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The LEP experiments ALEPH, DELPHI, L3, and OPAL measured \(e^+e^-\) collisions from Z peak energies between 89 and 93 GeV up to highest energies above the W-pair threshold between 161 and 209 GeV. The goal of their experimental program was the determination of the properties of W and Z bosons, like their mass, width and their couplings to fermions and gauge bosons. It was also hoped to discover new phenomena, like the discovery of the Standard Model Higgs boson or super-symmetric particles. This was, however, not achieved. This chapter introduces the LEP collider and the important aspect of the calibration of the collision energy. The main features of the LEP experiments are presented.

2.1 The LEP Collider

The LEP ring [1] at CERN was installed in a tunnel of 26.7 km circumference at 50–175 m under ground and crossing the Swiss–French border. It was composed of eight 2.9 km long arc sections and eight 210 m long straight sections. The accelerator lattice was made of focusing-defocusing quadrupole and dipole structures, so-called FODO elements. Each of the element was 79 m long and 31 elements were arranged into one octant. The magnet system was built of 3,368 bending dipoles, together with about 800 quadrupoles for focusing and defocusing, and 500 sextupoles and further 600 dipoles for orbit correction. A bending field of up to 0.134 T created by steel-concrete dipoles kept the electrons circulating in the LEP ring with an effective bending radius of 3,026 m.

The acceleration of the electrons and positrons started with the 600 MeV linear LINAC injector for LEP (LIL). Both particle types were accumulated in the Electron-Positron-Accumulator (EPA) and the particle bunches were further accelerated in the Proton Synchrotron (PS) and the Super-Proton-Synchrotron (SPS) before they are eventually injected with an energy of about 20 GeV into LEP. The bunches collided every \(22\,\upmu\textrm{s}\) at the interaction points (IP), where the experiments were installed. A collimator system protected the installation from synchrotron radiation. The CERN accelerator complex is shown in Fig. 2.1. The number of interactions at the IPs is proportional to the luminosity of the \(e^+\) and \(e^-\) beams, which is given by

$$L=\frac{N_b^2 n_b f_{rev}}{4\pi (\sigma^*)^2}\;,$$
((2.1))

where \(N_b\approx 10^{11}\) is the number of particles per bunch, n b the number of bunches per beam, \(f_{rev}=11,\!246\,s^{-1}\) the revolution frequency and \((\sigma^*)^2\) the transverse intersecting beam area at the IP. The luminosity is limited by electromagnetic beam–beam interactions between electron and positron bunches. They lead to a shift of the tune value Q, which describes the number of betatron oscillations per turn. For LEP, Q varied between 60 and 100, depending on the beam optics. The tune shift \(\Delta Q\) had to be kept below 0.04 to provide stable running and optimal luminosity.

Fig. 2.1
figure 1

The CERN accelerator complex with the various components for injection, storage and pre-acceleration of electrons, positrons, protons and heavy ions, and the LEP/LHC ring. The LEP experiments ALEPH, DELPHI, L3 and OPAL were installed in the underground caverns of IP 2, 4, 6, and 8. The LHC detectors ATLAS, CMS, LHC-b and ALICE are in IP 1, 5, 2, and 8, respectively

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The maximisation of luminosity was achieved by increasing the number of bunches, n b. In the first years, there were 4 bunches per beam, which was then changed to the “Pretzel” scheme with 8 equidistant bunches. Eventually, in the last years of LEP, a bunch train scheme with four trains of three bunches and later four trains of two bunches was employed.

The transverse betatron amplitude, \(\sigma^*\), is also an important luminosity parameter. It is usually expressed in terms of the transverse emittance \(\varepsilon\) and the \(\beta\)-function at the IP, \(\beta^*\):

$$\sigma^*=\sqrt{\varepsilon \beta^*}$$
((2.2))

In the horizontal plane, \(\beta^*\) reached 1.25 m mainly determined by horizontal oscillation damping due to emission of synchrotron radiation. In the vertical plane \(\beta^*\) was about 4 cm. Both values were achieved by installing strong focusing superconducting quadrupoles with a high gradient of 55 T/m at the IPs.

