Inclusive-jet cross sections in NC DIS at HERA and a comparison of the kT, anti-kT and SIScone jet algorithms

For the first time, differential inclusive-jet cross sections have been measured in neutral current deep inelastic ep scattering using the anti-kT and SIScone algorithms. The measurements were made for boson virtualities Q^2>125 GeV^2 with the ZEUS detector at HERA using an integrated luminosity of 82 pb^-1 and the jets were identified in the Breit frame. The performance and suitability of the jet algorithms for their use in hadron-like reactions were investigated by comparing the measurements to those performed with the kT algorithm. Next-to-leading-order QCD calculations give a good description of the measurements. Measurements of the ratios of cross sections using different jet algorithms are also presented; the measured ratios are well described by calculations including up to O(alphas^3) terms. Values of alphas(Mz) were extracted from the data; the results are compatible with and have similar precision to the value extracted from the kT analysis.

h now at Rockefeller University, New York, NY 10065, USA i now at DESY group FS-CFEL-1 j now at DESY group FEB, Hamburg, Germany k also at Moscow State University, Russia l now at University of Liverpool, United Kingdom m on leave of absence at CERN, Geneva, Switzerland n now at CERN, Geneva, Switzerland o now at Goldman Sachs, London, UK p also at Institute of Theoretical and Experimental Physics, Moscow, Russia q also at INP, Cracow, Poland r also at FPACS, AGH-UST, Cracow, Poland s partially supported by Warsaw University, Poland t partially supported by Moscow State University, Russia u also affiliated with DESY, Germany v now at Japan Synchrotron Radiation Research Institute (JASRI), Hyogo, Japan w also at University of Tokyo, Japan x now at Kobe University, Japan y supported by DESY, Germany z partially supported by Russian Foundation for Basic Research grant No. 05-02-39028-NSFC-a † deceased aa STFC Advanced Fellow ab nee Korcsak-Gorzo ac This material was based on work supported by the National Science Foundation, while working at the Foundation. ad also at Max Planck Institute, Munich, Germany, Alexander von Humboldt Research Award ae now at Nihon Institute of Medical Science, Japan af now at SunMelx Co. Ltd., Tokyo, Japan ag now at Osaka University, Osaka, Japan ah now at University of Bonn, Germany ai also at Lódź University, Poland aj member of Lódź University, Poland ak now at Lund University, Lund, Sweden al also at University of Podlasie, Siedlce, Poland

Introduction
Jet production at high transverse energies in neutral-current (NC) deep inelastic ep scattering (DIS) at HERA provides unique tests of perturbative QCD (pQCD) in a cleaner hadron-like enviroment than that encountered in hadron-hadron collisions. In NC DIS, the jet search is performed in the Breit frame [1] in which, at leading order (LO) in the strong coupling constant, α s , the boson-gluon-fusion (V * g → qq, with V = γ, Z 0 ) and QCD-Compton (V * q → qg) processes give rise to two hard jets with opposite transverse momenta.
In previous publications, the observables used to test pQCD included inclusive-jet [2][3][4][5][6][7][8][9], dijet [4,6,[9][10][11] and multijet [9,[12][13][14] cross sections and the internal structure of jets [15][16][17][18] defined using the k T cluster algorithm [19] in the longitudinally invariant inclusive mode [20]. These studies demonstrated that this jet algorithm results in the smallest uncertainties in the reconstruction of jets in ep collisions. The k T algorithm is well suited for ep collisions and yields infrared-and collinear-safe cross sections at any order of pQCD. However, it might not be best suited to reconstruct jets in hadronhadron collisions, such as those at the LHC. In order to optimise the reconstruction of jet observables in such environments, new algorithms were recently developed, like the anti-k T [21], a recombination-type jet algorithm, and the "Seedless Infrared-Safe" cone (SIScone) [22] algorithms. The measurements of jet cross sections in NC DIS using these algorithms provide additional tests of QCD and also a test of their performance with data in a well understood hadron-induced reaction.
