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Research article

Exclusive ρ⁰ production in deep inelastic scattering at HERA

ZEUS Collaboration

PMC Physics A 2007, 1:6doi:10.1186/1754-0410-1-6

The electronic version of this article is the complete one and can be found online at: http://www.physmathcentral.com/1754-0410/1/6

Received:	13  August  2007
Accepted:	12  November  2007
Published:	12  November  2007

© 2007 Zeus Collaboration; This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Exclusive ρ⁰ electroproduction at HERA has been studied with the ZEUS detector using 120 pb^-1 of integrated luminosity collected during 1996–2000. The analysis was carried out in the kinematic range of photon virtuality 2 <Q² < 160 GeV², and γ*p centre-of-mass energy 32 <W < 180 GeV. The results include the Q² and W dependence of the γ*p → ρ⁰p cross section and the distribution of the squared-four-momentum transfer to the proton. The helicity analysis of the decay-matrix elements of the ρ⁰ was used to study the ratio of the γ*p cross section for longitudinal and transverse photon as a function of Q² and W. Finally, an effective Pomeron trajectory was extracted. The results are compared to various theoretical predictions.

PACS Codes: 13.60.Hb, 13.60.Le

1 Introduction

Two of the most surprising aspects of high-energy deep inelastic scattering (DIS) observed at the HERA ep collider have been the sharp rise of the proton structure function, F₂, with decreasing value of Bjorken x and the abundance of events with a large rapidity gap in the hadronic final state [1]. The latter are identified as due to diffraction in the deep inelastic regime. A contribution to the diffractive cross section arises from the exclusive production of vector mesons (VM).

High-energy exclusive VM production in DIS has been postulated to proceed through two-gluon exchange [2,3], once the scale, usually taken as the virtuality Q² of the exchanged photon, is large enough for perturbative Quantum Chromodynamics (pQCD) to be applicable. The gluons in the proton, which lie at the origin of the sharp increase of F₂, are also expected to cause the VM cross section to increase with increasing photon proton centre-of-mass energy, W, with the rate of increase growing with Q². Moreover, the effective size of the virtual photon decreases with increasing Q², leading to a flatter distribution in t, the four-momentum-transfer squared at the proton vertex. All these features, with varying levels of significance, have been observed at HERA [4-10] in the exclusive production of ρ⁰, ω, φ, and J/ψ mesons.

This paper reports on an extensive study of the properties of exclusive ρ⁰-meson production,

γ*p → ρ⁰p,

based on a high statistics data sample collected with the ZEUS detector during the period 1996–2000, corresponding to an integrated luminosity of about 120 pb^-1.

2 Theoretical background

Calculations of the VM production cross section in DIS require knowledge of the wave-function of the virtual photon, specified by QED and which depends on the polarisation of the virtual photon. For longitudinally polarised photons, , pairs of small transverse size dominate [3]. The opposite holds for transversely polarised photons, , where configurations with large transverse size dominate. The favourable feature of exclusive VM production is that, at high Q², the longitudinal component of the virtual photon is dominant. The interaction cross section in this case can be fully calculated in pQCD [11], with two-gluon exchange as the leading process in the high-energy regime. For heavy vector mesons, such as the J/ψ or the ϒ, perturbative calculations apply even at Q² = 0, as the smallness of the dipole originating from the photon is guaranteed by the mass of the quarks.

Irrespective of particular calculations [12], in the region dominated by perturbative QCD the following features are predicted:

• the total γ*p → Vp cross section, σ_γ*p, exhibits a steep rise with W, which can be parameterised as σ ~ W^δ, with δ increasing with Q²;

• the Q² dependence of the cross-section, which for a longitudinally polarised photon is expected to behave as Q^-6, is moderated to become Q^-4 by the rapid increase of the gluon density with Q²;

• the distribution of t becomes universal, with little or no dependence on W or Q²;

• breaking of the s-channel helicity conservation (SCHC) is expected.

