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The Phenomenology of Extra Neutral Gauge Bosons

A. Leike

arXiv:hep-ph/9805494v1hep-ph

TL;DR

The paper reviews how to constrain and measure extra neutral gauge bosons at present and future colliders, accounting for model assumptions and experimental conditions. It derives transparent approximate estimates and compares them with existing results, finding complementary strengths across lepton and hadron colliders.

  • Problem

    The Z′ mass is not fixed by theory, so its possible collider signatures and constraints must be assessed across models and experimental settings.

    Source: Introduction · p. 4

  • Method

    The paper derives approximate formulas for Z′ constraints and measurements, makes their dependence on experimental conditions and model parameters explicit, and compares them with literature results.

    Source: Introduction · p. 5

  • Results

    Lepton colliders best constrain ZZ′ mixing and weakly interacting Z′s, while hadron colliders best constrain masses for popular GUT-predicted Z′s; their model-measurement strengths are complementary.

    Source: Summary and conclusions · p. 99

  • Takeaways & Limitations

    For a Z′ signal, hadron colliders measure Standard Model fermion couplings with smaller errors but a 16-fold sign ambiguity, while polarized lepton colliders have a 2-fold ambiguity.

    Source: Summary and conclusions · p. 99

    The review excludes supersymmetric-particle interactions in SUSY GUTs and does not discuss leptophobic Z′ models.

    Source: §1.1.1 · p. 7

Abstract

from arXiv · show

The phenomenological constraints on extra neutral gauge bosons at present and at future colliders are reviewed. Special attention is paid to the influence of radiative corrections, systematic errors, and kinematic cuts on the Z' constraints. Simple estimates of the Z' constraints from different reactions are derived. They make the physical origin of these constraints transparent. The results existing in the literature are summarized and compared with the estimates. The consequence of model assumptions on the Z' constraints is discussed. The paper starts with an overview of Z' parameters and the possible links between them by model assumptions. It continues with a discussion of Z' limits and Z' measurements in different reactions at e+e- and mu+mu- colliders. It follows an overview of the corresponding limits at proton colliders. Possible Z' constraints from other reactions as ep collisions, atomic parity violation, neutrino scattering and cosmology are briefly mentioned.

Introduction

The paper reviews Z′ phenomenology and the mechanisms behind its constraints, deriving approximate formulae to clarify their dependence on experimental conditions and model parameters. This section introduces the theoretical motivation and scope, parameterizes fermion couplings, discusses mixing and radiative effects in decay widths, and states the assumptions used to restrict the model parameters, decay channels, and couplings considered.

Introduction

  • GUTs larger than SU(5) predict at least one extra neutral gauge boson; SO(10) predicts one and requires a right-handed neutrino to complete its fermion multiplet.The Z′ mass is not fixed by theory and can range from Eweak to EGUT; some supersymmetric GUTs naturally give a mass of about one TeV, potentially observable at the next collider generation.Source: Introduction · p. 4
  • Approximate formulae clarify how Z′ constraints and measurements depend on experimental conditions and model parameters, and are compared with results in the literature.The review highlights radiative corrections, realistic cuts, systematic errors, and analyses both with and without model assumptions.Source: Introduction · p. 5

1.1.1 Assumptions and limitations

  • The analysis assumes one sufficiently light Z′ gives the first GUT signal, generation-universal fermion couplings, and kinematic suppression of decays to exotic fermions and Higgs bosons.It also neglects mixing with exotic fermions and supersymmetric-particle effects, and excludes leptophobic Z′ models.Source: §1.1.1 · pp. 6–7

1.1.2.1 Kinetic mixing

  • Kinetic mixing is described by an angle and can arise during evolution from GUT energies to the weak scale; it is important in some leptophobic models and can communicate SUSY breaking to the visible sector.A nonzero coupling term is needed for renormalizability when no additional symmetry sets it to zero.Source: §1.1.2.1 · p. 7

1.1.2.2 Mass mixing

  • For fixed Z1 and Z masses, the mass constraint gives θM ∼ 1/M2 at large M2, while the Higgs constraint is stronger and gives θM ∼ 1/M2^2.The Higgs constraint applies for a fixed Higgs sector; the mass constraint is independent of that sector.Source: §1.1.2.2 · p. 9

1.1.2.3 Alternative parametrization of the mixing

  • Left-right models can use a one-step orthogonal 3 by 3 mixing; the paper prefers two-step mixing for model-independent analyses, and the procedures coincide when gR = gL.For unequal chiral couplings, fermion loops can generate non-diagonal gauge-field terms after symmetry breaking.Source: §1.1.2.3 · p. 9

1.1.3 Couplings to SM fermions

  • Experiments probe the Z1 and Z2 mass-eigenstate couplings, which need not be the fixed GUT couplings of the symmetry eigenstate Z′.In the Sequential Standard Model, Z′ couplings to SM fermions equal those of the SM Z boson.Source: §1.1.3 · p. 10, §1.1.3.1 · p. 10

1.1.3.1 Couplings of the Symmetry Eigenstates

  • SU(2)L relations reduce the Z′ fermion couplings to five independent parameters; right-handed neutrinos are treated as heavy exotics and excluded.The SSM is described as a popular benchmark model that is hard to obtain in GUTs.Source: §1.1.3.1 · pp. 10–11

1.1.3.2 Couplings of the Mass eigenstates

  • Form factors describe ZZ′ mixing alongside electroweak corrections at the Z1 peak, and the mixing contributions are small for small θM.The form factors are proportional to θM in the small-mixing limit.Source: §1.1.3.2 · p. 11

1.1.4 Decay width

  • The analysis considers Z′ decays to Standard Model particles, while decays to nonstandard particles are assumed kinematically suppressed.Widths for decays to other GUT particles and to bosons are referenced separately.Source: §1.1.4 · p. 11, §1.1.4.1 · p. 12

1.1.4.1 Born approximation

  • With nonzero mass mixing, the Z component of the mass eigenstates couples to W bosons; nonstandard-particle decays are assumed suppressed.In GUTs, the potentially large M2^4 factor in Z2→W+W− decay is compensated by small mixing angles under the Higgs constraint.Source: §1.1.4.1 · p. 12

