Expectations for Electroweak Measurements at the LHC

Abstract

At the LHC, the large amount of luminosity expected will allow a determination of the masses of the W boson and of the top quark with even greater precision than present day measurements at LEP and the Tevatron. This is a very challenging task. In particular, the experimental and systematic uncertainties need to be under control. There is also the opportunity to improve the knowledge on the weak mixing angle and on triple gauge boson couplings. This will further constrain the theoretical models and will probe the consistency of the Standard Model in finer detail. This chapter focuses therefore on the prospects for improved electroweak measurements by the LHC experiments.

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At the LHC, the large amount of luminosity expected will allow a determination of the masses of the W boson and of the top quark with even greater precision than present day measurements at LEP and the Tevatron. This is a very challenging task. In particular, the experimental and systematic uncertainties need to be under control. There is also the opportunity to improve the knowledge on the weak mixing angle and on triple gauge boson couplings. This will further constrain the theoretical models and will probe the consistency of the Standard Model in finer detail. This chapter focuses therefore on the prospects for improved electroweak measurements by the LHC experiments.

6.1 W and Z Boson Production

The production of W and Z bosons is a process with high event rates at the LHC, with NNLO cross-sections of 20.5 nb [1] for $W\to e\nu_e,\mu\nu_\mu$ final states and about 10 times smaller, 2.02 nb [1], for $Z\to {\textrm{e}^+{\textrm e}^-},{\mu^+\mu^-}$ production at $\sqrt{s}=14\,{TeV}$. The theoretical predictions performed at NNLO have a rather high precision of about 1% and the cross-section measurements provide stringent tests of QCD. In addition, differential cross-sections like the rapidity distribution of W and Z bosons, $d\sigma/dy$, are sensitive to parton distribution functions (PDFs) of the proton (compare Sect. 1.7 and References [10,39]).

In the initial LHC phase, at low luminosities of $10^{31} {\textrm{cm}^{-2}{\textrm s}^{-1}}$, the trigger thresholds for $\textrm{W}\to{\ell \nu}$ and $\textrm{Z}\to{\ell^+\ell^-}$ production can be rather low. Single electron and muon signatures are planned to be accepted by the first level trigger at ${p_T}>10\,{GeV}$. Electron and muon pairs are required to pass at least ${p_T}>5\,{GeV}$ and ${p_T}>4\,{GeV}$, respectively. These thresholds are approximately doubled at $10^{33} {\textrm{cm}^{-2}{\textrm s}^{-1}}$ and will be further adapted for nominal LHC running at $10^{34} {\textrm{cm}^{-2}{\textrm s}^{-1}}$. The selection of $W\to e\nu$ events, as prepared by the ATLAS collaboration, is subsequently asking for a well identified electron reconstructed from an electromagnetic cluster in the calorimeter matched in angle to a track, such that ${E_T}>25\,{GeV}$ and $\eta$ values within the fiducial volume [2]. The missing transverse energy due to the neutrino has to pass ${E_T}miss>25\,{GeV}$ and the transverse mass of the lepton-neutrino system, ${m_T}^W$, must be larger than $40\,{\textrm GeV}$.

In $50\,{\textrm pb}^{-1}$ of data, 220 thousand signal events are expected. Main backgrounds are QCD jet production, followed by $W\to \tau\nu$ and $Z\to {\textrm{e}^+{\textrm e}^-}$ events, summing up to about 10% of the signal expectation. The transverse mass distribution is shown in Fig. 6.1 [2].

Fig. 6.1

(a) Simulated transverse mass distribution in the $W\to e\nu_e$ channel for an integrated luminosity of $50\,\textrm{pb}^{-1}$ [2]. (b) Invariant di-electron mass measured in $Z\to \textrm{e}^+{\textrm e}^-$ events and the corresponding background, again for $50\,\textrm{pb}^{-1}$ [2]

Since QCD jet production is the primary background and has at the same time relatively large theoretical uncertainties, a data driven method is developed for its estimation. Especially the modelling of the ${E_T}^{miss}$ background distribution before the final cut is important. A $\gamma+jets$ event sample, very similar to the signal process, is selected by requiring that no charged tracks are pointing to the electromagnetic cluster instead of having an angular match. Thus, the control sample has similar kinematics to the signal but a priori no missing energy like the background. Therefore, the background ${E_T}^{miss}$ spectrum can be derived from these events and systematic uncertainties are reduced. Table 6.1 summarises the event numbers, acceptances, A, and efficiencies, $\varepsilon$, as well as the prediction for the cross-section measurement in $50\,\textrm{pb}^{-1}$ of data.

Table 6.1 Expected number of signal and background events, N and B, overall selection efficiencies, $A\times\varepsilon$, and cross-section measurements, $\sigma$, together with their uncertainties, for an integrated luminosity of $50\,\textrm{pb}^{-1}$. The uncertainty on N is statistical, the other sources are systematic [2]. An overall luminosity uncertainty is not included

The $W\to \mu\nu$ selection follows a similar strategy. However, the jet backgrounds are dominated by $bb\to \mu X$ events which can be rejected by requiring muon isolation from hadronic activity and impact parameter cuts. A good understanding of the detector and the underlying event is necessary. Luminosity uncertainties will be significant during initial running, but can be removed by taking ratios of cross-sections $\sigma_W/\sigma_Z$. The situation will improve once the absolute luminosity calibration with the ALFA detector [3] will be available, for which dedicated runs are foreseen. The absolute luminosity calibration will then be projected to normal ATLAS running, using e.g. the relative measurement of the LUCID system.

The measurement of single gauge boson production thus represents a first test of the Standard Model predictions, testing both the electroweak and QCD part of the theory. Verifying the latter in the high energy hadronic environment is especially interesting and the basis for many other measurements and searches at the LHC. Controlling and measuring PDFs, as one aspect of understanding the hadronic part of the interactions, will be an important activity.

Insights into beyond leading-order QCD jet production can be learnt by selecting explicitly $Z\to {\mu^+\mu^-} + jets$ signatures from the $Z\to {\mu^+\mu^-}$ sample (and similarly for $Z\to \textrm{e}^+{\textrm e}^-$). This is interesting by itself and necessary to understand backgrounds to new particle searches. In an ATLAS study [2], di-muon events in the mass range $81\,{GeV} <m_{ee}<101\,{GeV}$ with isolated muons of high ${p_T}$ are selected. Jets are identified with a minimal angular distance $\varDelta R>0.4$ with respect to the muon, a minimal transverse momentum, ${p_T}>40\,{GeV}$, and a pseudo-rapidity range in the central part of the detector, $\eta <3.0$. The purity for $Z\to {\mu^+\mu^-}$ with additional 1-, 2- and 3-jets is found to be rather high, but to decrease with the jet multiplicity from about 96 to 90%.

