Fermion pair production in e+e- collisions at 189-209 GeV and constraints on physics beyond the Standard Model

Cross sections, angular distributions and forward-backward asymmetries are presented, of two-fermion events produced in e+e- collisions at centre-of-mass energies from 189 to 209 GeV at LEP, measured with the ALEPH detector. Results for e+e-, mu+mu-, tau+tau-, qq, bb and cc production are in agreement with the Standard Model predictions. Constraints are set on scenarios of new physics such as four-fermion contact interactions, leptoquarks, Z' bosons, TeV-scale quantum gravity and R-parity violating squarks and sneutrinos.


Introduction
In the years 1995-2000, the LEP collider delivered e + e − collisions at centre-of-mass energies from 130 to 209 GeV. Measurements of the e + e − → ff process with the ALEPH detector up to √ s = 183 GeV have been published in Ref. [1]. The results obtained at seven additional energy values are presented in this paper with analyses largely unchanged with respect to Ref. [1]. The seven centre-of-mass energies are listed in Table 1, together with the corresponding luminosities. In the year 2000 the luminosity was delivered in a range of energies. The 2000 data are divided into two energy bins, from 202.5 GeV to 205.5 GeV and above. This paper is organized as follows. Section 2 gives a brief description of the ALEPH detector, Section 3 presents the event generators used for the simulation of the signal and backgrounds, and Section 4 recalls some useful definitions. Measurements of hadronic, leptonic and heavy-flavour final states are discussed in Sections 5, 6 and 7, respectively. The results are used to set constraints on new physics in Section 8.

The ALEPH detector
The ALEPH detector and performance are described in Refs. [2,3], and only a short summary is given here.
Charged particles are detected in the central part, comprising a precision silicon vertex detector, a cylindrical drift chamber and a large time projection chamber, embedded in a 1.5 T axial magnetic field. The momentum p of charged particles is measured with a resolution of σ(p)/p = 6 × 10 −4 p T ⊕ 0.005 (where p T is the momentum component perpendicular to the beam axis in GeV/c). The three-dimensional impact parameter is measured with a resolution of (34 + 70/p) × (1 + 1.6 cos 4 θ)µm (where p is measured in GeV/c and θ is the polar angle with the beam axis). In addition, the time projection chamber provides up to 344 measurements of the specific energy loss by ionisation dE/dx.
In the following, only charged particle tracks reconstructed with at least four hits in the time projection chamber, originating from within a cylinder of length 20 cm and radius 2 cm coaxial with the beam and centred at the nominal collision point, and with a polar angle fulfilling | cos θ| < 0.95 are considered.
Electrons and photons are identified by the characteristic longitudinal and transverse developments of the associated showers in the electromagnetic calorimeter (ECAL), a 22 radiation length thick sandwich of lead planes and proportional wires chambers with fine read-out segmentation. The relative energy resolution achieved is 0.18/ E(GeV) for isolated electrons and photons.
Muons are identified by their characteristic penetration pattern in the hadron calorimeter (HCAL), a 1.5 m thick iron yoke interleaved with 23 layers of streamer tubes, together with two surrounding double-layers of muon chambers. In association with the electromagnetic calorimeter, the hadron calorimeter also provides a measurement of the hadronic energy with a relative resolution of 0.85/ E(GeV).
The total visible energy, and therefore the event missing energy, is measured with an energy-flow reconstruction algorithm [3] which combines all the above measurements, supplemented by the energy detected down to 34 mrad from the beam axis by two additional electromagnetic calorimeters, used for the luminosity determination [4,5]. The 1 relative resolution on the total visible energy is 0.60/ E(GeV) for high-multiplicity final states. This algorithm also provides a list of reconstructed energy-flow objects, classified as charged particles, photons and neutral hadrons.
The luminosity is determined with small-angle Bhabha events, detected with the leadwire luminosity calorimeter (LCAL), using the method described in Ref. [4]. The Bhabha cross section in the LCAL acceptance varies from 4.3 nb at 189 GeV to 3.6 nb at 207 GeV. The uncertainty on the measurement is smaller than 0.5%.

Event simulation and Standard Model predictions
Samples of simulated events are produced as follows. The generator BHWIDE version 1.01 [6] is used for the electron pair channel, and KK version 4.14 [7] for di-quark, di-tau and dimuon events. Interference between initial-state (ISR) and final-state (FSR) radiation is included in KK generator for the leptonic channels, whereas for the qq channel the FSR is introduced by PYTHIA in the parton shower and therefore interferences with ISR are not included. PYTHIA version 6.1 [8] is used for ZZ and Ze + e − production. Twophoton interactions (e + e − → e + e − X) are generated with PHOT02 [9] and HERWIG [10]. Finally, backgrounds from W-pair production are simulated with the KORALW generator version 1.51 [11]. Single-W processes are simulated with EXCALIBUR [12]. Hadronic final states were generated with hadronisation and fragmentation parameters described in Ref. [13].
Standard Model (SM) predictions in the electron-pair channel are obtained from BHWIDE. The analytic program ZFITTER [14] is used in all other cases, with the steering flags and main input parameters listed in the Appendix.

