Erratum to: Measurement of jet production cross sections in deep-inelastic ep scattering at HERA

The measurement of the jet cross sections by the H1 collaboration had been compared to various predictions including the next-to-next-to-leading order (NNLO) QCD calculations which are corrected in this erratum for an implementation error in one of the components of the NNLO calculations. The jet data and the other predictions remain unchanged. Eight ﬁgures, one table and conclusions are adapted accordingly, exhibiting even better agreement between the corrected NNLO predictions and the jet data.

Abstract The measurement of the jet cross sections by the H1 collaboration had been compared to various predictions including the next-to-next-to-leading order (NNLO) QCD calculations which are corrected in this erratum for an implementation error in one of the components of the NNLO calculations. The jet data and the other predictions remain unchanged. Eight figures, one table and conclusions are adapted accordingly, exhibiting even better agreement between the corrected NNLO predictions and the jet data. The measurement of absolute and normalised inclusive jet and dijet cross sections by the H1 collaboration [1] were compared to next-to-next-to-leading order QCD predictions by the NNLOJET program [2]. An implementation error of specific integrated initial-final antenna functions that has impact on the numerical predictions for jet production cross sections in DIS was found in this numeric calculation [3,4]. The data, the next-to-leading (NLO) and approximate next-to-next-toleading order (aNNLO) predictions remain unchanged, as well as figures for three-jet cross sections and the results on the strong coupling constant α s (M Z ), which are based on NLO predictions.
In this erratum, we provide corrected figures for the comparison of inclusive jet and dijet cross section data, both for absolute and normalised jet cross sections. The changes to the NNLO predictions are overall small, such that differences are only visible in figures of cross section ratios. The calculated values for χ 2 /n dof are corrected in Table 3. The discussion is corrected accordingly.

Cross section measurements
The differential cross sections are presented for absolute and normalised inclusive jet, dijet and trijet production at hadron level in Ref. [1]. The agreement of the various predictions with the data is judged by calculating values of χ 2 /n dof [5].
Here n dof is the number of data points in the calculation. The values of χ 2 /n dof for the absolute and the normalised jet cross sections are listed in Table 3. All calculations provide a reasonable value of χ 2 /n dof , taking into account the fact that uncertainties on the theory predictions, such as scale variations or the PDF uncertainties, are not included. The measured double-differential inclusive jet cross sections as function of P jet T and Q 2 for low and high values of Q 2 are compared to different theoretical predictions in Fig. 8. Ratios of the data and of the predictions in aNNLO and full NNLO to the NLO predictions are provided in Fig. 9.

Inclusive jet cross sections at low-Q
The conclusions drawn on the agreement between the NNLO predictions and the data remain largely unchanged with respect to Ref. [1]. The NNLO predictions give a good description of the P jet T -distributions following the excellent value of χ 2 /n dof ( Table 3).
The NNLO corrections to the cross section predictions, which are defined as ratios of NNLO to NLO predictions and are displayed in Fig. 9, are particularly large at low values of P jet T or at low values of Q 2 , equivalent to low values of the renormalisation and factorisation scales μ r and μ f . The NNLO predictions themselves have significantly smaller scale uncertainties than the NLO predictions. At low values of P jet T and small Q 2 , where the data are most precise, the uncertainties from scale variations of all predictions, however, are significantly larger than the experimental uncertainties. At higher values of P jet T the relative theoretical uncertainties are becoming smaller, but the data uncertainties, both statistical and systematic, increase and overshoot the uncertainties from scale variations.

Measurement of inclusive jets at high-Q
The phase space of additional inclusive jet cross sections at high values of Q 2 is extended to the region P jet T < 7 GeV by adding an extra bin at low P jet T [1]. These additional cross section points as a function of Q 2 for inclusive jet production in the range 5 < P jet T < 7 GeV are shown in Figs. 8 and 9.
The low-P jet T inclusive jet cross sections at high-Q 2 are underestimated by the NLO and aNNLO predictions, while the NNLO predictions give a good description of these data points. In the high-Q 2 domain the NNLO predictions have significantly smaller scale uncertainties than the NLO calculations, and the NNLO scale uncertainties typically are smaller than the experimental uncertainties. Figure 9 and the values of χ 2 /n dof in Table 3 indicate that the aNNLO predictions have difficulties describing the previously published high-Q 2 inclusive jet data [5] accurately. The NNLO predictions provide a good description of both, the low-and high-Q 2 inclusive jet data.

Normalised inclusive jet cross section
In order to obtain the normalised jet cross sections, cross sections for inclusive NC DIS are measured for 0.2 < y < 0.6 in the Q 2 bins in the range 5.5 < Q 2 < 80 GeV 2 . The normalised inclusive jet cross sections, derived using the inclusive NC DIS and the absolute inclusive jet cross sections, are displayed together with theoretical predictions in Fig. 11. The ratio of normalised inclusive jet cross sections to NLO predictions and the predictions in aNNLO and full NNLO to the NLO predictions is shown in Fig. 12. The dominating systematic uncertainties do not cancel in the normalisation, and the systematic uncertainty is significantly reduced only in bins where the overall systematic error is small, typically at low P jet T . The normalised jet cross sections confronted with theoretical predictions confirm the observations obtained using the absolute cross sections.

Dijet cross sections
The double-differential dijet cross sections as function of P T 2 and Q 2 are displayed in Fig. 13 and compared to theoretical predictions in NLO, aNNLO and NNLO. A comparison of the ratio of data to NLO predictions is provided in Fig. 14 together with the ratio of NNLO to NLO.
The aNNLO and NNLO predictions provide a better description of the shapes, while the NNLO predictions pro-  vide an overall accurate description of the normalisation of the dijet data. The uncertainty from scale variations of the NLO predictions is larger than the experimental uncertainty for P T 2 < 35 GeV, while the scale uncertainty of the NNLO calculations is reduced compared to the NLO predictions and is larger than the experimental uncertainties only at lower Q 2 values and for P T 2 < 11 GeV. The normalised dijet cross sections are displayed together with theoretical predictions in Fig. 15, and the ratio to NLO predictions is shown in Fig. 16. When comparing normalised dijet cross sections to theory predictions, the features observed with the absolute dijet cross sections are confirmed.

Summary
An error in the implementation of the next-to-next-to-leading order (NNLO) predictions is corrected. While the data, the NLO and the aNNLO predictions remain unchanged, some conclusions drawn from the comparison of the data to the NNLO predictions are revisited. The predictions in NNLO perturbative QCD improve the descriptions of the inclusive jet and dijet cross sections compared to NLO predictions, and give an overall good description of the new data at low and high Q 2 .