Jet Production in ep Collisions at Low Q^2 and Determination of alpha_s

The production of jets is studied in deep-inelastic e+p scattering at low negative four momentum transfer squared 5<Q^2<100 GeV^2 and at inelasticity 0.2<y<0.7 using data recorded by the H1 detector at HERA in the years 1999 and 2000, corresponding to an integrated luminosity of 43.5 pb^-1. Inclusive jet, 2-jet and 3-jet cross sections as well as the ratio of 3-jet to 2-jet cross sections are measured as a function of Q^2 and jet transverse momentum. The 2-jet cross section is also measured as a function of the proton momentum fraction xi. The measurements are well described by perturbative quantum chromodynamics at next-to-leading order corrected for hadronisation effects and are subsequently used to extract the strong coupling alpha_s.


Introduction
Jet production in neutral current (NC) deep-inelastic scattering (DIS) at HERA provides an important testing ground for Quantum Chromodynamics (QCD). While inclusive DIS gives only indirect information on the strong coupling via scaling violations of the proton structure functions, the production of jets allows a direct measurement of α s . The Born level contribution to DIS ( Fig. 1(a)) generates no transverse momentum in the Breit frame, where the virtual boson and the proton collide head on [1]. Significant transverse momentum P T in the Breit frame is produced at leading order (LO) in the strong coupling α s by the QCD Compton ( Fig. 1(b)) and boson-gluon fusion ( Fig. 1(c)) processes. The latter dominates jet production for the range of the negative four momentum transfer squared of this analysis, 5 < Q 2 < 100 GeV 2 , and provides direct sensitivity to the gluon density function of the proton [2].
Analyses of inclusive and multi-jet production in DIS were previously performed at high Q 2 ( 100 GeV 2 ) [2][3][4][5] and at low Q 2 ( 100 GeV 2 ) [2,6,7] by the H1 and ZEUS Collaborations at HERA. In this paper new measurements of the inclusive jet, 2-jet and 3-jet production cross sections, as well as the ratio of 3-jet to 2-jet cross sections, are presented as a function of Q 2 and the jet transverse momenta in the Breit frame, P T , in the ranges 5 < Q 2 < 100 GeV 2 and P T > 5 GeV. The 2-jet cross section is also presented as a function of ξ = x Bj (1 + M 2 12 /Q 2 ), which in LO corresponds to the momentum fraction of the proton carried by the interacting parton (see Figs.1(b) and 1(c)). The variable x Bj denotes the Bjorken scaling variable and M 12 the invariant mass of the two jets of highest P T . The data correspond to higher integrated luminosity and a higher centre-of-mass energy than in the previous H1 analyses at low Q 2 [2,6]. The larger data set together with improved understanding of the hadronic energy measurement significantly reduces the total uncertainty of the cross section measurements. The results are compared with perturbative QCD predictions at next-to-leading order (NLO) corrected for hadronisation effects, and α s is extracted from a fit of the predictions to the data. These measurements allow the running of the strong coupling to be tested down to the limits of the perturbative calculation. Together with the high Q 2 measurements [4] the data test the running of α s in the range of renormalisation scale µ r between about 6 and 70 GeV.

Experimental Method
The data used for this analysis were recorded with H1 detector in the years 1999 and 2000, when HERA collided positrons of energy E e = 27.6 GeV with protons of energy E p = 920 GeV giving a centre-of-mass energy √ s = 319 GeV. The corresponding integrated luminosity is 43.5 pb −1 .