The spectrum of the jets is determined and then corrected back to Monte Carlo generator level, using the ALPGEN [4] calculation as a reference. Figure 6.2 compares the different predictions at parton level. The actual comparison is done on hadron-level but reveals similar features: the LO prediction of the PYTHIA program differs from the NLO MCFM [5] and ALPGEN calculations, especially in the high jet ${p_T}$ region. Fig. 6.2 shows also the expected precision for different systematics due to the jet energy scale . Initially, this scale will not be known better than 10% (with ${\cal L}=1\,\textrm{fb}^{-1}$). But with more data, a 5% precision is expected to be reached, providing sensitivity to LO vs. NLO differences.

Fig. 6.2

(a) Parton level comparison of the ${p_T}$ of the leading jet in $Z\to {\mu^+\mu^-} + jets$ Monte Carlo samples for LO and NLO predictions [2]. (b) Systematic uncertainty on hadron level ${p_T}$ of the leading jet in $Z\to \textrm{e}^+{\textrm e}^- + jets$ events for ${\cal L}=1\,\textrm{fb}^{-1}$ [2]. If the dominant jet energy scale uncertainty can be reduced below 10%, sensitivity to NLO predictions is possible

6.2 W Mass Measurement at the LHC

The techniques foreseen to determine the mass of the W boson at the LHC are very similar to those applied at the Tevatron . The data samples consist of leptonically decaying W boson with ${\textrm{e} \nu}$ and ${\mu \nu}$ final states. Events with high-${p_T}$ leptons, typically above 20 GeV, and missing energy of more than 20 GeV due to the unmeasured neutrino are selected. Activity from QCD jets is suppressed by limiting the hadronic recoil to the reconstructed W boson to 30 GeV. This retains about 40 million W decays per experiment in $10\,\textrm{fb}^{-1}$ of data with efficiencies of 20% in each decay channel $W\to{\textrm{e} \nu}$ and $W\to{\mu \nu}$ [6]. With this number of signal events the statistical precision is only a subordinate uncertainty. Controlling the different sources of systematics is therefore the main task in this measurement. The goal is to push the experimental precision close to the current theoretical uncertainty of 4 MeV [7], calculated at electroweak two-loop order (and for ${M_\textrm{H}} <300\,{GeV}$).

In general, many effects can be determined from the closely related single gauge boson channels, $Z\to\textrm{e}^+{\textrm e}^-$ and $Z\to{\mu^+\mu^-}$. It could be shown [6] that from Z control samples, the uncertainties on the W rapidity spectrum, the W ${p_T}$ spectrum, as well as the lepton energy scale and resolution as a function of lepton energy can be derived. Like for the Tevatron measurements, the $Z\to\ell^+\ell^-$ samples can be well calibrated by applying the nominal Z mass [8] as an external constraint. Similarly, the lepton energy scale can be determined and calibrated for the lower lying di-lepton resonances, like $J/\psi$ and $\varUpsilon$. CMS had proposed a method to remove one of the leptons of the Z decay and adjust the ${p_T}$, respectively E _T, and the transverse mass spectra such that the resulting distributions fit the measured ones in the corresponding $W\to{\ell \nu}$ channel. The adjustment is done with the following approximation:

$$\left.\frac{d\sigma^W}{d{p_T}}\right|_{pred}= \frac{{M_\textrm{Z}}}{{M_\textrm{W}}} \left.\left(\frac{d\sigma^W}{M_W d{p_T}}\left/\frac{d\sigma^Z}{M_Z d{p_T}}\right.\right)\right|_{theo} \left. \left\{\frac{d\sigma^W}{d{p_T}} \left( {p_T}^Z-\frac{{M_\textrm{Z}}}{{M_\textrm{W}}}{p_T}^W) \right)\right\} \right|_{meas},$$

((6.1))

where ${M_\textrm{W}}$ is a free fit parameter and many theoretical uncertainties are absorbed in the ratio, $\left.\left(\dfrac{d\sigma^W}{M_W d{p_T}}/\dfrac{d\sigma^Z}{M_Z d{p_T}}\right)\right|_{theo}$, which is used as theoretical input. This is illustrated in Fig. 6.3. Since then, more refined methods like the template method , also applied in top and W mass analyses at Tevatron and elsewhere, are used. They yield a very high statistical precision of about 2 MeV per channel in $10\,\textrm{fb}^{-1}$ of data. The method relies on Monte Carlo samples of signal events which are reweighted to different underlying masses using theoretical matrix element predictions which are adjusted to fit the data spectra of transverse lepton momentum and transverse mass.

Fig. 6.3

(a) Comparison of the rescaled E _T spectrum of electrons in $Z\to\textrm{e}^+{\textrm e}^-$ events $1\,\textrm{fb}^{-1}$ [9]. (b) The corresponding $\chi^2$ distribution of the W mass derived from this spectrum

A detailed study of the different sources of systematics [6] shows that many uncertainties can be controlled by studying event samples of leptonically decaying Z bosons. The size of these samples is a factor 10 smaller than the number of W boson decays in the respective channels (compare Table 6.1). The leptonic Z final state can however be fully reconstructed and the kinematic properties of the Z can be determined. Assuming that each leptonic energy, E _i ($i=1,2$), scales with a factor $\alpha_i$ the reconstructed Z mass, m _ij, varies as

$$\alpha_{ij}m_{ij}=\sqrt{2\alpha_iE_i\alpha_jE_j(1-\cos\varDelta\phi)}$$

((6.2))

$$\alpha_{ij} \approx \frac{1}{2}(\alpha_i+\alpha_j),$$

((6.3))

with the angle between the leptons, $\varDelta\phi$, and an approximative expression for the mass scale-factor, $\alpha_{ij}$. In this way, the measured Z mass spectrum is adjusted to follow the theoretical expectations. The energy scale and resolution parameters are then derived as a function of the lepton energy itself with high precision. The impact of energy scale and resolution on the W mass are thus reduced to 4 MeV and 1 MeV, respectively, for both electrons and muons.

Furthermore, energy and pseudo-rapidity dependent trigger, reconstruction and identification efficiencies, as well as the recoil energy scale is extracted from Z decay events. Uncertainties on efficiencies may alter the ${p_T}$ and m _T spectra, and influence eventually ${M_\textrm{W}}$ by 4.5 MeV and 1 MeV in the electron and muon channel, respectively. When studying the recoil system, the leptons are properly removed from the event to simulate and then measure missing energy signatures, very similar to the neutrino in $W\to{\ell \nu}$ decays. The extraction of ${M_\textrm{W}}$ from the transverse mass has a corresponding systematic uncertainty of 5 MeV. These estimates are performed based on Monte Carlo simulations and exploit experience from the Tevatron precision measurements. However, the detailed understanding of the detectors of ATLAS and CMS is primordial to achieve such a performance.

The modelling of the W production mechanism also must be understood, in particular the ${p_T}$ and rapidity distribution of the W bosons. Here, the Z decay studies and Drell-Yan di-lepton events help to constrain these. Extrapolating the current knowledge on PDFs, e.g. by analysing the systematically varied PDF sets provided by the CTEQ group [10], and assuming an significant improvement, the corresponding uncertainty on ${M_\textrm{W}}$ can be reduced to below 3 MeV level. Modelling of QED photon radiation from final state leptons contributes with only 1 MeV, expecting a similar level of understanding of the photon spectra and their influence on the lepton reconstruction as for the LEP Z mass measurements.

