Measurement of the Photon Structure Function F2gamma with the L3 Detector at LEP

The e+e- ->e+e- hadrons reaction, where one of the two electrons is detected in a low polar-angle calorimeter, is analysed in order to measure the hadronic photon structure function F2gamma . The full high-energy and high-luminosity data set, collected with the L3 detector at centre-of-mass energies 189-209GeV, corresponding to an integrated luminosity of 608/pb is used. The Q^2 range 11-34GeV^2 and the x range 0.006-0.556 are considered. The data are compared with recent parton density functions.


Introduction
Photons are ideal tools for probing the structure of more complex objects such as the proton in deep-inelastic scattering experiments.At LEP, in the e + e − → e + e − γ * γ → e + e − hadrons reaction, two virtual photons are produced by the incoming electrons 1) and their interaction yields hadrons.If the scattering angle of one of the electrons, θ tag , is sufficiently large, it is observed in the low polar-angle electromagnetic BGO calorimeter [1] of the L3 detector [2], originally devised to detect low angle Bhabha scattering in order to measure the LEP luminosity.This allows to measure the four-momentum, k ′ , of this "tagged" electron.For "single-tagged" events the second electron is undetected, its polar angle is small and the virtual photon radiated from this electron is quasi-real.In the framework of a deep-inelastic scattering formalism the process e + e − → e + e − hadrons is written as the convolution of the target photon flux with the reaction e(k) + γ(p) → e(k ′ ) + hadrons.The photon, γ * , with four-momentum q = k − k ′ and a large virtuality Q 2 = −q 2 ∼ 2E tag E beam (1 − cos θ tag ), is considered as a point-like probe investigating the structure of the target photon, γ, with four-momentum p and virtuality P 2 = −p 2 ≃ 0. E tag is the energy of the tagged electron and E beam the energy of the beam.The differential cross section is written in terms of the scaling variables γγ + P 2 ) and y = (q • p)/(k • p) = 1 − (E tag /E beam cos 2 θ tag ) as [3,4]: The variable x depends on the two-photon centre-of-mass energy, W γγ , equal to the effective mass of the produced hadrons.The inelasticity y is small (y < 0.3) in the kinematic region of this study and consequently only F γ 2 (x, Q 2 ) contributes appreciably to the cross section.By convention, F γ 2 /α is measured, where α is the fine-structure constant.Using this approach, the photon structure function has been extensively studied at low-energy e + e − colliders [5] and at LEP [5][6][7].
A virtual photon can interact as a point-like particle in "direct processes"; it can fluctuate into a vector meson (ρ, ω, φ) inducing soft hadronic interactions in "VDM processes" or it can interact via its partonic content of quarks or gluons in "resolved processes".High Q 2 single-tag events favour perturbative QED and QCD diagrams such as γγ → qq , γq → gq or γg → qq.The two resolved processes, γq → gq and γg → qq, are described using parton density functions extracted from the photon structure functions measured in previous experiments at PEP, PETRA and TRISTAN.Reviews of the existing parameterisations may be found in References 8 and 9. Recently, a new parametrisation was obtained adding published LEP data [10].
This analysis uses the 608.1 pb −1 of high-energy LEP data, collected at e + e − centre-ofmass energies 189 GeV ≤ √ s ≤ 209 GeV.The data are grouped in four average √ s values, presented in Table 1.These high-energy data allow to extend our previous measurements [6] at √ s ≃ 91 GeV and √ s = 183 GeV in the small-x region down to 0.006 and in the medium-x region up to 0.556 for the Q 2 range 11 GeV 2 ≤ Q 2 ≤ 34 GeV 2 .The e + e − → e + e − hadrons cross section and the F γ 2 photon structure function are studied as a function of x in the three 2 is also studied combining the values at < Q 2 >= 12.4, 16.7 and 25.5 GeV 2 with our previous measurements. 1)Throughout this Letter, the term "electron" indicates both electron and positron.

