Precise determination of $\alpha _{S}(M_Z)$ from a global fit of energy–energy correlation to NNLO+NNLL predictions

Abstract

We present a comparison of the computation of energy–energy correlation in $e^{+}e^{-}$ collisions in the back-to-back region at next-to-next-to-leading logarithmic accuracy matched with the next-to-next-to-leading order perturbative prediction to LEP, PEP, PETRA, SLC and TRISTAN data. With these predictions we perform an extraction of the strong coupling constant taking into account non-perturbative effects modelled with Monte Carlo event generators. The final result at NNLO+NNLL precision is $\alpha _{S}(M_{Z})= 0.11750\pm 0.00018 {\text{( } exp.)}\pm 0.00102{\text{( }hadr.)}\pm 0.00257{\text{( }ren.)}\pm 0.00078{\text{( }res.)}$.

1 Introduction

The strong interaction in the Standard Model (SM) is described by quantum chromodynamics (QCD) [1,2,3,4]. The theory successfully models the interactions between quarks and gluons and is a source of numerous predictions. Verifying the predictions of QCD is instrumental for searches for physics beyond the SM at the LHC, since the reliable prediction of SM processes as sources of backgrounds for searches is essential.

Precision measurements of event shape distributions in $e^+e^-$ annihilation have provided detailed experimental tests of QCD and remain one of the most precise tools used for extracting the strong coupling $\alpha _{\mathrm{S}}$ from data [5, 6]. Quantities related to three-jet events are particularly well suited for this task.

The state of the art for QCD for event shape observables currently includes exact fixed-order next-to-next-to-leading order (NNLO) corrections for the six standard three-jet event shapes of thrust, heavy jet mass, total and wide jet broadening, C-parameter and the two-to-three jet transition variable $y_{23}$ [7,8,9] as well as jet cone energy fraction [9], oblateness and energy–energy correlation [10]. The numerical matrix element integration codes described in the references allow the straightforward computation of any suitable, i.e. collinear and infrared safe event shape or jet observable.

However, fixed-order predictions have a limited kinematical range of applicability. For small values of an event shape observable y corresponding to events with two-jet-like topologies the fixed-order predictions do not converge well. This is due to terms where each power of the strong coupling $\alpha _{\mathrm{S}}^n$ is enhanced by a factor $(\ln y)^{n+1}$ (leading logs), $(\ln y)^{n}$ (next-to-leading logs) etc. For three-jet event shapes such logarithmically enhanced terms can be resummed at next-to-next-to-leading logarithmic (NNLL) accuracy [11,12,13,14,15,16,17], i.e. up to terms $\sim (\ln y)^{n-1}$. Resummation in next-to-next-to-next-to-leading logarithmic $(\hbox {N}^3\mathrm{LL})$ accuracy has been achieved for the C-parameter [18] and thrust [19]. A prediction incorporating the complete perturbative knowledge about the observable can be derived by matching the fixed-order and resummed calculations.

For the frequently used event shapes of thrust, heavy jet mass, total and wide jet broadening, C-parameter and $y_{23}$, NNLO predictions matched to NLL resummation were presented in [20]. Predictions at NNLO matched to N${}^3$LL resummation are also known for thrust [12, 19] and the C-parameter [18].

In this paper we consider the energy–energy correlation (EEC) in $e^+e^-$ annihilation and present NNLO predictions matched to NNLL resummation for the back-to-back region. EEC was the first event shape for which a complete NNLL resummation was performed [11] while the fixed-order NNLO corrections to this observable were computed recently [10]. Moreover, EEC is the first event shape observable for which an analytic fixed-order NLO correction was computed [21].

The agreement between the predictions at NNLO+NNLL accuracy and the measured data is still not perfect. The discrepancy can be attributed mainly to non-perturbative hadronization corrections. We extract these corrections from data by comparison to state-of-the-art Monte Carlo predictions and determine the value of the strong coupling by comparing our results to measurements over a wide range of centre-of-mass energies. Our analysis allows us to target the highest precision of $\alpha _{\mathrm{S}}$ determination and we present the first global fit of the strong coupling to EEC at NNLO+NNLL accuracy. Our analysis also represents the first extraction of $\alpha _{\mathrm{S}}$ based on Monte Carlo hadronization corrections obtained from NLO Monte Carlo setups at NNLO+NNLL precision.

