Determination of the strong-coupling constant from the Z-boson transverse-momentum distribution

Abstract

The strong-coupling constant is determined from the low-momentum region of the transverse-momentum distribution of Z bosons produced through the Drell–Yan process, using predictions at third order in perturbative QCD. The analysis employs a measurement performed in proton-antiproton collisions at a centre-of-mass energy of $\sqrt{s} = 1.96$ TeV with the CDF experiment. The determined value of the strong coupling at the reference scale corresponding to the Z-boson mass is $\alpha _S(m_Z)= 0.1191^{+0.0013}_{-0.0016}$.

1 Introduction

The coupling constant of the strong interaction is one of the fundamental parameters of the standard model, and is the least precisely known among the fundamental couplings in nature. The most recent world average of the strong-coupling constant at the scale of the Z-boson mass yields $\alpha _S(m_Z)= 0.1179 \pm 0.0009$, with a relative uncertainty of $0.8\%$ [1]. Various different determinations contribute to the world average, and are categorised according to their methodological approach [2]. The most precise determinations of $\alpha _S(m_Z)$ are from lattice QCD, with a result of $\alpha _S(m_Z)= 0.1184 \pm 0.0008$ [3], and hadronic tau decays, with a result of $\alpha _S(m_Z)= 0.1177 \pm 0.0019$ [1]. Tensions exist between some of the most precise determinations of $\alpha _S(m_Z)$. For instance, several determinations from deep inelastic lepton-nucleon scattering [4,5,6] and from hadronic final states of electron-positron annihilation [7,8,9] are significantly lower than the lattice QCD determination. Some of these determinations are performed at next-to-next-to-next-to-leading order (N$^3$LO) in QCD, namely from hadronic tau decays and low $Q^2$ continuum [10], from non-singlet structure functions in deep inelastic scattering [4], from heavy quarkonia decays [11, 12], and from the global fit of the electroweak observables [13, 14]. At hadron colliders, the strong-coupling constant has been determined in final states with jets [15, 16] from inclusive top quark pairs production [17,18,19], and more recently from inclusive W and Z bosons production [20]. The high-momentum region of the Z-boson transverse-momentum ($p_{\textrm{T}}$) distribution measured at the LHC [21,22,23] was included in parton distribution function (PDF) determinations [24], and contributed to the simultaneous determination of PDFs and strong-coupling constant in Refs. [25,26,27]. Some of these determinations, in particular those with jets in the final state, allow probing the strong coupling at high values of momentum transfer.

In this context, it is highly desirable to perform alternative determinations of $\alpha _S(m_Z)$ based on new observables and high-order theory predictions, which can help improving the precision in the determination of the strong coupling and resolving existing tensions. This paper presents a new methodology for a precise determination of $\alpha _S(m_Z)$ at hadron colliders from a semi-inclusive (i.e. radiation inhibited) observable, namely the low-momentum Sudakov [28] region of the transverse-momentum distribution of Z bosons produced through the Drell–Yan process [29]. The strong force is responsible for the recoil of the Z bosons, which acquire non-zero transverse momentum from QCD radiation off the initial-state partons, and from non-perturbative intrinsic $k_T$ effects. The hardness of the transverse-momentum distribution is a measure of the strength of the recoil of the Z bosons, which in turn is proportional to the strong coupling. Compared to other determinations of $\alpha _S(m_Z)$ at hadron colliders based on either exclusive or inclusive observables, this determination gathers all desirable features for a precise determination: large observable sensitivity to $\alpha _S(m_Z)$ compared to the experimental precision, high perturbative accuracy of the theoretical prediction [30,31,32,33,34], and in-situ controllable non-perturbative QCD effects [35,36,37,38,39,40,41,42,43,44,45].

The proposed methodology can be applied to proton-antiproton and proton-proton colliders. In this paper we consider proton-antiproton collisions data from the Tevatron collider, because the Drell–Yan process has reduced contribution from heavy-flavour-initiated production, compared to the proton-proton collisions of the LHC. The application to proton-proton collisions can profit from the large high-quality datasets already collected at the LHC experiments, which will be further increased in the future, but could require a more careful study of heavy-flavour-initiated production, and is left to future work.

