Scaled momentum spectra in deep inelastic scattering at HERA

Charged particle production has been studied in neutral current deep inelastic ep scattering with the ZEUS detector at HERA using an integrated luminosity of 0.44 fb^-1. Distributions of scaled momenta in the Breit frame are presented for particles in the current fragmentation region. The evolution of these spectra with the photon virtuality, Q^2, is described in the kinematic region 10<Q^2<41000 GeV^2. Next-to-leading-order and modified leading-log-approximation QCD calculations as well as predictions from Monte Carlo models are compared to the data. The results are also compared to e+e- annihilation data. The dependences of the pseudorapidity distribution of the particles on Q^2 and on the energy in the \gamma p system, W, are presented and interpreted in the context of the hypothesis of limiting fragmentation.


Introduction
Quark fragmentation has previously been studied experimentally in deep inelastic ep scattering (DIS) at HERA using observables such as multiplicity moments, scaled momentum distributions and fragmentation functions [1][2][3]. The results were compared to those obtained in e + e − and pp collisions. In general, universal behaviour has been established and scaling violations of the fragmentation functions [4] observed. It has also been observed that perturbative Quantum Chromodynamics (pQCD) calculations using the modified leading-log-approximation (MLLA) [5,6] and assuming local parton-hadron duality (LPHD) [7] do not provide a full description of the data.
In this paper, multiplicity distributions of charged hadrons in the current region in the Breit frame 1 are presented as functions of the virtuality of the exchanged boson, Q 2 per unit of the scaled momentum, x p = 2P Breit /Q, and the variable ln(1/x p ) in bins of Q 2 . Here, P Breit denotes the momentum of a hadron in the Breit frame. The data sample collected with the ZEUS detector between 1996-2007, comprising 0.44 fb −1 , enables the study to be extended to Q 2 as high as 41 000 GeV 2 . Predictions from next-to-leading-order (NLO) QCD calculations that combine full NLO matrix elements with fragmentation functions (FF) obtained from fits to e + e − annihilation data [8][9][10][11] are compared to the measurements. Predictions from MLLA+LPHD [6,12,13] and leading-order plus partonshower Monte Carlo programs are also considered.
In addition, the measurements are compared to previous ep results [1][2][3] and to e + e − annihilation data [14]. The hadronisation in the current region in the Breit frame in ep scattering can be compared directly to the hadronisation in one hemisphere of e + e − annihilation events. There, particle momenta are scaled to half of the centre-of-mass energy, E * = √ s/2. Previous studies on DIS hadronisation [1][2][3]15] have shown good agreement with e + e − annihilation at medium Q 2 . At lower Q 2 , Q 2 < 40 GeV 2 , the two processes were observed to behave differently [2]. This can be explained by higher-order QCD processes such as boson-gluon fusion (BGF) and initial-state QCD Compton radiation occurring as part of the hard interaction in ep scattering but not in e + e − annihilation.
Finally, the density of charged particles is studied as a function of the particle pseudorapidity, η Breit , in bins of Q 2 , and as a function of the total γ ⋆ p centre-of-mass energy, W . The data are used to test the hypothesis of limiting fragmentation [16], in which there has recently been renewed interest, most notably in relativistic heavy ion collisions [17].

Experimental setup
A detailed description of the ZEUS detector can be found elsewhere [18]. A brief outline of the components most relevant for this analysis is given below. Charged particles were tracked in the central tracking detector (CTD) [19], which operated in a magnetic field of 1.43 T provided by a thin superconducting solenoid. The CTD consisted of 72 cylindrical drift-chamber layers, organized in nine superlayers covering the polar-angle 2 region 15 • < θ < 164 • and the radial range from 18.2 cm to 79.4 cm. Before the 2003-2007 running period, the ZEUS tracking system was upgraded with a silicon microvertex detector (MVD) [20].
The high-resolution uranium-scintillator calorimeter (CAL) [21] covered 99.7% of the total solid angle and 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 CAL was called a cell. Under test-beam conditions, the CAL single-particle relative energy The position of the scattered electron was determined by combining information from the CAL, the small-angle rear tracking detector (SRTD) [22] and the presampler (PRES) [23], both mounted on the face of the RCAL.
