Erratum to: Determination of the strong coupling constant αs(mZ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\varvec{\alpha _{\mathrm{s}} (m_{\mathrm{Z}})}}}$$\end{document} in next-to-next-to-leading order QCD using H1 jet cross section measurements

The determination of the strong coupling constant αs(mZ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{\mathrm{s}} (m_{\mathrm{Z}})$$\end{document} from H1 inclusive and dijet cross section data [1] exploits perturbative QCD predictions in next-to-next-to-leading order (NNLO) [2–4]. An implementation error in the NNLO predictions was found [4] which changes the numerical values of the predictions and the resulting values of the fits. Using the corrected NNLO predictions together with inclusive jet and dijet data, the strong coupling constant is determined to be αs(mZ)=0.1166(19)exp(24)th\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{\mathrm{s}} (m_{\mathrm{Z}}) =0.1166\,(19)_{\mathrm{exp}}\,(24)_{\mathrm{th}}$$\end{document}. Complementarily, αs(mZ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{\mathrm{s}} (m_{\mathrm{Z}})$$\end{document} is determined together with parton distribution functions of the proton (PDFs) from jet and inclusive DIS data measured by the H1 experiment. The value αs(mZ)=0.1147(25)tot\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{\mathrm{s}} (m_{\mathrm{Z}}) =0.1147\,(25)_{\mathrm{tot}}$$\end{document} obtained is consistent with the determination from jet data alone. Corrected figures and numerical results are provided and the discussion is adapted accordingly.

error in the NNLO predictions was found [4] which changes the numerical values of the predictions and the resulting values of the fits. Using the corrected NNLO predictions together with inclusive jet and dijet data, the strong coupling constant is determined to be α s (m Z ) = 0.1166 (19) exp (24) th . Complementarily, α s (m Z ) is determined together with parton distribution functions of the proton (PDFs) from jet and inclusive DIS data measured by the H1 experiment. The value α s (m Z ) = 0.1147 (25) tot obtained is consistent with the determination from jet data alone. Corrected figures and numerical results are provided and the discussion is adapted accordingly. The determination of the strong coupling constant α s (m Z ) from H1 inclusive and dijet cross section data [1] exploits perturbative QCD predictions in next-to-next-to-leading order (NNLO) [2][3][4]. An implementation error of specific integrated initial-final antenna functions in the NNLO predictions that has impact on the numerical predictions for jet production cross sections in DIS was found in this numeric calculation [4]. This changes the values of the predictions and consequently the resulting values of the fits. The employed data cross sections and the α s -extraction methodology remain unchanged. In this erratum we provide corrections for two

Determination of α s (m Z ) from H1 jet cross sections
The strong coupling constant α s (m Z ) is determined from inclusive jet and dijet cross sections in NC DIS measured by the H1 collaboration and using NNLO QCD predictions. Fits are performed for each individual data set, for all inclusive jet measurements, for all dijet measurements, and for all H1 jet data taken together. The latter is denoted as 'H1 jets' in the following. In the case of fits to 'H1 jets', dijet data from the HERA-I running period however are excluded, since their statistical correlations to the respective inclusive jet data are not known.
For several studies, and to restrict the data to higher scales, a representative scale valueμ is assigned to each data point.
The value foμ is calculated from the geometric mean of the bin boundaries in Q 2 and P T [1].

Predictions
The inclusive jet and dijet NNLO predictions as a functions of the renormalisation scale μ R and the factorisation scale μ F are studied for selected phase space regions in Fig. 1. The dependence on the scale factor is strongest for cross sections at lower values μ R , i.e. lower values of Q 2 and P T . The NNLO predictions depend less on the scale factor than the NLO predictions.

Sensitivity of the fit to input parameters
Sensitivity to α s (m Z ) The sensitivity of the data to α s (m Z ) and the consistency of the calculations are investigated by performing fits with two free parameters representing the two distinct appearances of α s (m Z ) in equation (1) of Ref. [1], i.e. in the PDF evolution, α s (m Z ), and in the partonic cross sections, ασ s (m Z ). The result of such a fit performed for H1 jets is displayed in Fig. 3.  Fig. 4 for fits to inclusive jet and dijet cross sections, and in Fig. 5 for H1 jets.

Dependence on the choice of PDF
In Fig. 5 (right) only H1 jets withμ > 28 GeV are used. The predictions using NNPDF3.1 [5], determined with α PDF s (m Z ) = 0.118, provide a good description of the data with χ 2 /n dof smaller than unity (Fig. 4), where n dof denotes the number of data points minus one.

