Gauge Boson Production at LEP

Straessner, Arno

doi:10.1007/978-3-642-05169-2_3

Arno Straessner²

Part of the book series: Springer Tracts in Modern Physics ((STMP,volume 235))

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Abstract

The main research goals of the LEP program were the detailed study of the Z and W boson properties. In the first phase of LEP, the Z line-shape was explored to exactly determine the Z mass, width and the Z-fermion couplings. The data collected by the four LEP experiments at energies around the Z peak consists of 17 million Z decays, $\textrm{e}^+\textrm{e}^-\to Z\to \textrm{f}\bar{\textrm{f}}(\gamma)$, completed by 600 thousand Z decays measured by the SLD experiment at the SLC. A short summary of the main results shall be given here.

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The main research goals of the LEP program were the detailed study of the Z and W boson properties. In the first phase of LEP, the Z line-shape was explored to exactly determine the Z mass, width and the Z-fermion couplings. The data collected by the four LEP experiments at energies around the Z peak consists of 17 million Z decays, $\textrm{e}^+\textrm{e}^-\to Z\to \textrm{f}\bar{\textrm{f}}(\gamma)$, completed by 600 thousand Z decays measured by the SLD experiment at the SLC. A short summary of the main results shall be given here.

At energies above the Z peak, single-Z production, $\textrm{e}^+\textrm{e}^-\to Z\textrm{e}^+\textrm{e}^-$, and Z-pair production, $\textrm{e}^+\textrm{e}^-\to ZZ$, are kinematically accessible. The reaction $\textrm{e}^+\textrm{e}^-\to Z+\gamma$ was studied as a possible calibration process for the LEP beam energy. This calibration is important for the determination of the mass of the other heavy gauge boson, the W.

The massless photon is ubiquitous [1] in all reactions as it is radiated by charged particles and therefore usually included in the definition of the physics process, like in $\textrm{e}^+\textrm{e}^- \to Z \to \textrm{f}\bar{\textrm{f}}(\gamma)$, for example. More interesting are the non-inclusive processes, like Compton scattering, $\textrm{e}^+\textrm{e}^- \to \gamma\textrm{e}^+\textrm{e}^-$ and photon pair-production $\textrm{e}^+\textrm{e}^-\to\gamma\gamma(\gamma)$.

At LEP energies above 161 GeV, W bosons are produced in pairs, $\textrm{e}^+\textrm{e}^-\rightarrow \textrm{WW}$ and singly in the process $\textrm{e}^+\textrm{e}^-\rightarrow \textit{W e}\nu$. The measurement of the corresponding cross-sections gives insight into the non-abelian structure of the boson couplings in the Standard Model. Pair production probes the WW$\gamma$ and WWZ vertex, while the single-W process involves only the WW$\gamma$ vertex. In the following, the measurements at the LEP collider are described.

3.1 Z Pole Measurements at LEP and SLD

At centre-of-mass energies around the Z pole the properties of the Z boson were determined with high precision. At LEP, the Z mass and width were derived from the line-shape of the Z resonance [2]. Forward-backward asymmetries of the Z decay products as well as decay branching fractions were determined by LEP and SLD for leptons, hadrons, and also separately for heavy quarks flavours. The polarised beam of the SLC allowed a measurement of the left-right asymmetry in leptonic Z decays.

To measure the mass and width of the Z boson, the $\textrm{e}^+\textrm{e}^-$ centre-of-mass energy of LEP was varied over a range of $\pm 3 \textrm{GeV}$ around $\sqrt{s}=M_\textrm{Z}$. At the different scan points the hadronic and leptonic cross-sections were measured to derive the resonance curve. The evolution of the hadronic cross-section in $\textrm{e}^+\textrm{e}^-$ collisions is illustrated in Fig. 3.1.

By scanning the line-shape of the Z resonance, the mass and width of the Z boson were determined by the four LEP experiments to be [2]

$$M_\textrm{Z}=91.1875\pm 0.0021\textrm{GeV}$$

((3.1))

$$\varGamma_\textrm{Z}=2.4952\pm 0.0023\textrm{GeV}$$

((3.2))

assuming lepton universality. The precision of the Z mass is at the ppm level and is comparable to that of the muon decay constant $G_\textrm{F}$ [3].

Further properties of the Z boson describe the production and decay at the resonance peak. They are usually summarised in a few observables that are input to the global electroweak analysis. The hadronic peak cross-section

$$\sigma^0_{\textrm{had}}=\frac{12\pi}{M_{\textrm{Z}}^2} \frac{\varGamma_{\textrm{ee}}\varGamma_{\textrm{had}}}{\varGamma_{\textrm{Z}}^2}= \frac{12\pi}{M_{\textrm{Z}}^2}\frac{\varGamma_{\textrm{ee}}\sum_{q\ne t}\varGamma_{\textrm{q}\bar{\textrm{q}}}}{\varGamma_{\textrm{Z}}^2}$$

((3.3))

is measured as [2]:

$${\sigma^0_\textrm{had}}=41.450\pm0.037 \textrm{nb}.$$

((3.4))

The analysis of leptonic decays into $\textrm{e}^{+}\textrm{e}^{-}, {\mu^{+}\mu^{-}}$ and $\tau^+\tau^-$ pairs yields the hadronic to leptonic decay width:

$${{R}^0_\ell}=\frac{\varGamma_\textrm{had}}{\varGamma_{\ell\ell}}=20.767\pm 0.025,$$

((3.5))

assuming lepton universality, which is experimentally confirmed by the good agreement of the measurement of the individual ratios, ${R}_\textrm{e}, {R}_\mu$, and ${R}_\tau$ [2]. Furthermore, the invisible Z decay width, $\varGamma_\textrm{inv} = \varGamma_\textrm{Z} -\varGamma_\textrm{had}-\varGamma_\textrm{ee} - \varGamma_{\mu \mu} -\varGamma_{\tau \tau}$, can be derived from ${{R}^0_\ell}$ as:

$${R}^0_\textrm{inv}=\frac{\varGamma_\textrm{inv}}{\varGamma_{\ell\ell}}=\left(\frac{12\pi{{R}^0_\ell}}{{\sigma^0_\textrm{had}}M_\textrm{Z}^2}\right)^{\tfrac{1}{2}}-{{R}^0_\ell}-(3+\delta_\tau),$$

((3.6))

where $\delta_\tau$ is correcting for the mass of the $\tau$ lepton. From this ratio the number of light neutrino species, $N_\nu$, is extracted assuming Standard Model couplings of the neutrinos to the Z boson:

$${R}^0_\textrm{inv}=N_\nu\left(\frac{\varGamma_{\nu \nu}}{\varGamma_{\ell\ell}}\right)_\textrm{SM}.$$

((3.7))

The analysis of the LEP data yields:

$$N_\nu=2.9840\pm 0.0082,$$

((3.8))

which is deviating from three by only 2 standard deviations. This is nicely illustrated in Fig. 3.2. A more direct determination of the invisible Z width is obtained from studies of single- and multi-photon events produced in the reaction $\textrm{e}^+\textrm{e}^-\to\nu\bar{\nu}\gamma(\gamma)$, which results in $N_\nu=2.92\pm 0.05$ [3,4], again compatible with three light neutrino generations. The corresponding cross-section measurements from which this number is derived is shown in Fig. 3.2.

Additional information on the Z coupling structure to the fermions is contained in the asymmetry measurements. They depend on the helicity of the colliding electrons and positrons and on the polarisation of the produced particles. The forward-backward asymmetry is defined as

$${A}_\textrm{FB}=\frac{N_{ F}-N_B}{N_{ F}+N_B},$$

((3.9))

where $N_{ F} (N_B)$ denotes the number of events in which the fermion is produced in the forward (backward) hemisphere, with polar angles $\theta<\pi/2 (\theta>\pi/2)$ with respect to the incoming electron beam. The leptonic asymmetry at the Z pole is measured to be:

$${A}_\textrm{FB}^{0,\ell}=0.0171\pm 0.0010.$$

((3.10))

For $\tau$ leptons, also the polarisation can be determined by studying the kinematic of the observed $\tau$ decay products. For example, in $\tau\rightarrow \pi\nu_\tau$ decays, the energy spectrum of pions in the tau rest frame depends on the tau helicity. These measurements yield

$${\cal A}_\ell({\cal P}_\tau)=0.1465\pm 0.0033,$$

((3.11))

assuming universality of taus and electrons.

The leptonic data are completed by the measurement of the left-right asymmetry at SLD

$${A}_\textrm{LR}=\frac{N_L-N_R}{N_L+N_R}\frac{1}{\langle {\cal P}_\textrm{e} \rangle},$$

((3.12))

from which the leptonic asymmetry parameter is derived as

$${\cal A}_\ell=0.1514\pm 0.0022,$$

((3.13))

using Eq. (1.79). After combination with the determination of the left-right forward-backward asymmetry, ${A}_\textrm{LR,FB}$, the measured value of ${\cal A}_\ell$ is only slightly changed:

$${\cal A}_\ell=0.1513\pm 0.0021.$$

((3.14))

The leptonic asymmetry measurements rely on correct charge tagging, which, in case of $e, \mu$ and $\tau$, is rather precise, because the single-track charge confusion is usually small and elementary leptons have unit charge. This is however not the case for quarks and more refined methods are introduced to determine the $\textrm{q} \overline{\textrm{q}}$ charge asymmetry in Z decays. By weighting the charged tracks in quarks jets according to their momentum, the forward and backward hemispheres can be assigned to the quark jets. The combined result of these measurements is expressed in terms of the effective leptonic weak mixing angle:

$${\sin^2{\theta_\textrm{eff}^\ell}}\left(Q^\textrm{had}_\textrm{FB}\right)=0.2324\pm 0.0010$$

((3.15))

Identifying the heavy quark flavours yields additional information on the details of the Z decays. The combination of LEP and SLD data results in the following branching fractions and asymmetries for c and b quarks separately:

$${R}^0_\textrm{b}=0.21629\pm 0.00066$$

((3.16))

$${R}^0_\textrm{c}=0.1721\pm 0.0030$$

((3.17))

$${A}_\textrm{FB}^\textrm{0,b}=0.0992\pm 0.0016$$

((3.18))

$${A}_\textrm{FB}^\textrm{0,c}=0.0707\pm 0.0035$$

((3.19))

$${\cal A}_\textrm{b}=0.923\pm 0.020$$

((3.20))

$${\cal A}_\textrm{c}=0.670\pm 0.027$$

((3.21))

The asymmetry parameters, ${\cal A}_\textrm{f}$, depend only on the ratio of the effective vector and axial-vector coupling constants, ${g}_\textrm{V}^\textrm{f}/{g}_\textrm{A}^\textrm{f}$, while the square-root of their squared sum enters the partial Z decay widths. At LEP and SLD, especially the leptonic coupling constants are determined with high precision:

$${g}_\textrm{A}^\nu={g}_\textrm{V}^\nu=+0.50076\pm 0.00076$$

((3.22))

$${g}_\textrm{A}^\ell=-0.50123\pm 0.00026$$

((3.23))

$${g}_\textrm{V}^\ell=-0.03783\pm 0.00041$$

((3.24))

with an anti-correlation of 48% between ${g}_\textrm{A}^\nu$ and ${g}_\textrm{A}^\ell$, respectively ${g}_\textrm{V}^\nu$ and ${g}_\textrm{A}^\ell$, and only small correlations around 5% between ${g}_\textrm{V}^\ell$ and the other couplings. A comparison between the individual results of the three leptons, e, $\mu$ and $\tau$, shows the universality of the leptonic couplings which is illustrated in Fig. 3.3. The variation of the Standard Model prediction is also indicated, which agrees best with data if the Higgs boson is light. The value of ${g}_\textrm{A}^\ell$ differs from the tree level value of $T_3=-\frac{1}{2}$ by 4.7 standard deviations, showing clearly the presence of electroweak radiative corrections .

The various asymmetry measurements allow an extraction of the effective weak mixing angle, $\sin^2{\theta_\textrm{eff}}$, independent of the value of the $\rho$ parameter, when exploiting Eq. (1.92), since only the ratio of the coupling constants, ${g}_\textrm{V}^\textrm{f}/{g}_\textrm{A}^\textrm{f}$, appears. The results are shown graphically in Fig. 3.3, where the measurements are compared in terms of the leptonic ${\sin^2{\theta_\textrm{eff}^\ell}}$. The two groups of leptonic and hadronic measurements each show very good agreement, while the comparison between the two is somewhat less consistent, although not with a very strong significance. This may hint to deviation in the hadronic sector, which is enhanced when looking at the right-handed b-coupling, as done in Fig. 3.4. This will however remain an unanswered question until new data may become available, possibly from Z boson decays at the hadron colliders Tevatron and LHC or from an international linear collider (ILC). For the latter, the option of a high-luminosity running at the Z pole, called GIGAZ [5], is discussed.

With the measured values of ${g}_\textrm{V}^\textrm{f}$ and ${g}_\textrm{A}^\textrm{f}$ by LEP and SLC the combined leptonic effective weak mixing angle and the $\rho_\ell$ parameter are extracted according to Eq. (1.53) [2], which yields:

$${\sin^2{\theta_\textrm{eff}^\ell}} = 0.23153\pm 0.00016$$

((3.25))

and

$$\rho_\ell = 1.0050\pm 0.0010.$$

((3.26))

The Standard Model predictions at tree level

$$\sin^2{\theta_\textrm{w}}=\frac{1}{2}\left(1-\sqrt{1-4\frac{\pi\alpha_\textrm{QED}\left(M_\textrm{Z}^2\right)}{\sqrt{2}G_\textrm{F}M_\textrm{Z}^2}} \right)=0.23098\pm 0.00012$$

((3.27))

$$\rho=1$$

((3.28))

deviate from these by 2.8 and 5.0 standard deviations. This is again a clear evidence for the existence of significant radiative corrections . Figure 3.4 shows the measured and predicted values in the $\rho_\ell-{\sin^2{\theta_\textrm{eff}^\ell}}$ plane. It is interesting to note that best agreement between theory and experiment is achieved for light Higgs masses, as mentioned before, but also for top masses around 174 GeV, which agrees better with the recent $m_\textrm{t}$ result [76] than with the slightly higher measurement used at the time the graphic was produced [2].