In the region where perturbative calculations are applicable, exclusive vector-meson production could become a complementary source of information on the gluon content of the proton. At present, the following theoretical uncertainties have been identified:

• the calculation of σ(γ*p → Vp) involves the generalised parton distributions [13,14], which are not well tested; in addition [15], it involves gluon densities outside the range constrained by global QCD analyses of parton densities;

• higher-order corrections have not been fully calculated [16]; therefore the overall normalisation is uncertain and the scale at which the gluons are probed is not known;

• the rapid rise of σ_γ*p with W implies a non-zero real part of the scattering amplitude, which is not known;

• the wave-functions of the vector mesons are not fully known.

In spite of all these problems, precise measurements of differential cross sections separated into longitudinal and transverse components [17], should help to resolve the above theoretical uncertainties.

It is important in these studies to establish a region of phase space where hard interactions dominate over the non-perturbative soft component. If the relative transverse momentum of the pair is small, the colour dipole is large and perturbative calculations do not apply. In this case the interaction looks similar to hadron-hadron elastic scattering, described by soft Pomeron exchange as in Regge phenomenology [18].

The parameters of the soft Pomeron are known from measurements of total cross sections for hadron-hadron interactions and elastic proton-proton measurements. It is usually assumed that the Pomeron trajectory is linear in t:

(1)

The parameter α_ℙ(0) determines the energy behaviour of the total cross section,

and describes the increase of the slope b of the t distribution with increasing W. The value of is inversely proportional to the square of the typical transverse momenta participating in the exchanged trajectory. A large value of suggests the presence of low transverse momenta typical of soft interactions. The accepted values of α_ℙ(0) [19] and [20] are

The non-universality of α_ℙ(0) has been established in inclusive DIS, where the slope of the γ*p total cross section with W has a pronounced Q² dependence [21]. The value of can be determined from exclusive VM production at HERA via the W dependence of the exponential b slope of the t distribution for fixed values of W, where b is expected to behave as

where b₀ and W₀ are free parameters. The value of can also be derived from the W dependence of dσ/dt at fixed t,

(2)

where F(t) is an arbitrary function. This approach has the advantage that no assumption needs to be made about the t dependence. The first indications from measurements of α_ℙ(t) in exclusive J/ψ photoproduction [8,22] are that α_ℙ(0) is larger and is smaller than those of the above soft Pomeron trajectory.

3 Experimental set-up

The present measurement is based on data taken with the ZEUS detector during two running periods of the HERA ep collider. During 1996–1997, protons with energy 820 GeV collided with 27.5 GeV positrons, while during 1998–2000, 920 GeV protons collided with 27.5 GeV electrons or positrons. The sample used for this study corresponds to an integrated luminosity of 118.9 pb^-1, consisting of 37.2 pb^-1 e⁺ p sample from 1996–1997 and 81.7 pb^-1 from the 1998–2000 sample (16.7 pb^-1 e^- and 65.0 pb^-1 e⁺)¹.

A detailed description of the ZEUS detector can be found elsewhere [23,24]. A brief outline of the components that are most relevant for this analysis is given below.

Charged particles are tracked in the central tracking detector (CTD) [25-27]. The CTD consists of 72 cylindrical drift chamber layers, organised in nine superlayers covering the polar-angle² region 15° <θ <164°. The CTD operates in a magnetic field of 1.43 T provided by a thin solenoid. The transverse-momentum resolution for full-length tracks is σ(p_T)/p_T = 0.0058p_T ⊕ 0.0065 ⊕ 0.0014/p_T, with p_T in GeV.

The high-resolution uranium-scintillator calorimeter (CAL) [28-31] covers 99.7% of the total solid angle and consists of three parts: the forward (FCAL), the barrel (BCAL) and the rear (RCAL) calorimeters. Each part is subdivided transversely into towers and longitudinally into one electromagnetic section (EMC) and either one (in RCAL) or two (in BCAL and FCAL) hadronic sections. The CAL energy resolutions, as measured under test-beam conditions, are σ(E)/E = 0.18/ for electrons and σ(E)/E = 0.35/ for hadrons, with E in GeV.

The position of the scattered electron was determined by combining information from the CAL, the small-angle rear tracking detector [32] and the hadron-electron separator [33].

