Measurement of the energy dependence of the total photon-proton cross section at HERA

The energy dependence of the photon-proton total cross section, sigma-tot, was determined from ep scattering data collected with the ZEUS detector at HERA at three values of the center-of-mass energy, W, of the gamma-p system in the range 194<W<296 GeV. This is the first determination of the W dependence of sigma-tot from a single experiment at high W. Parameterizing sigma-tot proportional to W^(2 epsilon), epsilon = 0.111 +/- 0.009 (stat.) +/- 0.036 (syst.) was obtained.


Introduction
The soft hadronic nature of the photon observed in γp collisions [1] is well described by the vector meson dominance model [2], in which the photon is considered to be a superposition of vector mesons interacting with the proton. Therefore, the energy dependence above the resonance region of the total γp cross section, σ γp tot , is expected to be similar in form to that of the total hadronic cross sections, σ tot , for pp,pp, πp and Kp interactions. [3] demonstrated that the energy dependences of all hadronhadron total cross sections may be described by a simple Regge-motivated form,

Donnachie and Landshoff
where W is the hadron-hadron center-of-mass energy, A and B are process-dependent constants, and α IP (0) (α IR (0)) is process-independent and interpreted as the Pomeron (Reggeon) trajectory intercept.
This observation together with the interest in estimating the total cross sections at high energies, well beyond the range probed experimentally (for example for pp scattering at the LHC or for cosmic-ray physics), prompted further Regge-type fits of the energy dependence of the total hadron-proton cross sections [4,5]. At sufficiently high energies, the power-like behavior of the energy dependence is expected to be modified by the Froissart bound [6] and the total cross section is expected to behave as ln 2 (W 2 ). Recent analyses of hadron-proton and photon-proton cross sections indicate that already at present energies a ln 2 (W 2 ) dependence is observed [7][8][9]. The data from many experiments must be combined in such fits and the evaluation of the influence of systematic uncertainties is complex.
At the ep collider HERA, σ γp tot can be extracted from ep scattering at very low squared momentum transferred at the electron vertex, Q 2 10 −3 GeV 2 . The measurements of the total γp cross section at HERA for W ≃ 200 GeV [10][11][12][13][14] combined with measurements at low W confirmed that the total photoproduction cross section has a W dependence similar to that of hadron-hadron reactions. This similarity extends to virtualities Q 2 of the photon up to ≈ 1 GeV 2 [15]. This paper presents a determination of the W dependence of σ γp tot from ZEUS data alone, in the range 194-296 GeV. This was made possible because in the final months of operation, the HERA collider was run with constant nominal positron energy, and switched to two additional proton energies, lower than the nominal value of 920 GeV. Many of the systematic uncertainties arising in the extraction of σ γp tot are now common and do not affect the relative values of σ γp tot at different W . As the Reggeon term is expected to be small, the function in Eq. (1) can be simplified to the form This is the first extraction of the logarithmic derivative of the cross section in W 2 from a single experiment.