3.2 Neutral Boson Production Above the Z Peak

3.2.1 Photon Production

The QED part of the neutral boson sector is tested at low energy to high precision [6]. At LEP energies, pure QED processes are as well found to be in nice agreement with the theoretical predictions. Figure 3.5 documents the measurements of quasi-real Compton scattering $\textrm{e}^+\textrm{e}^-\to\gamma \textrm{e}^+\textrm{e}^-$. In this reaction, one beam electron (or positron) emits a quasi-real photon with low virtuality, $Q^2 <2 \textrm{GeV}$, and escapes along the beam pipe. The photon and the beam positron (or electron) are scattered and measured in the detector. The effective centre-of-mass energy of this process can be calculated assuming three-particle kinematics:

$$\sqrt{s^{\prime}}=\sqrt{1-\frac{2E_\textrm{miss}}{\sqrt{s}}} ,\textrm{with}$$

((3.29))

$$E_\textrm{miss}=\sqrt{s}\frac{|\sin(\theta_e+\theta_\gamma)|}{\sin\theta_e+\sin\theta_\gamma+|\sin(\theta_e+\theta_\gamma)|}.$$

((3.30))

To reduce background from Bhabha scattering the effective scattering angle, $\cos\theta^*$, is constrained to the central region

$$|\cos\theta^*|=\left|\frac{\sin(\theta_\gamma-\theta_e)}{\sin\theta_\gamma+\sin\theta_e}\right| <0.8.$$

((3.31))

The typical backward scattering angular distribution, which is given to lowest order by

$$\frac{d\sigma}{d\cos\theta^*}=\frac{\alpha_\textrm{QED}^2\pi}{s'}\left(\frac{1+\cos\theta^*}{2}+\frac{2}{1+\cos\theta^*}\right),$$

((3.32))

is shown in Fig. 3.5. The LEP measurement is sensitive to very high values of $\sqrt{s^{\prime}}=175 \textrm{GeV}$, that were never reached before.

A similar good agreement with QED predictions is found for real photon production $\textrm{e}^+\textrm{e}^-\to\gamma\gamma(\gamma)$, where at least two high energetic photons are required with strict cuts on activity in the tracking system. The cross-section measurements at the highest LEP energies are compared to the theory in Fig. 3.6. The angular distribution is used to test possible deviations from QED, also shown in Fig. 3.6. The lowest order expression

$$\frac{d\sigma}{d\cos\theta}=\frac{\alpha_\textrm{QED}^2 2\pi}{s}\frac{1+\cos^2\theta}{1-\cos^2\theta},$$

((3.33))

is extended by a term

$$\ldots +\frac{\alpha_\textrm{QED}^2\pi s}{\varLambda_\pm}(1+\cos^2\theta),$$

((3.34))

which corresponds to a short-range exponential deviation from the Coulomb potential with a cut-off parameter $\varLambda_\pm$. The LEP combined limits for this parameter are $\varLambda_+ >392$ and $\varLambda_- >364 \textrm{GeV}$ at 95% confidence level [8], verifying the QED prediction nearly up to the TeV range.

3.2.2 Single Z and Z-Pair Production

At LEP energies above the Z resonance, the heavy neutral gauge boson is detected also in single Z and Z-pair production, $\textrm{e}^+\textrm{e}^-\to Z/\gamma^*\textrm{e}^+\textrm{e}^-\to \textrm{f}\bar{\textrm{f}}\textrm{e}^+\textrm{e}^-$ and $\textrm{e}^+\textrm{e}^-\to ZZ\to \textrm{ffff}$, respectively. They can be imagined as the extension of QED Compton scattering and photon pair-production to virtual, off-shell photons, $\gamma^*$, whose mass eventually reaches the Z boson mass. The final state objects are therefore not the massless photons but the decay fermions of the Z.

On-shell ZZ production is described by two neutral current Feynman graphs with t- and u-channel electron exchange, the so-called NC02 set. In the fully hadronic channel $\textrm{ZZ}\to\textrm{q q q q}$, likelihood-based analyses are applied to separate signal from the most important $\textrm{e}^+\textrm{e}^-\to\textrm{q} \overline{\textrm{q}}(\gamma)$ and $\textrm{e}^+\textrm{e}^-\to\textrm{WW}$ backgrounds. Variables exploiting event shape and kinematics are combined, like, for example, the event sphericity, jet energy differences, inter-jet opening angles, jet resolution parameters, and the reconstructed Z boson masses. W-pairs are efficiently rejected by b-jet tagging. Figure 3.7 gives an example for the discrimination between ZZ signal and background on the basis of the b-quark content and the final probability variable.

In the decay channel with two neutrinos, $\textrm{q} \overline{\textrm{q}}\nu\nu$ and $\ell^+\ell^-\nu\nu$, jet and lepton pairs are selected without any other activity in the detectors. Only electron and muon pairs are accepted. The visible $\textrm{q} \overline{\textrm{q}}$ and $\ell^+\ell^-$ mass as well as the recoil mass are required to be compatible with $M_\textrm{Z}$. To reject $\textrm{q} \overline{\textrm{q}}(\gamma)$ events with hard ISR photons, the missing momentum should not point along the beam direction.

In the semi-leptonic channel, $\textrm{q} \overline{\textrm{q}}\ell^+\ell^-$, with $\ell=\textrm{e},\mu \tau$, similar kinematic criteria are imposed as in the $\textrm{q q q q}$ final state, for example: compatibility with the Z boson masses, transverse momentum balance, large effective centre-of-mass energy.

The channel with the least number of ZZ candidates is the fully leptonic decay $\ell^+\ell^-\ell^+\ell^-$. The sample is however very clean with the practically only background from non-resonant $\textrm{e}^+\textrm{e}^-\ell^+\ell^-$ production. Z mass constraints are imposed either on both lepton pairs, or on the better reconstructed one and its recoil system.

In general, the cross-section is derived from a fit to the final selection variable, usually a likelihood or neural network distribution. The ZZ event rate at the highest centre-of-mass energy of 207 GeV yields, e.g., 358 candidates selected by L3 [11] in a data sample of 138.9 pb⁻¹. The signal expectation is $80.4\pm 0.1$ events and the background $278.4\pm 0.6$ events. The relative contributions to the ZZ signal from $\textrm{q q q q}$ is about 58%, 24% from $\textrm{q} \overline{\textrm{q}}\nu\nu$, 15% from $\textrm{q} \overline{\textrm{q}}\ell^+\ell^-$, 2% from $\ell^+\ell^-\nu\nu$, and only 1% from $\ell^+\ell^-\ell^+\ell^-$.

The LEP combined ZZ cross-section [8] is illustrated in Fig. 3.8. It agrees very well with the NC02 calculation of the ZZTO [12] and YFSZZ [13] programs, which have a 2% theoretical uncertainty. The ratio of all measurements at the different centre-of-mass energies to the expectation is concentrated into a single number, $R_{ZZ}$, taking correlations into account, which yields [8]:

$$R_\textrm{ZZ}^\textrm{ZZTO}=0.952\pm0.052$$

$$R_\textrm{ZZ}^\textrm{YFSZZ}=0.945\pm0.052 .$$

Within the 5% measurement precision, there is good agreement with the theoretical predictions. The uncertainty is dominated by statistics, and the main systematic uncertainties are from the understanding of the b-tagging, the lepton and jet energy scale, jet rates, and background cross-sections in the ZZ signal region.

Single Z production is measured at LEP in $\textrm{e}^+\textrm{e}^-\textrm{q} \overline{\textrm{q}}$ and $\textrm{e}^+\textrm{e}^-{\mu^+\mu^-}$ final states. The signal cross-section is defined as the Compton-like four-fermion process $\textrm{e}^+\textrm{e}^- \to\textrm{e}^+\textrm{e}^-\textrm{f}\bar{\textrm{f}}$ with the following criteria on the phase space: $M_{\textrm{f}\bar{\textrm{f}}} >60 \textrm{GeV}, \theta_{e^+} <12^\circ, 12^\circ<\theta_{e^-}<120^\circ$ and $E_{e^-}>3 \textrm{GeV}$. The positron is assumed to have emitted the scattering quasi-real photon, and the electron is interacting with this photon and is scattered into the detector acceptance region. The corresponding criteria apply similarly to the charged conjugate reaction.

The detector signal is thus a pair of jets or muons, compatible with an on-shell Z boson, and a single scattered electron. In addition, large missing momentum pointing along the beam direction is required. The main backgrounds in the hadronic channel are from single W’s, $\textrm{e}^+\textrm{e}^-\to\textrm{q q} \textrm{e} \nu$, quark pair production and W pairs. In the $\textrm{e}^+\textrm{e}^-{\mu^+\mu^-}$ channel, mainly muon-pair and two-photon production, $\textrm{e}^+\textrm{e}^-{\mu^+\mu^-}$, need to be rejected. Figure 3.8 shows the measured cross-section as a function of centre-of-mass energy. Like for ZZ production, the measurement uncertainty is dominated by statistics, and the main systematic uncertainties are from lepton and jet energy measurement, selection efficiencies, background and signal modelling. Very good agreement with the theoretical calculations using the four-fermion programs WPHACT [14] and GRC4f [15] is obtained.

3.2.3 Z+$\gamma$ Production

The simultaneous production of the neutral gauge bosons , photon and Z, is investigated at LEP in the decay channels $\textrm{q} \overline{\textrm{q}}\gamma$ and $\nu\nu\gamma$ which are the dominant Z+$\gamma$ decays. In both final states, events containing an energetic photon with a recoil mass compatible with the Z boson mass are selected. In the hadronic channel, the mass of the jet-jet system must as well be close to $M_\textrm{Z}$ and there should be little energy imbalance in the event. Figure 3.9 shows an example of the L3 measurement, with a distribution of the photon recoil mass in $\nu\nu\gamma$ events, and the Z$\gamma$ cross-section as a function of centre-of-mass energy.

The Z$\gamma$ process with an initial state photon, $\gamma_{ISR}$, escaping invisibly along the beam pipe is of special interest as a calibration method of the LEP beam energy. In those events, the detector signature is a jet or lepton pair, from $\textrm{e}^+\textrm{e}^-\to Z+\gamma_{ISR}\to\textrm{q} \overline{\textrm{q}}+\gamma_{ISR}$ or $\textrm{e}^+\textrm{e}^-\to Z+\gamma_{ISR}\to\ell^+\ell^-+\gamma_{ISR}$, with large missing momentum along the beam direction. The Z mass can be determined precisely from the fermion directions, according to Eq. (3.29), if $\sqrt{s^{\prime}}$ is identified with the mass of the produced Z boson. The so-called radiative return to the Z is seen nicely in Fig. 3.10. The Z mass determined from the radiative events, $M_\textrm{Z}^{\textrm{f}\bar{\textrm{f}}}$, can be compared to the precision Z mass, $M_\textrm{Z}$, from the Z peak. Equivalently, this comparison can be translated into a test of the LEP centre-of-mass energy using:

$$\varDelta\sqrt{s}=\sqrt{s}-\sqrt{s}_\textrm{LEP}=\sqrt{s} \frac{M_\textrm{Z}^{\textrm{f}\bar{\textrm{f}}}-M_\textrm{Z}} {M_\textrm{Z}},$$

((3.35))

with the nominal value of $\sqrt{s}_\textrm{LEP}$ [19] provided by the LEP energy working group. The LEP combined value is [18]

$$\varDelta\sqrt{s}=0.054\pm 0.054 \textrm{MeV}$$

((3.36))

in good agreement with no shift with respect to the more precise standard LEP energy calibration. This is a nice confirmation of the calibration procedure and an interesting cross-check for the precise determination of $M_\textrm{W}$.

3.2.4 Anomalous Neutral Gauge Boson Couplings

From the measurement of ZZ and Z$\gamma$ production, one can infer on the coupling structure of the neutral triple gauge boson vertex . The most general ZZV vertex, with V = Z,$\gamma$, for on-shell Z’s which respects Bose symmetry can be written as [20,21,22]:

$$\varGamma^{\alpha\beta\mu}_{ZZV}(q_1,q_2,P)=\frac{s-m_V^2}{M_\textrm{Z}^2}\left\{ i f_4^{ZZV}\left(P^\alpha g^{\mu\beta}+i f_5^{ZZV} \varepsilon^{\mu\alpha\beta\rho}(q_1-q_2)_\rho\right. \right\},$$

((3.37))

where P is the four-momentum of the incoming V boson, while q ₁ and q ₂ are the four-momenta of the produced Z boson pair. The f ₅ coupling is CP violating while f ₄ is CP conserving . In the Standard Model, both are zero at tree level.

Similarly, the Z$\gamma$V vertex is described by

$$\begin{aligned} \varGamma^{\alpha \beta \mu}_{ZZV}(q_1,q_2,P)&{}=\frac{s-m_V^2}{M_\textrm{Z}^2}\left\{ h_1^V\left(q_2^\mu g^{\alpha\beta}-q_2^\alpha g^{\mu\beta}\right)\right.\\ &{}+\frac{h_2^V}{M_\textrm{Z}^2}P^\alpha\left(P\cdot q_2 g^{\mu\beta}-q_2^\mu P^\beta\right)\\ &{}+h_3^V \varepsilon^{\mu\alpha\beta\rho}q_{2,\rho}\\ &{}\left.+h_4^Vp^\alpha \varepsilon^{\mu\beta\rho\sigma} P_\rho q_{2,\sigma} \right\}, \end{aligned}$$

((3.38))

where V = Z,$\gamma$ is again the incoming virtual boson in the s-channel. Terms proportional to $P^\mu$ and $q_1^\alpha$ are omitted because they do not contribute in $\textrm{e}^+\textrm{e}^-$ annihilation. The couplings $h_1^V$ and $h_2^V$ are even under parity, and $h_3^V$ and $h_4^V$ are CP even . All couplings are C-odd, $h_1^V$ and $h_2^V$ are therefore CP violating. Because of gauge invariance for V = $\gamma$ both anomalous contributions vanish for $s=M_V^2$. In the terminology of an effective Lagrangian, the interaction is induced by operators of dimension six and higher.

The $h_i^V$ couplings are determined from the kinematic observables in Z+$\gamma$ events, like the photon energies and angles and, if measured, the fermion energies and angles. Similarly, the full ZZ event kinematics is exploited to determine the $f_i^V$ couplings. The results of all LEP data combined [8] are shown in Figs. 3.11 and 3.12. All couplings are found to be consistent with zero. The gauge boson coupling structure in the neutral sector fully corresponds to the Standard Model expectations.

3.3 Measurement of the W-Pair Cross-Section

The three main final state categories for W-pair production at LEP are fully leptonic, ${\ell \nu \ell \nu}$, semi-leptonic, ${\textrm{q q} \ell \nu (\gamma)}$, and fully hadronic, $\textrm{q q q q }(\gamma)$, with branching fractions of 43.5, 46.2 and 10.3%, respectively. All events may be accompanied by photon radiation, indicated by $(\gamma)$, either in the initial or final state. The events are detected by identifying the visible W decay products, the leptons and quark jets. All lepton flavours, including leptonic and hadronic tau decays, are considered in the selection channels.