The luminosity was measured using the Bethe-Heitler process ep → eγp by a luminosity detector which consisted of a lead-scintillator [24][25][26] calorimeter and, after 2002, an independent magnetic spectrometer [27]. The fractional systematic uncertainty on the measured luminosity was 2% and 2.6% for the 1996-2000 and 2004-2007 running periods, respectively.

Event sample
The data presented here were collected with the ZEUS detector at HERA between 1996 and 2007 and correspond to an integrated luminosity of 0.44 fb −1 . During 1995-97 (1998-2007) HERA operated with protons with an energy of E p = 820 GeV (920 GeV) and electrons 3 with an energy of E e = 27.5 GeV, resulting in a centre-of-mass energy of √ s = 300 GeV (318 GeV).
A three-level trigger system [18,28,29] was used to select events online. It relied on the presence of an energy deposition in the CAL compatible with that of a scattered electron. At the third level, an electron, identified using the pattern of energy deposits in the CAL [30] and having an energy larger than 4 GeV, was required.
The kinematic variables, Q 2 and the Bjorken scaling variable, x, as well as the boost vector to the Breit frame were reconstructed using the double angle (DA) method [31] based on the angles of the scattered electron and of the hadronic system. The total energy of the γp-system, W , was calculated using W 2 = Q 2 DA (1 − x DA )/x DA . The tracks used in the analysis had to be associated with the primary interaction vertex and were required to be in the region of high CTD acceptance, |η| < 1.75, where η = − ln(tan θ/2) is the pseudorapidity of the track in the laboratory frame with θ being the polar angle of the measured track with respect to the proton direction. The tracks had to pass through at least three CTD superlayers and were required to have a transverse momentum, P track T > 150 MeV. The details of the event reconstruction are similar to those in a previous ZEUS publication [2] and are described in detail elsewhere [32].
To reconstruct x p , the momentum four-vector of each track was boosted to the Breit frame assuming the particle to have the pion mass.
The analysis of the scaled momenta was restricted to events with Q 2 > 160 GeV 2 . A well reconstructed neutral current DIS sample was selected by requiring the following: • E ′ e > 10 GeV, where E ′ e is the energy of the scattered electron; • y e ≤ 0.95, where y e is the inelasticity, y = Q 2 /sx, estimated from the energy and angle of the scattered electron; this reduces the photoproduction background; • y JB ≥ 0.04, where y JB is estimated by the Jacquet-Blondel method [33]; this rejects events for which the DA method gives a poor reconstruction; and E i is the energy of the i-th calorimeter cell, P Z i is its momentum along the Z axis and the sum runs over all cells; this removes photoproduction and events with initial-state radiation; • |Z vertex | < 50 cm, where Z vertex is the Z component of the position of the primary interaction vertex; this reduces background from events not originating from ep collisions; • the position (X, Y ) of the scattered electron candidates in the RCAL was required to satisfy √ X 2 + Y 2 > 35 cm.
The limiting-fragmentation analysis was extended to events with lower Q 2 values, Q 2 > 10 GeV 2 , using data collected during 1996-2000 corresponding to an integrated luminosity of 77 pb −1 . The same selection as described above was used but, for events with 10 < Q 2 < 160 GeV 2 , the last requirement was modified and one additional requirement introduced: • depending on experimental conditions, the position (X, Y ) of the scattered electron candidate in the RCAL was required to satisfy |X| > 12 cm and |Y | > 6 cm or |X| > 14 cm and |Y | > 14 cm or √ X 2 + Y 2 > 35 cm.
• η max > 3.2, where η max is the pseudorapidity in the laboratory frame of the most forward energy deposit with at least 400 MeV; this removes diffractive events. Even before the cut, for Q 2 > 100 GeV 2 , the contamination with diffractive events is well below 5%, and the resulting corrections to the multiplicity are below 2%. In the lowest Q 2 bin, the contribution of diffractive events would grow up to 10% without the cut.
Only tracks with P Breit Z < 0 enter the analysis of the scaled momenta, whereas this restriction does not apply to the limiting-fragmentation studies. The total number of events is listed in Table 1.

