Determination of the strong coupling constant using inclusive jet cross section data from multiple experiments

Inclusive jet cross section measurements from the ATLAS, CDF, CMS, D0, H1, STAR, and ZEUS experiments are explored for determinations 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}$$\alpha _{\text {s}} (M_{\text {Z}})$$\end{document}. Various jet cross section data sets are reviewed, their consistency is examined, and the benefit of their simultaneous inclusion in the αs(MZ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{\text {s}} (M_{\text {Z}})$$\end{document} determination is demonstrated. Different methods for the statistical analysis of these data are compared and one method is proposed for a coherent treatment of all data sets. While the presented studies are based on next-to-leading order in perturbative quantum chromodynamics (pQCD), they lay the groundwork for determinations of αs(MZ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{\text {s}} (M_{\text {Z}})$$\end{document} at next-to-next-to-leading order.


Introduction
The strong coupling constant, α s , is one of the least precisely known fundamental parameters in the Standard Model of particle physics. Because of its importance for precision phenomenology at the LHC and elsewhere, large efforts have been undertaken in the past decades to reduce uncertainties in determinations of α s [1,2,3,4].
With the advent of modern particle detectors and sophisticated algorithms for their simulation and calibration, jet measurements have become very precise. Many determinations of α s in deep-inelastic scattering (DIS) and in hadron-hadron collisions are therefore based on measurements of the inclusive jet cross section, which is directly proportional to α s in DIS in the Breit frame and α 2 s in hadron-hadron collisions. Using the most precise predictions of perturbative quantum chromodynamics (pQCD) available at the time, all previous α s extractions (except for ref. [5]) were performed at next-to-leading order (NLO) in α s . Their total uncertainty is dominated by the contribution related to the renormalisation scale dependence of the NLO pQCD results. The recent completion of next-tonext-to-leading order (NNLO) predictions for the inclusive jet cross section [6,7] promises a considerable reduction of the renormalisation scale dependence and will allow the inclusion of α s results from inclusive jet data in future determinations of the world average value of α s [1].
A determination of α s at NNLO from jet measurements in hadron-hadron collisions is still not readily achievable, because the new NNLO pQCD calculations are com-putationally very demanding and cannot yet be repeated quickly for different parton distribution functions (PDFs) or values of α s (M Z ). In preparation of such a determination, it is desirable to study a simultaneous analysis of data sets from different processes and experiments. This study includes an investigation of the consistency of the various data sets and an estimation of the reduction of the experimental contributions to the α s uncertainty. The groundwork for these two aspects is presented in this article.
We review inclusive jet cross section data over a wide kinematic range, from different experiments for various initial states and centre-of-mass energies, and study their potential for determinations of α s . The consistency of the diverse data sets is examined and the benefit of their simultaneous inclusion is demonstrated. Different methods for the statistical analysis of the data are compared and one method is proposed for a coherent treatment of all data sets in an extraction of α s (M Z ).
The article is structured as follows: The experimental data sets and the theoretical predictions are introduced in sections 2 and 3, respectively. Methods and results from previous α s determinations by different experimental collaborations are discussed and employed in section 4. The strategy for a determination of α s from multiple data sets and the final result are presented in section 5.

Experimental data
The first measurement of the inclusive jet cross section has been performed in 1982 by the UA2 Collaboration at the SppS collider at a centre-of-mass energy of 540 GeV [8]. Further measurements have been conducted at centre-ofmass energies of -540 GeV, 546 GeV, and 630 GeV at the pp collider SppS by the UA1 [9,10] and UA2 experiments [11], -546 GeV, 630 GeV, 1.8 TeV, and 1.96 TeV at the Tev-atron pp collider by the CDF [12,13,14,15,16,17] and D0 experiments [18,19,20], -300 GeV and 320 GeV at the ep collider HERA by the H1 [21,22,23,24,25,26,27,28] and ZEUS experiments [29,30,31,32,33,34,35], -200 GeV in pp collisions at RHIC by the STAR experiment [36], and of 2. 76  Previously, α s (M Z ) determinations were based on inclusive jet cross section data from individual experiments, as summarised in table 1. An extraction of α s (M Z ) from multiple inclusive jet cross section data sets has not been performed so far, except in the context of global PDF analyses, in which PDFs and α s (M Z ) are determined simultaneously. These analyses, however, require data for a variety of measured quantities [21,57,58,59,60]. In this article, α s (M Z ) is determined in a fit to multiple inclusive jet cross section measurements from experiments at HERA, RHIC, the Tevatron, and the LHC. The analysis is based on one selected measurement from each, the H1, ZEUS, STAR, CDF, D0, ATLAS, and CMS collaborations, as listed in table 2.