Cross section definition
Cross section results are provided for inclusive and exclusive processes. The inclusive processes include events with hard ISR, which correspond to about 85% of the selected events, while the exclusive processes exclude these final states.
The inclusive processes correspond to a cut s ′ /s> 0.1, where √ s is the centre-ofmass energy and √ s ′ is defined as the invariant mass of the outgoing lepton pair for leptonic final states, and as the mass of the s channel propagator for hadronic final states. Differently from Ref. [1], exclusive processes are defined by a cut s ′ /s> 0.85 .
Here θ 1,2 are the angles of the outgoing fermions measured with respect to the direction of the incoming electron beam or with respect to the direction of the most energetic photon seen in the apparatus and consistent with ISR [1]. In order to reduce the uncertainties related to interferences between ISR and FSR, the exclusive cross sections and asymmetries are not extrapolated to the full acceptance. They are evaluated over the reduced angular range corresponding to | cos θ| < 0.95, where θ is the polar angle of the outgoing fermion for the hadronic cross section measurements. For the leptonic cross section and the forward-backward asymmetry measurements | cos θ| < 0.95 cut applies to both outgoing fermion and anti-fermion polar angles.

Hadronic final states
The selection of hadronic final states is described in Ref. [1]; events with high chargedtrack multiplicity are required. For inclusive processes, the cross sections are determined, after background subtraction, using a global efficiency correction. Backgrounds and selection efficiencies, which are both obtained from Monte Carlo studies, are listed in Table 2 as a function of centre-of-mass energy. The main background arises from W pair and Z pair production. The contribution from γγ interactions is suppressed by requiring the event visible mass to be larger than 50 GeV/c 2 . The measured cross sections are presented in Table 3, together with the ZFITTER predictions over the same acceptance as the experimental measurements.
For the exclusive cross sections the events are divided into two hemispheres (hereafter called jets) with respect to the thrust axis, determined after removing the ISR photons. The quantity s ′ m /s is measured from the reconstructed jet directions and a cut s ′ m /s > 0.85 is applied. The s ′ m /s distribution for the data collected at √ s=207 GeV is displayed in Fig. 1, together with the expected background. In the exclusive region, the latter is dominated by: • W-pair production. For these events, the thrust (T ) distribution extends to lower values than for qq events, as shown in Fig. 2a. A cut T > 0.85 rejects approximately 80% of this background.
• Fermion-pair events where, due to photon radiation by both colliding electrons, the measured s ′ m /s from the jets directions is above 0.85. This background is reduced by requiring that the event visible mass, calculated excluding ISR photons with energies above 10 GeV, is greater than 70% of the centre-of-mass energy. The residual background is called "radiative background". Figure 2b shows the visible mass distribution for events with s ′ /s > 0.85 and thrust value exceeding 0.85. The systematic uncertainty on this radiative background accounts for the small discrepancy visible in Fig. 1.
The contribution from four-fermion processes other than WW production is found to be small. It is taken into account by including an additional 0.1% systematic uncertainty on the exclusive cross section measurements. Other systematic uncertainties arise from the knowledge of the calorimeter calibration and of the detector response to the hadronization process. These uncertainties are taken as fully correlated between years. The evaluation of the detector response uncertainties includes the calorimeter effects described in Ref. [15], which were shown to have negligible impact on this measurement.
The efficiencies for the exclusive process and the background contributions are summarized in Table 2 and the measured cross sections are presented in Table 3.
The systematic uncertainties for the inclusive and exclusive processes are listed in Table 4. Figures 3 and 4 show the measured inclusive and exclusive qq cross sections as a function of energy. The exclusive differential cross sections as a function of the thrust axis polar angle are shown in Fig. 5 (in this case the selection efficiencies have been determined in angular bins).