As regards backgrounds, their contribution to the selected event sample is only small, mainly dominated by irreducible $W\to{\tau \nu}\to{\textrm{e} \nu}\nu$ and $W\to{\tau \nu}\to{\mu \nu}\nu$ events with the same signature as the signal process. An uncertainty of about 2 MeV is attributed to systematic variations of the background level and its spectral shape. Effects from the underlying event, in-time and out-of-time pile-up , and variations of the beam crossing angle will influence the W mass by only 1 MeV.

In total, the systematic uncertainties sum up to 6–7 MeV in the muon channel and to 7–8 MeV in the electron channel for each of the mass measurements from the ${p_T}$ and m _T spectra. Taking correlations into account this yields an accuracy of 7 MeV in both channels [6]. As mentioned above, this estimate assumes $10\,\textrm{fb}^{-1}$ of well understood data. Only data taken with close-to-ideal detector conditions and under well controlled beam conditions with negligible background will enter the W mass analysis.

Since both experiments, ATLAS and CMS, will independently measure ${M_\textrm{W}}$, a combined uncertainty (including correlations) very close to the current theoretical precision of 5 MeV will therefore be possible, improving the current world average by a factor of 5. This will then match well with the uncertainty on ${M_\textrm{Z}}$ of 2.1 MeV. Theoretical models are thus tested in future also in the W mass sector at the electroweak two-loop level, and possibly beyond.

6.3 Top Physics and Determination of the Top Quark Mass

The most prominent Standard Model process at the LHC is the top quark production, making the LHC a top factory. About 83,000 top pairs are expected in $100\,\textrm{pb}^{-1}$ [11] at a centre-of-mass energy of 14 TeV. They are produced through gluon fusion diagrams $gg\to {\textrm{t} \overline{\textrm{t}}}$ (90%) and quark annihilation $qq\to g \to {\textrm{t} \overline{\textrm{t}}}$ (10%). The cross-section depends on the exact value of the top quark mass, ${m_\textrm{t}}$, but can be calculated at NLO order including NLL soft gluon resummation. The renormalisation scale uncertainty is however non-negligible, in the order of 10% when the scale is varied by a factor of two [12]. When including NNLO calculations and using the minimal subtraction ($\overline{\textrm{MS}}$) renormalisation scheme to define the top quark mass, ${m_\textrm{t}}(\mu_r)$, an improved theoretical uncertainty of below 5% is achieved [13].

The top quarks decay practically exclusively to W+b since the CKM matrix element V _tb is close to unity. The ${\textrm{t} \overline{\textrm{t}}}$ events are therefore measured in three topologies according to the W decay final states: fully hadronic (46.2%), semi-leptonic (43.5%) and fully leptonic (10.3%). The trigger systems of ATLAS and CMS identify those events by multiple signatures: high-${p_T}$ jets, isolated high-${p_T}$ electrons and muons in the leptonic channels, and multi-jets in the fully hadronic channel. Typical efficiencies normalised to the total event rate are in the order of 50–60% for the lepton triggers with ${p_T}>20-25\,{GeV}$, nearly 100% for low threshold jet triggers with ${p_T}>20\,{GeV}$ and about 10% for multi-jet triggers. Especially in the semi-leptonic final state there is a large redundancy.

The event selection in the single lepton channel requires a high ${p_T}$ lepton of 20 GeV, ${E_T}miss>20\,{GeV}$, four jets of ${p_T}>20\,{GeV}$ with three jets passing ${p_T}>40\,{GeV}$. This results in combined trigger and selection efficiencies of 18% in the electron and 24% in the muon channel. Furthermore, additional kinematic cuts can be applied like W mass constraints and a top-mass window, as well as b-tagging. The latter is however considered as not applicable in very early data since it requires a thorough understanding of the ATLAS tracking. Without asking for a b-tag, the electron analysis expects a signal-to-background ratio of $N_S/N_B=561/96$ events in $100\,\textrm{pb}^{-1}$ and the muon analysis is expecting a ratio of $N_S/N_B=755/143$. Events from $W+jet$ production represent the main background. The cross-section is extracted from a likelihood fit to the three-jet mass spectrum, as shown in Fig. 6.4, which yields a relative precision of $\varDelta\sigma/\sigma=(7(\mbox{stat.})\pm 15(\mbox{syst.})\pm 3(\mbox{PDF})\pm 4(\mbox{lumi.}))\%$ for both channels combined. The systematics are dominated by initial and final state radiation (ISR/FSR) of gluons and photons, as well as the shape of the fit function used.

Fig. 6.4

(a) Reconstructed top mass in the decay $t \to Wb \to jjb$ in semi-leptonic ${\textrm{t} \overline{\textrm{t}}}$ events for ${\cal L}=100\,\textrm{pb}^{-1}$ [2]. From this distribution the ${\textrm{t} \overline{\textrm{t}}}$ cross-section is determined by fitting a parameterised signal and background function to the simulated data. (b) A high purity ${\textrm{t} \overline{\textrm{t}}}$ sample is used to measure the top quark mass [2]. An increased width of the jjb mass due to the light jet energy scale is corrected by adding the jj mass peak-value, $M_{jj}^{peak}$, instead of the actual jj mass, M _jj, in each event. This corrects the jet energy scale uncertainty to first order. The statistical uncertainty on ${m_\textrm{t}}$ in $1\,\textrm{fb}^{-1}$ obtained in this case is $0.3\,{GeV}$

In the di-lepton channel, the typical signature are two high ${p_T}$ leptons, ${E_T}^{miss}$ due to two neutrinos which escape detection and two high ${p_T}$ b-jets. Combining ee, $e\mu$ and $\mu\mu$ channels, the signal to background ratio is $N_S/N_B=987/228$. Here, the leptonic decays of Z and W bosons are dominating the background rate. The expected precision in $100\,\textrm{pb}^{-1}$ is $\varDelta\sigma/\sigma=\left(4(\mbox{stat.})^{+5}_{-2}(\mbox{syst.})\pm 2(\mbox{PDF})\pm 5(\mbox{lumi.})\right)\!\%$ using a simple event counting method [2]. In this case, the jet energy scale is expected to be the main source of systematic uncertainties.

The top events themselves, and in particular the hadronic W decays, can actually be explored to calibrate the jet energy scale in data. Knowing the W mass value, the invariant jj mass spectrum can be adjusted to the expectations. Iterative energy rescaling and template methods are used . As an example, in 4,000 semi-leptonic ${\textrm{t} \overline{\textrm{t}}}$ events from $1\,\textrm{fb}^{-1}$ of simulated data, an overall scale factor of $K={M_\textrm{W}}^{PDG}/M_{jj}=1.014\pm 0.003$ is achieved, reproducing well the expected value of the absolute jet energy scale of $K_{exp}=E_{parton}/E_{jet}=1.014\pm 0.002$ for the specific sample analysed.

