Total Cross Section in gamma gamma Collisions at LEP

The reaction e+e- ->e+e- gamma* gamma* ->e+e- hadrons for quasi-real photons is studied using data from root(s) = 183 GeV up to 202 GeV. Results on the total cross sections sigma(e+e- ->e+e- hadrons) and sigma(+e- gamma* gamma* ->e+e- hadrons) are given for the two-photon centre-of-mass energies 5 GeV<Wgammagamma<185 GeV. The total cross section of two real photons is described by a Regge parametrisation. We observe a steeper rise with the two-photon centre-of-mass energy as compared to the hadron-hadron and the photon-proton cross sections. The data are also compared to the expectations of different theoretical models.


Event selection
The analysis is based on the central tracking system, the high resolution electromagnetic calorimeter, the hadron calorimeter and the luminosity monitor.
Two-photon events are collected predominantly by the track triggers [10]. The trigger efficiency is studied separately for each data sample by comparing the number of events accepted by the track trigger and the calorimetric energy trigger. The efficiencies of higher level triggers are measured using prescaled events. The trigger efficiency increases from 80 % at W vis = 5 GeV to 94 % above 80 GeV.
Hadronic two-photon events are selected by the following criteria : • To exclude scattered electrons, events with clusters in the luminosity monitor having energy greater than 30 GeV, in a fiducial region of 33 mrad ≤ θ ≤ 64 mrad are rejected. The virtuality of the interacting photons, Q 2 , is thus less than 8 GeV 2 , with an average value < Q 2 >∼ 1.5 ×10 −2 GeV 2 . The distribution of low energy clusters in the luminosity monitor, presented in Figure 1a, shows a good agreement with both Monte Carlo programs. When the scattered electron reaches the detector, the agreement is maintained with the 1) Electron stands for electron and positron throughout this paper.
PHOJET Monte Carlo, while these configurations are missing in PYTHIA because of a ρ−mass cutoff, Q 2 ≤ m 2 ρ , applied to the two-photon luminosity function in the generation of the events.
• The total energy in the electromagnetic calorimeter is required to be greater than 0.5 GeV, in order to suppress beam-gas and beam-wall backgrounds, and less than 50 GeV, to exclude radiative events, e + e − → Zγ. The total energy deposited in the electromagnetic and hadronic calorimeters, E cal , must be less than 40% of √ s, to exclude annihilation events, as shown in Figure 1b.
• At least six particles must be detected, in order to exclude events containing τ . A particle is defined [2] as either a track, a photon in the electromagnetic calorimeter, or a cluster in the hadron calorimeter or in the luminosity monitor. Clusters in the luminosity monitor are considered as pions if their energy is below 5 GeV, as photons otherwise. The distribution of the number of particles is presented in Figure 1c. This distribution is not well reproduced by the Monte Carlo simulations.
After selection, the background from beam-gas and beam-wall interactions is found to be negligible. The visible effective mass of the event, W vis , is calculated from the four-momenta of the measured particles. The analysis is limited to events with W vis ≥ 5 GeV. Almost 2 million events are selected, 1.6 × 10 5 at √ s= 183 GeV, 7.8 × 10 5 at √ s= 189 GeV and 1 × 10 6 at √ s= 192 − 202 GeV. The average centre-of-mass energy of this last sample is √ s= 198 GeV.
The W vis spectrum is shown in Figure 1d for the total data sample. The background is below 1% at low masses, where it is dominated by two-photon τ -pair production. It increases at high masses, due mainly to annihilation processes and reaches a maximum of 15%.
The distributions of the rapidity, y, of the particles and of their energy flow are compared to the Monte Carlo expectations in Figure 2. A good agreement is observed also in the regions where | y | ≃ 3, between the luminosity monitor and the hadron calorimeter.

Unfolding and efficiency
The distribution of the two-photon effective mass W γγ is obtained from the visible effective mass W vis by the same unfolding procedure [11] used in Reference 2. For each data sample, the W vis spectrum is subdivided in 16 intervals, presented in Figure 3a, and the resulting W γγ distribution in 8 intervals, presented in Figure 3b. The result of the unfolding procedure depends on the Monte Carlo used. Data unfolded with PYTHIA are in general higher than if unfolded with PHOJET. After unfolding, the events are corrected for the efficiency, using the ratio between selected and generated events in each W γγ interval. This includes the purely geometrical acceptance as well as the efficiencies of the detector and the analysis procedure. For W γγ > 30 GeV, the efficiency is rather constant, with a value of about 80%. The efficiency obtained with PYTHIA is lower by about 10%, which may be attributed to a different modeling of the diffractive interactions, of difficult detection.