The LEP experiments apply different strategies to retain the signal events in data and to reject the different backgrounds. Cut-based and likelihood-based selections are applied, as well as neural network techniques [23,24,25,26]. They all exploit the kinematic properties of the events.

The main selection criteria for ${\ell \nu \ell \nu}$ events are two charged leptons and missing energy. The leptons are required to be acoplanar to reduce the dominating backgrounds from lepton-pair production and possible cosmic rays. The latter is further suppressed by requiring an event timing compatible with the beam crossing. A veto on other activity in the detector is furthermore applied. The main remaining background is from two-photon collision processes. Figure 3.13 shows an example for the measured acoplanarity distribution in L3 data. The purity of the event samples are around 70–80% and the selection efficiencies between 20 and 30% in the hadronic ${\tau \nu \tau \nu}$ channel up to 60% for $\textrm{e} \nu \textrm{e} \nu$ and ${\textrm{e} \nu \mu \nu}$ events. The cross-efficiencies, especially for events with $W\rightarrow\tau{\nu_\tau}$ decays, can be up to 10%.

Semi-leptonic $\textrm{q q} \ell \nu$ events are selected by their hadronic activity, two jets and one energetic and isolated lepton in the final state, and high invariant masses of the jet-jet system. The neutrino momentum is actually well reconstructed in $\textrm{q q} \textrm{e} \nu$ and $\textrm{q q} \mu \nu$ events by identification with the missing momentum. A large ${\ell \nu}$ mass is used as selection criteria as well. The missing momentum is required to not point along the beam direction. An example for a variable combining lepton-jet separation and missing momentum direction is shown in Fig. 3.13. In the $\textrm{q q} \tau \nu$ channel with leptonic tau decays, a low ${\ell \nu}$ mass is used to separate this decay from the $\textrm{q q} \textrm{e} \nu$ and $\textrm{q q} \mu \nu$ events. This minimises the overlap between the semi-leptonic channels, which is at the 5–10% level. Hence, it reduces correlations in the measurement of the W branching fractions. The main background in the semi-leptonic channels is from $\textrm{e}^+\textrm{e}^-\rightarrow\textrm{q} \overline{\textrm{q}}(\gamma)$ and Z-boson pair production. Efficiencies are between 50 and 65% in the tau channel and 75–90% in the electron and muon channel. The experiments with a larger angular coverage for efficient lepton detection generally obtain higher efficiencies. The purity is as high as 98% and above for $\textrm{q q} \textrm{e} \nu$ and $\textrm{q q} \mu \nu$ events and reaches 65–85% in the $\textrm{q q} \tau \nu$ channel.

The selection of fully hadronic W-pair decays is based on little missing energy, high multiplicity and four-jet topology. Jet resolution parameters, like the Durham y ₃₄ variable [27], discriminate against quark-pair production with additional gluon radiation, $\textrm{e}^+\textrm{e}^-\rightarrow \textrm{q} \overline{\textrm{q}} g$. By applying a neural network based selection which uses different kinematic and topological quantities the main background from $\textrm{e}^+\textrm{e}^-\rightarrow \textrm{q} \overline{\textrm{q}} gg$ and hadronic Z-pairs is further reduced. At a selection efficiency of 85–90% a purity of 80% is reached. Figure 3.14 gives examples for the spherocity distribution and the neural network output measured by L3.

Each experiment retains 700–1,000 ${\ell \nu \ell \nu}$ candidates, 1,200–1,500 $\textrm{q q} \ell \nu$ candidates, and 5,000–5,500 $\textrm{q q q q}$ candidates, from which the W-pair cross-section is derived. The total production cross-section for W-pairs is defined as the corresponding CC03 process, which means that other diagrams that lead to the same final state are considered as background. The efficiencies that enter the calculation are calculated with the state-of-the art Monte Carlo programs that include full $O(\alpha)$ electroweak corrections [28], like the KandY [29], WPHACT [14] and RacoonWW [30] event generators. Cross-checks between the different calculations are performed and yield consistent results.

The W-pair cross-sections are determined in each decay channel separately and cross-efficiencies for other channels are taken into account. Systematic uncertainties arise mainly from the modelling of the detector response to the measured leptons and quark jets, which is between 1 and 2% of the cross-section. Smaller contributions are from hadronisation modelling, where the string-fragmentation model Pythia, and the colour-dipole models Herwig and Ariadne are compared. These models are tuned on Z decay data depleted in $\textrm{b} \overline{\textrm{b}}$ final states, which are the most similar to the hadronic W decays. Furthermore, there are uncertainties due to Final State Interactions (see Sect. 3.6), photon radiation and the variation of the W mass and width within their measurement uncertainties. Luminosity uncertainties are considered as well. Altogether, the systematic uncertainties are in the order of 1.5–2.5%, much smaller than the statistical uncertainty for each channel and energy point.

The total W-pair cross-section is determined by combining all decay channels at each energy points and assuming the W branching fractions to follow the Standard Model prediction. When combining data of different experiments, correlations between systematic uncertainties are properly considered. Already with the data close to the $\textrm{WW}$ threshold, there is clear evidence for the contribution of all three lowest order diagrams to the W-pair production, as can be seen in Fig. 3.15.

Each single contribution from t-channel neutrino exchange, or s-channel $\gamma$ or Z boson exchange, shown in Fig. 1.8, would lead to a steadily increasing cross-section with centre-of-mass energy and would eventually violate unitarity. Only the coupling of the W boson to the other gauge bosons, $\gamma$ and Z, caused by the non-abelian nature of the SU(2) gauge group, and the interference between the three contributions guarantees correct high-energy behaviour and agrees with the measured W-pair cross-section.

The result of the combined LEP measurement for $\sqrt{s}=161-209 \textrm{GeV}$ is shown in Fig. 3.15. The accuracy of the theoretical prediction increases from 0.7% at $\sqrt{s}=170$ to 0.4% at $\sqrt{s}$ above 200 GeV. At the threshold, where the W bosons are practically on-shell, the latest LPA/DPA techniques can not be applied and the uncertainty on the prediction reaches 2%. All LEP data are in excellent agreement with the Standard Model prediction over the whole energy range.

The agreement between the theoretical prediction is summarised in a single R parameter, which is the combined ratio of measurement to theory for the whole energy range:

$$R_\textrm{WW}=\frac{\sigma^\textrm{meas}_\textrm{WW}}{\sigma^\textrm{theo}_\textrm{WW}}$$

((3.39))

The extracted R values are [8]:

$$R^\textrm{YFSWW}_\textrm{WW}=0.994\pm0.009$$

((3.40))

$$R^\textrm{RACOONWW}_\textrm{WW}=0.996\pm0.009$$

((3.41))

when comparing with the YFSWW and RacoonWW predictions. A very good agreement is observed within the experimental precision of 1%, which is close to the theoretical uncertainty of 0.5% [28].

From the measurements of the cross-sections in the individual decay channels the W branching fractions are determined, where the total sum is assumed to be unity. The results of the LEP experiments are shown in Fig. 3.16. The Standard Model expectation of the hadronic branching fraction, $Br(\textrm{W}\rightarrow\textrm{q} \overline{\textrm{q}})$, is 67.51%. The leptonic ones are expected to be all equal with $Br(\textrm{W}\rightarrow{\ell \nu})=10.83\%$. As it is evident from the results of all LEP experiments, there is an excess of the branching ratio $W\rightarrow{\tau \nu}$, which is at a level of 2.8 standard deviations:

$$\frac{2Br(\textrm{W}\rightarrow{\tau \nu})}{Br(\textrm{W}\rightarrow\textrm{e} \nu)Br(\textrm{W}\rightarrow\mu \nu)}=1.077\pm 0.026.$$

((3.42))

Assuming lepton universality, the combined results of:

$$Br(\textrm{W}\rightarrow{\ell \nu})=10.84\pm 0.06\textrm{(stat.)}\pm 0.07\textrm{(syst.)}$$

((3.43))

$$Br(\textrm{W}\rightarrow\textrm{q} \overline{\textrm{q}})=67.48\pm 0.18\textrm{(stat.)}\pm 0.21\textrm{(syst.)}$$

((3.44))

are however in good agreement with the expectations [8].

3.4 Measurement of Single-W Production

The production of single W bosons , $\textrm{e}^+\textrm{e}^-\rightarrow \textit{W e}\nu$, is defined as the complete t-channel subset of Feynman diagrams leading to the $e \nu\mathrm{f f}$ final state [8]. To suppress multi-peripheral graphs the process is further characterised by the following phase space requirements on the $\textrm{q} \overline{\textrm{q}}$ mass, the lepton energy and the electron and positron scattering angles:

$$\begin{aligned} m_\textrm{q} \overline{\textrm{q}}>45\textrm{GeV}&{}\textrm{for} W\textrm{e} \nu\rightarrow{\textrm{e} \nu}\textrm{q q} ,\\ E_\ell >20\textrm{GeV}&{}\textrm{for} W\textrm{e} \nu\rightarrow{\textrm{e} \nu \tau \nu}\\ &{}\textrm{and} W\textrm{e} \nu\rightarrow{\textrm{e} \nu \mu \nu} ,\\ E_{e^+}>20\textrm{GeV}, |\cos\theta_{e^+}| <0.95, |\cos\theta_{e^-}|>0.95&{}\textrm{for} W\textrm{e} \nu\rightarrow\textrm{e} \nu \textrm{e} \nu .\\ &{}\textrm{(and the charge conjugate)} \end{aligned}$$

The main signature of single-W production is the forward scattered electron, which remains invisible in the detector, and the decay products of the W that can be measured [31,32,33]. In L3, the hadronic W decay is therefore selected by requiring a visible energy in the calorimeters compatible with a W decay, $0.30 <E_\textrm{vis}/\sqrt{s}<0.65$. The missing momentum vector should not point along the beam direction, $|\cos\theta_\textrm{miss}| <0.92$, because this is the typical signature of radiative $\textrm{e}^+\textrm{e}^-\rightarrow\textrm{q} \overline{\textrm{q}}(\gamma)$ events with ISR photons escaping along the beam pipe. Events with three-jet topology from $\textrm{q q} \tau \nu$ W-pair production are also removed. Kinematic criteria like the invariant jet-jet mass and velocity are combined in a likelihood variable to suppress $\textrm{e}^+\textrm{e}^-\rightarrow\textrm{ZZ}\to \nu\nu\textrm{q} \overline{\textrm{q}}$ events. Eventually, a neural network based on visible energy, visible mass, single-W kinematics and 2/3-jet topology is applied to select the final event sample.

Leptonic single-W final states have the striking signature of single leptons in the detector without any other visible activity. The recoil mass to the lepton must be compatible with single-W production, which also rejects $\textrm{e}^+\textrm{e}^-\rightarrow \nu\nu\gamma$ events with converted photons.

Figure 3.17 shows two selection criteria for hadronic and leptonic W decays. The hadronic W final states are triggered redundantly, while trigger efficiencies for leptonic final states are about 90% for muons, 95% for electrons and close to 100% for taus. They are determined directly from data.

The measured cross-sections combined for the LEP experiments is shown in Fig. 3.18 for the hadronic W decays and all channels combined. The theoretical predictions are less precise than for W-pair production and reach a precision of 5%. This is estimated from two different calculations in fixed-width and fermion-loop scheme [34], which consists in including all fermionic one-loop corrections in tree-level amplitudes and resumming the self-energies. Data and theory agree well over the whole LEP energy range. The global agreement is expressed in the measurement-to-theory ratio R as [8]:

$$R^\textrm{GRC4f}_\textit{W e}\nu=1.051\pm0.075$$

((3.45))

$$R^\textrm{WPHACT}_\textit{W e}\nu=1.083\pm0.078$$

((3.46))

for the two theoretical programs GRC4f [15] and WPHACT [14]. The measurements show again that the W coupling structure is in agreement with the Standard Model, especially the WW$\gamma$ vertex contributions. This is further investigated in the next chapter.

3.5 Determination of Triple Gauge Boson Couplings

In the Standard Model, W bosons couple to the other gauge bosons, $\gamma$ and Z, by means of their charge and the weak coupling constant, respectively. To test the coupling structure it is helpful to extend the Standard Model to a general WWV vertex, with V = $\gamma$,Z. In this way, additional coupling constants are introduced which describe possible low energy manifestations of new physics beyond the Standard Model. Both, the Standard Model and anomalous couplings can then be measured or constrained by data.

The WWV vertex is generally parameterised in a phenomenological effective Lagrangian [20,21] :

$$\begin{aligned} i{\cal L}^{WWV}_{eff}&{}=g_{WWV}\left[g_1^V V^\mu\left(W_{\mu\nu}^-W^{+\nu}-W_{\mu\nu}^+W^{-\nu}\right)+\kappa_V W_\mu^+W_\nu^-V^{\mu\nu}\right.\\ &{}+\frac{\lambda_V}{M_\textrm{W}^2}V^{\mu\nu}W_\nu^{+\rho}W_{\mu\rho}^- +ig_5^V\varepsilon_{\mu\nu\rho\sigma}\left\{\left(\partial^\rho W^{-\mu}\right)W^{+\nu}-W^{-\mu}\left(\partial^\rho W^{+\nu}\right)\right\}V^\sigma\\ &{}+ig_4^VW_\mu^-W_\nu^+\left(\partial^\mu V^\nu+\partial^\nu V^\mu\right)\\ &{}\left.-\frac{\tilde{\kappa}_V}{2}W_\mu^-W_\nu^+\varepsilon^{\mu\nu\rho\sigma}V_{\rho\sigma} -\frac{\tilde{\lambda}_V}{2M_\textrm{W}^2}W_{\rho\mu}^-W^{+\mu}_\nu\varepsilon^{\nu\rho\alpha\beta}V_{\alpha\beta}\right] \end{aligned}$$

The overall couplings are defined as $g_{WW\gamma}=e$ and $g_{WWZ}=e\cot{\theta_\textrm{w}}$, which are the W electromagnetic charge and weak coupling to the Z. The W and V field strengths are here defined as: $W_{\mu\nu}=\partial_\mu W_\nu-\partial_\nu W_\mu$ and $V_{\mu\nu}=\partial_\mu V_\nu-\partial_\nu V_\mu$. For real photons $(Q^2=0), g_1^\gamma$ and $g_5^\gamma$ are fixed by gauge invariance to 1 and 0, respectively. In the Standard Model the only non-zero couplings are

$$g_1^Z=g_1^\gamma=\kappa_Z=\kappa_\gamma=1$$

((3.47))

at tree level. The terms proportional to $g_1^V, \kappa_V$ and $\lambda_V$ are C and P conserving, while $g_5^V$ violates C and P but conserves CP. Violation of CP is parameterised by $g_4^V, \tilde{\kappa}_V$ and $\tilde{\lambda}_V$.

