# Determination of the strong coupling constant \({\varvec{{\alpha _\mathrm{s} (m_\mathrm{Z})}}}\) in next-to-next-to-leading order QCD using H1 jet cross section measurements

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## Abstract

The strong coupling constant \(\alpha _\mathrm{s}\) is determined from inclusive jet and dijet cross sections in neutral-current deep-inelastic *ep* scattering (DIS) measured at HERA by the H1 collaboration using next-to-next-to-leading order (NNLO) QCD predictions. The dependence of the NNLO predictions and of the resulting value of \(\alpha _\mathrm{s} (m_\mathrm{Z})\) at the *Z*-boson mass \(m_Z\) are studied as a function of the choice of the renormalisation and factorisation scales. Using inclusive jet and dijet data together, the strong coupling constant is determined to be \(\alpha _\mathrm{s} (m_\mathrm{Z}) =0.1157\,(20)_\mathrm{exp}\,(29)_\mathrm{th}\). Complementary, \(\alpha _\mathrm{s} (m_\mathrm{Z})\) is determined together with parton distribution functions of the proton (PDFs) from jet and inclusive DIS data measured by the H1 experiment. The value \(\alpha _\mathrm{s} (m_\mathrm{Z}) =0.1142\,(28)_\mathrm{tot}\) obtained is consistent with the determination from jet data alone. The impact of the jet data on the PDFs is studied. The running of the strong coupling is tested at different values of the renormalisation scale and the results are found to be in agreement with expectations.

## 1 Introduction

The strong coupling constant is one of the least well known parameters of the Standard Model of particle physics (SM) and a precise knowledge of this coupling is crucial for precision measurements, consistency tests of the SM and searches for physics beyond the SM. It has been determined in a large variety of processes and using different techniques [1, 2]. Jet production in the Breit frame [3] in neutral-current deep-inelastic *ep* scattering (NC DIS) is directly sensitive to the strong coupling and has a clean experimental signature with sizable cross sections. It is thus ideally suited for the precision determination of the strong coupling constant \(\alpha _\mathrm{s} (m_\mathrm{Z})\) at the *Z*-boson mass \(m_Z\).

Cross section predictions for inclusive jet and dijet production in NC DIS are obtained within the framework of perturbative QCD (pQCD) [4], where for the past 25 years only next-to-leading order (NLO) calculations have been available [5, 6]. Continuous developments enabled the advancement of these calculations [7, 8, 9, 10], and next-to-next-to-leading order (NNLO) predictions for jet production in DIS [11, 12] and hadron-hadron collisions [13, 14] have become available recently. The theoretical uncertainties of the NNLO predictions are substantially reduced compared to those of the NLO predictions. It is observed [11, 12, 15] that the NNLO predictions and the current experimental data are of comparable precision for large parts of the measured phase space.

Measurements of inclusive jet and dijet cross sections in NC DIS have been performed at HERA by the H1 [15, 16, 17, 18, 19, 20, 21, 22, 23, 24] and ZEUS [25, 26, 27, 28, 29, 30, 31, 32] collaborations during different data taking periods and for different centre-of-mass energies. In general, the predictions in pQCD provide a good description of these data.

The strong coupling constant has been determined from jet cross sections in DIS at NLO accuracy [15, 17, 21, 22, 23, 24, 27, 30, 33, 34, 35] and the precision of \(\alpha _\mathrm{s} (m_\mathrm{Z})\) of these determinations is typically limited by the scale uncertainty of the NLO calculations. Only recently an \(\alpha _\mathrm{s}\) determination was performed using inclusive jet cross sections, where NLO calculations have been supplemented with contributions beyond NLO in the threshold resummation formalism, and a moderate reduction of the scale uncertainty was achieved [36].

Measurements of jet production cross sections in processes other than NC DIS, such as photoproduction [37, 38] or in \(e^+e^-\) [39, 40, 41, 42, 43, 44], \(p\bar{p}\) [45, 46, 47] and *pp* collisions [48, 49, 50, 51, 52], have also been employed for the determination of the strong coupling constant. The corresponding predictions were at NLO accuracy in most cases, possibly supplemented with 2-loop threshold corrections or matched with next-to-leading logarithmic approximations (NLLA). An exception are 3-jet observables in \(e^+e^-\) collisions using predictions in NNLO accuracy [42], which are also matched to NLLA contributions [43, 44]. In contrast to variables such as hadronic event shape observables [53, 54] where only limited regions of the corresponding distributions are described by fixed order pQCD calculations, jet observables such as their transverse momenta typically are well described by such calculations over the full experimentally accessible range.

The presence of a proton in the initial state in lepton-hadron or hadron-hadron collisions complicates the determination of \(\alpha _\mathrm{s} (m_\mathrm{Z})\) and therefore \(\alpha _\mathrm{s} (m_\mathrm{Z})\) is often determined together with parton distribution functions of the proton (PDFs). Such simultaneous determinations of \(\alpha _\mathrm{s} (m_\mathrm{Z})\) and PDFs were performed using jet cross sections in DIS [17, 55, 56] or jet cross sections at either the LHC or Tevatron [50, 52, 57, 58, 59]. However, the absence of full NNLO corrections for jet production cross sections limited the theoretical precision of these approaches.

This article presents the first determination of \(\alpha _\mathrm{s} (m_\mathrm{Z})\) making use of the recent calculations of jet production at NNLO [11, 12, 13, 14]. These calculations are also used in this paper for the first time for the determination of PDFs. The jet cross section calculations are performed using the program Open image in new window [11, 12, 60].

Two strategies for the extraction of \(\alpha _\mathrm{s} (m_\mathrm{Z})\) are investigated. First, described in Sect. 3, the value of \(\alpha _\mathrm{s} (m_\mathrm{Z})\) is determined in NNLO from inclusive jet and dijet cross sections [15, 17, 21, 23, 24] using pre-determined PDFs as input. In a second approach described in Sect. 4, the value of \(\alpha _\mathrm{s} (m_\mathrm{Z})\) is determined together with the PDFs. This approach is denoted as ‘PDF+\(\alpha _\mathrm{s}\)-fit’ in the following and uses inclusive DIS data [61, 62, 63, 64, 65, 66] in addition to normalised jet cross section data [15, 21, 24], both measured by the H1 experiment [67, 68, 69, 70].

## 2 Cross section measurements

For the present analysis, measurements of jet cross sections and inclusive DIS cross sections in lepton-proton collisions performed by the H1 experiment at HERA are exploited.

*Jet cross sections*Cross sections for jet production in lepton-proton collisions have been measured by H1 at two different centre-of-mass energies using data from different periods of data taking. In the present analysis, inclusive jet and dijet cross sections measured in the range of negative four-momentum transfer squared \(5<Q^{2}<15\,000\,\mathrm {GeV}^2 \) and inelasticities \(0.2<y<0.7\) are considered. An overview of the individual measurements [15, 17, 21, 23, 24] is given in Table 1. Common to all data, jets are defined in the Breit frame [3] using the \(k_t\) clustering algorithm [71] with a resolution parameter \(R=1\). The jet four-vectors are restricted to the pseudorapidity range \(-1<\eta _\mathrm{lab}^\mathrm{jet}<2.5\) in the laboratory frame. The data sets ‘\(300\,\mathrm {GeV} \)’, ‘HERA-I’ and ‘HERA-II’ correspond to different data taking periods and are subdivided into two kinematic ranges, the low-\(Q^{2}\) (\(Q^{2}\lesssim 100\,\mathrm {GeV}^2 \)) and high-\(Q^{2}\) (\(Q^{2}\gtrsim 150\,\mathrm {GeV}^2 \)) domains, where different components of the H1 detector were used for the measurement of the scattered lepton.