1.1.4.2 Higher order effects

  • Radiative corrections to Γ2 are necessary for precision measurements at the Z2 peak in e+e− or μ+μ− collisions.Full one-loop GUT corrections require the relevant new parameters to be known from independent experiments.Source: §1.1.4.2 · pp. 12–13

Mass mixing

  • Mass mixing changes Z2 partial widths through coupling shifts of order θM, an effect relevant to precision measurements despite current limits on mixing.For fermion-pair production, the width should depend on s; finite-width treatments can violate gauge invariance in some four-fermion processes unless additional higher-order terms restore it.Source: Mass mixing · p. 13

1.1.5 Summary of Z′ parameters

  • The summarized Z′ parameters are M2, Γ2, θM, g2, and the fermion vector and axial couplings; unpolarized proton collisions cannot determine their relative sign.An alternative parameter set is therefore used for unpolarized proton collisions.Source: §1.1.5 · p. 14

1.2 Extra neutral gauge bosons in E6 models

E6 symmetry-breaking assumptions constrain the extra-boson charges and couplings, and experimental tests of the resulting relations can assess whether a Z′ fits the scheme. Its width depends on which decay channels are kinematically allowed.

1.2 Extra neutral gauge bosons in E6 models

  • Five independent E6 charges are constrained, and the conditions yield two relations among Z′ couplings when the coupling ratio is unknown.Testing the conditions can determine whether a Z′ belongs to the E6 breaking scheme.Source: §1.2 · pp. 14–15
  • E6 permits multiple breaking patterns; the χ and ψ examples arise from SO(10) to SU(5) and E6 to SO(10), respectively.The couplings to Standard Model fermions are specified for these cases.Source: §1.2 · p. 15
  • For E6 models with only Standard Model fermion decays open, typical Γ2/MZ′ values range from 0.5% to 2%; allowing exotic fermion and Higgs decays makes Γ2 roughly three times larger.The table gives vector and axial-vector Z′ couplings to Standard Model fermions in E6 and left-right models.Source: Table 1.1 · p. 15, §1.2 · p. 16
  • In a model with two Higgs doublets and one Higgs singlet, the mass-matrix entries are fixed, while the doublet charges depend on the symmetry breaking.The example provides a Higgs constraint involving those charges.Source: §1.2 · p. 16

1.3 Extra neutral gauge bosons and contact interactions

Far below the Z′ resonance, its effects can be represented by four-fermion contact interactions, but this representation does not apply near resonance. Form factors incorporate Z′ contributions into the Standard Model Z couplings and can also combine additional amplitudes.

1.3 Extra neutral gauge bosons and contact interactions

  • Far below the Z′ resonance, its effects can be described by four-fermion contact interactions, but this approximation cannot describe effects near the resonance.Such interactions constrain coupling ratios and the Z′ mass separately, not the quantities individually.Source: §1.3 · pp. 16–17
  • Four-fermion contact interactions allow more parameters than Z′ interactions; if deviations are due to a Z′, the contact-interaction parameters must satisfy additional relations.Under the stated gauge-group assumption, the Z′ couplings also obey additional relations.Source: §1.3 · p. 17

1.4 Four fermion interactions and form factors

  • The Z′ amplitude can be absorbed into the Standard Model Z couplings while leaving photon couplings unchanged; the resulting form factors include the Z′ contribution without approximation.They are generally energy-dependent and can resonate near the Z′ peak.Source: §1.4 · pp. 18–19
  • Exact form-factor summation rules combine additional amplitudes, while approximate rules apply when form factors are near one and neglect products of the epsilon terms.The framework also covers simultaneous ZZ′ mixing and electroweak corrections.Source: §1.4 · p. 19

1.5 Model dependence of Z′ constraints

This section examines how model assumptions determine whether extra neutral gauge boson constraints apply to independent coupling combinations or to mass and coupling parameters linked by a model. It describes how measurement precision, systematic and radiative effects, observables, and collider conditions shape limits and coupling measurements, including analyses at and away from the Z′ resonance.

1.5 Model dependence of Z′ constraints

  • Assumptions progressively narrow interpretation from a gauge-group Z′ to E6 and SO(10) breaking or a definite model such as Z′ = χ, enabling more detailed tests.These increasingly specific interpretations are justified when consistent with experimental data.Source: §1.5 · p. 20
  • Model-independent analyses usually constrain parameter combinations using limited observables; converting their results to a specified model generally gives weaker constraints than a direct model-dependent fit.The two approaches are complementary, and the paper identifies which type produced each constraint.Source: §1.5 · p. 20

1.6.1 Indirect Z′ limits

  • Indirect limits use deviations exceeding experimental errors in observables; assuming Gaussian errors and neglecting correlations, a chi-square criterion determines model exclusion at a confidence level.This applies, for example, below the Z′ resonance, where the Standard Model predicts many events.Source: §1.6.1 · p. 21

1.6.2 Direct Z′ limits

  • Direct lepton-collider limits use Gaussian chi-square analyses and can exploit precision measurements at the Z′ peak, whereas hadron-collider limits use event-count criteria when backgrounds are small.With zero background and zero observed events, theories predicting 3 Z′ events are excluded at 95% confidence.Source: §1.6.2 · p. 22

colliders

  • Fermion-pair production can separately constrain charged-fermion Z′ couplings, while W-pair production is especially sensitive to ZZ′ mixing at high energies; other reactions offer no useful added information.Bhabha and Møller scattering probe electron couplings only, with sensitivity competing against fermion-pair production.Source: colliders · p. 23

2.1 Z′ search in e+e−→f ¯f

  • Across four energy regimes, the Z1 peak best constrains mixing, below-resonance measurements can set fixed-model mass bounds, the Z2 peak enables detailed studies, and above-resonance searches probe weakly coupled Z′ bosons.Below the Z2 resonance, sensitivity to mixing is suppressed relative to measurements at the Z1 peak.Source: §2.1 · p. 24

2.1.1.1 Amplitude

  • Away from resonances, fermion-pair production constrains normalized couplings rather than mass and couplings separately, and cannot distinguish models differing only by the signs of all fermion couplings.Several widths below resonance, indirect limits from this process are independent of the Z′ width, unlike hadron-collider limits that depend on branching ratios.Source: §2.1.1.1 · p. 25