2 Methodology

The experimental data used in the analysis is the Z-boson transverse-momentum distribution measured with the CDF detector at a centre-of-mass energy of $\sqrt{s} = 1.96$ TeV with 2.1 fb$^{-1}$ of integrated luminosity [46]. The measurement was performed in the electron decay channel, and extrapolated to a kinematic region without requirements on the transverse-momentum and pseudorapidity of the electrons. The extrapolation to full-lepton phase space, which was based on the measured decay lepton angular distributions [47] to avoid significant theoretical uncertainties, enables the usage of fast analytic predictions. In the low-momentum region below 25 GeV, the measurement was performed in bins of Z-boson transverse momentum of 0.5 GeV. The electron resolution for electrons of transverse momentum of 45 GeV was approximately 1 GeV in the central region $|\eta _e| < 1.05$, and 1.5 GeV in the forward region $1.2< |\eta _e| < 2.8$, enabling small bin-to-bin correlations at the level of $30\%$ for neighbouring bins.

The theoretical predictions are computed with the public numerical program DYTurbo [48], which implements the resummation of logarithmically-enhanced contributions in the small-$p_{\textrm{T}}$ region of the leptons pairs at next-to-next-to-next-to-leading-logarithmic (N$^3$LL) accuracy, combined with the hard-collinear contributions at N$^3$LO in powers of the QCD coupling [30]. We briefly review the resummation formalism implemented in DYTurbo and developed in Refs. [49,50,51]. The transverse-momentum resummed cross section for Z-boson¹ production can be written as

$$\begin{aligned} \text {d}\sigma ^{\text {V}}= & {} \text {d}\sigma ^{\text {res}} -\text {d}\sigma ^{\text {asy}} +\text {d}\sigma ^{\text {f.o.}}\,, \end{aligned}$$

(1)

where $\text {d}\sigma ^{\text {res}}$ is the resummed component of the cross-section, $\text {d}\sigma ^{\text {asy}}$ is the asymptotic term that represents the fixed-order expansion of $\text {d}\sigma ^{\text {res}}$, and $\text {d}\sigma ^{\text {f.o.}}$ is the Z+jet finite-order cross section integrated over final-state QCD radiation. All the cross sections are differential in $p_{\textrm{T}}^2$. The resummed component $\text {d}\sigma ^{\text {res}}$ is the most important term at small $p_{\textrm{T}}$ (i.e. $p_{\textrm{T}}\ll m_Z$). The finite-order term $\text {d}\sigma ^{\text {f.o.}}$ gives the larger net contribution at large $p_{\textrm{T}}$ (i.e. $p_{\textrm{T}}\sim m_Z$). The fixed-order expansion of the resummed component $\text {d}\sigma ^{\text {asy}}$ embodies the singular behaviour of the finite-order term, providing a smooth behaviour of Eq. (1) as $p_{\textrm{T}}$ approaches zero. The resummed component is given by²

$$\begin{aligned} \text {d}\sigma ^{\text {res}}= & {} \text {d}{\hat{\sigma }}^{\text {V}}_{\text {LO}} \times \mathcal {H}_{\text {V}} \times \exp \{\mathcal {G}\} \times S_{\text {NP}}. \end{aligned}$$

(2)

The term $\text {d}{\hat{\sigma }}^{\text {V}}_{\text {LO}}$ is the leading-order (LO) cross section.

The function $\mathcal{H}_V$ [52, 53] includes the hard-collinear contributions and it can be expanded in powers of $\alpha _S$ as

$$\begin{aligned} \mathcal{H}_V(\alpha _S)= 1+ \frac{\alpha _S}{\pi } \,\mathcal{H}_V^{(1)} + \left( \frac{\alpha _S}{\pi }\right) ^2 \,\mathcal{H}_V^{(2)} + \left( \frac{\alpha _S}{\pi }\right) ^3 \,\mathcal{H}_V^{(3)}+\cdots \;. \end{aligned}$$

(3)

The universal (process independent) form factor $\exp \{\mathcal{G}\}$ contains all the terms that order-by-order in $\alpha _S$ are logarithmically divergent as $p_{\textrm{T}}\rightarrow 0$. The resummed logarithmic expansion of $\mathcal{G}$ reads

$$\begin{aligned} \mathcal{G}(\alpha _S,L){} & {} = \nonumber L\, g^{(1)}(\alpha _SL) +g^{(2)}(\alpha _SL) \\{} & {} \quad +\frac{\alpha _S}{\pi } \;g^{(3)}(\alpha _SL)+ \left( \frac{\alpha _S}{\pi }\right) ^2 \;g^{(4)}(\alpha _SL)+\cdots ,\nonumber \\ \end{aligned}$$

(4)

where $L$ is the logarithmic expansion parameter, the functions $g^{(n)}$ control and resum the $\alpha _S^kL^{k}$ (with $k\ge 1$) logarithmic terms in the exponent of Eq. (2) due to soft and collinear radiation.