This letter presents measurements of differential inclusive-jet cross sections as a function of the jet transverse energy in the Breit frame, E jet T,B , and the virtuality of the exchanged boson, Q 2 , based on the anti-k T and SIScone jet algorithms. Differential inclusive-jet cross sections as functions of E jet T,B in different regions of Q 2 are also presented. The analysis is based on the same data sample which was used in recent publications [4,5] to make crosssection measurements with the k T jet algorithm. The results are compared with nextto-leading-order (NLO) QCD calculations using recent parameterisations of the parton distribution functions (PDFs) of the proton [23][24][25] and with previous measurements based on the k T algorithm. Furthermore, measurements of the ratios of inclusive-jet cross sections using different jet algorithms are presented and compared to pQCD calculations including up to O(α 3 s ) terms. In addition, new determinations of α s (M Z ) were obtained so as to quantify the performance of the anti-k T and SIScone algorithms in comparison with that of the k T analysis.

Jet algorithms
The study of jet production in hadronic reactions has been well established as a testing ground of pQCD. Such tests rely on a trustworthy reconstruction of the topology of the final-state partons. This is best done by using jet algorithms with analogous implementation in experiment and theory. On the experimental side, the results of the application of a jet algorithm should not depend significantly on the presence of soft particles or particles which undergo strong decays. On the theoretical side, they should be infrared and collinear safe, so that order-by-order pQCD calculations can be performed. A close correspondence between the jets and the final-state partons, so that hadronisation corrections are small, and suppression of beam-remnant jet contributions, when necessary, are also required.
There are two different methods to reconstruct jets from the final-state particles: clusterand cone-type algorithms. Cluster algorithms, such as JADE [26] or k T , are based on successive recombinations of particles. They are usually applied in e + e − experiments, where the initial state is governed by QED and the final-state hadrons arise uniquely from the short-distance interaction so that all hadrons observed should be associated with the hard process and thus clustered. Cone algorithms, such as the iterative cone algorithm [27], are based on a maximisation of the energy density within a cone of fixed size and are usually applied in hadron collisions, where the initial state consists of coloured partons. The initial parton carries only a fraction of the momentum of the parent hadron, and the spectator partons lead to the presence of remnant jets and the underlying event, which leads to the production of soft hadrons not correlated with the hard interaction. Thus, not all final-state particles would be associated to jets.
In NC DIS, jets are usually defined using the transverse-energy flow in the pseudorapidity (η B )−azimuth (φ B ) plane of the Breit frame. The procedure to reconstruct jets with the k T algorithm from an initial list of objects (e.g. final-state partons, final-state hadrons or energy deposits in the calorimeter) is described below in some detail. In the following discussion, E i T,B denotes the transverse energy, η i B the pseudorapidity and φ i B the azimuthal angle of object i in the Breit frame. For each pair of objects, the quantity is calculated. For each individual object, the distance to the beam, d i = (E i T,B ) 2 , is also calculated. If, of all the values {d ij , d i }, d kl is the smallest, then objects k and l are combined into a single new object. If, however, d k is the smallest, then object k is considered a jet and removed from the sample. The procedure is repeated until all objects are assigned to jets.
The k T algorithm is infrared and collinear safe to all orders in pQCD. As a result of the distance between objects defined in Eq. (1), the jets thus obtained have irregular shapes [28], in contrast to jets defined using a cone algorithm. From an experimental point of view, it may be desirable for jets to have a well-defined shape, e.g. to calibrate the jet-energy scale or to estimate the underlying-event contribution. On the other hand, cone algorithms produce jets that have an approximate circular shape in the pseudorapidityazimuth plane. However, most implementations of cone algorithms rely on "seeds" to start the search for stable cones, which results in unsafe infrared and collinear behaviour beyond a given order in pQCD.
Recently, new jet algorithms have been proposed which produce jets with an approximate circular shape but maintain the infrared and collinear safety to all orders in pQCD. Two approaches have been followed. The SIScone jet algorithm has been developed to overcome the seed problem. A new recombination-type algorithm, the anti-k T , has been devised which, after a modification of the distance as defined in Eq. (1), yields circular jets.