Scale variants and comparison of NLO and NNLO predictions
The dependence of the results on μ R and μ F is studied by applying scale factors to the definition of μ R and μ F . The values of α s (m Z ) and χ 2 /n dof resulting from the fits to inclusive jet and to dijet cross sections are displayed in Fig. 6 indicating that the standard choice for the scales (unity scale factor) yields good values of χ 2 /n dof . Figure 7 displays the resulting α s (m Z ) for fits to H1 jets.
Alternative choices for μ R and μ F are investigated and the results for α s (m Z ) with values of χ 2 /n dof are displayed in T results in good agreement of theory and data in terms of χ 2 /n dof . The results obtained with alternative scale choices typically vary within the assigned scale uncertainty. This is also observed for fits to H1 jets, presented in Fig. 9.
The NLO calculations exhibit an enhanced sensitivity to the choice of the scale and to scale variations, as com-pared to NNLO, resulting in scale uncertainties of α s (m Z ) of 0.0077, 0.0081 and 0.0083 for inclusive jets, dijet and H1 jets, respectively, as compared to uncertainties of 0.0034, 0.0033 and 0.0038 in NNLO, respectively. The previously observed reduction of scale uncertainties of the cross section predictions at NNLO [3,4,6] is reflected in a corresponding reduction of the α s (m Z ) scale uncertainties.   Restricting the scaleμ In order to study the size of the uncertainties as a function ofμ, the fits to inclusive jet and to dijet cross sections are repeated using only those data points exceeding a given valueμ cut . The resulting uncertainties are displayed in Fig. 10.

Results
The values of α s (m Z ) obtained from the fits to the data are collected in Table 4 and displayed in Fig. 11. Good agreement between theory and data is found.
For the fits to the individual data sets the χ 2 /n dof is below unity in most cases. The α s (m Z ) values are all found to be consistent, in particular between inclusive jet and dijet measurements.
The fits to the inclusive jet data exhibit very reasonable χ 2 /n dof values, thus indicating the consistency of the individual data sets. The value of α s (m Z ) from 'H1 inclusive jets' has a significantly reduced experimental uncertainty compared to the results for the individual data sets. The cutμ > 28 GeV results for inclusive jets in 0.1158 (19) exp (23) th , which is consistent with the world average [7,8].
Value of χ 2 /n dof lower than unity are obtained for fits to all dijet cross sections confirming their consistency. The results agree with those from inclusive jet cross sections and the world average. At high scalesμ > 28 GeV, a value α s (m Z ) = 0.1157 (22) exp (23) th is found.
The α s (m Z ) value obtained from H1 jet data restricted tõ μ > 28 GeV is In the present analysis, the value with the smallest total uncertainty is obtained in a fit to H1 jets restricted toμ > 42 GeV with the result α s (m Z ) = 0.1172 (23) exp (18) theo and a value of χ 2 /n dof = 37.0/40.
The ratio of all H1 jet cross section measurements to the NNLO predictions is displayed in Fig. 12. Overall good agreement between data and predictions is observed.

Running of the strong coupling constant
The strong coupling is determined in fits to data points grouped into intervals [μ lo ;μ up ] ofμ. The results for fits to inclusive jet and to dijet cross sections, as well as to H1 jets, are presented for the ten selected intervals inμ in Table 5 and are displayed Table 4 Summary of values of α s (m Z ) from fits to H1 jet cross section measurements using NNLO predictions. The uncertainties denote the experimental (exp), hadronisation (had), PDF, PDFα s , PDFset and scale uncertainties as described in the text. The rightmost three columns denote the quadratic sum of the theoretical uncertainties (th), the total (tot) uncertainties and the value of χ 2 /n dof of the corresponding fit. Along the vertical direction, the table data are segmented into five parts. The uppermost part summarises fits to individual inclusive jet datasets. The second part corresponds to fits of the individual dijet datasets. The third part summarises fits to all inclusive jets or all dijets together, with different choices of the lower cut on the scaleμ cut . The fourth group of fits, labelled H1 jets, is made using all available dijet and inclusive jet data together, for three different choices ofμ cut . The bottom row corresponds to a combined fit of inclusive data and normalised jet data. For that fit, theoretical uncertainties related to the PDF determination interfere with the experimental uncertainties and overall theoretical uncertainty is quoted in Fig. 13. Consistency is found for the fits to inclusive jets, dijets, and H1 jets, and the running of the strong coupling is confirmed in the accessible range of approximately 7 to 90 GeV. The values obtained from fits to H1 jets are compared to other determinations of at least NNLO accuracy [9][10][11][12] and to results at NLO at very high scale [13] in Fig. 14, and consistency with the other experiments is found.