Summary of the kinematic ranges of the studied inclusive jet and dijet data sets. The *ep* centre-of-mass energy \(\sqrt{s}\) and the integrated luminosity \(\mathcal {L}\) are shown. Kinematic restrictions are made on the negative four-momentum transfer squared \(Q^{2}\), the inelasticity *y* and the jet transverse momenta \(P_\mathrm{T}^\mathrm{jet}\) as indicated. Common to all data sets is a requirement on the pseudorapidity of the jets, \(-1<\eta _\mathrm{lab}^\mathrm{jet}<2.5\), not shown in the table. Dijet events are defined by extra cuts or on the average jet transverse momentum \(\langle P_\mathrm{T} \rangle \) or the invariant mass of the two leading jets \(m_{12}\). The asterisk denotes a cut not present in the original work [23] but imposed for the present analysis

Data set [ref.] | \(\sqrt{s}\) \([\mathrm {GeV} ]\) | \(\mathcal {L}\) \([\mathrm{pb}^{-1}]\) | DIS kinematic range | Inclusive jets | Dijets \(n_\mathrm{jets}\ge 2 \) |
---|---|---|---|---|---|

\(300\,\mathrm {GeV} \) [17] | 300 | 33 | \(150<Q^{2}<5000\,\mathrm {GeV}^2 \) | \(7<P_\mathrm{T}^\mathrm{jet}<50\,\mathrm {GeV} \) | \(P_\mathrm{T}^\mathrm{jet}>7\,\mathrm {GeV} \) |

\(0.2<y<0.6\) | \(8.5<\langle P_\mathrm{T} \rangle <35\,\mathrm {GeV} \) | ||||

HERA-I [23] | 319 | 43.5 | \(5<Q^{2}<100\,\mathrm {GeV}^2 \) | \(5<P_\mathrm{T}^\mathrm{jet}<80\,\mathrm {GeV} \) | \(5<P_\mathrm{T}^\mathrm{jet}<50\,\mathrm {GeV} \) |

\(0.2<y<0.7\) | \(5<\langle P_\mathrm{T} \rangle <80\,\mathrm {GeV} \) | ||||

\(m_{12}>18\,\mathrm {GeV} \) | |||||

\((\langle P_\mathrm{T} \rangle >7\,\mathrm {GeV} )^*\) | |||||

HERA-I [21] | 319 | 65.4 | \(150<Q^{2}<15000\,\mathrm {GeV}^2 \) | \(5<P_\mathrm{T}^\mathrm{jet}<50\,\mathrm {GeV} \) | − |

\(0.2<y<0.7\) | |||||

HERA-II [15] | 319 | 290 | \(5.5<Q^{2}<80\,\mathrm {GeV}^2 \) | \(4.5<P_\mathrm{T}^\mathrm{jet}<50\,\mathrm {GeV} \) | \(P_\mathrm{T}^\mathrm{jet}>4\,\mathrm {GeV} \) |

\(0.2<y<0.6\) | \(5<\langle P_\mathrm{T} \rangle <50\,\mathrm {GeV} \) | ||||

319 | 351 | \(150<Q^{2}<15000\,\mathrm {GeV}^2 \) | \(5<P_\mathrm{T}^\mathrm{jet}<50\,\mathrm {GeV} \) | \(5<P_\mathrm{T}^\mathrm{jet}<50\,\mathrm {GeV} \) | |

\(0.2<y<0.7\) | \(7<\langle P_\mathrm{T} \rangle <50\,\mathrm {GeV} \) | ||||

\(m_{12}>16\,\mathrm {GeV} \) |

Data set [ref.] | \(Q^{2}\) domain | Inclusive jets | Dijets | Normalised inclusive jets | Normalised dijets | Stat. corr. between samples |
---|---|---|---|---|---|---|

\(300\,\mathrm {GeV} \) [17] | High-\(Q^{2}\) | \(\checkmark \) | \(\checkmark \) | – | – | – |

HERA-I [23] | Low-\(Q^{2}\) | \(\checkmark \) | \(\checkmark \) | – | – | – |

HERA-I [21] | High-\(Q^{2}\) | \(\checkmark \) | – | \(\checkmark \) | – | – |

HERA-II [15] | Low-\(Q^{2}\) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) |

High-\(Q^{2}\) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) |

The inclusive jet cross sections are measured double-differentially as functions of \(Q^{2}\) and the jet transverse momentum in the Breit frame, \(P_\mathrm{T}^\mathrm{jet}\), where the phase space is constrained by \(Q^{2}\), *y*, \(\eta _\mathrm{lab}^\mathrm{jet}\) and \(P_\mathrm{T}^\mathrm{jet}\), as specified in Table 1.

Summary of the inclusive NC and CC DIS data sets. The lepton type, the *ep* centre-of-mass energy \(\sqrt{s}\) and the considered \(Q^{2}\) range are shown. The numbers in parenthesis show the whole kinematic range of the data prior to applying the \(Q^2\) cut specific for this analysis. The check-marks indicate the available measurements. The last column indicates cross sections determined with longitudinally polarised leptons

Data set [ref.] | Lepton type | \(\sqrt{s}\) \([\mathrm {GeV} ]\) | \(Q^{2}\) range \([\mathrm {GeV}^2 ]\) | NC cross sections | CC cross sections | Lepton beam polarisation |
---|---|---|---|---|---|---|

Combined low-\(Q^{2}\) [64] | \(e^+\) | 301,319 | (0.5) 12–150 | \(\checkmark \) | – | – |

Combined low-\(E_p\) [64] | \(e^+\) | 225,252 | (1.5) 12–90 | \(\checkmark \) | – | – |

94–97 [61] | \(e^+\) | 301 | 150–30 000 | \(\checkmark \) | \(\checkmark \) | – |

\(e^-\) | 319 | 150–30 000 | \(\checkmark \) | \(\checkmark \) | – | |

99–00 [63] | \(e^+\) | 319 | 150–30 000 | \(\checkmark \) | \(\checkmark \) | – |

HERA-II [65] | \(e^+\) | 319 | 120–30 000 | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) |

HERA-II [65] | \(e^-\) | 319 | 120–50 000 | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) |

Data from different periods and \(Q^{2}\) ranges are statistically independent, whereas dijet and inclusive jet data of the same data set are statistically correlated. These correlations have been determined for the HERA-II data sets [15, 24]. Different data sets, as well as inclusive jet and dijet data of the same data set, may furthermore share individual sources of experimental uncertainties [15, 56] and thus correlations are present for all data points considered.

*Normalised jet cross sections* The more recent data sets [15, 21, 24] also include measurements where the jet cross sections are normalised to the inclusive NC DIS cross section of the respective \(Q^{2}\) interval, as indicated in Table 2. Correlations of systematic and statistical uncertainties partially cancel for the ratio of jet cross sections and inclusive NC DIS cross sections. Therefore, normalised jet cross sections are ideally suited for studies together with inclusive NC DIS data.

*Inclusive DIS cross sections* In order to constrain the parameters of the PDFs in the PDF+\(\alpha _\mathrm{s}\)-fit, polarised and unpolarised inclusive NC and CC (charged current) DIS cross sections [61, 62, 63, 64, 65, 66] measured by the H1 experiment are used in addition. Data taken during different data taking periods and with different centre-of-mass energies are considered and a summary of these measurements is given in Table 3. This data sample is identical to the one used in the H1PDF2012 PDF fit [65], where correlations of experimental uncertainties have been quantified. Inclusive DIS and jet cross sections are statistically and experimentally correlated. These correlations are taken into account by using normalised jet cross sections.

## 3 Determination of \({\varvec{\alpha _\mathrm{s} (m_\mathrm{Z})}}\) from H1 jet cross sections

The strong coupling constant \(\alpha _\mathrm{s} (m_\mathrm{Z})\) is determined from inclusive jet and dijet cross sections in NC DIS measured by the H1 collaboration and using NNLO QCD predictions.