2 EEC distribution in perturbation theory

EEC is the normalized energy-weighted cross section defined in terms of the angle between two particles i and j in an event [22]:

$$\begin{aligned} \frac{1}{\sigma _{\mathrm{t}}} \frac{{\mathrm{d}}\varSigma (\chi )}{{\mathrm{d}}\cos \chi } \equiv \frac{1}{\sigma _{\mathrm{t}}} \int \sum _{i,j} \frac{E_i E_j}{Q^2} {\mathrm{d}}\sigma ^{\mathrm{}}_{e^+e^- \rightarrow \, i j + X} \delta (\cos \chi - \cos \theta _{ij}), \end{aligned}$$

(1)

where $E_i$ and $E_j$ are the particle energies, Q is the centre-of-mass energy, $\theta _{ij} = \chi$ is the angle between the two particles and $\sigma _{\mathrm{t}}$ is the total hadronic cross section. The back-to-back region $\theta _{ij} \rightarrow 180^{\circ }$ corresponds to $\chi \rightarrow \pi$, while the normalization ensures that the integral of the EEC distribution from $\chi = 0^{\circ }$ to $\chi = 180^{\circ }$ is unity.¹

2.1 Fixed-order and resummed calculations

The differential EEC distribution has been computed numerically at NLO accuracy in perturbation theory some time ago [24,25,26,27,28,29,30,31,32,33,34] and efforts towards obtaining an analytic result at this order [35, 36] have culminated in a complete calculation very recently [21]. The NNLO prediction has also been obtained in ref.  [10] using the CoLoRFulNNLO method [9, 37, 38]. At the default renormalization scale² of $\mu =Q$ the fixed-order prediction reads

$$\begin{aligned} \bigg [\frac{1}{\sigma ^{\mathrm{}}_{0}} \frac{{\mathrm{d}}\varSigma (\chi ,Q)}{{\mathrm{d}}\cos \chi }\bigg ]_{\mathrm {f.o.}}= & {} \frac{\alpha _{\mathrm{S}}(Q)}{2\pi } \frac{{\mathrm{d}}A(\chi )}{{\mathrm{d}}\cos \chi }+ \left( \frac{\alpha _{\mathrm{S}}(Q)}{2\pi }\right) ^2 \frac{{\mathrm{d}}B(\chi ) }{{\mathrm{d}}\cos \chi } \nonumber \\&+\left( \frac{\alpha _{\mathrm{S}}(Q)}{2\pi }\right) ^3 \frac{{\mathrm{d}}C(\chi )}{{\mathrm{d}}\cos \chi } +\mathcal O(\alpha _{\mathrm{S}}^{4}), \end{aligned}$$

(2)

where A, B and C are the perturbative coefficients at LO, NLO and NNLO, normalized to the LO cross section for $e^+e^- \rightarrow \text{ hadrons }$, $\sigma ^{\mathrm{}}_{0}$. In massless QCD this normalization cancels all electroweak coupling factors, and the dependence on the collision energy enters only through $\alpha _{\mathrm{S}}(Q)$. However, experiments measure the distribution normalized to the total hadronic cross section, so physical predictions must be normalized to $\sigma _{\mathrm{t}}$. The distribution normalized to the total hadronic cross section can be obtained from the expansion in Eq. (2) through multiplying by $\sigma ^{\mathrm{}}_{0}/\sigma _{\mathrm{t}}$. For massless quarks, this ratio is independent of all electroweak couplings and reads

$$\begin{aligned} \frac{\sigma ^{\mathrm{}}_{0}}{\sigma _{\mathrm{t}}} = 1 - \frac{\alpha _{\mathrm{S}}(Q)}{2\pi } A_{\mathrm{t}} + \left( \frac{\alpha _{\mathrm{S}}(Q)}{2\pi }\right) ^2 \left( A_{\mathrm{t}}^2 - B_{\mathrm{t}}\right) + \mathcal O(\alpha _{\mathrm{S}}^{3}), \end{aligned}$$

