# Calculations for deep inelastic scattering using fast interpolation grid techniques at NNLO in QCD and the extraction of \(\alpha _{\mathrm {s}}\) from HERA data

## Abstract

The extension of interpolation-grid frameworks for perturbative QCD calculations at next-to-next-to-leading order (NNLO) is presented for deep inelastic scattering (DIS) processes. A fast and flexible evaluation of higher-order predictions for any a posteriori choice of parton distribution functions (PDFs) or value of the strong coupling constant is essential in iterative fitting procedures to extract PDFs and Standard Model parameters as well as for a detailed study of the scale dependence. The APPLfast project, described here, provides a generic interface between the parton-level Monte Carlo program NNLOjet and both the APPLgrid and fastNLO libraries for the production of interpolation grids at NNLO accuracy. Details of the interface for DIS processes are presented together with the required interpolation grids at NNLO, which are made available. They cover numerous inclusive jet measurements by the H1 and ZEUS experiments at HERA. An extraction of the strong coupling constant is performed as an application of the use of such grids and a best-fit value of \(\alpha _{\mathrm {s}} (M_{{\mathrm {Z}}}) = 0.1170\,(15)_\text {exp}\,(25)_\text {th}\) is obtained using the HERA inclusive jet cross section data.

## 1 Introduction

Modern calculations of higher-order corrections in perturbative QCD for predictions of cross sections from collider experiments are computationally very demanding. In particular, complicated measurement functions and fiducial phase-space definitions associated with differential cross sections prevent an analytic integration over the final-state kinematics, thus calling for numerical approaches. Next-to-next-to-leading order computations for differential cross-section predictions, for example, often require \(\mathcal {O}(10^5)\) CPU hours due to the complicated singularity structure of the real-emission amplitudes and the delicate numerical cancellations they entail. Further challenges arise from the requirement of high precision for important benchmark processes. Common examples are jet production cross sections in both electron–proton collisions or \({\mathrm {p}} {\mathrm {p}} \) collisions, the Drell–Yan production of \({\mathrm {Z}} \) and \({\mathrm {W}} \) bosons, and gauge-boson production in association with jets.

The NNLOjet program [1] is a recent and continuously developing framework for the calculation of fully differential cross sections for collider experiments. It includes a large number of processes calculated at NNLO in perturbative QCD, implemented in a unified and holistic manner.

For a detailed study of NNLO predictions and the estimation of theoretical uncertainties, these calculations must be repeated with different input conditions. This includes, for example, using different values for the strong coupling \(\alpha _{\mathrm {s}} (M_{{\mathrm {Z}}})\), different parametrisations for the PDFs, or different choices for the factorisation or renormalisation scales. Computationally even more demanding are fits for the determination of the strong coupling constant and the parton densities in the proton.

In such fits, comparisons must be performed between the data and the NNLO predictions for the many thousands of points that are drawn from the multidimensional parameter space used in the minimisation. As such, it is computationally prohibitive to run the full calculation at NNLO for each required input condition encountered in such a fit. Applications of this nature therefore critically require an efficient approach to perform the convolution of the partonic hard scattering with PDFs, change the value of the strong coupling constant, and vary the scales.

The technique of using a grid to store the perturbative coefficients stripped of the parton luminosity and factors of the strong coupling constant \(\alpha _{\mathrm {s}}\), during the full Monte Carlo integration allows the convolution with arbitrary PDFs to be performed later with essentially no additional computational cost. Variation of \(\alpha _{\mathrm {s}} (M_{{\mathrm {Z}}})\), and the renormalisation and factorisation scales is also possible. The grid technique, used in Ref. [2], is implemented independently in the APPLgrid [3, 4] and fastNLO [5, 6] packages. The technique works by using interpolation functions to distribute each single weight from the *x* and \(\mu ^2\) phase space of the integration, over a number of discrete *a priori* determined nodes in that phase space along with the relevant interpolating function coefficients. Subsequently summing over those discrete nodes will therefore reproduce the original value for the weight, or any product of the weight with some function of the phase space parameters for that specific phase space point. One dimension in the grid is required for each parameter upon which the subsequently varied parameters will depend. For instance, for DIS processes, a dimension for *x* and \(\mu ^2\) will be required. For \({\mathrm {p}} {\mathrm {p}} \) collisions, a third dimension must be added to account for the momentum fraction \(x_2\) of the second proton.