The lowest order terms of the multipole expansion of the W-photon interaction are directly related to the couplings. The charge, Q _W, the magnetic dipole moment, $\mu_W$, and the electric quadrupole moment, q _W, of the W ⁺ are given by:

$$Q_W=eg_1^\gamma$$

((3.48))

$$\mu_W=\frac{e}{2M_\textrm{W}}\left(g_1^\gamma+\kappa_\gamma+\lambda_\gamma\right)$$

((3.49))

$$q_W=-\frac{e}{M_\mathrm{W}^2}\left(\kappa_\gamma-\lambda_\gamma\right)$$

((3.50))

One can obtain theoretical constraints on the triple-gauge boson couplings (TGC) by asking for a global “custodial” SU(2) symmetry [35] of the effective Lagrangian. This is supported by experimental data since it avoids deviations of the $\rho$ parameter from the well established value close to 1. The additional symmetry leads to the following relations between the C and P violating couplings:

$$\kappa_Z=g_1^Z-\tan^2{\theta_\textrm{w}}(\kappa_\gamma-1)$$

((3.51))

$$\lambda_Z=\lambda_\gamma$$

((3.52))

With this assumption the parameter space becomes more restricted and constraints are more stringent with the given amount of data.

The charged TGCs are extracted from LEP data in an analysis of the multi-differential cross-section of W-pair production [36,37,38,39]. Neglecting photon radiation and fixing the mass of the W boson, five angles describe the four-fermion final state in W-pair decays: the polar decay angle of the W ⁻ boson, $\varTheta_W$, and the polar and azimuthal decay angles of the fermions in the W rest frames. The TGCs influence the total production cross-section, the W ⁻ production angle and the fermion angles by changing the W polarisation.

To improve the resolution on the angles, a kinematic fit is applied to the events, asking for four-momentum conservation and equality of the two reconstructed W masses. In $\textrm{q q} \tau \nu$ events the two hadronic jet energies are rescaled such that their sum equals $\sqrt{s}/2$. This yields nearly the same performance as a kinematic fit, which can be less constrained due to the tau neutrino. In fully hadronic events, the assignment of two jet pairs to two W decays, the so-called “pairing”, can be done using neural network techniques. They yield a correct pairing typically in the order of 75–80% of the cases [38]. Input to the neural network are kinematic variables like the difference and sum of the masses of the jet pairs, and the sum and minimum of the angles between paired jets. Another quantity is the value of the matrix element for the reaction $\textrm{e}^+\textrm{e}^-\rightarrow\textrm{WW}\rightarrow\textrm{ffff}$ applied to the four reconstructed jets and calculated, e.g., with the EXCALIBUR [40] program. Finally, the difference between the W charges is used as calculated from the sum of jet charges.

The jet-charge is assigned by applying a track based technique. The tracks belonging to each quark jet are weighted by their momenta:

$$q_\textrm{jet}=\frac{1}{N_\textrm{tracks}}\sum_\textrm{tracks} q_i|\textbf{p}_i\cdot\textbf{d}_\textrm{jet}|^\kappa ,$$

((3.53))

with the number of tracks, $N_\textrm{tracks}$, their charge, $q_i=\pm1$, and the track momentum $\textbf{p}_i$ projected on the jet direction $\textbf{d}_\textrm{jet}$. The momentum weight $\kappa$ is chosen to be unity. With this definition, the charge is found to be correct in about 70% [38] of the Monte Carlo events under the condition that the pairing is correct. Alternatively, if the jet charge is not used in the analysis [36] the corresponding angular distribution are folded and no sign is determined.

The W ⁻ direction, $\cos\varTheta_W$, in semi-leptonic events is determined from the hadronic W decay. The sign of $\cos\varTheta_W$ is derived from the lepton charge. In fully hadronic events, the direction comes from either W pair, which are back-to-back after the kinematic fit. Sign information is provided by the charges of the jet-pairs. Figure 3.19 shows the W boson production angle in W-pairs as measured by the ALEPH experiment, and the sensitivity of the data to the anomalous $\lambda_\gamma$ coupling.

Additional sensitivity to the WW$\gamma$ vertex is also in single-W production, where the hadronic cross-section and the leptonic energy spectrum are used as input. Overlap with W-pair events is carefully removed to avoid double-counting. Single photon events, $\textrm{e}^+\textrm{e}^-\to\nu_{e}\bar{\nu}_e\gamma$, are mainly produced through initial-state radiation in s-channel Z-boson exchange or t-channel W-boson exchange, but the process contains also a small contribution from W-boson fusion through the WW$\gamma$ vertex.

All measured final states are input to a combined analysis of the angular distributions, energy spectra and the production rates at the different centre-of-mass energies. The dependence of the observables on the couplings is derived by mainly using two different techniques, the optimal observable [41] and matrix-element reweighting methods . The former is exploiting the fact that the differential cross-section, $d\sigma/d\varOmega$, in any set of measurable variables, $\varOmega$, depends only quadratically on the couplings, g _i:

$$\frac{d\sigma}{d\varOmega}=S_0+S_ig_i+S_{ij}g_ig_j .$$

((3.54))

It can then be shown that the set of observables ${\cal O}_i=S_i/S_0$ and ${\cal O}_{ij}=S_{ij}/S_0$ are optimal in the sense that they contain the maximum information about the couplings g _i. The coupling parameters, $g_i (i=1,\ldots,n)$, are determined by comparing the measured mean values of ${\cal O}_i$ and ${\cal O}_{ij}$ to their expectation values, $E[{\cal O}_i]$ and $E[{\cal O}_{ij}]$, for a certain set of couplings. The optimal observables can be either calculated analytically or by Monte Carlo simulations. Correlations between them are properly taken into account.

The expectation values, $E[{\cal O}(g_1,\ldots,g_n)]$, are usually constructed by reweighting of simulated Monte Carlo events . This reweighting is based on the matrix element squared, $|M|^2$, calculated for the given production process, e.g. $\textrm{e}^+\textrm{e}^-\to\textrm{WW}\to\textrm{ffff}(\gamma)$. A weight for each simulated signal event j is then determined as the ratio

$$w_j(g_1,\ldots,g_n)=\frac{\left|M\left(p^j_1,\ldots,p^j_n;g_1,\ldots,g_n\right)\right|^2}{\left|M_{SM}\left(p^j_1,\ldots,p^j_n\right)\right|^2},$$

((3.55))

where the four-momenta of the final state particles are denoted as $p^j_1,\ldots,p^j_n$. The matrix element M contains anomalous contribution parameterised by $g_1,\ldots,g_n$, while M _SM is the matrix element for the Standard Model expectation.

The second extraction method is based directly on matrix element reweighting and compares the measured differential cross-section with the predicted ones. A likelihood is constructed as the Poisson probability to observe N _i events in a certain phase space interval (bin), when $\mu_i(g_1,\ldots,g_n)$ events are expected:

$$L=\prod\limits_{i=1}^{N_\textrm{bins}} \frac{e^{-\mu_i(g_1,\ldots,g_n)}\mu_i(g_1,\ldots,g_n)^{N_i}}{N_i!}.$$

((3.56))

The expectation value $\mu_i$ is calculated by taking the normalised sum of the simulated Monte Carlo events, rescaled by the event weights $w_j(g_1,\ldots,g_n)$.

The ALEPH and OPAL experiments apply an optimal observable analysis [36,39] and the L3 experiment uses the reweighting method to determine the charged TGCs [38]. DELPHI derives the couplings from the spin-density matrix [37]. All experiments exploit furthermore the dependency of the total signal cross-section sections on the couplings.

Systematic uncertainties of the TGC measurement are from various sources, like background estimation, modelling of hadronisation, LEP beam energy, lepton and jet measurement, charge confusion, and final state interaction in W-pairs, none of them actually dominating. With the more accurate theoretical predictions of the W-pair production cross-section, $\sigma_{WW}$, applying LPA/DPA techniques, the uncertainty on $\sigma_{WW}$ are much reduced, as well as the uncertainties on the $O(\alpha)$ corrections to the distribution of W production angle.

The likelihood curves of the LEP experiment for the different gauge couplings are combined taking correlations between systematic uncertainties into account. The current preliminary results of 1-parameter fits to the data are [8]

$$\begin{aligned} g_1^Z&{}= 0.991^{+0.022}_{-0.021}\\ \kappa_\gamma&{}= 0.984^{+0.042}_{-0.047}\\ \lambda_\gamma&{}=-0.016^{+0.021}_{-0.023} . \end{aligned}$$

The main systematic uncertainty is from $O(\alpha)$ corrections which contributes with an uncertainty of 0.010 to $g_1^Z$ and $\lambda_\gamma$ and with 0.010 to $\kappa_\gamma$. The LEP TGC measurements are however dominated by statistical uncertainties.

All couplings agree well with the theoretical expectations, as can be seen in Fig. 3.20 for two-parameter fits. This is a direct evidence for the non-abelian structure of the theory. The W boson couples indeed to the photon and the Z boson. The measurements are fully compatible with the predicted vertex structure and coupling strength of the Standard Model. Furthermore, no evidence for neutral three-boson coupling is found, neither at LEP [8] nor in pp processes at the Tevatron [42].

Anomalous contributions to the quartic couplings of the W boson, with four-boson vertices WW$\gamma\gamma$ and WWZ$\gamma$, are as well constrained by measurements [43] of the photon spectra in WW$\gamma$ and $\nu\nu\gamma\gamma$ events. The combined LEP results for the WW$\gamma$ cross-sections are shown in Fig. 3.21. The theoretical framework is given by the following Lagrangian containing dimension-6 operators [44]:

$${\cal L}^0_{6}=-\frac{\pi\alpha}{4}\frac{a_0}{\varLambda^2}F_{\mu\nu}F^{\mu\nu}\textbf{W}^\alpha\cdot\textbf{W}_\alpha$$

((3.57))

$${\cal L}^c_{6}=-\frac{\pi\alpha}{4}\frac{a_c}{\varLambda^2}F_{\mu\alpha}F^{\mu\beta}\textbf{W}^\alpha\cdot\textbf{W}_\beta$$

((3.58))

$${\cal L}^n_{6}=-\frac{\pi\alpha}{4}\frac{a_n}{\varLambda^2}\varepsilon_{ijk}W^{(i)}_{\mu\alpha}W^{(j)}_\nu W^{(k)\alpha}F^{\mu\nu},$$

((3.59))

where a ₀, a _c, and a _n are the anomalous quartic couplings of the W boson and $\varLambda$ the energy scale of the effective theory. The influence of the anomalous couplings on the photon spectrum in WW$\gamma$ events is demonstrated in Fig. 3.21. Limits are set by the LEP experiments, which are all compatible with the non-existence of anomalous contributions. They are in the order of

$$-0.020 \textrm{GeV}^{-2}< a_0/\varLambda^2 <0.020 \textrm{GeV}^{-2}$$

((3.60))

$$-0.064 \textrm{GeV}^{-2}< a_c/\varLambda^2 <0.032 \textrm{GeV}^{-2}$$

((3.61))

$$-0.18 \textrm{GeV}^{-2}< a_n/\varLambda^2 <0.14 \textrm{GeV}^{-2},$$

((3.62))

citing the DELPHI result [43] as a typical example.

The WWZZ and WWWW scattering vertices can not be directly probed at LEP. But anomalous behaviour would show up in the neutral and charged TGC measurements. The ALEPH collaboration, for example, interpreted the measurements of W-pair production and corresponding angular distributions in terms of strong W _L W _L scattering with an intermediate techni-$\rho$ exchange. Since data were found to be in agreement with Standard Model expectations a lower limit on the mass of the techni-$\rho$ could be set at 600 GeV (95% C.L.) assuming that the width of the new particle it at most as large as its mass [36].

In summary, measurements of the electroweak boson couplings at LEP as well as at other collider experiments agree very well with the Standard Model predictions, both in their structure as well as in their magnitude.

3.6 Final State Interactions in W Boson Decays

When W boson pairs decay hadronically, so-called Bose-Einstein correlations and colour reconnection effects may alter the hadronic final state. These phenomena also influence measurements of W boson properties, especially the determination of the mass of the W.

3.6.1 Bose-Einstein Correlations

The origin of Bose-Einstein correlations (BEC) is the quantum mechanical requirement that the decay amplitude is symmetric under the exchange of identical bosons in the final state. The idea was developed in astronomy as intensity interferometry, also known as Hanbury/Brown/Twiss interferometry [45]. In particle physics, charged pion correlations were first observed in $p\bar{p}$ collisions [46,47,48].

In the plane-wave approximation, the amplitude to observe two identical bosons, that are produced at space-time point x ₁ and x ₂ with momenta p ₁ and p ₂ and which are measured by two detectors at points x _A and x _B is given by [47,49]:

$$\begin{aligned} |p_1p_2\rangle&{}=\frac{1}{\sqrt{2}}\left\{ e^{i p_1 (x_1-x_A)+i\phi_1} e^{i p_2 (x_2-x_B)+i\phi_2}\right.\\ &{}\left.+e^{i p_1 (x_2-x_A)+i\phi_1} e^{i p_2 (x_1-x_B)+i\phi_2}\right\}\\ &{}=\frac{1}{\sqrt{2}} e^{i (p_1 x_A p_2x_B-\phi_1-\phi_2)}\left\{ e^{ip_1x_1} e^{ip_2x_2}+e^{ip_1x_2} e^{ip_2x_1} \right\}, \end{aligned}$$

((3.63))

where $\phi_1$ and $\phi_2$ are the phases. The last term arises due to the symmetrisation of the amplitude, as required by Bose-Einstein statistics. Squaring the amplitude yields the two-particle correlation:

$$R_2(p_1,p_2)=R_2(Q)=1+\cos(Q\varDelta x),$$

((3.64))

with the four-momentum difference $Q=p_1-p_2$ and the space-time difference of the sources $\varDelta x=x_1-x_2$. One obtains an enhancement of particles produced close in phase-space. Note that the dependence of R ₂ on the location of the detectors dropped out. The correlation function therefore contains also information about the source of the bosons. In more realistic calculations, the particle source is modelled with various shapes and may also move in space-time [47].