with

$$\begin{aligned} A_{\mathrm{t}} = \frac{3}{2}C_{\mathrm{F}}\end{aligned}$$

and

$$\begin{aligned} B_{\mathrm{t}}= & {} C_{\mathrm{F}}\left[ \left( \frac{123}{8} - 11 \zeta _3\right) \right. C_{\mathrm{A}}-\frac{3}{8} C_{\mathrm{F}}- \left. \left( \frac{11}{2} - 4 \zeta _3\right) n_{\mathrm{f}}T_{\mathrm{R}}\right] . \end{aligned}$$

The colour factors which appear above are given by

$$\begin{aligned} C_{\mathrm{A}}= 2N_{\mathrm{c}}T_{\mathrm{R}}, \quad C_{\mathrm{F}}= \frac{N_{\mathrm{c}}^2-1}{N_{\mathrm{c}}}T_{\mathrm{R}}\quad \text{ and }\quad T_{\mathrm{R}}= \frac{1}{2}, \end{aligned}$$

while $n_{\mathrm{f}}$ denotes the number of light quark flavours.

The renormalization scale dependence of the fixed-order prediction can be restored using the renormalization group equation for $\alpha _{\mathrm{S}}$ and one finds

$$\begin{aligned} \begin{aligned} \bigg [\frac{1}{\sigma _{\mathrm{t}}} \frac{{\mathrm{d}}\varSigma (\chi ,\mu )}{{\mathrm{d}}\cos \chi }\bigg ]_{\mathrm {f.o.}}&= \frac{\alpha _{\mathrm{S}}(\mu )}{2\pi } \frac{{\mathrm{d}}{\bar{A}}(\chi ,x_{R})}{{\mathrm{d}}\cos \chi } \\&\quad + \left( \frac{\alpha _{\mathrm{S}}(\mu )}{2\pi }\right) ^2 \frac{{\mathrm{d}}{\bar{B}}(\chi ,x_{R}) }{{\mathrm{d}}\cos \chi }\\&\quad +\left( \frac{\alpha _{\mathrm{S}}(\mu )}{2\pi }\right) ^3 \frac{{\mathrm{d}}{\bar{C}}(\chi ,x_{R})}{{\mathrm{d}}\cos \chi } + \mathcal O(\alpha _{\mathrm{S}}^{4}), \end{aligned}\end{aligned}$$

(3)

where

$$\begin{aligned}\begin{aligned} {\bar{A}}(\chi ,x_{R})&= A(\chi ), \\ {\bar{B}}(\chi ,x_{R})&= B(\chi ) + \left( \frac{1}{2}\beta _0 \ln (x_{R}^2)- A_{\mathrm{t}}\right) A(\chi ), \\ {\bar{C}}(\chi ,x_{R})&= C(\chi ) + \left( \beta _0 \ln (x_{R}^2)- A_{\mathrm{t}}\right) B(\chi )\\&\quad + \bigg (\frac{1}{4} \beta _1 \ln (x_{R}^2)+ \frac{1}{4}\beta _0^2 \ln ^2(x_{R}^2)- A_{\mathrm{t}} \beta _0 \ln (x_{R}^2)\\&\quad + A_{\mathrm{t}}^2 - B_{\mathrm{t}} \bigg ) A(\chi ), \end{aligned}\end{aligned}$$

with $x_{R}= \mu /Q$. Finally, using three-loop running the scale dependence of the strong coupling is given by

$$\begin{aligned} \begin{aligned} \alpha _{\mathrm{S}}(\mu )&= \frac{4\pi }{\beta _0 t} \left[ 1 - \frac{\beta _1}{\beta _0^2 t} \ln t\right. \\&\quad + \left. \left( \frac{\beta _1}{\beta _0^2 t}\right) ^2 \left( \ln ^2 t - \ln t - 1 + \frac{\beta _0 \beta _2}{\beta _1^2}\right) \right] . \end{aligned}\end{aligned}$$