Whenever experiments provide multiple measurements, we include those measured with a collinear-and infraredsafe jet algorithm (k T [68] or anti-k T [69]) and with a larger jet size parameter R, which improves the stability of fixed-order pQCD calculations. The STAR experiment published inclusive jet data collected by two different triggers with partially overlapping jet p T ranges. We choose the data set collected with the trigger covering the higher jet p T range from 7.6 GeV up to 50 GeV. The measurements from the STAR and D0 experiments are using the midpoint cone jet algorithms (MP) [55]. The infraredunsafety of this jet algorithm [54] prohibits NNLO pQCD predictions for these data sets, but it does not affect calculations at NLO. Four new measurements [28,50,42,43] could not be included in this study; they are left for a future extension.

Theoretical predictions and tools
Predictions for the inclusive jet cross section in processes with initial-state hadrons are calculated as the convolution of the partonic cross sectionσ (computed in pQCD) and the PDFs of the hadron(s). The inclusive jet cross section in hadron-hadron collisions can be written as [70,1] where the sum is over all combinations of parton flavors i and j (quarks, anti-quarks, and the gluon). The f i,j/h1,2 denote the PDFs for the parton flavours i or j in the initial-state hadrons h 1 and h 2 , and x 1 and x 2 correspond to the fractional hadron momenta carried by the partons i and j, respectively. The partonic cross sectionσ ij→jet+X is computed as a perturbative expansion in α s aŝ where the c (n) ij→jet+X are computed from the pQCD matrix elements and the sum is over all orders of α s taken into account in the perturbative calculation. The renormalisation and factorisation scales are labelled µ r and µ f , respectively. For inclusive jet production in hadron-hadron collisions, the first non-vanishing order (i.e. the leading order, LO) is given by n = 2, while n = 3 corresponds to the NLO corrections. For inclusive jet production in DIS in the Breit frame the partonic cross sections are convoluted with a single PDF and the LO (NLO) contribution is given by n = 1 (n = 2). Hence, inclusive jet production in pp, pp, and ep collisions is sensitive to α s already at LO.
For transverse jet momenta at the TeV scale accessible at the LHC, electroweak (EW) tree-level effects of O αα s , α 2 and loop effects of O αα 2 s become sizeable [71]. A recent study of the complete set of QCD and EW NLO corrections has been presented in ref. [72].
Non-perturbative (NP) corrections to the cross section due to multiparton interactions and hadronisation can be estimated by using Monte Carlo (MC) event generators. An overview of MC event generators for the LHC is presented in ref. [73]. The size of this correction depends on the jet size R, shrinks with increasing jet p T , and becomes negligible at the TeV scale. The total theory prediction for the inclusive jet cross section is given by where c EW and c NP are the correction factors for electroweak and non-perturbative corrections, respectively. The partonic cross section is computed at NLO accuracy for five massless quark flavours using the NLOJet++ program version 4.1.3 [74,75] within the fastNLO framework at version 2 [76,77] to allow us fast recalculations for varying PDFs, scales µ r and µ f , and assumptions on  Table 2. Overview of the inclusive jet data sets used in the αs determinations. For each data set the process (proc), the centre-of-mass energy √ s, the integrated luminosity L, the number of data points, and the jet algorithm are listed. In case of ep collider data, the kinematic range may be defined by the four-momentum transfer squared Q 2 , the inelasticity yDIS, or the angle of the hadronic final state | cos γ h | of the NC DIS process. In all cases, jets are required to be within a given range of pseudorapidity η or rapidity y in the laboratory frame. α s (M Z ). Jet algorithms are taken either from the FastJet software library [78] or, for jet cross sections in DIS, from NLOJet++ . The PDFs are evaluated via the LHAPDF interface [79,80] at version 6. The running of α s (µ r ) is performed at 2-loop order using the package CRunDec with five massless quark flavours [81,82]. The minimal subtraction (MS) scheme [83,84,85] has been adopted for the renormalisation procedure in these calculations.