Kinematics
The photon-proton total cross section was measured in the process e + p → e + γp → e + X, where the interacting photon is almost real. The event kinematics may be described in terms of Lorentz-invariant variables: the photon virtuality, Q 2 , the event inelasticity, y, and the square of the photon-proton center-of-mass energy, W , defined by where k, k ′ and p are the four-momenta of the incoming positron, scattered positron and incident proton, respectively, and q = k − k ′ . These variables can be expressed in terms of the experimentally measured quantities where Q 2 min = m 2 e y 2 1 − y , E e , E ′ e and E p are the energies of the incoming positron, scattered positron and incident proton, respectively, θ e is the positron scattering angle with respect to the initial positron direction and m e is the positron mass. The scattered positron was detected in a positron tagger close to the beam line, restricting θ e (and hence Q 2 ) to small values. The photon virtuality ranged from the kinematic minimum, Q 2 min ≃ 10 −6 GeV 2 , up to Q 2 max ≃ 10 −3 GeV 2 , determined by the acceptance of the positron tagger. The equivalent photon approximation [16] relates the electroproduction cross section to the photoproduction cross section. The doubly-differential ep cross section can be written as where φ(y, Q 2 ) is the doubly differential photon flux. The longitudinal cross section is small (σ γp L /σ γp T < 0.1% [17]), and can be neglected. Then the transverse component of the flux has the form φ(y, Q 2 ) = α 2π For each of the incident proton energies, σ γp (y, Q 2 ) has a small variation as a function of y and Q 2 over the range of the measurement (< 1.5% over y and <0.1% over Q 2 [1,2]) and may be taken to be a constant, σ γp tot . Thus, the flux may be integrated over the range of measurement to give a total flux F γ , which, when multiplied by the total γp cross section gives σ ep tot , the ep cross section integrated over the measured range, 3 Experimental conditions HERA operated with a positron beam energy of approximately 27.5 GeV for all of the data used in this analysis. The proton beam energies, in chronological order, were 920 GeV for the high-energy run (HER), 460 GeV for the low-energy run (LER), and 575 GeV for the medium-energy run (MER).
A detailed description of the ZEUS detector can be found elsewhere [18]. A brief outline of the components that are most relevant for this analysis is given below.
Charged particles were tracked in the central tracking detector (CTD) [19] and the microvertex detector (MVD) [20]. The CTD and the MVD operated in a magnetic field of 1.43 T provided by a thin superconducting solenoid. The CTD drift chamber covered the polar-angle 1 region 15 • < θ < 164 • . The MVD silicon tracker provided polar angle coverage for tracks from 7 • to 150 • .
The high-resolution uranium-scintillator calorimeter (CAL) [21] consisted of three parts: the forward (FCAL), the barrel (BCAL) and the rear (RCAL) calorimeters. Each part was subdivided transversely into towers and longitudinally into one electromagnetic section (EMC) and either one (in RCAL) or two (in BCAL and FCAL) hadronic sections (HAC). The smallest subdivision of the calorimeter is called a cell. The CAL energy resolutions, as measured under test-beam conditions, were σ(E)/E = 0.18/ √ E for electrons and σ(E)/E = 0.35/ √ E for hadrons (E in GeV). Timing information from the CAL was available for identification of out-of-time beam-gas events. The energy scale of RCAL had an uncertainty of 1%.
The luminosity-measuring system consisted of three components. They were all used for this analysis and are described in some detail here. Their layout relative to the ZEUS central detector is shown in Fig. 1.
A positron tagger (TAG6) was positioned at approximately Z = −6 m, shown in detail in the inset in Fig. 1. It consisted of a tungsten-scintillator spaghetti calorimeter, segmented into an array of 14 (5) cells with size 6 (4.7) mm in the horizontal (vertical) direction. Scattered positrons were bent into it by the first HERA dipole and quadrupole magnets after the interaction region, with full acceptance for positrons with zero transverse momentum in the approximate energy range 3.8-7.1 GeV with a y range of 0.74-0.86.
At Z = −92 m, photons from the interaction point exited the HERA vacuum system; approximately 9% of photons converted into e + e − pairs in the exit window. Converted pairs were separated vertically by a dipole magnet at Z = −95 m. Pairs from photons in the approximate energy range 15-25 GeV were bent into the luminosity spectrometer (SPEC) [22], located at Z = −104 m. It consisted of a pair of tungsten-scintillator sandwich calorimeters located ≈ 10 cm above and below the plane of the HERA electron ring.
Photons which did not convert in the exit window were detected in the lead-scintillator sandwich photon calorimeter (PCAL) [23], located at Z = −107 m. It was shielded from primary synchrotron radiation by two carbon filters, each approximately two radiation lengths deep. Each filter was followed by an aerogel Cherenkov detector (AERO) to measure the energy of showers starting in the filters.
The luminosity detectors were calibrated using photons and positrons from the bremsstrahlung reaction ep → epγ. The SPEC calorimeters were calibrated at the end of HERA fills by inserting a collimator which constrained the vertical position of e + e − pairs; their energies were then determined by their vertical positions in the calorimeter and the magnetic spectrometer geometry. The energy scale was checked using the endpoint of the bremsstrahlung photon spectrum and agreed with the HERA positron beam energy within 1%. The TAG6 was calibrated using coincidences of TAG6 positrons with calibrated SPEC photons and by constraining the sum of the photon and positron energies to the HERA positron beam energy [24]. The energy ranges of bremsstrahlung positrons accepted by the TAG6 for different running periods were determined with uncertainties of 0.01-0.03 GeV. The PCAL and PCAL+AERO assembly were calibrated using coincidences of PCAL(+AERO) photons with calibrated TAG6 positrons and constraining the sum of their energies to the beam energy.
Using photons from the bremsstrahlung reaction, the luminosity was measured independently with the PCAL and with the SPEC. The systematic uncertainty on the measured luminosity was 1.8%, including a relative uncertainty between different running periods of 1%. The integrated luminosities used for the σ γp tot measurement are listed in Table 1.