This paper describes developments in the APPLfast project which provides a common interface for the APPLgrid and fastNLO grid libraries to link to the NNLOjet program for the calculation of the perturbative coefficients. The generation and application of interpolation grids for DIS jet production at NNLO [7, 8] is discussed. Grids are made publicly available on the ploughshare website [9]. A subset of these grids have previously been employed for a determination of the strong coupling constant, \(\alpha _{\mathrm {s}} (M_{{\mathrm {Z}}})\) [10]. Here, additional details of the grid methodology for DIS are discussed, together with the NNLO extraction of \(\alpha _{\mathrm {s}} (M_{{\mathrm {Z}}})\) using data on inclusive jet production from both H1 and ZEUS.

## 2 DIS at NNLO and the NNLOjet framework

*a*which denotes the incoming parton flavour. In perturbative QCD, the hard-scattering cross section can be expanded in the coupling constant

*k*corresponds to the power in \(\alpha _{\mathrm {s}} \) at leading order (LO). Jet cross section measurements in DIS commonly employ a reconstruction in the Breit frame of reference, in which the proton and the gauge boson of virtuality \(Q^{2}\) collide head-on. This is further assumed in the remainder of this work. As a consequence, jet production proceeds through the basic scattering processes \(\gamma ^*g\rightarrow q\bar{q}\) and \(\gamma ^*q\rightarrow qg\), thus requiring at least two partons in the final state. This choice not only gives a direct sensitivity to \(\alpha _{\mathrm {s}}\) (\(k=1\)) but also a rare handle on the gluon density already at LO.

## 3 The APPLgrid and fastNLO packages

*f*(

*x*) to be obtained from the knowledge of its value at discrete nodes \(a\equiv x^{[0]}< x^{[1]}< \ldots < x^{[N]}\equiv b\) that partition the interval \([x_{\mathrm{min}},x_{\mathrm{max}}]\) into

*N*disjoint sub-intervals. To this end, interpolation kernels \(E_i(x)\) are introduced for each node

*i*, which are constructed from polynomials of degree

*n*and satisfy \(E_i(x^{[j]})=\delta _i^j\). The set of interpolation kernels further form a partition of unity,

*f*(

*x*) can be approximated as

*f*(

*x*) varies more rapidly. In this case, nodes are chosen with respect to

*y*(

*x*) and the corresponding interpolation kernels are denoted by \(E^y_i(x)\).

*f*(

*x*) appears under an integral, the integration can be approximated by a sum over the nodes

*i*,

*f*(

*x*) using the sum from the right hand side, which can be evaluated very quickly.

### 3.1 Application to the DIS cross section

*x*and \(\mu _{\mathrm{F}} \) is constructed. The respective interpolation kernels \(E^y_i(x)\) and \(E^\tau _j(\mu _{\mathrm{F}})\) can be chosen independently for the two variables, introducing the additional transformation in the scale variable, \(\mu _{\mathrm{F}} \longmapsto \tau (\mu _{\mathrm{F}})\). Typical transformations for DIS are for instance

*x*or \(\mu \), and \(\varLambda \) can be chosen of the order of \(\varLambda _{\mathrm {QCD}}\), but need not necessarily be identical. Additional transforms are available in both APPLgrid and fastNLO.

*x*and \(\mu \), both the PDFs and the running of the strong coupling can then be represented by a sum over the interpolation nodes,

*p*has been defined as

### 3.2 Renormalisation and factorisation scale dependence

## 4 The APPLfast project

The APPLfast project provides a library of code written in C++ with Fortran callable components. It is a lightweight interface used to bridge between the NNLOjet code and the specific code for booking and filling the grids themselves using either APPLgrid or fastNLO.