The final luminosity achieved was \(4.3\times 10^{31} {\textrm cm}^{-2}s^{-1}\) at 46 GeV beam energy and about \(10^{32} {\textrm cm}^{-2}s^{-1}\) at 100 GeV, with a beam current of 1 mA per bunch.

The record beam energy of 104.5 GeV that could be reached was limited by synchrotron radiation. The energy loss per turn is in good approximation given by

$$\Delta E = \frac{{4\pi \alpha _{{\textrm{QED}}} }}{3}\frac{1}{{m_e^4 }}\frac{{E_b^4 }}{R},$$
((2.3))

with the electron mass, m e, the beam energy, E b, and the effective bending radius, \(R\approx 3026\,m\). At \(E_b=104.5 {\textrm GeV}\) the loss was therefore about 3.3% of the beam energy per turn, which had to be compensated by the accelerating radio-frequency (RF) power. The RF cavity system [2] was installed in the straight sections. For Z pole energies in the LEP1 phase, 128 five-cell copper cavities were sufficient to supply the acceleration power. For high energy operation in the LEP2 phase, the cavities were replaced by 288 superconducting four-cell cavities running at 352 MHz, 31,320 times the revolution frequency, \(f_{rev}\). To reach the highest energies 56 copper cavities were added to finally achieve a total voltage of 3,630 MV, corresponding to an average gradient of 7.5 MV/m. This dramatically exceeded the original cavity design value of 6 MV/m and was only possible by special cavity conditioning.

2.2 LEP Energy Calibration

The calibration of the beam energy [3,4] was of primordial importance during both LEP phases to determine MZ and MW with high precision . At LEP1, the Z boson mass was derived from the measurement of the fermion-pair cross-sections mainly at the 91.2 GeV peak of the resonance shape and at two off-peak points \(\pm 1.8 {\textrm Ge V}\) above and below the peak. The contribution of the LEP energy uncertainty to the Z mass and width error is approximatively given by [13]:

$$\Delta M_Z \approx 0.5\,\Delta \left( {E_{p + 2} + E_{p - 2} } \right),$$
((2.4))
$$\Delta \Gamma _Z \approx \frac{{\Gamma _Z }}{{E_{p + 2} - E_{p - 2} }}\Delta \left( {E_{p + 2} - E_{p - 2} } \right),$$
((2.5))

where \(E_{p-2}\) and \(E_{p+2}\) are the two off-peak centre-of-mass energies.

The best method to measure the beam energy is by resonant depolarisation. The Sokolov–Ternov effect [5,6] provides the mechanism for transverse polarisation of the beam electrons. Due to synchrotron radiation the electron spin is aligned in the magnetic dipole field. The degree of polarisation is measured with a Compton polarimeter using polarised laser light. The beam polarisation is disturbed by a transverse oscillating magnetic field of a certain frequency, \(\nu_r\). Depolarisation resonance appears if the ratio \(\nu_r/\nu_{rev}\) is equal to the non-integer part of the number of spin precessions per turn, also called spin tune, \(\nu_s\). For the three energy scan points, \(E_{p-2}\), \(E_{p}\), \(E_{p+2}\), the spin tunes were 101.5, 103.5 and 105.5 respectively. The corresponding beam energy can then be determined from the relation

$$E_b=\frac{\nu_s m_e}{(g_e-2)/2}\;,$$
((2.6))

where m e is the electron mass and \((g_e-2)/2\) the anomalous magnetic moment of the electron. The resonant depolarisation method yields a beam energy precision below 1 MeV. The main measurements were performed on the electron beam only, but a few cross-calibration measurements with the positron beam showed that both beam energies agree well within less than 0.4 MeV.