In the LEP analyses [50,51,52,53], the PYBOEI model implemented in the PYTHIA Monte Carlo generator [54] is used to describe BEC effects with a Gaussian parameterisation of the correlation function:

$$R_2(Q)=1+\lambda e^{-Q^2R^2}.$$

((3.65))

The parameter $\lambda$ is the BE correlation strength, while the parameter R corresponds to the radius of the source. The correlations alter the momentum distribution of the final state particles, but the total momentum must be conserved. This is achieved by a local energy compensation with a negative BE enhancement with 1/3 of the radius R. This compensation is further constrained to vanish at $Q=0$ by introducing an additional $1-e^{Q^2R^2/4}$ factor. This corresponds to the BE ₃₂ option of the PYTHIA generator . The LEP experiments have tuned their hadronisation models including BEC to hadronic data taken at the Z resonance, where BEC was clearly established [55] in $\pi^\pm\pi^\pm$ and $\pi^0\pi^0$ data samples.

The correlation function R ₂ is determined from data by taking the ratio of the measured and background corrected two-particle density, $\rho_2(Q)=1/N_\textrm{pairs}\frac{dN_\textrm{pairs}}{dQ}$ to the two-particle density of a reference Monte Carlo simulation without BEC, $\rho_0(Q)$:

$$R_2(Q)=\frac{\rho_2(Q)}{\rho_0(Q)}$$

((3.66))

Using this method, a good agreement between BEC in hadronically decaying W bosons in ${\textrm{q q} \ell \nu}$ events and hadronic Z boson decays is observed, as can be seen in Fig. 3.22. The selected Z decays are depleted in their $Z\to {\textrm{b} \overline{\textrm{b}}}$ content to resemble the W decay branching fractions, which mainly decay to ud and cs quark pairs.

In fully hadronic W-pair decays, correlations may also appear between final state particles from different W bosons. It would have an impact on the measurement of the mass of the W boson because it may lead to momentum transfer between the two hadronically decaying W’s.

This interesting scenario of BEC between two different W bosons (inter-W BEC) is studied by comparing the two-particle density in $\textrm{q q q q}$ events, $\rho_2^{WW}(Q)$, with those expected for an event sample without interference between the two decaying W bosons. This correlation function is constructed in the following way:

$$\rho_2^{WW}(Q)=2\rho_2^{W}(Q)+2\rho_\textrm{mix}^{WW}(Q),$$

((3.67))

where $\rho_2^{W}(Q)$ is the background-corrected density measured in semi-leptonic events and $\rho_\textrm{mix}^{WW}(Q)$ the density constructed by mixing uncorrelated particle pairs of independent semi-leptonic events. The mixed events are carefully arranged so that their kinematic properties agree with the $\textrm{q q q q}$ event sample, including the corresponding event selection procedure.

In absence of interference between the W bosons, Eq. (3.67) holds and the difference

$$\delta\rho(Q)=\rho_2^{WW}(Q)-2\rho_2^{W}(Q)-2\rho_\textrm{mix}^{WW}(Q)$$

((3.68))

is expected to vanish. Figure 3.22 shows the measured $\delta\rho(Q)$ distribution for the OPAL experiment. The agreement with zero can be further quantified by determining the integral of the $\delta\rho(Q)$ distribution from 0 to $Q_{\max}$:

$$J=\int_0^{Q_{\max}}\delta\rho(Q) dQ.$$

((3.69))

Sensitivity to genuine correlations between the W bosons (inter-W correlations) is obtained in the inter-source correlation function $\delta_I(Q)=\delta\rho(Q)/\rho_\textrm{mix}^{WW}(Q)$, as well as from the ratio

$$D(Q)=\frac{\rho_2^{WW}(Q)}{2\rho_2^{W}(Q)+2\rho_\textrm{mix}^{WW}(Q)}.$$

((3.70))

To correct possible distortions of the ratio due to the event mixing procedure or detector effects, the double ratio

$$D^{\prime}(Q)=\frac{D(Q)_\textrm{data}}{D(Q)_{MC,\textrm{no-inter}}}$$

((3.71))

is introduced, where $D(Q)_{MC,\textrm{no-inter}}$ is derived from a Monte Carlo sample without BEC between W bosons. $D'(Q)$ can then be parameterised as

$$D^{\prime}(Q)=(1+\delta Q)(1+\varLambda e^{-k^2Q^2}),$$

((3.72))

where $\varLambda$ corresponds to the correlation strength. In the measurement of like-sign final state hadrons, the L3 experiment finds values compatible with no inter-W correlations [52]:

$$\begin{aligned} J(\pm,\pm)&{}=0.03\pm 0.33\pm 0.15,\textrm{and}\\ \varLambda&{}=0.008\pm 0.018\pm 0.012 \end{aligned}$$

where the first uncertainty is statistical and the second systematic. The variation of track and event selection, as well as the mixing procedure contribute most to the systematic uncertainty. As a consistency check, the J integral for unlike-sign pairs yields $J(+,-)=0.01\pm 0.36\pm 0.16$, consistent with zero. The expectations for a scenario with inter-W BEC are $J(\pm,\pm)=1.38\pm 0.10$ and $\varLambda=0.098\pm 0.008$, respectively, which means the inter-W BEC model disagrees with data by more than 3.5 standard deviations.

Figure 3.23 summarises the various LEP measurement on BEC, where various variables and methods are compared in terms of the relative fraction of the inter-W BEC model that is seen in data. Combining the results of all LEP experiments, this fraction is found to be $0.17\pm 0.13$ [8]. This means data are consistent with the absence of inter-W BEC and significant BEC effects according to the BE ₃₂ model are disfavoured.

3.6.2 Colour Reconnection

Colour reconnection (CR) is summarising QCD interference effects which lead to strong interaction between two colourless states. This phenomenon is observed, for example, in the colour-suppressed decay of B mesons, $B\to J/\psi+X$, where the c and $\bar{c}$ quarks of the $c\bar{c}$ colour singlet state of the $J/\psi$ meson are each originating from separate colour singlet states, which have to be colour reconnected. It is therefore interesting to study if this effect is also present in hadronically decaying W’s, whose decay products are necessarily colour singlets. Due to the short but non-zero lifetime of the W, the decays are separated in space. The distance is however much shorter than the typical hadronic interaction length of 1 fm, so that CR effects may alter the two W decays.

Like inter-W BEC, CR strongly affects the W mass measurement in fully hadronic W-pair events because the reconstructed mass of the W bosons would not be identical to their propagator mass. At LEP, this was the initial motivation to search for signatures of CR. The W mass is actually the observable that is most affected by CR, and without refinement of the data analysis it leads to significant systematic uncertainties on $M_\textrm{W}$ in the fully hadronic decay channel.

The description of CR [56,57] in perturbative QCD is considering pure gluon exchange between the quarks from the W decays. But the perturbative effect is suppressed by the probability $1/N_c^2$ for accidental colour reconnection. Furthermore, the momenta, k, of primary gluons generated by the reconnected system is expected to be limited by the finite width of the W boson to $k\le k_{\max}\approx\varGamma_\textrm{W}M_\textrm{W}/E_W$. It was indeed found that the effect on the rate of charged final state hadrons, on their momentum spectrum and their distribution in phase space is only small. Not far from the W-pair threshold the CR effects on the particle-flow distribution is expected to be [58]:

$$\frac{\varDelta N^{CR}}{N_{\textrm{no}-CR}}\le\frac{\alpha_s(\varGamma_\textrm{W})} {N_c^2}\frac{N^{\prime}_q\left(k^{CR}_{\max}\right)}{N^{\prime}_q(M_\textrm{W}/2)}\approx {\cal O}(10^{-2}),$$

((3.73))

where

$$N^{\prime}_q(E)=\frac{d N_q(E)}{d\log E}$$

((3.74))

with the multiplicity, N _q(E), inside a QCD jet of energy E. Therefore, only few low-energy particles are affected by perturbative CR.

Ad-hoc models [57,58,59,60,61,62] for the non-perturbative phase predict, however, a much larger influence on the hadronic observables, in particular on the reconstructed W mass in $\textrm{q q q q}$ events. Several models are interfaced to the standard Monte Carlo generators that are used to simulate the fragmentation and hadronisation phase of the hadronic W decays.

The Sjöstrand-Khose model SK I [57] is the most popular CR model since it has a free parameter, k _I, to adjust the CR probability. The model is implemented in the Pythia program and is based on string fragmentation concepts. In the SK I model, the colour field strength, $\varOmega_0(\textbf{x},t,\textbf{u})$, of strings stretched out along the unit vector $\textbf{u}$, are given an extension in phase-space with a cylindrical shape:

$$\varOmega_0(\textbf{x},t,\textbf{u})=\exp\left(\frac{-(\textbf{x}^2-(\textbf{u}\textbf{x})^2)}{2r_\textrm{had}^2}\right)\theta(t-|\textbf{x}|)\exp\left(\frac{-(t^2-(\textbf{u}\textbf{x})^2)}{2\tau_\textrm{frag}^2}\right),$$

((3.75))

with the radial extension of the string $r_\textrm{had}$ and proper time scale of the string fragmentation, $\tau_\textrm{frag}\approx 1.5 \textrm{fm}/c\approx 3 r_\textrm{had}/c$. When the string is produced at space-time coordinate $(t_0,\textbf{x}_0)$ and is propagating with $\boldsymbol{\beta}$ perpendicular to $\textbf{u}$ the colour field is travelling according to

$$\varOmega(\textbf{x}-\textbf{x}_0,t-t_0,\boldsymbol{\beta},\textbf{u})=\varOmega_0((\textbf{x}-\textbf{x}_0)',(t-t_0)',\textbf{u})$$

((3.76))

with the boosted coordinates $(\textbf{x}-\textbf{x}_0)'$ and $(t-t_0)'$.

The reconnection occurs when two colour fields of the two $W^\pm$ bosons have sufficient overlap. In general, many strings are present in each fragmenting $W^\pm$, but only the one with maximum $\varOmega^\pm$ value is assumed to contribute to the reconnection probability. The calculation of the overlap integral

$$I^\pm=\int d^3x dt \varOmega^+_{\max}\left(\textbf{x},t,\textbf{x}_0^+,t_0^+\right)\varOmega^-_{\max}\left(\textbf{x},t,\textbf{x}_0^-,t_0^-\right)$$

((3.77))

is however time-consuming if performed numerically during the Monte Carlo generation of each event. It is therefore approximated by the sampling sum

$$I^\pm=\frac{1}{N_\textrm{samp}}\sum_{N_\textrm{samp}}\frac{\varOmega_{\max}^+\varOmega_{\max}^-}{\varOmega_\textrm{trial}}$$

((3.78))

with a corresponding trial distribution

$$\varOmega_\textrm{trial}=\exp\left(\frac{-\textbf{x}^2}{f_r^2r_\textrm{had}^2}\right)\exp\left(\frac{-t'^2}{f_t^2\tau_\textrm{frag}^2}\right)\theta(t'),$$

((3.79))

with $t'=t-\textrm{max}(t^-,t^+)$. The parameters $N_\textrm{samp}=1000, f_r=2.5$ and $f_t=2.0$ are chosen to yield sufficient efficiency and numerical accuracy. When two strings overlap they are eventually reconnected according to the probability:

$$p_\textrm{reco}=1-e^{-k_I f_r^3 f_t I^\pm},$$

((3.80))

with the free parameter k _I. At most one reconnection per event is allowed. Figure 3.24 shows the reconnection probability obtained from a reference Monte Carlo sample of $\textrm{WW}\to\textrm{q q q q}$ events at $\sqrt{s}=189 \textrm{GeV}$ as a function of k _I.

The models SK II and SK II’ apply a string picture that corresponds to vortex lines like in type II superconductors. The topological information is concentrated in the core region around the string and reconnection occurs with unit probability when two of these regions overlap. The transverse extent of the strings can thus be neglected, which was not the case in the SK I model. There is also no free parameter.

The Ariadne program introduces CR in the dipole-cascade model [58]. The two quark pairs of the $W^\pm$ decays form two colour dipoles from which two cascades start. A randomly chosen colour index in the range 1–9 is assigned to each colour dipole in every step of the cascades. If two dipoles happen to have the same index they are allowed to reconnect. Special care is taken so that no unphysical colour flows appear. The actual reconnection is eventually taking place when the string related $\lambda$ measure is reduced. This measure is defined as

$$\lambda=\sum\limits_1^{n-1}\log\frac{(p_i+p_{i+1})^2}{m_0^2}$$

((3.81))

with the four momenta p _i of the n partons along the colour string, and the mass scale $m_0\approx 1 \textrm{GeV}$. This condition corresponds to the reasonable assumption that partons close in phase-space are more likely to reconnect.

In W pair events the two dipole cascades first evolve independently from large gluon energies down to $E_g\approx\varGamma_\textrm{W}$. Only in the low energy regime $E_g<\varGamma_\textrm{W}$ inter-W colour reconnection between the two cascades is turned on. The matching of the two stages of the cascade is properly taken care of. In the Ariadne AR 2 model, the inter-W reconnection probability is about 22% at a centre-of-mass energy of 189 GeV. To compare with the Ariadne model with only intra-W CR, AR 1, also this cascade is run in two stages, first down to $E_g\approx\varGamma_\textrm{W}$ and then for lower gluon energies.

Colour reconnection is as well implemented in the Herwig cluster fragmentation model [62]. In the limit of a large number of colours, every quark or anti-quark produced in the parton shower has a unique colour-connected partner with which it can be clustered. In addition, every gluon has a colour and an anti-colour index, each uniquely connected to another parton, and after showering the gluons split into quark pairs. Each quark pair is eventually forming a cluster with its colour connected partner. Like in perturbative QCD, local colour reconnection can then occur with the probability $1/N_c^2=1/9$, which also defines the inter-W reconnection rate.

Some of the models can also be tested at the Z peak in three-jet event topologies, where two jets originate mainly from $\mathrm{b} \overline{\mathrm{b}}$ pairs and the third from gluon radiation. This allowed the gluon jet to be identified with an anti-b-quark tag. In the subsequent analysis gluon jets with a rapidity gap were studied with respect to the total charged particle multiplicity [63,64]. Similarly, the asymmetry in particle-flow between the inter-quark jet region and the quark-gluon region was measured [65]. Both type of analyses are sensitive to a rearrangement of the colour flow in the event. All LEP measurements show that the Ariadne model type 1 [58] is not consistent with data. Also the Rathsman/GAL model [61] with default parameter settings fails to describe the measurements. Other models like SK I and AR 2 could not be tested because they predict CR effects only in WW decays, not in $\mathrm{q} \overline{\mathrm{q}}(\gamma)$ events.