### 3.1 Predictions

*i*(for instance a ‘bin’ in the relevant physical observables) are calculated [4, 74] as a convolution in the variable

*x*of the PDFs \(f_k\) and perturbatively calculated partonic cross sections \({\hat{\sigma }}_{i,k}\),

*k*. The calculations depend on the renormalisation scale \(\mu _\mathrm{R}\) and the factorisation scale \(\mu _\mathrm{F}\). The factors \(c_{\mathrm{had},i}\) account for non-perturbative effects (hadronisation corrections).

*Z*-boson, \(m_\mathrm{Z} =91.1876\,\mathrm {GeV} \) [2]. Here, the calculations are performed in the modified minimal subtraction (\(\overline{\mathrm{MS}}\)) scheme in 3-loop accuracy and using 5 flavors, \(\alpha _\mathrm{s}(\mu _\mathrm{R}) =\alpha ^{(5)}_{\overline{\mathrm{MS}}}(\mu _\mathrm{R})\).

*x*-dependence of the PDFs \(f_k\) at a scale \(\mu _{0} \) and setting \(\mu _\mathrm{R} =\mu _\mathrm{F} \), the PDF at any factorisation scale \(\mu _\mathrm{F}\) is calculated as

The evolution starting scale is chosen to be \(\mu _{0} =20\,\mathrm {GeV} \). This is a typical scale of the jet data studied. As a consequence, the influence of the evolution of Eq. (5) on the \(\alpha _\mathrm{s}\) determination is moderate, because \(\mu _\mathrm{F} \approx \mu _{0} \). The PDFs at that scale are well known, in particular the quark densities. Moreover, the latter are to a large extent insensitive to the assumption made on the strong coupling \(\alpha _\mathrm{s} ^\mathrm{PDF}(m_\mathrm{Z})\) during their determination, because in leading order QCD inclusive DIS is independent of \(\alpha _\mathrm{s}\) Ṫhe gluon density is constrained due to QCD sum-rules and the precisely known quark densities. In the vicinity of a scale of \(20\,\mathrm {GeV} \) threshold effects from heavy quarks are not relevant. The PDFs at \(\mu _{0} =20\,\mathrm {GeV} \) are provided by the NNPDF3.1 PDF set [82] which was obtained with a nominal value of \(\alpha _\mathrm{s} ^\mathrm{PDF}(m_\mathrm{Z}) =0.118\). The influence of those choices is quantified in Sect. 3.3.

The inclusive jet and dijet NNLO predictions as a function of \(\mu _\mathrm{R}\) and \(\mu _\mathrm{F}\) are studied for selected phase space regions in Fig. 1. The dependence on the scale factor is strongest for cross sections at lower values \(\mu _\mathrm{R}\), i.e. lower values of \(Q^{2}\) and \(P_{T}\). The NNLO predictions depend less on the scale factor than the NLO predictions. Other choices of \(\mu _\mathrm{R}\) and \(\mu _\mathrm{F}\) are studied with the \(\alpha _\mathrm{s}\) fit in Sect. 3.3.

The hard coefficients \(\hat{\sigma }^{(n)}_{i,k}\) are calculated using the program Open image in new window [11, 12, 60], which is interfaced to fastNLO [92] to allow for computationally efficient, repeated calculations with different values of \(\alpha _\mathrm{s} (m_\mathrm{Z})\), different scale choices and different PDF sets. The PDFs are included in the LHAPDF package [93]. The evolution kernels are calculated using the program APFEL++ [94] and all results are validated with the programs APFEL [95] and QCDNUM [96, 97]. The \(\alpha _\mathrm{s}\) evolution is calculated using the APFEL++ code and validated with the CRunDec code [98], and the running of the electromagnetic coupling with \(Q^{2}\) is calculated using the package EPRC [99, 100]. The fits are performed using the Alpos fitting framework [101].

### 3.2 Methodology

*b*-quark.

The uncertainty calculated by TMinuit contains the experimental (exp), hadronisation (had) and PDF uncertainties (PDF). The breakdown of the uncertainties into these three components is obtained from repeated fits with \(V_\mathrm{had}\) and/or \(V_\mathrm{PDF}\) set to zero. Further uncertainties are defined in Sect. 3.3 and will be denoted as PDFset, PDF\(\alpha _\mathrm{s}\), and scale uncertainties. The theory uncertainty (‘th’) is defined as the quadratic sum of the PDF, PDFset, PDF\(\alpha _\mathrm{s}\), hadronisation and scale uncertainties, and the ‘total’ uncertainty considers additionally the experimental uncertainty.

The value of \(\alpha _\mathrm{s} (m_\mathrm{Z})\) is determined separately 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 (Table 2).

### 3.3 Sensitivity of the fit to input parameters

*Sensitivity to*\({{\alpha _\mathrm{s} (m_\mathrm{Z})}}\) The sensitivity of the data to \(\alpha _\mathrm{s} (m_\mathrm{Z})\) and the consistency of the calculations are investigated by performing fits with two free parameters representing the two distinct appearances of \(\alpha _\mathrm{s} (m_\mathrm{Z})\) in Eq. (1), i.e. in the PDF evolution, \(\alpha _\mathrm{s} ^{\Gamma }(m_\mathrm{Z}) \), and in the partonic cross sections, \(\alpha _\mathrm{s} ^{\hat{\sigma }}(m_Z)\). The cross sections with the \(\alpha _\mathrm{s}\) contributions identified separately are schematically expressed by

*Dependence on the choice of PDF* Values of \(\alpha _\mathrm{s} (m_\mathrm{Z})\) are determined for other PDF sets and for alternative values \(\alpha _\mathrm{s} ^\mathrm{PDF}(m_\mathrm{Z})\).

Three PDF related uncertainties are assigned to the fitted \(\alpha _\mathrm{s} (m_\mathrm{Z})\) results. The ‘PDF’ uncertainty originates from the data used for the PDF extraction [82]. A ‘PDFset’ uncertainty is defined as half of the maximum difference of the results from fits using the ABMP [104], CT14 [105], HERAPDF2.0 [56], MMHT [58] or NNPDF3.1 PDF set [82]. The ‘PDF\(\alpha _\mathrm{s}\) ’ uncertainty is defined as the difference of results from repeated fits using PDFs of the NNPDF3.1 series determined with \(\alpha _\mathrm{s} ^\mathrm{PDF}(m_\mathrm{Z})\) values differing by 0.002 [106]. This uncertainty can be considered to be uncorrelated to the PDF uncertainty [106, 107]. The size of the variation includes the NNPDF3.1 PDF set determined with \(\alpha _\mathrm{s} ^\mathrm{PDF}(m_\mathrm{Z}) =0.116\), where \(\alpha _\mathrm{s} ^\mathrm{PDF}(m_\mathrm{Z})\) is close to the fitted \(\alpha _\mathrm{s} (m_\mathrm{Z})\), in particular when restricting H1 jets to \(\tilde{\mu } >28\,\mathrm {GeV} \) (Fig. 5). The variation of \(\mu _{0} \) in the range 10 to 90 GeV is also studied but has negligible effect on the results.

*Scale variants and comparison of NLO and NNLO predictions* Studies of different choices for \(\mu _\mathrm{R}\) and \(\mu _\mathrm{F}\) are commonly used to estimate contributions of higher orders beyond NNLO.

Scale uncertainties are estimated through repeated fits with scale factors applied simultaneously to \(\mu _\mathrm{R}\) and \(\mu _\mathrm{F}\). Instead of varying the scales up and down by conventional factors, in this analysis a linear error propagation to the scale factors of 0.5 and 2 is performed using the derivative determined at the nominal scale. This is justified by the almost linear dependence on the logarithm of the scale factor (Figs. 6 and 7) and thus symmetric scale uncertainties are presented.