Monte Carlo Models
The value of the Q 2 variable is accurately determined by measuring the four-momentum of the scattered electron.However, the effective mass of the final state hadrons is only partially reconstructed, as these are often produced at low polar angles where no tracking system can be installed.A Monte Carlo modelling of the final state hadrons is therefore necessary [11] to determine the x variable.
PHOJET describes hadron-hadron, photon-hadron and photon-photon collisions.It is based on the Dual Parton Model combined with the QCD-improved parton model [15].In order to have a continuous transition between hard and soft processes, the distribution of the transverse momentum, p t , of the soft partons is matched to the one predicted by QCD.The two-photon luminosity is calculated from the flux of transversely polarised photons; corrections for the longitudinally polarised photons are thus incorporated into an effective two-photon cross section.The transition from real-photon to virtual-photon scattering is obtained by a change of the relative weight of all partial cross sections.
PYTHIA is a general purpose Monte Carlo.For two-photon interactions it incorporates leading order (LO) hard-scattering processes as well as elastic, diffractive and low p t events.The classification of the photon interactions into three different components, direct, resolved and VDM, results in six different classes of events.Events are also classified according to the hard scales involved in the process: photon virtualities and parton transverse momenta.
TWOGAM generates three different processes separately: point-like photon-photon interactions, resolved processes, and non-perturbative soft processes described by the Generalised Vector Dominance Model (GVDM).The structure of the program is modular and the photon flux is calculated with an exact LO formula.The cross sections of the three different processes are adjusted to fit the x-distribution of the data.The cross section of the direct process is fixed to the value expected in our kinematic range, σ = 41 pb.The QCD and the VDM cross sections are then adjusted to σ = 5 pb and σ = 28 pb, respectively.
All Monte Carlo samples are generated with a luminosity at least five times greater than the experimental one.All events are passed through a full detector simulation which uses the GEANT [18] and GHEISHA [19] programs and takes into account detector efficiencies and time-dependent effects.Monte Carlo events are then reconstructed in the same way as the data.

Data Analysis
Events are mainly accepted by two independent triggers: the single-tag trigger and the central track-trigger.The single-tag trigger requires at least 70% of the beam energy to be deposited in one of the low polar-angle calorimeters, in coincidence with at least one track in the central tracking chamber.The central track-trigger requires at least two tracks back-to-back in the transverse plane within ±60 • , each with p t > 150 MeV.The average trigger efficiency is about 97%.There is a single-tag trigger signal for about 90% of the selected events and a central track-trigger signal for about 85% of the selected events.

Event Selection
Events are selected by requiring a single scattered electron in the low polar-angle calorimeter and a hadronic final state.A tagged electron candidate is the highest energy cluster with a shape consistent with an electromagnetic shower, E tag /E beam > 0.7, as shown in Figure 1a, and a polar angle in the fiducial region 0.0325 rad ≤ θ ≤ 0.0637 rad inside the geometrical acceptance 0.030 rad ≤ θ ≤ 0.066 rad.To ensure that the virtuality of the target photon is small, the highest-energy cluster in the low polar-angle calorimeter opposite to the tagged electron must have an energy less than 20% of the beam energy, as shown in Figure 1b.
At least four additional particles must be detected.A particle can be a track or a photon.A track must have p t > 100 MeV and a distance of closest approach in the transverse plane to the interaction vertex of less than 10 mm.A photon is a cluster in the electromagnetic BGO calorimeters with energy above 100 MeV, not associated with a charged track.
To reduce the background from the process e + e − → qq(γ), the total energy deposited in the electromagnetic and hadronic calorimeters must be less than 40% of the center-of-mass energy, as shown in Figure 1c.The events with a large value of the total energy are due to the e + e − → Zγ → qqγ process, where the radiative photon is misidentified as the tagged electron.
The mass of the hadronic final state, W vis , is calculated from all tracks and calorimetric clusters.Additional clusters detected in the low polar-angle calorimeter are assigned the pion mass and included in the calculation of W vis .To avoid the hadronic resonance region, W vis is required to be greater than 4 GeV, as presented in Figure 1d.
Figure 2 shows the Q 2 distribution for each √ s sample.Only events with 11 GeV 2 ≤ Q 2 ≤ 34 GeV 2 are studied.The number of selected events and the backgrounds from the e + e − → e + e − τ + τ − and e + e − → qq(γ) processes are given in Table 1.The background is dominated by the e + e − → e + e − τ + τ − production.The contribution from the e + e − → τ + τ − and e + e − → W + W − processes is negligible.The background from beam-gas and beam-wall events is found to be negligible by inspection of the radial distribution of track intersections.The W vis and ) distributions are presented in Figure 3 for all selected data.The PYTHIA and TWOGAM model reproduce the data rather well, except at large values of W vis .PHOJET presents a harder mass spectrum and predicts too many events for x vis < 0.1 and is therefore not used in the following.
The total acceptance is calculated for each data sample separately.It takes into account the trigger efficiency, the geometrical acceptance and the selection cuts.An example is presented in Figure 4a for the data at √ s= 189 GeV.