Here, $t = \ln (\mu ^2/\varLambda _{\mathrm{QCD}}^2)$ and the $\beta _i$ are the $\overline{\mathrm{MS}}$-scheme coefficients of the QCD beta function,

$$\begin{aligned} \begin{aligned} \beta _0&= \frac{11C_{\mathrm{A}}}{3} - \frac{4n_{\mathrm{f}}T_{\mathrm{R}}}{3}, \\ \beta _1&= \frac{34}{3} C_{\mathrm{A}}^2 - \frac{20}{3} C_{\mathrm{A}}T_{\mathrm{R}}n_{\mathrm{f}}- 4 C_{\mathrm{F}}T_{\mathrm{R}}n_{\mathrm{f}}, \\ \beta _2&= \frac{2857}{54} C_{\mathrm{A}}^3 - \left( \frac{1415}{27} C_{\mathrm{A}}^2 + \frac{205}{9} C_{\mathrm{A}}C_{\mathrm{F}}- 2 C_{\mathrm{F}}^2\right) T_{\mathrm{R}}n_{\mathrm{f}}\\&\quad + \left( \frac{158}{27} C_{\mathrm{A}}+ \frac{44}{9} C_{\mathrm{F}}\right) T_{\mathrm{R}}^2 n_{\mathrm{f}}^2. \end{aligned} \end{aligned}$$

The fixed-order perturbative predictions diverge for both small and large values of $\chi$, due to the presence of large logarithmic contributions of infrared origin. Concentrating on the back-to-back region $\chi \rightarrow 180^{\circ }$, these contributions take the form $\alpha _{\mathrm{S}}^n \log ^{2n-1} y$, where

$$\begin{aligned} y = \cos ^2 \frac{\chi }{2}. \end{aligned}$$

As y decreases, the logarithms become large and invalidate the use of the fixed-order perturbative expansion. In order to obtain a description of EEC in this limit, the logarithmic contributions must be resummed to all orders. This resummation has been computed at NNLL accuracy in Ref.  [11]³ while in Ref.  [40] a factorization theorem for EEC was derived based on soft-collinear effective theory which will allow to preform the resummation at N$^3$LL accuracy once the corresponding NNLO jet function is computed. Since the complete jet function is currently not available, we use the NNLL results and formalism of Ref.  [11] in the following. The resummed prediction at the default scale of $\mu =Q$ can be written as

$$\begin{aligned} \bigg [\frac{1}{\sigma _{\mathrm{t}}}\frac{{\mathrm{d}}\varSigma (\chi ,Q)}{{\mathrm{d}}\cos \chi }\bigg ]_{\mathrm {res.}}= & {} \frac{Q^2}{8} H(\alpha _{\mathrm{S}}(Q)) \nonumber \\&\times \int \limits _0^\infty {\mathrm{d}}b\, b\, J_0(b\, Q \sqrt{y}) S(Q,b). \end{aligned}$$

(4)

The large logarithmic corrections are exponentiated in the Sudakov form factor,

$$\begin{aligned} S(Q,b)= & {} \exp \Bigg \{-\int \limits _{b_0^2/b^2}^{Q^2} \frac{{\mathrm{d}}q^2}{q^2}\left[ A(\alpha _{\mathrm{S}}(q^2)) \ln \frac{Q^2}{q^2} \right. \nonumber \\&+\left. B(\alpha _{\mathrm{S}}(q^2))\right] \Bigg \}. \end{aligned}$$

(5)

The zeroth order Bessel function $J_0$ in Eq. (4) and $b_0 = 2 e^{-\gamma _\mathrm{E}}$ in Eq. (5) have a kinematic origin. The functions A, B (not to be confused with the fixed-order expansion coefficients appearing in Eq. (2)) and H in Eqs. (4) and (5) are free of logarithmic corrections and can be computed as perturbative expansions in $\alpha _{\mathrm{S}}$,

$$\begin{aligned} A(\alpha _{\mathrm{S}})&= \sum _{n=1}^{\infty } \left( \frac{\alpha _{\mathrm{S}}}{4\pi }\right) ^n A^{(n)}, \end{aligned}$$

(6)

$$\begin{aligned} B(\alpha _{\mathrm{S}})&= \sum _{n=1}^{\infty } \left( \frac{\alpha _{\mathrm{S}}}{4\pi }\right) ^n B^{(n)}, \end{aligned}$$

(7)

$$\begin{aligned} H(\alpha _{\mathrm{S}})&= 1+ \sum _{n=1}^{\infty } \left( \frac{\alpha _{\mathrm{S}}}{4\pi }\right) ^n H^{(n)}. \end{aligned}$$

(8)

Explicit expressions for the expansion coefficients (up to NNLL accuracy) in our normalization conventions can be found in Ref.  [23].