Leptonic final states
For the e + e − → µ + µ − and e + e − → τ + τ − channels, cross section measurements are provided for the inclusive and exclusive processes as defined in Section 4. The inclusive cross sections are determined after background subtraction and a global efficiency correction, while the exclusive cross sections are computed as the sum of the measured cross sections in bins of cos θ. Asymmetries are extracted by a counting method from the cos θ * distributions, where θ * is the scattering angle between the incoming e − and the outgoing ℓ − in the ℓ + ℓ − rest frame. The asymmetry A FB is defined as: where N F and N B are the numbers of events with the negative lepton in the forward and backward regions, respectively. Acceptance corrections, as well as corrections for asymmetric distributions of the main backgrounds, are determined with Monte Carlo samples.
For the e + e − → e + e − channel, because of the dominant contribution from the t-channel photon exchange, the cross section is provided only for s ′ /s > 0.85 over two angular ranges: −0.9 < cos θ * < 0.9 and −0.9 < cos θ * < 0.7.
For all leptonic channels, the background contamination, estimated from simulation, stems from γγ processes, four-fermion final states W + W − , ZZ, Ze + e − and production of other di-fermion species. As for the hadronic final state, for the exclusive selection only, events reconstructed with s ′ m /s > 0.85 but with a ℓ + ℓ − invariant mass below 0.85 √ s are called radiative event background.

The µ + µ − channel
The selection of muon pairs is described in Ref. [1]. For the inclusive selection, the main background comes from γγ → µ + µ − and is largely reduced by requiring that the invariant mass of the muon pair exceeds 60 GeV/c 2 . For the exclusive selection the background from radiative events is removed by asking that the invariant mass of the muon pair exceeds 0.8 √ s. The s ′ m /s distribution for the data collected at √ s=207 GeV is displayed in Fig. 6. The µ + µ − selection efficiencies, evaluated using the KK Monte Carlo, are listed in Table 5. The main systematic uncertainty is due to the simulation of the muon identification efficiency and is estimated from the difference between data and simulation for the muon identification efficiency in muon-pair events recorded at the Z peak.
The background contamination is also given in Table 5. For the inclusive selection, a major contribution to the systematic uncertainty on the estimated background comes from the normalization of the γγ → µ + µ − process, and is determined by comparing data and Monte Carlo in the µ + µ − mass range 15 GeV/c 2 < M µ + µ − < 50 GeV/c 2 . Other systematic uncertainties on the inclusive background arise from the knowledge of the τ + τ − , W + W − , ZZ and Ze + e − cross sections, and are at the level of 3%, 1%, 5% and 10%, respectively. For the exclusive selection, the dominant background systematic uncertainty comes from radiative events, and is estimated from the difference between the data and the Monte Carlo prediction in the region 60 < M µ + µ − < 150 GeV/c 2 .
The measured cross sections are presented in Table 6 and in Figs. 7 and 8, and compared to the SM prediction from ZFITTER. The dominant contributions to the systematic uncertainties on the cross sections (Table 7) come from the limited statistics of the Monte Carlo samples and from the knowledge of the integrated luminosity, of the muon identification efficiency and of the background contamination.
The differential cross sections are shown in Table 8 and Fig. 9, while the asymmetry results are presented in Table 9 and in Fig. 10. The dominant systematic uncertainty on the asymmetry comes from the statistical error on the Monte Carlo based corrections to the µ + µ − acceptance.

6.2
The τ + τ − channel As described in Ref. [1], the selection of tau pairs requires two collimated jets with low charged-track multiplicity. Each event is divided into two hemispheres and is accepted if at least one hemisphere contains a tau candidate decaying into either a muon, or charged hadrons, or charged hadrons plus one or more π 0 .
The dominant backgrounds are reduced in the following way. Criteria against the Bhabha process are applied to events containing two high-momentum charged tracks. An additional cut on the polar angle of both tau-jet candidates is introduced (| cos θ| < 0.92), in order to accept only tracks for which the ionisation estimator dE/dx, used to reject electron candidates, is accurately determined. WW events are rejected by requiring the acoplanarity angle between the two tau candidates to be smaller than 250 mrad. Di-muon events are removed by demanding that one of the two hemispheres does not contain a muon. Finally, the tau-pair visible invariant mass is required to exceed 25 GeV/c 2 in order to reduce the γγ → ℓ + ℓ − contamination. The s ′ m /s distribution for the data collected at √ s=207 GeV is displayed in Fig. 11.
The resulting selection efficiencies and the total background contamination are listed in Table 10. For the inclusive selection, the systematic uncertainty on the dominant γγ → ℓ + ℓ − background is estimated by comparing data and Monte Carlo in the τ + τ − mass range 15 GeV/c 2 < M τ + τ − < 50 GeV/c 2 . Bhabha and WW cross section uncertainties amount to 3% and 1% respectively. The systematic uncertainty for the exclusive selection is dominated by the limited knowledge of the radiative background cross section. The uncertainty on the latter is determined as the relative difference between the simulated and the observed numbers of τ + τ − events selected with a value of s ′ m /s between 0.5 and 0.8, assumed to be identical for values in excess of 0.85.
The measured cross sections are presented in Table 11 and Figs. 7 and 8, together with the SM prediction. The systematic uncertainties on these measurements are listed in Table 12. Table 13 and Fig. 12 show the differential cross sections, while the asymmetry results are given in Table 14 and in Fig. 10. Asymmetric contributions from the main backgrounds (Bhabha and radiative events) are significant, and the statistical error on the estimated Bhabha asymmetry yields the dominant systematic uncertainty on the τ + τ − asymmetry.