For the computation of the inclusive jet cross section in hadron-hadron collisions, the renormalisation and factorisation scales, µ r and µ f , are identified with each jet's p T , i.e. µ r = µ f = p jet T . In neutral current (NC) DIS, the scales are chosen to be µ r 2 = 1 2 Q 2 + (p jet T ) 2 and µ f 2 = Q 2 as used by the H1 Collaboration [27]. Alternative scale choices have been discussed with respect to NNLO predictions [7,86,87,5], but are beyond the scope of this article. The EW corrections, c EW , relevant for the LHC data are provided by the experimental collaborations together with the data, based on ref. [71]. These are considered to have negligible uncertainties. Due to restrictions of the scale choices in this calculation, the leading jet's transverse momentum, p max T , is used to define the scales µ r and µ f . The NP correction factors c NP , except for the STAR data [88], are also provided by the experimental collaborations, together with an estimate of the corresponding uncertainty [27,32,33,16,20,61,41,46,62].

Comparison of three extraction methods for α s (M Z )
Commonly, the value of α s (M Z ) is determined from inclusive jet cross sections in a comparison of pQCD predictions to the measurements. These α s (M Z ) results therefore depend on details of the extraction method such as the treatment of uncertainties in the characterisation of differences between theory and data, or the evaluation and propagation of theoretical uncertainties to the final result. An overview of previous determinations of α s (M Z ) from fits to inclusive jet cross section data is provided in table 1. We choose the three α s (M Z ) determinations performed by the CMS [62], D0 [61], and H1 [27] collaborations listed in the upper part of table 1 for further study. The three extraction methods differ in the following aspects: the definition of the χ 2 function to quantify the agreement between theory and data, the uncertainties considered in the χ 2 function, the strategy to determine the central result for α s (M Z ), the propagation of the uncertainties to the value of α s (M Z ), the choice of PDF sets, the consideration of the α s (M Z ) dependence of the PDFs, and the treatment of further theoretical uncertainties.
To study the impact of these differences, we have implemented the three methods in our computational framework and will refer to them as "CMS-type", "D0-type", and "H1-type", respectively. Each method is employed to extract α s (M Z ) from each of the individual data sets selected in section 2, cf. also table 2. The experimental uncertainties and their correlations are treated according to the respective prescriptions by the experiments. The CMS result was obtained with the CT10 PDF set [89], and the D0 and H1 results with MSTW2008 PDFs [90]. The CMStype and D0-type methods use the entire α PDF s (M Z ) series available for the PDF set, whereas the H1-type method uses a PDF determined with a value of α PDF s (M Z ) = 0.1180. The resulting α s (M Z ) values are listed in table 3.
In a first step, these results are compared to the ones obtained by the CMS [62], D0 [61], and H1 [27] collaborations as listed in table 1. All three central results are reproduced, the H1 result exactly, and the CMS and D0 results within +0.0003 and +0.0001. Such small differences can easily be caused already by using different versions of LHAPDF (e.g. changes from version 5 to version 6). The experimental uncertainties of the CMS and H1 analyses are exactly reproduced. 1 In a second step, the α s (M Z ) results and their experimental uncertainties are compared to each other and their dependencies on the extraction method and PDFs are studied. The α s (M Z ) results determined for each data set are displayed in figure 1 (top row) for the three different extraction methods using CT10 PDFs (left) and MSTW2008 PDFs (right). For the STAR data, α s (M Z ) results cannot be determined in case of the CMS-type and D0-type methods with MSTW2008 PDFs, since no local χ 2 minima are found. In all other cases the α s (M Z ) results obtained with MSTW2008 PDFs are rather independent of the extraction method for all data sets. This is different when using CT10 PDFs: While in this case the extraction method has little impact on the α s (M Z ) results from HERA data (H1 and ZEUS), it notably affects the results for the LHC data (ATLAS and CMS), and has large effects for the Tevatron data (CDF and D0). In the latter cases, the D0-type method produces significantly lower α s (M Z ) results as compared to the other two methods.
The χ 2 /n dof values for the α s (M Z ) extractions are displayed in figure 1 (bottom row) for the three extraction methods using CT10 PDFs (left) and MSTW2008 PDFs (right). Overall, the fits exhibit reasonable values of χ 2 /n dof , thus indicating agreement between theory and data. Exceptions are observed for the ZEUS data with rather low values of χ 2 /n dof , and for the ATLAS data, where the values of χ 2 /n dof are large as also observed elsewhere [42, 91,92].
