Measurement of the Running of the Electromagnetic Coupling at Large Momentum-Transfer at LEP

The evolution of the electromagnetic coupling, alpha, in the momentum-transfer range 1800GeV^2<-Q^2<21600GeV^2 is studied with about 40000 Bhabha-scattering events collected with the L3 detector at LEP at centre-of-mass energies 189-209GeV. The running of alpha is parametrised as: alpha(Q^2) = alpha_0/(1-C Delta alpha(Q^2)), where alpha_0=\alpha(Q^2=0) is the fine-structure constant and C=1 corresponds to the evolution expected in QED. A fit to the differential cross section of the e+e- ->e+e- process for scattering angles in the range |cos theta|<0.9 excludes the hypothesis of a constant value of alpha, C=0, and validates the QED prediction with the result: C = 1.05 +/- 0.07 +/- 0.14, where the first uncertainty is statistical and the second systematic.


Introduction
A fundamental consequence of quantum field theory is that the value of the electromagnetic coupling, α, depends on, or runs with, the squared momentum transfer, Q 2 . This phenomenon is due to higher momentum-transfers probing virtual-loop corrections to the photon propagator. This process of vacuum polarisation is sketched in Figure 1. In QED, the dependence of α on Q 2 is described as [2]: where the fine-structure constant, α 0 ≡ α(Q 2 = 0), is a fundamental quantity of Physics. It is measured with high accuracy in solid-state processes and via the study of the anomalous magnetic moment of the electron to be 1/α 0 = 137.03599911±0.00000046 [1]. The contributions to ∆α(Q 2 ) from lepton loops are precisely predicted [3], while those from quark loops are difficult to calculate due to non-perturbative QCD effects. They are estimated using dispersionintegral techniques [4] and information from the e + e − → hadrons cross section. At the scale of the Z-boson mass, recent calculations yield α −1 (m 2 Z ) = 128.936 ±0.046 [5]. Similar results, with smaller uncertainty, are found by other evaluations using stronger theoretical assumptions. For example, Reference 6 obtains α −1 (m 2 Z ) = 128.962 ± 0.016. The running of α was studied at e + e − colliders both in the time-like region, Q 2 > 0, and the space-like region, Q 2 < 0. The first measurement in the time-like region was performed by the TOPAZ Collaboration at TRISTAN for Q 2 = 3338 GeV 2 by comparing the cross sections of the e + e − → e + e − and e + e − → e + e − µ + µ − processes [7]. The OPAL Collaboration at LEP exploited the different sensitivity to α(Q 2 ) of the cross sections of the e + e − → µ + µ − , e + e − → τ + τ − and e + e − → qq processes above the Z resonance to determine α(37236 GeV 2 ) [8]. Information on α(m 2 Z ) is also extracted from the couplings of the Z boson to fermion pairs [9]. Bhabha scattering at e + e − colliders, e + e − → e + e − , gives access to the running of α in the space-like region. In addition, like other processes dominated by t-channel photon exchange, it has little dependence on weak corrections. The four-momentum transfer in Bhabha scattering depends on s and on the scattering angle, θ: Q 2 = t ≃ −s(1 − cos θ)/2 < 0. Small-angle and large-angle Bhabha scattering allow to probe the running of α in different Q 2 ranges.
LEP detectors were equipped with luminosity monitors, high-precision calorimeters located close to the beam pipe and designed to measure small-angle Bhabha scattering in order to determine the integrated luminosity collected by the experiments. The L3 collaboration first established the running of α in the range 2.10 GeV 2 < −Q 2 < 6.25 GeV 2 [10] by comparing event counts in different regions of its luminosity monitor. More recently, the OPAL Collaboration studied the similar range 1.81 GeV 2 < −Q 2 < 6.07 GeV 2 [11].
The running of α in large-angle Bhabha scattering was first investigated by the VENUS Collaboration at TRISTAN in the range 100 GeV 2 < −Q 2 < 2916 GeV 2 [12]. Later, the L3 Collaboration studied the same process at √ s = 189 GeV for scattering angles 0.81 < | cos θ| < 0.94, probing the range 12.25 GeV 2 < −Q 2 < 3434 GeV 2 [10]. This Letter investigates the running of α by studying the differential cross section for Bhabha scattering at LEP at √ s = 189 − 209 GeV for scattering angles such that | cos θ| < 0.9. Less than 1% of the events scatter backwards, cos θ < 0, and this analysis effectively probes the region 1800 GeV 2 < −Q 2 < 21600 GeV 2 , extending and complementing previous space-like studies.