Monte Carlo simulation
Monte Carlo (MC) programs were used to simulate physics processes in the ZEUS detector. The Pythia 6.416 [25] generator was used for checking the acceptance of the hadronic final state. The generated events were passed through the Geant 3.21-based [26] ZEUS detector-and trigger-simulation programs [18]. They were reconstructed and analyzed by the same program chain as the data. The mixture of photoproduction processes generated by Pythia was adjusted to describe the CAL energy distributions in the totalcross-section data. The optimized Pythia was also used in the TAG6 flux measurement described in Section 8. That study also used the Djangoh 1.6 [27] generator to simulate deep inelastic processes, where the positron was measured in the CAL.
5 Event selection

Online Selection
Events for the measurement of σ γp tot were collected during special runs with a dedicated trigger requiring activity in RCAL and a positron hit in TAG6. The RCAL requirement was a summed energy deposit in the EMC cells of either more than 464 MeV (excluding the 8 towers immediately adjacent to the beampipe) or 1250 MeV (including those towers). The TAG6 portion of the trigger required at least one cell in the fiducial region of the tagger to have an energy more than 8 times larger than the RMS noise above the pedestal [28]. To reduce the background from events with energy in RCAL and a TAG6 hit caused by a random coincidence with a bremsstrahlung event in the same HERA bunch, the energy in the PCAL, E PCAL , was restricted to E PCAL 14 GeV.

Offline Selection
Offline, clean positron hits in the TAG6 were selected by requiring that the highestenergy cell was not at the edge of the detector. Showers from inactive material in front of the tagger were rejected by a cut on the energy sharing among towers surrounding the tower with highest energy. The position of the positron was reconstructed by a neural network trained on an MC simulation of the TAG6 [29]. The neural-network method was also used to correct the energy of the positrons for a small number of noisy cells, which were excluded. Events from the bremsstrahlung process, selected by requiring a positron in the TAG6 in coincidence with a photon in the SPEC, were used to calibrate the TAG6 with positrons with very small transverse momentum. The energy, E, was determined as a function of the horizontal position, X, and the correlation between X and the vertical position, Y , was also measured. Cuts were placed on E(X) and Y (X) for the photoproduction events to reject positrons with transverse momentum p T 10 MeV, off-momentum beam positrons, and background from beam-gas interactions [29]. The (X, Y ) distribution of positrons from a sample of bremsstrahlung events from the MER, and the Y (X) cuts, are shown in the inset in Fig. 1.
In RCAL, the towers immediately horizontally adjacent to the beam-pipe hole had a large rate from off-momentum beam positrons and debris from beam-gas interactions which satisfied the trigger conditions. In events in which the RCAL cell with highest energy was in one of these towers, the fraction of total RCAL energy, E RCAL , in that tower was required to be below an E RCAL -dependent threshold [29]. This eliminated most of the background and resulted in only about 2.9% loss of signal events.