Unfolding and Differential Cross Sections
The cross section ∆σ/∆x as a function of x for the reaction e + e − → e + e − hadrons is measured for three Each data set is subdivided into bins of x vis of similar statistics, as listed in Table 2.A Bayesian unfolding procedure [20] is used to relate the measured x vis to the true value of x and to correct the data for the detector acceptance and efficiency.This procedure is applied using, in turn, the PYTHIA and TWOGAM generators.The average of the cross sections obtained in the two cases is retained.The correlation between the generated value of x and x vis is similar for the two models.The one obtained with PYTHIA is shown in Figure 4b.The cross sections measured for each x interval of average value x and for each value of √ s are given in Table 2 with their statistical and systematic uncertainties.The bin-to-bin correlation matrices obtained with PYTHIA and TWOGAM for √ s = 189 GeV are shown in Tables 3 and 4, respectively.Similar matrices are obtained for the other values of √ s.

Systematic Uncertainties
The systematic uncertainties on the cross sections are estimated for each data sample, for each x bin and for each Q 2 interval.Three main sources of systematic uncertainties are considered: the selection procedure, the trigger efficiency and the Monte Carlo model.Their average effects over the full data sample are listed in Table 5.
The uncertainties from the selection procedure are estimated by varying the selection cuts.The fiducial value of the polar angle in the low polar-angle calorimeter is varied from 0.0325 rad to 0.0360 rad and from 0.0637 rad to 0.060 rad.These changes result in a Q 2 -dependent uncertainty, as the highest and lowest values of the Q 2 are affected by the fiducial cut.The cut on E tag is varied from 0.70E beam to 0.65E beam and 0.75E beam .The anti-tag cut is changed from 0.20E beam to 0.15E beam and 0.25E beam .The numbers of particles is varied from four to three and five.The cut on the total energy in the calorimeters relative to √ s is varied from 0.40 to 0.35 and 0.45.The uncertainty on the trigger efficiency, as determined from the data, is 1.5%.An additional uncertainty comes from the limited Monte Carlo statistics.
The discrepancies of the results obtained with the PYTHIA and TWOGAM generators are considered as systematic uncertainties related to the Monte Carlo modelling.This difference is due to the calculated acceptance as well as to the unfolding procedure.An additional contribution to this modelling uncertainty is evaluated by repeating the analysis doubling or halving the VDM component of the TWOGAM generator and is found to be negligible.