The measurement of CR in WW decay data was first investigated by analysing charged particle rates in $\textrm{q q q q}$ events compared to $\textrm{q q} \ell \nu$ events. The OPAL experiment determined, for example, the mean number of charged particles in the two decay classes. Well reconstructed tracks in the OPAL jet chamber with a minimal transverse momentum of 0.15 GeV pointing to the interaction vertex were used in the analysis. Using all high-energy data at $\sqrt{s}=189-209 \textrm{GeV}$ the following values were found [66]:

$$\begin{aligned} \left\langle n_{ch}^\textrm{q q q q}\right\rangle&{}=38.74\pm 0.12\pm 0.26\\ \left\langle n_{ch}^\textrm{q q} \ell \nu\right\rangle&{}=19.39\pm 0.11\pm 0.09\\ \varDelta\langle n_{ch}\rangle&{}=\left\langle n_{ch}^\textrm{q q q q}\right\rangle-2\left\langle n_{ch}^\textrm{q q} \ell \nu\right\rangle=-0.04\pm 0.25\pm 0.16 , \end{aligned}$$

where the first uncertainty is statistical and the second systematic, mainly originating from uncertainties on the hadronisation model and on the comparison of $\textrm{e}^+\textrm{e}^-\to\textrm{q} \overline{\textrm{q}}$ data to simulations, which were used to refine the detector simulation. The CR models predict the values of

$$\varDelta\langle n_{ch}\rangle= \begin{cases} -0.42 & (\textrm{SK I, k}_\textrm{I}=100)\\ -0.29& (\textrm{SK I, k}_\textrm{I}=0.9)\\ -0.14& (\textrm{SK II})\\ -0.19& \textrm{(AR 2)}\\ +0.32& \textrm{(Herwig CR)} \end{cases}$$

((3.82))

The mean value differences are all compatible with OPAL data, which is also visible in Fig. 3.24. Thus, the charge multiplicity is not sensitive enough to decide on the validity of the CR models.

A much more sensitive variable is constructed from the particle-flow [67,68] in the $\textrm{q q q q}$ event, based on the string fragmentation picture [69]. It leads to the expectation that CR changes the fragmentation in the regions between the primary quark jets. Hadronic activity between jets from different W bosons should be slightly enhanced, while the one between jets coming from the same W boson should be reduced. Furthermore, CR should mainly alter the low-momentum jet particles.

In the L3 analysis [70], a well defined sub-sample of the $\textrm{q q q q}$ events is selected by requiring that the quark-jet association is optimal. The two largest inter-jet angles are required to be between $100^\circ$ and $140^\circ$ and not adjacent. The two other inter-jet angles must be less than $100^\circ$. In this way, the two strings between the $\textrm{q} \overline{\textrm{q}^{\prime}}$ pairs evolve in opposite directions. The selection efficiency is low, in the order of 15%, but the event sample has a high purity of 85%. The rate of correct pairing of the jets to two W bosons is estimated to be 91%.

In the second step of the analysis, a plane is defined by the direction of the most energetic jet (jet 1) and the direction of the closest jet that has an angle larger than $100^\circ$ with respect to jet 1. These jets are likely to originate from the same W. Since the events are in general not planar, three more planes are defined, spanned by the directions of each other jet pair 2+3, 3+4, 4+1. All particles, i, are projected on the planes and for those in the inter-jet region an angle $\phi_{i,j}$ with respect to jet j is calculated. Eventually, the analysis uses the rescaled angles

$$\phi_i^\textrm{resc}=j-1+\frac{\phi_{i,j}}{\psi_{j,j+1}},$$

((3.83))

where $\psi_{j,j+1}$ is the angle between jet j and jet $j+1$. Figure 3.25 shows the rescaled angular distribution after background subtraction. The directions of the jets are at angles 0, 1, 2, and 3. By construction, the regions $A\in [0,1]$ and $B\in [2,3]$ contain preferentially particles originating from the same W boson, while the regions $C\in [1,2]$ and $D\in [3,4]$ contain particles which are between jets of different W bosons. To enhance the CR sensitivity, a ratio of the integrated particle-flow distributions is built:

$$R_N=\int_{0.2}^{0.8}\frac{1}{2}\left(\frac{dn _A}{d\phi}+\frac{dn _B}{d\phi}\right)\left\{\int_{0.2}^{0.8}\frac{1}{2}\left(\frac{dn_C}{d\phi}+\frac{dn _D}{d\phi}\right)\right\}^{-1}$$

((3.84))

The ratio of the non-integrated distributions is shown in Fig. 3.25. The ratio R _N depends on the centre-of-mass energy and for the combination of all data the values are scaled back to $\sqrt{s}=189 \textrm{GeV}$. Possible reconnection effects in the remaining ZZ$\to\textrm{q q q q}$ background is neglected. L3 obtains [70]:

$$R_N=0.915\pm 0.023 \textrm{(stat.)}\pm 0.021 \textrm{(syst.)}.$$

((3.85))

The systematic uncertainty is mainly due to the experimental definition of the energy flow objects, where calorimetric cluster and track based analyses are compared. The measurement agrees well with the expectation from the Pythia fragmentation without CR, $R_N(\textrm{Pythia, no-CR})=0.918\pm 0.003$.

Similar analyses are performed by the DELPHI and OPAL experiments [66,71]. The DELPHI analysis is based on selecting W-pair events with a particular jet configuration, like the L3 data sample, while OPAL applies an event selection used in the W mass analysis. The OPAL event selection has therefore a higher efficiency of 86% but accepts jet topologies with a more complicate colour flow. Since the measured values of R _N are not corrected for detector acceptance, resolution or efficiency the following ratio is constructed [8]:

$$r^\textrm{data}=\frac{R_N^\textrm{data}}{R_N^\textrm{no-CR}},$$

((3.86))

where $R_N^\textrm{no-CR}$ is a reference value from a Monte-Carlo sample without CR. The measured $r^\textrm{data}$ can then be compared to a model prediction, for example the r(k _I) determined for the SK I model, which depends on the reconnection parameter k _I as shown in Fig. 3.26.

As mentioned above, the W mass measured in the LEP detectors is also a very sensitive variable to CR. In a series of studies, the effect of CR on $M_\textrm{W}$ was found to be reduced if low-momentum tracks and low-energy cluster are removed in the jet reconstruction. Only those above a certain $p_\textrm{cut}$ or $E_\textrm{cut}$ are considered. Alternatively, the jets are restricted to a certain angular cone, and the new jet direction and momentum is determined in an iterative procedure. Excluding particles in the inter-jet regions outside the cones reduces the effect of CR on M _W. A third method applies weights, $w=p^{\kappa}$, to the momentum contribution of each particle in the jet, again suppressing low-momentum tracks, and therefore CR effects, if $\kappa>0$, but enhancing CR effects if $\kappa <0$.

Figure 3.26 shows the shift in $M_\textrm{W}$ that is expected to be induced by CR to the nominal W mass value as a function of the particle momentum cut applied in the jet reconstruction. As can be clearly seen in this ALEPH study [72], the systematic shift of $M_\textrm{W}$ is much reduced by this method. It is also remarkable that all CR models show the same trend. However, at the same time the jet resolution is worsening the more particles are removed from the jets, and so is the statistical uncertainty on $M_\textrm{W}$ in the qqqq channel. Eventually, the best method and its corresponding cut value must be found in an optimisation procedure taking statistical and systematic uncertainties and their correlations into account. More details are given in the mass measurement section.

Important for the measurement of CR is the fact that, if CR exists, the measured $M_\textrm{W}^\textrm{q q q q}$ should vary when the jet reconstruction parameters are varied. The ALEPH experiment measured the variation of $M_\textrm{W}$ as a function of $p_\textrm{cut}$ and of the cone radius R, and a linear fit yields values compatible with no $M_\textrm{W}$ shift:

$$\begin{aligned} \varDelta M_\textrm{W}&{}=(-11\pm 16) \frac{\textrm{MeV}}{\textrm{GeV}} p_\textrm{cut}\\ \varDelta M_\textrm{W}&{}=(+9\pm 19) \frac{\textrm{MeV}}{\textrm{rad}^{-1}} \frac{1}{R} . \end{aligned}$$

DELPHI compared the standard $M_\textrm{W}$ value with alternative estimators applying a cone cut at $R=0.5 \textrm{rad}$ and a momentum cut at 2 GeV:

$$\begin{aligned} \varDelta M_\textrm{W}(\textrm{std},R=0.5 \textrm{rad})&{}=(59\pm 35 \textrm{(stat.)}\pm 14 \textrm{(syst.)}) \textrm{MeV}\\ \varDelta M_\textrm{W}(\textrm{std},p_\textrm{cut}=2.5 \textrm{GeV})&{}=(143\pm 61 \textrm{(stat.)}\pm 29 \textrm{(syst.)}) \textrm{MeV} \end{aligned}$$

The former is well consistent with zero, the latter, however, differs from no shift by about two standard deviations. And finally, the OPAL experiment uses their most sensitive estimator, the $M_\textrm{W}$ difference between the mass reconstructed applying a $p_\textrm{cut}$ of 2.5 GeV and applying a negative momentum weight with a $\kappa$ value of −0.5:

$$\varDelta M_\textrm{W}(p_\textrm{cut}=2.5 \textrm{GeV},\kappa=-0.5)=(-152\pm 68 \textrm{(stat.)}\pm 61 \textrm{(syst.)}) \textrm{MeV}.$$

((3.87))

The OPAL result is compatible with zero at the 1.5 standard deviation level. All experiments performed a systematic cross-check using semi-leptonic events, in which the same jet variations are applied. The $\textrm{q q} \ell \nu$ analyses all gave results consistent with no effect. The observed shifts are compared to the SK I model predictions, as it is shown in Fig. 3.27 for the OPAL result.

The LEP data is eventually combined by constructing a total $\varDelta\chi^2$ function which takes correlated uncertainties between the individual measurements into account. Sources of systematic uncertainties are from hadronisation and BEC effects, as well as from the modelling of the background scale and shape. The previously mentioned limit on the BEC strength is used to constrain the BEC systematics. The SK I model is taken as the main reference and k _I is the main parameter varied in the $\chi^2$ minimisation. Figure 3.28 shows the resulting curves of the four LEP experiments together with the combined LEP measurement using information from the particle-flow analysis and the $M_\textrm{W}$-shift studies. The best value for the parameter k _I is found to be

$$k_I=1.26^{+0.84}_{-0.64}$$

((3.88))

$$k_I\in [0.62,2.10] (68\% \textrm{C.L.}).$$

((3.89))

This corresponds to a preferred reconnection probability of 51% in the SK I model , evaluated at a centre-of-mass energy of 189 GeV. The absence of CR can not be excluded, but it is disfavoured with 2.8 standard deviations. On the other hand, an extreme CR scenario with 100% reconnection fraction is ruled out with a significance of 6.9 standard deviations. The LEP data therefore supports the existence of CR as modelled in SK I. Other CR scenarios like AR 2 and Herwig predict smaller reconnection probabilities of 22 and 11%, respectively. In dedicated studies of these two CR models by the LEP experiments [72, 71, 70, 66] both show only small deviations from the no-CR scenario, which is not favoured by the combined LEP result. When re-interpreting the SK I measurement, the AR-2 model agrees with data at the 2.0 standard deviation level, while for Herwig the consistency is at the 2.4 standard deviation level.

The measurement of CR in hadronic W-pair events is an interesting physics result by itself because it confirms the existence of colour rearrangement as observed in colour-suppressed meson decays. The reconnection probability is even beyond the naive expectation when simply counting the number of colour combinations, $1/N^2_c$. The determination of CR effects directly from data is a very important ingredient for the W-mass measurement at LEP. As discussed below, CR is one of the main sources of systematics on $M_\textrm{W}$ and the result is used to estimate the corresponding uncertainties.

The CR measurement at LEP may also be helpful for physics at $p\bar{p}$ and pp colliders, Tevatron and LHC. In the simulation of the underlying event CR effects play a significant role [73]. The CR Monte-Carlo parameters are usually tuned to measured event shape variables of the underlying event and thus constrained [74]. The CR models used in these estimations are implemented in Pythia [54] and typically assume a 33% direct reconnection probability with nearest neighbours in momentum space. New multiple interaction (MI) models like Colour Annealing [75] are defining a probability for a given string to not participate in the reconnection process as:

$$p_\textrm{keep} = (1-\eta)^{n_\textrm{MI}}$$

((3.90))

where $\eta$ is a free parameter and $n_\textrm{MI}$ the number of interactions that occurred in each event, which increases the reconnection probability for any given string in events with many interactions. In addition, the model tends to perform colour connection to the partons closest in momentum space, and therefore minimising the total string length. This model would in principle be valid also in WW decays, but it was not studied by the LEP collaborations. However, it may well be directly compared to the result in the SK I framework. This may give an additional handle to control CR systematics at Tevatron and the LHC. This is of importance for the measurement of the top quark mass, where the CR uncertainty currently amounts to 0.41 GeV, being among the three largest single contributions to the total systematic uncertainty on $m_\textrm{t}$ [76].

3.7 Measurement of the W Boson Mass

The mass of the W boson , $M_\textrm{W}$, is a central parameter in the Standard Model, and a precise determination of $M_\textrm{W}$ is one of the main tasks of the LEP experiments. There are two methods used to measure this quantity at LEP: by determining the W-pair production rate at the threshold and by direct reconstruction of the W decay spectrum. Since the LEP physics goals were not only the determination of W parameters but also the searches for new particles, only a fraction of the total data, about $4\times 20 \textrm{pb}^{-1}$, were recorded at threshold energies. The largest data samples of about $4\times 680 \textrm{pb}^{-1}$, are available at energies between 183 and 209 GeV.

At the W pair threshold, the cross-section of W pair production is dominated by t-channel neutrino exchange and it is proportional to the velocity of the W bosons:

$$\sigma_{WW}\propto \beta = \frac{p_W}{E_W}=\sqrt{1-\frac{4M_\mathrm{W}^2}{s}}$$

((3.91))

The first determination of $M_\textrm{W}$ at LEP was therefore derived from the cross-section measurement at the optimal $\sqrt{s}$ value of 161 GeV, and at 172 GeV. They yield a W-mass value of [8]:

$$M_\textrm{W}=80.40 \pm 0.20 \textrm{(stat.)} \pm 0.07 \textrm{(syst.)} \pm 0.03 (E_\textrm{beam}) \textrm{GeV}$$

((3.92))

which has a rather large uncertainty, dominated by the statistical uncertainty on $\sigma_{WW}$. In principle, $M_\textrm{W}$ could have been measured more precisely if more data would have been collected in the threshold energy range. With the same amount of data, a precision very similar to the one from fully reconstructed events can be reached, with different systematic uncertainties. This method may therefore be used again at a future linear $\textrm{e}^+\textrm{e}^-$ colliders [5]. As mentioned earlier, an improved accuracy of the theoretical predictions for $\sigma_{WW}$ at the threshold is being worked on [77].