The function $\mathcal{G}$ is singular in the region of transverse-momenta of the order of the scale of the QCD coupling $\Lambda _{\text {QCD}}$. This signals that a truly non-perturbative region is approached and perturbative results are not reliable. The singular behaviour of the perturbative form factor is removed by using the so-called $b_*$ [35, 54] regularisation procedure, in which the dependence of $\exp \{\mathcal{G}\}$ on the impact parameter b, that is the Fourier-conjugate variable to $p_{\textrm{T}}$, is frozen before reaching the singular point by performing the replacement $b^2 \rightarrow b_*^2 = b^2 b_{\text {lim}}^2/( b^2 + b_{\text {lim}}^2)$. In the calculation the default value of $b_{\text {lim}} = 3$ GeV$^{-1}$ is used. The minimal prescription [41, 55, 56] is considered as alternative regularisation procedure.

Concerning non-perturbative corrections of the type $\Lambda ^p/M^p$, where $\Lambda$ is the non-perturbative scale of QCD and M is the order of magnitude of the momentum transfer in the process, we note that the dominant power corrections are linear, for instance, in the case of hadronic final states of electron-positron annihilation, whereas they are expected to be quadratic for the Drell–Yan $p_{\textrm{T}}$ distribution at large $p_T$ [57, 58], or, equivalently, in the limit of small b [59]. In the small $p_{\textrm{T}}$ region, the non-perturbative corrections are expected to become linear below some scale [44, 60], which is estimated of the order $\mathcal {O}$(1 GeV) in Ref. [61]. Determinations of non-perturbative TMD functions from fits to Drell–Yan and semi-inclusive deep inelastic scattering (SIDIS) data further confirm a transition from quadratic to linear behaviour below a scale which is of order $\mathcal {O}$(1.5 GeV) for Z-boson production at the Tevatron [62]. The Z-boson $p_{\textrm{T}}$ distribution has negligible sensitivity to non-perturbative corrections below such a small scale. Accordingly, non-perturbative QCD effects are included in this analysis in the form of a Gaussian form factor $S_{\text {NP}}=\exp \{-g\,b^2\}$, which corresponds to a quadratic ansatz for the non-perturbative corrections.

At N$^3$LL+$\mathcal {O}(\alpha _S^3)$ accuracy in the small-$p_{\textrm{T}}$ region (i.e. including all the $\mathcal {O}(\alpha _S^3)$ terms) we have included in the calculation the functions $g^{(4)}$ and $\mathcal{H}_V^{(3)}$ in Eqs. (3) and (4). The asymptotic term $\text {d}\sigma ^{\text {asy}}$ and the $Z$+jet finite-order cross section $\text {d}\sigma ^{\text {f.o.}}$ are evaluated at $\mathcal {O}(\alpha _S^3)$. The $\mathcal {O}(\alpha _S^3)$ term of the $Z$+jet cross section predictions was computed with MCFM [34, 63], using a lower cutoff of $p_{\textrm{T}}= 5$ GeV, and the corresponding $\text {d}\sigma ^{\text {f.o.}}-\text {d}\sigma ^{\text {asy}}$ matching correction, which is as large as $-1\%$ in the Sudakov region, was extrapolated down to $p_{\textrm{T}}= 0$ by interpolating the corrections with their expected quadratic dependence on $p_{\textrm{T}}/m_Z$ [64], i.e. with the function $(p_{\textrm{T}}/m_Z)^2\sum _i c_i \log ^i(p_{\textrm{T}}/m_Z)$ including a set of free parameters $c_i$ (see also Refs. [33, 65] for similar parametrisations).