The e + e − channel
The selection of electron pairs [1] requires that the two most energetic tracks with opposite sign in the event satisfy the following conditions: where p i , E i and E i are the track momentum, the ECAL energy deposition associated to the track, and the total calorimeter energy associated to the track (including the ECAL and HCAL energies as well as the energy from a radiated photon), respectively. The previous cuts reduce significantly the contamination from tau and muon pairs. In addition, events with both tracks identified as muons are discarded. Finally, the requirement on the invariant mass of the e + e − pair candidate (M e + e − > 80 GeV/c 2 ) suppresses most of the residual radiative background. The M e + e − distribution for the data collected at √ s=207 GeV is displayed in Fig. 13.
The resulting selection efficiencies and the total background contamination are listed in Table 15. The background is dominated by radiative events whose normalization is extracted from fits to the M e + e − and (Σp + ΣE) experimental distributions using the expected shapes for the e + e − signal and radiative background. For both selections, −0.9 < cos θ * < 0.9 and −0.9 < cos θ * < 0.7, the statistical uncertainty on the fit result contributes the dominant systematic uncertainty on the background estimation.
The cross section measurements are compared to the SM prediction from the BHWIDE generator in Table 16 and Fig. 8. The main contributions to the systematic uncertainties are listed in Table 17. Finally, Table 18 and Fig. 14 show the measured differential cross section.

Heavy-flavour production
Measurements with heavy-flavour final states are described in this section. The ratios of the bb and cc cross sections to the hadronic cross section, indicated as R b and R c respectively, are discussed in Sections 7.1 and 7.2. The charge forward-backward asymmetry is measured on a b-enriched (A b FB ) and a b-depleted ( Q FB ) event sample, as presented in Section 7.3.
Results are given for the signal definition as in Ref. [1], with s ′ /s > 0.9 and an angular range restricted to | cos θ| < 0.95. An additional acceptance cut requiring that both jets have | cos θ jet | < 0.9 is applied to ensure that they are contained in the vertex detector acceptance. Table 19 gives the number of selected hadronic events at each centre-of-mass energy. The resulting efficiency is typically 78%, with a dependence on the quark flavour of less than ±1%. The background from qq events produced outside the acceptance, but reconstructed inside, is of the order of 2.6% and varies within 0.5% depending on the quark flavour. This variation is taken as systematic uncertainty on the contribution of the radiative background. The total uncertainty of the hadronic selection efficiency in the considered angular range is of the order of 1%.
The long lifetime and large decay multiplicity of b hadrons allow the separation of bb final states from other quarks. The same tagging variables, complemented by additional variables, can be used to separate cc final states from light quarks. These selections have a 6 moderate dependence on b-quark and c-quark physics modeling uncertainties [17,18,19], listed in Tables 20 and 21.