Models and Monte Carlo Simulations
The NLO perturbative QCD calculations considered, combine the full NLO matrix elements with NLO fragmentation functions obtained from fits to e + e − data [8][9][10][11]. The resulting predictions were obtained using the Cyclops program [34].
The MLLA calculations [6,12,[35][36][37] describe parton production in terms of a shower evolution. They depend on two parameters only, the effective QCD scale, Λ eff , and the infrared cutoff scale, Q 0 , at which the parton cascade is stopped. The calculations intrinsically include colour coherence and gluon interference effects. Leading collinear and infrared singularities are removed and energy-momentum conservation is obeyed.
To connect predictions at the parton level to the hadron-level data, LPHD [7] was assumed, which leaves only one free parameter, the hadronisation constant, K h . The conversion from energy to momentum spectra for the final-state hadrons was performed assuming an effective hadron mass, m eff = Q 0 [13]. The input parameters for the calculations were obtained by fits to LEP e + e − data. Conservative uncertainties, equivalent to three standard deviations of the experimental uncertainty, were assumed for the parameters: Q 0 = Λ eff = 270 ± 20 MeV and K h = 1.31 ± 0.03. The Λ eff value agrees with the value Λ eff = 275 ± 4(stat.) +4 −8 (syst.) MeV deduced from a ZEUS analysis of scaled momenta in dijet photoproduction [38]. The usage of input parameters deduced from LEP data is justified by the assumed equivalence of hadronisation in one hemisphere of e + e − annihilation and in the current region of ep interactions in the Breit frame.
The predictions from several Monte Carlo (MC) models were compared to the measurements. Neutral current DIS events were generated using the leading-order QCD Ariadne 4.12 program [39]. The QCD cascade was simulated using the colour-dipole model (CDM) [40] inside Ariadne. Additional samples were generated with the MEPS model of Lepto 6.5 [41]. Both MC programs, Ariadne and Lepto, were also used to calculate detector acceptances and to correct the data to the hadron level. For this purpose they are used with the Djangoh 1.1 [42] interface and QED radiative effects are included using the Heracles 4.6.1 [43] program. Both MC programs use the Lund string model [44] for hadronisation as implemented in Jetset 7.4 [45,46]. Hadrons are assumed stable if their lifetime is larger than 3 × 10 −11 s; their decay products are not considered. This excludes in particular the charged decay products of K 0 and Λ particles.
All generated events were passed through the ZEUS detector-and trigger-simulation programs which are based on GEANT 3 [47]. They were reconstructed and analysed by the same program chain as the data.

Correction procedure and systematic uncertainties
All measured distributions were corrected to the hadron level. The correction factors were calculated bin by bin using MC events. For the scaled momentum spectra, the factors are typically below 1.2 for Q 2 < 5000 GeV 2 , but rise up to 1.5 for higher Q 2 . The corrections for charged-particle densities as a function of the pseudorapidity are of a similar magnitude and approach 1.5 as η Breit increases towards the most positive values measured.
Cross sections, measured separately for each data-taking period, were combined using a standard weighted average [48]. The dependence of the scaled momentum distributions on the variation of the proton-beam energy in the different data samples was determined using MC events; the resulting changes were found to be smaller than 0.5% and were neglected. Finally, the results were corrected to the QED Born level using correction factors obtained from MC, reducing the charged hadron multiplicities by up to 4%.
The systematic uncertainties were investigated separately for data with Q 2 above and below 160 GeV 2 . The numbers in parentheses correspond to the largest variations observed in the scaled momentum spectra. The uncertainties in the limiting-fragmentation distributions are of similar magnitude. For data with Q 2 > 160 GeV 2 , the systematic uncertainties are due to: • imperfections in the simulation affecting the determination of the efficiency of event reconstruction and event selection ( +1 −2 %). This was evaluated by modifying the selection cuts within the experimental resolutions.
• track reconstruction uncertainties close to the borders of acceptance ( +6 −3 %). This was evaluated by − raising (lowering) the cut on P track − including tracks not associated to the primary vertex.