2.1.1.3 ZZ′ mixing

  • ZZ′ mixing can be incorporated through form factors, a formalism useful for describing mixing and electroweak corrections together at the Z1 peak.Only four cross sections and asymmetries are independent; measuring them may require flavor tagging, particle–antiparticle discrimination, beam polarization, or final-state polarization, while transverse asymmetries face experimental difficulties.Source: §2.1.1.2 · p. 26, §2.1.1.3 · p. 27, §2.1.1.4 · p. 27

2.1.1.4 Observables

  • At the Z1 peak, three independent observables measure the charged-lepton axial and vector Z′ couplings, and inconsistent signals can disprove a Z′ explanation.Relations among the four additional observables must also hold if the interaction is due to a Z′; the stated relations are approximate for light final-state fermions with appropriate cuts.Source: §2.1.1.4 · p. 28

2.1.2.1 QED corrections

  • Below the Z′ resonance, radiative return enhances SM cross sections but not the Z′ signal, so hard-photon events must be rejected; suitable cuts can make corrected cross sections resemble Born predictions.Radiative corrections still need to be included in fits to real data.Source: §2.1.2.1 · pp. 30–31

2.1.2.2 Weak corrections

  • Weak corrections matter for precision e+e− predictions, while non-standard one-loop corrections cannot be calculated without the underlying theory and are expected to be small if the Z′ is the first new-physics signal.Source: §2.1.2.2 · p. 32

2.1.2.3 QCD corrections

  • For mf=0 and ∆g=1, lowest-order QCD corrections vanish for forward-backward asymmetries and give total cross sections the factor 1+αs/π; spin asymmetries require care because of gluon-induced spin flips.The massless calculation does not match the mf→0 limit of the massive calculation for spin asymmetries.Source: §2.1.2.3 · p. 33

2.1.2.4 Corrections to Γ1

  • Measurements at the Z1 resonance are sensitive to radiative corrections to Γ1, and nonzero ZZ′ mixing requires additional replacements in its prediction.Source: §2.1.2.4 · p. 33

2.1.3 Z′ constraints at s ≈M2

  • Direct simultaneous fits of SM and Z′ parameters at the Z1 peak and indirect two-stage fits are alternative procedures, and the reported indirect analyses agree with the direct analysis.Indirect fits first extract peak observables from data around the resonance, then determine SM and Z′ parameters.Source: §2.1.3 · p. 34

2.1.3.1 Model independent constraints on vM

  • Observables are blind to Z′ models whose vM lie between two parallel lines; related equations also allow an independent constraint on aM_e and a test of the relations (1.27).The test is described as an experimental check.Source: §2.1.3.1 · pp. 34–35
  • Model-independent constraints agree very well with estimates; ρmix=1±0.003 removes M2 dependence from Γf1, but its uncertainty must be added and the result assumes one Higgs doublet.The uncertainty in ρmix is expected to retain a small model dependence for extended Higgs sectors.Source: §2.1.3.1 · p. 35
  • The Z1-peak exclusion region shifts with the experimental deviation and ρmix uncertainty, and the ρmix shift is one-directional because M1 < MZ.Improved measurements of MW and Mt and a determination of MH would reduce the shift; measurements at M2 cannot improve constraints from the Z1 peak, while a Z2-peak measurement could separately determine its couplings and θM.Source: §2.1.3.1 · pp. 35, 37

2.1.3.2 Model independent constraint on θM

  • For g2a′ l/(g1al) = 0.62, the approximate model-independent bound is |θM| < 0.003; the exact calculation gives |θM| < 0.035 for zero Z′ coupling and c = al.The exact calculation also constrains θM when the Z′ coupling is zero, because mixing shifts the couplings beyond experimental errors.Source: §2.1.3.2 · p. 37

2.1.3.3 Model dependent constraint on θM and M2

  • For a specified Higgs sector, the Higgs constraint relates θM and M2, converting tight θM limits into much stronger M2 limits than for unconstrained Higgs sectors.In L3 data, the θM bound is dominated by measurements at the Z1 peak, while the M2 bound is dominated by measurements above it; for unconstrained Higgs sectors, Z1-peak precision limits on M2 are poor.Source: §2.1.3.3 · pp. 38, 40
  • Linking Z′ couplings through model parameters yields tighter mixing-angle limits than model-independent analyses, which use only final-state leptons.The linkage permits both leptonic and hadronic observables to enter the fit.Source: §2.1.3.3 · p. 40
  • For model-dependent M2 limits, future e+e− and pp(p̄p) experiments will improve the bounds, while indirect Z1-peak limits with constrained Higgs sectors can compete with other experiments.The limits on θM from existing data could be improved only by future e+e− → W+W− measurements.Source: §2.1.3.3 · p. 41

2.1.4 Z′ limits at M2

  • Off the Z1 peak and below the Z2 resonance, Z′ effects arise from interference with the photon or Z1; θM sensitivity is suppressed by Γ1/M1, motivating θM = 0 for M2 limits.The transverse-normal spin asymmetry avoids this suppression but loses statistics, leaving its M2 sensitivity weaker than the estimate in equation (2.46).Source: §2.1.3.3 · p. 38, §2.1.4 · pp. 41–42

2.1.4.1 Model independent constraints on vN

  • For model-independent off-resonance limits, σlT, the forward–backward asymmetry, and the left–right asymmetry exclude different ellipses or hyperbolas in coupling space.With appropriate kinematic cuts, radiative corrections change the quantitative prediction only a little; the Cramer–Rao bound gives the theoretical limit for the considered observables, while the alternative expression is worse. Present constraints are available from TRISTAN and LEP 2.Source: §2.1.4.1 · pp. 42–43
  • At a 500 GeV collider, allowed regions are approximately elliptical for σ_l^T and between hyperbolas for A_l^FB and A_l^LR; A_LR adds only marginal improvement.The Born prediction for exclusion regions is accurate to below 10%, although this depends on kinematic cuts.Source: §2.1.4.1 · p. 44
  • With the stated coupling relations, future-collider measurements constrain five couplings, and agreement with Standard Model predictions leaves allowed regions containing zero.Quark couplings can be constrained only if Z′ couplings to leptons are nonzero.Source: §2.1.4.1 · p. 44