Analysis Strategy
In the following, the running of α is described by a free parameter, C, defined according to: where the parametrisation of Reference 5 is used for the term ∆α(Q 2 ). A value of C consistent with C = 1 would indicate that data follow the behaviour predicted by QED, while the hypothesis α = α 0 , with no dependence on Q 2 , corresponds to C = 0.
The value of C is derived by a study of the measured differential cross section of the e + e − → e + e − process, dσ/d cos θ. This quantity depends on C through the measured integrated luminosity, L(C), which is calculated from the expected cross section of the e + e − → e + e − process for small scattering angles. The measurements used in the following are obtained under the Standard Model hypothesis, C = 1, as: where N(cos θ) is the number of events observed in a given cos θ range, of width ∆ cos θ, with average acceptance ε(cos θ). The measured integrated luminosity depends on C as: where N L is the number of events observed in the fiducial volume of the luminosity monitor, σ L (C) is the corresponding e + e − → e + e − cross section for a given value of C and ε L (C) is the detector acceptance. This acceptance may depend on C due to the combined effect of small angular anisotropies of detector efficiencies and the dependence of the predicted differential cross section on C. These changes in the acceptance are found to have negligible impact on the results presented below. The value of the parameter C is extracted by comparing the measured differential cross section to the theoretical prediction as a function of C, dσ th (C)/d cos θ, derived as: where dσ th (1)/d cos θ is the Standard Model prediction, discussed in Reference 13. The value of L(1) is derived by using the BHLUMI Monte Carlo program [14]. The dependence of dσ th (C)/d cos θ and L(C) on C is implemented by means of the BHWIDE Monte Carlo program [15]. The differential cross section is factorised as: where dσ Born (C)/d cos θ is the tree-level differential cross section, which has a simple analytical form. The term F rad (cos θ) parametrises initial-state and final-state radiation effects, dominated by real-photon emission, as implemented in BHWIDE. It is verified that F rad (cos θ) has a negligible dependence on the spread of √ s considered in this analysis and, most important, on C.

Cross Section Measurement
The data were collected at LEP by the L3 detector [16,17] in the years from 1998 through 2000. They correspond to an integrated luminosity of 607.4 pb −1 and are grouped in eight intervals of √ s with the average values and corresponding integrated luminosities listed in Table 1.
Events from the e + e − → e + e − process are selected as described in Reference 18. Electrons and positrons are identified as clusters in the BGO electromagnetic calorimeter, matched with tracks in the central tracker. In the barrel region of the detector, | cos θ| < 0.72, the energy of the most energetic cluster must satisfy E 1 > 0. 25 √ s, while the energy of the other cluster must satisfy E 2 > 20 GeV. In the endcap region, 0.81 < | cos θ| < 0.98, these criteria are relaxed to E 1 > 0.2 √ s and E 2 > 10 GeV. Events with clusters in the transition region between the barrel and endcap regions, 0.72 < | cos θ| < 0.81, instrumented with a lead and scintillating-fiber calorimeter [17], are rejected. To suppress contributions from events with high-energy initialstate radiation, the complement to 180 • of the angle between the two clusters, the acollinearity, ζ, is required to be less than 25 o . The number of events observed at different values of √ s is shown in Table 1 together with the Monte Carlo expectations for signal and background. The e + e − → e + e − process is simulated with the BHWIDE Monte Carlo generator assuming C = 1. Background processes are described with the following Monte Carlo generators: KORALZ [19] for e + e − → τ + τ − , KORALW [20] for e + e − → W + W − , PYTHIA [21] for e + e − → Ze + e − , DIAG36 [22] for e + e − → e + e − e + e − , GGG [23] for e + e − → γγγ and TEEGG [24] for e + e − → e + e − γ events where one fermion is scattered into the beam pipe and the photon is in the detector. The L3 detector response is simulated using the GEANT package [25], which describes effects of energy loss, multiple scattering and showering in the detector. Timedependent detector inefficiencies, as monitored during the data-taking period, are included in the simulation.
Systematic effects, such as charge confusion, are reduced by folding the differential cross section into dσ/d| cos θ|, which is defined as: This differential cross section is measured in the fiducial volume defined by: where θ e − and θ e + are the polar angles of the electron and the positron, respectively. The value of cos θ is derived as: Ten intervals of | cos θ| are considered for each of the eight values of √ s, for a total of 80 independent measurements. Table 2 and Figure 2 present the measurements of dσ/d| cos θ| and the Standard Model expectations. The larger uncertainties in the interval 0.72 − 0.81 are due to the transition region between the barrel and the endcap regions.