The fits are repeated with the partonic cross sections \(\hat{\sigma }_{i,k}\) calculated only up to NLO where for better comparisons identical scale definitions and identical PDFs determined in NNLO fits are used. For inclusive jets, the values of \(\chi ^{2}\)/\(n_\mathrm{dof}\) of the NLO fits are of comparable size for some of the studied scale choices, but are significantly worse for certain choices such as \(\mu _\mathrm{R} ^2=\mu _\mathrm{F} ^2=Q^{2}\). For dijets, the values of \(\chi ^{2}\)/\(n_\mathrm{dof}\) are always higher for NLO than for NNLO calculations. The NLO calculations exhibit an enhanced sensitivity to the choice of the scale and to scale variations, as compared to NNLO, resulting in scale uncertainties of \(\alpha _\mathrm{s} (m_\mathrm{Z})\) of 0.0077, 0.0081 and 0.0083 for inclusive jets, dijet and H1 jets, respectively, as compared to uncertainties of 0.0040, 0.0040 and 0.0042 in NNLO, respectively. The previously observed reduction of scale uncertainties of the cross section predictions at NNLO [11, 12, 15] is reflected in a corresponding reduction of the \(\alpha _\mathrm{s} (m_\mathrm{Z})\) scale uncertainties.

*Restricting the scale*\({{\tilde{\mu }}}\) In order to study the size of the uncertainties as a function of \(\tilde{\mu } \), the fits to inclusive jet and to dijet cross sections are repeated using data points exceeding a given value \(\tilde{\mu } _\mathrm{cut}\). The resulting uncertainties are displayed in Fig. 10. The experimental uncertainties are smaller for lower \(\tilde{\mu } \). This is because more data are considered in the fit, but also since the data at lower values of \(\tilde{\mu } \) have an enhanced sensitivity to \(\alpha _\mathrm{s} (m_\mathrm{Z})\) due to the running of the strong coupling. In contrast, the scale uncertainties of the NNLO cross section predictions are largest for low values of \(\tilde{\mu } \), and thus decrease with increasing \(\tilde{\mu } \). Considering only data with values of \(\tilde{\mu } \) above approximately \(30\,\mathrm {GeV} \) the experimental and scale uncertainty become similar in size.

The result obtained with \(\tilde{\mu } >28\,\mathrm {GeV} \) is considered as the main result of this article.

At values of \(\tilde{\mu } _\mathrm{cut}\) around \(20\,\mathrm {GeV} \) the PDF\(\alpha _\mathrm{s}\) uncertainty effectively vanishes. In other words, the fit result is insensitive to the \(\alpha _\mathrm{s} ^\mathrm{PDF}(m_\mathrm{Z})\) assumptions made for the PDF determination. A possible explanation is the gradual change of the fraction of gluon and quark induced processes with \(\tilde{\mu }\): data at lower values of \(\tilde{\mu }\) have contributions from low-*x* where the gluon PDF is dominating, whereas data at higher values of \(\tilde{\mu }\) have a successively higher fraction of quark induced processes. The quark PDFs are less dependent on \(\alpha _\mathrm{s} ^\mathrm{PDF}(m_\mathrm{Z})\) than the gluon PDF, and are well determined by inclusive DIS data.

### 3.4 Results

*The value of the strong coupling constant*\({{\alpha _\mathrm{s} (m_\mathrm{Z})}}\) The values of \(\alpha _\mathrm{s} (m_\mathrm{Z})\) 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.

Summary of values of \(\alpha _\mathrm{s} (m_\mathrm{Z})\) from fits to H1 jet cross section measurements using NNLO predictions. The uncertainties denote the experimental (exp), hadronisation (had), PDF, PDF\(\alpha _\mathrm{s}\), 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 \(\chi ^{2}/n_\mathrm{dof}\) 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 scale \(\tilde{\mu } _{\mathrm{cut}}\). The fourth group of fits, labelled H1 jets, is made using all available dijet and inclusive jet data together, for three different choices of \(\tilde{\mu } _{\mathrm{cut}}\). 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 thus no overall theoretical uncertainty is quoted

\(\alpha _\mathrm{s} (m_\mathrm{Z})\) values from H1 jet cross sections | |||||
---|---|---|---|---|---|

Data | \(\tilde{\mu } _{\mathrm{cut}}\) | \(\alpha _\mathrm{s} (m_\mathrm{Z})\) with uncertainties | th | tot | \(\chi ^{2}/n_\mathrm{dof}\) |

| |||||

\(300\,\mathrm {GeV} \) high-\(Q^{2}\) | \(2m_b\) | \( 0.1221\,(31)_\mathrm{exp}\,(22)_\mathrm{had}\,(5)_\mathrm{PDF}\,(3)_\mathrm{PDF\alpha _\mathrm{s}}\,(4)_\mathrm{PDFset}\,(36)_\mathrm{scale}\) | \((43)_\mathrm{th}\) | \((53)_\mathrm{tot}\) | 6.5 / 15 |

HERA-I low-\(Q^{2}\) | \(2m_b\) | \( 0.1093\,(17)_\mathrm{exp}\,(8)_\mathrm{had}\,(5)_\mathrm{PDF}\,(5)_\mathrm{PDF\alpha _\mathrm{s}}\,(7)_\mathrm{PDFset}\,(33)_\mathrm{scale}\) | \((35)_\mathrm{th}\) | \((39)_\mathrm{tot}\) | 17.5 / 22 |

HERA-I high-\(Q^{2}\) | \(2m_b\) | \( 0.1136\,(24)_\mathrm{exp}\,(9)_\mathrm{had}\,(6)_\mathrm{PDF}\,(4)_\mathrm{PDF\alpha _\mathrm{s}}\,(4)_\mathrm{PDFset}\,(31)_\mathrm{scale}\) | \((33)_\mathrm{th}\) | \((41)_\mathrm{tot}\) | 14.7 / 23 |

HERA-II low-\(Q^{2}\) | \(2m_b\) | \( 0.1187\,(18)_\mathrm{exp}\,(8)_\mathrm{had}\,(4)_\mathrm{PDF}\,(4)_\mathrm{PDF\alpha _\mathrm{s}}\,(3)_\mathrm{PDFset}\,(45)_\mathrm{scale}\) | \((46)_\mathrm{th}\) | \((50)_\mathrm{tot}\) | 29.6 / 40 |

HERA-II high-\(Q^{2}\) | \(2m_b\) | \( 0.1121\,(18)_\mathrm{exp}\,(9)_\mathrm{had}\,(5)_\mathrm{PDF}\,(4)_\mathrm{PDF\alpha _\mathrm{s}}\,(2)_\mathrm{PDFset}\,(35)_\mathrm{scale}\) | \((37)_\mathrm{th}\) | \((41)_\mathrm{tot}\) | 42.5 / 29 |

| |||||

\(300\,\mathrm {GeV} \) high-\(Q^{2}\) | \(2m_b\) | \( 0.1213\,(39)_\mathrm{exp}\,(17)_\mathrm{had}\,(5)_\mathrm{PDF}\,(2)_\mathrm{PDF\alpha _\mathrm{s}}\,(3)_\mathrm{PDFset}\,(31)_\mathrm{scale}\) | \((35)_\mathrm{th}\) | \((52)_\mathrm{tot}\) | 13.6 / 15 |

HERA-I low-\(Q^{2}\) | \(2m_b\) | \( 0.1101\,(23)_\mathrm{exp}\,(8)_\mathrm{had}\,(5)_\mathrm{PDF}\,(4)_\mathrm{PDF\alpha _\mathrm{s}}\,(5)_\mathrm{PDFset}\,(36)_\mathrm{scale}\) | \((38)_\mathrm{th}\) | \((45)_\mathrm{tot}\) | 10.4 / 20 |