Simultaneous α s and PDF determination
In addition to the fits described above also a fit in NNLO accuracy of α s (m Z ) together with the non-perturbative PDFs is performed which takes jet data and inclusive DIS data as input. This fit is denoted as 'PDF+α s -fit' in the following.

Fit results and the value of α s (m Z )
The results of the PDF+α s -fit are presented in Table 6. The fit yields χ 2 /n dof = 1518.6/(1529−13), confirming good agreement between the predictions and the data. The resulting PDF is able to describe 141 jet data points and the inclusive DIS data simultaneously.
The value of α s (m Z ) is determined to and is determined to an overall precision of 2.2%. The α s (m Z ) value is consistent with the main result of the 'H1 jets' fit. The result is compared to values from the PDF fitting groups ABM [14], ABMP [15], BBG [16], HERAPDF [17], JR [18], NNPDF [19] and MMHT [20] in Fig. 15 and consistency is found. The result exhibits a competitive experimental uncertainty to other determinations [15,19,20], which is achieved Fig. 11 Summary of α s (m Z ) values obtained from fits to individual and multiple H1 jet data sets. The inner error bars indicate the experimental uncertainty and the outer error bars the total uncertainty by using H1 normalised jet cross sections in addition to the H1 inclusive DIS data.

PDF parametrisation results
The PDF and α s (m Z ) parameters determined together in this fit (Table 6) are denoted as H1PDF2017 [NNLO]. It is released [21] in the LHAPDF [22] format with experimental, hadronisation and α s (m Z ) uncertainties included. The gluon and singlet momentum distributions, xg and x , the latter defined as the sum of all quark and anti-quark densities, are compared to NNPDF3.1 at a scale μ F = 20 GeV in Fig. 16.

The impact of H1 jet data on PDF fits
The PDF+α s -fit is repeated with the normalised jet data excluded, i.e. only inclusive DIS data are considered. The resulting Hessian error ellipses are displayed in Fig. 17 at a confidence level of 68%. Compared to the fit without jet data, the inclusion of jet data significantly reduces the uncertainties of α s (m Z ) and xg, as well as their correlation. The correlation coefficient is −0.92 and reduce to −0.85 if jet data is included.

Summary
The corrected new next-to-next-to-leading order pQCD calculations (NNLO) for jet production cross sections in neutralcurrent DIS are exploited for a determination of the strong coupling constant α s (m Z ) using inclusive jet and dijet cross section measurements published by the H1 collaboration. Two methods are explored to determine the value of α s (m Z ).
In the first approach H1 inclusive jet and dijet data are analysed. The strong coupling constant is determined to be α s (m Z ) = 0.1166 (19) exp (24) th , where the jet data are restricted to high scalesμ > 28 GeV. Uncertainties due to the input PDFs or the hadronisation corrections are found to be small, and the largest source of uncertainty is from scale variations of the NNLO calculations. The smallest total uncertainty on α s (m Z ) of 2.2% is obtained when restricting the data toμ > 42 GeV.
In a second approach a combined determination of PDF parameters and α s (m Z ) in NNLO accuracy is performed. In this fit all normalised inclusive jet and dijet cross sections published by H1 are analysed together with all inclusive neutral-current and charged-current DIS cross sections determined by H1. Using the data with Q 2 > 10 GeV 2 , the value of α s (m Z ) is determined to be α s (m Z ) = 0.1147 (25) tot . Consistency with the other results and the world average is found. The resulting PDF set H1PDF2017 [NNLO] is found to be consistent with the NNPDF3.1 PDF set at sufficiently large x > 0.01, albeit there are differences at lower x.
All jet cross section measurements are found to be well described by the NNLO predictions.  Table 5 Values of the strong coupling constant α s (μ R ) and at the Zboson mass, α s (m Z ), obtained from fits to groups of data points with comparable values of μ R . The first (second) uncertainty of each point corresponds to the experimental (theory) uncertainty. The theory uncertainties include PDF related uncertainties and the dominating scale uncertainty Running of the strong coupling μ R Inclusive jets Dijets H1 jets    Fig. 15 Comparison of the value of α s (m Z ) obtained in the H1PDF2017 [NNLO] PDF+α s -fit and in the H1 jets fit in NNLO accuracy to other α s determinations from DIS data. The pre-average of structure function data and the world average [8] are also indicated