The PDF dependence is displayed in figure 2, where the α s (M Z ) results for CT10 and MSTW2008 PDFs are compared to each other, both obtained using the H1-type method. While the PDF choice has no significant effect for the results from the H1, ZEUS, and D0 data, smaller variations are seen for the CDF data, and a large dependence for the ATLAS and CMS data.  World average H1-type Fit method: Figure 2. Comparison of the αs(MZ) results with their experimental uncertainties obtained using the H1-type extraction methods for CT10 and MSTW2008 PDFs. The world average value [1] is shown together with a band representing its uncertainty.
PDF dependence in the context of a common determination of α s (M Z ) as described in the next section, we observe that differences with respect to the updated PDF sets, CT14 [93] and MMHT2014 [60], are reduced.

Determination of α s (M Z ) from multiple inclusive jet data sets
The analysis of multiple data sets requires their correlations to be taken into account. For the present study, measurements from different colliders are considered to be uncorrelated because of the largely complementary kinematic ranges of the data sets and different detector calibration techniques. Furthermore, investigations with respect to H1 and ZEUS data [58], CDF and D0 data [94], or ATLAS and CMS data [95,96], did not identify a relevant source of experimental correlation. This only leaves theoretical uncertainties as a source of potential correlations in this study. For the determination of NP effects and their uncertainties various methods and MC event generators have been employed [27,32,36,88,16,20,61,41,62]. While a consistent derivation of these corrections with corresponding correlations is desirable, this is beyond the scope of this analysis. Hence, the NP correction factors and their uncertainties are considered to be uncorrelated between the different data sets. In contrast, the PDF uncertainties and the uncertainties due to the renormalisation and factorisation scale variations are treated as fully correlated; the relative variations with respect to the nominal scales are performed simultaneously for all data sets.
The method employed for the simultaneous α s extraction combines components of the individual methods outlined in the previous section and is referred to as "CMStype method". The central α s (M Z ) result is found in an iterative χ 2 minimisation procedure adopted from the H1type method, where a normal distribution is assumed for the relative uncertainties. The exact χ 2 formula is given by equation (4) of appendix A. Whereas in the H1-type χ 2 expression, only experimental uncertainties are taken into account, the common-type method also accounts for the NP and PDF uncertainties in the χ 2 expression, as in the CMS-type and D0-type methods. This χ 2 definition treats variances as relative values and thus has advantages, e.g. when numerically inverting the covariance matrix. Moreover, uncertainties of experimental and theoretical origin are put on an equal footing. As in the H1type method, and in contrast to the CMS-type and D0type ones, only PDF sets obtained with a fixed value of α s (M Z ) = 0.1180 are employed in the determination of the central α s (M Z ) result, leaving the α s dependence of the PDFs to be treated as a separate uncertainty.
In summary, the individual contributions to the total uncertainty of the α s (M Z ) result are evaluated as follows: The experimental uncertainty (exp) is obtained from the Hesse algorithm [97] when performing the α s (M Z ) extraction with only the uncertainties of the measurements included. The NP and PDF uncertainties are derived by repeating the α s (M Z ) extraction while successively including the corresponding uncertainty contributions and calculating the quadratic differences. Further sources of systematic effects are considered as follows: -The "PDFα s " uncertainty accounts for the initial assumption of α PDF s (M Z ) = 0.1180 made in the PDF extraction, which is not necessarily consistent with the value of α s (M Z ) used in the pQCD calculation. It is calculated as the maximal difference between any of the results obtained with PDF sets determined for α PDF s (M Z ) = 0.1170, 0.1180, and 0.1190, and therefore covers a difference of ∆α PDF s (M Z ) = 0.0020, which is somewhat more conservative than the recommendation in ref. [98].