The basic structure for the filling of either grid technology is essentially the same, and as such, much of the functionality for the interface exists as common code that is used for filling both, with only the code that actually fills the weights needing to be specific to either technology. Efforts are under way to implement a common filling API for both fastNLO and APPLgrid, which will allow significantly more of the specific filling code to be shared.

A design principle, applied from the outset, was that the interface should be as unobtrusive as possible in the NNLOjet code, and should provide no additional performance overhead in terms of execution time when not filling a grid. When filling a grid, any additional overhead should be kept as low as possible. This is achieved by the use of a minimal set of hook functions that can be called from within the NNLOjet code itself and which can be left within the code with no impact on performance if the grid filling functionality is not required. The original proof-of-concept implementation accessed the required variables for the weights, scales and momentum fractions via the NNLOjet data structures directly, but following this it was decided to instead implement custom access functions that allow, e.g., for a full decomposition of the event weights as described by Eq. (4), thus enabling a more straightforward design for the filling code.

Each process in NNLOjet consists of a large number of subprocesses. In order to fill the grids, during the configuration stage the internal list of NNLOjet processes is mapped to a minimal set of the unique parton luminosities that are used for the grid. When filling, these internal NNLOjet process identifiers are used to determine which parton luminosity terms in the grid should be filled on the interface side.

- 1.
*Vegas adaption*This is the first stage in the standard NNLOjet workflow and is used to generate an optimised Vegas phase-space grid for the subsequent production runs. At this stage the grid filling is not enabled and NNLOjet can run in multi-threaded mode. - 2.
*Grid warm-up*This is required in order to optimise the limits for the phase space in*x*and \(\mu _{\mathrm{F}} \) for the grids. During this stage, the NNLOjet code runs in a custom mode intended solely to sample the phase-space volume, thus skipping the costly evaluation of the Matrix Elements. - 3.
*Grid production*Here, the grids from stage 2 are filled with the weights generated from a full NNLOjet run, using the optimised phase-space sampling determined in stage 1. The calculation can be run in parallel using many independent jobs to achieve the desired statistical precision. - 4.
*Grid combination*In this stage, the grids from the individual jobs are combined, first merging the results for each of the LO, NLO (R and V), and NNLO (RR, VV, RV) terms separately, and subsequently assembling the respective grids into a final master grid.

The stabilisation of higher-order cross sections with respect to statistical fluctuations demands a substantial number of events to be generated. This is particularly true for the double-real contribution, since the large number of final-state partons lead to a complex pattern of infrared divergences that need to be compensated. Typically, computing times of the order of hundreds of thousands of CPU hours are required. In stage 3 it is therefore mandatory to run hundreds to thousands of separate jobs in parallel, in particular for the NNLO sub-contributions. The resulting interpolation grids for each cross section and job typically are about 10–100 MBytes in size. The final master grid obtained by summing the output from all jobs then is somewhat larger than the largest single grid, because it contains at least one weight grid for each order in \(\alpha _s\).

The interpolation accuracy must be evaluated to ensure that the results of the full calculation can be reproduced with the desired precision. For sufficiently well-behaved functions, as usually the case for PDFs, it is always possible to reach such precision by increasing the number of nodes in the fractional momentum *x* and scale \(\mu \) at the cost of larger grid sizes. For proton-proton scattering, because of the additional momentum fraction associated with the second proton, the grid size grows quadratically with the number of *x* nodes.

*x*and \(\mu \), and the accessed ranges in

*x*and \(\mu \), as determined in the grid warm-up stage 3, can be chosen such that the number of nodes can be reduced significantly while retaining the same approximation accuracy. Figure 1 shows the root mean square (RMS) of the fractional difference of the fast grid convolution with respect to the corresponding reference for HERA inclusive jet production data. This uses a third order interpolation in the transformed

*y*(

*x*) variable and the transform from Eq. (10) and shows that the precision is better than one per mille for grids with 20

*x*nodes, and better than 0.1 per mille for grids with more than 30

*x*nodes.