The SIScone algorithm consists of two steps. First, for a given set of initial objects, all stable cones are identified; cones are classified as stable by the coincidence of the cone axis with that defined by the total momentum of the objects contained in the given cone of radius R in the η B −φ B plane of the Breit frame. In this procedure, no seed is used. Stable cones are then discarded if their transverse momentum is below a given threshold, p t,min . For each selected stable cone, the scalar sum of the transverse momentum of the objects associated to it,p t , is defined. Second, overlapping cones are identified and subsequently split or merged according to the following procedure. Two cones are merged if the scalar sum of the transverse momentum of the objects shared by the two cones exceeds a certain fraction f of the lowest-p t cone; otherwise, two different cones are considered and the common objects are assigned to the nearest cone. This jet algorithm is infrared and collinear safe to all orders in pQCD.
The k T and anti-k T algorithms are identical except for a modified distance measure, and the distance to the beam, which is defined as This procedure preserves the infrared and collinear safety of the k T algorithm and, in addition, gives rise to circular jets.
For the measurements presented in this letter, the parameter R was set to unity and the jet variables were defined according to the Snowmass convention [29]. In the application of the SIScone algorithm, the fraction f was set to 0.75 and the transverse momentum threshold p t,min was set to zero.

Experimental set-up
A detailed description of the ZEUS detector can be found elsewhere [30,31]. 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) [32], 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, organised in nine superlayers covering the polar-angle 1 region 15 • < θ < 164 • . The transverse-momentum resolution for full-length tracks can be parameterised as σ(p T )/p T = 0.0058p T ⊕ 0.0065 ⊕ 0.0014/p T , with p T in GeV. The tracking system was used to measure the interaction vertex with a typical resolution along (transverse to) the beam direction of 0.4 (0.1) cm and to cross-check the energy scale of the calorimeter.
The high-resolution uranium-scintillator calorimeter (CAL) [33] 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 and either one (in RCAL) or two (in BCAL and FCAL) hadronic sections. The smallest subdivision of the calorimeter was called a cell. Under test-beam conditions, the CAL single-particle relative energy resolutions were σ(E)/E = 0.18/ √ E for electrons and σ(E)/E = 0.35/ √ E for hadrons, with E in GeV.
The luminosity was measured from the rate of the bremsstrahlung process ep → eγp. The resulting small-angle energetic photons were measured by the luminosity monitor [34], a lead-scintillator calorimeter placed in the HERA tunnel at Z = −107 m.

Data selection
The data were collected during the running period 1998-2000, when HERA operated with protons of energy E p = 920 GeV and electrons 2 of energy E e = 27.5 GeV, and correspond to an integrated luminosity of 81.7 ± 1.8 pb −1 , of which 16.7 pb −1 (65.0 pb −1 ) was for e − p (e + p) collisions.
Neutral current DIS events were selected using the same criteria as reported in recent publications [4,5]. The main steps are briefly listed below.
The scattered-electron candidate was identified from the pattern of energy deposits in the CAL [35]. The energy (E ′ e ) and polar angle of the electron candidate were determined from the CAL measurements. The Q 2 variable was reconstructed using the double-angle method [36]. The angle γ h , which corresponds to the angle of the scattered quark in the quark-parton model, was reconstructed using the hadronic final state [36].
The main requirements imposed on the data sample were: an electron candidate with E ′ e > 10 GeV; a vertex position along the beam axis in the range |Z| < 34 cm; 38 < (E − P Z ) < 65 GeV, where E is the total energy as measured by the CAL, E = i E i , and P Z is the Z component of the vector P = i E iri ; in both cases the sum runs over all CAL cells, E i is the energy of the CAL cell i andr i is a unit vector along the line joining the reconstructed vertex and the geometric centre of the cell i; Q 2 > 125 GeV 2 ; and | cos γ h | < 0.65. In this selected sample, contamination from non-ep interactions and other physics processes was negligible.