The H1 detector
A detailed description of the H1 detector can be found in [8,9]. Here, a brief account of the components most relevant to the present analysis is given. The origin of the H1 coordinate system is the nominal ep interaction point. The direction of the proton beam defines the positive z-axis (forward direction). The polar angle θ is measured with respect to this direction. The pseudorapidity is defined as η = − ln tan(θ/2).
In the central region (20 • < θ < 160 • ) the ep interaction region is surrounded by a twolayered silicon strip detector [10] and two large concentric drift chambers (CJCs), operated inside a 1.16 T solenoidal magnetic field. The trajectories of charged particles are measured in the central tracker with a transverse momentum resolution of σ(p T )/p T = 0.006 p T /GeV ⊕ 0.02 [11]. Two additional drift chambers complement the CJCs by precisely measuring the z-coordinates of track segments and hence improve the determination of the polar angle. The central tracking detectors also provide triggering information based on track segments measured in the r-φ plane of the central jet chambers and on the z position of the event vertex obtained from the double layers of two multiwire proportional chambers (MWPCs). The forward tracking detector and the backward drift chamber (BDC) measure tracks of charged particles at smaller (7 • < θ < 25 • ) and larger (155 • < θ < 175 • ) polar angle than the central tracker, respectively.
The luminosity is measured via the Bethe-Heitler Bremsstrahlung process ep → epγ, the final state photon being detected in a crystal calorimeter at z = −103 m.

Event and jet selection
The data sample of this analysis was collected using a combination of triggers which require the scattered positron to be measured in the SpaCal, at least one high transverse momentum track (p T > 800 MeV) to be reconstructed in the central tracking chambers and an event vertex to be identified by the MWPCs. The trigger efficiency is close to 100% for the whole analysis phase space as determined from the data using independent triggers as a reference.
The selection of NC DIS events is based on the identification of the scattered positron as the most energetic compact calorimetric deposit in the SpaCal with an energy E ′ e > 7.5 GeV and a polar angle 156 • < θ ′ e < 175 • . The energy weighted radius of this cluster is required to be less than 4 cm, as expected for an electromagnetic shower. The cluster must be geometrically associated with a track candidate in the BDC. The z-coordinate of the primary event vertex is required to be within ±35 cm of the nominal position of the interaction point.
The remaining clusters in the calorimeters and the charged tracks are combined to reconstruct the hadronic final state, using an algorithm which avoids double counting of energy [14,15]. The total longitudinal energy balance, determined as the difference of the total energy E and the longitudinal component of the total momentum P z , calculated from all detected particles including the scattered positron, must satisfy 45 < E − P z < 65 GeV. This requirement reduces contributions of DIS events with hard initial state photon radiation. For the latter events, the undetected photons propagating in the negative z direction lead to values of E − P z significantly lower than the expected value 2E e = 55.2 GeV. After this selection the contribution from photoproduction is negligible as estimated using Monte Carlo simulations.
The kinematic region covered by this analysis is defined by 5 < Q 2 < 100 GeV 2 and 0.2 < y < 0.7 , where y = Q 2 /(s · x Bj ) quantifies the inelasticity of the interaction. These two variables are reconstructed from the four momenta of the scattered positron and the hadronic final state particles using the electron-sigma method [16].
Jet finding is performed in the Breit frame. The boost from the laboratory system is determined by Q 2 , y and by the azimuthal angle of the scattered positron. Particles of the hadronic final state are clustered into jets using the inclusive k T algorithm [17,18] with the massless P T recombination scheme and with the distance parameter in the η − φ plane R 0 = 1. The cut −1 < η jet Lab < 2.5, where η jet Lab is the jet pseudorapidity in the laboratory frame, ensures that jets are contained within the acceptance of the LAr calorimeter. The transverse energy of jets in the Breit frame is required to be above 5 GeV. Jets are ordered by decreasing transverse momentum P T in the Breit frame, which is identical to the transverse energy E T for massless jets. The jet with highest P T is referred to as the "leading jet".
Three jet samples are defined: the inclusive jet sample contains all jets which satisfy the jet selection criteria; the 2-jet and 3-jet samples contain events with at least 2 and 3 jets, respectively. In addition, to avoid regions of phase space where fixed order perturbation theory is not reliable [19], 2-jet events are accepted only if the invariant mass M 12 of the two leading jets exceeds 18 GeV. The same requirement is applied for 3-jet events such that the 3-jet sample is a subset of the 2-jet sample.
The selection criteria are summarised in Table 1. The final inclusive jet sample contains 164522 events with 230140 jets. The 2-jet and 3-jet samples contain 31550 and 4879 events respectively.