For a specific process, observable, and phase space selection, an initial indication of the level of precision can be gained already using a single job by comparing the interpolated result with the reference calculation for the chosen PDF set for each bin in the observable.

Since identical events are filled both into the grid and into the reference cross section, then any statistical fluctuations should be reproduced and thus a limited number of events is usually sufficient for this validation. Subsequently, a similar level of precision should be possible for each of the contributions for the full calculation. In future, this could be exploited to avoid the time consuming access to the reference PDF during the full NNLOjet calculation itself during the mass production of interpolation grids at a previously validated level of precision.

Additional cross checks can be performed, for example, comparing the interpolated result of the final grid using an alternative PDF from the reference cross section, with an independent reference calculation for this same alternative PDF set. Here, of course, agreement can only be confirmed within the statistical precision of the two independent calculations. Moreover, it can be verified that the fast convolution with a change in scale, \(\mu \), is consistent with the full calculation performed at that scale.

A significant benefit of using such interpolation grids is that the detailed uncertainties can be calculated without the need to rerun the calculation. This is illustrated in Fig. 4, which shows the full seven point scale variation and the PDF uncertainties derived for the \(p_\mathrm {T,jet}\) dependent cross sections of the same H1 and ZEUS measurements from before. The seven point scale uncertainty is a conventional means of estimating the possible effect of uncalculated higher orders. It is defined by the maximal upward and downward changes in the cross section when varying the renormalisation and factorisation scales by factors of two around the nominal scale in the following six combinations of \((\mu _{\mathrm{R}}/\mu _0, \mu _{\mathrm{F}}/\mu _0)\): (1 / 2, 1 / 2), (2, 2), (1 / 2, 1), (1, 1 / 2), (2, 1), and (1, 2). The PDF uncertainties at the \(1\,\sigma \) level are evaluated as prescribed for the respective PDF sets^{1}: NNPDF31 [33], CT14 [34], MMHT2014 [35], and ABMP16 [36]. In all plots PDFs at NNLO have been used with \(\alpha _{\mathrm {s}} (M_{{\mathrm {Z}}}) =0.118\).

## 5 Application: determination of the strong coupling constant

As an application in using the DIS jet grids at NNLO, an extraction of the strong coupling constant, \(\alpha _{\mathrm {s}} (M_{{\mathrm {Z}}}) \), is performed using a fit of the NNLO QCD predictions from NNLOjet to the HERA inclusive jet cross-section data.

Seven sets of cross section measurements by the HERA experiments are considered for the \(\alpha _{\mathrm {s}} (M_{{\mathrm {Z}}})\) determination: Five from H1 and two from ZEUS, each given by an inclusive jet cross section measurement as a function of \(p_\mathrm {T,jet}\) and \(Q^{2}\). The H1 results include measurements at \(\sqrt{s}=300~\,\mathrm {GeV} \) [2] and \(\sqrt{s}=320~\,\mathrm {GeV} \) [26, 27, 28, 29], in the ranges \(Q^{2} \lesssim 120~\,\mathrm {GeV}^2 \) [26, 28] and \(Q^{2} \gtrsim 120~\,\mathrm {GeV}^2 \) [2, 27, 29], where jets are measured within a kinematic range between \(4.5<p_\mathrm {T,jet} <80~\,\mathrm {GeV} \). For ZEUS, the data are similarly comprised of measurements at \(\sqrt{s}=300~\,\mathrm {GeV} \) [30] and \(\sqrt{s}=320~\,\mathrm {GeV} \) [31], but in the range \(Q^{2} > 125~\,\mathrm {GeV}^2 \) and with jets having \(p_\mathrm {T,jet} >8~\,\mathrm {GeV} \). For all data sets jets are defined in the Breit frame of reference using the \(k_T\) jet algorithm with a jet-resolution parameter \(R=1\).