Measurement of R b
Events containing b hadrons are tagged using the procedure developed by ALEPH at LEP1 [20]. For each charged track, a probability (P T ) that it originates at the primary vertex is evaluated using the measured impact parameter significance. This is defined as the signed distance of closest approach of the track to the interaction point divided by the uncertainty on that distance. By taking all tracks or by grouping them according to which hemisphere or which jet they populate, the probability that the event (P E ), the hemisphere (P H ) or the jet (P J ) contains only light-quark jets can be determined. A low value of the probability indicates the presence of long-lived states, which arise dominantly from b-quark production. The b tagging therefore corresponds to a cut on the appropriate probability.
In order to reproduce the detector resolution in the simulation, the procedure to smear the impact parameter significance used for the LEP1 analyses [21] is employed. This is based on the ∼ 3 pb −1 calibration data taken at the Z peak each year, in order to optimize the smearing parameters for that year's data (Fig. 15).
The crucial factor in the determination of R b is the b-tag efficiency. The highly accurate measurements of R b at LEP1 were made possible by the use of a double-tag method, relying on the fact that b-quarks are produced in pairs which populate opposite hemispheres [21]. The use of single-and double-hemisphere tags enables the efficiency, as well as the rate of bb production, to be determined from the data, leaving only the level of background to be obtained from the simulation. Furthermore, uncertainties arising from the background knowledge can be minimized with hard cuts.
Unlike at LEP1, the double-tag method is not practical at LEP2 because of the much smaller statistics. For this reason, previous ALEPH measurements of R b at 130-183 GeV [1] were made with a single overall event tag. The efficiency was then determined either directly from the simulation, or by correcting the simulated efficiency by the ratio of the R b value measured with each year Z peak data to the world average. Neither method was satisfactory as they both require an extrapolation (either from the basic simulation or from the Z to LEP2 energies), with mostly unknown related systematic uncertainties.
The full LEP2 data sample, however, has become sufficiently large for an average R b value to be measured with the double-tag procedure, with reduced and controlled systematic uncertainty. An overall efficiency correction can therefore be obtained by taking the ratio of the average values of R b over all centre-of-mass energies, measured with the double-and single-tag methods respectively, so that where R k is the final value of R b at energy k, R k s is the value of R b determined by the single-tag procedure at energy k, and R d and R s are the values of R b , averaged over all energy points, as measured with the double-or single-tag method respectively. The above correction, which amounts to about 1.05, assumes that the ratio between the double-and single-tag efficiencies is energy independent, which is true as long as the cuts are not changed on an energy-by-energy basis. It does not require the b-tag cut to be the same for both methods. The optimal selection cut for both the event and hemisphere tags is taken to be the point where the total fractional error on R b is minimized. The b-tag cut corresponds to a b-selection efficiency of 49% (69%) and to a purity of 80% (72%) for the event (hemisphere) tag. The correlation between the single-and double-tag methods is estimated to be 0.95 from the simulation.
The final statistical uncertainty is dominated by the statistical precision on R d . To determine the systematic uncertainty, it is assumed that both the uncertainty for each method and the correlation between them are energy independent. It can then be shown that the relative systematic uncertainty at each energy is given to a good approximation by the relative systematic uncertainty on the average double-tag determination. The systematic uncertainties for the double-tag method are calculated over the full data set, and the contributions are given in Table 22. The dominant sources come from the btagging parameters (used to define the track selection and the jet reconstruction) and from the smearing procedure [22]. In addition, by comparing the average efficiency obtained with the double-tag method between data and simulation, the uncertainty on the uds and c backgrounds is found to be smaller than 11%.
The measured average value of R b is The individual values determined at each energy point are presented in Table 23 and in Fig. 16.

Measurement of R c
The ratio of the cc cross section to the hadronic cross section, R c , is measured from the hadronic sample pre-selected as described above.
In a first step, the background from bb events is reduced to 4% of the hadronic sample with a cut on P E (log P E > −2), which retains 86% of the cc events and close to 100% of the light-quark events. The efficiency of this cut is controlled on a sample of WW events, and the resulting systematic uncertainty is about 1%.
The final selection of cc events uses a Neural Network (NN) algorithm trained to separate jets originating from c quarks from jets originating from light quarks. The nine input variables, exploiting the lifetime of D mesons, their masses and their decays into leptons or kaons, are: • P J , as defined in Section 7.1.
• The probability that tracks having a high rapidity with respect to the jet axis originate from the primary vertex.
• The decay length significance of a reconstructed secondary vertex. [23] • The p T , with respect to the jet axis, of the last track used to build a 2 GeV/c 2 mass system, tracks being ordered by increasing P T .
• The sum of the rapidities, with respect to the jet axis, of all energy-flow objects within 40 degrees of this axis.
• The total energy of the four most energetic energy-flow objects in the jet.

8
• The missing energy per jet defined as the difference between the beam energy and the reconstructed jet energy.
• The largest rapidity of lepton candidates with respect to the jet axis.
• The largest momentum of kaon candidates. Here a charged particle track is identified as a kaon candidate if its ionization energy loss (dE/dx) is compatible with that expected from a kaon within three standard deviations, and more compatible with that expected from a kaon than with that expected from a pion.
The distribution of the NN output for light-quark jets in the simulation is corrected with the data by comparing enriched samples of light-quark jets selected with a cut applied to the opposite hemisphere. The correction is applied energy by energy. The statistical error on this correction is taken as systematic uncertainty; an additional uncertainty originates from the residual cc background in the selected sample. An example of the distributions used to derive the correction and the correction itself are shown in Fig. 17. A bb-enriched sample is used to control the fraction of bb background in the final event sample to 5%. Other sources of systematic uncertainties come from the limited statistics of the Monte Carlo samples, the knowledge of the luminosity, detector effects (smearing and momentum corrections), the hadronic pre-selection, and the modeling of c-quark physics. They are listed in Table 24.
The distribution of the sum of the NN outputs for both jets in the event is shown in Fig. 18. At each energy point, the NN cuts (indicated in Fig. 18 for the √ s= 189 GeV case) are chosen so as to minimize the total uncertainty. The upper cut suppresses about 5% of the remaining b background with a signal loss of less than 1%. The typical efficiency is 75% with a signal-to-background ratio of 50%. The resulting R c measurements are listed in Table 25.