2.1.4.2 Model dependent constraints on MZ′

  • The Z′ mass limit scales approximately as (sL)^1/4 without systematic errors; comparable systematic and statistical errors reduce M_lim by only 16%.The estimate predicts exact exclusion limits well but does not distinguish among E6 models.Source: §2.1.4.2 · pp. 45, 47
  • The best present mass limits for extra neutral gauge bosons in E6 models come from proton collisions, while e+e− analyses provide an example of present limits.The excerpt does not provide the mass-limit values.Source: §2.1.4.2 · p. 46

2.1.4.3 Constraints on g2

  • With measurements accurate to 1%, a Z′ with g2 > g1/7 is excluded below its resonance at 95% confidence.Sensitivity to a weakly coupled Z′ increases as center-of-mass energy approaches its mass.Source: §2.1.4.3 · p. 48
  • The g2 constraint is not very sensitive to Γ2/M2, unlike Z′ limits from hadron collisions; PEP and PETRA data would improve the bound at small MZ′.The cited comparison notes that the 220 GeV ψ mass limit is better because it also uses hadronic observables.Source: §2.1.4.3 · p. 49
  • For MZ′ below the collision energy, searching for photons from radiative return to the Z′ resonance can improve the constraint on g2.Source: §2.1.4.3 · p. 49

2.1.4.4 Errors of model measurements

  • Systematic errors have a large influence on model-measurement uncertainties, more pronounced than on exclusion limits; scaling estimates reproduce most tested contour-size changes, with one exception.For the five scenarios in Figure 2.14, normalized region sizes are 1.9, 5.6, 2.5, 1, and 2.9; the estimates give 1.8, 4.2, 2.5, 1, and 2.8, except for the second curve.Source: §2.1.4.4 · pp. 50, 52, 53, Figure 2.14 · p. 54
  • For AMY, estimates (2.62) and (2.63) predict ΔMZ/MZ and Δg/g at approximately 3.4%, to be compared with the AMY analysis result.Across PEP, PETRA, and TRISTAN, the estimates predict experimental Z-coupling measurement errors within a factor of two.Source: §2.1.4.4 · pp. 50–51
  • Fermion-pair production alone leaves a two-fold sign ambiguity in Z′ measurements, while ALR-sensitive observables and beam polarization are essential for resolving coupling signs.Without ALR or related observables, a four-fold ambiguity remains; fit errors without beam polarization are several times larger.Source: §2.1.4.4 · pp. 51–52

2.1.5 Z′ measurements at s ≈M2

  • A precision scan of the Z′ peak favors a muon collider: its expected beam-energy spread is 0.04–0.08%, compared with 0.6–2.5% at discussed 500 GeV electron colliders.Some precision measurements require a spread as low as 0.2%.Source: §2.1.5 · p. 55

2.1.5.1 Model independent Z′ measurements

  • A model-independent Z′-resonance fit can fix the QED and Z1 terms because the resonances are well separated, and the required radiative corrections preserve the pole structure.The approach needs no unknown radiative corrections; QED and QCD corrections can be absorbed into effective coefficients without changing the poles.Source: §2.1.5.1 · pp. 55, 57

2.1.5.2 Model dependent Z′ measurements

  • Expected Z2 event yields and systematic errors could allow fermion couplings to be measured with precision similar to Z1 couplings, despite uncertain radiative corrections.The measurements constrain ZZ′ mixing, with sensitivity to θM ≈ 10^-4 reported in reference, and can constrain GUT parameters such as cos β.Source: §2.1.5.2 · pp. 57–58

2.1.5.3 Limit on g2

  • At M2 = √s, the Z2 peak cross section is independent of g2/g1, but finite beam energy spread limits sensitivity to that coupling ratio.The best limit on a weakly interacting Z′ is obtained at the peak.Source: §2.1.5.3 · p. 58

2.1.6 Z′ limits at s > M2

  • Above the Z2 resonance, photon energy measurement error limits bounds at muon colliders, while e+e− colliders also face large beam energy spread.The bounds use events with a fermion pair and one hard photon in the final state.Source: §2.1.6 · p. 59

2.1.6.1 Model independent limits on g2

  • A 1% e+e−→f ¯f cross-section measurement can exclude models with g2 > g1/7 at 95% confidence for all MZ′ < √s.The estimates are general and can be improved when more model details are known.Source: §2.1.6.1 · p. 59
  • With 1% photon energy resolution and √s/MZ′ ≈ 1, hard-photon events yield a 95% limit of g2 ≈ g1/24, improving on off-resonance fermion-pair production.The estimate assumes nσ = 2.Source: §2.1.6.1 · p. 60

2.1.6.2 Measurements of MZ′

  • If a Z′ signal with MZ′ < √s is found, its mass can be measured, and radiative-return photons can then be used to tune the collider to the resonance for precision measurements.Source: §2.1.6.2 · p. 61

2.2 Z′ search in e+e−→e+e−and e−e−→e−e−

Bhabha and Møller scattering provide complementary probes of Z′ couplings to electrons, with polarized beams particularly strengthening Møller exclusion limits. Model-dependent mass reach depends on which final states are included.

2.2.1.1 Amplitude

  • Bhabha and Møller scattering constrain the model-independent parameter a_e^N and are relatively insensitive to ZZ′ mixing; their angular distributions have distinct channel and forward-angle patterns.Bhabha exchange occurs in the s and t channels and is forward-peaked; Møller exchange occurs in the t and u channels and is symmetric.Source: §2.2.1.1 · p. 61

2.2.1.3 Observables

  • Unpolarized Bhabha scattering probes f0, polarized-electron asymmetries probe f1, and two polarized beams add f2 sensitivity; Møller angular distributions combine three contributions.The two-beam observables provide information unavailable from electron polarization alone in fermion-pair and W-pair production.Source: §2.2.1.3 · pp. 62–63

2.2.2 Radiative Corrections

  • QED corrections to Bhabha scattering are universal and can be generalized to include Z′ contributions; Møller QED and electroweak corrections, including initial-state corrections, are also discussed.The section notes that no essential new problems arise for Z′ exchange in the t or u channel.Source: §2.2.2 · pp. 63–64

