Strong coupling $\alpha_s(m_Z)$ extraction from a combined NNLO analysis of inclusive electroweak boson cross sections at hadron colliders

The inclusive cross sections of W$^+$, W$^-$, and Z boson production from 34 different measurements performed in proton-(anti)proton collisions at center-of-mass energies $\sqrt{s}$ = 1.8--13 TeV, are compared to perturbative QCD calculations at next-to-next-to-leading-order (NNLO) accuracy with four sets of parton distributions functions (CT14, HERAPDF2.0, MMHT14, and NNPDF3.0 PDFs) and varying values of the strong coupling constant at the Z mass pole, $\alpha_s(m_Z)$. The data-theory agreement is good within the experimental and theoretical uncertainties, with the CT14 and MMHT14 parton densities providing the most overall consistent description of all cross section data. A value of $\alpha_s(m_Z) = 0.1188^{+0.0019}_{-0.0013}$ is extracted from a combined fit of the 28 experimental LHC measurements to the corresponding NNLO theoretical predictions obtained with the MMHT14 PDF set, which provides the most robust and stable QCD coupling extraction of this analysis.

The quality of the experimental and theoretical knowledge of the DY cross sections is such that they constitute today a key tool to accurately constrain the quark and gluon PDFs, through fits of their experimental differential distributions [22][23][24][25][26][27]. Recently, we proposed to exploit also the absolute inclusive W ± and Z cross sections, σ W,Z , as a means to precisely derive the value of α s (m Z ) through detailed comparisons of the data to pQCD predictions at NNLO accuracy [21]. Based on this approach, the CMS collaboration has recently analyzed [28] twelve measurements of W ± and Z cross sections in pp collisions at center-ofmass (c.m.) energies of √ s = 7 and 8 TeV [5,6] to extract α s (m Z ) = 0.1175 +0.0025 −0.0028 , a result with a final propagated ∼2.3% uncertainty that is comparable to that previously obtained in a similar analysis of the inclusive top-quark pair (tt) cross sections in pp collisions at the LHC [29]. Since the QCD coupling is known today through a combination of six types of observables, with a relatively poor ∼1% precision [30], the inclusion of new independent extraction methods is one of the suggested paths towards the reduction of the α s (m Z ) world average uncertainty [31]. In this context, the purpose of our work is twofold. First, a total of 34 electroweak boson cross section measurements performed at the LHC and Tevatron are compared to the corresponding state-of-the-art NNLO pQCD theoretical predictions. Second, we combine the results of the CMS analysis of Ref. [28] with those derived here from the W ± and Z cross sections measured at √ s = 7, 8, and 13 TeV by ATLAS [7][8][9] and LHCb [10][11][12][13] in order to more accurately determine α s (m Z ) by exploiting the maximum LHC data possible 1 . A single value of the QCD coupling constant is finally extracted through a proper combination of all individual LHC α s (m Z ) values derived as explained below.
The paper is organized as follows. In Section 2, the theoretical tools used to compute the W ± and Z cross sections are outlined. The list of Tevatron and LHC measurements, including fiducial criteria applied on their decay leptons, are collected in Section 3. The data are compared to the NNLO predictions in Section 4, and the preferred α s (m Z ) value for each measurement is derived in Section 5. The final QCD coupling constant is determined by combining all α s (m Z ) values through a χ 2 -minimization procedure that takes properly into account all propagated experimental and theoretical uncertainties and their correlations, as described in Section 6. The main conclusions of the work are summarized in Section 7.
corrections are then applied as a multiplicative correction factor, K EW = σ(NNLO,EWon)/σ(NNLO,EW-off), on top of the mcfm cross sections. They all represent a negative correction, in the range of 0.1-4%, of the (pure-QCD) DY cross section. Additional higherorder terms, from other photon-induced and mixed QCD⊗QED NLO processes, estimated to represent a few permille correction of the inclusive DY cross section [38,39], are neglected here.
