55. Z Boson

1

55. Z Boson
Revised August 2018 by M. Grünewald (University Coll. Dublin) and A. Gurtu (CERN; TIFR
Mumbai).
Precision measurements at the Z-boson resonance using electron–positron colliding beams began
in 1989 at the SLC and at LEP. During 1989–95, the four LEP experiments (ALEPH, DELPHI,
L3, OPAL) made high-statistics studies of the production and decay properties of the Z. Although
the SLD experiment at the SLC collected much lower statistics, it was able to match the precisionof LEP experiments in determining the effective electroweak mixing angle sin2 θW and the rates ofZ decay to b- and c-quarks, owing to availability of polarized electron beams, small beam size, andstable beam spot.
The Z-boson properties reported in this section may broadly be categorized as:• The standard ‘lineshape’ parameters of the Z consisting of its mass, MZ , its total width, ΓZ ,and its partial decay widths, Γ (hadrons), and Γ (``) where ` = e, µ, τ, ν;• Z asymmetries in leptonic decays and extraction of Z couplings to charged and neutral
leptons;• The b- and c-quark-related partial widths and charge asymmetries which require special techniques;• Determination of Z decay modes and the search for modes that violate known conservation
laws;• Average particle multiplicities in hadronic Z decay;• Z anomalous couplings.
The effective vector and axial-vector coupling constants describing the Z-to-fermion coupling
are also measured in pp̄ and ep collisions at the Tevatron and at HERA. The corresponding crosssection formulae are given in Section 39 (Cross-section formulae for specific processes) and Section
16 (Structure Functions) in this Review. In this minireview, we concentrate on the measurementsin e+ e− collisions at LEP and SLC.
The standard ‘lineshape’ parameters of the Z are determined from an analysis of the production
cross sections of these final states in e+ e− collisions. The Z → νν(γ) state is identified directly
by detecting single photon production and indirectly by subtracting the visible partial widths
from the total width. Inclusion in this analysis of the forward-backward asymmetry of charged(0,`)
leptons, AFB , of the τ polarization, P (τ ), and its forward-backward asymmetry, P (τ )f b , enables
the separate determination of the effective vector (g V ) and axial vector (g A ) couplings of the Z
to these leptons and the ratio (g V /g A ), which is related to the effective electroweak mixing angle
sin2 θW (see the “Electroweak Model and Constraints on New Physics” review).
Determination of the b- and c-quark-related partial widths and charge asymmetries involves
tagging the b and c quarks for which various methods are employed: requiring the presence ofa high momentum prompt lepton in the event with high transverse momentum with respect tothe accompanying jet; impact parameter and lifetime tagging using precision vertex measurementwith high-resolution detectors; application of neural-network techniques to classify events as b or
non-b on a statistical basis using event–shape variables; and using the presence of a charmed meson(D/D∗ ) or a kaon as a tag.

55.1 Z-parameter determination
LEP was run at energy points on and around the Z mass (88–94 GeV) constituting an energy‘scan’. The shape of the cross-section variation around the Z peak can be described by a BreitS. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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Wigner ansatz with an energy-dependent total width [1–3]. The three main properties of this
distribution, viz., the position of the peak, the width of the distribution, and the height ofthe peak, determine respectively the values of MZ , ΓZ , and Γ (e+ e− ) × Γ (f f ), where Γ (e+ e− )and Γ (f f ) are the electron and fermion partial widths of the Z. The quantitative determinationof these parameters is done by writing analytic expressions for these cross sections in terms of
the parameters, and fitting the calculated cross sections to the measured ones by varying theseparameters, taking properly into account all the errors. Single-photon exchange (σγ0 ) and γ-Z0 ) are included, and the large (∼25 %) initial-state radiation (ISR) effects are takeninterference (σγZ
into account by convoluting the analytic expressions over a ‘Radiator Function’ [1–5] H(s, s0 ). Thus
for the process e+ e− → f f :Z

σf (s) =H(s, s0 ) σf0 (s0 ) ds0(55.1)0
σf0 (s) =σZ0 + σγ0 + σγZ(55.2)