Determination of the jet cross sections
In order to extract the jet cross sections at hadron level, the experimental data are corrected binby-bin for effects of limited detector acceptance and resolution and for QED radiation effects. The following leading order Monte Carlo event generators are used for the correction procedure: DJANGOH [20], which uses the Colour Dipole Model with QCD matrix element corrections as implemented in ARIADNE [21], and RAPGAP [22], based on QCD matrix elements matched with parton showers in leading log approximation. The effects of QED radiation are included using the HERACLES [23] program interfaced with RAPGAP and DJANGOH. In both Monte Carlo generators the hadronisation is modelled with Lund string fragmentation [24]. The generated events are passed through a GEANT3 [25] based simulation of the H1 apparatus and are reconstructed using the same program chain as for the data. Both the RAPGAP and DJANGOH simulations provide a good overall description of the shapes of all relevant data distributions. To further improve the agreement between Monte Carlo and data, a reweighting as a function of Q 2 and P T of the leading jet is applied to the Monte Carlo events.
The bin dependent correction factors are determined from Monte Carlo simulations as the ratios of the cross sections obtained from particles at hadron level without QED radiation to the cross section calculated using reconstructed particles and including QED radiation effects. The mean values of the correction factors determined by RAPGAP and DJANGOH are used, and half of the difference is assigned as a model uncertainty. The typical value of these factors is between 1.2 and 1.4.
The binnings in Q 2 , P T and ξ used to measure the jet cross sections are given in Table 2. The bin purities, defined as the fraction of events reconstructed in a particular bin that originate from that bin on hadron level, is found to be typically 70% and larger than 50% in all analysis bins. The bin stabilities, defined as the fraction of events originated from a particular bin on hadron level that are reconstructed in that bin, is typically 60% and larger than 40% in all analysis bins.

Experimental uncertainties
Several sources of experimental uncertainties are considered. The systematic uncertainties of the jet cross sections are determined by propagating the corresponding estimated measurement errors through the full analysis: • The relative uncertainty of the positron energy calibration is better than 1%. The absolute uncertainty of the positron polar angle is about 1 mrad. Uncertainties in the positron reconstruction affect the event kinematics and thus the boost to the Breit frame. This in turn leads to a relative error of up to 2% on the jet cross section for each of the two sources.
• The relative uncertainty on the energy of the reconstructed hadronic final state as well as of the jet energy is estimated to be 2%. It is dominated by the uncertainty of the hadronic energy scale of the LAr calorimeter. Two different calibration methods are used for jet transverse momentum below and above 10 GeV, respectively. The resulting uncertainty on the cross sections is typically in the range of 4% to 10%. The uncertainty of the SpaCal hadronic energy scale of 7% contributes less than 1% to the uncertainty of the cross section.
• The model dependence of the detector correction factors is estimated as described in section 2.3. It reflects the sensitivity of the detector simulation to the details of the model, especially the parton showering and its impact on the migration between adjacent bins in P T . The model dependence is below 10% in most of the bins and typically 4%.
• The luminosity measurement uncertainty leads to an overall normalisation error of the jet cross sections of 1.5%.
The uncertainty of the luminosity measurement is assumed to be fully correlated between the bins. The remaining sources of systematics, namely the positron energy scale and polar angle, the hadronic final state energy scale and the model dependence are assumed to be equally shared between correlated and uncorrelated parts.
The dominant experimental uncertainties on the jet cross sections arise from the model dependence of the data correction and from the LAr hadronic energy scale uncertainty. The individual contributions are added in quadrature to obtain the total systematic uncertainty.