• alignment uncertainties affecting the calculation of the boost vector to the Breit frame ( +3 −2 %). This was evaluated by − varying the polar angle for the scattered electron by ±2 mrad; − varying the polar angles for the hadrons by ±4 mrad.
• assumptions concerning the details of the simulation of the hadronic final state (−4%). This was estimated by using Lepto instead of Ariadne.
For data with Q 2 < 160 GeV 2 , the systematic uncertainties are slightly different and are due to: • imperfections in the simulation affecting the determination of the efficiency of event reconstruction and event selection ( +3 −1 %). • track reconstruction uncertainties ( +5 −0.5 %). • assumptions concerning the details of the simulation of the hadronic final state (+7%).
• an uncertainty about the size of the contribution of diffractive events ( +2 −1 %). This was estimated by varying the η max cut by ±0.2 units.
Further details can be found elsewhere [32]. All individual uncertainties were added in quadrature.

Scaled momentum spectra
Scaled momentum spectra were measured in the current region in the Breit frame as a function of Q 2 in the kinematic range 160 < Q 2 < 40960 GeV 2 and 0.002 < x < 0.75. Results are presented in Figs. 1-4 and Tables 2-9. Also shown in Figs. 1-4 are previously published results for 10 < Q 2 < 160 GeV 2 . In addition, in Fig. 1 data from a previous ZEUS publication [2] are given for 160 < Q 2 < 320 GeV 2 . They agree well with the measurements presented here. The same is true for previously obtained results up to Q 2 = 5120 GeV 2 which are not shown.
The normalised spectrum, 1/N dn ± /d ln(1/x p ), with N being the number of events and n ± being the number of charged particles, is shown in Figs. 1-2. These scaled momentum spectra exhibit a hump-backed form with an approximately Gaussian shape around the peak. The mean charged multiplicities are given by the integrals of the spectra. As Q 2 increases, the multiplicity increases and, in addition, the peak of the spectrum moves to larger values of ln(1/x p ).
In Fig. 1, the predictions of Ariadne and Lepto are compared to the data. They reproduce the main features of the data but do not agree in detail. For the highest Q 2 bin, both models predict too many charged particles at medium and low values of ln(1/x p ). Lepto also predicts too many particles for medium-Q 2 bins while Ariadne predicts too few for low-Q 2 bins.
In Fig. 2, the MLLA+LPHD predictions [6,12] are compared to the data. Too many particles are predicted for the highest-and lowest-Q 2 bins, while at medium Q 2 the data is reasonably well described. At low Q 2 , the observed particle deficit can be interpreted as a significant migration of particles to the target region of the Breit frame; this was also previously observed [1]. At medium Q 2 , the agreement is surprising, because BGF contributes significantly to the cross section and the observed particles should reflect the qq final state which is not included in the prediction. At the highest Q 2 available, the failure of the MLLA prediction probably reflects the fact that the e + e − data used to obtain the input parameters are dominated by Z 0 exchange while, in ep collisions, photon exchange still dominates.
The MLLA+LPHD calculations predict long tails towards large values of ln(1/x p ) over the complete range of Q 2 . These tails are sensitive to the mass correction applied in the calculation [13,49]. The data do not show such tails in the predicted size. A better description of the large ln (

Scaling violation
As the energy scale, Q, increases, the phase space for soft gluon radiation increases, leading to a rise of the number of soft particles with small x p . These scaling violations can be seen when the data are plotted in bins of x p as a function of Q 2 . Figure 3 and Tables 7-9 show that the number of charged particles increases with Q 2 at low x p and decreases with Q 2 at high x p . Neither Lepto nor Ariadne provides a good description of this Q 2 dependence over the whole range of x p . Figure 4 shows the data together with four NLO+FF QCD predictions [8][9][10][11] for x p > 0.1, where theoretical uncertainties are small and the predictions not too strongly affected by hadron-mass effects which are not included in the calculations [34]. The fragmentation functions (FF) used in all four calculations were extracted from e + e − data. The four predictions are similar in shape and have similar uncertainties. The uncertainties are only illustrated for the calculation of Kretzer [8]. The NLO calculations also do not provide a good description of the data. Too many particles are predicted at small x p and too few at large x p . In general, the scaling violations predicted are not strong enough. Figure 5 shows the same data as Fig. 4 together with results from H1 [3] and from e + e − experiments [14]. For a proper comparison, the the particle momenta from e + e − data were scaled to half of the centre-of-mass energy as discussed in the introduction and the scale was set to Q = 2 E beam , where E beam is the beam energy. In addition, corrections for the different treatment of K 0 and Λ decays were applied. The overall agreement between the different data sets supports fragmentation universality. The presentation of the data using a linear scale as presented in Fig. 6 does, however, show some significant differences between e + e − and ep, in particular around the Z 0 mass at 0.02 < x p < 0.2 and at low Q 2 at 0.1 < x p < 0.2.