The detailed fiducial cuts on the leptonic final state of each of the 34 experimental measurements, of which 22 are listed in the next section (the same information for CMS appears in Table 1 of Ref. [28]), are implemented into mcfm. All theoretical W ± and Z boson production cross sections quoted in the paper are multiplied by their measured (flavour-averaged) leptonic branching fractions [30], and are simply referred to as "cross sections" hereafter. A total of ∼20 000 computing jobs are run, taking into account all combinations of eigenvectors/replicas of each one of the 4 PDF sets, the 5 to 7 α s (m Z ) values of each PDF set available among (0.115, 0.116, 0.117, 0.118, 0.119, 0.120, and 0.121), and the 34 fiducial cross sections considered. Using at the time the longest 2-week computing queue at the CERN computing center, we reach 0.2-0.6% numerical accuracy on the theoretical predictions.

Experimental data sets
The study presented here is based on a combined analysis of most existing measurements of W ± and Z boson fiducial cross sections carried out by the ATLAS, CMS, and LHCb experiments at the LHC. The twelve CMS measurements, in separated electron (Z e , W ± e ) and muon (Z µ , W ± µ ) final states, appear compiled in Ref. [28]. Here, in addition we collect the seven ATLAS and nine LHCb data sets listed in Tables 1 and 2, respectively. The ATLAS results are for the combined leptonic (average of electron and muon) final states (hereafter labeled by default simply as Z and W ± , without any subindex, in all tables and plots), whereas those from LHCb are a mixture of electron, muon, and combined-lepton decays. The LHCb measurements are complementary to the ATLAS/CMS ones as they cover different rapidity phase spaces, the central region (|η| 2.5) for the latter, and the forward hemisphere (2 < η 4.5) for the former. In addition, despite not being used in the final α s (m Z ) extraction, but only compared to the NNLO predictions, we also list ( Table 3) the CDF and D0 measurements 2 , which feature 2-3 times larger experimental uncertainties (4.5-6.5%) than those at the LHC. All tables include the individual acceptance criteria applied on the transverse momentum (p T for charged leptons, p ν T for neutrinos) and rapidity η of the decay leptons for each measurement, the mass window around the Z peak considered (or the transverse mass m T in the W ± case), as well as the breakdown of the experimental uncertainties. The ATLAS cross sections have typical uncertainties in the range 1.8-2.8%, smaller than the 3.4-4.6% range of CMS [5,6], whereas the LHCb data are slightly more precise (1.6-2.7%) and include the knowledge of the pp collision energy as an 2 The Tevatron measurements of the W ± and Z boson cross sections at √ s = 630 GeV, and similarly the older CERN SppS and the most recent RHIC ones, are not discussed here, as their precision is even worse than that from the 1.8 and 1.96 TeV results.
-4 -extra ∼1% source of systematic uncertainty. In all cases, the main source of uncertainty is related to the integrated luminosity as shown in Table 4 that summarizes the range of all individual experimental (and theoretical) uncertainties and their correlations, which is an important source of information when combining all α s (m Z ) results as discussed in Section 6.    [2][3][4], in the various leptonic final states. The three sources of experimental uncertainties, and the selection criteria on the (W) Z (transverse) masses, are indicated.