HERA-II low-\(Q^{2}\) | \(2m_b\) | \( 0.1173\,(14)_\mathrm{exp}\,(9)_\mathrm{had}\,(5)_\mathrm{PDF}\,(5)_\mathrm{PDF\alpha _\mathrm{s}}\,(3)_\mathrm{PDFset}\,(44)_\mathrm{scale}\) | \((45)_\mathrm{th}\) | \((47)_\mathrm{tot}\) | 17.4 / 41 |

| |||||

HERA-II high-\(Q^{2}\) | \(2m_b\) | \( 0.1089\,(21)_\mathrm{exp}\,(7)_\mathrm{had}\,(5)_\mathrm{PDF}\,(3)_\mathrm{PDF\alpha _\mathrm{s}}\,(3)_\mathrm{PDFset}\,(25)_\mathrm{scale}\) | \((27)_\mathrm{th}\) | \((34)_\mathrm{tot}\) | 28.0 / 23 |

H1 inclusive jets | \(2m_b\) | \( 0.1132\,(10)_\mathrm{exp}\,(5)_\mathrm{had}\,(4)_\mathrm{PDF}\,(4)_\mathrm{PDF\alpha _\mathrm{s}}\,(2)_\mathrm{PDFset}\,(40)_\mathrm{scale}\) | \((40)_\mathrm{th}\) | \((42)_\mathrm{tot}\) | 134.0 / 133 |

H1 inclusive jets | \(28\,\mathrm {GeV} \) | \( 0.1152\,(20)_\mathrm{exp}\,(6)_\mathrm{had}\,(2)_\mathrm{PDF}\,(2)_\mathrm{PDF\alpha _\mathrm{s}}\,(3)_\mathrm{PDFset}\,(26)_\mathrm{scale}\) | \((27)_\mathrm{th}\) | \((33)_\mathrm{tot}\) | 44.1 / 60 |

H1 dijets | \(2m_b\) | \( 0.1148\,(11)_\mathrm{exp}\,(6)_\mathrm{had}\,(5)_\mathrm{PDF}\,(4)_\mathrm{PDF\alpha _\mathrm{s}}\,(4)_\mathrm{PDFset}\,(40)_\mathrm{scale}\) | \((41)_\mathrm{th}\) | \((42)_\mathrm{tot}\) | 93.9 / 102 |

H1 dijets | \(28\,\mathrm {GeV} \) | \( 0.1147\,(24)_\mathrm{exp}\,(5)_\mathrm{had}\,(3)_\mathrm{PDF}\,(2)_\mathrm{PDF\alpha _\mathrm{s}}\,(3)_\mathrm{PDFset}\,(24)_\mathrm{scale}\) | \((25)_\mathrm{th}\) | \((35)_\mathrm{tot}\) | 30.8 / 43 |

H1 jets | \(2m_b\) | \( 0.1143\,(9)_\mathrm{exp}\,(6)_\mathrm{had}\,(5)_\mathrm{PDF}\,(5)_\mathrm{PDF\alpha _\mathrm{s}}\,(4)_\mathrm{PDFset}\,(42)_\mathrm{scale}\) | \((43)_\mathrm{th}\) | \((44)_\mathrm{tot}\) | 195.0 / 199 |

H1 jets | \(28\,\mathrm {GeV} \) | \( 0.1157\,(20)_\mathrm{exp}\,(6)_\mathrm{had}\,(3)_\mathrm{PDF}\,(2)_\mathrm{PDF\alpha _\mathrm{s}}\,(3)_\mathrm{PDFset}\,(27)_\mathrm{scale}\) | \((28)_\mathrm{th}\) | \((34)_\mathrm{tot}\) | 63.2 / 90 |

H1 jets | \(42\,\mathrm {GeV} \) | \( 0.1168\,(22)_\mathrm{exp}\,(7)_\mathrm{had}\,(2)_\mathrm{PDF}\,(2)_\mathrm{PDF\alpha _\mathrm{s}}\,(5)_\mathrm{PDFset}\,(17)_\mathrm{scale}\) | \((20)_\mathrm{th}\) | \((30)_\mathrm{tot}\) | 37.6 / 40 |

| |||||

\(2m_b\) | \(0.1142\,(11)_\mathrm{exp,NP,PDF}\,(2)_\mathrm{mod}\,(2)_\mathrm{par}\,(26)_\mathrm{scale}\) | \((28)_\mathrm{tot}\) | 1539.7 / 1516 |

For the fits to the individual data sets the \(\chi ^{2}\)/\(n_\mathrm{dof}\) is around unity in most cases. The \(\alpha _\mathrm{s} (m_\mathrm{Z})\) values are all found to be consistent, in particular between inclusive jet and dijet measurements.

The fits to the inclusive jet data exhibit \(\chi ^{2}\)/\(n_\mathrm{dof}\) values around unity, thus indicating the consistency of the individual data sets. The value of \(\alpha _\mathrm{s} (m_\mathrm{Z})\) from ‘H1 inclusive jets’ has a significantly reduced experimental uncertainty compared to the results for the individual data sets. The cut \(\tilde{\mu } >28\,\mathrm {GeV} \) results for inclusive jets in \(\alpha _\mathrm{s} (m_\mathrm{Z}) =0.1152\,(20)_\mathrm{exp}\,(27)_\mathrm{th}\), which is consistent with the world average [2, 108].

Value of \(\chi ^{2}\)/\(n_\mathrm{dof}\) around 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 scales \(\tilde{\mu } >28\,\mathrm {GeV} \), a value \(\alpha _\mathrm{s} (m_\mathrm{Z}) =0.1147\,(24)_\mathrm{exp}\,(25)_\mathrm{th}\) is found.

The fit to H1 jets yields \(\chi ^{2}/n_\mathrm{dof}= 0.98\) for 200 data points and \(\alpha _\mathrm{s} (m_\mathrm{Z}) =0.1143\,(9)_\mathrm{exp}\,(43)_\mathrm{th}\). The scale uncertainty is the largest among the theoretical uncertainties and all other uncertainties are negligible in comparison.

^{1}Therefore, this \(\alpha _\mathrm{s} (m_\mathrm{Z})\) determination is taken as the main result. This result as well as those results obtained from the inclusive jet and dijet data separately are consistent with the world average.

The main result is also found to be consistent with \(\alpha _\mathrm{s} (m_\mathrm{Z}) =0.1165(8)_{\text {exp}}(38)_{\text {pdf,theo}}\) determined previously in NLO accuracy from normalised H1 HERA-II high-\(Q^{2}\) jet cross section data [24]. That result is experimentally more precise, mainly because data at somewhat lower scales and three-jet data are included.^{2} The scale uncertainty of the previous NLO fit is larger than for the present analysis in NNLO, despite of the fact that it was considered to be partially uncorrelated bin-to-bin in the previous NLO fit, whereas the present approach is more conservative.

In the present analysis, the value with the smallest total uncertainty is obtained in a fit to H1 jets restricted to \(\tilde{\mu } >42\,\mathrm {GeV} \) with the result \(\alpha _\mathrm{s} (m_\mathrm{Z}) =0.1168\,(22)_\mathrm{exp}\,(20)_\mathrm{theo}\) and a value of \(\chi ^{2}/n_\mathrm{dof}=37.6/40\). This result, however, is obtained from a very limited number of measurements, the precision of which is limited by statistical uncertainties.

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 \([\tilde{\mu } _\mathrm{lo};\tilde{\mu } _\mathrm{up}]\) of \(\tilde{\mu }\). The data point grouping and the interval boundaries can be read off Fig. 12. The assumptions on the running of \(\alpha _\mathrm{s}(\mu _\mathrm{R})\) thus are for each fit restricted to a limited \(\mu _\mathrm{R}\) range.^{3} For a given data point its \(\tilde{\mu }\) value is representative for the \(\mu _\mathrm{R}\) range probed by the corresponding prediction, see Eqs. (8) and (9). The fit results are for each interval shown at the representative scale \(\mu _\mathrm{R} =\sqrt{\tilde{\mu } _\mathrm{lo}\tilde{\mu } _\mathrm{up}}\).