-The "PDFset" uncertainty covers differences due to the considered PDF set. These are caused by assumptions made on the data selection, parameterisation, parameter values, theoretical assumptions, or the analysis method for the PDF determination. It is defined as half of the width of the envelope of the results obtained with the PDF sets CT14 [93], HERAPDF2.0 [58], MMHT2014 [60], NNPDF3.0 [99] and ABMP16 [100,101]. -The uncertainty due to variations of the renormalisation and factorisation scales customarily is taken as an estimate for the error of a fixed-order calculation caused by the truncation of the perturbative series. It is obtained using six additional α s (M Z ) determinations, in which the nominal scales (µ r , µ f ) are varied by the conventional factors of (1/2, 1/2), (1/2, 1), (1, 1/2), (1, 2), (2, 1), and (2, 2). The scale factor combinations of (1/2, 2) and (2, 1/2) are customarily omitted [102,103,104]. The NP, PDF, PDFα s , PDFset, and scale uncertainties are added in quadrature to give the theoretical uncertainty (theo). The total uncertainty (tot) further includes the experimental uncertainty.
In the previous section, cf. figure 1, it was found that the χ 2 /n dof values differ significantly from unity for some of the data sets. This necessitates to investigate in further detail the consistency of the data within an individual data set as well as among the different data sets. Moreover, new PDF sets have become available. Therefore, the common-type method is employed to extract For a given data set, χ 2 /n dof is rather independent of the PDF set used for the predictions and varies between 0.8 and 1.2. These values indicate reasonable agreement of the predictions with the data. Exceptions are rather low values of χ 2 /n dof around 0.54 found for all PDF sets with the ZEUS data and large χ 2 /n dof values between 1.9 and 3.5 exhibited by the ATLAS data, also for all PDFs. Exceptionally large χ 2 /n dof values appear for the Tevatron or LHC data together with theory predictions using the HERAPDF2.0 set, and for the STAR data in conjunction with the ABMP16 PDF set.
To further investigate the consistency among the data sets, a series of α s (M Z ) extractions is performed, in which α s (M Z ) is determined simultaneously from all data sets but one. This is repeated for each PDF set. The resulting χ 2 /n dof values are displayed in figure 3 right. Apparently, the exclusion of the ATLAS data leads to significantly smaller χ 2 /n dof values independent of the PDF set used. This hints at a compatibility issue when using all data sets together, which is not present when the ATLAS data set is ignored. Therefore, we choose to exclude the ATLAS data for our main result, which is thus obtained from the CDF, CMS, D0, H1, STAR, and ZEUS inclusive jet data. The choice of the NNPDF3.0 set for the central result yields with χ 2 = 328 for 381 data points. This result is consistent with the world average value of 0.1181 (11) [1]. The experimental uncertainty for the extraction from multiple data sets is significantly smaller than each of the experimental uncertainties reported previously for the separate α s (M Z ) determinations. Results obtained with different PDF sets constitute the PDFset uncertainty. They are listed in table 4 together with the PDF 2 and PDFα s uncertainties as appropriate for the respective PDF set. The corresponding values of χ 2 /n dof can be read off from row six of figure 3 right. Other uncertainties remain unchanged in the leading digit as compared to the ones obtained for the NNPDF3.0 PDFs. The α s (M Z ) values from fits using the various PDF sets given in table 4 are found to be consistent within the experimental uncertainty. The NP, PDF, and PDFα s uncertainties are smaller than the experimental uncertainty, while the PDFset uncertainty is of a similar size as the experimental one. The scale uncertainty is the largest individual uncertainty and is more than three times larger than any other uncertainty. Results of the α s (M Z ) extractions from single data sets, cf. appendix B, from the simultaneous α s (M Z ) extraction from all data sets, and the world average [1] are compared in figure 4 and are seen to be consistent with each other.
The ratio of data to the predictions as a function of jet p T for all selected data sets is presented in figure 5. The predictions are computed for α s (M Z ) = 0.1192 as obtained in this analysis. Visually, all data sets are well described by the theory predictions.   Table 4. Values of αs(MZ) for the simultaneous fit to the H1, ZEUS, STAR, CDF, D0, and CMS data using the common-type method for various PDF sets. The experimental, NP, PDFset, and scale uncertainties remain mostly unchanged under a change of the PDF set and are quoted only once for NNPDF3.0.

Summary and outlook
Inclusive jet cross section data from different experiments at various particle colliders with jet transverse momenta ranging from 7 GeV up to 2 TeV are explored for determinations of α s (M Z ) using next-to-leading order predictions. Previous α s (M Z ) determinations reported by the CMS, D0, and H1 collaborations [62,61,27] are taken as a baseline, and these α s (M Z ) extraction methods, which differ in various aspects, are applied to inclusive jet cross section data measured by the ATLAS, CDF, CMS, D0, H1, STAR, and ZEUS experiments [41,16,62,20,27,36,32]. Differences among the α s (M Z ) results due to the extraction technique are found to be negligible in most cases. A new extraction method is proposed, which combines aspects of the baseline approaches above.