The running of the strong coupling is evaluated at four loops [66, 67] consistently in all parts of the calculation. The PDFs are interpolated with LHAPDF [68] at the factorisation scale $\mu _F$, and evolved backward using the next-to-next-to-leading order (NNLO) solution of the evolution equation, as implemented in Ref. [69], and four-loops running of the strong coupling. As shown in Appendix A of Ref. [49], such a procedure consistently resums the N${^3}$LL contributions to the form factor. The number of active flavours is set to five in all the coefficients entering the calculation, and in the evolution of the PDFs. In order to assess the impact of charm and bottom thresholds in the PDF evolution, an alternative forward PDF evolution with variable-flavour number scheme is used, and the difference with respect to the nominal five-flavour backward evolution is considered as an uncertainty. The predicted cross sections depend on three unphysical scales: the renormalisation scale $\mu _R$, the factorisation scale $\mu _F$, and the resummation scale Q, which parametrises the arbitrariness in the resummation procedure. The central value of the scales is set to the invariant mass of the lepton pair $m_{\ell \ell }$. We note that within the transverse-momentum resummation formalism of Refs. [49,50,51] the $\mu _R$, $\mu _F$, and Q scales have to be set of the order of the hard scale of the process $m_{\ell \ell }$ and do not depend on the transverse momentum of the Z boson. The electroweak parameters are set according to the $G_\mu$ scheme, in which the Fermi coupling constant $G_\text {F}$, the W-boson mass $m_W$, and the Z-boson mass $m_Z$ are set to the input values $G_\text {F} = 1.1663787 \cdot 10^{-5}$ GeV$^{-2}$, $m_W = 80.385$ GeV, $m_Z = 91.1876$ GeV [1], whereas the weak-mixing angle and the QED coupling are calculated at tree level.

The statistical analysis for the determination of $\alpha _S(m_Z)$ is performed with the xFitter framework [70]. The dependence of PDFs on the value of $\alpha _S(m_Z)$ is accounted for by using corresponding $\alpha _S$-series of PDF sets. The value of $\alpha _S(m_Z)$ is determined by minimising a $\chi ^2$ function which includes both the experimental uncertainties and the theoretical uncertainties arising from PDF variations:

$$\begin{aligned}{} & {} {\chi ^2(\beta _{\text {exp}},\beta _{\text {th}}) } \nonumber \\{} & {} \quad = \sum _{i=1}^{N_{\text {data}}} \frac{\textstyle \left( \sigma ^{\text {exp}}_i + \sum _j \Gamma ^{\text {exp}}_{ij} \beta _{j,\text {exp}} - \sigma ^{\text {th}}_i - \sum _k \Gamma ^{\text {th}}_{ik}\beta _{k,\text {th}} \right) ^2}{\Delta _i^2}\nonumber \\{} & {} \qquad + \sum _j \beta _{j,\text {exp}}^2 + \sum _k \beta _{k,\text {th}}^2. \end{aligned}$$

(5)

The correlated experimental and theoretical uncertainties are included using the nuisance parameter vectors $\beta _{\text {exp}}$ and $\beta _{\text {th}}$, respectively. Their influence on the data and theory predictions is described by the $\Gamma ^{\text {exp}}_{ij}$ and $\Gamma ^{\text {th}}_{ik}$ matrices. The index i runs over all $N_{\text {data}}$ data points, whereas the index j (k) corresponds to the experimental (theoretical) uncertainty nuisance parameters. The measurements and the uncorrelated experimental uncertainties are given by $\sigma ^{\text {exp}}_i$ and $\Delta _i$ , respectively, and the theory predictions are $\sigma _i^{\text {th}}$. At each value of $\alpha _S(m_Z)$, the PDF uncertainties are Hessian profiled according to Eq. (5) [71]. The parameter g of the Gaussian non-perturbative form factor is left free in the fit by adding g variations in Eq. (5) as an unconstrained nuisance parameter. The region of Z-boson transverse momentum $p_{\textrm{T}}< 30$ GeV is considered in the fit. Initial-state radiation of photons (QED ISR) is estimated at leading-logarithmic accuracy with Pythia8 [72] and the AZ tune of parton shower parameters [21], and the predictions are corrected with a bin-by-bin multiplicative factor. The effect on $\alpha _S(m_Z)$ of including these corrections is $\delta \alpha _S(m_Z)= -0.0006$. Uncertainties are estimated with initial-state photon radiation at next-to-leading logarithmic accuracy [73].

3 Results

The determination of $\alpha _S(m_Z)$ with the Hessian conversion [74] of the NNLO PDF set NNPDF4.0 [75] yields $\alpha _S(m_Z)= 0.1192$, with a statistical uncertainty of $\pm 0.0007$, a systematic experimental uncertainty of $\pm 0.0001$, and a PDF uncertainty of $\pm 0.0004$. The value of g determined in the fit is $g = 0.66 \pm 0.05$ GeV$^2$, with a correlation to $\alpha _S(m_Z)$ of $-0.8$. When performing a fit with fixed value of g, the uncertainties on $\alpha _S(m_Z)$ are reduced by $30\%$, yielding an estimate for the uncertainty contribution from non-perturbative QCD effects of $\pm 0.0006$. The value of the $\chi ^2$ function at minimum is 41 per 53 degrees of freedom. The pre- and post-fit predictions are compared to the measured Z-boson transverse-momentum distribution in Fig. 1.