The values obtained from fits to H1 jets are compared to other determinations of at least NNLO accuracy [41, 44, 54, 109] and to results at NLO at very high scale [52] in Fig. 14, and consistency with the other experiments is found.

The results are consistent with results obtained from an alternative method used as a cross check, where in a single fit with ten free parameters the \(\alpha _\mathrm{s}\) values in the ten bins are determined simultaneously.

## 4 Simultaneous \({\varvec{\alpha _\mathrm{s}}}\) and PDF determination

In addition to the fits described above also a fit in NNLO accuracy of \(\alpha _\mathrm{s} (m_\mathrm{Z})\) together with the non-perturbative PDFs is performed which takes jet data and inclusive DIS data as input. This fit is denoted as ‘PDF+\(\alpha _\mathrm{s}\)-fit’ in the following.

### 4.1 Methodology

The methodology of the PDF+\(\alpha _\mathrm{s}\)-fit is closely related to PDF determinations as performed by other groups [56, 58, 82, 104, 105]. The PDFs are parametrised at a low starting scale \(\mu _0\) which is below the charm-quark mass. Heavy-quark PDFs are generated dynamically and only light-quark PDFs and the gluon distribution have to be determined in the fit.

In order to have constraints on the PDFs, polarised and unpolarised inclusive NC and CC DIS cross sections [61, 62, 63, 64, 65, 66] are used (Table 3). This data sample is identical to the one used in the H1PDF2012 PDF fit [65]. In addition, normalised inclusive jet and dijet cross sections [15, 21, 24] are used (Table 2).

The calculations of the splitting kernels are performed in NNLO using the program QCDNUM [96, 97]. The predictions for the inclusive DIS cross sections are calculated using structure function calculations in NNLO using the zero-mass variable flavour number scheme (ZM-VFNS) [65] as implemented in QCDNUM [96, 97]. Normalised jet cross sections are calculated as a ratio of jet cross sections to inclusive NC DIS, where the former are calculated as outlined in Sect. 3.1 and the latter are calculated using ZM-VFNS structure functions using QCDNUM. For inclusive DIS predictions the scales \(\mu _\mathrm{R} ^2\) and \(\mu _\mathrm{F} ^2\) are both set to \(Q^{2}\) and for jet predictions to \(Q^{2}+P_{T}^2\), as specified in Eq. (6).

*x*-range down to 0.003, whereas without these cuts it would be 0.002. Major contributions to the data points at highest values of \(\tilde{\mu }\) are within the

*x*-range 0.1 to 0.5.

*f*is one of

*g*, \(\tilde{u}\), \(\tilde{d}\), \(\bar{U}\), \(\bar{D}\), denoting the density of the gluon, up-valence, down-valence, up-sea, down-sea in the proton, respectively. The strange sea is set to \(\bar{s}(x)=f_s\bar{D}\), where \(f_s=0.4\). Parameters \(f_D\) and \(f_E\) are set to zero by default, but are added for specific flavours in order to improve the fit. The parameters \(g_A\), \(\tilde{u}_A\) and \(\tilde{d}_A\) are constrained by sum rules. The parameter \(\bar{U}_A\) is set equal to \(\bar{D}_A(1-f_s)\). The parameter \(\bar{U}_B\) is set equal to \(\bar{D}_B\). A total of 12 fit parameters are used to describe the PDFs.

Values of the strong coupling constant \(\alpha _\mathrm{s}(\mu _\mathrm{R})\) and at the *Z*-boson mass, \(\alpha _\mathrm{s} (m_\mathrm{Z})\), obtained from fits to groups of data points with comparable values of \(\mu _\mathrm{R} \). 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 | ||||||
---|---|---|---|---|---|---|

\(\mu _\mathrm{R}\) \([\hbox {GeV}]\) | Inclusive jets | Dijets | H1 jets | |||

\(\alpha _\mathrm{s} (m_\mathrm{Z})\) | \(\alpha _\mathrm{s}(\mu _\mathrm{R})\) | \(\alpha _\mathrm{s} (m_\mathrm{Z})\) | \(\alpha _\mathrm{s}(\mu _\mathrm{R})\) | \(\alpha _\mathrm{s} (m_\mathrm{Z})\) | \(\alpha _\mathrm{s}(\mu _\mathrm{R})\) | |

7.4 | \( 0.1148\,(13)\,(42)\) | \(0.1830\,(34)\,(114) \) | \( 0.1182\,(28)\,(41)\) | \(0.1923\,(77)\,(116) \) | \( 0.1147\,(13)\,(43)\) | \(0.1829\,(34)\,(114) \) |

10.1 | \( 0.1136\,(17)\,(36)\) | \(0.1678\,(39)\,(81) \) | \( 0.1169\,(14)\,(42)\) | \(0.1751\,(34)\,(99) \) | \( 0.1148\,(14)\,(40)\) | \(0.1705\,(31)\,(91) \) |

13.3 | \( 0.1147\,(15)\,(43)\) | \(0.1605\,(30)\,(88) \) | \( 0.1131\,(18)\,(38)\) | \(0.1573\,(36)\,(76) \) | \( 0.1144\,(15)\,(42)\) | \(0.1600\,(30)\,(86) \) |

17.2 | \( 0.1130\,(15)\,(33)\) | \(0.1492\,(26)\,(59) \) | \( 0.1104\,(19)\,(30)\) | \(0.1445\,(33)\,(53) \) | \( 0.1127\,(15)\,(33)\) | \(0.1486\,(27)\,(59) \) |

20.1 | \( 0.1136\,(17)\,(33)\) | \(0.1457\,(29)\,(56) \) | \( 0.1116\,(22)\,(31)\) | \(0.1425\,(36)\,(52) \) | \( 0.1134\,(17)\,(33)\) | \(0.1454\,(29)\,(55) \) |

24.5 | \( 0.1173\,(17)\,(30)\) | \(0.1463\,(26)\,(48) \) | \( 0.1147\,(23)\,(24)\) | \(0.1423\,(36)\,(38) \) | \( 0.1171\,(17)\,(29)\) | \(0.1460\,(27)\,(46) \) |

29.3 | \( 0.1084\,(36)\,(29)\) | \(0.1287\,(51)\,(41) \) | \( 0.1163\,(34)\,(34)\) | \(0.1401\,(50)\,(50) \) | \( 0.1134\,(30)\,(32)\) | \(0.1358\,(44)\,(46) \) |

36.0 | \( 0.1153\,(32)\,(37)\) | \(0.1338\,(43)\,(50) \) | \( 0.1135\,(37)\,(29)\) | \(0.1314\,(50)\,(39) \) | \( 0.1146\,(30)\,(33)\) | \(0.1328\,(41)\,(44) \) |

49.0 | \( 0.1170\,(22)\,(20)\) | \( 0.1290\,(27)\,(25) \) | \( 0.1127\,(31)\,(15)\) | \( 0.1238\,(37)\,(18) \) | \( 0.1169\,(23)\,(19)\) | \( 0.1290\,(28)\,(24) \) |

77.5 | \( 0.1111\,(55)\,(19)\) | \(0.1137\,(58)\,(20) \) | \( 0.1074\,(84)\,(19)\) | \(0.1099\,(88)\,(20) \) | \( 0.1113\,(55)\,(19)\) | \(0.1139\,(58)\,(20) \) |

The uncertainty obtained from the fit comprises experimental uncertainties of the data and hadronisation uncertainties of the jet cross section predictions. The resulting uncertainty of \(\alpha _\mathrm{s} (m_\mathrm{Z})\) from the PDF+\(\alpha _\mathrm{s}\)-fit is denoted as ‘exp,had,PDF’. In order to determine also model (‘mod’) and parametrisation (‘par’) uncertainties, an additional error estimation similar to HERAPDF2.0 [56] is performed. The model uncertainty is estimated as the quadratic sum of the differences of the nominal result to the resulting values of \(\alpha _\mathrm{s}\) when repeating the PDF+\(\alpha _\mathrm{s}\)-fit with alternative parameters, such as the charm or beauty masses or the sea quark suppression factor \(f_\mathrm{s}\) [56]. Parametrisation uncertainties are attributed by adding extra \(f_D\) or \(f_E\) parameters to the fit or by varying the starting scale. In addition, a more flexible functional form is allowed for the gluon, similar to the PDF parametrisation used for the default HERAPDF2.0 [56] fit.^{4} A total of eight parametric forms different from the default are considered.