Data analysis
The number of selected events must be corrected to take into account beam-gas interactions as well as various effects due to random coincidences (overlaps) with bremsstrahlung interactions.
Background from positron beam-gas interactions passing the trigger requirement was determined from non-colliding HERA positron bunches. This sample was subtracted statistically from the colliding HERA bunches by the ratio of currents of ep bunches to e-only bunches. Higher instantaneous luminosity during the HER resulted in a lower fraction of beam-gas backgrounds relative to the LER and MER. The fraction of events subtracted was ≈ 0.2% for the HER and ≈ 1% for the LER and MER data samples.
Photoproduction events associated with the TAG6 hit could have a random coincidence with an event in the same HERA bunch from the bremsstrahlung process, with the bremsstrahlung photon depositing more than 14 GeV in the PCAL and therefore vetoing the event. To account for this loss, accepted events were weighted by a factor determined from the rate of overlaps at the time the event was accepted. The fraction of overlaps is proportional to the instantaneous luminosity, which was higher during the HER relative to the LER and MER. The correction for this effect was ≈ +2.6% for the HER and ≈ +1.2% for the LER and MER data samples.
Another background came from photoproduction events outside the W range of the TAG6 but satisfying the RCAL trigger, with a random coincidence from bremsstrahlung hitting the TAG6. The photon from the bremsstrahlung event may not have been vetoed by the E PCAL 14 GeV requirement due to the limited acceptance and resolution of the PCAL. Such overlaps were studied using the distribution of the energy of the PCAL+AERO, E PCKV ; this offered greatly improved photon energy resolution over the PCAL alone. In addition to the bremsstrahlung events which produced a TAG6 hit, this spectrum also contains photoproduction events associated with the TAG6 hit overlapping in the same HERA bunch with a photon from a random bremsstrahlung event whose positron did not hit the TAG6.
The measured E PCKV distribution from the MER photoproduction data is shown in Fig. 2a, with and without the constraint E PCAL 4 GeV. The large peak near E PCKV = 0 contains most of the tagged photoproduction events. Figure 2b shows the constrained photoproduction data along with two distributions from independent samples of bremsstrahlung events recorded simultaneously with the photoproduction data. One sample required also the TAG6 trigger with all TAG6 cuts applied and provides a sample of TAG6 bremsstrahlung overlaps. The other sample was selected with a trigger requiring E PCAL 4 GeV and provides a sample of the bremsstrahlung overlaps independent of a TAG6 hit. Only signals from the PCAL were available at the trigger level. This results in the smeared thresholds in the E PCKV distributions of Fig. 2b. Note that the quoted thresholds in E PCAL are only approximate, since the trigger conditions were based on uncorrected E PCAL values. All distributions are restricted to 4 E PCAL 14 GeV to account for the threshold of the various data samples. The two distributions of bremsstrahlung events were used to fit the distribution from the photoproduction events; the component from the tagged bremsstrahlung events is the number of tagged bremsstrahlung overlaps in the sample where the photon reached the PCAL. The acceptance of photons in the PCAL was ≈ 85%, with losses due to conversions in the exit window and the limited geometric acceptance from the aperture defined by the HERA beamline elements. The number of overlaps seen in the PCAL, corrected for the PCAL acceptance, is the number of bremsstrahlung overlaps to subtract from the selected photoproduction sample. The uncertainty of 1% on the PCAL acceptance produces a systematic uncertainty of ≈ 0.3% on the subtraction, shown in Table 1.
This subtraction procedure was performed in bins of E RCAL . The measured E RCAL distribution before and after the subtraction is shown in Fig. 3a for the MER sample, together with the systematic uncertainty from the subtraction procedure. The amount subtracted is largest at low values of E RCAL . To reduce the statistical and systematic uncertainties from the subtraction procedure, the signal region for the σ γp tot measurement was restricted to E RCAL > 5 GeV. The fraction of selected events subtracted was 3.6-4.1%. The final numbers of events and their uncertainties are listed in Table 1.

Acceptance of the hadronic final state
The acceptance of the hadronic final state, mainly determined by the trigger requirement of energy deposit in RCAL, is expected to be the same for the three energy settings since the positron beam energy, and thus the photon energy, remained approximately the same throughout. The trigger covers the photon-fragmentation region, which is expected to be W independent due to the phenomenon of limiting fragmentation [30]. Figure 3b shows the measured E RCAL distributions, after all selections and corrections, for all three proton energies. The HER and LER distributions were normalized to the MER distribution for E RCAL > 5 GeV. The three distributions are very similar in shape. The acceptance of the hadronic final state was further investigated using the Pythia MC described in Section 4. Figure 3b shows the E RCAL distribution from the simulation for all three proton energies, normalized to the MER data for E RCAL > 5 GeV. The differences between Pythia and the data are similar for all proton energies. The acceptance for the hadronic final state determined from Pythia was found to be fairly high (above 80% for most of the processes) and as expected W independent within small statistical uncertainties.