Fig. 1

Comparison of N$^3$LL+$\mathcal {O}(\alpha _S^3)$ DYTurbo predictions to the measured Z-boson transverse-momentum distribution. The settings of the pre- and post-fit predictions are $\alpha _S(m_Z)= 0.118$, $g = 0$ GeV$^{2}$, and $\alpha _S(m_Z)= 0.1190$, $g = 0.66$ GeV$^{2}$, respectively. The dashed bands represent the PDF uncertainty of the NNPDF4.0 PDF set

Various alternative NNLO PDF sets are considered: CT18 [26], CT18Z, MSHT20 [76], HERAPDF2.0 [77], and ABMP16 [6]. The determined values of $\alpha _S(m_Z)$ range from a minimum of 0.1185 with the MSHT20 PDF set to a maximum of 0.1198 with the CT18Z PDF set. The midpoint value in this range of $\alpha _S(m_Z)= 0.1191$ is considered as nominal result, and the PDF envelope of $\pm 0.0007$ as an additional source of uncertainty. The determination of $\alpha _S(m_Z)$ from the various different NNLO PDF sets is shown in Fig. 2. The approximate N$^3$LO MSHT20 PDF set [78] is also considered, using predictions at approximate next-to-next-to-next-to-next-to-leading-logarithmic (N$^4$LL) accuracy [79], yielding a value of $\alpha _S(m_Z)= 0.1184$. Determinations of $\alpha _S(m_Z)$ at hadron colliders are exposed to possible biases unless the PDFs are determined simultaneously along with $\alpha _S(m_Z)$ [80]. Nonetheless, $\alpha _S(m_Z)$ determinations from single or limited hadron collider datasets based on existing PDF sets, are interesting to study in detail the sensitivity to $\alpha _S(m_Z)$ of a particular observable and the associated theoretical uncertainties. The Hessian profiling employed in this analysis provides an approximation to a PDF determination which relies on the accuracy of the quadratic approximation around the minimum [81] (see Appendix B for details). In all the cases considered in this analysis, pulls and constraints of the nuisance parameters associated to the PDF uncertainties are below $20\%$ and $10\%$, respectively, indicating that the new minimum of the profiled PDFs is very close to the original minimum, which gives confidence in the validity of the quadratic approximation.

Fig. 2

Comparison of the $\alpha _S(m_Z)$ determination from the Z-boson transverse-momentum distribution with varying fit range, with various different PDF sets, and with measurements performed with the D0 detector

With the aim of further testing the validity of the Hessian profiling approximation, a simultaneous fit of PDFs, $\alpha _S(m_Z)$, and the non-perturbative parameter g is performed. The combined neutral and charged current deep inelastic scattering (DIS) cross-section data from the H1 and ZEUS experiments at the HERA collider [77] are included in the fit, with a minimum squared four-momentum transfer $Q^2$ of 3.5 GeV$^2$, together with the Z-boson transverse-momentum distribution measured by CDF. The light-quark coefficient functions of the DIS cross sections are calculated in the $\overline{MS}$ scheme [82], and with the renormalisation and factorisation scales set to the squared four-momentum transfer $Q^2$. The heavy quarks c and b are dynamically generated, and the corresponding coefficient functions for the neutral-current processes with $\gamma ^*$ exchange are calculated in the general-mass variable-flavour-number (VFN) scheme [83,84,85], with up to five active quark flavours. The charm mass is set to $m_c = 1.43$ GeV, and the bottom mass to $m_b = 4.50$ GeV [77]. For the charged-current processes the heavy quarks are treated as massless. The PDFs for the gluon, u-valence, d-valence, $\bar{u}$, $\bar{d}$ quark densities are parameterised at the input scale $Q^2_0=1.9$ GeV$^2$ with the parametrisation of Ref. [77]. The contribution of the s-quark density is taken to be proportional to the $\bar{d}$-quark density by setting $x\bar{s}(x) = r_s x\bar{d}(x)$, with $r_s=0.67$. The determined value of $\alpha _S(m_Z)$ from this fit is $0.1184 \pm 0.0006$, where the quoted uncertainty is the uncertainty from the fit, which includes experimental and PDF uncertainties. The value of $\alpha _S(m_Z)$ is in agreement with the determinations based on the Hessian profiling approach.