The scale uncertainty of \(\alpha _\mathrm{s} (m_\mathrm{Z})\) from this fit is determined by repeating fits with scale factors 0.5 and 2 applied to \(\mu _\mathrm{R}\) and \(\mu _\mathrm{F}\) simultaneously to all calculations involved. The larger of the two deviations from the central fit, corresponding to a scale factor of 0.5, is taken as symmetric scale uncertainty. A more detailed study is beyond the scope of this paper.

The PDF+\(\alpha _\mathrm{s}\)-fit differs from the \(\alpha _\mathrm{s}\)-fit outlined in Sect. 3 in the following aspects: the usage of normalised jet cross sections, the inclusion of NC and CC DIS cross sections and the low starting scale \(\mu _0\) of the DGLAP evolution, thus assuming the validity of the running coupling and the PDF evolution down to lower scale values.

### 4.2 Results

*Fit results and the value of* \({{\alpha _\mathrm{s} (m_\mathrm{Z})}}\) The results of the PDF+\(\alpha _\mathrm{s}\)-fit are presented in Table 6. The fit yields \(\chi ^{2}/n_\mathrm{dof}=1539.7/(1529-13)\), 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.

Results of the PDF+\(\alpha _\mathrm{s}\) fit. The columns denote the resulting fit value, its uncertainty and the correlations to the other parameters

Results for the PDF+\({{{\alpha _\mathrm{s}}}}\)-fit | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Parameter | Fit result | Correlation coefficients | ||||||||||||

\(\alpha _\mathrm{s} (m_\mathrm{Z})\) | \(g_B\) | \(g_C\) | \(g_D\) | \(\tilde{u}_B\) | \(\tilde{u}_C\) | \(\tilde{u}_E\) | \(\tilde{d}_B\) | \(\tilde{d}_C\) | \(\bar{U}_C\) | \(\bar{D}_A\) | \(\bar{D}_B\) | \(\bar{D}_C\) | ||

\(\alpha _\mathrm{s} (m_\mathrm{Z})\) | \(0.1142\pm 0.0011 \) | 1 | ||||||||||||

\(g_B\) | \( -0.023 \pm 0.035 \) | 0.25 | 1 | |||||||||||

\(g_C\) | \( 5.69 \pm 4.09 \) | \(-0.08\) | 0.01 | 1 | ||||||||||

\(g_D\) | \( -0.44 \pm 4.20 \) | \(-0.03\) | \(-0.10\) | 0.99 | 1 | |||||||||

\(\tilde{u}_B\) | \( 0.707 \pm 0.036 \) | 0.39 | 0.25 | 0.05 | 0.04 | 1 | ||||||||

\(\tilde{u}_C\) | \( 4.909 \pm 0.096 \) | \(-0.09\) | \(-0.13\) | 0.02 | 0.03 | \(-0.08\) | 1 | |||||||

\(\tilde{u}_E\) | \( 12.7 \pm 1.8 \) | \(-0.03\) | \(-0.25\) | \(-0.04\) | \(-0.01\) | \(-0.75\) | 0.57 | 1 | ||||||

\(\tilde{d}_B\) | \( 1.036 \pm 0.098 \) | 0.24 | \(-0.02\) | 0.06 | 0.08 | 0.32 | \(-0.24\) | \(-0.24\) | 1 | |||||

\(\tilde{d}_C\) | \( 5.35 \pm 0.49 \) | \(-0.10\) | \(-0.07\) | 0.03 | 0.05 | \(-0.08\) | \(-0.24\) | 0.00 | 0.80 | 1 | ||||

\(\bar{U}_C\) | \( 4.96 \pm 0.86 \) | 0.32 | \(-0.28\) | \(-0.01\) | 0.05 | 0.76 | 0.09 | \(-0.39\) | 0.53 | 0.11 | 1 | |||

\(\bar{D}_A\) | \( 0.299 \pm 0.032 \) | 0.29 | \(-0.71\) | \(-0.04\) | 0.07 | 0.32 | 0.01 | \(-0.08\) | 0.38 | 0.13 | 0.71 | 1 | ||

\(\bar{D}_B\) | \( -0.091 \pm 0.017 \) | 0.22 | \(-0.79\) | \(-0.05\) | 0.06 | 0.19 | 0.03 | 0.01 | 0.29 | 0.09 | 0.61 | 0.97 | 1 | |

\(\bar{D}_C\) | \( 16.1 \pm 3.8 \) | \(-0.13\) | \(-0.51\) | \(-0.01\) | 0.08 | \(-0.15\) | \(-0.24\) | \(-0.06\) | 0.14 | 0.24 | 0.05 | 0.48 | 0.46 | 1 |

\(g_A\) | 2.84 | Constrained by sum-rules | ||||||||||||

\(\tilde{u}_A\) | 4.11 | Constrained by sum-rules | ||||||||||||

\(\tilde{d}_A\) | 6.94 | Constrained by sum-rules | ||||||||||||

\(\bar{U}_A\) | 1.80 | Set equal to \(\bar{D}_A(1-f_s)\) | ||||||||||||

\(\bar{U}_B\) | \(-0.091\) | Set equal to \(\bar{D}_B\) |

*PDF parametrisation results*The PDF and \(\alpha _\mathrm{s} (m_\mathrm{Z})\) parameters determined together in this fit (Table 6) are denoted as Open image in new window . It is released [114] in the LHAPDF [93] format with experimental, hadronisation and \(\alpha _\mathrm{s} (m_\mathrm{Z})\) uncertainties included. The gluon and singlet momentum distributions,

*xg*and \(x\Sigma \), the latter defined as the sum of all quark and anti-quark densities, are compared to NNPDF3.1 at a scale \(\mu _\mathrm{F} =20\,\mathrm {GeV} \) in Fig. 16. The uncertainties of the fitted PDFs are somewhat larger than the uncertainties of NNPDF3.1. For NNPDF3.1, \(\alpha _\mathrm{s} ^\mathrm{PDF}(m_\mathrm{Z})\) is fixed while it is a free parameter in the Open image in new window fit. Within uncertainties, the singlet distribution obtained for Open image in new window is in fair agreement with NNPDF3.1 over a large range in

*x*, whereas the gluon density is consistent with NNPDF3.1 only for \(x>0.01\) and is significantly higher than NNPDF3.1 at lower

*x*. This difference can not be explained by the assumptions made on the strong coupling in NNPDF3.1, as can be seen from the NNPDF3.1 distributions obtained for \(\alpha _\mathrm{s} ^\mathrm{PDF}(m_\mathrm{Z}) =0.114\). However, there are differences in the datasets used for the fits. For Open image in new window only H1 data are considered, restricted to the range \(Q^2>10\,\text {GeV}^2\). For NNPDF3.1 the combined HERA DIS data [56] are used, starting from \(Q^2>3.5\,\text {GeV}^2\). Data from other processes and experiments are also included, but no DIS jet data.

The PDFs obtained for each of the model and parametrisation variations (not shown in Fig. 16) are contained in the exp,had,PDF uncertainty band for \(x>0.0004\) and thus do not explain the differences to NNDPF3.1.