Results
Figures 3 and 4 compare the combined differential cross section at the average centre-of-mass energy √ s = 198 GeV with the Standard Model prediction, corresponding to C = 1, and with a constant value of α, corresponding to C = 0. The data favour the hypothesis C = 1 over the hypothesis C = 0, as also presented in Table 3.
The value of C is extracted by comparing the 80 measurements of dσ/d| cos θ| with the theoretical expectations dσ th (C)/d cos θ in a χ 2 fit with the result: where the quoted uncertainty is statistical only. Several sources of systematic uncertainties are then considered.
• The theoretical expectations for dσ th (1)/d cos θ have an uncertainty which varies from 0.5% in the endcap region to 1.5% in the barrel region [13,15].
• The measurements of dσ/d| cos θ| are affected by a systematic uncertainty, dominated by the event-selection procedure, which varies between 1% and 10%, as listed in Table 2 [18].
• An uncertainty between 0.2% and 1.5% is assigned to F rad (cos θ), as a function of cos θ, in order to account for possible higher-order effects not included in the BHWIDE parametrisation.
• Migration effects among the different cos θ bins are found to be negligible due to the large bin size and the good detector resolution.
Systematic uncertainties are conservatively treated as fully correlated and the fit is repeated including both statistical and systematic uncertainties with the result: where the first uncertainty is statistical and the second systematic. A breakdown of the systematic uncertainty is presented in Table 4. This result is in agreement with the Standard Model expectation, C = 1. The quality of the fit is satisfactory, with a χ 2 of 91.9 for 79 degrees of freedom, corresponding to a confidence level of 17%. The hypothesis of a value of α which does not depend on Q 2 , C = 0, is totally excluded with a χ 2 of 316 for 80 degrees of freedom, corresponding to a a confidence level of 10 −29 .

Discussion
The result presented above establishes the evolution of the electromagnetic coupling with −Q 2 in the range 1800 GeV 2 < −Q 2 < 21600 GeV 2 . This finding extends and complements studies based on small-angle Bhabha scattering by the L3 [10] and OPAL [11] Collaborations, which studied the regions 2.10 GeV 2 < −Q 2 < 6.25 GeV 2 and 1.81 GeV 2 < −Q 2 < 6.07 GeV 2 , respectively. The advantage of large-angle Bhabha scattering, investigated in this Letter, is to probe large values of −Q 2 , while studies of small-angle Bhabha scattering at lower values of −Q 2 benefit from a larger cross section and thus statistical accuracy. The experimental systematic uncertainties of measurements in the two −Q 2 regions are implicitly different. At large −Q 2 , they are dominated by the selection of Bhabha events in the large-angle calorimeters, while at low −Q 2 they mostly arise from the event reconstruction in the luminosity monitors and from effects of the material traversed by electrons and positrons before their detection. Both studies, at large and low −Q 2 , are affected by theoretical uncertainties on the differential cross section of Bhabha scattering, although in different angular regions Figures 5 and 6 present the evolution of the electromagnetic coupling with −Q 2 . A band for 1800 GeV 2 < −Q 2 < 21600 GeV 2 shows the 68% confidence level result from this analysis. It is derived by inserting the measured value of C with its errors in Equation (2) together with the QED predictions of Reference 5. The results from previous L3 data for Bhabha scattering at 2.10 GeV 2 < −Q 2 < 6.25 GeV 2 and 12.25 GeV 2 < −Q 2 < 3434 GeV 2 [10] are also shown. These two measurements are not absolute measurements of the electromagnetic coupling but differences between the values of α(Q 2 ) at the extreme of the Q 2 ranges [10]: The results in Figure 5 are obtained by fixing the values of α(−2.10 GeV 2 ) and α(−12.25 GeV 2 ) to the QED predictions of Reference 5 in order to extract the values of α(−6.25 GeV 2 ) and α(−3434 GeV 2 ) from Equations (12) and (13). The results shown in Figure 6 are obtained by first determining the values of α(−2.10 GeV 2 ) and α(−12.25 GeV 2 ) from the measured value of C and from Equation (2) and then extracting the values of α(−6.25 GeV 2 ) and α(−3434 GeV 2 ) from Equations (12) and (13). This procedure relies on the assumption that the measured value of C also describes the running of the electromagnetic coupling for lower values of −Q 2 . Both figures provide an impressive evidence of the running of the electromagnetic coupling in the energy range accessible at LEP.            Figure 5: Evolution of the electromagnetic coupling with Q 2 determined from the present measurement of C for 1800 GeV 2 < −Q 2 < 21600 GeV 2 , yellow band, and from previous data for Bhabha scattering at 2.10 GeV 2 < −Q 2 < 6.25 GeV 2 and 12.25 GeV 2 < −Q 2 < 3434 GeV 2 [10]. The open symbols indicate the values of Q 2 where α(Q 2 ) was fixed to the QED predictions [5] in order to infer the values of α(Q 2 ) shown by the full symbols. These QED predictions are shown by the solid line.  Figure 6: Evolution of the electromagnetic coupling with Q 2 determined from the present measurement of C for 1800 GeV 2 < −Q 2 < 21600 GeV 2 , yellow band, and from previous data for Bhabha scattering at 2.10 GeV 2 < −Q 2 < 6.25 GeV 2 and 12.25 GeV 2 < −Q 2 < 3434 GeV 2 [10], full symbols. The solid line represent the QED predictions [5].