In 1998, the forward plug calorimeter (FPC) [34] was installed in the 20 × 20 cm² beam hole of the FCAL with a small hole of radius 3.15 cm in the centre to accommodate the beam pipe. The FPC increased the forward calorimeter coverage by about one unit in pseudorapidity to η ≤ 5.

The leading-proton spectrometer (LPS) [35] detected positively charged particles scattered at small angles and carrying a substantial fraction, x_L, of the incoming proton momentum; these particles remained in the beam-pipe and their trajectories were measured by a system of silicon microstrip detectors, located between 23.8 m and 90.0 m from the interaction point. The particle deflections induced by the magnets of the proton beam-line allowed a momentum analysis of the scattered proton.

During the 1996–1997 data taking, a proton-remnant tagger (PRT1) was used to tag events in which the proton dissociates. It consisted of two layers of scintillation counters perpendicular to the beam at Z = 5.15 m. The two layers were separated by a 2 mm-thick lead absorber. The pseudorapidity range covered by the PRT1 was 4.3 <η < 5.8.

The luminosity was measured from the rate of the bremsstrahlung process ep → eγp. The photon was measured in a lead-scintillator calorimeter [36-38] placed in the HERA tunnel at Z = -107 m.

4 Data selection and reconstruction

The following kinematic variables are used to describe exclusive ρ⁰ production and its subsequent decay into a π⁺π^- pair:

• the four-momenta of the incident electron (k), scattered electron (k'), incident proton (P), scattered proton (P') and virtual photon (q);

• Q² = -q² = -(k - k')², the negative squared four-momentum of the virtual photon;

• W² = (q + P)², the squared centre-of-mass energy of the photon-proton system;

• y = (P·q)/(P·k), the fraction of the electron energy transferred to the proton in its rest frame;

• M_ππ, the invariant mass of the two decay pions;

• t = (P - P')², the squared four-momentum transfer at the proton vertex;

• three helicity angles, Φ_h, θ_h and φ_h (see Section 9).

The kinematic variables were reconstructed using the so-called "constrained" method [10,39], which uses the momenta of the decay particles measured in the CTD and the reconstructed polar and azimuthal angles of the scattered electron.

The online event selection required an electron candidate in the CAL, along with the detection of at least one and not more than six tracks in the CTD.

In the offline selection, the following further requirements were imposed:

• the presence of a scattered electron, with energy in the CAL greater than 10 GeV and with an impact point on the face of the RCAL outside a rectangular area of 26.4 × 16 cm²;

• E - P_Z > 45 GeV, where E - P_Z = ∑_i(E_i - ) and the summation is over the energies and longitudinal momenta of the final-state electron and pions, was imposed. This cut excludes events with high energy photons radiated in the initial state;

• the Z coordinate of the interaction vertex within ± 50 cm of the nominal interaction point;

• in addition to the scattered electron, exactly two oppositely charged tracks, each associated with the reconstructed vertex, and each having pseudorapidity |η| less than 1.75 and transverse momentum greater than 150 MeV; this excluded regions of low reconstruction efficiency and poor momentum resolution in the CTD. These tracks were treated in the following analysis as a π⁺π^- pair;

• events with any energy deposit larger than 300 MeV in the CAL and not associated with the pion tracks (so-called 'unmatched islands') were rejected [40-42].

In addition, the following requirements were applied to select kinematic regions of high acceptance:

• the analysis was restricted to the kinematic regions 2 <Q² < 80 GeV² and 32 <W < 160 GeV in the 1996–1997 data and 2 <Q² < 160 GeV² and 32 <W < 180 GeV in the 1998–2000 sample;

• only events in the π⁺π^- mass interval 0.65 <M_ππ < 1.1 GeV and with |t| < 1 GeV² were taken. The mass interval is slightly narrower than that used previously [10], in order to reduce the effect of the background from non-resonant π⁺π^- production. In the selected M_ππ range, the resonant contribution is ≈ 100% (see Section 8).