NLO QCD Calculations
Reliable quantitative predictions of jet cross sections in DIS require the perturbative calculations to be performed at least to NLO in the strong coupling. The NLO calculations are used for comparison to data and for the α s extraction. By using the inclusive k T jet algorithm, the observables in the present analysis are infrared and collinear safe. Application of this algorithm in the Breit frame allows the initial state singularities to be absorbed in the definition of the proton parton densities, as needed for the calculation of factorised jet cross sections [26].
Jet cross sections are predicted at the parton level using the NLOJET++ program [27] at NLO in the strong coupling using the same jet definition as in the data analysis. When comparing data and theory predictions the strong coupling is taken to be α s (M Z ) = 0.118 at the Z 0 boson mass and is evolved as a function of the renormalisation scale with two loop precision. No QED radiation is included in the calculations, but the running of the electromagnetic coupling with Q 2 is taken into account in the theoretical predictions. The calculations are performed in the MS scheme [28] for five massless quark flavours. The parton density functions (PDFs) of the proton are taken from the CTEQ6.5M set [29]. The factorisation scale µ f and the renormalisation scale µ r are taken to be (Q 2 + P 2 T , obs )/2 for the NLO predictions, with P T,obs denoting the P T of the jet for inclusive jet cross sections and the average transverse momentum of the two leading jets P T for the 2-jet and 3-jet cross sections. This choice of the scales is motivated by the presence of two hard scales in jet production in DIS, P T and Q, the latter being smaller in most of the analysis bins. The calculations were also performed using µ r = P T,obs . With this choice of renormalisation scale the NLO QCD prediction decreases by 10 − 20% at lowest Q 2 and P T and is disfavoured by the data.
Hadronisation corrections are calculated for each bin using Monte Carlo event generators DJANGOH and RAPGAP which implement different models for parton showering. These corrections are determined as the ratio of the cross section at hadron level to the cross section at the parton level after parton showers. It was verified that the parton level jet cross sections obtained from DJANGOH and RAPGAP are in agreement with those from the NLO calculation within the systematic uncertainties considered here. The hadronisation correction factors are determined as the average of values obtained from DJANGOH and RAPGAP. Half of the difference is assigned as hadronisation uncertainty and included as a part of the theoretical uncertainty. For inclusive and 2-jet cross sections the hadronisation correction factors differ typically by less than 10% from unity and agree at the level of 2 to 5% between the two Monte Carlo simulations. For 3-jet cross sections, as well as for the ratios of 3-jet to 2-jet cross sections, the hadronisation correction factors differ from unity by about 20% with up to 10% difference between the two MC models.
The dominant theoretical uncertainty is related to the missing higher orders in the perturbative calculation, and is conventionally estimated by separately varying the scales µ f and µ r by factors in the arbitrary range 0.5 to 2. The contributions from the two scale variations are similar and are added in quadrature to obtain the total scale dependence uncertainty. The uncertainty originating from the PDFs is taken into account for the α s extraction using the variations of the CTEQ6.5M set of parton densities.

Cross Section Measurements
In the following, the differential cross sections, corresponding to the phase space given in Table 1, are presented for inclusive jet, 2-jet and 3-jet production at hadron level. Ratios of 3-jet to 2-jet hadron level cross sections are also presented. The measurements are shown in Tables  3 to 9 and Figs. 2 to 8.

Inclusive jet cross section
The measured inclusive jet cross section, corrected for detector and radiative QED effects, is presented as a function of Q 2 and P T of the jet, as single differential distributions in Figs. 2(a,b) and double differentially in Fig. 3. Each jet which satisfies the jet selection criteria described in section 2.2 enters these distributions.
The measurements are well described by NLO QCD predictions corrected for hadronisation effects as explained in section 3. The theoretical uncertainty, dominated by the scale variation, reaches 30% for the lowest Q 2 and P T bins and decreases to 10% for the highest Q 2 and P T values. The relative contribution of hadronisation corrections to this error is small. The PDF uncertainty is about 6% at the lowest Q 2 and P T and decreases to 2% for the highest Q 2 .

2-jet and 3-jet cross sections
The measured single differential cross sections for 2-jet and 3-jet production as functions of Q 2 and of the average transverse momentum of the two leading jets P T are shown in Figs. 2(c,e) and Figs. 2(d,f) and are well described by the NLO QCD calculations corrected for hadronisation. The relative uncertainties on the NLO QCD calculations of 2-jet, 3-jet and inclusive cross sections are all of similar size.
The double differential 2-jet cross sections are presented in seven Q 2 bins as functions of the variables P T and ξ in Fig. 4 and Fig. 5 respectively. The 3-jet cross section is shown in four Q 2 bins as a function of P T in Fig. 6. The NLO QCD calculation provides an overall good description of the measured distributions within the quoted theoretical and experimental uncertainties. Requirements in P T and M 12 suppress the cross section at low ξ where a rise is expected due to the increase of the gluon density.
The present results for the inclusive and the 2-jet cross sections were compared to the previous H1 results in [2] and [6], respectively. Taking into account the difference between proton beam energies and differences in the kinematic region studied, the results are found to be consistent with each other.
The 3-jet cross section normalised to the 2-jet cross section is presented in Fig. 7 for single differential and in Fig. 8 for double differential distributions. This observable benefits from cancellation of the normalisation uncertainties and reduction of the other systematic uncertainties by about 50%. It is described by the NLO cross section except for the lowest P T bin, as seen in Fig. 7(b) and Fig. 8, and shows a reduced sensitivity to the renormalisation scale variation which is done simultaneously for 2-jet and 3-jet cross sections. The ratio shows no significant dependence on Q 2 ( Fig. 7(a)), but increases with P T (Figs. 7(b),8) mainly due to the increasing phase space.