It is possible to perform the $q^2$ integration in Eq. (5) analytically and the Sudakov form factor can be written as

$$\begin{aligned} \begin{aligned} S(Q,b)&= \exp [L g_1(a_{\mathrm{S}} \beta _0 L) + g_2(a_{\mathrm{S}} \beta _0 L) \\&\quad + a_{\mathrm{S}} g_3(a_{\mathrm{S}} \beta _0 L) + \ldots ], \end{aligned} \end{aligned}$$

(9)

where $a_{\mathrm{S}} = \alpha _{\mathrm{S}}(Q)/(4\pi )$ and $L = \ln (Q^2 b^2/b_0^2)$ corresponds to $\ln y$ at large b (the $y \ll 1$ limit corresponds to $Q b \gg 1$ through a Fourier transformation). Writing the Sudakov form factor this way clearly shows that S(Q, b) depends on its variables only through the dimensionless combination $b\,Q$. The functions $g_1$, $g_2$ and $g_3$ correspond to the LL, NLL and NNLL contributions. Their explicit expressions can be found in Refs.  [11, 23].

So far, we have not considered the dependence of the resummed prediction on the renormalization scale. Besides the replacement of $\alpha _{\mathrm{S}}(Q)$ by $\alpha _{\mathrm{S}}(\mu )$ in Eqs. (4) and (9), the resummation functions $g_i(\lambda )$ also acquire renormalization scale dependence,

$$\begin{aligned} g_1(\lambda ,x_{R})= & {} g_1(\lambda ),\\ g_2(\lambda ,x_{R})= & {} g_2(\lambda ) + \lambda ^2 g_1'(\lambda ) \ln (x_{R}^2),\\ g_3(\lambda ,x_{R})= & {} g_3(\lambda ) + \left[ \frac{\beta _1}{\beta _0} \lambda ^2 g_1'(\lambda ) + \beta _0 \lambda g_2'(\lambda )\right] \ln (x_{R}^2)\\&+\left[ \frac{\beta _0}{2} \lambda ^3 g_1''(\lambda ) + \beta _0 \lambda ^2 g_1'(\lambda )\right] \ln ^2 (x_{R}^2), \end{aligned}$$

where the prime denotes differentiation with respect to $\lambda$.

The factorization between the constant and logarithmic terms $H(\alpha _{\mathrm{S}})$ and S(Q, b) in Eq. (4) also involves some arbitrariness, since the argument of the large logarithm L can always be rescaled as

$$\begin{aligned} L = \ln (Q^2 b^2/b_0^2) = \ln (x_{L}^2Q^2 b^2/b_0^2) - \ln (x_{L}^2), \end{aligned}$$

provided that $x_{L}$ is independent of b and that $x_{L}= {{\mathcal {O}}}(1)$ when $Q b \gg 1$. This arbitrariness is parametrized by $x_{L}$, which plays a role in the resummed computation which is analogous to the role played by the renormalization scale in the fixed-order calculation. This rescaling of the logarithm introduces some modifications of the resummed formulae and the expansion coefficients in Eqs. (6)–(8). We find

$$\begin{aligned} {\tilde{A}}^{(n)}(x_{L})&= A^{(n)}, \\ {\tilde{B}}^{(n)}(x_{L})&= B^{(n)} - A^{(n)} \ln (x_{L}^2), \\ {\tilde{H}}^{(1)}(x_{L})&= H^{(1)} - \beta _0 g_2'(0) \ln (x_{L}^2) + \beta _0 g_1'(0) \ln ^2(x_{L}^2), \end{aligned}$$