The alternative fits with different PDF sets and the simultaneous fit of PDFs and $\alpha _S(m_Z)$ are summarised in Table 1.

Table 1 Alternative fits of $\alpha _S(m_Z)$ with different PDF sets

Missing higher order uncertainties are estimated through independent variations of $\mu _R$, $\mu _F$ and Q in the range $m_{\ell \ell }/2 \le \{ \mu _R, \mu _F, Q \} \le 2m_{\ell \ell }$ with the constraints $0.5\le \{ \mu _F/\mu _R, Q/\mu _R, Q/\mu _F \}\le 2$, leading to 14 variations. The determined values of $\alpha _S(m_Z)$ range from a minimum of 0.1183 to a maximum of 0.1196 with respect to the value at the central scale choice of $\alpha _S(m_Z)= 0.1192$, yielding a scale-variation envelope of $^{+0.0004}_{-0.0009}$. The alternative fits with different choices of the QCD scales are summarised in Table 2.

Table 2 Alternative fits of $\alpha _S(m_Z)$ with different choices of the renormalisation ($\mu _R$), factorisation ($\mu _F$) and resummation (Q) scales

Fits without the $\mathcal {O}(\alpha _S^3)$ matching corrections yield a central value which is 0.0005 lower, and an increase in the half envelope of scale variations from 0.0007 to 0.0009, which is consistent with the observed shift. Systematic uncertainties in the $\mathcal {O}(\alpha _S^3)$ matching corrections are estimated by raising the lower cutoff from $p_{\textrm{T}}= 5$ GeV to $p_{\textrm{T}}= 10$ GeV. The difference of 0.0001 with respect to the nominal fit is considered as a source of uncertainty. Statistical uncertainties in the $\mathcal {O}(\alpha _S^3)$ matching corrections are estimated with a set of 1000 replicas of the matching corrections generated by fluctuating them within their numerical uncertainties. The upper and lower limits of the 68$\%$ confidence level envelope of interpolations to the replicas are used for the estimate of the statistical uncertainty, yielding less than $\pm 0.0001$. Further details are provided in Appendix A.

Uncertainties in the modelling of the non-perturbative form factor are estimated by performing four alternative fits with: a value of $b_{\text {lim}} = 2$ GeV$^{-1}$ in the $b_*$ regularisation procedure; the minimal prescription, which corresponds to the limit $b_{\text {lim}} \rightarrow \infty$; using an additional quartic term $\exp \{-q\,b^4\}$ with $q = 0.1$ GeV$^4$; using the additional term $\exp \{-g_{k}\}$ with $g_k = g_0\big (1-\exp \big [-\frac{\text {C}_{\text {F}}}{\pi g_0 b_{\text {lim}}^2}\big ]\big )\log (m_{\ell \ell }^2/Q_0^2)$ with $g_0 = 0.3$, $Q_0 = 1$ GeV, and $b_{\text {lim}} = 2$ GeV$^{-1}$ [44], where $C_F$ is the colour-factor associated with gluon emission from a quark. The alternative fits yield variations of $\alpha _S(m_Z)$ in the range of $\pm 0.0007$, which is considered as an uncertainty. In the alternative fits, the parameter of the Gaussian non-perturbative form factor ranges from $g = 0.42$ GeV$^{2}$ in the case of the minimal prescription to $g = 0.83$ GeV$^{2}$ in the case of the fit with $b_{\text {lim}} = 2$ GeV$^{-1}$, in agreement with values obtained by global fits [42, 86, 87], and corresponding to values of average primordial $k_T^2$ of the partons, $\langle k_T^2 \rangle = 2\,g$ [88, 89], in the range 0.8–1.7 GeV$^2$. Such values are generally large for non-perturbative effects within a bound state with a mass of 1 GeV as the proton. However the fitted values of g also accounts for power corrections related to the regularisation procedure of the perturbative form factor, to the perturbative evolution of the non-perturbative form factor from low scales to $m_Z$, and to yet uncalculated higher-order corrections. A fit in which the NNPDF4.0 PDF set is evolved with a variable-flavour number scheme yields $\delta \alpha _S(m_Z)= -0.0003$, which is considered as an additional source of uncertainty. The alternative fits with different non-perturbative and heavy flavour models are summarised in Table 3.