Sufficient beam polarisation could however only be achieved for E b up to 61 GeV. Also, the energy calibration could only be performed in dedicated calibration runs and not during physics data taking. The precise energy values had therefore to be extrapolated to physics runs and to other beam energies. This is performed by means of the strength of the magnetic dipole field B which, after integration over the whole LEP ring, is proportional to the beam energy

$$E_b = \frac{ec}{2\pi} \oint B \, ds \;.$$
((2.7))

A continuous measurement of the B field was therefore performed using nuclear magnetic resonance (NMR) probes installed inside the magnets, as shown in Fig. 2.2. In the LEP1 phase, 4 NMR probes were read out and 12 more probes were added for LEP2. A LEP energy model [4,7] was developed to derive the beam energy from the NMR measurements. Many time-dependent details were taken into account, for example the variation of the bending field due to parasitic currents flowing along the beam pipe (the “TGV effect”), the monitored dipole temperature, corrections due to tidal movement of the LEP ring (the “moon effect”), as well as corrections due to the beam orbit position.

Fig. 2.2
figure 2

(a) Schematic view of the LEP reference dipole magnet with NMR probe and flux loop installed. (b) Comparison of the beam energy measurements using the LEP spectrometer, the flux loop and the tune shift, Q s, with the measurement by the NMR probes. Also shown is the result of the global calibration fit [4]

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Three further and complementary measurement methods were applied to estimate the uncertainty of the reference energy determination with NMR. A magnetic flux-loop was installed in one special dipole magnet. It determined the magnetic field induced in a large copper loop during the ramping of the magnet currents. In the last year of LEP running, a beam spectrometer made of a steel dipole and a triplet of beam-position monitors provided a second alternative energy measurement. Finally, the beam energy can be determined from the synchrotron tune, Q s. It is defined as the ratio of the longitudinal beam oscillation frequency to the revolution frequency. The longitudinal beam oscillations are a combined effect of energy loss due to synchrotron radiation and the acceleration in the RF fields. From the relative phase of a bunch and the RF voltage Q s was measured and, knowing the RF peak voltage, E b could be calculated. Figure 2.2 compares the three alternative methods to the nominal NMR energy calibration as a function of beam energy. The methods yield consistent results.

The systematic uncertainties in the final calibration originated mainly from this comparison, which contributes with about 20 MeV to the uncertainty on the centre-of-mass energy at LEP2. Additional 10 MeV are from the modelling of the energy loss between the RF stations, which is needed to determine the exact energy at each IP and also to relate the LEP spectrometer measurements to those of the NMR method. A special situation was in the last year of LEP running, where previously unused horizontal correction dipoles were used. With these additional magnets the bending field was spread over a longer trajectory which leads to an increase in attainable beam energy. The gain was about 120 MeV per beam, which was pushing the discovery potential of LEP to higher particle masses. The downside was an increased systematic uncertainty of about 30 MeV on \(\sqrt{s}\), however, only for the highest centre-of-mass energies \(\sqrt{s}>205 {\textrm Ge V}\). Sources of smaller systematic uncertainties were also studied like the \(e^+e^-\) energy difference, the beam energy variation during the fill, the variation of the RF frequency, the precision of the resonant depolarisation measurement, and the additional dipole field component due an imbalance in the current feeding the focusing and defocusing quadrupoles.

Eventually, an IP-dependent calibrated centre-of-mass energy was provided in time steps of 15 minutes. A precision of 2–3 MeV and 3–7 MeV was reached for each off-peak and on-peak point of the Z resonance scan, respectively. The beam energy spread was in the order of 55 MeV. At higher energies above the W pair threshold, the centre-of-mass energy was calibrated for most of the energy points to better than 25 MeV, while the beam energy spread was about 250 MeV. Since the calibration procedure applies common corrections for the energy points, also correlations are determined and taken into account. The very good understanding of the LEP accelerator is eventually the basis for the precise measurements of mass and width of the Z and W bosons performed by the LEP experiments.