Determination of the photon flux
The photon flux accepted by the TAG6, F TAG6 γ , is the integral of the doubly differential flux weighted by the acceptance of the TAG6, A TAG6 , as a function of (y, Q 2 ) The HERA magnets closest to the interaction region provided fields guiding both the proton and positron beams. Accommodation of the different proton energies required changes in the fields. These magnets determined the range of positron energies and scattering angles accepted by the TAG6. The changes in accepted kinematic region required a determination of the photon flux in Eq. (4) separately for each of the proton energies.
In order to measure F TAG6 γ , a sample of photoproduction events with and without a TAG6 tag is needed. This was provided by an independent sample of photoproduction events, selected by a trigger based on E −P Z , explained in detail below. The total ep cross section measured for such a sample is where σ γp (y, Q 2 ) is the photoproduction cross section and A inc (y, Q 2 ) is the acceptance for the selection of the inclusive photoproduction sample. The ep cross section measured for the subset of this sample with a TAG6 tag is The last step follows from the assumption that σ γp (y, Q 2 ) = σ 0 γp and A inc (y, Q 2 ) = A 0 inc are constant over the small (y, Q 2 ) region selected by the TAG6. Then, the fraction of selected events with a TAG6 tag is A MC sample of photoproduction events was then selected in the same way as these data; it has the same total ep cross section as in Eq. (5). A well defined test region in (y, Q 2 ), corresponding to the TAG6 region, was used to select a subset of the MC events. The integrated flux of the test region, F test γ , was evaluated by integrating the function in Eq. (3) numerically over the test region. The cross section for the events in this region has a form similar to that of Eq. (6). The fraction of selected MC events in the test region is Then, from Eqs. (7) and (8) The photoproduction data used for this measurement of the TAG6 flux were collected simultaneously with the total-cross-section data (LER/MER), or during a similar running period (HER). They were collected with a trigger requiring E − P Z > 30 GeV, where E − P Z = i E i (1 − cos θ i ), with the sum running over all CAL cells with energy E i and polar angle θ i . Offline, E − P Z > 31 GeV was required. The cut on RCAL towers adjacent to the beam-pipe hole described in Section 5 was applied. A good tracking vertex was required with |Z vtx | < 25 cm, and timing in RCAL, and FCAL if available, was required to be within 3 ns of that of an ep collision; these cuts reduced beam-induced backgrounds. Scattered positrons in events with Q 2 1 GeV 2 which hit the CAL, with E − P Z ≈ 55 GeV, were identified using a neural network [31]; events with an identified positron were rejected.
A subsample with a positron in the TAG6 was selected following the same procedure described in Section 5; the same bremsstrahlung background correction described in Section 6 was applied. For both the inclusive and tagged samples, a small contribution from positron beam-gas events was subtracted statistically in the same manner as described in Section 6; this amounted to a 1-2.5% correction for the inclusive sample.
The Pythia and Djangoh programs described in Section 4 were used to produce the MC samples. The Pythia samples were restricted to Q 2 < 1.5 GeV 2 , and the Djangoh samples to Q 2 > 1.5 GeV 2 . The MC events were selected with the same criteria as for the data, except for the timing cuts. The Pythia and Djangoh samples were added to give the same fraction of events with and without an identified positron as in the data. The TAG6 test region in Pythia had the same y range as the corresponding data set and Q 2 < 10 −3 GeV 2 . Figure 4 shows the E −P Z distributions for the MER sample. Here E −P Z was calculated using energy-flow objects [32]. The MC distribution was normalized to the data in the region 35 < E − P Z < 50 GeV. The MC gives a fair description of the data; discrepancies between the data and MC are similar for all three proton energies, and have a negligible effect on the relative fluxes determined. The region 35 < E − P Z < 50 GeV was used to determine the ratios in Eqs. (7) and (8) for the flux measurement, avoiding triggerthreshold effects on the low side and unidentified positrons with E − P Z ≈ 55 GeV on the high side.
The experimental data with a TAG6 tag and the MC in the TAG6 test region in Fig. 4 are for the full y range of the TAG6. The MC shows that there is a change in the acceptance of inclusive events (A inc (y) in Eq. (5)) across this range, whereas A inc is taken to be constant in Eq. (6). To minimize the error of this acceptance variation, the TAG6 data were divided into 12 bins according to the horizontal position of the TAG6 cell with highest energy; the MC test region was divided into the corresponding 12 regions of y, based on the TAG6 E(X) relation described in Section 5.2. The flux measurement was performed for these 12 regions and summed. The results are listed in Table 1. The statistical uncertainties on the flux, dominated by the number of TAG6 events, are also shown; the systematic uncertainties are described in the next section. The flux-weighted mean photon energy was calculated over the 12 bins. The mean and ranges of photon energies and W are also listed in Table 1.