Γ (e+ e− )Γ (f f )s ΓZ2(s − MZ2 )2 + s2 ΓZ2 /MZ2

12π
MZ2
ΓZ2
4πα2 (s) 2 f
σγ0 =
Qf Nc
3s√
2 2α(s)(s − MZ2 )MZ20σγZ=−
(Qf GF Ncf GVe GVf )3(s − MZ2 )2 + s2 ΓZ2 /MZ2
σZ0 =(55.3)(55.4)(55.5)

where Qf is the charge of the fermion, Ncf = 3 for quarks and 1 for leptons, and GVf is the vector
coupling of the Z to the fermion-antifermion pair f f .
0 is expected to be much less than σ 0 , the LEP Collaborations have generally calculatedSince σγZZ
0 leads to a tighter
the interference term in the framework of the Standard Model. This fixing of σγZ
constraint on MZ , and consequently a smaller error on its fitted value. It is possible to relax thisconstraint and carry out the fit within the S-matrix framework, which is briefly described in thenext section.
In the above framework, the QED radiative corrections have been explicitly taken into accountby convoluting over the ISR and allowing the electromagnetic coupling constant to run [6]: α(s) =α/(1 − ∆α). On the other hand, weak radiative corrections that depend upon the assumptions of
the electroweak theory and on the values of Mtop and MHiggs are accounted for by absorbing theminto the couplings, which are then called the effective couplings GV and GA (or alternatively theeffective parameters of the ? scheme of Kennedy and Lynn [7].)f
GVf and GA
are complex numbers with small imaginary parts. As experimental data does
not allow simultaneous extraction of both real and imaginary parts of the effective couplings, theffconvention gA= Re(GA) and gVf = Re(GVf ) is used and the imaginary parts are added in the fittingcode [4].
Defining
gf · gf
Af = 2 f V A f(55.6)(gV )2 + (gA )2
the lowest-order expressions for the various lepton-related asymmetries on the Z pole are [8–10](0,`)
AFB = (3/4)Ae Af , P (τ ) = −Aτ , P (τ )f b = −(3/4)Ae , ALR = Ae . The full analysis takes into account the energy-dependence of the asymmetries. Experimentally ALR is defined as (σL −σR )/(σL +σR ), where σL(R) are the e+ e− → Z production cross sections with left- (right)-handed electrons.
The definition of the partial decay width of the Z to f f includes the effects of QED and QCD
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55. Z Boson

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final-state corrections, as well as the contribution due to the imaginary parts of the couplings:
Γ (f f ) =2
GF MZ3 f f 2 f√
Nc ( GA RA + GVf RVf ) + ∆ew/QCD
6 2π(55.7)fwhere RVf and RA
are radiator factors to account for final state QED and QCD corrections, aswell as effects due to nonzero fermion masses, and ∆ew/QCD represents the non-factorizable electroweak/QCD corrections.

55.2 S-matrix approach to the Z
While most experimental analyses of LEP/SLC data have followed the ‘Breit-Wigner’ approach,an alternative S-matrix-based analysis is also possible. The Z, like all unstable particles, is associated with a complex pole in the S matrix. The pole position is process-independent and gaugeinvariant. The mass, M Z , and width, Γ Z , can be defined in terms of the pole in the energy plane
via [11–14]
2
s = M Z − iM Z Γ Z(55.8)
leading to the relationsq

M Z = MZ / 1 + ΓZ2 /MZ2
≈ MZ − 34.1 MeV(55.9)qΓ Z = ΓZ / 1 + ΓZ2 /MZ2
≈ ΓZ − 0.9 MeV .

(55.10)

The LEP collaborations [15] have analyzed their data using the S–matrix approach as definedin Eq. (55.8), in addition to the conventional one. They observe a downward shift in the Z mass
as expected.