while the Sudakov form factor in Eq. (9) is also modified as follows

$$\begin{aligned}&S(Q,b,x_{R},x_{L}) = \exp \nonumber \\&\quad \times \bigg [ {\tilde{L}} g_1\left( a_{\mathrm{S}} \beta _0 {\tilde{L}}, \frac{x_{R}}{x_{L}}\right) + g_2\left( a_{\mathrm{S}} \beta _0 {\tilde{L}}, \frac{x_{R}}{x_{L}}\right) \nonumber \\&\quad + \alpha _{\mathrm{S}}g_3\left( a_{\mathrm{S}} \beta _0 {\tilde{L}}, \frac{x_{R}}{x_{L}}\right) + \cdots \bigg ], \end{aligned}$$

(10)

where ${\tilde{L}} = \ln (x_{L}^2 Q^2 b^2/b_0^2)$.

2.2 Matching the fixed-order and resummed predictions

In order to obtain a prediction which is valid over a wide kinematical range⁴ the fixed-order and resummed calculations must be matched. Here we employ the log-R matching scheme as worked out for EEC in Ref. [23], and limit ourselves to recalling the final results.

In the log-R matching scheme for EEC we consider the cumulative distribution

$$\begin{aligned} \begin{aligned} \frac{1}{\sigma _{\mathrm{t}}} {\widetilde{\varSigma }}(\chi ,\mu )&\equiv \frac{1}{\sigma _{\mathrm{t}}} \int \limits _0^\chi {\mathrm{d}}\chi '\, (1-\cos \chi ') \frac{{\mathrm{d}}\varSigma (\chi ',\mu )}{{\mathrm{d}}\chi '} \\&= \frac{1}{\sigma _{\mathrm{t}}} \int \limits _0^{y(\chi )} {\mathrm{d}}y'\, 2(1-y') \frac{{\mathrm{d}}\varSigma (y',\mu )}{{\mathrm{d}}y'}. \end{aligned} \end{aligned}$$

(11)

The differential EEC distribution is easily recovered from ${\widetilde{\varSigma }}(\chi ,\mu )$,

$$\begin{aligned} \frac{1}{\sigma _{\mathrm{t}}} \frac{{\mathrm{d}}\varSigma (\chi ,\mu )}{{\mathrm{d}}\chi } = \frac{1}{1-\cos \chi }\frac{{\mathrm{d}}}{{\mathrm{d}}\chi } \left[ \frac{1}{\sigma _{\mathrm{t}}} {\widetilde{\varSigma }}(\chi ,\mu )\right] . \end{aligned}$$

The particular linear combination of moments introduced in Eq. (11) has the property that the divergence of the differential EEC distribution in the forward region ($\chi \rightarrow 0$) is suppressed by the factor of $(1-\cos \chi )$. Hence, in contrast to EEC itself, the fixed-order cumulative coefficients of ${\widetilde{\varSigma }}(\chi ,\mu )$ can be computed reliably. Furthermore, one can show that in massless QCD this cumulative distribution is unity when $\chi = 180^{\circ }$. Hence, we can integrate the fixed-order differential distribution in Eq. (3) and use the unitarity constraint ${\widetilde{\varSigma }}(\pi ,\mu )/\sigma _{\mathrm{t}}= 1$ to all orders in $\alpha _{\mathrm{S}}$ to fix the constants of integration,

$$\begin{aligned} \bigg [\frac{1}{\sigma _{\mathrm{t}}} {\widetilde{\varSigma }}(\chi ,\mu )\bigg ]_{\mathrm {f.o.}}= & {} 1 + \frac{\alpha _{\mathrm{S}}(\mu )}{2\pi } \bar{{{\mathcal {A}}}}(\chi ,x_{R}) \nonumber \\&+ \left( \frac{\alpha _{\mathrm{S}}(\mu )}{2\pi }\right) ^2 \bar{{{\mathcal {B}}}}(\chi ,x_{R}) \nonumber \\&+ \left( \frac{\alpha _{\mathrm{S}}(\mu )}{2\pi }\right) ^3 \bar{{{\mathcal {C}}}}(\chi ,x_{R}) + \mathcal O(\alpha _{\mathrm{S}}^{4}). \end{aligned}$$