Extraction of the Strong Coupling
The QCD predictions for jet production depend on α s and on the PDFs of the proton. The strong coupling α s is determined from the measured jet cross sections using the PDFs as obtained from inclusive DIS data and other measurements.

Data and QCD predictions
The α s determination is performed from individual bins of the double differential inclusive jet cross section, d 2 σ/dQ 2 dP T , and the 2-jet and 3-jet cross sections, d 2 σ/dQ 2 d P T . Only bins are used in which the size of the k-factor, defined as the ratio of the cross sections calculated in NLO and LO (both obtained with NLOJET++), is below 2.5. The other bins most likely are affected by slow convergence of perturbation series and exhibit a high scale dependence, up to 30% at NLO.
The requirement that the k-factor be less than 2.5 corresponds to removing all points with P T,obs < 10 GeV for Q 2 < 20 GeV 2 from the inclusive jet and the 2-jet cross sections and points with 10 < P T < 15 GeV for Q 2 < 10 GeV 2 from the inclusive jet cross section 1 . In total, 62 cross sections measurements are used for α s extraction: 22 inclusive jet, 24 2-jet and all 16 3-jet points.
QCD predictions of the jet cross sections are calculated as a function of α s (µ r ) with the FastNLO package [30] using the CTEQ6.5M proton PDFs and applying the hadronisation corrections as described in section 3.

The χ 2 definition
Measurements and theory predictions are used to calculate a χ 2 (α s ) with the Hessian method [31], where parameters representing the systematic shifts of detector related observables, described in section 2.4, are left free in the fit. The shifts found by the fit are consistent with the a priori estimated experimental uncertainties. Due to different calibration strategies for jets with P T above and below 10 GeV, two different parameters are used for the hadronic final state energy scale for bins with P T,obs < 10 GeV and P T,obs ≥ 10 GeV. The Hessian method used here takes into account correlations of experimental uncertainties and has also been used in global data analyses [31,32] and in previous H1 publications [3,33]. The statistical correlations among the different bins and different observables are treated as described in [4,34]. The experimental uncertainty of α s is defined by the change in α s which gives an increase in χ 2 of one unit with respect to the minimal value.

Theory and PDF uncertainties
The theoretical uncertainty is estimated by the offset method as the difference between the value of α s from the nominal fit to the value when the fit is repeated with independent variations of different sources of theoretical uncertainties as described in section 3. The resulting uncertainties due to the different sources are summed in quadrature. The up (or down) variations are applied simultaneously to all bins in the fit. The impact of hadronisation corrections and the factorisation scale uncertainty on α s typically amounts to 1% to 2% for each source. The largest uncertainty, of typically 8%, corresponds to the accuracy of the NLO approximation to the jet cross section estimated by varying the renormalisation scale as described in section 3.
The uncertainty due to PDFs is estimated by propagating the CTEQ6.5M errors. The typical size of the resulting error is 2% for α s determined from the inclusive jet or 2-jet cross sections and 1% when measured with the 3-jet cross sections. This uncertainty is twice as large as that estimated with the uncertainties given for the MSTW2008nlo90cl set [35] which in turn exceeds the difference between α s values extracted with the central sets of CTEQ6.5M, CTEQ6.6M [36] and MSTW2008nlo.
The CTEQ6.5M PDF parameterisation was obtained assuming α s (M Z ) = 0.118. In order to test whether this value of α s (M Z ) biases the results obtained using the nominal method presented above, a method, similar to the one used in [3], is employed using the PDFs from the CTEQ6.6 series, which were obtained assuming different values for α s (M Z ). The cross section as a function of the strong coupling is interpolated with a polynomial and this interpolation is used to determine the best fit of the strong coupling to the data. The result obtained with this alternative fit method is found to be compatible, well inside one standard deviation of the experimental error, with the value determined by the nominal method. Hence there is no indication for a bias due to the value of the strong coupling assumed for the CTEQ6.5M PDFs.

