Study of the azimuthal asymmetry of jets in neutral current deep inelastic scattering at HERA

The azimuthal distribution of jets produced in the Breit frame in high-Q**2 deep inelastic e+p scattering has been studied with the ZEUS detector at HERA using an integrated luminosity of 38.6 pb-1. The measured azimuthal distribution shows a structure that is well described by next-to-leading-order QCD predictions over the Q**2 range considered, Q**2>125 GeV**2.

1 on leave of absence at University of Erlangen-Nürnberg, Germany 2 supported by the GIF, contract I-523-13.7/97 3 PPARC Advanced fellow 4 supported by the Portuguese Foundation for Science and Technology (FCT) 5

Introduction
Jet production in neutral current (NC) deep inelastic scattering (DIS) at high Q 2 (where Q 2 is the negative of the virtuality of the exchanged boson) provides a testing ground for the theory of the strong interaction between quarks and gluons, namely quantum chromodynamics (QCD). An observable of interest is φ B jet , the azimuthal angle in the Breit frame [1] between the lepton scattering plane, defined by the incoming and outgoing lepton momenta, and the jets produced with high transverse energy (E B T,jet ) in that frame. In the Standard Model, azimuthal asymmetries arising from perturbative QCD effects [2,3,4] are expected in the φ B jet distribution. At leading order (LO), the azimuthal dependence for unpolarised NC DIS at The current-current form of the electromagnetic interactions makes the cross section linear in cos(φ B jet ), cos(2φ B jet ), sin(φ B jet ) and sin(2φ B jet ). However, the coefficients of the terms in sin(φ B jet ) and sin(2φ B jet ) vanish due to time-reversal invariance and the absence of finalstate interactions at the quark-gluon level at LO. The coefficients A, B and C result from the convolution of the matrix elements for the partonic processes with the parton distribution functions (PDFs) of the proton [3,4]. The cos(2φ B jet ) term is expected from the interference of amplitudes arising from the +1 and −1 helicity components of the transversely polarised part of the exchanged photon, whereas the interference between the transverse and longitudinal components gives rise to the cos(φ B jet ) term. In addition, a non-perturbative contribution to the asymmetry arises from the intrinsic transverse momentum of partons in the proton. Since such intrinsic transverse momenta are small [5], this contribution is expected to be negligible for jet production at high E B T,jet [6]. Previous studies of single hadron production in NC DIS observed a cos φ term that was attributed to non-perturbative effects [7]. However, more recently, a ZEUS measurement of the azimuthal dependence of charged hadrons with high transverse momentum in the centre-of-mass system gave evidence for perturbative contributions to the azimuthal asymmetry [8]. This paper presents the first study of the azimuthal distribution of jets with high transverse energy in the Breit frame and the comparison with LO and next-toleading-order (NLO) QCD predictions.

Experimental details
These results are based on data collected in 1996-1997 with the ZEUS detector at HERA, corresponding to an integrated luminosity of 38.6 ± 0.6 pb −1 . The HERA rings were op-erated with protons of energy E p = 820 GeV and positrons of energy E e = 27.5 GeV. The ZEUS detector is described elsewhere [9,10]. The main components used in the present analysis are the central tracking detector [11], positioned in a 1.43 T solenoidal magnetic field, and the uranium-scintillator sampling calorimeter (CAL) [12]. The tracking detector was used to establish an interaction vertex. The CAL covers 99.7% of the total solid angle. It is divided into three parts with a corresponding division in the polar angle 1 , θ, as viewed from the nominal interaction point: forward (FCAL, 2.6 • < θ < 36.7 • ), barrel (BCAL, 36.7 • < θ < 129.1 • ), and rear (RCAL, 129.1 • < θ < 176.2 • ). The smallest subdivision of the CAL is called a cell. Under test-beam conditions, the CAL relative energy resolution is 18%/ E(GeV) for electrons and 35%/ E(GeV) for hadrons. A three-level trigger was used to select the events online [10].
As the analysis follows very closely that of the inclusive jet cross sections in the Breit frame [13], details about the event selection, jet finding, systematic uncertainties and theoretical predictions are not repeated here.
The scattered-positron candidate was identified from the pattern of energy deposits in the CAL [14]. The kinematic region of the analysis was selected by the requirements Q 2 > 125 GeV 2 and −0.7 < cos γ < 0.5, where γ is the angle of the scattered quark in the quark-parton model. Cuts on this angle restrict the phase-space selection in Bjorken x and the inelasticity y due to the relation The longitudinally invariant k T cluster algorithm [15] was used in the inclusive mode [16] to reconstruct the jets in the hadronic final state both in data and in events simulated by Monte Carlo (MC) techniques. In data, the algorithm was applied to the energy deposits measured in the CAL cells after excluding those associated with the scattered-positron candidate. The jet search was performed in the pseudorapidity (η B )-azimuth (φ B ) plane of the Breit frame, where φ B = 0 corresponds to the direction of the scattered positron. The transverse energy of the jets in the Breit frame, E B T,jet , was required to be larger than 8 GeV and the pseudorapidity range was restricted to −2 < η B jet < 1.8. The data sample contained 8523 events, 5073 of which were one-jet, 3262 two-jet, 182 three-jet and 6 fourjet events. The Q 2 range covered by the data sample extended up to Q 2 ∼ 16 000 GeV 2 ; measurements of the azimuthal distribution are presented up to a mean Q 2 value of ∼ 2300 GeV 2 .