Measurement
Inclusive cross section pp at √ s = 1.8 TeV (CDF) [2] (extrapolated to full acceptance) W ± e 2490 ± 20 (stat) ± 80 (syst) ± 90 (lumi) pb = 2490 ± 120 pb Z e (66 GeV < m Z < 116 GeV) 231 ± 6 (stat) ± 7 (syst) ± 8 (lumi) pb = 231 ± 12 pb pp at √ s = 1.8 TeV (D0) [3] (extrapolated to full acceptance) W ± e (40 GeV < m T < 120 GeV) 2310 ± 10 (stat) ± 50 (syst) ± 100 (lumi) pb = 2310 ± 110 pb Z e (66 GeV < m Z < 116 GeV) 221 ± 3 (stat) ± 4 (syst) ± 10 (lumi) pb = 221 ± 11 pb pp at √ s = 1.96 TeV (CDF) [4] (extrapolated to full acceptance) 254.9 ± 3.3 (stat) ± 4.6 (syst) ± 15.2 (lumi) pb = 254.9 ± 16.2 pb   Tables 1, 2, and 3, we have computed the corresponding theoretical NNLO pQCD predictions using the four PDF sets, and 5 to 7 α s (m Z ) variations per PDF aforementioned. The NLO EW corrections are computed with mcsanc just using one single PDF set and QCD coupling value (NNPDF3.0 and α s (m Z ) = 0.118), as those are used to adjust each one of the mcfm cross sections via a multiplicative K EW ratio, previously defined, where any PDF and α s (m Z ) dependencies largely cancel out. The numerical comparisons of all experimental and theoretical fiducial cross sections for W + , W − , and Z production in pp and pp collisions are listed in Tables 5-10. For each measurement, the fiducial cross section definition and the experimental result are tabulated along with their uncertainties. The mcfm NNLO predictions computed with each of the four PDF sets, with the NLO EW corrections already applied, are listed in--6 -cluding their associated PDF, α s (obtained from the cross section change when the default α s (m Z ) = 0.118 value is modified by ±0.002), and scale uncertainties. Similarly, the (negative) NLO EW (absolute and relative) correction factors are given. For various systems, the theoretical predictions obtained with alternative dynnlo [15] and fewz [16] NNLO pQCD calculators using various PDF sets, are also listed as provided in the original experimental references. For the ATLAS cases, the dynnlo and/or fewz predictions include also the NLO EW corrections as computed with mcsanc or fewz itself. The LHCb fewz theoretical predictions do not, however, include any EW correction. The level of agreement among the three NNLO pQCD calculators predictions for the same system(s) is good, at the percent level, except for LHCb at √ s = 13 TeV where relative differences of 3-5% are observed between the mcfm and fewz calculations.   Table 7: Comparison of the LHCb fiducial cross sections for W + , W − , and Z boson production in pp collisions at √ s = 7 and 13 TeV to the NNLO pQCD results obtained with mcfm using the CT14, HERAPDF2.0, MMHT14, and NNPDF3.0 PDF sets, including NLO EW corrections computed with mcsanc (also listed independently in the last row). Alternative fewz NNLO results (without EW corrections) quoted in the original LHCb works are also given for comparison.
The first observation from Figs. 1-3 is that not all theoretical predictions obtained with the default value of the QCD coupling constant consistently agree with all experimental ATLAS, LHCb, CDF, and D0 (as well as CMS, see Ref. [28]) measurements. Namely, although almost all theory predictions for all PDF sets are consistent with the data within the (relatively large) experimental and theoretical uncertainties, they do not always yield the same preferred value of α s (m Z ). Inspecting the figures in more detail (as well as the similar figures presented in Ref. [28] for CMS), one can see that among the theoretical predictions, those obtained with HERAPDF2.0 (NNPDF.3.0) tend to be mostly shifted to the left (right) of the σ W,Z vs. α s (m Z ) plots, i.e. they prefer comparatively smaller (larger) values of the QCD coupling. Or, otherwise said, since larger α s (m Z ) values trivially imply larger W ± and Z cross sections, the HERAPDF2.0 (NNPDF3.0) predictions for the default α s (m Z ) = 0.118 value tend to be mostly above (below) the data. The results computed with CT14 and MMHT14, on the other hand, appear mostly in-between those from the other two PDF sets. Thus, our first conclusions are that, in order to reproduce the experimental cross sections, HERAPDF2.0 (NNPDF3.0) tends in general to prefer a smaller (larger) value of α s (m Z ) than other PDFs, and that the predictions from CT14 and MMHT14 tend to be less scattered over the α s (m Z ) axis than those from HERAPDF2.0. A second result to point out is that, in general, the HERAPDF2.0 (MMHT14) filled ellipses have the smallest (largest) relative slope as a function of α s (m Z ) (a result also observed with the CMS data alone). A larger slope is advantageous for extracting the QCD coupling constant, as it indicates that the underlying α s (m Z ) value in the calculations has a larger impact on the computed cross sections, also leading to a lower propagated uncertainty in the α s (m Z ) value eventually derived by comparing the theoretical prediction to the experimental result.
The data-theory comparison plots shown in Fig. 3 for the Tevatron measurements feature about twice larger experimental uncertainties, and also much shallower dependence of the theoretical cross section on α s (m Z ), compared to their LHC counterpart measurements and calculations. The apparently smaller slopes in the Tevatron, compared to the LHC, -14 -figures is mostly due to the fact that the more imprecise experimental cross sections "flatten out" (via the Jpdfs convolution) the plotted ellipses.