(12)

Moreover, starting from Eq. (4) and using the definition of ${\widetilde{\varSigma }}$, Eq. (11), we obtain the following expression for the resummed prediction⁵:

$$\begin{aligned} \bigg [\frac{1}{\sigma _{\mathrm{t}}} {\widetilde{\varSigma }}(\chi ,\mu )\bigg ]_{\mathrm {res.}}= & {} H(\alpha _{\mathrm{S}}(\mu )) \int \limits _0^\infty \bigg [ Q\sqrt{y}(1-y) J_1(b\, Q\sqrt{y}) \nonumber \\&+ \frac{2 y}{b} J_2(b\, Q\sqrt{y}) \bigg ] S(Q,b) {\mathrm{d}}b, \end{aligned}$$

(13)

where the Sudakov form factor S(Q, b) is the one given in Eq. (9).

The final expression for the matched prediction was derived in Ref.  [23] and reads

$$\begin{aligned} \ln \bigg [\frac{1}{\sigma _{\mathrm{t}}} {\widetilde{\varSigma }}(\chi ,\mu )\bigg ]= & {} \ln \bigg \{ \frac{1}{H(\alpha _{\mathrm{S}}(\mu ))} \bigg [ \frac{1}{\sigma _{\mathrm{t}}} {\widetilde{\varSigma }}(\chi ,\mu ) \bigg ]_{\mathrm {res.}} \bigg \} \nonumber \\&-\ln \bigg \{ \frac{1}{H(\alpha _{\mathrm{S}}(\mu ))} \bigg [ \frac{1}{\sigma _{\mathrm{t}}} {\widetilde{\varSigma }}(\chi ,\mu ) \bigg ]_{\mathrm {res.}} \bigg \}_{\mathrm {f.o.}} \nonumber \\&+ \frac{\alpha _{\mathrm{S}}(\mu )}{2\pi } \bar{{\mathcal {A}}}(\chi ,\mu )\, \nonumber \\&+\left( \frac{\alpha _{\mathrm{S}}(\mu )}{2\pi }\right) ^2 \bigg [\bar{{\mathcal {B}}}(\chi ,\mu ) - \frac{1}{2}\bar{{\mathcal {A}}}^2(\chi ,\mu )\bigg ]\nonumber \\&+\left( \frac{\alpha _{\mathrm{S}}(\mu )}{2\pi }\right) ^3 \left[ \bar{{\mathcal {C}}}(\chi ,\mu ) - \bar{{\mathcal {A}}}(\chi ,\mu )\bar{{\mathcal {B}}}(\chi ,\mu )\right. \nonumber \\&\left. +\frac{1}{3}\bar{{\mathcal {A}}}^3(\chi ,\mu ) \right] . \end{aligned}$$

(14)

Here $\bigg \{ \frac{1}{H(\alpha _{\mathrm{S}}(\mu ))} \bigg [ \frac{1}{\sigma _{\mathrm{t}}} {\widetilde{\varSigma }}(\chi ,\mu ) \bigg ]_{\mathrm {res.}} \bigg \}_{\mathrm {f.o.}}$ is the fixed-order expansion of the resummed result,

$$\begin{aligned} \frac{1}{H(\alpha _{\mathrm{S}}(\mu ))} \bigg [\frac{1}{\sigma _{\mathrm{t}}} {\widetilde{\varSigma }}(\chi ,\mu )\bigg ]_{\mathrm {res.}}= & {} 1 + \frac{\alpha _{\mathrm{S}}(\mu )}{2\pi } \bar{{\mathcal {A}}}_{\mathrm {res.}}(\chi ,\mu ) \nonumber \\&+ \left( \frac{\alpha _{\mathrm{S}}(\mu )}{2\pi }\right) ^2 \bar{\mathcal B}_{\mathrm {res.}}(\chi ,\mu ) \nonumber \\&+ \left( \frac{\alpha _{\mathrm{S}}(\mu )}{2\pi }\right) ^3 \bar{\mathcal C}_{\mathrm {res.}}(\chi ,\mu ) \nonumber \\&+ \mathcal O(\alpha _{\mathrm{S}}^{4}). \end{aligned}$$

(15)

The expansion coefficients $\bar{{\mathcal {A}}}_{\mathrm {res.}}$, $\bar{{\mathcal {B}}}_{\mathrm {res.}}$ and $\bar{\mathcal C}_{\mathrm {res.}}$ can be found in Ref.  [23].