Limiting fragmentation
The concept of limiting fragmentation [16] is based on the assumption that a Lorentzcontracted object passes through another object at rest, leaving behind an excited state with properties depending neither on the energy nor the identity of the passing object. This excited state fragments into particles in a restricted window of rapidity, called the limiting-fragmentation region. In this region, the limiting-fragmentation hypothesis predicts that the density of charged particles per unit of rapidity depends only on W .
Limiting fragmentation has been observed in a variety of hadronic collisions [50,51], including nucleus-nucleus interactions [17]. It was observed that in the region of limiting fragmentation the particle density increases linearly with the rapidity before reaching a plateau. The slope of the increase did not show a W dependence, but the height of the plateau increased with W . These features are illustrated in Fig. 7. Bialas and Jezabek [52] proposed a statistical model to explain the missing W dependence of the slopes. In this model, soft particle production in hadronic collisions is described in terms of multiple gluon exchanges between partons of the colliding hadrons and by the subsequent radiation of hadronic clusters.
The application of the limiting-fragmentation hypothesis to e + e − annihilations is not straight-forward. However, again a behaviour as illustrated in Fig. 7 was observed, only in this case the slopes increase with W [51,53].
In the case of ep collisions, the passing object is the proton while the virtual photon exchanged in the interaction is the object "at rest". It is assumed to be the excited hadron which fragments [53].  and Tables 10-15 present the density of charged particles per unit of pseudorapidity, η Breit , for 10 < Q 2 < 10240 GeV 2 in bins of Q 2 and W as listed in Table 1. A region of linear rise and the onset of a plateau are observed in all bins. This supports the applicability of the hypothesis of limiting fragmentation to the case of ep collisions. However, for low Q 2 , the plateau is only reached in the target region, η Breit > 0.
The slopes in the region of linear rise do not depend significantly on either Q 2 or W , as also demonstrated in Figs. 10 and 11. The lack of a W dependence indicates that the model of Bialas and Jezabek is also applicable for ep collisions.
Figures. 8 and 9 also show predictions from Ariadne and Lepto. Overall, Ariadne provides reasonable predictions for the whole range in Q 2 and W . Lepto, however, predicts a sizeable increase in the height of the plateau with Q 2 and W which is not observed in the data. The predictions in the plateau region are sensitive to the input parameters used in the fragmentation functions. The usage of input parameters derived from SMC data [54] in Lepto results in the prediction of a softer spectrum, reflected in a charged-particle density of up to 30 % too high.
The hypothesis of limiting fragmentation was further tested by studying the chargedparticle densities in the rest frame of the fragmenting object, i.e. the virtual photon. The η Breit distributions were rebinned by shifting event by event all entries by ln Q/m π , thus scaling the available energy to the pion mass. The resulting distributions are shown in Fig. 11. The distributions are very similar but for Q 2 > 160 GeV 2 a slightly larger slope is observed. This is a region where the BGF contribution is decreasing. In general, the observations support the hypothesis of limiting fragmentation and the model of Bialas and Jezabek. This indicates that, even at high Q 2 , soft processes are involved in the fragmentation and a statistical approach is justified.

Conclusions
Scaled momentum spectra have been measured in NC DIS for the current region in the Breit frame over the large range of Q 2 from 10 GeV 2 to 40960 GeV 2 . Large scaling violations are observed. Comparing the data to e + e − results generally supports the concept of quark-fragmentation universality. Neither MLLA+LPHD nor NLO+FF calculations describe the data well. A better, albeit not prefect description is provided by the Ariadne program.