Systematic uncertainties
Several sources of systematic uncertainty were investigated besides the uncertainty on the background subtraction already discussed in Section 6. Any uncertainty correlated for all three proton energies largely cancels when ratios of cross sections are determined. The following list provides a summary of the uncertainties and in parentheses the maximum effects on the ratios of cross sections: • uncorrelated uncertainty on the PCAL acceptance affecting the bremsstrahlung background subtraction: 1% (0.3%); • uncertainty on the change of the acceptance of the hadronic final state: As discussed in Section 7, the acceptance has negligible differences for different center-of-mass energies as it is mostly sensitive to the positron energy and hence cancels in the ratios of cross sections at different proton energies. This variation is ignored here: < 0.1% (< 0.1%); • uncertainties on the photon flux: uncorrelated statistical uncertainties from event samples used for flux determination: 1-1.1% (1.1%); uncorrelated uncertainties on the TAG6 photon energy ranges, which result in uncertainties on the flux caused by a steep y dependence of A inc (y, Q 2 ) as discussed in Section 8: 0.01-0.03 GeV (1.1%); correlated uncertainty on the SPEC photon energy scale, introducing uncertainties on the flux through the y dependence of A inc (y, Q 2 ) in Section 8: 1% (0.7%); correlated uncertainty on the CAL energy scale: 1% (0.5%); correlated uncertainty on W and Q 2 dependences of the photoproduction cross section as modeled in Pythia, determined by varying the power of the W dependence and the cutoff mass for Q 2 [33]: 0.2-2% (0.03%); uncorrelated uncertainty due to the statistical uncertainties in the procedure to determine the flux: 1-1.2% (1.2%); • uncorrelated uncertainty on luminosity as described in Section 3: 1% (1%).
All uncorrelated systematic uncertainties were added in quadrature; the largest contributions were from the statistical uncertainties of the flux determination and the luminosity uncertainty. The uncertainties are summarized in Table 1.

Energy dependence of the total cross section
The total photon-proton cross section for one proton energy is given by where N is the measured number of events, L is the integrated luminosity, F TAG6 γ is the fraction of the photon flux tagged by the TAG6, and A RCAL is the acceptance of the hadronic final state for tagged events. Figure 5 shows the measured relative values of σ γp tot as a function of W , where the cross section for HER is normalized to unity. The functional form of Eq. (2) was fit to the relative cross sections, with the parameter W 0 chosen to minimize correlations between the fit parameters A ′ and ǫ. The fit was performed using only the statistical uncertainties, and separately with all the uncorrelated systematic uncertainties (as in Section 9) added in quadrature. The correlated shifts discussed in Section 9 were then applied to the data and the fit repeated; the change in ǫ was negligible. All uncertainties are listed in Table 1. The result for the logarithmic derivative in W 2 of the energy dependence is ǫ = 0.111 ± 0.009 (stat.) ± 0.036 (syst.) .
In the picture in which the photoproduction cross section is ∝ ln 2 (W 2 ) as required by the Froissart bound [7], ǫ ≈ 0.11 is expected, in agreement with the present measurement.
The most recent analysis of all hadronic cross sections using a fit taking into account Pomeron and Reggeon terms [5] yielded a Pomeron intercept of 0.0959 ± 0.0021. This is in agreement with the result presented here.

Summary
The energy dependence of the total photon-proton cross section has been measured using three different center-of-mass energies in the range 194≤W ≤296 GeV. A simple W 2ǫ dependence was assumed and a value of ǫ = 0.111 ± 0.009 (stat.) ± 0.036 (syst.) was determined from a fit to the data. This is the first determination of the energy dependence of the total cross section at high energy in a single experiment. The possible Reggeon contribution, though model-dependent, is expected to be at most a few percent and therefore the measured value of ǫ is compatible with the energy dependence observed in hadron-hadron interactions.
DESY computing staff are especially acknowledged for their success in providing excellent operation of the collider and the data-analysis environment. We thank the DESY directorate for their strong support and encouragement.