The methodology for the \(\alpha _{\mathrm {s}} (M_{{\mathrm {Z}}})\) determination employs the same technique as Refs. [10] and [37]. In brief, a goodness-of-fit quantifier between data and prediction that depends on \(\alpha _{\mathrm {s}} (M_{{\mathrm {Z}}})\) is defined in terms of a \(\chi ^{2}\) function, which is based on normally-distributed relative uncertainties and accounts for all experimental, hadronisation, and PDF uncertainties. The experimental uncertainties, and the hadronisation corrections and their uncertainties are provided together with the data by the H1 and ZEUS collaborations. The PDF uncertainties are calculated using the prescriptions provided by the respective PDF fitting groups. The \(\chi ^{2}\) function is then minimised using Minuit [38]. The \(\alpha _{\mathrm {s}} (M_{{\mathrm {Z}}})\) dependence in the predictions takes into account the contributions from both the hard coefficients and the PDFs. The latter is evaluated using the DGLAP evolution as implemented in the Apfel++ package [39, 40], using the PDFs evaluated at a scale of \(\mu _0=20~\,\mathrm {GeV} \). A different choice for the value of \(\mu _0\) is found to have negligible impact on the results. The uncertainties on the fit quantity are obtained by the HESSE algorithm and validated by comparison with results obtained using the MINOS algorithm [38]. The uncertainties are separated into experimental (exp), hadronisation (had), and PDF uncertainties (PDF) by repeating the fit excluding uncertainty components.

*dn*and

*up*) as

*b*-quark [41], i.e. \(\tilde{\mu }>2m_b\).

A summary of values of \(\alpha _{\mathrm {s}} (M_{{\mathrm {Z}}})\) from fits to HERA inclusive 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

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

| |||||

\(300\,\,\mathrm {GeV} \) high-\(Q^{2}\) | \(2m_b\) | \( 0.1217\,(31)_{\mathrm{exp}}\,(22)_{\mathrm{had}}\,(5)_{\mathrm{PDF}}\,(3)_{\mathrm{PDF\alpha _{\mathrm {s}}}}\,(5)_{\mathrm{PDFset}}\,(35)_{\mathrm{scale}}\) | \((42)_{\mathrm{th}}\) | \((52)_{\mathrm{tot}}\) | 5.6 / 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-\(Q^{2}\) | \(2m_b\) | \( 0.1136\,(24)_{\mathrm{exp}}\,(9)_{\mathrm{had}}\,(6)_{\mathrm{PDF}}\,(4)_{\mathrm{PDF\alpha _{\mathrm {s}}}}\,(4)_{\mathrm{PDFset}}\,(28)_{\mathrm{scale}}\) | \((31)_{\mathrm{th}}\) | \((39)_{\mathrm{tot}}\) | 15.5 / 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-\(Q^{2}\) | \(2m_b\) | \( 0.1126\,(19)_{\mathrm{exp}}\,(9)_{\mathrm{had}}\,(6)_{\mathrm{PDF}}\,(4)_{\mathrm{PDF\alpha _{\mathrm {s}}}}\,(2)_{\mathrm{PDFset}}\,(32)_{\mathrm{scale}}\) | \((34)_{\mathrm{th}}\) | \((39)_{\mathrm{tot}}\) | 34.7 / 29 |

| |||||

\(300\,\,\mathrm {GeV} \) high-\(Q^{2}\) | \(2m_b\) | \( 0.1213\,(28)_{\mathrm{exp}}\,(3)_{\mathrm{had}}\,(5)_{\mathrm{PDF}}\,(2)_{\mathrm{PDF\alpha _{\mathrm {s}}}}\,(3)_{\mathrm{PDFset}}\,(26)_{\mathrm{scale}}\) | \((27)_{\mathrm{th}}\) | \((39)_{\mathrm{tot}}\) | 28.6 / 29 |