The above selection yielded 22,400 events in the 1996–1997 sample and 49,300 events in the 1998–2000 sample, giving a total of 71,700 events for this analysis.

5 Monte Carlo simulation

The relevant Monte Carlo (MC) generators have been described in detail previously [10]. Here their main features are summarised.

The program ZEUSVM [43] interfaced to HERACLES4.4 [44] was used. The effective Q², W and t dependences of the cross section were parameterised to reproduce the data [42].

The decay angular distributions were generated uniformly and the MC events were then iteratively reweighted using the results of the present analysis for the 15 combinations of matrix elements , (see Section 9).

The contribution of the proton-dissociative process was studied with the EPSOFT [45] generator for the 1996–1997 data and with PYTHIA [46] for the 1998–2000 data. The Q², W and t dependences were parameterised to reproduce the control samples in the data. The decay angular distributions were generated as in the ZEUSVM sample.

The generated events were processed through the same chain of selection and reconstruction procedures as the data, thus accounting for trigger as well as detector acceptance and smearing effects. For both MC sets, the number of simulated events after reconstruction was about a factor of seven greater than the number of reconstructed data events.

All measured distributions are well described by the MC simulations. Some examples are shown in Fig. 1, for the W, Q², t variables, and the three helicity angles, θ_h, φ_h, and Φ_h, and in Fig. 2 for the transverse momentum p_T of the pions, for different Q² bins.

6 Systematics

The systematic uncertainties of the cross section were evaluated by varying the selection cuts and the MC simulation parameters. The following selection cuts were varied:

• the E - P_Z cut was changed within the appropriate resolution of ±3 GeV;

• the p_T of the pion tracks (default 0.15 GeV) was increased to 0.2 GeV;

• the distance of closest approach of the extrapolated track to the matched island in the CAL was changed from 30 cm to 20 cm;

• the π⁺π^--mass window was changed to 0.65–1.2 GeV;

• the Z vertex cut was varied by ±10 cm;

• the rectangular area of the electron impact point on the CAL was increased by 0.5 cm in X and Y ;

• the energy of an unmatched island was lowered to 0.25 GeV and then raised to 0.35 GeV.

The dependence of the results on the precision with which the MC reproduces the performance of the detector and the data was checked by varying the following inputs within their estimated uncertainty:

• the reconstructed position of the electron was shifted with respect to the MC by ±1 mm;

• the electron-position resolution was varied by ±10% in the MC;

• the W^δ-dependence in the MC was changed by varying δ by ±0.03;

• the exponential t-distribution in the MC was reweighted by changing the nominal slope parameter b by ±0.5 GeV^-2;

• the angular distributions in the MC were reweighted assuming SCHC;

• the Q²-distribution in the MC was reweighted by (Q² + )^k, where k = ±0.05.

The largest uncertainty of about ± 4% originated from the variation of the energy of the unmatched islands. All the other checks resulted on average in a 0.5% change in the measured cross sections. All the systematic uncertainties were added in quadrature. In addition, the cross-section measurements have an overall normalisation uncertainty of ±2% due to the luminosity measurement.

7 Proton dissociation

The production of ρ⁰ mesons may be accompanied by the proton-dissociation process, γ*p → ρ⁰N. For low masses M_N of the dissociative system N, the hadronisation products may remain inside the beam-pipe, leaving no signals in the main detector. The contribution of these events to the exclusive ρ⁰ cross section was estimated from MC generators for proton-dissociative processes.

A class of proton dissociative events for which the final-state particles leave observed signals in the surrounding detectors was used to tune the M_N and the t distribution in the MC. In the 1998–2000 running period, these events were selected by requiring a signal in the FPC detector with energy above 1 GeV. The comparison of the data with PYTHIA expectations for the energy distribution in the FPC is shown in Fig. 3(a). The same procedure was repeated with a sample of ρ⁰ events for which the FPC energy was less than 1 GeV and a leading proton was measured in the LPS detector, with the fraction of the incoming proton momentum x_L < 0.95. The comparison between the x_L distribution measured in the data and that expected from PYTHIA is shown in Fig. 3(b), where the elastic peak in the data (x_L > 0.95) is also observed. Also shown in Fig. 3(c–e) is the fraction of proton-dissociative events expected in the selected ρ⁰ sample as a function of Q², W and t. The fraction is at the level of 19%, independent of Q² and W, but increasing with increasing |t|. The combined use of the FPC and LPS methods leads to an estimate of the proton dissociative contribution for |t| < 1 GeV² of 0.19 ± 0.02(stat.) ± 0.03(syst.). The systematic uncertainty was estimated by varying the parameters of the M_N distribution and by changing the FPC cut.