The jet energy scale is also the main uncertainty in the determination of the top quark mass . A first study for this measurement was performed in the semi-leptonic channel. The purity of the ${\textrm{t} \overline{\textrm{t}}}$ sample is increased by additional kinematic constraints, e.g., on the reconstructed hadronic W mass, the b-quark energy, $E_b^*$, and the difference of b and W energies, $E_W^*-E_b^*$, in the top rest frame. Eventually, top, W and b purities of $(86.4\pm 0.9)\%$, $(86.9\pm 0.9)\%$ and $(94.0\pm 0.6)\%$ are reached. The selection efficiency is $(0.57\pm 0.05)\%$. From the top mass spectrum, shown in Fig. 6.4, ${m_\textrm{t}}$ is derived with a very good precision of $0.3\,{GeV}$, assuming ${\cal L}=1\,\textrm{fb}^{-1}$, with practically no bias. This means that the systematic uncertainties dominate, as there are: b-jet energy scale with $0.7\,{GeV}$ per %, light jet energy scale $0.2\,{GeV}$ per %, and ISR/FSR systematics of $0.4\,{GeV}$. The ultimate goal is therefore the reduction of the jet energy scale uncertainties to at least 1%, which is one of the main challenges in this measurement [2,9]. The Tevatron measurements of ${m_\textrm{t}}$ (Sect. 4.2) already push forward into these regions of systematic uncertainties. An improvement by the LHC experiments of the top quark mass will therefore require a detailed understanding also of the modelling of the underlying event, of measurements and modelling of the much larger pile-up , and of a measurement and control of QCD effects like Colour Reconnection. Here, the large data samples which will be collected by the LHC experiments may give a handle to possibly select a set of $t\bar{t}$ events in which Colour Reconnection can either be measured or in which this effect is suppressed. Then the LHC will eventually be able to achieve a precision of the top quarks mass at or even below the 1 GeV level.

An alternative method to the direct determination of the top quark mass from the reconstruction of the top decay is the extraction of ${m_\textrm{t}}$ from the top-pair production cross-section, $\sigma_{t\bar{t}}$. This method is already applied at the Tevatron [18] with uncertainties of $5.5--6 {GeV}$ on the top quark mass. The dependence of $\sigma_{t\bar{t}}$ on ${m_\textrm{t}}$, needed for this measurement, was calculated at NLO [14,15] and NNLO accuracy [16,17]. At this level of theoretical accuracy, it turns out that the definition of ${m_\textrm{t}}$ as the pole mass is not safe with respect to the order of perturbation theory that is applied. A running mass with renormalisation scale, $m(\mu_r)$, is theoretically better defined since the top is a coloured object. QCD confinement prevents to extract the pole mass in the top-quark channel. The pole mass is usually converted into $m(\mu_r)$ using the following NNLO relation [16]:

$${m_\textrm{t}}=m(\mu_r)\left(1+\frac{{\alpha_s}(\mu_r)}{\pi}d_1+\left(\frac{{\alpha_s}(\mu_r)}{\pi}\right)^2d_2\right),$$

((6.4))

with $\mu_r$-dependent coefficients d ₁ and d ₂. If the cross-section predictions are compared to ${\textrm{D\O}}$ measurements at the Tevatron of $\sigma_{t\bar{t}}=8.18^{+0.96}_{-0.87}$ [18] one can extract a pole mass of $168.2\pm 3.6 {GeV}$ [16] (The more precise CDF result of $\sigma_{t\bar{t}}=7.0\pm 0.6$ [19] is not used in the analysis) . The $\overline{\textrm{MS}}$ mass, m(m), yields a lower value of $160\pm 3.3 {GeV}$, however, much more stable against contributions from higher order effects. These arguments need to be taken into account also for a more precise determination of ${m_\textrm{t}}$ from the top decay spectrum. Currently, improved LO and NLO Monte Carlo predictions are used to derive the pole top mass using template or other unfolding methods [20]. The renormalisation scheme must therefore be much better studied and defined if top masses from top decay measurements and from top-pair cross-section data shall be compared or even combined. This is important in particular for the presumably more precise measurements at the LHC.

In proton–proton collisions, top quarks are not only produced in pairs but also in single-top processes , where the electroweak t-channel production, $qg\to q'+t\bar{b}$ and $qb\to q't$, dominates with $\sigma_t=246\pm 12 {pb}$ [21]. The Wt-channel, $gb\to b \to Wt$, is contributing with $66\pm 2 {pb}$ [22] and the s-channel, $q\bar{q}'\to W \to t\bar{b}$, with $11\pm 1 {pb}$ [21]. Single-top production is especially interesting because the cross-section is directly proportional to the CKM matrix element $|V_{tb}|^2$. The backgrounds from ${\textrm{t} \overline{\textrm{t}}}$, $W+bb$ and $W+jets$ are very difficult to reject. Multivariate analyses are therefore applied using variables like b-jet ${p_T}$ and $\eta$, $\varDelta R$ between jets and leptons, ${m_T}^W$, etc. Figure 6.5 shows the output of a so-called boosted decision tree (BDT) analysis [23] and the top mass spectrum for high purity events. Assuming ${\cal L}=1 \textrm{fb}^{-1}$, a relative precision in the cross-section measurement of $\varDelta\sigma/\sigma=(5.6(\mbox{stat.})\pm 22(\mbox{syst.}))\%$ can be achieved. Systematic effects from b-tagging, jet energy scale, and PDFs contribute the most to the total uncertainty. Translated into a measurement of $|V_{tb}|$ one can derive $\varDelta|V_{tb}|/|V_{tb}|=(11(\mbox{stat.+syst.})\pm 4(\mbox{theo.}))\%$, where the theory uncertainty takes strong scale and PDF dependencies into account. The estimated precision is very much compatible with the one obtained in about $3 \textrm{fb}^{-1}$ by the Tevatron experiments [24,25], which yield, in case of the CDF measurement, $|V_{tb}| = 0.91\pm 0.11(\mbox{exp.})\pm 0.07(\mbox{theo.})$ . The s- and Wt-channels are studied as well for the early LHC scenario, but a few $\textrm{fb}^{-1}$ will be needed to establish a signal with more than 3 standard deviations.

Fig. 6.5

(a) Boosted decision tree (BDT) output for single-top signal and background [2]. (b) Leptonic top quark mass distribution applying cut on the BDT output at 0.6 [2]

Fig. 6.6

Lowest order Feynman diagrams for di-boson production at hadron colliders. The s-channel diagram (left) involves a triple gauge boson vertex, which may differ from the Standard Model coupling structure. In the Standard Model, only WWZ and WW$\gamma$ vertices are allowed at tree level

6.4 Measurement of Triple Gauge Boson Couplings

Similar to $\textrm{e}^+{\textrm e}^-$ colliders, the gauge bosons W, Z and $\gamma$ can be produced in pairs at hadron colliders like the LHC. Possible final states include all di-boson combinations, from which the most interesting are WW, ZZ, WZ, ${\textrm{W}\gamma}$, and ${\textrm{Z}\gamma}$, since they may involve triple gauge boson couplings (TGCs) . The generic lowest order Feynman diagrams are shown in Fig. 6.6.