HERA-I high-\(Q^{2}\) | \(2m_b\) | \( 0.1181\,(27)_{\mathrm{exp}}\,(16)_{\mathrm{had}}\,(6)_{\mathrm{PDF}}\,(2)_{\mathrm{PDF\alpha _{\mathrm {s}}}}\,(6)_{\mathrm{PDFset}}\,(25)_{\mathrm{scale}}\) | \((31)_{\mathrm{th}}\) | \((41)_{\mathrm{tot}}\) | 20.8 / 29 |

| |||||

H1 inclusive jets | \(2m_b\) | \( 0.1133\,(10)_{\mathrm{exp}}\,(6)_{\mathrm{had}}\,(5)_{\mathrm{PDF}}\,(4)_{\mathrm{PDF\alpha _{\mathrm {s}}}}\,(2)_{\mathrm{PDFset}}\,(39)_{\mathrm{scale}}\) | \((40)_{\mathrm{th}}\) | \((41)_{\mathrm{tot}}\) | 125.8 / 133 |

H1 inclusive jets | \(28\,\,\mathrm {GeV} \) | \( 0.1153\,(19)_{\mathrm{exp}}\,(9)_{\mathrm{had}}\,(2)_{\mathrm{PDF}}\,(2)_{\mathrm{PDF\alpha _{\mathrm {s}}}}\,(3)_{\mathrm{PDFset}}\,(26)_{\mathrm{scale}}\) | \((28)_{\mathrm{th}}\) | \((33)_{\mathrm{tot}}\) | 44.1 / 60 |

| |||||

ZEUS inclusive jets | \(2m_b\) | \( 0.1199\,(20)_{\mathrm{exp}}\,(8)_{\mathrm{had}}\,(6)_{\mathrm{PDF}}\,(1)_{\mathrm{PDF\alpha _{\mathrm {s}}}}\,(5)_{\mathrm{PDFset}}\,(26)_{\mathrm{scale}}\) | \((29)_{\mathrm{th}}\) | \((35)_{\mathrm{tot}}\) | 49.8 / 59 |

ZEUS inclusive jets | \(28\,\,\mathrm {GeV} \) | \( 0.1194\,(24)_{\mathrm{exp}}\,(7)_{\mathrm{had}}\,(6)_{\mathrm{PDF}}\,(1)_{\mathrm{PDF\alpha _{\mathrm {s}}}}\,(5)_{\mathrm{PDFset}}\,(25)_{\mathrm{scale}}\) | \((27)_{\mathrm{th}}\) | \((34)_{\mathrm{tot}}\) | 39.3 / 43 |

| |||||

HERA inclusive jets | \(2m_b\) | \( 0.1149\,(9)_{\mathrm{exp}}\,(5)_{\mathrm{had}}\,(4)_{\mathrm{PDF}}\,(3)_{\mathrm{PDF\alpha _{\mathrm {s}}}}\,(2)_{\mathrm{PDFset}}\,(37)_{\mathrm{scale}}\) | \((38)_{\mathrm{th}}\) | \((39)_{\mathrm{tot}}\) | 182.9 / 193 |

HERA inclusive jets | \(28\,\,\mathrm {GeV} \) | \( 0.1170\,(15)_{\mathrm{exp}}\,(7)_{\mathrm{had}}\,(3)_{\mathrm{PDF}}\,(2)_{\mathrm{PDF\alpha _{\mathrm {s}}}}\,(3)_{\mathrm{PDFset}}\,(24)_{\mathrm{scale}}\) | \((25)_{\mathrm{th}}\) | \((29)_{\mathrm{tot}}\) | 85.7 / 104 |

Results for the values of \(\alpha _{\mathrm {s}} (M_{{\mathrm {Z}}})\) as obtained from the individual fits to the inclusive jet cross section data are collected in Table 1. The entries for the H1 data sets correspond to values previously reported in Ref. [10] but some have been updated using NNLO predictions with higher statistical precision. New results are presented for the fits to the ZEUS inclusive jet cross section data [30, 31] and fits to all the H1 and ZEUS inclusive jet cross section data, which are the principle results of this current study. The \(\alpha _{\mathrm {s}} (M_{{\mathrm {Z}}})\) values from the individual data sets are found to be mutually compatible within their respective errors. Figure 5 summarises the values for a visual comparison, and includes the world average [41, 43], which is seen to be consistent with the value extracted here. All the H1 and ZEUS inclusive jet cross section data are found to be in good agreement with the NNLO predictions, as indicated by the individual \(\chi ^{2}/n_\mathrm {dof} \) values in Table 1. From the fit to all HERA inclusive jet data a value of \(\alpha _{\mathrm {s}} (M_{{\mathrm {Z}}}) =0.1149\,(9)_{\mathrm{exp}}\,(38)_{\mathrm{th}}\) is obtained, where *exp* and *th* denote the experimental and theoretical uncertainties, respectively, and where the latter is obtained by combining individual theory uncertainties in quadrature. A detailed description of the uncertainty evaluation procedure can be found in Ref. [10]. The fit yields \(\chi ^{2}/n_\mathrm {dof} =182.9/193\), thus indicating an excellent description of the data by the NNLO predictions. Furthermore, an overall high degree of consistency for all of the HERA inclusive jet cross section data is found.