Measurements of A b FB and Q FB
The A b FB and Q FB measurements are both extracted from hadronic events pre-selected as described above. A b-enriched sample and a b-depleted sample are obtained using appropriate cuts on P E (log P E < −4.3 and log P E > −2, respectively). The cuts are set with the aid of the simulation, and correspond to a b content of the order of 90% and 4% for the two samples, respectively. The selection efficiencies vary only slightly with the centre-of-mass energy.
The jet charge Q jet of each jet is defined as where the sums extend over the reconstructed charged tracks in the jet and q i and p ,i are the track charge and track momentum parallel to the jet axis, respectively. The parameter κ is optimized with simulated events so as to maximize the charge separation between b jets andb jets. It is found to be fairly independent of the centre-of-mass energy and the average value of 0.36 is used. The same κ value is also used for the b-depleted event sample.
The mean charge asymmetry Q FB = Q F jet − Q B jet is measured on both the benriched and b-depleted samples as the average of the jet charge difference between the forward and backward hemispheres, defined with respect to the thrust axis. It is related to the quark forward-backward asymmetries (A q FB ) as follows where the index q indicates the quark flavours (u, d, s, c and b) and the index x indicates the various background components (WW, ZZ and radiative qq). In this expression Q x FB indicates the background mean charge asymmetry , ǫ q,x the selection efficiencies and δ q the charge separation (defined as the mean of the Q q − Qq distribution).
The asymmetry A b FB is obtained from the b-enriched sample; it is extracted from Q enr FB , the charge asymmetry measured from the data, using the previous formula. The background mean charge asymmetry, the selection efficiencies and the charge separations are taken from the simulation. The non-b quark cross sections σ q and asymmetries A q FB are computed with ZFITTER for the signal definition s ′ /s> 0.9, with | cos θ| < 0.9 for both quark and anti-quark. It is possible to reduce the dependence of this measurement on the b efficiency estimated from the simulation by replacing the product data is the number of b events in the data and L is the integrated luminosity. It follows: MC is the number of background events predicted by the simulation. The measurement is corrected by a factor 1.03 to extrapolate from the range | cos θ| < 0.9 to the nominal range | cos θ| < 0.95. The potentially large uncertainty originating from the cc contamination in the b-enriched sample is reduced to a negligible level by a tight cut on P E .
In order to evaluate the systematic uncertainty on the jet charge separation, δ q is measured with the data, using semileptonic b decays for b quarks and semileptonic WW events for light and c quarks. Semileptonic b decays are selected by requiring an electron or muon with high transverse momentum in one jet. The charge of the opposite b jet is then known. Because of the low event statistics surviving this selection, data taken at all energies must be combined. The difference between the jet charge distributions in data and simulation (Fig. 19) is propagated as systematic uncertainty to the A b FB measurement, representing the dominant systematic effect. A similar procedure is applied to a selected sample of semileptonic W-pair events to measure the average lighter quarks charge separation.
These and other sources of systematic uncertainties are summarized in Table 26. The A b FB measurements are presented in Table 27. Finally, the difference ∆ = Q depl FB − Q MC FB , measured with b-depleted samples, constrains simultaneously A q FB and σ q (q=u,d,s or c), providing additional sensitivity to physics beyond the Standard Model. Assuming small deviations from the SM, the linearized equation is used to constrain the deviations of A q FB and σ q from the SM with the measured values of ∆ at each centre-of-mass energy, as described in Ref. [1]. Examples of the coefficients of the above equations are shown in Table 28.
The dominant systematic uncertainty originates from the jet charge, determined as explained above. This and other sources are listed in Table 29, while the measurement  results are reported in Table 30.

Interpretation in terms of new physics
New physics, if present, could give rise to deviations of the measured cross sections and asymmetries from the Standard Model expectations. The results presented in the previous Sections indicate good agreement between the data and the SM predictions. As an example the global fit of the muon, tau and hadronic exclusive cross sections and of the muon and tau asymmetries at the seven energies gives χ 2 /d.o.f. = 29.79/35. Stringent limits can be placed on scenarios beyond the Standard Model.
Predictions of several models of new physics are fitted to the data using binned maximum likelihoods, as explained in [1]. For this purpose, the measurements described in this paper are combined with those at lower energies reported in [1].