2.2.3 Z′ Constraints

  • The Cramer–Rao bound describes ideal Møller sensitivity with infinitely many angular bins and no systematic errors, while the estimate predicts allowed coupling-band widths.Systematic effects from polarization, angular resolution, and luminosity are also studied.Source: §2.2.3 · p. 64
  • Møller measurements with equal-handed polarized beams independently restrict the two electron couplings, and polarized beams improve exclusion limits over unpolarized beams.Figure 2.15 shows 95% confidence exclusion regions for the L′ and R′ couplings under different beam polarizations.Source: §2.2.3 · p. 64, Figure 2.15 · p. 65
  • Under the stated assumptions, Møller scattering gives the strongest model-independent exclusion limits; adding electron and τ observables makes them comparable to fermion-pair limits, while polarized positrons would improve Bhabha limits.Figure 2.16 compares the exclusion limits including systematic errors.Source: §2.2.3 · p. 64, Figure 2.16 · p. 65

2.2.3.2 Model dependent constraints on MZ′

  • Bhabha and Møller mass limits can compete with fermion-pair production when only leptonic final states are used, but fermion-pair production gives better limits when quark observables contribute.The same distinction is stated for future mass limits.Source: §2.2.3.2 · p. 66

2.2.3.3 Errors of model measurements

  • Fermion-pair production and Møller scattering are complementary in model measurements: fermion-pair production removes a sign ambiguity in Møller measurements.Figure 2.17 presents the corresponding resolvability contours around possible true coupling values.Source: §2.2.3.3 · p. 66, Figure 2.17 · p. 66

2.3 Z′ search in e+e−→W +W −

The section examines how e⁺e⁻→W⁺W⁻ constrains Z′–Z mixing through anomalous couplings, with sensitivity shaped by collision energy and proximity to the Z′ resonance. It distinguishes below-resonance exclusion from near-resonance measurement, where interference can strengthen mixing constraints, while noting that mass bounds are weaker than those from fermion-pair production.

2.3 Z′ search in e+e−→W +W −

  • At large energies, nonzero ZZ′ mixing destroys the SM gauge cancellation in W-pair production, greatly magnifying new-physics effects; unitarity is restored for s ≫ M2_2.In the SM, the sum of interferences scales as ln s/s at large s.Source: §2.3 · p. 67
  • Case 1 yields stringent exclusions on ZZ′ mixing, whereas case 2 permits the best exclusion limit or the most accurate mixing-angle measurement if a Z′ exists.Case 2 is the interval (M2 − Γ2)2 < s < (M2 + Γ2)2.Source: §2.3 · p. 67

2.3.1.1 Amplitude

  • Because the W-pair amplitude is linear in the Z′ electron couplings, its measurement can determine their absolute sign and remove the sign ambiguity of fermion-pair production.In fermion-pair production, Z′ fermion couplings appear in pairs.Source: §2.3.1.1 · p. 68
  • Below the Z′ peak, W-pair measurements constrain the same parameter combinations as fermion-pair measurements on the Z1 resonance, but cannot separately determine the Z′ mass, mixing angle, and fermion couplings.The model-independent anomalous couplings δγ and δZ constrain combinations of Z′ parameters.Source: §2.3.1.1 · p. 69

2.3.1.3 Observables

  • Unpolarized W-pair production is dominated by left-handed electrons; right-handed-electron scattering is angle-symmetric and obeys ALR,FB = AFB.Same-helicity electron and positron beam cross sections vanish.Source: §2.3.1.3 · p. 70
  • Only some produced W pairs can be reconstructed; one leptonic and one hadronic decay gives the most complete information, but reduces effective luminosity.The hadronic decay measures the W energy-momentum but not its charge, while the leptonic decay identifies charge but loses momentum information to the neutrino.Source: §2.3.1.3 · p. 70

2.3.2 Background and Radiative Corrections

  • For off-shell W pairs, non-resonant diagrams to the same final state must be added coherently to obtain a gauge-invariant result; QCD factors applied to cut cross sections or backgrounds are only rough estimates.The complete SM one-loop correction to the off-shell process is not yet calculated, and the present Z′ analyses use the Born approximation.Source: §2.3.2.1 · p. 71, §2.3.2 · p. 71, §2.3.2.4 · p. 72

2.3.2.2 QED corrections

  • Initial-state QED corrections to off-shell W-pair production reach several percent near threshold, while the Coulomb correction can exceed 6% near threshold.The finite W width and slight off-shell production regularize the Coulomb singularity; initial-state corrections calculated with flux- or structure-function approaches are adequate within expected data accuracy.Source: §2.3.2.2 · pp. 71–72

2.3.3.1 Model independent constraints on g∗

  • The σ+T and σT observables alone leave bands of anomalous couplings unconstrained, while their combined bounds agree well with analytic estimates.For a fixed Z′ model, varying θM traces a straight line in the (δγ, δZ) plane; the model-independent bounds can be translated into mixing-angle limits, though W-pair production is less sensitive for models with small electron couplings. A fermion-pair cross-check can test whether a deviation is attributable to a Z′ alone.Source: §2.3.3.1 · p. 75, Figure 2.18 · p. 76

2.3.3.2 Constraints on θM

  • At 500 GeV, W-pair production adds little to ψ-model mixing limits from fermion pairs, but estimates predict considerable sensitivity improvement at higher energies.The estimate gives a reasonable prediction for the future θM constraint as M₂ → ∞, despite neglecting model details and approximating the excluded region.Source: §2.3.3.2 · p. 76, Figure 2.19 · p. 77

2.3.3.3 Constraints on M2

  • W-pair production yields weaker Z′ mass bounds than fermion-pair production because its sensitivity is suppressed by the small ZZ′ mixing angle.Indirect mass bounds combining the mixing-angle and Higgs constraints also depend on the extended Higgs sector and are worse than fermion-pair constraints.Source: §2.3.3.3 · p. 78

2.3.3.4 Model measurements

  • If anomalous couplings δγ and δZ are nonzero, their measurement errors can be estimated because cross-section deviations are linear in those couplings.The supplied passage introduces the estimate but provides no resulting error values.Source: §2.3.3.4 · p. 78

2.3.4 Z′ measurements at s ≈M2

  • On the Z₂ resonance, fermion-pair production best determines the Z′ mass, width, and fermion couplings because it has large statistics, while W-pair radiative corrections depend on unknown GUT parameters.Source: §2.3.4 · p. 78

2.3.4.1 Constraints on θM

  • Near the Z₂ resonance, Z₁Z₂ interference makes W-pair production much more sensitive to θM than fermion-pair production.The enhancement factor is approximately 2Γ₂/M₂ ≈ 1/20–1/50, depending on the Z′ model and exotic-fermion decays; it overcomes poorer statistics. Beam polarization is important, and sensitivity strengthens at higher energies.Source: §2.3.4.1 · pp. 78–79