In the 1996–1997 data-taking period, a similar procedure was applied, after tuning the EPSOFT MC to reproduce events with hits in the PRT1 or energy deposits in the FCAL. The proton-dissociative contribution for |t| < 1 GeV² was determined to be 0.07 ± 0.02 after rejecting events with hits in the PRT1 or energy deposits in the FCAL. This number is consistent with that determined from the LPS and FPC because of the different angular coverage of the PRT1.

After subtraction of the proton-dissociative contribution, a good agreement between the cross sections derived from the two data-taking periods was found. For all the quoted cross sections integrated over t, the overall normalisation uncertainty due to the subtraction of the proton-dissociative contributions was estimated to be ± 4% and was not included in the systematic uncertainty. The proton-dissociative contribution was statistically subtracted in each analysed bin, unless stated otherwise.

8 Mass distributions

The π⁺π^--invariant-mass distribution is presented in Fig. 4. A clear enhancement in the ρ⁰ region is observed. Background coming from the decay φ → K⁺ K^-, where the kaons are misidentified as pions, is expected [42] in the region M_ππ < 0.55 GeV. That coming from ω events in the decay channel ω → π⁺π^-π⁰, where the π⁰ remains undetected, contributes [42] in the region M_ππ < 0.65 GeV. Therefore defining the selected ρ⁰ events to be in the window 0.65 <M_ππ < 1.1 GeV ensures no background from these two channels.

In order to estimate the non-resonant π⁺π^- background under the ρ⁰, the Söding parameterisation [47] was fitted to the data, with results shown in the figure. The resulting mass and width values are in agreement with those given in the Particle Data Group [48] compilation. The integrated non-resonant background is of the order of 1% and is thus neglected.

The π⁺π^- mass distributions in different regions of Q² and t are shown in Fig. 5 and Fig. 6, respectively. The shape of the mass distribution changes neither with Q² nor with t. The results of the fit to the Söding parameterisation are also shown. Note that the interference term decreases with Q² as expected but is independent of t, indicating that the non-exclusive background is negligible.

9 Angular distributions and decay-matrix density

The exclusive electroproduction and decay of ρ⁰ mesons is described, at fixed W, Q², M_ππ and t, by three helicity angles: Φ_h is the angle between the ρ⁰ production plane and the electron scattering plane in the γ*p centre-of-mass frame; θ_h and φ_h are the polar and azimuthal angles of the positively charged decay pion in the s-channel helicity frame. In this frame, the spin-quantisation axis is defined as the direction opposite to the momentum of the final-state proton in the ρ⁰ rest frame. In the γ*p centre-of-mass system, φ_h is the angle between the decay plane and the ρ⁰ production plane. The angular distribution as a function of these three angles, W(cos θ_h, φ_h, Φ_h), is parameterised by the ρ⁰ spin-density matrix elements, , where i, k = -1, 0, 1 and by convention α = 0, 1, 2, 4, 5, 6 for an unpolarised charged-lepton beam [49]. The superscript denotes the decomposition of the spin-density matrix into contributions from the following photon-polarisation states: unpolarised transverse photons (0); linearly polarised transverse photons (1,2); longitudinally polarised photons (4); and from the interference of the longitudinal and transverse amplitudes (5,6).

The decay angular distribution can be expressed in terms of combinations, and , of the density matrix elements

where ε is the ratio of the longitudinal- to transverse-photon fluxes and R = σ_L/σ_T, with σ_L and σ_T the cross sections for exclusive ρ⁰ production from longitudinal and transverse virtual photons, respectively. In the kinematic range of this analysis, the value of ε varies between 0.96 and 1 with an average value of 0.996; hence and cannot be distinguished.