Table 3 Alternative fits of $\alpha _S(m_Z)$ with different non-perturbative and heavy flavour models

A fit with NLL initial-state radiation of photons yields a difference on $\alpha _S(m_Z)$with respect to the Pythia8 modelling of less than 0.0001, which is considered as an additional source of uncertainty.

The stability of the results upon variations of the fit range is tested by performing fits in the regions of Z-boson transverse momentum $p_{\textrm{T}}< 20$ GeV and $p_{\textrm{T}}< 40$ GeV. The spread in the determined values of $\alpha _S(m_Z)$ is at the level of $\pm 0.0001$ and the uncertainty of the fit increases from $\pm 0.0007$ to $\pm 0.0008$. Since the region $20< p_{\textrm{T}}< 40$ GeV is sensitive to the matching of the resummed cross section to the fixed order prediction, this test provides a confirmation that the result is largely independent from the matching corrections in this region. Uncertainties associated to the stability of the fit results with respect to variations of the upper limit of the fit range are considered negligible. The fit range is also varied by excluding the low transverse-momentum region. The range is reduced up to $4< p_{\textrm{T}}< 30$ GeV, with a spread in the values of $\alpha _S(m_Z)$ at the level of $\pm 0.0002$, and an increase in the uncertainty of the fit from $\pm 0.0008$ to $\pm 0.0016$. For the fit in the range $4< p_{\textrm{T}}< 30$ GeV the value of g is determined as $0.3 \pm 0.3$ GeV$^2$ and the correlation between $\alpha _S(m_Z)$ and g is reduced from $-0.8$ to $-0.4$. Since the low transverse-momentum region is the most sensitive to the non-perturbative QCD effects, this test provides a validation of the model for the non-perturbative form factor. The spread of $\pm 0.0002$ from variations of the lower limit of the fit range is considered as an additional source of uncertainty.

A consistency check of the $\alpha _S(m_Z)$ determination was performed using cross sections measured with the D0 detector [90]. The fit to the D0 data in the Z-boson rapidity range $|y| < 1$ yields value of $\alpha _S(m_Z)= 0.1190 \pm 0.0013$ in the electron decay channel and $\alpha _S(m_Z)= 0.1192 \pm 0.0013$ in the muon decay channel, where the quoted uncertainties include experimental and PDF uncertainties. The D0 measurement, which was performed on the variable $\phi ^{*}_{\eta }$, is extrapolated to the transverse-momentum $p_{\textrm{T}}$. The extrapolation procedure has associated uncertainties which were not estimated in the analysis. The determined values of $\alpha _S(m_Z)$ are compatible with the CDF result within experimental uncertainties. Determinations of $\alpha _S(m_Z)$ with varying fit range and with cross sections measured with the D0 detector are shown in Fig. 2.

Table 4 Summary of the uncertainties for the determination of $\alpha _S(m_Z)$, in units of $10^{-3}$

A summary of the uncertainties in the determination of $\alpha _S(m_Z)$ is shown in Table 4.

4 Conclusions

In summary, the value of the strong-coupling constant determined in this analysis is $\alpha _S(m_Z)= 0.1191^{+0.0013}_{-0.0016}$, with a statistical uncertainty of $\pm 0.0007$, an experimental systematic uncertainty of $\pm 0.0001$, a PDF uncertainty of $\pm 0.0008$, missing higher order uncertainties of $^{+0.0004}_{-0.0009}$, and additional theory uncertainties (non-perturbative model, flavour scheme, matching corrections, photon initial-state radiation) of $\pm {0.0008}$. The strong-coupling constant is also determined in a simultaneous PDF-fit determination including DIS cross-section data from the H1 and ZEUS experiments at the HERA collider. When considering the fit uncertainties of $\pm 0.0006$ and all the other relevant uncertainties listed in Table 4, the result of this determination is $\alpha _S(m_Z)= 0.1184 ^{+0.0013}_{-0.0015}$.