2.3.4.2 Measurements of θM

  • For a detectable θM signal, its measurement error is estimated using the sensitivity limit derived for W-pair production near the Z₂ resonance.The analysis assumes the mixing is large enough to produce a signal; the limit used in the error estimate is taken from the near-resonance estimate.Source: §2.3.4.2 · p. 79

2.3.5 Z′ Constraints at s > M2

  • Far above the Z₂ resonance, θM sensitivity is much worse, and radiative-return signals cannot compete with fermion-pair production because of lower statistics.At s ≫ M₂², the cross section behaves like the Standard Model and unitarity is restored independently of the large gauge group’s details.Source: §2.3.5 · p. 79

2.4 Z′ search in other reactions

Other two-body channels are less sensitive to a Z′ than W-pair production, and higher-order channels generally add little usable sensitivity after cuts. WW scattering remains a possible TeV-scale probe of Z′ mixing.

2.4 Z′ search in other reactions

  • e+e−→ZZ and e+e−→ZH lack W-pair production’s enhancement factors and are therefore much less sensitive to a Z′.Source: §2.4 · p. 79
  • Higher-order processes do not enhance the Z′ contribution; after cuts remove their background-like enhancements, their cross sections are too small to compete with two-particle final states.Resonating gauge bosons and collinear radiation can enhance these processes without enhancing the Z′ contribution.Source: §2.4 · p. 80
  • Invariant-mass cuts on the muon and tau pairs suppress photon exchange and, above the Z mass, Z exchange too, leaving a cross section about α^2 smaller than particle-pair production.The effect is not altered by radiative corrections.Source: §2.4 · p. 80
  • The e+e−→Z′Z′ process requires very high energies, while e+e−→e+e−Z′ is difficult to observe above background; measurements below the Z′ peak can already resolve the stated θ ambiguity.Source: §2.4 · p. 80
  • WW scattering could be worth investigating at TeV colliders because nonzero ZZ′ mixing would disrupt the gauge cancellation that suppresses its high-energy cross section.Individual WW-scattering interferences scale like s^2, while the SM total cross section scales like 1/s at very high energies.Source: §2.4 · p. 80

Z′ search at pp and p¯p colliders

Hadron-collider searches rely mainly on direct Z′ production and its decays, with reach limited by parton energies and challenging backgrounds. Other production and decay channels can add complementary model information, but have specific constraints.

Z′ search at pp and p¯p colliders

  • A detectable Z′ must be lighter than the proton collision energy and, in practice, at least two times lighter because production occurs in parton collisions.Its decay products must also be separated from the hadronic SM background, which can make some observables difficult or impossible to measure.Source: Z′ search at pp and p¯p colliders · p. 81
  • A fermion-pair decay is a likely first hadron-collider signal: its invariant-mass peak can separate it from background and measure the Z′ mass.Tagging different final-state fermions provides cross sections and asymmetries as observables.Source: Z′ search at pp and p¯p colliders · p. 81
  • Rare Z′→f1 f̄2 V decays are higher-order channels enhanced by large soft and collinear logarithms, providing complementary information about a Z′ model.Source: Z′ search at pp and p¯p colliders · p. 81
  • Z′ decay to W pairs requires ZZ′ mixing and suffers from background from associated W production with two jets; no other useful hadron-collider processes are identified.Source: Z′ search at pp and p¯p colliders · p. 81
  • Associated production pp→Z′V, with V=Z, W, or γ, is higher order but logarithmically enhanced when s≫M_Z^2 and adds independent information about Z′ models.Source: Z′ search at pp and p¯p colliders · p. 81

3.1 Born cross section of pp(p¯p) →Z′ →f ¯f

For the hadron-collider Born cross section, the treatment neglects ZZ′ mixing and expresses production observables through quark structure functions. The section also describes additional observables enabled by final-state or proton polarization.

3.1 Born cross section of pp(p¯p) →Z′ →f f̄

  • ZZ′ mixing is neglected for this reaction because its sensitivity is much weaker than that of e+e− experiments at the Z1 peak.Source: §3.1 · p. 81
  • The expressions for total and forward-backward cross sections depend on quark structure functions and include the helicities of the initial and final fermions.Source: §3.1 · p. 82
  • If final-fermion polarization is measurable, it provides an additional observable; polarized proton beams also allow left-right cross sections and left-right differences in forward-backward cross sections.Source: §3.1 · p. 82
  • With both proton beams polarized, more complicated cross sections can be measured.Source: §3.1 · p. 82

3.2 Higher order processes and background

The section surveys rare and associated Z′ processes, radiative corrections, and the backgrounds that shape hadron-collider searches.

3.2.1 Rare Z′ decays

  • Rare Z′ decays to ordinary fermions plus a gauge boson are logarithmically enhanced; Z′→Wlνl provides complementary coupling information, while Z′→l+l−l+l− is a consistency check.The latter has only weak dependence on Z′ couplings; rare decays to quarks are measurable with larger systematic errors.Source: §3.2.1 · pp. 82–83

3.2.2 Associated Z′ production

  • Associated pp→Z′V production is logarithmically enhanced at high energies, is free of Standard Model backgrounds, and can compete in determining Z′ model parameters.The enhancement arises in a kinematically constrained region of Q², with structure functions suppressing larger Q².Source: §3.2.2 · p. 83

3.2.3 Radiative Corrections

  • A first-discovery approximation adds tree-level Z′ contributions to the Standard Model cross section, but its adequacy must be checked for model measurements.Radiative corrections to new physics are treated as small corrections to a small effect in this approximation.Source: §3.2.3 · p. 83

3.2.3.1 QCD corrections

  • QCD corrections raise the Tevatron’s lowest-order vector-boson production cross section by 20–30%.These corrections are often called K factors.Source: §3.2.3.1 · p. 83

3.2.3.2 QED and weak corrections

  • Final-state radiation dominates QED corrections, while initial-state and interference corrections are small after collinear singularities are factorized into parton distributions.Pure weak corrections to Z′ production are expected to be even smaller than those for Standard Model Z production.Source: §3.2.3.2 · pp. 83–84

3.2.4 Background

  • Quark-pair searches have reduced Z′ sensitivity relative to electron and muon pairs because of QCD backgrounds.Tau-pair backgrounds for an E6-GUT Z′ were found manageable, including backgrounds from W pairs, misidentified jets, top pairs, and Drell–Yan production.Source: §3.2.4 · p. 84

3.3 Observables

The section uses cross sections, asymmetries, and rapidity-based observables to probe Z′ properties, while noting that not all are experimentally accessible.