55.3 Handling the large-angle e+ e− final state
Unlike other f f decay final states of the Z, the e+ e− final state has a contribution not only
from the s-channel but also from the t-channel and s-t interference. The full amplitude is not
amenable to fast calculation, which is essential if one has to carry out minimization fits withinreasonable computer time. The usual procedure is to calculate the non-s channel part of the crosssection separately using the Standard Model programs ALIBABA [16] or TOPAZ0 [17], with themeasured value of Mtop , and MHiggs = 150 GeV, and add it to the s-channel cross section calculated
as for other channels. This leads to two additional sources of error in the analysis: firstly, thetheoretical calculation in ALIBABA itself is known to be accurate to ∼ 0.5%, and secondly, there
is uncertainty due to the error on Mtop and the unknown value of MHiggs (100–1000 GeV). Theseerrors are propagated into the analysis by including them in the systematic error on the e+ e− final
state. As these errors are common to the four LEP experiments, this is taken into account when
performing the LEP average.

55.4 Errors due to uncertainty in LEP energy determination
The systematic errors related to the LEP energy measurement, see [18–23], can be classified as:• The absolute energy scale error;• Energy-point-to-energy-point errors due to the nonlinear response of the magnets to theexciting currents;• Energy-point-to-energy-point errors due to possible higher-order effects in the relationshipbetween the dipole field and beam energy;
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55. Z Boson

4• Energy reproducibility errors due to various unknown uncertainties in temperatures, tidal
effects, corrector settings, RF status, etc.
Precise energy calibration was done outside normal data-taking using the resonant depolarization
technique. Run-time energies were determined every 10 minutes by measuring the relevant machineparameters and using a model which takes into account all the known effects, including leakage
currents produced by trains in the Geneva area and the tidal effects due to gravitational forces of
the Sun and the Moon. The LEP Energy Working Group has provided a covariance matrix fromthe determination of LEP energies for the different running periods during 1993–1995 [18].

55.5 Choice of fit parameters
The LEP Collaborations have chosen the following primary set of parameters for fitting:(0,`)

0
MZ , ΓZ , σhadron, R(lepton), AFB ,where
0
R(lepton) = Γ (hadrons)/Γ (lepton), σhadron= 12πΓ (e+ e− )Γ (hadrons)/MZ2 ΓZ2 .

With a knowledge of these fitted parameters and their covariance matrix, any other parameter canbe derived. The main advantage of these parameters is that they form a physics motivated set ofparameters with much reduced correlations.
Thus, the most general fit carried out to cross section and asymmetry data determines the
(0,e)
(0,µ)
(0,τ )
0nine parameters: MZ , ΓZ , σhadron, R(e), R(µ), R(τ ), AFB , AFB , AFB . Assumption of lepton(0,`)
0universality leads to a five-parameter fit determining MZ , ΓZ , σhadron, R(lepton), AFB .

55.6 Combining results from LEP and SLC experiments
With a steady increase in statistics over the years and improved understanding of the common
systematic errors between LEP experiments, the procedures for combining results have evolved
continuously [24]. The Line Shape Sub-group of the LEP Electroweak Working Group investigatedthe effects of these common errors, and devised a combination procedure for the precise determination of the Z parameters from LEP experiments. Using these procedures, this note also gives theresults after combining the final parameter sets from the four experiments, and these are the results quoted as the fit results in the Z listings below. Transformation of variables leads to values ofderived parameters like partial decay widths and branching ratios to hadrons and leptons. Finally,f◦transforming the LEP combined nine parameter set to (MZ , ΓZ , σhadron, gA, gVf , f = e, µ, τ ) using
the average values of lepton asymmetry parameters (Ae , Aµ , Aτ ) as constraints, leads to the best
fitted values of the vector and axial-vector couplings (gV , gA ) of the charged leptons to the Z.
Brief remarks on the handling of common errors and their magnitudes are given below. The
identified common errors are those coming from(a) LEP energy-calibration uncertainties, and(b) the theoretical uncertainties in (i) the luminosity determination using small angle Bhabha
scattering, (ii) estimating the non-s channel contribution to large angle Bhabha scattering, (iii) the
calculation of QED radiative effects, and (iv) the parametrization of the cross section in terms of
the parameter set used.