Monte Carlo studies and systematic uncertainties
Samples of events were generated to determine the response of the detector to jets of hadrons and to calculate the correction factors necessary to obtain the hadron-level jet cross sections. The generated events were passed through the GEANT 3.13-based [17] ZEUS detector-and trigger-simulation programs [10] and were reconstructed and analysed by the same program chain as the data. The NC DIS events were generated using the LEPTO 6.5 program [18] interfaced to HERACLES 4.6.1 [19] via DJANGOH 1.1 [20]. The HERACLES program includes photon and Z exchanges and first-order electroweak radiative corrections. The QCD cascade was modelled with the ARIADNE 4.08 program [21]. The CTEQ4D [22] parameterisations of the proton PDFs were used. As an alternative, samples of events were generated using the model of LEPTO based on firstorder QCD matrix elements plus parton showers (MEPS). In both cases, fragmentation into hadrons was performed using the JETSET 7.4 program [23]. In these programs, the azimuthal distribution was generated according to the LO QCD calculation.
The comparison of the reconstructed jet variables for the hadronic and the calorimetric jets in simulated events showed that no correction was necessary for φ B jet and that the average resolution was 0.09 radians. The sample of events generated with either ARIADNE or LEPTO-MEPS, after applying the same offline selection as for the data, gave a good description of the measured distributions for both the event and jet variables [13,24]. However, a MC sample of events generated with a uniform azimuthal distribution did not describe the observed φ B jet distribution at detector level. These comparisons establish the presence of an azimuthal modulation in the data.
The cross sections presented here were corrected to the hadron level by applying bin-by-bin corrections to the measured distributions. The correction factors had some dependence on φ B jet due to the cuts applied to remove the effects of QED radiation that lead to a radiated photon from the positron being misidentified as a hadronic jet. The observed φ B jet dependence of the correction factor was not sensitive to the assumed azimuthal distribution in the generator; this was confirmed by the observation that the correction factors based on the MC sample generated with a uniform azimuthal distribution had the same dependence on φ B jet . The MC programs were also used to evaluate the corrections for QED radiative effects, which were negligible for the normalised cross sections.
A detailed study of the systematic uncertainties was carried out. Those that had an effect on the shape of the azimuthal distribution were: • the uncertainty in the absolute energy scale of the jets; • the uncertainty in the MC modelling of the hadronic final state, which was estimated from the differences between ARIADNE and LEPTO-MEPS in correcting the data for detector effects; • the uncertainty in the positron identification, which was estimated by repeating the analysis using an alternative technique [25] to select the scattered-positron candidate.
The relative changes in the normalised differential cross section induced by the variations mentioned above were typically smaller than the statistical uncertainties, which ranged from ∼ 2% at Q 2 ∼ 125 GeV 2 up to 6% at Q 2 ∼ 1000 GeV 2 .

Perturbative QCD calculations
The LO and NLO QCD predictions were obtained using the program DISENT [26]. The number of flavours was set to five and the renormalisation (µ R ) and factorisation (µ F ) scales were chosen to be µ R = E B T,jet and µ F = Q, respectively. The strong coupling constant, α s , was calculated at two loops with Λ  [27] parameterisations of the proton PDFs. The results obtained with DISENT were cross-checked by using the program DISASTER++ [28]. The differences were always smaller than 1%.
The perturbative QCD contribution to the terms B and C in Eq. (1) is large. At LO in α s , two processes contribute to jet production in the Breit frame: QCD-Compton scattering (QCDC, γ * q → qg) and photon-gluon fusion (PGF, γ * g → qq). For the former, the scattered gluon (quark) preferentially appears at φ B jet = 0 (π), whilst for the PGF process, the φ B jet dependence is dominated by the cos(2φ B jet ) term and is very similar for quarks and antiquarks. Thus, the different contributions to the cos(φ B jet ) term from quarks and gluons tend to cancel in the cos(φ B jet ) asymmetry and the predicted azimuthal distribution is very close to A + C cos(2φ B jet ). The NLO QCD correction mainly modifies the normalisation and slightly affects the shape of this prediction. In order to test the QCD prediction for the azimuthal distribution, it is desirable that no cut be applied to the jets in the laboratory frame; otherwise, the azimuthal distribution can be strongly distorted by kinematic effects [4]. For this reason, no such cut was used in the definition of the cross sections.
Since the measurements refer to jets of hadrons, whereas the perturbative QCD calculations refer to partons, the hadronisation effects were investigated by using the models of ARIADNE, LEPTO-MEPS and HERWIG [29]. These effects were negligible [24].