The anti-k T and SIScone algorithms 3 were both used to reconstruct jets in the hadronic final state both in data and in Monte Carlo (MC) simulated events (see Section 5). In the data, the algorithms were applied to the energy deposits measured in the CAL cells after excluding those associated with the scattered-electron candidate. After reconstructing the jet variables in the Breit frame, the massless four-momenta were boosted into the laboratory frame, where the transverse energy (E jet T,LAB ) and the pseudorapidity (η jet LAB ) of each jet were calculated. Energy corrections [2,15,38] were then applied to the jets in the laboratory frame, separately for the anti-k T and SIScone jet samples. In addition, events were removed from the samples if any of the jets was in the backward region of the detector (η jet LAB < −2) and jets were not included in the final samples if E jet T,LAB < 2.5 GeV. Only events with at least one jet in the pseudorapidity range −2 < η jet B < 1.5 were kept for further analysis. The final data samples with at least one jet satisfying E jet T,B > 8 GeV contained 18847 events (12575 one-jet, 6126 two-jet, 145 three-jet and one four-jet) in the anti-k T sample and 19486 events (13081 one-jet, 6252 two-jet, 152 three-jet and one four-jet) in the SIScone sample.

Monte Carlo simulation
Samples of events were generated to determine the response of the detector to jets of hadrons and to compute the correction factors necessary to obtain the hadron-level jet cross sections. The hadron level is defined by those hadrons with lifetime τ ≥ 10 ps.
The generated events were passed through the Geant 3.13-based [39] ZEUS detectorand trigger-simulation programs [31]. They were reconstructed and analysed by the same program chain as the data.
Neutral current DIS events including electroweak radiative effects were simulated using the Heracles 4.6.1 [40] program with the Djangoh 1.1 [41] interface to the QCD programs. The QCD cascade was simulated using the colour-dipole model (CDM) [42] including the LO QCD diagrams as implemented in Ariadne 4.08 [43] and, alternatively, with the MEPS model of Lepto 6.5 [44]. The CTEQ5D [45] parameterisations of the proton PDFs were used for these simulations. Fragmentation into hadrons was performed using the Lund string model [46] as implemented in Jetset [47,48].
The jet search was performed on the simulated events using the energy measured in the CAL cells in the same way as for the data. The same jet algorithms were also applied to the final-state particles (hadron level) and to the partons available after the parton shower (parton level). Additional MC samples were used to correct the measured cross sections for QED radiative effects and the running of α em .

Acceptance corrections and experimental uncertainties
To measure the cross sections, the E jet T,B and Q 2 distributions in the data were corrected for detector effects using bin-by-bin correction factors determined with the MC samples. These correction factors took into account the efficiency of the selection criteria and the purity and efficiency of the jet reconstruction. For this approach to be valid, the uncorrected distributions of the data must be well described by the MC simulations at the detector level. This condition was satisfied by both the Ariadne and Lepto-MEPS MC samples. The average between the acceptance-correction values obtained with Ariadne and Lepto-MEPS was used to correct the data to the hadron level. The deviations in the results obtained by using either Ariadne or Lepto-MEPS to correct the data from their average were taken to represent systematic uncertainties of the effect of the QCDcascade model in the corrections (see below). The acceptance-correction factors differed from unity by typically less than 10%.
The following sources of systematic uncertainty were considered for the measured cross sections: • the uncertainty in the absolute energy scale of the jets was estimated to be ±1% for E jet T,LAB > 10 GeV and ±3% for lower E jet T,LAB values [38,49,50]. The resulting uncertainty on the cross sections was about ±5%; • the differences in the results obtained by using either Ariadne or Lepto-MEPS to correct the data for detector effects were taken to represent systematic uncertainties. The resulting uncertainty on the cross sections was typically below ±3%; • the uncertainty due to the selection cuts was estimated by varying the values of the cuts within the resolution of each variable; the effect on the cross sections was typically below ±3%; • the uncertainty on the reconstruction of the boost to the Breit frame was estimated by using the direction of the track associated with the scattered electron instead of that derived from its impact position in the CAL. The effect on the cross sections was typically below ±1%; • the uncertainty in the absolute energy scale of the electron candidate was estimated to be ±1% [51]. The resulting uncertainty on the cross sections was below ±1%; • the uncertainty in the cross sections due to that in the simulation of the trigger was negligible.
The systematic uncertainties were similar for both jet algorithms. Those not associated with the absolute energy scale of the jets were added in quadrature to the statistical uncertainties and are shown in the figures as error bars. The uncertainty due to the absolute energy scale of the jets is shown separately as a shaded band due to the large bin-to-bin correlation. In addition, there was an overall normalisation uncertainty of 2.2% from the luminosity determination, which is not included in the figures.