At a centre-of-mass energy of 14 TeV, the W-pair, WZ and Z-pair production cross-sections amount to 111.6, 47.8 and 14.8 pb [26], respectively, where the first also takes finite width effects of the intermediate bosons into account. These cross-sections are a factor 10 higher than at the Tevatron at $\sqrt{s}=1.96 {TeV}$. All di-boson processes have been observed at the Tevatron and cross-sections are measured, including ZZ production which has the lowest rate. The photons in ${\textrm{W}\gamma}$ and ${\textrm{Z}\gamma}$ production are typically required to have high transverse energy with respect to the beam directions and large angle to the final state fermions from W and Z decays, to separate the di-boson process process from QED photon radiation. Asking for ${E_T}^\gamma>7 {GeV}$ and $\varDelta R(f,\gamma)>0.7$ yields cross-sections of 451 pb [27] for ${\textrm{W}\gamma}$ and 219 pb [28] for ${\textrm{Z}\gamma}$ production.

Deviations from the Standard Model predictions due to anomalous TGCs are described with the same effective Lagrangian used in TGC analysis at LEP. The charged couplings terms, with couplings $g_1^Z$, $\kappa_Z$, $\kappa_\gamma$, $\lambda_\gamma$ and $\lambda_Z$, are given in Eq. (3.47). The coupling structure of the neutral sector, detailed in Eq. (3.37), contains two anomalous couplings, f ₄ and f ₅. The terms beyond the Standard Model have the common characteristic to increase with momentum scale of the process. The resulting cross-section will therefore violate unitarity at some high scale $\Lambda^2$, where new, yet unknown physics must set in to regulate this behaviour. To incorporate this potential extension of the Standard Model, a cut-off term is introduced in the anomalous parts of the couplings:

$$\varDelta g (\hat{s}) = \frac{\varDelta g}{\left(1+\frac{\hat{s}}{\Lambda^2}\right)^n},$$

((6.5))

where g stands generically for one of the couplings listed above, and $\varDelta g$ describes its deviation from the Standard Model value. The cut-off scales with the ratio of four-momentum transfer $\hat{s}$ to $\Lambda^2$. For charged couplings the power n is chosen to be equal to 2, while for neutral couplings n equals 3. This compensates different powers of momenta in the interaction terms.

The various di-boson final states have different sensitivities to the charged couplings due to the given coupling structure. The $\varDelta \kappa_\gamma$ and $\varDelta \kappa_Z$ terms in WW production scale with di-boson mass squared, $\hat{s}$, while it is only proportional to $\sqrt{\hat{s}}$ in WZ and ${\textrm{W}\gamma}$ production. Using the analogue argument, the WZ final state is more sensitive to $\varDelta g_1^Z$ than WW production. In case of $\lambda_\gamma$ and $\lambda_Z$, the corresponding terms scale with $\hat{s}$ in all WW, WZ and ${\textrm{W}\gamma}$ channels. For this reason, the LHC experiments are expected to have an enhanced sensitivity to anomalous TGCs due to the high centre-of-mass energy of 14 TeV.

For numerical evaluation of the LHC performance, the Monte Carlo programs BosoMC [27] and BHO [28] are used. These are accurate to NLO, and compare well to standard generators [2] if anomalous couplings are turned off. In case of a variation of neutral couplings in the ZZ channel, a ${p_T}$ dependent K-factor is applied to reach NLO accuracy.

The selection of the di-boson events is concentrating on the final states where W and Z bosons decay leptonically: $\textrm{W}\to{\ell \nu}$ and $\textrm{Z}\to\ell^+\ell^-$, where the leptons are either muons or electrons. The ${\textrm{WZ}}\to{\ell \nu}\ell^+\ell^-$ events are characterised by three high ${p_T}$ leptons, two of the same flavour, and missing transverse energy due to the neutrino. The main backgrounds consist of ${\textrm{ZZ}}\to\ell^+\ell^-\ell^+\ell^-$ events, where one lepton escapes undetected, of $\textrm{Z}+ jet/\gamma$, where the jet or photon is misidentified as a lepton, and of top-pair events with missing energy and three final state leptons coming from W and b quark decays. Strict cuts on a di-lepton mass window and a transverse mass formed with the third lepton and missing transverse energy consistent with Z and W mass, respectively, are applied. Events containing activity from hadronic jets are furthermore suppressed. In this way, a signal-to-background ratio of 7 with a number of signal events of about 50 in $1 \textrm{fb}^{-1}$ of data are expected. This can be further improved by using multi-variate analysis (MVA) techniques [2] .

Production of ${\textrm{W}\gamma}$ events with sensitivity to ${\textrm{WW}}\gamma$ couplings are enhanced by the kinematic cuts mentioned above, which define the signal process. The selection follows mainly the standard W selection with additional criteria to identify isolated photons. Background from final state photon radiation and misidentified photons is however difficult to suppress. With optimised MVA analyses it is expected to find about 1,600 signal and 1,200 background events in $1 \textrm{fb}^{-1}$ of data in the ${\textrm{W}\gamma}\to{\textrm{e} \nu}\gamma$ channel. The selection of ${\textrm{W}\gamma}\to{\mu \nu}\gamma$ events is about 50% more efficient, similar to the inclusive W boson selection, listed in Table 6.1.

Final states with W bosons are completed by the ${\textrm{WW}}\to{\ell \nu}LN$ channel . Two isolated, high-${p_T}$ leptons above 25 GeV with opposite sign and measured in the central detector, $|\eta| <2.5$, are require in the event selection. A jet veto reduces top-pair background, and a minimal ${E_T}miss>50 {GeV}$ suppressed Drell-Yann production, $\textrm{Z}/\gamma^*\to\ell^+\ell^-$. When the WW cross-section is measured, a cut on the angle between the two leptons, $\varDelta\phi_{\ell\ell} <2$, improves the purity. However, for TGC extraction it is more advantageous to require a minimal separation angle between the ${p_T}miss$ vector and the momentum of the di-lepton system, $\varDelta\phi_{p_{T}^{\ell\ell},{p_T}miss}>175^\circ$. This retains more events in the high $p_T^{\ell\ell}$ and ${p_T}L$ range where the sensitivity to TGC is largest. The WW signal will be selected with a purity of better than 85%. When optimized for the cross-section measurement, 110 signal events are expected to be identified at an integrated luminosity of $1 \textrm{fb}^{-1}$ [2]. In the TGC analysis the number is reduced to 75 signal events with the same amount of data.