Results
The cross sections presented here include every jet of hadrons in an event with E B T,jet > 8 GeV and −2 < η B jet < 1.8. A detailed comparison of the differential cross sections as functions of Q 2 , E B T,jet and η B jet for inclusive jet production in the same kinematic region as used here was presented in a previous publication [13]. At low Q 2 and low E B T,jet , the NLO QCD calculations fall below the data by ∼ 10%. Nonetheless, the differences between the measurements and calculations are of the same size as the theoretical uncertainties [13]. The comparison of the shape of interest in this publication is facilitated by normalising the predicted cross section and the data to unity. The normalised differential cross-section (1/σ) dσ/dφ B jet for inclusive jet production as a function of φ B jet is shown in Fig. 1 and in Table 1. This distribution has clear enhancements at φ B jet = 0 and φ B jet = π. The NLO QCD calculations with either µ R = E B T,jet or Q reproduce the asymmetry. This comparison constitutes a precise test of the perturbative QCD prediction for the azimuthal distribution since the theoretical uncertainties are small. The dominant theoretical uncertainty arose from terms beyond NLO and was estimated by varying µ R between E B T,jet /2 and 2E B T,jet ; the effect on the amplitude of the modulation of the distribution was ∼ ±1%. This observation complements the ZEUS measurement of the azimuthal dependence of charged hadrons with high transverse momentum in NC DIS [8].
The measurements folded about π, |φ B jet |, in different regions of Q 2 are presented in Fig. 2 and in Table 2. The LO and NLO QCD predictions are compared to the data. The NLO QCD prediction describes the data well, whereas the LO QCD calculations predict a larger asymmetry, particularly in the lower Q 2 intervals. In both cases, the asymmetry is predicted to decrease as Q 2 increases, as a result of the progressive decline of the contribution from the PGF process.
In order to perform a more quantitative study of the asymmetry and its dependence on Q 2 , a fit was performed to the values of (1/σ) dσ/d|φ B jet | both in the data and in the QCD predictions. The functional form was used. The parameter f 1 (f 2 ) represents the contribution of the cos φ B jet (cos 2φ B jet ) term to the total asymmetry. The fitted values of f 1 and f 2 as functions of Q 2 and for the entire sample with Q 2 > 125 GeV 2 are shown in Fig. 3 and listed in Table 3, together with the LO and NLO QCD predictions and their uncertainties. The comparison of the LO QCD calculations for the QCDC and PGF process shows that the asymmetry is predicted to arise mostly from the gluon-induced interactions. The LO QCD predictions do not reproduce the measurements. However, the uncertainty at LO is rather large. The difference between the LO and NLO calculations has been assigned as the theoretical uncertainty of the LO predictions and is ∼ ±0.04 (±0.01) for f 2 (f 1 ). At NLO, the dominant theoretical uncertainty on f 2 (f 1 ) was that due to terms beyond NLO and was estimated by fitting the predictions obtained with µ R = E B T,jet /2 and 2E B T,jet ; it amounted to ∼ ±0.01 (±0.005). The NLO predictions for f 1 and f 2 based on calculations using µ R = Q differed from those using µ R = E B T,jet by as much as the estimated theoretical uncertainty. The NLO QCD predictions are in good agreement with the measured values of f 2 . For f 1 , the observed asymmetry tends to be slightly larger and more negative than that predicted by perturbative QCD. The measurements are consistent with the Q 2 dependence of f 1 and f 2 predicted by NLO QCD.

Summary
A study of the azimuthal asymmetry for inclusive jet production in neutral current deep inelastic e + p scattering in the Breit frame at a centre-of-mass energy of 300 GeV has been presented. Jets of hadrons were identified with the longitudinally invariant k T cluster algorithm in the Breit frame. The normalised cross sections as a function of the azimuthal angle of the jets in the Breit frame are given in the kinematic region Q 2 > 125 GeV 2 and −0.7 < cos γ < 0.5. The cross sections include every jet of hadrons in the event with E B T,jet > 8 GeV and −2 < η B jet < 1.8. The measured azimuthal distribution peaks in the directions along, and opposite to, that of the scattered positron in the Breit frame. The NLO QCD calculations give a good description of the observed azimuthal variation. The dependence of the azimuthal asymmetry on Q 2 is also compatible with NLO QCD.
These measurements constitute a precise test of the perturbative QCD prediction for the azimuthal distribution since the theoretical uncertainties are small.      The fitted values of a) f 1 and b) f 2 from the folded normalised differential cross-section (1/σ) dσ/d|φ B jet | for inclusive jet production with E B T,jet > 8 GeV and −2 < η B jet < 1.8 as a function of Q 2 (points). The fitted values for the entire sample, Q 2 > 125 GeV 2 , are shown on the left-hand side. The inner error bars represent the statistical uncertainty. The outer error bars show the statistical and systematic uncertainties added in quadrature. The results of the fits to the LO and NLO QCD predictions using DISENT and the MRST99 parameterisations of the proton PDFs are shown. The shaded bands represent the uncertainty of the calculations due to the dependence on the renormalisation scale.