Fit results
The fits of the strong coupling α s are performed individually for each of the 62 cross section measurements as described in section 5.1. These measurements constrain the value of the strong coupling at the Z 0 mass, α s (M Z ). As an example, α s (M Z ) values determined from the inclusive jet cross section are shown in Fig. 9. Different α s (M Z ) values agree within experimental errors. For each of the three observables and each Q 2 region, fits of α s (M Z ) to all P T or P T bins in that region are performed. The fit results are evolved from M Z to the average scale µ r in the respective Q 2 regions, where the average renormalisation scale µ r is calculated using NLO predictions. Figs. 10(a-c) show the obtained α s (µ r ) values for fits to inclusive, 2-jet and 3-jet cross-sections, respectively. Also shown in these figures are QCD predictions α s (µ r ), derived from common fits of α s (M Z ) to all respective Q 2 and P T or P T bins. The results of these three common fits are summarised in Table 10. If the points with k-factor above 2.5 are also included, the α s (M Z ) values obtained are changed by less than one standard deviation of the total experimental uncertainty. As the α s (M Z ) measurements derived from inclusive, 2-jet or 3-jet observables agree within uncertainties in any of the Q 2 regions, they are combined within four Q 2 regions, taking into account statistical and experimental systematic correlations. These results, evolved from the scale M Z to the average µ r in each region, are shown in Fig. 11(a). As compared to the case of extracting α s from only one variable, the experimental uncertainties are reduced significantly.
Finally, all 62 data points are used in a common fit of the strong coupling taking the correlations into account with a fit quality χ 2 /ndf = 49.8/61: The experimental error on α s (M Z ) measured with each observable typically amounts to 1.5%. The combination of different observables, even though partially correlated, gives rise to additional constraints on the strong coupling and leads to an improved experimental uncertainty of 1.2%. This error changes by at most 20% when the scales, hadronisation factors and PDF parameterisation are changed within the limits defined by their uncertainties. The total error is dominated by the theoretical uncertainty of about 7% mainly due to scale variations.
The determination of the strong coupling from the ratio of the 3-jet to the 2-jet cross section provides an alternative approach to combining the different cross section data. On the one hand the sensitivity of this observable to α s , which is O(α s ), is reduced with respect to the 3-jet cross section, which is O(α 2 s ). On the other hand this observable benefits from reduced experimental and theoretical uncertainties (see section 4.2). The common fit of the strong coupling to the 14 ratio points, for which the k-factors are below 2.5 for both the 3-jet and 2-jet cross sections, is given in Table 10. The extracted α s as a function of µ r is shown in Fig. 11(b). The experimental uncertainty on α s increases to 3% with respect to the combined fit from the cross sections. The theoretical uncertainty is reduced to 5% and is dominated by the hadronisation uncertainty.
The strong coupling extracted from all 62 data points (1) agrees well with that obtained from jet cross sections in the higher Q 2 range between 150 and 15000 GeV 2 : α s (M Z ) = 0.1168 ± 0.0007(exp.) +0.0046 −0.0030 (th.)±0.0016(PDF) [4]. This agreement is remarkable, given the sensitivity of the NLO prediction to the renormalisation scale in the low Q 2 regime. At high Q 2 [4] the fit was done using the factorisation scale Q, instead of (Q 2 + P 2 T , obs )/2 used in this analysis. However, as was shown in [4], the choice of factorisation scale has only little impact on extracted value of α s at high Q 2 . The value of α s (M Z ) obtained in this analysis is also consistent with the world averages α s (M Z ) = 0.1176 ± 0.0020 [37] and α s (M Z ) = 0.1184 ± 0.0007 [38].
The new low Q 2 measurement together with data from the high Q 2 analysis provides a test of the running of the strong coupling for µ r between 6 and 70 GeV as illustrated in Fig. 12. A simultaneous fit of α s from low and high Q 2 data was also performed. It did not lead to an improved precision with respect to the high Q 2 determination alone due to the large theoretical uncertainties at low Q 2 .