Extraction of F γ 2
The photon structure function F γ 2 /α is derived from the cross section of Equation 1 and the target-photon flux calculated by the program GALUGA [21] as: The program calculates the theoretical value ∆σ Galuga in the given Q 2 and x range setting F γ 2 = 1 and F γ L to the QPM value [22].In practice the F γ L contribution to the cross section is smaller than 1%, due to the small value of y.The running of the fine-structure constant with Q 2 is included.A GVDM form factor is used in the calculation for the target photon virtuality whose average value is of the order of 0.07 GeV 2 .The low polar-angle calorimeter acceptance for the tagged and the anti-tagged electron and the W γγ > 4 GeV requirement are taken into account.The uncertainty on ∆σ Galuga , as estimated by comparing the GVDM to a ρ form factor, is 2%.
The contribution of radiative corrections to the cross section is evaluated by using the program RADCOR [23], which includes initial and final state radiation for the reaction e + e − → e + e − µ + µ − .The corrections are mainly due to initial state radiation from the electron scattered at large angle.Final state radiation is detected together with the scattered electron due to the finite granularity of the calorimeter.Initial state radiation from the electron producing the quasi-real target photon is very small.The calculations are performed at the generator level using the Q 2 from the electron variables and W γγ from the muon pair.The measured F γ 2 /α is multiplied by the ratio, R, of the non-radiative and the total cross section, shown in Tables 6  and 7 for different values of x and Q 2 .
The F γ 2 values, averaged over the x intervals, are first obtained for each individual data set.The results are statistically compatible and, consequently, a weighted average of F γ 2 is calculated for the Q 2 ranges with average values 12.4 GeV 2 , 16.7 GeV 2 and 25.5 GeV 2 .This procedure is applied to data unfolded separately with PYTHIA and TWOGAM and the two different values are shown in Figure 5. Their average value is given in Table 6 and in Figure 6 for each x interval of expected average value x .In addition to the systematic uncertainty on the cross section, presented in Table 5, two systematic uncertainties are further considered in the extraction of F γ 2 .The first uncertainty, of 2%, is associated to the GALUGA calculations.The second uncertainty, also of 2%, covers the estimation of the initial-state and final-state radiative corrections.The uncertainty on initial-state radiation is assessed by changing the angular and momentum criteria which separate soft from hard photons in the Monte Carlo programs.The uncertainty on final-state radiation is estimated by varying the cone angle of the calorimeter for which final state radiation is detected together with the scattered electron.
A comparison of the data with the existing parameterisations as obtained with the PDFLIB library [24] shows that our data are not well described by the leading-order parton density functions.In Figure 6 the data are compared with the predictions of the leading-and higherorder parton density functions GRV [25] and the higher-order parton density functions CJK [10].The best agreement is found for the higher-order GRV [25] predictions.In all cases four quarks, u, d, s and c are used.The pure QPM prediction for γγ → qq is also indicated, as calculated by using GALUGA with a mass of 0.32 GeV for the u and d quarks, 0.5 GeV for the s quark and 1.4 GeV for the c quark.It is clearly insufficient to describe the data.
2 , is studied in four x bins, x = 0.01 − 0.1, x = 0.1 − 0.2, x = 0.2 − 0.3 and x = 0.3 − 0.5 and the results are given in Table 7.In Figure 7 the F γ 2 /α values are presented for the lowest x bin and for a combined bin x = 0.1 − 0.5, together with our previous results [6].Corrections for radiative effects are applied.The new measurements at 11 GeV 2 ≤ Q 2 ≤ 14 GeV 2 , 14 GeV 2 ≤ Q 2 ≤ 20 GeV 2 and 20 GeV 2 ≤ Q 2 ≤ 34 GeV 2 are in good agreement with our previous results.The expected linear growth with ln Q 2 is observed in both x intervals.The function a + b ln(Q 2 / GeV 2 ) is fitted to the data, taking into account the total uncertainty calculated from the quadratic sum of statistical and systematic uncertainties.The fit results are: a = 0.141 ± 0.007 and b = 0.060 ± 0.005 for x = 0.01 − 0.1 with a confidence level of 44% and a = 0.05 ± 0.11 and b = 0.13 ± 0.04 for x = 0.1 − 0.5 with a confidence level of 71%.
The predictions of the leading-and higher-order parton density functions GRV [25] and the higher-order parton density functions CJK [10] are also indicated in Figure 7.The evolution is different for the different models and the data are better described by the higher-order GRV model.