In a statistical analysis, data measured by the CDF, CMS, D0, H1, STAR, and ZEUS experiments are found to be well described by pQCD predictions at next-to-leading order, and hence are considered to be mutually consistent. Moreover, the values of α s (M Z ) determined from each individual data set are found to be consistent among each other. By determining α s (M Z ) simultaneously from these data, the experimental uncertainty of α s (M Z ) is reduced to 1.0 %, as compared to 1.9 % when only the single most precise data set of that selection is considered.
The largest contribution to the uncertainty of α s (M Z ) originates from the renormalisation scale dependence of the next-to-leading order pQCD calculation. This uncertainty is expected to be reduced once the next-to-next-toleading order predictions become available for such studies. Furthermore, a reevaluation of the non-perturbative corrections and their uncertainties for all data sets in a consistent manner is recommended for a determination of α s (M Z ) at high precision. The presented study and the developed analysis framework provide a solid basis for fu- ture determinations of α s (M Z ) and facilitate the inclusion of additional data sets, further observables, and improved theory predictions.
We thank our colleagues in the CMS, D0, and H1 collaborations for fruitful discussions, and K. Bjørke and D.       Detailed results of the common-type method applied to the individual data sets are given in table 5. The result for the H1 data agrees with the value published in ref. [27]. Even though using the full D0 data set with 110 points, the extracted α s (M Z ) value is consistent with the value achieved by the D0 Collaboration at NLO for a subset of 22 points in ref. [61]. For the CMS measurement, the common-type method leads to a consistent but somewhat lower result than reported in ref. [62] for various PDFs. Our result for the ZEUS data is compatible with the value obtained by the ZEUS Collaboration from a single-differential variant of the measurement in a reduced phase space as published in ref. [33]. With respect to the ATLAS, CDF, and STAR inclusive jet data, this study constitutes the first α s (M Z ) determination from either data set. Within uncertainties, all α s (M Z ) values are consistent with each other and with the world average. The individual uncertainties compare as follows: -The experimental uncertainty of α s (M Z ) is roughly comparable between experiments at the same collider. It is largest for the STAR data, and smallest for the ATLAS data. -The NP uncertainties are found to vary significantly, even between data sets in similar kinematic regions, for instance between CDF and D0. In case of the LHC experiments the NP uncertainties appear to be negligible. -The PDF uncertainty as estimated with the NNPDF3.0 PDF set is smaller than the experimental uncertainty. For the HERA data, the PDF uncertainty is found to be moderately smaller than for Tevatron or LHC data as observed also with other PDF sets. -For all data sets, the PDFα s uncertainty is rather small.
This observation justifies to neglect the α s dependence of the PDFs in the α s (M Z ) determinations and to assign a separately derived uncertainty instead. -The PDFset uncertainty constitutes the largest contribution of the PDF related ones. -The largely dominating scale uncertainty is of similar size in case of HERA and LHC data and somewhat larger for Tevatron data or the STAR experiment. The results of α s (M Z ) determinations from single measurements for the alternative PDF sets ABMP16, CT14, HERAPDF2.0, and MMHT2014 PDF sets are provided in columns 2-5 of table 6. The envelope constructed from these four values together with the NNPDF3.0 result constitutes the PDFset uncertainty shown in column seven of table 5. The further columns in table 6 present the PDF and PDFα s uncertainty for the respective PDF sets.
The spread among the α s (M Z ) determinations from a single data set with varying PDF sets is illustrated in figure 6. For each of the individual data sets, the results are mostly consistent. Larger deviations are observed for the Tevatron data when using the ABMP16 and HERAPDF2.0 sets, and for the STAR data in conjunction with the ABMP16 PDF set.
The PDF uncertainty obtained with different PDF sets for the same data set is largest for CT14 and smallest for HERAPDF2.0. These numbers can differ by a factor of up to almost four. Moreover, we observe that in particular for Tevatron and LHC data the ABMP16 and HERAPDF2.0 sets give significantly larger PDFα s uncertainties than the nominal PDF NNPDF3.0, whereas the CT14 or MMHT2014