Cross Section Determinations
The measured cross sections ∆σ(e + e − → e + e − hadrons) are given in Table 1 for the three data sets, as a function of the W γγ intervals. The average of the results obtained by unfolding the data with PYTHIA or PHOJET is used. Due to the unfolding procedure, the measurements are highly correlated. The correlation matrix, similar for the three data sets, is given in Table 2 for √ s= 189 GeV. The differential cross section ∆σ/∆W γγ is shown in Figure 4, together with our measurements at lower LEP collision energies [2]. The fast decrease of the cross section as a function of W γγ is due to the two photon luminosity function, L γγ , which depends on W 2 γγ /s. The systematic uncertainties are evaluated for each W γγ bin. They are independent of the data sample, inside statistical accuracy. They are evaluated as follows and their contribution is listed in Table 3.
• Trigger efficiencies: by varying this quantity within the accuracy of its determination, of about 10%.
• Energy scale of the calorimeters and contribution of the annihilation background: by varying the E cal cut by ± 10% of √ s.
• Uncertainties on the rejection of scattered electrons: by changing the E lumi cut from 30 GeV to 50 GeV.
• Uncertainties on the particle multiplicity: by accepting a minimum number of four or eight particles instead of six.
• Uncertainties due to Monte Carlo statistics are negligible for W γγ < 65 GeV, but important in the higher W γγ bins.
Uncertainties on the energy scale of the small angle calorimeter, evaluated by varying the gain by a factor two, are negligible. The total experimental systematic uncertainty, obtained by adding in quadrature all contributions, is also given in Table 3. The uncertainty related to the Monte Carlo model is given in the last column of Table 3. It is half of the difference between the results obtained by unfolding the data with PHOJET or PYTHIA and exceeds the experimental uncertainty in almost all bins.
To extract the total cross section of two real photons, the luminosity function L γγ [12] is calculated and the hadronic two-photon process is extrapolated to Q 2 = 0. This is done as in Reference [2] by using an analytical program [13]. Depending on the choice of photon form factors, this calculation varies of ±5%.
The cross sections obtained for the three data sets are compatible within the experimental uncertainties and are presented in Table 4. As expected from the study of the experimental systematics, the largest differences are observed in the first and last bins. The combined value is also given in Table 4 and in Figure 5a with the statistical uncertainties obtained from the unfolding and the experimental systematics. The values obtained by unfolding the data with the two Monte Carlo programs separately are shown in Figure 5b and can be obtained from the last column of Table 4.

Regge parametrisation
The total cross sections for hadron-hadron, σ pp , photon-hadron, σ γp , and photon-photon, σ γγ , production of hadrons show a characteristic steep decrease in the region of low centre-of-mass energy, followed by a slow rise at high energies. From Regge theory [15] this behaviour is understood as the consequence of the exchange of Regge trajectories, α(t), in the t-channel. The total cross section takes the form σ tot ∝ s α(0)−1 . The low energy region is sensitive to the exchange of a Reggeon R (R = ρ, ω, f, a ..), with α R (0) ≃ 0.5. At high energies, the Pomeron exchange dominates, with α P (0) ≃ 1. A parametrisation of the form accounts for the energy behaviour of all hadronic and photoproduction total cross sections, the powers of s being universal [16]. This is confirmed by the recent compilation of the total cross section data [17] where a fit of Equation 1 for all hadron total cross sections gives a result compatible with the universal values ǫ = 0.093 ± 0.002 and η = 0.358 ± 0.015. The coefficients A and B are process and Q 2 dependent. If photons behave predominantly like hadrons, this expression may also be valid for the two-photon total hadronic cross section, with s = W 2 γγ . Considering only the experimental uncertainties, statistical and systematic, several Regge fits are performed on the data and their results are presented in Table 5. The exponent η is fixed to the universal value, since the low mass range is too small to be sensitive to this parameter. When the W γγ interval is restricted to 5 GeV − 65 GeV, a range similar to the one covered by our previous data [2], similar values of the parameters A and B are obtained. In this limited interval the data are compatible with the universal value of ǫ. Extending the range to the whole W γγ interval, the fit with the exponents ǫ and η fixed to the universal value, dashed line in Figure 5a, does not represent the σ γγ energy dependence. A fit with A, B and ǫ as free parameters, represented as a full line in Figure 5a, gives ǫ = 0.225 ± 0.021 with a confidence level of 4%. This value is more than a factor two higher than the universal value. It is independent of the Monte Carlo model used to correct the data, as shown in Table 5 and in Figure 5b.
The fitted value of ǫ is strongly correlated to the Reggeon component. To avoid this correlation, we fit only the Pomeron exchange for sufficiently high W γγ values. The results, using a different initial value of W γγ , are listed in the second part of Table 5 . The value of ǫ increases by increasing the lower mass cutoff, thus indicating that its value is not universal, but it reflects the onset of QCD phenomena, as ǫ increases with increasing W γγ .