Contact interactions
Four-fermion contact interactions, expected to occur for example if fermions are composite, are characterized by a scale Λ, interpreted as the mass of a new heavy particle exchanged between the incoming and outgoing fermions, and a coupling g giving the strength of the interaction. Conventionally, g is assumed to satisfy g/ √ 4π = 1. Following the notation of Ref. [25], the effective Lagrangian for the four-fermion contact interaction in the process e + e − → ff is given by with δ = 1 if f = e and δ = 0 otherwise. The fields e L,R (f L,R ) are left-or right-handed projections of electron (fermion) spinors, and the coefficients η ij specify the relative contribution of the different chirality combinations. New physics can add constructively or destructively to the SM Lagrangian, according to the sign of η sign . Several models, defined in Table 31, are considered in this analysis.
In the presence of contact interactions, the differential cross section for e + e − → ff as a function of the polar angle θ of the outgoing fermion with respect to the e − beam can be written as dσ where s, t are the Mandelstam variables and ǫ = g 2 η sign /(4πΛ 2 ). F SM is the Standard Model cross section. F Born IF and F Born CI are the contributions from the interference between the Standard Model and the contact interaction and from the pure contact interaction, respectively. The above formula is fitted to the data using a binned maximum likelihood 11 function, as described in Ref. [1]. Limits are quoted at the 95% C.L. for Λ ± corresponding to η sign = ±1.
For leptonic final states, limits on the scale Λ are derived from the leptonic differential cross sections. The results are shown in Table 32 and Fig. 20.
For generic hadronic final states, limits on Λ are obtained from fits to the hadronic cross sections assuming that the contact interaction affects all flavours with equal strength. In addition, limits on models involving only couplings to c or b quarks can be derived from the R c and Q depl FB or R b and A b FB measurements respectively. The results are shown in Table 33 and Fig. 21. Combining hadronic cross section measurements with observables in the charm sector improves the overall sensitivity, whereas the gain is marginal for the bottom sector.
In summary, the ALEPH limits on the scale of contact interactions Λ are in the range 2-17 TeV, and most stringent for the VV and AA models. Constraints on e + e − ℓ + ℓ − , e + e − bb and e + e − cc contact interactions are of particular interest because these couplings are not accessible at pp and ep colliders.

R-parity violating sneutrinos
Supersymmetric theories with R-parity violation have terms in the Lagrangian of the form λ ijk L i L jĒk , where L denotes a left-handed lepton doublet superfield andĒ a righthanded lepton singlet superfield [26]. The parameters λ are Yukawa couplings and i, j, k are generation indices. The couplings λ ijk , assumed to be real in this analysis, are nonvanishing only for i < j. These terms allow for single production of sleptons in e + e − collisions.
At LEP, R-parity violating sneutrinos could be exchanged in the s or t channel, leading to deviations of di-lepton production from the SM expectations. Table 34 shows the most interesting cases. Sneutrino exchange in the s channel gives rise to a resonant state, assumed here to have a width of 1 GeV/c 2 [26]. Limits on couplings are extracted as explained in Ref.
[1] using leptonic differential cross section measurements. Figures 22-24 show the resulting constraints for processes involving sneutrino exchange.

Leptoquarks and R-Parity violating squarks
At LEP, the t channel exchange of a leptoquark can modify the qq cross section and jet charge asymmetry. In scenarios where leptoquarks couple to the first-generation leptons and to the second-or third-generation quarks, more stringent limits can be placed by using in addition the relevant heavy-flavour observables R b , R c and A b FB . Comparisons of the measurements with the predictions given in Ref. [27] allow upper limits to be set on the leptoquark coupling as a function of its mass.
Leptoquarks are classified according to the spin, weak isospin I and hypercharge. Scalar and vector leptoquarks are denoted by symbols S I and V I , and different hypercharge states are indicated by a tilde. In addition, "L" or "R" specifies if the leptoquark couples to the right-or left-handed leptons exclusively. TheS1 2 (L) and S 0 (L) leptoquarks are equivalent to up-type anti-squarks and down-type squarks, respectively, in supersymmetric theories with an R-parity breaking term λ ′ 1jk L 1 Q jDk (j, k=1,2,3). Limits on leptoquark couplings are then equivalent to limits on λ 1jk . Table 35 gives, for various leptoquark type, the 95% C.L. lower limit on the mass M LQ assuming the leptoquark couples with strength g 2 = 4πα 2 em . Figure 25 shows the exclusion contour in the plane coupling-mass for leptoquarks coupling to the third quark generation.