QCD calculations and theoretical uncertainties
Next-to-leading-order (O(α 2 s )) QCD calculations were compared to the measured inclusivejet cross sections. The calculations were obtained using the program Disent [52], in the MS renormalisation and factorisation schemes using a generalised version [52] of the subtraction method [53]. 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 jet T,B and µ F = Q, respectively. The strong coupling constant was calculated at two loops with Λ Measurements are also presented of the ratio of inclusive-jet cross sections based on different jet algorithms. Although the calculations for inclusive-jet cross sections can be made at present only up to O(α 2 s ) (see Fig. 1), the differences 5 of cross sections using different jet algorithms can be predicted up to O(α 3 s ) using the program Nlojet++ [55]. The same parameter settings as defined above were used for the Nlojet++ predictions. The predicted ratio (dσ anti−k T /dX)/(dσ k T /dX), where X = Q 2 or E jet T,B , was calculated as since differences between the anti-k T and k T algorithms appear first for final states with four partons and were evaluated using the tree-level eq → eqggg, eq → eqgqq, eg → eqqqq and eg → eggqq subprocesses (see Fig. 1). The predicted ratio (dσ SIScone /dX)/(dσ k T /dX) was calculated as since differences between the SIScone and k T algorithms appear first for final states with three partons and were evaluated using the tree-level three-parton and four-parton subprocesses and the one-loop three-parton configurations (see Fig. 1). Equation (4) also applies to the predicted ratio (dσ anti−k T /dX)/(dσ SIScone /dX).
Since the measurements refer to jets of hadrons, whereas the NLO QCD calculations refer to jets of partons, the predictions were corrected to the hadron level using the MC models. The multiplicative correction factor (C had ) was defined as the ratio of the cross section for jets of hadrons over that for jets of partons, estimated by using the MC programs described in Section 5. The mean of the ratios obtained with Ariadne and Lepto-MEPS was taken as the value of C had . The value of C had for the inclusive-jet cross sections differs from unity by less than 5% [5], 6% and 11% for the k T , anti-k T and SIScone jet algorithms, respectively, in the region Q 2 ≥ 500 GeV 2 .
Neither Disent nor Nlojet++ include the contribution from Z 0 exchange; MC simulated events with and without Z 0 exchange were used to include this effect in the pQCD predictions. In the following, pQCD calculations refer to the fully corrected predictions.
Several sources of uncertainty in the theoretical predictions for the inclusive-jet cross sections were considered: • the uncertainty on the NLO QCD calculations due to terms beyond NLO, estimated by varying µ R between E jet T,B /2 and 2E jet T,B , was below ±7 (±10)% at low Q 2 and low E jet T,B and decreased to less than ±5 (±7)% for Q 2 > 250 GeV 2 for the k T [5] and anti-k T (SIScone) algorithms; • the uncertainty on the NLO QCD calculations due to those on the proton PDFs was estimated by repeating the calculations using 22 additional sets from the ZEUS-S analysis, which takes into account the statistical and correlated systematic experimental uncertainties of each data set used in the determination of the proton PDFs. The resulting uncertainty in the cross sections for all three jet algorithms was below ±3%, except in the high-E jet T,B region where it reached ±4%; • the uncertainty on the NLO QCD calculations due to that on α s (M Z ) was estimated by repeating the calculations using two additional sets of proton PDFs, for which different values of α s (M Z ) were assumed in the fits. The difference between the calculations using these various sets was scaled by a factor to reflect the uncertainty on the current world average of α s [56]. The resulting uncertainty in the cross sections was below ±2% for all three jet algorithms; • the uncertainty from the modelling of the QCD cascade was assumed to be half the difference between the hadronisation corrections obtained using the Ariadne and Lepto-MEPS models. The resulting uncertainty on the cross sections was less than ±1.4% [5], ±1.7% and ±2.3% for the k T , anti-k T and SIScone algorithms, respectively; • the uncertainty of the calculations due to the choice of µ F was estimated by repeating the calculations with µ F = Q/2 and 2Q. The effect was negligible.