Models for γγ total cross sections
Several models [18][19][20] were recently compared to the L3 and OPAL [21] measurements. Their predictions for the two-photon total cross section are typically derived from measurements of proton-proton and photoproduction total cross sections via the factorization relation: σ γγ ≈ σ 2 γp /σ pp [22]. In general, these models give an energy dependence of the cross section similar to the universal fit discussed above. Two examples [19,20] are shown in Figure 6a in comparison with the results of previous experiments [23], those presented in this letter and those of OPAL. While the measurements at the low energy colliders present a wide spread, a good agreement is found between L3 and OPAL in the common W γγ range, 10 GeV ≤ W γγ ≤ 110 GeV. Good agreement is also found if the data, unfolded separately with either PHOJET or PYTHIA, are compared. In this W γγ region, a model [20] reproduces well the data and the predictions of the other [19] are too high by 20%. However, for both lower and higher values of W γγ , the L3 data show a much steeper energy dependence than the theoretical predictions.
In the Regge theory, the Pomeron intercept is 1, yielding a constant total cross section. When the rise of the proton-proton total cross section was first observed, it was explained [24] with an increase of the number of hard partonic interactions. The predictions of a model [25] that calculates such effects, using an eikonalized prescription to enforce unitarity, are shown in Figure 6b. The parameters of the model are determined from photoproduction data and the L3 results are well inside the uncertainty related to this extrapolation.
Models with two Pomerons were recently proposed [26] to explain the fast energy increase of charm production at HERA. In this model, the 'soft' and the 'hard' Pomeron have different intercepts. Because of the qq component in the photon wave-function, the 'hard' Pomeron can contribute to the two-photon cross section even at Q 2 = 0. Thus a more rapid energy dependence for σ γγ is expected. The increase in ǫ with larger values of W γγ , as listed in the second part of Table 5, is consistent with such a contribution of the 'hard' Pomeron.   Table 3: Evaluation of systematic uncertainties due to the trigger, the analysis cuts and the Monte Carlo statistics. All values are per-cent uncertainties on the cross sections ∆σ(e + e − → e + e − hadrons) and σ(γγ → hadrons). The uncertainty introduced by unfolding the data with PYTHIA or PHOJET is considered separately in the last column. A further scale uncertainty of 5% must be added for the σ(γγ → hadrons) cross sections, due to the two-photon luminosity function.  Table 4: The σ(γγ → hadrons) cross sections as a function of the average γγ centre-of-mass energy, < W γγ >, for the three data sets and for their combination. The statistical uncertainties, obtained after unfolding, are given for each data set. The experimental systematic uncertainty, ∆σ exp γγ , and the difference, ∆σ MC γγ , between the average value and the result unfolded with PHO-JET (lower sign) and with PYTHIA (upper sign) are also given. A further scale uncertainty of 5% must be added, due to the two-photon luminosity function.   Table 4 of the form σ γγ = A s ǫ + B s −η [16], where s = W 2 γγ . PHOJET and PYTHIA indicates that only this Monte Carlo is used to unfold the data. In all other cases the average unfolding result of the two generators is used. The statistical and experimental uncertainties and the correlation matrix between the data points are used. The fitted parameters are strongly correlated. The second set of fits evaluates only the increase of σ γγ with s, i.e. the Pomeron part of the fit. The values of the χ 2 and the corresponding confidence level are given.    Events/GeV raw data PHOJET unfolding PYTHIA unfolding  202 GeV. a) The average result, obtained by unfolding the data with the two Monte Carlo models, is used. Two Regge fits, described in the text, are superimposed to the data. The continous line corresponds to the fit with the coefficient ǫ left as a free parameter, the dashed line is the fit with ǫ fixed to 0.093. b) The two-photon total cross section obtained by correcting the data sample with PHOJET (full points) and with PYTHIA (open points). The Regge fits of Table 5 are superimposed to the data. The statistical and experimental systematic uncertainties are added in quadrature. σ γγ [nb] Figure 6: The two-photon total cross sections compared to various models. a) The predictions of References 19 and 20 are compared to all two-photon total cross section data [21,23]. b) Predictions of the minijet model [25]; the two lines correspond to different choices of the model parameters. The statistical and experimental systematic uncertainties are added in quadrature. The uncertainties due to Monte Carlo models and to the two-photon luminosity function are included in the dashed lines in b).