In the direct reconstruction method, the masses of each decaying W is determined from the measured leptons and jets. All LEP experiments analysed the $\mathrm{q q} \ell \nu$ and $\textrm{q q q q}$ final states, OPAL also determined $M_\textrm{W}$ in $\ell \nu \ell \nu$ events.

In the fully leptonic final state the W masses can not be completely reconstructed because of the two neutrinos of the W decay. However, neglecting the lepton masses and the finite W width the dependence of lepton energy, $E_\ell$, on $M_\textrm{W}$ is given by

$$E_\ell=\frac{\sqrt{s}}{4}+\cos\theta^*_\ell\sqrt{\frac{s}{16}-\frac{M_\mathrm{W}^2}{4}},$$

((3.93))

where $\cos\theta^*_\ell$ is the angle between the lepton momentum in the W rest frame and the direction of the W in the laboratory frame. This angle is not measurable, and the sensitivity of the lepton energy to $M_\textrm{W}$ is mainly from the endpoints of the spectrum, where $\cos\theta^*_\ell=\pm 1$. When assuming that the neutrinos are in the same plane as the leptons the event kinematics can be solved with respect to a pseudo-mass, $M_\pm$, up to a two-fold ambiguity. The solutions are [78]:

$$\begin{aligned} M_\pm^2&{}=\frac{2}{|\textbf{p}_{\ell^{\prime}}+\textbf{p}_\ell|^2}\left\{(P\textbf{p}_{\ell^{\prime}}-Q\textbf{p}_\ell)(\textbf{p}_{\ell^{\prime}}+\textbf{p}_\ell)\right.\\ &{}\left.\pm \sqrt{|\textbf{p}_{\ell^{\prime}}\times\textbf{p}_\ell|^2[|\textbf{p}_{\ell^{\prime}}+\textbf{p}_\ell|^2(E_\textrm{beam}-E_\ell)^2-(P+Q)^2]} \right\}, \end{aligned}$$

((3.94))

with $P=E_\textrm{beam}E_\ell-E_\ell^2+\frac{1}{2}m_\ell^2$, $Q=E_\textrm{beam}E_{\ell'}-\textbf{p}_{\ell'}\textbf{p}_\ell+\frac{1}{2}m_{\ell'}^2$, the beam energy $E_\textrm{beam}$ and the lepton masses and momenta, $\textbf{p}_{\ell/\ell'}$ and $m_{\ell/\ell'}$. OPAL performed a likelihood analysis fitting parameterised spectra to the data distributions of the two quantities. The fit results in

$$M_\textrm{W}(\ell \nu \ell \nu)=80.41\pm 0.41\pm 0.13 \textrm{GeV},$$

((3.95))

where the first uncertainty is statistical and the second systematic. Here, the beam energy uncertainty, QED radiative corrections, as well as background modelling are taken into account. The main systematic uncertainties are due to the lepton momentum scale, which is determined from $Z\to\ell\ell$ events, both at the Z peak and at higher energies. Although the precision is not extraordinary, the analysis of this decay channel is important because systematic effects from hadronisation and FSI are completely absent.

Precise knowledge of the lepton momentum scale and resolution is also important when extracting $M_\textrm{W}$ in the $\textrm{q q} \textrm{e} \nu$ and $\textrm{q q} \mu \nu (\gamma)$ channels. The hadronically decaying W can be fully reconstructed from the two quark jets. Using momentum conservation, also the neutrino four-momentum can be calculated and thus the four-momentum of the leptonically decaying W. Final state photon radiation (FSR) is predominantly emitted along the charged lepton or quark jet and included in the quark jets and reconstructed leptons. Photons radiated from the initial state electrons (ISR) are detected in about 5% of the events as isolated clusters in the calorimeters. When identified, the photon clusters are excluded in the formation of the jets.

The resolution for the W masses is improved by applying a kinematic fit to the event. The measured lepton energies and angles, as well as the jet energies and directions are varied within their resolution until energy-momentum conservation is fulfilled. The variation of the jet momenta (or jet masses) in the kinematic fit is done by keeping the jet velocity $\beta$ (or the boost $\gamma$) of the jets constant because many systematic effects cancel in the corresponding ratios with the jet energy. Since the momentum conservation was already exploited to calculate the neutrino momentum, this results in a fit with one constraint (1C). Furthermore, a second constraint is applied requiring the two W masses in the event to be equal within the W width. Figure 3.29 shows an example of the L3 data analysis, where the mass resolution is reduced by the 2C fit by about a factor of two in the $\textrm{q q} \textrm{e} \nu$ and $\textrm{q q} \mu \nu$ channels. Information form both 1C and 2C masses are usually used in the subsequent mass analyses.

In the $\textrm{q q} \tau \nu$ channel, the kinematic constraints are spoilt by the additional neutrino from the $\tau$ decay. Only the hadronically decaying W boson contains mass information. The mass resolution can however be improved by applying a rescaling of the sum of the jet energies to the beam energy, where a factor two can be gained in resolution. Overlap of the leptonic $\tau$ decays with the $\textrm{q q} \textrm{e} \nu$ and $\textrm{q q} \mu \nu$ channels is avoided by applying strict separation cuts, for example on the $M_{\ell\nu}$ mass.

In the fully hadronic channel the complete final state can be reconstructed, including isolated ISR photons in the forward detectors. QCD gluon radiation is taken into account by splitting the event samples in four-jet and five-jet events, using, for example, the Durham [27] jet resolution parameter y ₄₅ as a measure to separate the two topologies. Energy-momentum conservation in the kinematic fit corresponds to four constraints (4C) and the equality of the W masses adds another constraint (5C). The improvements in mass resolution are in the order of a factor two to three, for the 4C and 5C fits, as illustrated in Fig. 3.29. Like in the $\textrm{q q} \ell \nu$ case, information from both fit classes are typically used to measure $M_\textrm{W}$.

The four and five jets in the $\textrm{q q q q}(+g)$ events can be paired in three, respectively ten, ways to form two W decays. They are distinguished by ordering the jets in energy, so that a probability for each combination can be calculated. In the 4-jet case all three combinations have at least a 5–10% probability to be correct and they are all considered in the mass analysis, assuming that the combinatorial background is well described by Monte Carlo. Figure 3.30 gives an impression of the size of the combinatorial background with respect to the correctly reconstructed signal and the real background from non-WW decays. In 5-jet events, some combinations have a negligible probability to be correct, e.g. the case in which the second most energetic jet is combined with the least energetic one. The ALEPH experiment selects only one combination in their analyses using a pairing probability that is based on the CC03 matrix element calculated for the reconstructed jets [72]. The other experiments use a W mass estimator that combines all pairings that have a high probability to be correct. They are weighted accordingly in the combined mass likelihood. The weights are based on the polar angle of the reconstructed W boson, the sum of jet charges of each jet combination and the transverse momentum of the gluon jet in 5-jet events (DELPHI [80]), the probability of the kinematic fit (L3 [80]), or a neural network variable (OPAL [81]) trained with the reconstructed mass differences as input.

The extraction of $M_\textrm{W}$ and $\varGamma_\textrm{W}$ from the reconstructed mass spectra is performed with various methods. ALEPH and L3 apply a Monte Carlo template method in which the measured spectra are compared as 2- or 3-dimensional distributions to Monte Carlo samples with different underlying $M_\textrm{W}$ and $\varGamma_\textrm{W}$ values. The test statistics for the data to Monte Carlo comparison is either a unbinned likelihood (L3) or binned histograms (ALEPH), where the binning is optimised to obtain a bias-free measurement. The unbinned likelihood is, for example, constructed as [79]

$$L(M_\textrm{W},\varGamma_\textrm{W})=\prod\limits_{i=0}^{N_\textrm{data}}\frac{1}{\sigma_{s}(M_\textrm{W},\varGamma_\textrm{W})+\sigma_b}\left\{\frac{d\sigma_s\left(M_\textrm{W},\varGamma_\textrm{W},m_1^i,m_2^i\right)}{dm_1 dm_2}+\frac{d\sigma_s\left(m_1^i,m_2^i\right)}{dm_1 dm_2}\right\},$$

((3.96))

where $\sigma_s$ and $\sigma_b$ are the signal and background cross-sections, and m ₁ and m ₂ the mass estimators, like the 1C and 2C, or the 4C and 5C mass pairs. ALEPH uses in addition the uncertainty on the 2C and 5C masses as a third variable. The likelihoods are evaluated for each decay channel and each centre-of-mass energy separately. The differential cross-sections is calculated from Monte Carlo by the sum of template events in an interval I in the (m ₁,m ₂) space close to the measured $\left(m_1^i,m_2^i\right)$ point of event i, divided by the size of the interval $\varDelta I$ and normalised to the Monte Carlo luminosity L _MC:

$$\frac{d^2\sigma}{dm_1 dm_2}\approx \frac{1}{L_{MC}} \sum_I \frac{1}{\varDelta I}.$$

((3.97))

The Monte Carlo templates are of large statistics, usually 10⁶ events, to reduce statistical fluctuations in the calculation of $d^2\sigma/(dm_1 dm_2)$. If the Monte Carlo describes all detector effects and physics phenomena correctly, the W parameters can then be extracted without any bias.

The generation of templates with all $M_\textrm{W}$ and $\varGamma_\textrm{W}$ values that are needed to perform the likelihood maximisation is however impossible, in the sense that it would take a lot of computing time. The virtue of the template method is therefore in the reweighting of the Monte Carlo samples . A weight is attributed to each simulated event j according to the ratio of the matrix element squared:

$$w_j=\frac{\left|M\left(p_k^j,M_\textrm{W},\varGamma_\textrm{W}\right)\right|^2}{\left|M\left(p_k^j,M_\mathrm{W}^{MC},\varGamma_\textrm{W}^{MC}\right)\right|^2}$$

((3.98))

with the $M_\textrm{W}$ and $\varGamma_\textrm{W}$ values that are to be determined and the nominal $M_\textrm{W}^{MC}$ and $\varGamma_\textrm{W}^{MC}$ values of the original Monte Carlo sample. The matrix element also depends on the four-momenta $p_k^j$ of the generated final state fermions of event j, and possibly on the four-momenta of ISR or FSR photons. The matrix elements are calculated using four-fermion Monte Carlo programs, like EXCALIBUR [40] or KORALW/KandY [29]. The extraction of the correct W mass and width value from a given data set is verified in tests with high statistics Monte Carlo samples and also using many samples, each with the expected size of the data set.

The OPAL experiment uses a convolution fit to determine $M_\textrm{W}$ and $\varGamma_\textrm{W}$ from data [81]. A normalised physics function P describes the double-differential cross-section in the two reconstructed W masses m ₁ and m ₂:

$$\begin{aligned} P(m_1,m_2,M_\textrm{W},\varGamma_\textrm{W})&{}= a_0 \left\{B(m_1,M_\textrm{W},\varGamma_\textrm{W}) \otimes B(m_2,M_\textrm{W},\varGamma_\textrm{W})\otimes S\left(m_1,m_2,\sqrt{s^{\prime}}\right)\right\}\\ &{}\otimes I(\sqrt{s},\sqrt{s^{\prime}}) \end{aligned}$$

((3.99))

with the normalisation factor a ₀, the Breit-Wigner distributions

$$B(m,M_\textrm{W},\varGamma_\textrm{W})=\frac{m^2}{\left(m^2-M_\mathrm{W}^2\right)^2+(m^2\varGamma_\textrm{W}/M_\textrm{W})^2}$$

((3.100))

a phase space term

$$S(m_1,m_2,\sqrt{s'})=\sqrt{(s'-(m_1+m_2)^2)(s'-(m_1-m_2)^2)}$$

((3.101))

and the radiator function

$$I(\sqrt{s'},\sqrt{s})=\beta x^{\beta-1}\frac{\sigma(\sqrt{s'},M_\textrm{W})}{\sigma(\sqrt{s},M_\textrm{W})},$$

((3.102))

which describes the reduction in centre-of-mass energy from $\sqrt{s}$ to $\sqrt{s'}$ due to ISR with energy fraction $x=E_\gamma/\sqrt{s}$ and $\beta=(2\alpha_\textrm{QED}/\pi)\log((\sqrt{s}/m_e)^2-1)$.

The physics function P is then folded with the resolution function R to obtain the likelihood $L^s_i$ of each event i to be compatible with the WW signal:

$$L^s_i(M_\textrm{W},\varGamma_\textrm{W})=R_i(m_1,m_2)\otimes P(m_1,m_2,M_\textrm{W},\varGamma_\textrm{W})$$

((3.103))

The total likelihood is taking also the parameterised background likelihood into account:

$$L_i=p_i^sL^s_i(M_\textrm{W},\varGamma_\textrm{W})+p_i^bL_i^b,$$

((3.104))

where the two likelihood terms are weighted with event-by-event probabilities to be signal or background, $p_i^s$ and $p_i^b$. In case of $\textrm{q q q q}$ events, information of each jet pairing is entering the likelihood as a separate estimator, weighted by the corresponding neural network output.

The DELPHI analysis [80] applies the convolution technique in the $\mathrm{q q} \ell \nu$ channels, similar to OPAL. In the $\textrm{q q q q}$ channel, however, DELPHI exploits a priori the most of the information in each event. The idea is to convolute the predicted probability density $P(m_1,m_2,M_\textrm{W},\varGamma_\textrm{W})$ with the probability density of the complete 4C fit. The latter is however difficult to compute in a time-efficient way for the whole (m ₁,m ₂) space. An approximation is therefore applied. The 4C $\chi^2$ function is evaluated at the minimum of the (m ₁,m ₂) space together with the covariance matrix V between m ₁ and m ₂. With this information a new $\chi^2$ is constructed as:

$$\chi^2(m_1,m_2)=\chi^2_{4C,\min}+(\textbf{m}-\textbf{m}^{\min})V^{-1}(\textbf{m}-\textbf{m}^{\min})^T$$

((3.105))

with $\textbf{m}=(m_1,m_2)$ and $\textbf{m}_{\min}=\left(m_1^{\min},m_2^{\min}\right)$. From this, the 4C probability density is easily calculated

$$P_{4C}(m_1,m_2)=e^{-\frac{1}{2}\chi^2(m_1,m_2)}$$

((3.106))

The probability density is determined for all possible jet pairings, for three different jet clustering algorithms (Durham [27], Cambridge [82], Diclus [83]) , and assuming an ISR photon escaping along the beam direction in the kinematic fit. A weighted sum of the 18 so-called ideograms is calculated in the 4-jet case, and 60 weighted ideograms are used in the 5-jet case. Figure 3.30 gives an example of these ideograms for a fully hadronic 4-jet event. Although technically complicated, the method obtained very good linearity when comparing fitted with underlying $M_\textrm{W}$ values in large statistics Monte Carlo samples. A global mass bias, in the order of 200 MeV with an uncertainty of less than 10 MeV, is observed and being corrected for.