The production of ${\textrm{Z}\gamma}\to\ell^+\ell^-\gamma$ events, within the above mentioned signal phase space, can be rather cleanly separated from all backgrounds, except those with Z bosons and real or misidentified photons in the final state. The signature of two high-${p_T}$ leptons and an isolated photon is identical. Photons outside the signal definition are however mainly from final state radiation and thus not sensitive to neutral TGCs. When rather advanced MVA techniques are applied, signal-to-background ratios of 1.7–2.0 can be reached in both the di-electron and di-muon channel with about 370 and 750 events expected in $1 \textrm{fb}^{-1}$ of data, respectively.

The process with the lowest Standard Model production cross-section is the Z boson pair production. In an ATLAS study, the ${\textrm{ZZ}}\to\ell^+\ell^-\ell^+\ell^-$ and ${\textrm{ZZ}}\to\ell^+\ell^-\nu\nu$ channels are analysed. The four-lepton final state is selected by identifying four isolated, high ${p_T}$ leptons which are combined to pairs of same flavour and opposite charge. The invariant mass of each lepton pair must be compatible with the nominal Z mass within about 20 GeV and the angular separation of the leptons is required to fulfil, $\varDelta R_{\ell^+\ell^-}>0.2$. Main background is from $\textrm{Z}{\textrm{b} \overline{\textrm{b}}}$ and ${\textrm{t} \overline{\textrm{t}}}$ events. For this reason, hadronic isolation criteria and proper identification of the production vertex of the leptons are important to suppress events with long-lived and leptonically decaying B hadrons. Eventually, the ZZ selection efficiency reaches 24, 41 and 28% in the $4\textrm{e}$, $4\mu$ and $2\textrm{e}2\mu$ channels. This corresponds to a signal expectation of 2.6, 4.5 and 6.2 events, again in the three channels and for $1 \textrm{fb}^{-1}$ of data, with total backgrounds estimated to be below 0.3 events [2]. Relaxing the di-lepton mass requirement for one of the lepton pair and accepting also one off-shell Z boson increases the efficiency and signal expectation by about 20%, but also increases the background level to about 2 events. The ZZ final state is completed by looking for the ${\textrm{ZZ}}\to\ell^+\ell^-\nu\nu$ signal. These events are selected by asking for exactly two measured leptons of opposite charge and an ${E_T}^{miss}$ value above 50 GeV. Since the ZZ pair is produced with only small ${p_T}$, the missing energy vector is pointing in opposite direction to the di-lepton momentum and is also of similar magnitude. Therefore, cuts on $|{E_T}miss-{p_T}^Z| <0.25{p_T}^Z$ and $145^\circ<\phi_{{E_T}miss}-\phi_{{p_T}^{\ell\ell}} <215^\circ$ efficiently suppresses background from WZ events. A jet veto and a minimal ${p_T}^{\ell\ell}$ of 100 GeV, reduces remaining top-pair and single-Z production. Eventually, a purity of 60% is achieved and the number of expected signal events amounts to about 10 in a data sample of $1 \textrm{fb}^{-1}$, combining the $\textrm{e}^+{\textrm e}^-\nu\nu$ and ${\mu^+\mu^-}\nu\nu$ channels.

The selected di-boson event samples will be used to measure the corresponding production cross-sections, and also to measure TGCs or to constrain anomalous contributions to them. When a total luminosity of $1 \textrm{fb}^{-1}$ is assumed, a signal significance of more than 5 standard deviations, typically even more than 10, are estimated in all channels, except ${\textrm{ZZ}}\to\ell^+\ell^-\nu\nu$. For the latter, about $2.5 \textrm{fb}^{-1}$ of data will be needed to establish a signal significance above 5 $\sigma$.

The main effect of anomalous TGC in di-boson production is an enhanced cross-section at high transverse momentum, ${p_T}$, of the individual gauge bosons and at high transverse masses, m _T. To test the sensitivity to TGCs, a reweighting scheme is applied . Event samples with anomalous couplings are generated with the BHO and BosoMC Monte Carlo programs in one- and two-dimensional grids of the coupling values, with step sizes in the range $10^{-4}$ to $10^{-3}$. The differential transverse mass distribution or the ${p_T}$ spectrum of one of the bosons is chosen as a reference. Each event of the main Monte Carlo sample, generated with MC@NLO, is multiplied by the ratio of the non-Standard Model and the Standard Model expectations for $d\sigma/d{m_T}$ (or $d\sigma/d{p_T}$) at a given generated m _T (${p_T}$). The reweighted, reconstructed transverse mass distribution is then fit to the expected differential distributions determined in many sets of Monte Carlo events, each corresponding to a given integrated luminosity. In this way, the expected sensitivity is derived. Each coupling is measured in a separate fit, setting the others to their Standard Model values. Examples of such data sets and the effect of anomalous TGCs on the m _T spectrum are shown in Fig. 6.7 for the WZ and WW channels.

Fig. 6.7

(a) Simulated transverse mass distribution of the WZ system for an integrated luminosity of $30 \textrm{fb}^{-1}$ [2]. The histogram shows the Standard Model expectation, including background. The data points are derived from a randomly selected sample of Monte Carlo events that corresponds to $30 \textrm{fb}^{-1}$ of data. The dashed lines indicate the variation of the cross-section in the high transverse mass region for two different values of anomalous gauge boson couplings, $\varDelta\kappa_Z=0.15$ and $\lambda_Z=0.02$ (b) The same distribution for WW events with expectations for anomalous contributions assuming $\varDelta\kappa_\gamma=0.16$ and $\lambda_Z=0.16$, reflecting the different sensitivity to the couplings compared to the WZ channel [2]

The Tables 6.2 and 6.3 summarise the ATLAS expectations for the determination of charged TGCs for $10 \textrm{fb}^{-1}$ of data and compares them to the LEP and Tevatron measurements. The results are based on the analysis of the m _T or ${p_T}$ spectra in the different di-boson channels. Systematic uncertainties at the LHC are conservatively assumed to be in the order of 10% on the signal scale and 18% on the background normalisation. In the WZ channel, a fit to the ${p_T}$ spectrum of the Z boson is slightly less sensitive. A combined multi-dimensional analysis would improve the final result but has not yet been performed.