In all plots of Figs. 1-3, the points where the filled ellipses cross the vertical dashed line at α s (m Z ) = 0.118 indicate the most likely cross section interval that would correspond to the baseline QCD coupling constant of all PDF sets, taking into account both the experimental and theoretical results. One can quantify the overall level of data-theory agreement through a goodness-of-fit test, χ 2 = ξ i (M −1 ) ij ξ j , where M is the covariance matrix accounting for all uncertainties and their correlations (Table 4), and ξ i = σ i,th − σ i,exp is the difference between the ATLAS, CMS, and LHCb experimental cross sections and their corresponding theoretical predictions for each PDF set. In the χ 2 calculation, the asymmetric uncertainties of the CT14, HERAPDF2.0, and MMHT14 PDF sets are symmetrized to the largest of the two values and also separately to the smallest of the two values. Table 11 lists the corresponding results. For the baseline QCD coupling constant value of α s (m Z ) = 0.118 of all PDF sets, the data-theory accord is better for the predictions calculated with CT14 and MMHT14 (χ 2 /ndf 1) compared to those obtained with the HERAPDF2.0 (when using the smaller symmetrized errors) and NNPDF3.0 sets (χ 2 /ndf 2.1).

Individual α s (m Z ) extraction per measurement
The cross sections calculated with different α s (m Z ) values are fitted (using χ 2 -minimization) to a first-order polynomial, and the corresponding slope k is extracted for each PDF and measurement. Over the considered α s (m Z ) range, the empirical linear fit describes well the observed σ th W,Z -versus-α s (m Z ) dependence for all PDF sets. The value of α s (m Z ) preferred by each individual measurement is determined by the crossing point of the fitted linear theoretical curve with the experimental horizontal line. It can be shown that all sources of uncertainty in the theoretical and experimental cross section, δσ, propagate as an α s (m Z ) uncertainty of δσ/k size, where k is the slope of the theoretical linear fit [21,28]. From this, it follows that since the luminosity (PDF) sources are the largest uncertainties in the cross section measurements (calculations), those propagate also as the dominant experimental (theoretical) uncertainties in each one of the α s (m Z ) values extracted.
The strong coupling values resulting from the fitting procedure described above are listed in Tables 15,16, and 17 of the Appendix for all individual ATLAS, LHCb, and Tevatron measurements, respectively, along with the uncertainty breakdowns from every source, for each PDF set. On the one hand, the LHC results feature extracted α s (m Z ) values with low overall uncertainty, in some cases as low as 2%. Similar results were obtained in Ref. [28] from the theoretical analysis of the CMS data. On the other hand, the α s (m Z ) extractions based on CDF and D0 data have propagated uncertainties above 7% in all cases, due to the less precise nature of the EW boson cross section measurements at the Tevatron. The latter extractions will, therefore, not be used in the final α s (m Z ) combination discussed in the next Section.  (Table 15), CMS [28], and LHCb (Table 16) W ± and Z cross section measurements, we determine a single α s (m Z ) per PDF by appropriately combining all results taking into account their uncorrelated, partially-, and fully-correlated experimental and theoretical uncertainties (Table 4). The following correlations among uncertainties are considered: • The integrated luminosity uncertainty is fully correlated for all α s (m Z ) results obtained at the same √ s for each experiment, but fully uncorrelated between different c.m. energies or experiments.
• The experimental systematic uncertainties among measurements at the same √ s are partially correlated, as quantified in Table 12 for the ATLAS and LHCb measurements. The correlation matrices used for the CMS results are those described in detail in Ref. [28]. For LHCb, the c.m. energy uncertainty is fully correlated at each √ s.
• The experimental statistical uncertainty is fully uncorrelated among all α s (m Z ) extractions.
• The scale uncertainty is partially correlated. Similarly as for the PDF uncertainties, for each pair of measurements a Pearson correlation coefficient is calculated using the results obtained from the theoretical scale variations. The scale correlations vary over 0.0-0.8. When combining the α s (m Z ) estimates, each specific correlation coefficient calculated for every specific pair of estimates is used.
