Z$28TRISTAN proposal uses M{\o}ller-like $Z$29 scattering to determine the weak mixing angle with percent to milli-level accuracy and to scan over interaction scales from about $Z$30 GeV up to several TeV in a single experiment (Chen et al., 2024).

Taken together, these developments place $Z$31 at the center of a broad precision program. At the $Z$32 pole it is a benchmark electroweak observable extracted from asymmetries with LEP-, SLD-, Tevatron-, and LHC-level precision; in SMEFT it is a technically useful input parameter; and in future collider and fixed-target programs it becomes a vehicle for tests of flavor dependence, PDF systematics, and the running of electroweak couplings across a wide range of scales.