2.3 The ALEPH, DELPHI, L3 and OPAL Experiments

The four LEP detectors, ALEPH [8], DELPHI [9], L3 [10] and OPAL [11], are multi-purpose detectors designed to measure the products of head-on \(e^+e^-\) collisions in their centre. A schematic view of the different sub-detector systems installed in the four experiments is shown in Figs. 2.3 and 2.4. The experiments are all equipped with silicon tracking detectors close to the interaction point. The silicon devices are arranged cylindrically around the beam pipe, typically at radii between 5 and \(15 {/textrm cm}\). Their main purpose is to resolve secondary vertices from B hadron decays, which travel about \(3 {\textrm mm}\) before decaying. With an impact parameter resolution below \(100\,\upmu\textrm{m}\), b quark decays of the Z boson can be separated from light quark decays, and b decay modes of a possible Higgs boson can be identified. The silicon detectors are surrounded by gas drift chambers, where different technologies are used. ALEPH and OPAL installed a tracking or vertex chamber at smaller radii, completed by a time projection or jet chamber used for tracking of charged particles at larger radii up to about \(2 {\textrm m}\). L3 had a single time expansion chamber [12] with an outer radius of \(60 {\textrm cm}\), while DELPHI used a time projection chamber. Identification of particles was done by determination of ionisation energy loss along the tracks, dE/dx. DELPHI used a Ring Image Cherenkov detector for separating relativistic particles of different mass. For the measurement of track momenta, solenoids provide a magnetic bending field between 0.5 and 1.5 T, which covers at least the inner tracking detectors.

Fig. 2.3
figure 3

The ALEPH and DELPHI detectors at the LEP collider

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Fig. 2.4
figure 4

Schematic view of the L3 and OPAL detectors

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The measurement of the energy of electromagnetic particles, like photons and electrons, is performed by electromagnetic calorimeters . The ALEPH detector used a lead/wire chamber sampling technique, while lead glass and bismuth germanate (BGO) crystals were installed in OPAL and L3, respectively. DELPHI used a high density projection chamber with lead absorber walls for electromagnetic calorimetry. Sufficient material density of in the order of 20 radiation lengths, X 0, guaranteed that the electromagnetic showers and energy depositions are contained in the calorimeters.

Jets from fragmented quarks and gluons usually traverse the electromagnetic detectors and were registered and eventually stopped in the hadronic calorimeters. Here, ALEPH, DELPHI and OPAL used the magnetic return yoke made of iron as absorber, equipped with streamer chambers or tubes. L3 had a depleted uranium and wire chamber sampling calorimeter. The minimal ionising muons are not stopped in the inner detector and calorimeter layers and were measured in a muon detection system in the outermost shell of the LEP experiments. The correct event timing and rejection of cosmic ray background was performed by scintillator time-of-flight systems.

The tracking performance for muons from Z peak decays, \(Z\to\mu^+\mu^-\), reaches a resolution between 1.5 and 2.5% for the LEP experiments . Electrons and photons are measured with about 1–2.5% energy resolution. The uncertainty on hadronic jet energies is in the order of 10% for 45 GeV jets.

Important for cross-section measurements of the different physics processes is the precise knowledge of the beam luminosity, L . At LEP, small-angle Bhabha scattering served as a reference process to determine L. This process is well described by QED and has small electroweak corrections when the acceptance region is restricted to small polar angles between \(\theta_{min}\) and \(\theta_{max}\). At lowest order the differential cross-section at small scattering angles is given by

$$\frac{d\sigma}{d\Omega}=\frac{\alpha^2}{s}\frac{1}{\sin^4(\theta/2)}\approx \frac{16\alpha^2}{s}\frac{1}{\theta^4}\;.$$
((2.8))

Integrating over the acceptance angles and using \(d\Omega\approx 2\pi\theta\,d\theta\), yields

$$\sigma_{acc}=\frac{16\pi\alpha^2}{s}\left(\frac{1}{\theta_{min}^2}-\frac{1}{\theta_{max}^2}\right)\;.$$
((2.9))

The electrons and positrons of the small-angle Bhabha process are detected in luminosity monitors installed in the very forward regions of the detectors. To obtain the 0.1% precision on the luminosity, the fiducial volumes have to be very well defined. Therefore a combination of electromagnetic calorimetry and silicon devices for exact angular measurement are used. The final uncertainties actually fully reached the expectations, and the dominating systematic effects on Z peak cross-section measurements were due to the limited precision of the theoretical prediction for the Bhabha cross-section in the angular range [13].