• The theoretical numerical uncertainty is fully uncorrelated among all α s (m Z ) extractions. All the individual 28 α s (m Z ) derived per PDF set, and the correlation matrices associated with all their uncertainties are given as inputs of the convino v1.2 [40] program that is employed to determine the best final estimate of all combined values. The Neyman χ 2 -minimization procedure is selected in the code. Identical results are obtained, when symmetrizing all uncertainties, if one uses the similar BLUE method [41] to carry out the combination. Figure 4 shows the individual results (symbols with horizontal error bars) and the final combined α s (m Z ) value (vertical coloured areas) extracted for each PDF set. The width of the vertical coloured areas in the plot indicates the size of the total propagated uncertainty in the final QCD coupling value derived for each PDF set. Table 13 lists the obtained α s (m Z ) values, along with the uncertainty breakdowns from every source, determined for each PDF set through the combination of the 28 individual extractions. The total α s (m Z ) uncertainties derived for NNPDF3.0 are symmetric by construction, and a small asymmetry propagates into the final extractions for the other PDF sets. The total (symmetrized) uncertainties amount to ∼1.4% for CT14, ∼2.1% for HERAPDF2.0, ∼1.3% for MMHT14 and ∼1.6% for NNPDF3.0.
The last column of Table 13 lists the goodness-of-fit per degree of freedom (χ 2 /ndf) of Table 13: Strong coupling constant α s (m Z ) values extracted per PDF set by combining all the individual results obtained for each W ± and Z boson production cross section measurements, listed along with their propagated total and individual uncertainties. The last column tabulates the goodness-of-fit per degree of freedom χ 2 /ndf of the final single combined result compared to the 28 individual α s (m Z ) extractions. Such a seemingly counterintuitive behaviour, also observed in the analysis of the CMS data alone [28], is due to the presence of strong correlations among individual extractions, with the lowest α s (m Z ) values derived having smaller uncertainties than the others. The underlying tension apparent between NNPDF3.0 and the weak boson measurements at the LHC studied here, has been solved in the latest NNPDF3.1 global fit [42].
The final α s (m Z ) extractions are plotted in Fig. 5 (left) compared with the current -22 -world average of α s (m Z ) = 0.1181 ± 0.0011 (orange band) for each individual PDF. The (asymmetric) parabolas are constructed to have a minimum at the combined value and are fitted to go through ∆χ 2 = 1 (horizontal black lines) at the one std. deviation uncertainties quoted in Table 13.   Table 14 and in the three top data points of each PDF grouping of Fig. 6. This separation of data sets yields final α s (m Z ) values consistent with those derived from the combined ones listed in Table 13, but with the 8 (13) TeV extractions often  Table 14. The different symbols with error bars indicate groups of α s (m Z ) values obtained using different subsets of the data, or symmetrizing the PDF uncertainties, or adding an extra 1% uncorrelated uncertainty to all individual estimates. The shaded areas and vertical lines show the final results of our default α s (m Z ) extraction (Table 13). systematically preferring lower (higher) values of α s (m Z ) compared to those derived at 7 TeV (except for MMHT14, where all three extractions yield values very close to each other). A second cross-check is carried out by using only data from each experiment separately. The ATLAS and LHCb data tend to prefer slightly higher and lower, respectively, values of the QCD coupling, with the CMS results appearing mostly in between. The last tests include repeating the full α s (m Z ) combination after symmetrizing the PDF uncertainties or after adding an uncorrelated 1% numerical uncertainty to all theoretical cross sections. The latter choice aims at evaluating the impact of small extra uncertainties coming e.g. from the use of different calculators for the theoretical pQCD cross sections (see Tables 6, 7 and 8), or from potentially overlooked experimental uncertainties (LHCb quotes a ∼1% uncertainty from the knowledge of the c.m. energy, which is not explicitly considered by ATLAS and CMS). As one can see in Fig. 6, the results of MMHT14 are the most stable against any variations in the analysis, whereas a few larger-than-1-standard-deviation changes appear for the rest of the PDF sets. In the bottom-right row of Table 14 −0.0026 and +0.0062 −0.0013 , respectively, are observed that are more than three standard-deviations away from the default extraction for each one of these PDF sets (Table 13).