Extra Z bosons
Several extensions to the Standard Model [28] predict the existence of at least one additional neutral gauge boson Z ′ . Two classes of models are considered here: E 6 models and Left-Right (LR) models. In E 6 models, the Z ′ properties depend on the breaking pattern of the gauge symmetry, governed by the parameter θ 6 . Limits on the Z ′ mass are derived here for the choices θ 6 = 0, π/2, ± arctan 5/3, known as the χ, ψ, η and I models. In LR models, right-handed extensions to the Standard Model gauge group are introduced. The Z ′ couplings to fermions depend on the parameter α LR , which is set here to the value α LR = 1.53 (as predicted in LR symmetric models). More details can be found in [1].
Limits on the Z ′ mass are derived using the method described in Ref. [1]. The theoretical predictions for the two-fermion exclusive cross-sections and asymmetries are obtained from ZFITTER 6.10 used together with ZEFIT [29] and they are compared to the corresponding measurements presented above.
The most conservative m Z ′ lower limits, with respect to the Z/Z ′ mixing angle, are presented in Table 36. Constraints on extra gauge boson have been also set at the Tevatron [30, 31, 32].

Limits on TeV-scale gravity
A solution to the hierarchy problem has been proposed in Ref. [33], where gravity is characterized by a fundamental scale M D which could be as low as 1 TeV, provided that space has δ extra dimensions compactified to a size R. The effective Gravitational constant is then given by g −1 N = 8πR δ M 2+δ D . Hence, gravity can become strong at small distances, leading for example to deviations of the e + e − → e + e − differential cross section from the SM expectation. These deviations are parametrized by a cut-off Λ T [34] of the same order of magnitude as M D . Figure 26 shows the e + e − → e + e − differential cross sections measured with data collected at √ s=189-209 GeV, normalized to the SM prediction, together with the expected deviations from TeV-scale gravity models. Using all data, a lower limit of 1.1 TeV (1.2 TeV) is obtained on Λ − T (Λ + T ), for destructive (constructive) interference with the SM prediction. In computing the limits the luminosity measurement was assumed unaffected and the theoretical errors of 0.5% (2.0%) assigned to the forward (central) e + e − cross sections were taken as uncorrelated.

Conclusions
Several measurements of di-fermion final states using data collected by ALEPH at √ s=189-209 GeV have been presented. In the leptonic sector, total and differential cross sections, as well as muon and tau forward-backward asymmetries, have been derived. In the hadronic sector, cross sections, forward-backward charge asymmetries for light and c quarks, b quark forward-backward asymmetries, and the R b and R c ratios have been measured. Similar measurements have been performed by the DELPHI [35], L3 [36] and OPAL [37] Collaborations.
The results are consistent with the Standard Model expectations and have been used to place constraints on several scenarios of new physics: four-fermion contact interactions, R-parity violating sneutrinos and squarks, leptoquarks, additional Z bosons and TeV-scale gravity. These constraints improve on previous ALEPH limits, and are similar to those obtained by the other LEP Collaborations.
Additional interpretations in terms of new physics, using measurements presented in this paper, can be found in Refs. [38,39]. Table 1: Luminosity weighted centre-of-mass energies and integrated luminosities for the data samples presented in this paper. The total (statistical and systematic combined) uncertainties on the integrated luminosities are given. The last column contains the data sample names used in this paper.
Year E cm (GeV) Luminosity (pb − Table 23: Measured values of R b (with their statistical and systematic uncertainties), as a function of the centre-of-mass energy, for √ s ′ /s > 0.9 and | cos θ| < 0.95. The SM prediction from ZFITTER is given in the last column.        Table 31: Four-fermion interaction models considered in this paper. Table 32: Limits on contact interactions coupling to di-lepton final states. The 68% C.L. range is given for the fitted variable ǫ, while 95% C.L. lower limits are given for Λ ± . The results for the e + e − → ℓ + ℓ − process assume lepton universality of the contact interactions.   Table 34: For the R-parity violating models considered in the analysis, and for each dilepton channel, the involved coupling and the type of exchanged sneutrino in the s or t channel. Table 35: The 95% C.L. lower limits (in GeV/c 2 ) on the mass of leptoquarks of various species, coupling to the first, second or third generation of quarks with strength g = e. A dash indicates that no limit can be set, while NA denotes leptoquarks coupling only to top quarks and hence not visible at LEP.
Quark generation S 0 (L) S 0 (R)S 0 (R) S1 2 (L) S1 2 (R)S1 2 (L) S         ZFITTER steering flags and input parameters The main flags used in the ZFITTER Monte Carlo are listed below: • General flags. As advised in Ref. [24], CONV=2 is used to properly take into account the angular dependence of the electroweak box diagrams; INTF=2 is used to include the contribution from initial and final state interferences; BOXD=2 is selected to take into account box contributions.
• Hadron flags. FINR=0 describes √ s ′ as the mass of the propagator excluding FSR.