*The impact of H1 jet data on PDF fits* The PDF+\(\alpha _\mathrm{s}\)-fit is repeated with the normalised jet data excluded, i.e. only inclusive DIS data are considered. For this fit and the
Open image in new window fit the gluon distribution \(xg(x,\mu _\mathrm{F})\) is evaluated at \(\mu _\mathrm{F} =20\,\mathrm {GeV} \) and \(x=0.01\) and its Hessian uncertainty together with its correlation coefficient with \(\alpha _\mathrm{s} (m_\mathrm{Z})\) are calculated. 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 \(\alpha _\mathrm{s} (m_\mathrm{Z})\) and *xg*, as well as their correlation. The correlation coefficient is −0.92 and reduces to −0.65 if jet data are included. Also shown is the gluon distribution of NNPDF3.1 determined for different values of \(\alpha _\mathrm{s} ^\mathrm{PDF}(m_\mathrm{Z})\). At this particular choice of *x* and \(\mu _\mathrm{F} \), the gluon density of
Open image in new window is found to be consistent with NNPDF3.1 in the range where \(\alpha _\mathrm{s} ^\mathrm{PDF}(m_\mathrm{Z})\) is close to the result of the
Open image in new window fit.

The two fits are repeated for each of the model and parametrisation variations (not shown in Fig. 17). For the
Open image in new window fit, only small variations of the results are observed, in accord with the small model and parametrisation uncertainties assigned to \(\alpha _\mathrm{s} (m_\mathrm{Z})\) . However, if the jet data are not included in the fit, the resulting \(\alpha _\mathrm{s} (m_\mathrm{Z})\) and *xg* are found to be strongly dependent on the assumptions made for the PDF parametrisation. This confirms previous observations [115], namely that *xg* and \(\alpha _\mathrm{s} (m_\mathrm{Z})\) together cannot be determined reliably from H1 inclusive DIS data alone.

*xg*and \(\alpha _\mathrm{s} (m_\mathrm{Z})\), with a precision on

*xg*competitive to global PDF fits obtained using fixed value of \(\alpha _\mathrm{s} ^\mathrm{PDF}(m_\mathrm{Z})\).

## 5 Summary

The new next-to-next-to-leading order pQCD calculations (NNLO) for jet production cross sections in neutral-current DIS are exploited for a determination of the strong coupling constant \(\alpha _\mathrm{s} (m_\mathrm{Z})\) using inclusive jet and dijet cross section measurements published by the H1 collaboration. Two methods are explored to determine the value of \(\alpha _\mathrm{s} (m_\mathrm{Z})\).

In the first approach H1 inclusive jet and dijet data are analysed. The cross section predictions account for the \(\alpha _\mathrm{s}\) dependence in the two components of the calculations, the partonic cross sections and the parton distribution functions (PDFs). The strong coupling constant is determined to be \(\alpha _\mathrm{s} (m_\mathrm{Z}) =0.1157\,(20)_\mathrm{exp}\,(29)_\mathrm{th}\), where the jet data are restricted to high scales \(\tilde{\mu } > 28\,\mathrm {GeV} \). 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 experimental uncertainty may be reduced to 0.8 %, if all inclusive jet and dijet data with \(\tilde{\mu } >2m_b\) are considered, but the scale uncertainties are increased significantly. The smallest total uncertainty on \(\alpha _\mathrm{s} (m_\mathrm{Z})\) of 2.5 % is obtained when restricting the data to \(\tilde{\mu } >42\,\mathrm {GeV} \). Values of \(\alpha _\mathrm{s} (m_\mathrm{Z})\) determined from inclusive jet data or dijet data alone are found to be consistent with the main result. All these results are found to be consistent with each other and with the world average value of \(\alpha _\mathrm{s} (m_\mathrm{Z})\).

The running of the strong coupling constant is tested in the range of approximately 7 to \(90\,\mathrm {GeV} \) by dividing the jet data into ten subsets of approximately constant scale. The scale dependence of the coupling is found to be consistent with the expectation.

In a second approach a combined determination of PDF parameters and \(\alpha _\mathrm{s} (m_\mathrm{Z})\) 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 \(Q^{2}>10\,\mathrm {GeV}^2 \), the value of \(\alpha _\mathrm{s} (m_\mathrm{Z})\) is determined to be \(\alpha _\mathrm{s} (m_\mathrm{Z}) =0.1142\,(28)_\mathrm{tot}\). Consistency with the other results and the world average is found. The resulting PDF set
Open image in new window is found to be consistent with the NNPDF3.1 PDF set at sufficiently large \(x>0.01\), albeit there are differences at lower *x*. It is demonstrated that the inclusion of H1 jet data into such a simultaneous PDF and \(\alpha _\mathrm{s} (m_\mathrm{Z})\) determination provides stringent constraints on \(\alpha _\mathrm{s} (m_\mathrm{Z})\) and the gluon density. The results and their uncertainties are found to be largely insensitive to the assumptions made for the PDF parametrisation.

Relevant phenomenological aspects of the NNLO calculations are studied for the first time. The NNLO calculations are repeated for a number of different scale choices and scale factors, as well as for a large variety of recent PDF sets. The level of agreement with H1 jet data is judged quantitatively. The NNLO calculations improve significantly the description of the data and reduce the dominating theoretical uncertainty on \(\alpha _\mathrm{s} (m_\mathrm{Z})\) in comparison to previously employed NLO calculations. All jet cross section measurements are found to be well described by the NNLO predictions. These NNLO calculations are employed for a PDF determination for the first time.

This is the first precision extraction of \(\alpha _\mathrm{s} (m_\mathrm{Z})\) from jet data at NNLO involving a hadron in the initial state. It opens a new chapter of precision QCD measurements at hadron colliders.

## Footnotes

- 1.
The difference of the main fit result to \(\alpha _\mathrm{s} ^\mathrm{PDF}(m_\mathrm{Z}) =0.118\) is covered by the systematic variation \(\alpha _\mathrm{s} ^\mathrm{PDF}(m_\mathrm{Z}) =0.118\pm 0.002\).

- 2.
No NNLO calculation is available for three-jet production in DIS to date.

- 3.
For purely technical reasons the fit parameter is \(\alpha _\mathrm{s} (m_\mathrm{Z})\), and thus the running is applied from \(\mu _\mathrm{R}\), as used in the calculation, to \(m_\mathrm{Z}\) and then ‘back’ to a representative value \(\mu _\mathrm{R}\) .

- 4.
The functional form referred to as “alternative gluon” in [56] actually corresponds to the default choice for this paper.

## Notes

### Acknowledgements

We are grateful to the HERA machine group whose outstanding efforts have made this experiment possible. We thank the engineers and technicians for their work in constructing and maintaining the H1 detector, our funding agencies for financial support, the DESY technical staff for continual assistance and the DESY directorate for support and for the hospitality which they extend to the non-DESY members of the collaboration. We would like to give credit to all partners contributing to the EGI computing infrastructure for their support for the H1 collaboration. We express our thanks to all those involved in securing not only the H1 data but also the software and working environment for long term use allowing the unique H1 data set to continue to be explored in the coming years. The transfer from experiment specific to central resources with long term support, including both storage and batch systems has also been crucial to this enterprise. We therefore also acknowledge the role played by DESY-IT and all people involved during this transition and their future role in the years to come. This research was supported by the Swiss National Science Foundation (SNF) under Contracts 200020-162487 and CRSII2-160814, in part by the UK Science and Technology Facilities Council as well as by the Research Executive Agency (REA) of the European Union under the Grant Agreement PITN-GA-2012-316704 (“HiggsTools”) and the ERC Advanced Grant MC@NNLO (340983). We gratefully express our thanks for support from the Institute for Particle Physics Phenomenology Durham (IPPP), in the form of an IPPP Associateship.

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