55.7 Common LEP energy errors
All the collaborations incorporate in their fit the full LEP energy error matrix as provided by
the LEP energy group for their intersection region [18]. The effect of these errors is separated out
from that of other errors by carrying out fits with energy errors scaled up and down by ∼ 10% andredoing the fits. From the observed changes in the overall error matrix, the covariance matrix of
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555. Z Boson

the common energy errors is determined. Common LEP energy errors lead to uncertainties on MZ ,◦ΓZ , and σhadronof 1.7, 1.2 MeV, and 0.011 nb, respectively.

55.8 Common luminosity errors
BHLUMI 4.04 [25] is used by all LEP collaborations for small-angle Bhabha scattering leading to
a common uncertainty in their measured cross sections of 0.061% [26]. BHLUMI does not include a
correction for production of light fermion pairs. OPAL explicitly corrects for this effect and reduces
their luminosity uncertainty to 0.054%, which is taken fully correlated with the other experiments.
The other three experiments among themselves have a common uncertainty of 0.061%.

55.9 Common non-s channel uncertainties
The same standard model programs ALIBABA [16] and TOPAZ0 [17] are used to calculate
the non-s channel contribution to the large angle Bhabha scattering [27]. As this contribution isa function of the Z mass, which itself is a variable in the fit, it is parameterized as a function of
MZ by each collaboration to properly track this contribution as MZ varies in the fit. The common(0,e)
errors on Re and AF B are 0.024 and 0.0014 respectively, and are correlated between them.

55.10 Common theoretical uncertainties: QED
There are large initial-state photon and fermion pair radiation effects near the Z resonance,
for which the best currently available evaluations include contributions up to O(α3 ). To estimate
the remaining uncertainties, different schemes are incorporated in the standard model programs
ZFITTER [5], TOPAZ0 [17], and MIZA [28]. Comparing the different options leads to error◦
estimates of 0.3 and 0.2 MeV on MZ and ΓZ respectively, and of 0.02% on σhadron.

55.11 Common theoretical uncertainties: parametrization of lineshape and asymmetries
To estimate uncertainties arising from ambiguities in the model-independent parametrizationof the differential cross-section near the Z resonance, results from TOPAZ0 and ZFITTER werecompared by using ZFITTER to fit the cross sections and asymmetries calculated using TOPAZ0.
(0,`)◦
The resulting uncertainties on MZ , ΓZ , σhadron, R(lepton), and AF B are 0.1 MeV, 0.1 MeV,
0.001 nb, 0.004, and 0.0001 respectively.
◦Thus, the overall theoretical errors on MZ , ΓZ , σhadron
are 0.3 MeV, 0.2 MeV, and 0.008 nb(0,`)
respectively; on each R(lepton) is 0.004 and on each AF B is 0.0001. Within the set of three(0,`)
R(lepton)’s and the set of three AF B ’s, the respective errors are fully correlated.
All the theory-related errors mentioned above utilize Standard Model programs which need the
Higgs mass and running electromagnetic coupling constant as inputs; uncertainties on these inputs
will also lead to common errors. All LEP collaborations used the same set of inputs for Standard
Model calculations: MZ = 91.187 GeV, the Fermi constant GF = (1.16637±0.00001)×10−5 GeV−2[29], α(5) (MZ ) = 1/128.877 ± 0.090 [30], αs (MZ ) = 0.119 [31], Mtop = 174.3 ± 5.1 GeV [31] andMHiggs = 150 GeV. The only observable effect, on MZ , is due to the variation of MHiggs between
100–1000 GeV (due to the variation of the γ/Z interference term which is taken from the Standard
Model): MZ changes by +0.23 MeV per unit change in log10 MHiggs /GeV, which is not an error but
a correction to be applied once MHiggs is determined. The effect is much smaller than the error on
MZ (±2.1 MeV).

55.12 Methodology of combining the LEP experimental results
The LEP experimental results actually used for combination are slightly modified from thosepublished by the experiments (which are given in the Listings below). This has been done in orderto facilitate the procedure by making the inputs more consistent. These modified results are given
explicitly in [24]. The main differences compared to the published results are (a) consistent use of
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ZFITTER 6.23 and TOPAZ0 (the published ALEPH results used ZFITTER 6.10); (b) use of the
combined energy-error matrix, which makes a difference of 0.1 MeV on the MZ and ΓZ for L3 only
as at that intersection the RF modeling uncertainties are the largest.
Thus, nine-parameter sets from all four experiments with their covariance matrices are used
together with all the common errors correlations. A grand covariance matrix, V , is constructed
and a combined nine-parameter set is obtained by minimizing χ2 = ∆T V −1 ∆, where ∆ is the vectorof residuals of the combined parameter set to the results of individual experiments. Imposing lepton
universality in the combination results in the combined five-parameter set.