We have performed a determination of $\alpha _S(m_Z)$ from the Z-boson transverse-momentum distribution measured at the Tevatron collider, in the low-momentum region of $p_{\textrm{T}}< 30$ GeV. This analysis represents the first determination using QCD resummed theory predictions based on a semi-inclusive observable at hadron-hadron colliders.³ The PDF uncertainties are estimated with a conservative approach, including the envelope of six different PDF sets, and with a Hessian profiling procedure, which avoids possible biases in the treatment of PDF uncertainties. Missing higher order uncertainties are estimated with the standard approach of computing an envelope of scale variations. The measured value of $\alpha _S(m_Z)$ has a relative uncertainty of $1.2\%$, and is compatible with other determinations and with the world-average value, as illustrated in Fig. 3.

Fig. 3

Comparison of the $\alpha _S(m_Z)$ determination from the Z-boson transverse-momentum distribution to other determinations and to the world-average value

Among hadron colliders determination, this is the most precise to date and the first based on N$^3$LL+$\mathcal {O}(\alpha _S^3)$ predictions in perturbative QCD.

Appendices

Matching corrections

In this Appendix we discuss the interpolation of the $\text {d}\sigma ^{\text {f.o.}}-\text {d}\sigma ^{\text {asy}}$ matching corrections of Eq. (1) at $\mathcal {O}(\alpha _S^3)$ with their expected quadratic dependence on $p_{\textrm{T}}/m_Z$ using the function

$$\begin{aligned} \frac{p_{\textrm{T}}^2}{m_Z^2}\sum _i c_i \ln ^i\left( \frac{p_{\textrm{T}}}{m_Z}\right) . \end{aligned}$$

(6)

Fits are performed in the region of $p_{\textrm{T}}< 50$ GeV, with 10 logarithmically spaced bins. The p-values for fits of the matching corrections with different choices of the renormalisation, factorisation, and resummation scales are in the range from 0.3 to 0.9.

We have considered two sources of uncertainties addressing the statistical and systematic uncertainties of the matching corrections. We have varied the lower cutoff from $p_T = 5$ GeV to $p_T = 10$ GeV. The difference in $\alpha _S(m_Z)$ of 0.0001 is considered as an additional systematic uncertainty. In order to estimate the statistical uncertainty, we have generated a set of 1000 Monte Carlo replicas of the matching corrections, by fluctuating them within their numerical uncertainties. The upper and lower limits of the 68$\%$ confidence level envelope of the extrapolation fits to the 1000 replicas are used for the estimate of the statistical uncertainty, yielding $\pm 0.00002$ on $\alpha _S(m_Z)$.

The difference between the NNLO Z+jet predictions and the expansion of the resummed calculation, showing the numerical accuracy in the matching procedure at $\alpha _S^3$ order, is presented in Fig. 4, which also shows the replicas and their 68$\%$ confidence level uncertainty band.

Comparable figures showing the difference of the asymptotic term $\text {d}\sigma ^{\text {asy}}$ and the $Z$+jet finite-order cross section $\text {d}\sigma ^{\text {f.o.}}$ at $\mathcal {O}(\alpha _S^3)$ can be found in Refs. [30, 33, 34, 64]

The studies of statistical and systematic uncertainties discussed above confirm that the $\mathcal {O}(\alpha _S^3)$ matching corrections are associated with small uncertainties, which are accounted for in the final result. The estimated uncertainties of $\pm 0.0001$ are consistent with the overall small impact of such corrections, which is estimated as $+0.0005$.

Fig. 4

The $\mathcal {O}(\alpha _S^3)$ matching corrections

Simultaneous PDF and $\alpha _S(m_Z)$ fit

The Hessian profiling employed in this analysis provides an approximation to a PDF determination which relies on the accuracy of the quadratic approximation around the minimum [81]. The validity of the Hessian profiling approximation, is verified by performing a simultaneous fit of PDFs, $\alpha _S(m_Z)$, and the the parameter g of the Gaussian non-perturbative form factor, with a setup similar to the one employed for the HERAPDF2.0 [77] PDF determination. In this Appendix we provide further quantitative details of the comparison of the Hessian profiling of HERAPDF2.0 with such a PDF fit. The PDF fit includes the combined neutral and charged current deep inelastic scattering (DIS) cross-section data from the H1 and ZEUS experiments at the HERA collider [77]. Table 5 shows the contribution to the total $\chi ^2$ of the various datasets used in the fit, compared to the $\chi ^2$ of the Hessian profiling, and a comparison of the determined values of $\alpha _S(m_Z)$ and g.

Table 5 Comparison of the Hessian profiling of HERAPDF2.0 with the PDF fit, including the contribution to the total $\chi ^2$ at minimum of the various datasets used in the fit