Values of the strong coupling constant at the \({\mathrm {Z}} \)-boson mass, \(\alpha _{\mathrm {s}} (M_{{\mathrm {Z}}})\), obtained from fits to groups of data with comparable values of \(\mu _{\mathrm{R}} \). The first (second) uncertainty of each point corresponds to the experimental (theory) uncertainties. The theory uncertainties include PDF related uncertainties and the dominating scale uncertainty

\(\mu _{\mathrm{R}}\) | H1 | ZEUS | HERA |
---|---|---|---|

(GeV) | \(\alpha _{\mathrm {s}} (M_{{\mathrm {Z}}})\) | \(\alpha _{\mathrm {s}} (M_{{\mathrm {Z}}})\) | \(\alpha _{\mathrm {s}} (M_{{\mathrm {Z}}})\) |

7.4 | \( 0.1148\,(12)\,(42)\) | \( - \) | \( 0.1148\,(12)\,(42)\) |

10.1 | \( 0.1136\,(17)\,(35)\) | \( - \) | \( 0.1136\,(17)\,(35)\) |

13.3 | \( 0.1147\,(14)\,(43)\) | \( -\) | \( 0.1147\,(14)\,(43)\) |

17.2 | \( 0.1133\,(15)\,(32)\) | \( 0.1183\,(26)\,(34)\) | \(0.1147\,(13)\,(33)\) |

20.1 | \( 0.1134\,(17)\,(34)\) | \( 0.1172\,(27)\,(28)\) | \(0.1145\,(14)\,(32)\) |

24.5 | \( 0.1163\,(16)\,(32)\) | \( 0.1192\,(25)\,(29)\) | \(0.1172\,(13)\,(32)\) |

29.3 | \( 0.1077\,(32)\,(34)\) | \( 0.1142\,(31)\,(24)\) | \(0.1113\,(22)\,(29)\) |

36.0 | \( 0.1152\,(26)\,(36)\) | \( 0.1209\,(28)\,(31)\) | \(0.1184\,(19)\,(31)\) |

49.0 | \( 0.1175\,(22)\,(19)\) | \( 0.1195\,(50)\,(29)\) | \(0.1179\,(20)\,(20)\) |

77.5 | \( 0.1099\,(53)\,(20)\) | \( 0.1286\,(46)\,(24)\) | \(0.1211\,(32)\,(20)\) |

The running of \(\alpha _{\mathrm {s}} (\mu _{\mathrm{R}})\) can be inferred from separate fits to groups of data points that share a similar value of the renormalisation scale, as estimated by \(\tilde{\mu }\) in Eq. (18). To this end, the \(\alpha _{\mathrm {s}} (M_{{\mathrm {Z}}})\) values are determined for each \(\tilde{\mu }\) collection individually, and are summarised in Table 2 and shown in the bottom panel of Fig. 6. All values are mutually compatible and in good agreement with the world average, and no significant dependence on \(\mu _{\mathrm{R}}\) is observed. The corresponding values for \(\alpha _{\mathrm {s}} (\mu _{\mathrm{R}}) \), as determined using the QCD renormalisation group equation, are displayed in the top panel of Fig. 6, illustrating the running of the strong coupling. The dashed line corresponds to the prediction for the \(\mu _{\mathrm{R}}\) dependence using the \(\alpha _{\mathrm {s}}\) value of Eq. (19). The predicted running is in excellent agreement with the individual \(\alpha _{\mathrm {s}} (\mu _{\mathrm{R}})\) determinations, further reflecting the internal consistency of the study.