The Hermitian nature of the spin-density matrix and the requirement of parity conservation reduces the number of independent parameters to 15 [49]. A 15-parameter fit was performed to the data and the obtained results are listed in Table 1 and shown in Fig. 7 as a function of Q². The published ZEUS results [50] at lower Q² values and the expectations of SCHC, when relevant, are also included. The observed Q² dependence, expected in some calculations [51] and previously reported by H1 [52], is driven by the R dependence on Q² under the assumption of helicity conservation and natural parity exchange. The significant deviation of from zero shows that SCHC does not hold [51] as was observed previously [50,52].

The angular distribution for the decay of the ρ⁰ meson, integrated over φ_h and Φ_h, reduces to

(3)

The element may be extracted from a one-dimensional fit to the cosθ_h distribution. The cosθ_h distributions, for different Q² intervals, are shown in Fig. 8, together with the results of a one-dimensional fit of the form (3). The data are well described by the fitted parameter at each value of Q².

10 Cross section

The measured γ*p cross sections are averaged over intervals listed in the appropriate tables and are quoted at fixed values of Q² and W. The cross sections are corrected for the mass range 0.28 <M_ππ < 1.5 GeV and integrated over the full t-range, where applicable.

10.1 t dependence of σ(γ*p → ρ⁰p)

The determination of σ(γ*p → ρ⁰p) as a function of t for W = 90 GeV was performed by averaging over 40 <W < 140 GeV. The differential cross-section dσ/dt(γ*p → ρ⁰p) is shown in Fig. 9 and listed in Table 2, for different ranges of Q². An exponential form proportional to e^-b|t| was fitted to the data in each range of Q²; the results are shown in Fig. 10. The exponent b, listed in Table 3, decreases as a function of Q². After including the previous results at lower Q² [10,53], a sharp decrease of b is observed at low Q²; the value of b then levels off at about 5 GeV^-2.

A compilation of the value of the slope b for exclusive VM electroproduction, as a function of Q² + M², is shown in Fig. 11. Here M is the mass of the corresponding final state. It also includes the exclusive production of a real photon, the deeply virtual Compton scattering (DVCS) measurement [54]. When b is plotted as a function of Q² + M², the trend of b decreasing with increasing scale to an asymptotic value of 5 GeV^-2, seems to be a universal property of exclusive processes, as expected in perturbative QCD [2].

10.2 Q² dependence of σ(γ*p → ρ⁰p)

The determination of σ(γ*p → ρ⁰p) as a function of Q² for W = 90 GeV was performed by averaging over 40 <W < 140 GeV. The results are shown in Fig. 12 with corresponding values given in Table 4. As expected, a steep decrease of the cross section with Q² is observed. The photoproduction and the low-Q² (< 1 GeV²) measurements are also shown in the figure. An attempt to fit the Q² dependence with a simple propagator term

with the normalisation and n as free parameters, failed to produce results with an acceptable χ². The data appear to favour an n value which increases with Q².

10.3 W dependence of σ(γ*p → ρ⁰p)

The values of the cross section σ(γ*p → ρ⁰p) as a function of W, for fixed values of Q², are plotted in Fig. 13 and given in Table 5. The cross sections increase with increasing W, with the rate of increase growing with increasing Q².

In order to quantify the rate of growth and its significance, the W dependence for each Q² value was fitted to the functional form

σ ~ W^δ.

The resulting δ values are presented as a function of Q² in Fig. 14 and listed in Table 6. For completeness, the δ values from lower Q² are also included. A clear increase of δ with Q² is observed. Such an increase is expected in pQCD, and reflects the change of the low-x gluon distribution of the proton with Q².

To facilitate the comparison, the ZEUS cross-section data as a function of W have been replotted in the Q² bins used by H1 [9]. The results are shown in Fig. 15. The agreement between the two measurements is reasonable. However, in some Q² bins the shape of the W dependence is somewhat different.