3.3 Observables

  • Total fermion cross sections and several asymmetries serve as hadron-collider observables, but hadronic-environment backgrounds prevent measuring all of them in practice.Final-state polarization asymmetry is independent of initial-fermion couplings and, in hadron collisions, quark structure functions.Source: §3.3 · p. 84
  • The rapidity ratio ry1 helps distinguish Z′ models, whereas AFBy1 is less model-sensitive.The text describes AFBy1 as “a refinement of a refinement.”Source: §3.3 · p. 84
  • Ratios involving rare Z′ decays can be defined, and rlνW depends only on Z′ couplings to leptons.The notation sums l over e and μ, and ν over the three neutrino flavors.Source: §3.3 · p. 85
  • Associated-production cross sections enter ratios, and the text distinguishes ALR,FB from the final-state polarization asymmetry notation Apol.ALR,FB corresponds to Apol,FB in reference, while Apol denotes final-state polarization asymmetries following reference.Source: §3.3 · pp. 84–85

3.4 Z′ constraints

The section derives approximate hadron-collider constraints and measurement sensitivities, then compares those estimates with present and projected analyses.

3.4.1.1 Estimate

  • The hadron-collider cross-section estimate uses an exponential approximation that fits f_u(rz) satisfactorily through rz = 10, but not Standard Model Z production at the Tevatron.The estimate is intended to make constraints and measurement sensitivities to collider conditions and model parameters transparent.Source: §3.4.1.1 · pp. 85–86

3.4.1.2 Present constraints

  • For CDF dilepton data, the estimate predicts a limiting cross section of 0.031 pb, in reasonable agreement with the exact result; QCD backgrounds prevented E6-Z′ constraints from dijets in the cited analysis.The CDF comparison accounts for detection efficiencies and the QCD K factor.Source: §3.4.1.2 · p. 87

3.4.2.1 Estimate

  • Model dependence enters the hadron-collider exclusion estimate under a logarithm; the ξ-normalized relation reproduces published limits to 10% for pp and p̄p collisions.The reported ξ values are 0.13 for pp and 0.10 for p̄p collisions.Source: §3.4.2.1 · p. 88

3.4.2.2 Present constraints

  • No Z′ signal was found in present experiments, and the quoted limits agree well with the prediction from relation (3.18).The section interprets the negative search result as model-dependent mass exclusion limits.Source: §3.4.2.2 · p. 88

3.4.2.3 Future constraints

  • The estimate (3.18) agrees with exact future-collider results within 10% across a wide range of integrated luminosities and collision energies.Analyses require luminosity, collision energy, fitted observables, and a demanded Z′ event count; cuts and radiative corrections must also be specified when used.Source: §3.4.2.3 · p. 89

3.4.3 Model independent constraint on g2/g1

  • For the SSM with MZ′ < 400 GeV, the model-independent coupling bound is g2 < g1/4.5.The bound is obtained graphically by shifting the signal cross section in figure 3.2.Source: §3.4.3 · p. 89

3.4.4.1 Estimate

  • Model-measurement errors share the exclusion estimate’s s and L scaling but are more model-dependent; the estimates fail for too-small fm and the approximation (3.15) becomes inaccurate at too-large √s/MZ′.The model dependence is stronger because cZ′ is no longer logarithmically suppressed.Source: §3.4.4.1 · p. 90

3.4.4.2 Future measurements

  • LHC measurements can determine symmetry-breaking parameters at around 10% under the assumptions of [45], while the stated γ parameters have a 16-fold sign ambiguity and a 5% expected accuracy for MZ′ = 1 TeV.A few Z′ events can suffice to measure MZ′ with accuracy ΓZ′; systematic errors become important with larger event samples.Source: §3.4.4.2 · p. 92

Z′ search in other experiments

The section briefly notes that other experiments can also constrain extra neutral gauge bosons.

Z′ search in other experiments

  • Other possible sources of bounds include electron–proton collisions, atomic parity violation, neutrino scattering, and cosmology.The paper briefly comments on these experiments in the following sections.Source: Z′ search in other experiments · p. 93

4.1 ep collisions

The review finds ep collisions limited by hadronic background and unable to match hadron colliders for usual GUT Z′ mass bounds. Sensitivity depends on radiative corrections, observables, kinematics, and model assumptions.

4.1.1 Born cross section

  • Most systematic errors cancel in the defined asymmetries, while total cross sections are relatively insensitive to Z′ effects because photon exchange is large.The massless-particle approximation is used for the cross-section treatment.Source: §4.1.1 · p. 94

4.1.2 Radiative Corrections

  • Electroweak corrections are the same size as Z′ effects and must be included; QED corrections to the small Z′ contribution can be treated model-independently in the leading-log approximation.The QED terms include initial- and final-state radiation and the Compton peak.Source: §4.1.2 · p. 94

4.1.3 Z′ constraints

  • A model-independent ep analysis involves six electron, up-quark, and down-quark Z′ couplings simultaneously and can constrain their relative signs.Under the stated condition, the analysis constrains five coupling combinations.Source: §4.1.1 · p. 93, §4.1.3 · p. 94
  • HERA cannot compete with hadron colliders for Z′ bounds in usual GUTs, and its anomalous high-Q² events cannot be explained by typical extended-gauge-theory Z′s.The excited weak boson and BESS models are not ruled out by the data discussed.Source: §4.1.3 · pp. 94–95
  • ep bounds scale with luminosity as in e+e− collisions and probe Z′ masses comparable to √s, but their model dependence is large.Reasonable statistics occur for Q² well below s, and the observable error depends on Q².Source: §4.1.3 · p. 95

4.2 Atomic parity violation

Atomic parity violation provides competitive constraints on extra neutral gauge bosons through parity-violating atomic transitions and the weak charge. Its constraints mainly address the Z′ mass ratio rather than mixing, and cancellations among new-physics contributions can relax the bounds.