The limiting-fragmentation hypothesis and the statistical model of Bialas and Jezabek were tested by studying the density of charged particles as a function of the pseudorapidity, η Breit , over the range of 10 < Q 2 < 10240 GeV 2 . A region of linear rise and the onset of a plateau are observed over the whole range in Q 2 and support the limiting fragmentation hypothesis. The independence of the slopes on W supports the statistical approach of Bialas and Jezabek.     Table 4: The bin-averaged scaled momentum spectra, 1/N dn ± /d ln(1/x p ), for 2560 < Q 2 < 10240 GeV 2 . The first uncertainty is statistical, the second systematic.  Table 5: The bin-averaged scaled momentum spectra, 1/N dn ± /d ln(1/x p ), for 10240 < Q 2 < 20480 GeV 2 . The first uncertainty is statistical, the second systematic.   Table 7: The number of charged particles per event and unit of x p , 1/N n ± /∆x p , as a function of Q 2 in bins of x p with widths ∆x p . The first uncertainty is statistical, the second systematic.   Table 9: The number of charged particles per event and unit of x p , 1/N n ± /∆x p , as a function of Q 2 in bins of x p with widths ∆x p . The first uncertainty is statistical, the second systematic.  Table 10: The charged-particle density, 1/N dn ± /dη Breit , for 5120 < Q 2 < 10240 GeV 2 and 3 bins in W . The first uncertainty is statistical, the second systematic.   Table 11: The charged-particle density, 1/N dn ± /dη Breit , for 2560 < Q 2 < 5120 GeV 2 and 3 bins in W . The first uncertainty is statistical, the second systematic.   Table 12: The charged-particle density, 1/N dn ± /dη Breit , for 1280 < Q 2 < 2560 GeV 2 and 4 bins in W . The first uncertainty is statistical, the second systematic.   Table 13: The charged-particle density, 1/N dn ± /dη Breit , for 640 < Q 2 < 1280 GeV 2 and 4 bins in W . The first uncertainty is statistical, the second systematic.  Table 14: The charged-particle density, 1/N dn ± /dη Breit , for 320 < Q 2 < 640 GeV 2 and 3 bins in W . The first uncertainty is statistical, the second systematic.   Table 15: The charged-particle density, 1/N dn ± /dη Breit , for 160 < Q 2 < 320 GeV 2 and 4 bins in W . The first uncertainty is statistical, the second systematic.
η Breit 80 < W < 120 GeV 120 < W < 160 GeV 160 < W < 200 GeV  Table 18: The charged-particle density, 1/N dn ± /dη Breit , for 40 < Q 2 < 50 GeV 2 and 3 bins in W . The first uncertainty is statistical, the second systematic.       Figure 5: The number of charged particles per event per unit of x p , 1/N n ± /∆x p , as a function of Q in x p bins with width ∆x p . Also shown are data from H1 [3] and e + e − [14]. The dots (triangles) represent the new (previous) ZEUS measurement, the squares the H1 data and the inverted triangles the e + e − data. The inner error bars, where visible, indicate statistical uncertainties, the outer statistical and systematic uncertainties added in quadrature. The three lowest x p bins are scaled by factors of 30, 5 and 2, respectively.  The normalised charged-particle density per unit of η Breit , 1/N dn ± /dη Breit , for different (W, Q 2 ) bins for Q 2 > 160 GeV 2 . The dots represent the ZEUS measurement. The error bars, where visible, indicate statistical and systematic uncertainties added in quadrature. The dashed and dotted lines represent the Lepto and Ariadne predictions, respectively. The normalised charged-particle density per unit of η Breit , 1/N dn ± /dη Breit , for different (W, Q 2 ) bins for 10 < Q 2 < 160 GeV 2 . Other details as in Fig. 8 Figure 11: The normalised charged-particle density per unit of η Breit , 1/N dn ± /dη Breit , for 2 bins in W and 5 bins in Q 2 . The distributions were rebinned by shifting the entries for each event by Y=ln Q/m π as explained in the text. The error bars, where visible, indicate statistical and systematic uncertainties added in quadrature.