A compilation of the value of the slope δ for exclusive VM electroproduction, as a function of Q² + M², is shown in Fig. 16. It also includes the DVCS result [54]. When plotted as a function of Q² + M², the value of δ and its increase with the scale are similar for all the exclusive processes, as expected in perturbative QCD [2].

11 R = σ_L/σ_T and

The SCHC hypothesis implies that and . In this case, the ratio R = σ_L/σ_T can be related to the matrix element,

(4)

and thus can be extracted from the θ_h distribution alone.

If the SCHC requirement is relaxed, then the relation between R and is modified,

with

In the kinematic range of the measurements presented in this paper, the non-zero value of Δ implies a correction of ~3% on R up to the highest Q² value, where it is ~10%, and is neglected.

Under the assumption that Eq. (4) is valid and for values of ε studied in this paper, <ε > = 0.996, the matrix element may be interpreted as

= σ_L/σ_tot,

where σ_tot = σ_L + σ_T. When the value of is close to one, as is the case for this analysis, the error on R becomes large and highly asymmetrical. It is then advantageous to study the properties of itself which carries the same information, rather than R.

The Q² dependence of for W = 90 GeV, averaged over the range 40 <W < 140 GeV, is shown in Fig. 17 and listed in Table 7 together with the corresponding R values. The figure includes three data points at lower Q² from previous studies [10,53]. An initial steep rise of with Q² is observed and above Q² ≃ 10 GeV², the rise with Q² becomes milder. At Q² = 40 GeV², σ_L constitutes about 90% of the total γ*p cross section.

The comparison of the H1 and ZEUS results is presented in Fig. 18 in terms of the ratio R. The H1 measurements are at W = 75 GeV and those of ZEUS at W = 90 GeV. Given the fact that R seems to be independent of W (see below), both data sets can be directly compared. The two measurements are in good agreement.

The dependence of R on M_ππ is presented in Fig. 19 for two Q² intervals. The value of R falls rapidly with M_ππ above the central ρ⁰ mass value. Although a change of R with M_ππ was anticipated to be ~10% [55], the effect seen in the data is much stronger. The effect remains strong also at higher Q², contrary to expectations [55]. Once averaged over the ρ⁰ mass region, the main contribution to R comes from the central ρ⁰ mass value.

The W dependence of , for different values of Q², is shown in Fig. 20 and listed in Table 8. Within the measurement uncertainties, is independent of W, for all Q² values. This implies that the W behaviour of σ_L is the same as that of σ_T, a result which is somewhat surprising. The configurations in the wave function of have typically a small transverse size, while the configurations contributing to may have large transverse size. The contribution to σ_T of large-size configurations, which are more hadron-like, is expected to lead to a shallower W dependence than in case of σ_L. Thus, the result presented in Fig. 20 suggests that the large-size configurations of the transversely polarised photon are suppressed.

The above conclusion can also explain the behaviour of as a function of t, shown in Fig. 21 and presented in Table 9 for two Q² values. Different sizes of interacting objects imply different t distributions, in particular a steeper dσ_T/dt compared to dσ_L/dt. This turns out not to be the case. In both Q² ranges, is independent of t, reinforcing the earlier conclusion about the suppression of the large-size configurations in the transversely polarised photon.

12 Effective Pomeron trajectory

An effective Pomeron trajectory can be determined from exclusive ρ⁰ electroproduction by using Eq. (2). Since the W dependence of the proton-dissociative contribution was established to be the same as the exclusive ρ⁰ sample, no subtraction for proton-dissociative events was performed.

A study of the W dependence of the differential dσ/dt cross section at fixed t results in values of α_ℙ(t), listed in Table 10 and displayed in Fig. 22, for Q² = 3 GeV² (upper plot) and 10 GeV² (lower plot). A linear fit of the form of Eq. (1), shown in the figures, yields values of α_ℙ(0) and shown in Fig. 23, and listed in Table 11. The value of α_ℙ(0) increases slightly with Q², while the value of is Q² independent, within the measurement uncertainties. Its value tends to be lower than that of the soft Pomeron [56].