Conclusions
The photon structure function F γ 2 is measured at LEP with the L3 detector at centre-of-mass energies 189 ≤ √ s ≤ 209 GeV in the Q 2 range 11 GeV 2 ≤ Q 2 ≤ 34 GeV 2 and the x range 0.006 ≤ x ≤ 0.556.The data are better reproduced by the higher-order parton density function of GRV than by other parton distribution functions determined from the low energy data.
Combining the present results with previous L3 measurements, the Q 2 evolution is studied from 1.5 GeV 2 to 120 GeV 2 in the low-x region, 0.01 ≤ x ≤ 0.1, and from 12.4 GeV 2 to 225 GeV 2 in the higher-x region, 0.1 < x ≤ 0.5.The measurements at different centre-of-mass energies are consistent and the ln Q 2 evolution of F γ 2 is clearly confirmed.
√ s (GeV) L(pb −1 ) Events e + e − → e + e − τ + τ − e + e − → qq(γ) Purity (%)    2 /α and the applied radiative correction factors, R, in bins of x for the three Q 2 ranges.The first uncertainty is statistic, the second systematic and the third is due to model dependence.

Figure 1 :Figure 2 :
Figure 1: Distribution of the highest energy clusters in the forward electromagnetic calorimeters for a) the tagged electron side and b) for the opposite side.c) Total energy in the central calorimeters.d) The visible mass of the hadronic final state.All distributions are presented after all other cuts are applied.The backgrounds from annihilation and two-photon leptonic events are indicated as shaded areas and added to the expectations of the PYTHIA, PHOJET and TWOGAM generators.The arrows indicate the position of the cuts.

Figure 3 :
Figure3: Distribution of the visible mass of the two-photon system and of the x vis for all selected events compared with Monte Carlo predictions.The backgrounds from annihilation and two-photon leptonic events are indicated as shaded areas and added to the expectations of the PYTHIA, PHOJET and TWOGAM generators.

Figure 5 :Figure 6 :Figure 7 :
Figure 5: The photon structure function F γ 2 /α obtained with PYTHIA and TWOGAM.Only the statistical uncertainties are shown.For clarity, the symbols corresponding to the two Monte Carlo generators are slightly offset.The measurements are correlated as indicated in Tables3 and 4

Table 1 :
The average e + e − centre-of-mass energies, √ s , and the corresponding luminosities for the four data samples together with the number of selected events in theQ 2 intervals 11 GeV 2 ≤ Q 2 ≤ 14 GeV 2 , 14 GeV 2 ≤ Q 2 ≤ 20 GeV 2 and 20 GeV 2 ≤ Q 2 ≤ 34 GeV 2 .The numbers of expected background events from the e + e − → e + e − τ + τ − and e + e − → qq(γ) processes and the signal purity are also listed.

Table 2 :
Cross sections ∆σ/∆x as a function of x for the reaction e + e − → e + e − hadrons for the four average values of √ s, in three Q 2 intervals.The first uncertainty is statistical, the second systematic.

Table 3 :
Correlation matrices obtained with the PYTHIA Monte Carlo for the data at √ s = 189 GeV for the three Q 2 intervals.

Table 4 :
Correlation matrices obtained with the TWOGAM Monte Carlo for the data at √ s = 189 GeV for the three Q 2 intervals.

Table 5 :
Systematic uncertainties on the measured cross sections.

Table 6 :
Measured values of F γ

Table 7 :
The values of F γ 2 /α in bins of Q 2 for four x ranges together with the radiative correction factor.The first uncertainty is statistic, the second systematic and the third is due to model dependence.