The total theoretical uncertainty was obtained by adding in quadrature the individual uncertainties listed above. As a function of Q 2 , the total theoretical uncertainty varies in the range 3 − 7% (3 − 10%) for the anti-k T (SIScone) algorithm; as a function of E jet T,B , the range of variation is 5 − 6% (6 − 8%) for the anti-k T (SIScone) algorithm.
It is concluded that the NLO QCD predictions for the inclusive-jet cross sections are of similar precision for the k T and anti-k T algorithms and somewhat less precise for the SIScone. A more detailed comparison of the theoretical uncertainties is presented in Section 8.

Inclusive-jet differential cross sections with different jet algorithms
The inclusive-jet differential cross sections were measured in the kinematic region Q 2 > 125 GeV 2 and | cos γ h | < 0.65. The jets were reconstructed using either the anti-k T or the SIScone jet algorithms and the cross sections refer to jets with E jet T,B > 8 GeV and −2 < η jet B < 1.5. These cross sections were corrected for detector and QED radiative effects and the running of α em .
The measurements of the inclusive-jet differential cross sections as functions of E jet T,B and Q 2 are presented in Fig. 2 and Tables 1 and 2 for the anti-k T and SIScone jet algorithms. For comparison, the measurements of inclusive-jet cross sections with the k T algorithm [5] are also shown. In Fig. 2, each data point is plotted at the abscissa at which the NLO QCD differential cross section was equal to its bin-averaged value. The measured dσ/dE jet T,B (dσ/dQ 2 ) exhibits a steep fall-off over three (five) orders of magnitude for the jet algorithms considered in the E jet T,B (Q 2 ) measured range. The measurements using the three jet algorithms have a very similar shape and normalisation.
The NLO QCD predictions with µ R = E jet T,B are compared to the measurements in Fig. 2. The hadronisation-correction factors and their uncertainties are also shown. The ratios of the measured differential cross sections to the NLO QCD calculations are shown in Figs. 3a and 3b separately for each jet algorithm. The measured differential cross sections are well reproduced by the calculations, with similar precision in all cases. To study the scale dependence, NLO QCD calculations using µ R = Q were also compared to the data (not shown); they also provide a good description of the data.
To compare in more detail the results of the different algorithms, the ratios of the measured cross-sections anti-k T /k T , SIScone/k T and anti-k T /SIScone were investigated (see Figs. 3c and 3d). In the ratios, the statistical correlations among the event samples as well as those among the jets in the same event were taken into account in the estimation of the statistical uncertainties. These ratios show that the cross sections with the three jet algorithms are similar: the ratios as functions of Q 2 differ from unity by less than 3.2%; as functions of E jet T,B , they differ by less than 3.6% except at high E jet T,B , where the difference is 10%. The pQCD predictions including up to O(α 3 s ) terms for the ratios of cross sections are also shown in Figs. 3c and 3d. In the estimation of the total theoretical uncertainty of the predicted ratio, all the theoretical contributions were assumed to be correlated except that due to terms beyond O(α 3 s ) and that from the modelling of the QCD cascade. The measured ratios are well described by the calculations including up to O(α 3 s ) terms within the small experimental and theoretical uncertainties, which are dominated by the uncertainty on the modelling of the QCD cascade. Figure 4 shows the contributions to the theoretical uncertainty of the inclusive-jet cross sections from terms beyond NLO, from the uncertainty in the PDFs, from that on the value of α s (M Z ) and from that on the modelling of the QCD cascade, separately for the anti-k T and SIScone algorithms. The uncertainty coming from the terms beyond NLO is dominant at low E jet T,B and low Q 2 in all cases and somewhat higher for the predictions based on SIScone than those for the anti-k T and k T [5] algorithms. At high E jet T,B , the PDF uncertainty is dominant in the case of the anti-k T algorithm and of the same order as that arising from terms beyond NLO for the SIScone algorithm. At high Q 2 , the uncertainty due to terms beyond NLO is of the same order for both algorithms and remains