To conclude this study it is worth commenting on the robustness of the procedure. On the theory side, the inclusive jet cross section represents an observable that is well defined in perturbative QCD and only moderately affected by non-perturbative effects and experimentally, this study rests on a solid basis, making use of measurements from two different experiments based on three separate data taking periods, which cover two different centre-of-mass energies and two kinematic regions in \(Q^{2}\). As a result, although only a single observable is used in the determination of \(\alpha _{\mathrm {s}}\), a highly competitive experimental and theoretical precision is achieved.

## 6 Conclusions and outlook

NNLO calculations in perturbative QCD are rapidly becoming the new standard for many important scattering processes. These calculations are critical in reducing theory uncertainties and often improve the description of the increasingly precise data, sometimes even resolving prior tensions. However, the computational resources required for such calculations prohibit their use in applications that require a frequent re-evaluation using different input conditions, e.g. fitting procedures for PDFs and Standard Model parameters.

Fast interpolations grid techniques circumvent these limitations by allowing for the a posteriori interchange of PDFs, values of the strong coupling \(\alpha _{\mathrm {s}}\), and scales in the prediction at essentially no cost. In this article the APPLfast project is discussed, which provides a generic interface for the APPLgrid and fastNLO grid libraries to produce interpolation tables where the hard coefficient functions are computed by the NNLOjet program. Details on the extension of the techniques to NNLO accuracy and their implementation for DIS are discussed, together with the public release of NNLO grid tables for jet cross-section measurements at HERA [9].

As an application of the grids, an extraction of the strong coupling constant \(\alpha _{\mathrm {s}}\) has been performed, based on jet data at HERA, closely following the methodology in Refs. [10, 37]. In contrast to Ref. [10], where the \(\alpha _{\mathrm {s}}\) determination considered both inclusive and di-jet cross section data from H1 alone, this current analysis includes data from both the H1 and ZEUS experiments, but \(\alpha _{\mathrm {s}}\) is fitted solely using the single jet inclusive data. The usage of a single observable facilitates the simultaneous determination of \(\alpha _{\mathrm {s}} (M_{{\mathrm {Z}}})\) from two experiments, as the observable is defined identically between both experiments and thus reduces ambiguities in the treatment of theory uncertainties. This work represents one of the first determinations of the strong coupling constant to include both H1 and ZEUS DIS jet data at NNLO accuracy, where such a determination is only possible using the foundational work presented in this paper. The determination of \(\alpha _{\mathrm {s}} (M_{{\mathrm {Z}}})\) from H1 and ZEUS data taken together provides a best-fit value of \(\alpha _{\mathrm {s}} (M_{{\mathrm {Z}}}) = 0.1170\,(15)_\text {exp}\,(25)_\text {th}\).

Although the discussion in the present work was limited to the DIS process, the implementation in both APPLfast and NNLOjet is fully generic and thus generalisable to hadron-hadron collider processes. This means that all NNLO calculations available from within NNLOjet , such as di-jet production and \(V+\text {jet} \) production in proton-proton scattering, are interfaced to grid-filling tools in a rather straightforward manner. This generalisation will be presented in a future publication.

## Footnotes

## Notes

### Acknowledgements

This research was supported in part by the UK Science and Technology Facilities Council, by the Swiss National Science Foundation (SNF) under contracts 200020-175595 and 200021-172478, by the Research Executive Agency (REA) of the European Union through the ERC Advanced Grant MC@NNLO (340983) and the Fundação para a Ciência e Tecnologia (FCT-Portugal), under projects UID/FIS/00777/2019, CERN/FIS-PAR/0022/2017. CG and MS were supported by the IPPP Associateship programme for this project. JP gratefully acknowledges the hospitality and financial support of the CERN theory group where work on this paper was conducted.

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