Table 6.2 Comparison of expected sensitivities to anomalous charged TGC between W and Z bosons at the LHC for an integrated luminosity of $10 \textrm{fb}^{-1}$. The projections are given for one experiment and the new-physics scale is chosen to be $\Lambda=2 {TeV}$. The LHC expectations are compared to the combined LEP measurements [29] and the results of CDF [30] and ${\textrm{D\O}}$ [31]. The Tevatron results are also evaluated for a $\Lambda$ cut-off at $2 {TeV}$

Table 6.3 Comparison of expected sensitivities to anomalous WW$\gamma$ couplings at the LHC for an integrated luminosity of $10 \textrm{fb}^{-1}$, again estimated for one experiment. The expectations are compared to the combined LEP measurements [29] and the results of CDF [31] and ${\textrm{D\O}}$ [31]. The LHC projections and the Tevatron results are evaluated for a $\Lambda$ cut-off at $2 {TeV}$

The LEP and Tevatron values, which are shown for comparison, are derived with the additional assumption of custodial symmetry, constraining $\lambda_Z$ and $\kappa_Z$ to:

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

((6.6))

$$\lambda_Z=\lambda_\gamma.$$

((6.7))

Table 6.4 Comparison of expected sensitivities to anomalous neutral TGC at the LHC for an integrated luminosity of $10 \textrm{fb}^{-1}$ analysed by one experiment using $\Lambda=2 {TeV}$. The expectations are compared to the combined LEP measurements [29] and the results of CDF [30]. The Tevatron limits are evaluated for a $\Lambda$ cut-off at $1.2 {TeV}$

Setting $g_1^Z=0$, the first condition is equivalent to $\varDelta\kappa_Z=-{\tan^2{\theta_\textrm{w}}}\varDelta\kappa_\gamma\approx -0.30\varDelta\kappa_\gamma$. This enhances the sensitivity of the LEP data to these couplings since s-channel exchange of both Z and photon is involved in $\textrm{e}^+{\textrm e}^-\to{\textrm{WW}}$ production. The coupling limits measured at the Tevatron are similarly improved by the symmetry assumption since the ${\textrm{W}\gamma}$, WZ and WW final states are combined. Furthermore, the LEP results do not take any scale dependence of the couplings into account and the couplings are thus given in the limit of zero energy scale, which is equivalent to setting the new-physics scale $\Lambda$ to infinity. Applying a luminosity-averaged scale, $\sqrt{\hat{s}_\textrm{LEP}}$, of about 196 GeV and a cut-off $\Lambda=2 {TeV}$ would increase the LEP uncertainties on charged couplings by $+2\%$ and in case of neutral couplings by $+3\%$.

The summary Tables 6.2, 6.3, and 6.4 show that due to higher di-boson masses reached at the hadron colliders, the LEP results will be improved in future, first by the Tevatron experiments when the final expected data set of at least $2\times 8 \textrm{fb}^{-1}$ will have been analysed, and later by the LHC. ATLAS and CMS will then profit from an even higher centre-of-mass energy and from the fact that the prospects for the integrated luminosity are in the order of several $100 \textrm{fb}^{-1}$. Deviations from the Standard Model couplings at the per-cent level or below will be accessible. The LHC estimates presented here are expected to improve further with more involved analysis techniques – a good understanding of the detector and of the measured data provided. Therefore, the sensitivity to possible new physics at the multi-TeV range, manifesting itself in terms of modified gauge boson self-couplings, is excellent at the LHC.

6.5 Prospects for the Weak Mixing Angle

The study of electron and muon pair production at the LHC allows a measurement of the forward-backward charge asymmetry , similar to the precision measurement performed at LEP at Z peak energies. The hard scattering process, ${\textrm{q} \overline{\textrm{q}}}\to \textrm{Z}/\gamma^*\to \ell^+\ell^-$, is the inverse of the LEP reaction, when setting $\ell=\textrm{e}$. In the ${\textrm{q} \overline{\textrm{q}}}$ rest frame the differential cross-section is therefore given by Eq. (1.62) which can be simplified to:

$$\frac{d\sigma({\textrm{q} \overline{\textrm{q}}}\to\ell^+\ell^-)}{d\cos\theta}\propto \left(\frac{3}{8}(1+\cos^2\theta)+{A}_\textrm{FB}\cos\theta\right),$$

((6.8))

with the scattering angle, $\theta$, of the lepton with respect to the incoming quark and with the forward-backward asymmetry, ${A}_\textrm{FB}$, defined in the usual way:

$${A}_\textrm{FB}=\frac{\int_0^{+1}\dfrac{d\sigma}{d\cos\theta}d\cos\theta-\int_{-1}^0\dfrac{d\sigma}{d\cos\theta}d\cos\theta}{\int_{-1}^{+1}\dfrac{d\sigma}{d\cos\theta}d\cos\theta}=\frac{\sigma_F-\sigma_B}{\sigma_F+\sigma_B}=\frac{N_F-N_B}{N_F+N_B}.$$

((6.9))

From a measurement of ${A}_\textrm{FB}$, the weak mixing angle, $\sin^2{\theta_\textrm{eff}^\ell}$, can be extracted. This is particularly interesting because the LEP and SLC data show some difference between the mixing angles measured in leptonic and hadronic final states (see Chap. 4). A similar precision as LEP/SLC of $\delta\sin^2{\theta_\textrm{eff}^\ell}\approx 2 \times 10^{-4}$ or better is therefore the ultimate goal. Currently, the Tevatron experiments reach accuracies of $\delta\sin^2{\theta_\textrm{eff}^\ell}=1.9\times 10^{-3}$ [32] using about $1 \textrm{fb}^{-1}$ of data, mainly dominated by statistical uncertainties and fully compatible with the LEP results.

The ${A}_\textrm{FB}$ measurement also gives a handle to constrain new physics beyond the Standard Model, like additional heavy vector bosons. Their existence may alter ${A}_\textrm{FB}$ in regions of invariant lepton pair masses which are below the mass of the new vector boson due to interference effects [33].

At the LHC, the scattering angle is not identical to the angle of the lepton with respect to the beam axis because the ${\textrm{q} \overline{\textrm{q}}}$ system is not produced at rest and carries transverse and longitudinal momentum with respect to the beam directions. The transverse momentum leads to an ambiguity when calculating the quark and anti-quark momenta from the kinematics of the $\ell^+\ell^-$ system. The scattering angle is therefore determined in the Collins–Soper frame [34] which reduces this ambiguity. The particle four-momenta are Lorentz boosted into the $\ell^+\ell^-$ rest frame and the polar axis is chosen to be the bisection of the two proton momentum vectors. The scattering angle, $\theta^*$, with respect to the polar axis is then given by [35]

$$\cos\theta^*=\frac{P_L}{|P_L|}\frac{2}{P^2\sqrt{P^2+P_T^2}}\left(P_1^+P_2^--P_1^-P_2^+\right),$$

((6.10))

with the four-momentum, P, and the transverse and longitudinal momenta, ${\it P}_{{\it T}} and {\it P}_{{\it L}}$, of the lepton pair. The quantities $P_i^\pm=\frac{1}{\sqrt{2}}(E_i\pm p_{L,i})$ ($i=1,2$) are calculated from the energies, E _i, and longitudinal momenta, $p_{L,i}$, of the lepton and the anti-lepton. The $\cos\theta^*$ distribution, expected for the ATLAS experiment, is shown in Fig. 6.8.