55.13 Study of Z → bb and Z → cc
In the sector of c- and b-physics, the LEP experiments have measured the ratios of partial
widths Rb = Γ (Z → bb)/Γ (Z → hadrons), and Rc = Γ (Z → cc)/Γ (Z → hadrons), and theccforward-backward (charge) asymmetries Abb
FB and AFB . The SLD experiment at SLC has measured
the ratios Rc and Rb and, utilizing the polarization of the electron beam, was able to obtain the
final state coupling parameters Ab and Ac from a measurement of the left-right forward-backward
asymmetry of b− and c−quarks. The high precision measurement of Rc at SLD was made possible
owing to the small beam size and very stable beam spot at SLC, coupled with a highly precise
CCD pixel detector. Several of the analyses have also determined other quantities, in particular
the semileptonic branching ratios, B(b → `− ), B(b → c → `+ ), and B(c → `+ ), the average time0
integrated B 0 B mixing parameter χ and the probabilities for a c–quark to fragment into a D+ ,a Ds , a D∗+ , or a charmed baryon. The latter measurements do not concern properties of theZ boson, and hence they do not appear in the Listing below. However, for completeness, we willreport at the end of this minireview their values as obtained fitting the data contained in the Z
section. All these quantities are correlated with the electroweak parameters, and since the mixtureof b hadrons is different from the one at the Υ (4S), their values might differ from those measured
at the Υ (4S).
All the above quantities are correlated to each other since:• Several analyses (for example the lepton fits) determine more than one parameter simultaneously;• Some of the electroweak parameters depend explicitly on the values of other parameters (forexample Rb depends on Rc );• Common tagging and analysis techniques produce common systematic uncertainties.
The LEP Electroweak Heavy Flavour Working Group has developed [32] a procedure for combining the measurements taking into account known sources of correlation. The combining procedure
determines fourteen parameters: the six parameters of interest in the electroweak sector, Rb , Rc ,bb , Acc , A and A and, in addition, B(b → `− ), B(b → c → `+ ), B(c → `+ ), χ, f (D + ), f (D ),
AFB
s
c
b
FB
f (cbaryon ) and P (c → D∗+ ) × B(D∗+ → π + D0 ), to take into account their correlations with theelectroweak parameters. Before the fit both the peak and off-peak asymmetries are translated to√the common energy s = 91.26 GeV using the predicted energy-dependence from ZFITTER [5].

55.14 Summary of the measurements and of the various kinds of analysis
The measurements of Rb and Rc fall into two classes. In the first, named single-tag measurement,
a method for selecting b and c events is applied and the number of tagged events is counted. A
second technique, named double-tag measurement, has the advantage that the tagging efficiency is
directly derived from the data thereby reducing the systematic error on the measurement.
The measurements in the b- and c-sector can be essentially grouped in the following categories:• Lifetime (and lepton) double-tagging measurements of Rb . These are the most precise mea31st May, 2024

55. Z Boson

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surements of Rb and obviously dominate the combined result. The main sources of systematics
come from the charm contamination and from estimating the hemisphere b-tagging efficiency
correlation;• Analyses with D/D∗± to measure Rc . These measurements make use of several different
tagging techniques (inclusive/exclusive double tag, exclusive double tag, reconstruction of all
weakly decaying charmed states) and no assumptions are made on the energy-dependence ofcharm fragmentation;• A measurement of Rc using single leptons and assuming B(b → c → `+ );• Lepton fits which use hadronic events with one or more leptons in the final state to measure thebb and Acc . Each analysis usually gives several other electroweak parameters.
asymmetries AFB
FBThe dominant sources of systematics are due to lepton identification, to other semileptonic
branching ratios and to the modeling of the semileptonic decay;• Measurements of Abb
FB using lifetime tagged events with a hemisphere charge measurement.
These measurements dominate the combined result;bb
cc• Analyses with D/D∗± to measure Acc
FB or simultaneously AFB and AFB ;• Measurements of Ab and Ac from SLD, using several tagging methods (lepton, kaon, D/D∗ ,and vertex mass). These quantities are directly extracted from a measurement of the left–rightforward–backward asymmetry in cc and bb production using a polarized electron beam.