4.2 Atomic parity violation

  • Atomic parity-violating measurements have reached precision that permits constraints competitive with collider experiments.The weak charge encodes the coherent interaction of an electron with the nucleus through axial-electron and vector-quark couplings.Source: §4.2 · p. 95
  • Agreement between the measured and Standard Model weak charge constrains new physics; for cesium, the cited prediction is Q^SM_W = −73.04.The cesium weak charge is expressed as −376C1u −422C1d.Source: §4.2 · p. 96
  • Atomic parity violation constrains mainly M1/M2 because Z1-peak measurements give stronger constraints on the ZZ′ mixing angle.For cesium, the first contribution is numerically negligible because of an accidental cancellation and the present M2 constraint.Source: §4.2 · p. 96
  • Compensating contributions from multiple new-physics sources can relax the Z′ limits inferred from the weak charge.Such cancellation effects are explicitly demonstrated in the cited analysis.Source: §4.2 · p. 96

4.3 Neutrino scattering

Neutrino scattering constrains extra neutral gauge bosons through neutrino couplings, with measurements at lower energies complementing Z-peak experiments. Existing bounds no longer rival collider limits, while proposed very-low-energy experiments may provide interesting Z′ mass bounds.

4.3 Neutrino scattering

  • High-energy neutrino-scattering limits were historically useful but cannot now compete with collider constraints.Neutrino-electron and neutrino-nucleon experiments measure neutrino couplings to the Z boson.Source: §4.3 · p. 97
  • Neutrino-scattering measurements probe couplings at lower center-of-mass energies than Z1-peak experiments, making them complementary.The cited values are vν = −0.035 ± 0.012 ± 0.012 and aν = −0.503 ± 0.006 ± 0.016.Source: §4.3 · p. 97
  • Agreement with the Standard Model constrains new physics, and model-independent neutrino-nucleon results can be converted into Z′ bounds for different models.The review gives CHARM II limits and constraints for an unconstrained Higgs sector.Source: §4.3 · p. 97, Table 4.3 · p. 97
  • Proposed sub-MeV neutrino-electron scattering experiments could constrain Z′ masses, although their possible mixing-angle bounds cannot match current LEP measurements.The quoted mass limits are extracted while neglecting systematic errors.Source: §4.3 · pp. 97–98

4.4 Cosmology

The review concludes that Z′ constraints depend strongly on the process, collider, and model assumptions. Cosmological bounds can be stronger but rely on more assumptions, while lepton and hadron colliders provide complementary constraints and measurements.

4.4 Cosmology

  • Big-bang nucleosynthesis gives 95% CL intervals Nν = 3.0–3.7 or 3.0–3.2 under two stated baryon-density assumptions; interpreting these as light-Z′ mass bounds yields stronger-than-collider limits with more model assumptions.The intervals assume YP = 0.242 ± 0.003 and the respective D+3He or D/H baryon-density bounds.Source: §4.4 · p. 98
  • Supernova constraints can exceed collider limits when additional light neutrinos are present, except when the Z′ coupling to right-handed neutrinos vanishes; nonzero mixing strengthens the bounds.The review says radiative corrections have little influence on these limits.Source: §4.4 · p. 98

Summary and conclusions

  • Lepton colliders best constrain ZZ′ mixing and weakly interacting Z′s, whereas hadron colliders best constrain popular-GUT Z′ masses; ep limits cannot compete with either.Hadron-collider mass bounds are relatively model-insensitive when the signal can be separated from background, but worsen for broader Z2 states.Source: Summary and conclusions · pp. 99, 102
  • For a Z′ signal, hadron colliders measure fermion couplings with smaller errors but a 16-fold sign ambiguity, while polarized-electron lepton colliders reduce that ambiguity to twofold.Source: Summary and conclusions · p. 99
  • Model-independent analyses apply broadly but may leave observables unusable or constraints incomparable; consistent model assumptions can tighten exclusions or reduce errors on remaining parameters.The review also treats QED corrections as model-independent and assumes new Z′ effects first enter at tree level for its higher-order discussion.Source: Summary and conclusions · pp. 99–100
  • Off-resonance lepton-collider limits reach above the collision energy but depend strongly on Z′ couplings; W-pair production offers strong constraints on mixing-related parameters through disrupted Standard Model gauge cancellation.Above the Z2 resonance, unitarity is restored and the large enhancement of Z′ effects ends.Source: Summary and conclusions · p. 102
  • At the Z2 resonance, experiments enable precise mass, width, and coupling measurements; muon colliders are favored because their beam-energy spread is smaller.Above the resonance, radiative-return limits depend on photon energy resolution.Source: Summary and conclusions · p. 102
  • Hadron colliders directly produce Z′ bosons only below their proton collision energy; raising that energy is favored over raising luminosity to improve exclusion limits.Muon-pair decays are favored for an initial Z′ discovery.Source: Summary and conclusions · pp. 102–103

Appendix A

The notation tables gather conventions used across the review. Appendix B catalogs officially released FORTRAN programs for Z′ studies and describes their uses.

Notation

  • The notation tables collect Z′ parameter definitions, kinematic parameters, observables, and notation used in the remaining chapters.The review says chapter one mainly defines Z′ parameters and chapter two introduces kinematic parameters and observables.Source: Notation · p. 120, Table A.1 · p. 120, Table A.2 · p. 121

Appendix B Available FORTRAN programs for Z′ fits

  • Appendix B catalogs officially released FORTRAN programs known to the author, with program names, references, and locations.Program-description references are bold, original physics papers are roman, and examples of use are italicized.Source: Appendix B · p. 123, Table B.1 · p. 124
  • ZCAMEL provides a fast standalone tool for theoretical e+e− fermion-pair studies with full O(α) QED corrections and initial-state soft-photon exponentiation.The appendix directs users to ZEFIT for fits to data.Source: Appendix B · p. 123
  • ZEFIT adds Z′ contributions to fermion-pair fits and treats electroweak corrections and ZZ′ mixing for data around and above the Z peak.ZEFIT must be used with ZFITTER, which supplies known SM corrections for e+e− fermion-pair production.Source: Appendix B · p. 123
  • GENTLE/4fan studies four-fermion e+e− production and Z′ effects in W-pair production through anomalous couplings.PYTHIA was used to obtain present Z′ limits at hadron colliders.Source: Appendix B · pp. 123–124
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