dominant. The uncertainty on the modelling of the QCD cascade is somewhat higher for the SIScone algorithm than for anti-k T , but significantly smaller than that from terms beyond NLO. The total theoretical uncertainty for the predictions using µ R = Q is slightly larger than that using µ R = E jet T,B . The inclusive-jet cross sections for different regions of Q 2 as a function of E jet T,B are presented in Fig. 5 and Tables 3 and 4 for both algorithms. The measured cross sections exhibit a steep fall-off within the E jet T,B range considered. As Q 2 increases, the E jet T,B dependence of the cross section becomes less steep. The measurements have a similar shape and normalisation for both jet algorithms and are similar to the results obtained with the k T algorithm [4]. The NLO QCD predictions are compared to the measurements in Fig. 5. The data are well described by the predictions. Figure 6 shows the ratios of the measured differential cross sections to the NLO QCD calculations. The uncertainty of the NLO QCD calculations is also shown: the uncertainties from the anti-k T predictions are of a similar size as those encountered for the k T algorithm [4], and those for SIScone are somewhat larger at low E jet T,B and low Q 2 . In summary, it is concluded that the data for the k T , anti-k T and SIScone jet algorithms are well described by the NLO QCD calculations with similar experimental and theoretical precision. Furthermore, the measured ratios are well described by the predictions including up to O(α 3 s ) terms, demonstrating the ability of the pQCD calculations with up to four partons in the final state to account adequately for the differences between the jet algorithms.

Determination of α s (M Z )
The measured differential cross-sections dσ/dQ 2 for Q 2 > 500 GeV 2 were used to determine values of α s (M Z ) using the method presented previously [2]. The NLO QCD calculations were performed using the program Disent with five sets of ZEUS-S proton PDFs which were determined from global fits assuming different values of α s (M Z ), namely α s (M Z ) = 0.115, 0.117, 0.119, 0.121 and 0.123. The value of α s (M Z ) used in each calculation was that associated with the corresponding set of PDFs. The α s (M Z ) dependence of the predicted cross sections in each bin i of Q 2 was parameterised according to where C i 1 and C i 2 were determined from a χ 2 fit to the NLO QCD calculations. The value of α s (M Z ) was determined by a χ 2 fit to the measured dσ/dQ 2 values.
The uncertainties on the extracted values of α s (M Z ) due to the experimental systematic uncertainties were evaluated by repeating the analysis for each systematic check presented in Section 6. The overall normalisation uncertainty from the luminosity determination was also considered. The largest contribution to the experimental uncertainty comes from the jet energy scale and amounts to ±1.9% on α s (M Z ) for both algorithms (±2% for k T [5]). The theoretical uncertainties were evaluated as described in Section 7. The largest contribution was the theoretical uncertainty arising from terms beyond NLO, which was estimated by using the method proposed by Jones et al. [57], and amounted to ±1.5% for both algorithms 6 (±1.5% for k T [5]). The uncertainty due to the proton PDFs was ±0.7 (±0.8)% for the anti-k T (SIScone) algorithm (±0.7% for k T [5]). The uncertainty arising from the hadronisation effects amounted to ±0.9 (±1.2)% for the anti-k T (SIScone) algorithm (±0.8% for k T [5]). Thus, the performance of the three jet algorithms is very similar.
As a cross-check, α s (M Z ) was determined by using NLO QCD calculations based on the CTEQ6.1 [24] (MSTW2008 [25] [56], with the results obtained in pp collisions [58] as well as with the HERA average of 0.1186±0.0051 [59]. It should be noted that the differences between the central values of α s (M Z ) obtained using the three jet algorithms are comparable to the uncertainties due to higher-order terms in the calculations. It is observed that the precision in α s (M Z ) obtained with the k T , anti-k T and SIScone jet algorithms is very similar, and comparable to those obtained in e + e − interactions [56]. 6 The theoretical uncertainty arising from terms beyond NLO estimated by refitting the data using calculations based on µ R = 2E jet T,B or E jet T,B /2 amounts to +2.1 −0.0 ( +3.1 −0.7 )% for the anti-k T (SIScone) algorithm.