Among all α s (m Z ) extractions, we consider the value obtained with MMHT14 as the most reliable in this analysis for several reasons, partially concomitant with those outlined in the CMS-only study [28]. First, the cross sections computed with the MMHT14 PDF for many measurements feature the largest sensitivity to α s (m Z ) variations, i.e. the σ th W,Z vs. α s (m Z ) dependencies observed for this PDF set have a larger k slope than for the others ( Figs. 1 and 2). Second, the level of agreement among the combined and individual α s (m Z ) extractions is the best of all PDFs (as indicated by the χ 2 /ndf = 19.3/27 value listed in Table 13). Third, MMHT14 features the lowest relative (symmetrized) propagated uncertainty of all final α s (m Z ) results (Table 13). Last but not least, the QCD coupling values extracted with MMHT14 show the largest stability and robustness of all PDFs with respect to variations in the data sets, and in the assumptions of underlying uncertainties. Using the MMHT14 extraction as our baseline result, we obtain a final value of the QCD coupling constant at the Z pole mass of α s (m Z ) = 0.1188 +0.0019 −0.0013 (symmetrized to α s (m Z ) = 0.1188 ± 0.0016), with a total uncertainty of ∼1.3%. The right plot of Fig. 5 shows the asymmetric α s (m Z ) parabola extracted from the MMHT14 results alone compared to the current world average (orange band). This final extraction is fully consistent with the PDG world average, and has an overall uncertainty that is better than that of other recent determinations at this level of (NNLO) theoretical accuracy, such as those from EW precision fits [43], and tt cross sections [29,44]. In terms of precision, our determination compares well with the α s (m Z ) = 0.1191 ± 0.0018 value extracted from the theoretical analysis of τ lepton hadronic decays, which has an uncertainty of ∼1.5% [30,45,46].
We have presented a study of the production cross sections of electroweak gauge (W ± and Z) bosons in proton-(anti)proton (pp, pp) collisions at the LHC and Tevatron colliders at center-of-mass energies of √ s = 1.8, 1.96, 7, 8, and 13 TeV, aiming at the extraction of the QCD coupling constant at the Z mass scale. Thirty-four different experimental data sets available, corresponding to different fiducial criteria on the electron and/or muon decay final states at the LHC and Tevatron, have been individually compared to theoretical pQCD predictions computed at next-to-next-to-leading-order (NNLO) accuracy with the CT14, HERAPDF2.0, MMHT14, and NNPDF3.0 parton distribution functions (PDFs). An overall good data-theory agreement, within the experimental and theoretical uncertainties, is found for all measurements. A more detailed analysis of the 28 LHC measurements (7 from ATLAS, 12 from CMS, and 9 from LHCb) has been carried out with the aim of extracting a precise value of α s (m Z ). From the quantitative comparison of the theoretical predictions to each one of the LHC measurements, we extract 28 "preferred" values of the QCD coupling per PDF set. The largest experimental (theoretical) propagated uncertainties are associated with the integrated luminosity (intra-PDF) uncertainties. A −0.0013 , has a ∼1.3% uncertainty that is better than any other observable currently measured at hadron colliders, and comparable to that of the analysis of hadronic τ lepton decay data.
This work confirms that the total inclusive W ± and Z boson cross sections at hadron colliders are new promising observables that can provide useful constraints on the value of the QCD coupling constant, and that can eventually help to reduce the current relatively large uncertainty of the α s (m Z ) world average. The future availability of N 3 LO codes, with one additional higher degree of theoretical accuracy than the current state-of-the-art, for the calculation of inclusive W ± and Z boson production cross sections will allow for further reductions of the propagated scale uncertainties. Such a result, combined with upcoming electroweak boson measurements at the LHC with ∼1% experimental uncertainties, thanks to further reduced integrated luminosity uncertainties, will enable future α s (m Z ) extractions with propagated uncertainties below the 1% level.
A Independent α s (m Z ) extractions per EW boson cross section measurement Tables 15, 16 and 17 list the α s (m Z ) values obtained (along with the propagated uncertainty breakdowns from every source) from each individual ATLAS, LHCb, and Tevatron electroweak cross section measurement, respectively, for each PDF set.