Conclusion
Measurements of the inclusive, 2-jet and 3-jet cross sections in deep-inelastic positron-proton scattering are presented in the range 5 < Q 2 < 100 GeV 2 and 0.2 < y < 0.7. Jets are reconstructed using the inclusive k T algorithm in the Breit frame and are required to have a minimum transverse momentum of 5 GeV. Calculations at NLO QCD, corrected for hadronisation effects, provide a good description of the single and double differential cross sections as functions of the jet transverse momentum P T , the boson virtuality Q 2 as well as of the proton momentum fraction ξ. The precision of the measurements is typically 6 to 10%.
The strong coupling α s is determined from a fit of the NLO prediction to the measured jet cross sections. The dominant source of uncertainties is related to the renormalisation scale dependence, which is used to estimate the effect of missing higher orders. The extracted value of the strong coupling is consistent with the value determined from high Q 2 jet cross sections. Both measurements test a running of the strong coupling for renormalisation scales µ r between 6 and 70 GeV.
NC DIS Selection 5 < Q 2 < 100 GeV 2 , 0.2 < y < 0.7 Inclusive jet P T > 5 GeV 40 < Q 2 < 100 GeV 2 bin number (3-jets and 3-jets/2-jets) corresponding Q 2 range bin letter corresponding P T or P T range a 5 < P T < 10 GeV b 10 < P T < 15 GeV c 15 < P T < 20 GeV d 20 < P T < 80 GeV bin letter corresponding ξ range Table 2: Nomenclature for the bins in negative four momentum transfer squared Q 2 , jet transverse momentum P T , average transverse momentum of the two leading jets P T and momentum fraction ξ used in the following tables.  The measurements refer to the phase-space given in table 1. In the columns 3 to 6 are shown the statistical uncertainty, the total experimental uncertainty, the total uncorrelated uncertainty including the statistical one and the total bin-to-bin correlated uncertainty calculated as the quadratic sum of the following five components: the model dependence, the electron and the hadronic energy scales, the electron polar angle and the luminosity measurement uncertainties. The contributions to correlated uncertainties are listed in columns 7 to 10. The sharing of the uncertainties between correlated and uncorrelated sources is described in detail in section 2.    Table 6: Double differential 2-jet cross sections as a function of Q 2 and P T of two leading jets together with their relative errors and hadronisation correction factors. Other details are given in the caption to table 3. The bin nomenclature is defined in table 2.     Determination of α s from jets              Figure 11: Values of α s (µ r = (Q 2 + P 2 T , obs )/2) obtained by a simultaneous fit of all jet cross sections in each Q 2 bin (a) and of the ratio of 3-jet cross section to 2-jet cross section (b). The solid lines show the two loop solution of the renormalisation group equation obtained by evolving the α s obtained from these measurements. For (a) the value of α s is extracted from a simultaneous fit of 62 measurements of inclusive jet, 2-jet and 3-jet double differential cross sections in bins of Q 2 and P T ( P T for 2-jets and 3-jets) with k-factor below 2.5, see table 10, 4th data row. For (b), α s is extracted from 14 measurements of the 3-jet cross section normalised to 2-jet cross section, using only data points with k-factor below 2.5 for both the 3-jet and 2-jet cross sections, see table 10, 5th data row. Other details are given in the caption to Fig. 9.  Figure 12: Values of α s (µ r = (Q 2 + P 2 T , obs )/2) obtained by a simultaneous fit of all jet cross sections in each Q 2 bin of this analysis (squares) together with the fit in different bins at high Q 2 (circles) [4]. The solid line shows the two loop solution of the renormalisation group equation obtained by evolving the α s (M Z ) extracted from jets at high Q 2 . Other details are given in the caption to Fig. 9.