55.15 Averaging procedure
All the measurements are provided by the LEP and SLD Collaborations in the form of tables
with a detailed breakdown of the systematic errors of each measurement and its dependence onother electroweak parameters.
The averaging proceeds via the following steps:• Define and propagate a consistent set of external inputs such as branching ratios, hadron
lifetimes, fragmentation models etc. All the measurements are checked to ensure that all
use a common set of assumptions (for instance, since the QCD corrections for the forward–
backward asymmetries are strongly dependent on the experimental conditions, the data arecorrected before combining);• Form the full (statistical and systematic) covariance matrix of the measurements. The systematic correlations between different analyses are calculated from the detailed error breakdown
in the measurement tables. The correlations relating several measurements made by the same
analysis are also used;• Take into account any explicit dependence of a measurement on the other electroweak parameters. As an example of this dependence, we illustrate the case of the double-tag measurementof Rb , where c-quarks constitute the main background. The normalization of the charm contribution is not usually fixed by the data and the measurement of Rb depends on the assumed
value of Rc , which can be written as:
Rb = Rbmeas + a(Rc )(Rc − Rcused ),Rc(55.11)

where Rbmeas is the result of the analysis which assumed a value of Rc = Rcused and a(Rc ) is
the constant which gives the dependence on Rc ;• Perform a χ2 minimization with respect to the combined electroweak parameters.
cc and Abb are corrected for the energy shift from
After the fit the average peak asymmetries AFB
FB
91.26 GeV to MZ and for QED (initial state radiation), γ exchange, and γZ interference effects, to31st May, 2024

55. Z Boson

80,bobtain the corresponding pole asymmetries A0,c
FB and AFB .
This averaging procedure, using the fourteen parameters described above, and applied to thedata contained in the Z particle listing below, gives the following results (where the last 8 parameters do not depend directly on the Z):

Rb0 = 0.21629 ± 0.00066
Rc0 = 0.1721 ± 0.0030
A0,b
FB = 0.0992 ± 0.0016
A0,c
FB = 0.0707 ± 0.0035
Ab = 0.923± 0.020

Ac = 0.670± 0.027

B(b → `− ) = 0.1071 ± 0.0022
B(b → c → `+ ) = 0.0801 ± 0.0018
B(c → `+ ) = 0.0969 ± 0.0031
χ = 0.1250 ± 0.0039+

P (c → D∗+) × B(D∗+f (D ) = 0.235± 0.016

f (Ds ) = 0.126± 0.026f (cbaryon ) = 0.093± 0.022+0→ π D ) = 0.1622 ± 0.0048(55.12)

Among the non–electroweak observables, the B semileptonic branching fraction B(b → `− )is of special interest, since the dominant error source on this quantity is the dependence on thesemileptonic decay model for b → `− , with ∆B(b → `− )b→`− −model = 0.0012. Extensive studies
have been made to understand the size of this error. Among the electroweak quantities, the quark
asymmetries with leptons depend also on the semileptonic decay model, while the asymmetries
using other methods usually do not. The fit implicitely requires that the different methods giveconsistent results and this effectively constrains the decay model, and thus reduces in principle theerror from this source in the fit result.
To obtain a conservative estimate of the modelling error, the above fit has been repeatedremoving all asymmetry measurements. The results of the fit on B–decay related observables
are [24]: B(b → `− ) = 0.1069 ± 0.0022, with ∆B(b → `− )b→`− −model = 0.0013, B(b → c → `+ ) =0.0802 ± 0.0019 and χ = 0.1259 ± 0.0042.
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