When only the W mass is a free parameter in the fit, the W width is assumed to follow the Standard Model relation:

$$\varGamma_\textrm{W}=\frac{3G_\textrm{F}M_\textrm{W}^3}{2\sqrt{2}\pi}\left(1+2\frac{\alpha_s}{3\pi}\right).$$

((3.107))

With the very refined measurement techniques, each experiment reaches a statistical precision on $M_\textrm{W}$ between 54 and 70 MeV in the combined $\textrm{q q} \ell \nu$ channels and between 59 and 70 MeV in the $\textrm{q q q q}$ channel. The precision on the latter is however reduced by applying a globally optimised jet reconstruction to account for FSI effects. The differences between the results of the LEP collaborations are mainly due to intrinsic experimental differences, like acceptance, resolution, and detection efficiencies.

The systematic error sources can be subdivided into correlated and uncorrelated systematics. Correlations can exist between analysis channels, between measurements at different centre-of-mass energies, and between experiments.

Each experiment studied the lepton energy scale and linearity, the angular measurement, and their resolutions in great detail. Usually, two-fermion events at the Z peak and at higher energies, measured in the same data taking periods as the W pair events, are used to determine remaining differences between Monte Carlo simulation and data. These differences were corrected for and their uncertainties translated into systematic uncertainties in the lepton and jet measurements. Effects that were rather negligible in any previous measurement at LEP became important, for example the detailed distribution of calorimetric clusters close to leptons which influences the hadronic mass in $\textrm{q q} \ell \nu$ events. Also the alignment across sub-detectors and their relative orientation to the beam were checked to avoid an angular bias. In the combined LEP measurement, the lepton systematics enter as correlated only between different data sets. Jet systematics are in addition correlated between the $\textrm{q q} \ell \nu$ and $\textrm{q q q q}$ channels, but uncorrelated between experiments. In total, the detector uncertainties contribute with 10 MeV in the $\textrm{q q} \ell \nu$ channel and 8 MeV in the $\textrm{q q q q}$ channel.

Backgrounds are mainly from Z pair and two-fermion production, whose cross-sections are also measured directly. The corresponding uncertainties are used to scale the background distributions globally by common factors over the whole mass spectrum. Also the slope of the background contributions in the measured spectra is changed. All experiments verified the background distributions in independent samples and apply missing corrections to the Monte Carlo predictions. In the LEP combination, this uncertainty is combined with the uncertainty due to limited Monte Carlo statistics and contributes with 3 MeV to the error in the $\textrm{q q} \ell \nu$ measurement, where backgrounds small, and with 11 MeV to the $\textrm{q q q q}$ error.

An uncertainty common to all measurements is the determination of the LEP beam energy. Since the beam energy constraint is applied in the kinematic fit, the beam energy error translates directly into an error on $M_\textrm{W}$. The W width is much less affected, also by the beam energy spread. The beam energy was determined by the LEP energy working group at each interaction point with time intervals of 10 min. Using the time information the correct beam energy is thus calculated for each individual W pair event. Data were grouped in different centre-of-mass energy bins, for which a global energy calibration was applied. This calibration is based on the flux-loop, beam spectrometer and tune shift measurements, as described in Chap. 2. When combining all data, the complete correlation matrix between these energy bins is used, resulting in an 8 MeV uncertainty on the LEP $M_\textrm{W}$ value.

Photon radiation evidently influences the reconstructed W mass spectra. The combined Monte Carlo programs KORALW and YFSWW3 (KandY) includes ISR effects in the YFS exponentiation scheme to $O(\alpha^3)$, full $O(\alpha)$ electroweak corrections, including interference between ISR, FSR and W radiation (WSR), as well as screened Coulomb corrections. The latter describe Coulomb interactions between the W bosons, which are potentially large but screened due to the limited lifetime of the W bosons. Higher order, leading-log FSR corrections are included using PHOTOS for leptons and Pythia for quarks. Alternatively, the RacoonWW Monte Carlo is used, which is also based on the complete $O(\alpha)$ matrix element completed with an ISR radiator function. ISR effects are generally estimated by comparing the $O(\alpha^3)$ with the $O(\alpha^2)$ calculation, yielding small shifts of about 1 MeV on $M_\textrm{W}$ [28,30,29]. The effect of Coulomb screening is studied by taking half of the difference between Monte Carlo samples with screened Coulomb effect and without any Coulomb effect, which amounts to about 7 MeV. To study the uncertainty on the non-leading $O(\alpha)$ electroweak corrections, a good estimate is derived from the direct comparison of the RacoonWW and the KandY generators. Some care has to be taken in this comparison since collinear photons are not explicitly generated in RacoonWW. The observed differences are in order of 10 MeV for $\textrm{q q} \ell \nu$ and 5 MeV for $\textrm{q q q q}$. Alternatively, the uncertainty on the $O(\alpha)$ corrections are derived from removing non infra-red photons radiated from W bosons and applying the $O(\alpha)$ corrections not only to the CC03 part of the event weight, but also on the difference between CC03 and the full 4f contribution. Both checks together give a 2 MeV uncertainty on $M_\textrm{W}$. Since some error estimates overlap and the experiments apply different strategies, the total LEP uncertainty due to radiative corrections is 8 MeV in the semi-leptonic channel and 5 MeV in the fully hadronic channel, assuming full correlation between all data sets [8].

Quark fragmentation and hadronisation is another common systematic error source that needs to be taken into account. The LEP experiments compare the Pythia, Herwig and Ariadne models, which are carefully tuned to Z decay events. The hadronic Z decay samples are depleted in b-quark jets, because these are practically absent in W decays at LEP. Additional attention was put on the rate of heavy hadrons produced in jets, like kaons and baryons. This is because the standard jet clustering algorithms assume hadrons with either zero mass or they attribute the pion mass to each cluster. If the rates of the more heavy hadrons is not exactly reproduced by Monte Carlo, this leads to a bias in the reconstructed W mass. In L3, for example, the baryon content of the simulation was compared to measurements [84] and good agreement was observed, as can be seen in Fig. 3.31. The uncertainty on the baryon rate translates into an systematic uncertainty on $M_\textrm{W}$ and $\varGamma_\textrm{W}$. Similar comparisons and adjustments of the baryon rate are applied by the other collaborations [72,80,81], generally reducing the hadronisation uncertainty. Eventually, in the LEP combination, these uncertainties are taken as fully correlated and contribute with 13 and 19 MeV in the $\textrm{q q} \ell \nu$ and $\textrm{q q q q}$ channel, respectively.

A systematic effect only present in the fully hadronic channel is from final state interactions, Bose-Einstein correlations (BEC) and colour reconnection (CR). The measurements of these effects are described in detail in the previous sections. LEP measurements are consistent with the absence of inter-W BEC, so that the baseline W mass and width analyses assume only intra-W BEC. The systematic uncertainty on $M_\textrm{W}$ is assessed by translating the BEC intra-W limit into a limit on the possible W mass shift. The BEC observables are actually found to have a linear correlation with $M_\textrm{W}$, as can be seen for the integral parameter J in Fig. 3.31. This is not trivial since J depends both on the modelled pion source radius and the correlation strength. The one standard deviation limit on the intra-W BEC strength of 30% corresponds to a 7 MeV uncertainty on $M_\textrm{W}$ in the hadronic channel.

Colour reconnection turned out to be a much harder problem than BEC. The effects on $M_\textrm{W}$ are potentially large in the SK I model, even using the limit $k_I <2.10$ from the direct measurement of CR. The corresponding shift of the hadronic W mass using the default jet clustering is in the order of 90 MeV at a centre-of-mass energy of 189 GeV [72]. However, CR affects mainly the inter-jet regions and low momentum particles, so that the CR systematics is much reduced when jets are limited to the high energy particles in the jet core. As shown in Fig. 3.26 the CR mass shift is only in the order of 30 MeV if for example a momentum cut at 3 GeV is applied. Each experiment is optimising the jet reconstruction with respect to the total uncertainty in the $\textrm{q q q q}$ channel. ALEPH is applying a particle momentum cut at 3 GeV, and DELPHI uses an iterative jet cone procedure with a cone radius of $R=0.5 \textrm{rad}$. The optimal cluster energy cut found by L3 is at 2 GeV, while OPAL removes particles with momenta below 2.5 GeV. The differences in the optimal analyses can be explained by the limits on CR that are obtained by the different experiments. L3, for example, finds a rather low limit on k _I at 1.1, while the ALEPH limit is around 2.0. For the same $k_I < 2.0$, also the L3 analysis would select 3 GeV as the optimal working point. In the LEP combination, this leads to a reduced weight for the mass measurements which are not close to the LEP combined limit of $k_I=2.10$. Overall, the CR uncertainty in the hadronic channel is 35 MeV, assuming CR according to SK I with the given LEP limit.^{Footnote 1}

Table 3.1 Decomposition of systematic and statistical uncertainties on the LEP combined W mass measurement [8]

Full size table

The final systematic uncertainties in the LEP combined W mass and width measurements are summarised in Table 3.1. The masses in the semi-leptonic and hadronic channel are found to be [8]:

$$\begin{aligned} M_\textrm{W}(\textrm{q q} \ell \nu) &{}= 80.372 \pm 0.030 \textrm{(stat.)} \pm 0.020 \textrm{(syst.)} \textrm{GeV}\\ M_\textrm{W}(\textrm{q q q q}) &{}= 80.387 \pm 0.040 \textrm{(stat.)} \pm 0.044 \textrm{(syst.)} \textrm{GeV} \end{aligned}$$

with a correlation coefficient of 0.20. The consistency of the two measurements is tested by taking the difference of the two, where CR and BEC errors are set to zero, which yields:

$$\varDelta M_\textrm{W}(\textrm{q q q q}-\textrm{q q} \ell \nu) = -12 \pm 45 \textrm{MeV},$$

((3.108))

a value well compatible with zero. This indicates that CR effects are efficiently reduced in the final LEP analyses and that they are not larger than expected from the various models. Eventually, combining all LEP data from threshold and direct reconstruction data, the W mass is found to be

$$M_\textrm{W}^{LEP} = 80.376 \pm 0.025 \textrm{(stat.)} \pm 0.022 \textrm{(syst.)} \textrm{GeV}.$$

((3.109))

This value agrees well with the Tevatron measurement of the W mass [85] of

$$M_\textrm{W}^{p\bar{p}} = 80.420 \pm 0.031 \textrm{GeV},$$

((3.110))

and also with the W mass from the analysis of all electroweak data, excluding the direct measurement [86]

$$M_\textrm{W}^{EW} = 80.364 \pm 0.020 \textrm{GeV}.$$

((3.111))

The LEP W-mass measurements are therefore still very competitive with the Tevatron results for this important Standard Model parameter.

The LEP experiments analysed the W decay spectra to also determine the W width. The same combination procedure was applied as in the W-mass measurement, including systematic uncertainties and their correlations. The W width data are not optimised with respect to the FSI effects, which are dominating the systematics together with the hadronisation effects. Taking the relatively large statistical uncertainty into account, an optimisation is however not needed. Combining all LEP data the combined fit yields:

$$\varGamma_\textrm{W}=2.196\pm 0.063 \textrm{(stat.)}\pm 0.055 \textrm{(syst.)} \textrm{GeV}.$$

((3.112))

This is again in good agreement with the combined CDF and $\textrm{D\O}$ measurement of [85]

$$\varGamma_\textrm{W}=2.050\pm 0.058 \textrm{GeV}.$$

((3.113))

Both are consistent with the Standard Model value of $\varGamma_\textrm{W}=2.091\pm 0.002 \textrm{GeV}$ [86].

Summarising the results of this chapter, electroweak boson production is in very good agreement with Standard Model predictions. The measurements of vector boson couplings are confirming the non-abelian $SU(2)\times U(1)$ gauge structure of the theory. Furthermore, the LEP era provides precise measurements of the vector boson couplings to fermions and of the Z and W boson masses. A discussion about the consistency with other precision measurements within the Standard Model and possible hints on new physics follows in the next chapter.

Notes

1.
The systematic uncertainty on M _W due to CR is actually derived for a k _I limit of 2.13, but no numerical difference is expected with respect to the current systematics of 35 MeV.

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Jun. Prof. Dr. Arno Straessner

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Straessner, A. (2010). Gauge Boson Production at LEP. In: Electroweak Physics at LEP and LHC. Springer Tracts in Modern Physics, vol 235. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-05169-2_3

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DOI: https://doi.org/10.1007/978-3-642-05169-2_3
Published: 05 January 2010
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-05168-5
Online ISBN: 978-3-642-05169-2
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Gauge Boson Production at LEP

Abstract

Keywords

3.1 Z Pole Measurements at LEP and SLD

3.2 Neutral Boson Production Above the Z Peak

3.2.1 Photon Production

3.2.2 Single Z and Z-Pair Production

3.2.3 Z+\(\gamma\) Production

3.2.4 Anomalous Neutral Gauge Boson Couplings

3.3 Measurement of the W-Pair Cross-Section

3.4 Measurement of Single-W Production

3.5 Determination of Triple Gauge Boson Couplings

3.6 Final State Interactions in W Boson Decays

3.6.1 Bose-Einstein Correlations

3.6.2 Colour Reconnection

3.7 Measurement of the W Boson Mass

Notes

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Gauge Boson Production at LEP

Abstract

Keywords

3.1 Z Pole Measurements at LEP and SLD

3.2 Neutral Boson Production Above the Z Peak

3.2.1 Photon Production

3.2.2 Single Z and Z-Pair Production

3.2.3 Z+\(\gamma\) Production

3.2.4 Anomalous Neutral Gauge Boson Couplings

3.3 Measurement of the W-Pair Cross-Section

3.4 Measurement of Single-W Production

3.5 Determination of Triple Gauge Boson Couplings

3.6 Final State Interactions in W Boson Decays

3.6.1 Bose-Einstein Correlations

3.6.2 Colour Reconnection

3.7 Measurement of the W Boson Mass

Notes

References

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