Fig 6.8

(a) Distribution of $\cos\theta^*$ for reconstructed electron pair events in the Z pole region [2]. The monte carlo events are generated with the PYTHIA [37] program and passed through the ATLAS detector simulation. (b) Simulated rapidity distribution of $\textrm{e}^+{\textrm e}^-$ pairs [2]. The upper line shows the distribution for all events, the lower line only events with correctly reconstructed quark direction

The sign of $\cos\theta^*$, respectively the quark direction, can only be inferred indirectly in pp collisions. This is different from the situation at $p\bar{p}$ colliders where the quark can be identified with one of the valence-quarks of the proton, and the quark and proton beam directions are thus the same. At the LHC, the quark is assumed similarly to be a valence-quark of the proton and sea contributions are small if the $\ell^+\ell^-$ pair is not produced centrally. The anti-quark, however, is necessarily from the sea-quarks inside the proton. It carries therefore a much lower momentum fraction than the quark in the ${\textrm{q} \overline{\textrm{q}}}$ reaction. Under these assumptions, the quark direction is preferably oriented along the direction of the boosted Z/$\gamma^*$ system [36]. In particular, the sign of the longitudinal component of the lepton-pair momentum indicates the hemisphere of the quark direction, which is taken into account by the additional factor $P_L/|P_L|$ in Eq. (6.10).

For a precise measurement of the charge asymmetry, a large angular coverage of the detector is necessary. In an ATLAS study [2], electron pairs identified in the central ($|\eta| <2.5$) and forward ($2.5<|\eta| <4.9$) calorimeters are used. Muons are not being studied since the corresponding detector acceptance is restricted to $|\eta| <2.7$ only.

The electrons are required to pass ${p_T}$ thresholds of about 20 GeV. To identify the charge and the direction of the $\textrm{e}^-$, respectively $\textrm{e}^+$, at least one of the electrons must have a corresponding track measured in the inner tracking detectors. Shower shape criteria optimised for central and forward electrons improve the purity of the selected event sample. The mass of the reconstructed $\textrm{e}^+{\textrm e}^-$ system is restricted to the Z mass region, $|\sqrt{P^2}-{M_\textrm{Z}}| <12 {GeV}$. Small missing transverse energy reduces background with neutrinos in the final state. The rapidity of the $\textrm{e}^+{\textrm e}^-$ pair should not be central, $|y_{\textrm{e}^+{\textrm e}^-}|>1$, to suppress ${\textrm{q} \overline{\textrm{q}}}$ reactions of only sea-quarks and to increase the probability of a correct identification of the quark direction, as illustrated in Fig. 6.8. Main backgrounds are QCD di-jet, W+jet, and ${\textrm{t} \overline{\textrm{t}}}$ events. The background events contribute at most with 0.3% to the final sample, mainly in the configuration with one central and one forward electron.

If the $\textrm{e}^+{\textrm e}^-$ mass interval is restricted to the Z pole region, the dependence of the forward-backward asymmetry on $\sin^2{\theta_\textrm{eff}^\ell}$ can be well approximated by a linear function [38]:

$${A}_\textrm{FB}=b\left(a-\sin^2{\theta_\textrm{eff}^\ell}\right),$$

((6.11))

with parameters a and b which include radiative corrections and depend in particular on the parton distribution functions. At LHC energies of 14 TeV and using the PYTHIA Monte Carlo interfaced to PDFs provided by the MRST group [39] one obtains $a=0.23\pm 0.03$ and $b=1.83\pm 0.26$ [2]. Using the above formula, the weak mixing angle can be directly extracted from ${A}_\textrm{FB}$.

The measurement of the charge asymmetry may be altered by systematic effects which are studied as well. The uncertainty on the charge determination can be derived from the number of electron pairs in the Z peak region which are measured with charges of equal sign. The expected charge misidentification rate is small, up to a maximum of 0.015 at $|\eta|=2.5$, decreasing with smaller pseudo-rapidity values. It results in an uncertainty on ${A}_\textrm{FB}$ of about $3\times 10^{-5}$. Experimental uncertainties on energy scale , energy resolution, and reconstruction efficiency contribute with $4\times 10^{-5}$ to the systematics. Since the background rates are small in the Z peak region, the corresponding uncertainty is below $10^{-5}$ on ${A}_\textrm{FB}$. The total systematics is therefore on the level of $5\times 10^{-5}$, to be compared with the expected statistical precision of $\delta{A}_\textrm{FB}\approx 3\times 10^{-4}$ calculated for a total luminosity of $100 \textrm{fb}^{-1}$. This corresponds to 2–3 years of high-luminosity running of LHC.

Applying relation (6.11) the weak mixing angle is derived from the ${A}_\textrm{FB}$ measurement. Due to the factor $b\approx 1.8$ the uncertainties on $\sin^2{\theta_\textrm{eff}^\ell}$ are practically halved with respect to those of ${A}_\textrm{FB}$. The expected statistical precision is therefore in the order of $1.5\times 10^{-4}$. The dominant systematic effect is however not one of those discussed above, but the uncertainties on the PDFs which are used to relate ${A}_\textrm{FB}$ to the underlying theory. The systematics is estimated by running the event simulation with different MRST data sets representing the uncertainties assigned to the PDFs. This yields an uncertainty of about $2\times 10^{-4}$ on $\sin^2{\theta_\textrm{eff}^\ell}$. The systematic uncertainty on the parton distributions inside the proton is also the dominant systematic effect in the corresponding measurement of $\sin^2{\theta_\textrm{eff}^\ell}$ at the Tevatron [32,40].

Improving the knowledge on the event modelling is therefore an important ingredient to control and possibly reduce the systematic effects on the weak mixing angle. In addition, higher order QCD corrections at NLO and beyond need to be applied to the signal prediction, which can be obtained, e.g., by event reweighting [32] . The LHC experiments will be able to perform a measurement of $\sin^2{\theta_\textrm{eff}^\ell}$ competitive with the LEP/SLC results if experimental and theoretical uncertainties can be mastered. The large data sets will help to study these in great detail by using different control and physics samples, for example $W\to{\textrm{e} \nu}$ events which may be used to narrow down the shape of the PDFs (see Chap. 1).

6.6 Prospects for Electroweak Measurements at the LHC

In summary, assuming a reasonable performance of the ATLAS and CMS detectors, many electroweak parameters will be measured more precisely. The experimental challenges in the top and W mass measurements are similar to those experienced at the Tevatron. High luminosity runs will help to reduce systematics by either studying the different sources better or by constraining the data sets to phase space regions which are less affected by those. This will also lead to a new precision measurement of the weak mixing angle. Due to the higher centre-of-mass range, improved determinations of the triple-gauge boson couplings will be possible. The LHC data will therefore push the experimental tests of the Standard Model predictions to even higher precision.

On the other hand, new physics may be discovered in this indirect way. Due to the LHC potential to observe new effects directly, like super-symmetric particles or heavy vector bosons, the precision measurements may be linked and compared with the new theoretical models which may manifest in proton–proton collisions at 14 TeV. But even if the Standard Model does not need to be extended in the energy range reached at the LHC, the Higgs boson should be detected by the LHC experiments, as the last missing building block of the Standard Model and manifestation of electroweak symmetry breaking. The indirect and direct Higgs mass determination could then be compared again as a further proof or disproof of the consistency of the theoretical model. The search for the Standard Model Higgs boson at the LHC is thus very important and subject of the following chapter.