Summary and conclusions
For the first time, differential cross sections for inclusive-jet production in neutral current deep inelastic ep scattering were measured using the anti-k T and SIScone jet algorithms with R = 1. The cross sections correspond to a centre-of-mass energy of 318 GeV and refer to jets of hadrons with E jet T,B > 8 GeV and −2 < η jet B < 1.5 identified in the Breit frame with the anti-k T or SIScone jet algorithms. The measurements are given in the kinematic region of Q 2 > 125 GeV 2 and | cos γ h | < 0.65.
A detailed comparison between the measurements as functions of E jet T,B and Q 2 for both algorithms and those from a previous analysis based on the k T jet reconstruction was performed. The measured cross sections for the three jet algorithms have similar shapes and normalisations.
The NLO QCD calculations of inclusive-jet cross sections and their uncertainties for the different algorithms were also compared: the data are well described by the predictions; the calculations for the anti-k T algorithm have a similar precision as that of the k T whereas those for the SIScone are somewhat less precise due to the contribution from terms beyond NLO.
Measurements of the ratios of cross sections using different jet algorithms were also presented and compared to calculations including up to O(α 3 s ) terms. The measured ratios are well reproduced by the predictions, demonstrating the ability of the pQCD calculations including up to four partons in the final state to account adequately for the differences between the jet algorithms.
Values of α s (M Z ) were extracted from the measured inclusive-jet differential cross sections using each algorithm. QCD fits of the cross-sections dσ/dQ 2 for Q 2 > 500 GeV 2 yield the following values of α s (M Z ): These values are consistent with each other and with that obtained from the k T analysis with a similar precision. The measured differential cross-sections dσ/dE jet T,B for inclusive-jet production using different jet algorithms. The statistical, uncorrelated systematic and jet-energy-scale (ES) uncertainties are shown separately. The multiplicative corrections applied to the data to correct for QED radiative effects, C QED , and the corrections for hadronisation and Z 0 effects to be applied to the parton-level NLO QCD calculations, C had · C Z 0 , are shown in the last two columns.
anti-k T  Table 2: The measured differential cross-sections dσ/dQ 2 for inclusive-jet production using different jet algorithms. Other details as in the caption to Table 1.  Table 3: The measured differential cross-section dσ/dE jet T,B for inclusive-jet production in different regions of Q 2 using the anti-k T jet algorithm. Other details as in the caption to Table 1.     Figure 2: The measured differential cross-sections (a) dσ/dE jet T,B and (b) dσ/dQ 2 for inclusive-jet production (dots) using different jet algorithms. The NLO QCD calculations with µ R = E jet T,B (solid lines) are also shown. The cross sections for the anti-k T and k T [5] algorithms were multiplied by the scale factors indicated in brackets to aid visibility. The error bars on the data points are smaller than the marker size and are therefore not visible. The shaded bands display the uncertainty due to the absolute energy scale of the jets. The lower part of the figures shows the hadronisation-correction factor applied to the NLO calculations together with its uncertainty (hatched bands) for each jet algorithm; the hadronisation-correction factor for the k T algorithm was shifted by the value indicated in brackets to aid visibility. (c) (d) Figure 3: The ratios between the measured cross-sections (a) dσ/dE jet T,B and (b) dσ/dQ 2 and the NLO QCD calculations (dots). The inner error bars represent the statistical uncertainty. The outer error bars show the statistical and systematic uncertainties, not associated with the uncertainty of the absolute energy scale of the jets, added in quadrature. The hatched bands display the total theoretical uncertainty and the shaded bands display the jet-energy scale uncertainty. The ratios of the measured cross sections (dots) anti-k T /k T , SIScone/k T and anti-k T /SIScone as functions of (c) E jet T,B and (d) Q 2 . In these plots, the outer error bars include also the uncertainty of the absolute energy scale of the jets. The predicted ratios based on calculations which include up to O(α 3 s ) terms are also shown (solid lines). The hatched bands display the theoretical uncertainty on the ratio.  Figure 4: Overview of the theoretical relative uncertainties for the inclusivejet cross sections in the kinematic range of the measurements as functions of (a) E jet T,B and (b) Q 2 for the anti-k T (shaded areas) and SIScone (hatched areas) jet algorithms. Shown are the relative uncertainties induced by the terms beyond NLO, those on the proton PDFs, that on the value of α s (M Z ) and that on the modelling of the QCD cascade.