Notice that the function $H(\alpha _{\mathrm{S}})$ does not appear in Eq. (14) at all. In the log-R matching scheme such non-logarithmically enhanced contributions should not be exponentiated, instead these terms, as well as subdominant logarithmic contributions, are all implicit in the unsubtracted parts of the fixed-order coefficients $\bar{{\mathcal {A}}}$, $\bar{{\mathcal {B}}}$ and $\bar{{\mathcal {C}}}$ [41]. Thus the log-R matched prediction can be computed without the explicit knowledge of $H^{(n)}$.

Finally, we comment on our implementation of the unitarity constraint ${\widetilde{\varSigma }}(\pi ,\mu )/\sigma _{\mathrm{t}}= 1$. It can be shown that this constraint can be satisfied by modifying the resummation formula in Eq. (4) such that in the kinematical limit $y=1$ the Sudakov form factor is unity. This may be achieved in several ways and here we choose a very simple solution and modify the resummation coefficients ${\tilde{A}}^{(n)}$ and ${\tilde{B}}^{(n)}$ according to

$$\begin{aligned} \begin{aligned} {\tilde{A}}^{(n)}(x_{L})&\rightarrow {\tilde{A}}^{(n)}(y,x_{L}) = {\tilde{A}}^{(n)}(x_{L})(1-y)^p, \\ {\tilde{B}}^{(n)}(x_{L})&\rightarrow {\tilde{B}}^{(n)}(y,x_{L}) = {\tilde{B}}^{(n)}(x_{L})(1-y)^p, \end{aligned} \end{aligned}$$

(16)

where p is a positive number.⁶ This modification is fully legitimate since it does not modify the logarithmic structure of the result and introduces only power-suppressed terms. In practice, we set $p=1$ and quantify the impact of this modification by comparing the results to those obtained with $p=2$.

2.3 Finite b-quark mass corrections

The theoretical prediction presented above was computed in massless QCD. However, the assumption of vanishing quark masses is not fully justified, especially at lower energies, where b-quark mass effects are relevant at the percent level [43]. In order to take b-quark mass corrections into account, we subtract the fraction of b-quark events, $r_b(Q)$ from the massless result and add back the corresponding massive contribution. Hence, we include mass effects directly at the level of matched distributions,

$$\begin{aligned} \frac{1}{\sigma _{\mathrm{t}}} \frac{{\mathrm{d}}\varSigma (\chi ,Q)}{{\mathrm{d}}\cos \chi }= & {} (1 - r_b(Q))\bigg [\frac{1}{\sigma _{\mathrm{t}}} \frac{{\mathrm{d}}\varSigma (\chi ,Q)}{{\mathrm{d}}\cos \chi }\bigg ]_{\mathrm {massless}} \nonumber \\&+ r_b(Q)\bigg [\frac{1}{\sigma _{\mathrm{t}}} \frac{{\mathrm{d}}\varSigma (\chi ,Q)}{{\mathrm{d}}\cos \chi }\bigg ]^{NNLO^{*}}_{\mathrm {massive}} . \end{aligned}$$

(17)

Here $\Big [\frac{1}{\sigma _{\mathrm{t}}} \frac{{\mathrm{d}}\varSigma (\chi ,Q)}{{\mathrm{d}}\cos \chi }\Big ]_{\mathrm {massless}}$ is the NNLO+NNLL matched distribution, computed in the log-R matching scheme in massless QCD as outlined above, while $\Big [\frac{1}{\sigma _{\mathrm{t}}} \frac{{\mathrm{d}}\varSigma (\chi ,Q)}{{\mathrm{d}}\cos \chi }\Big ]^{NNLO^{*}}_{\mathrm {massive}}$ is the fixed-order massive distribution. As the complete NNLO correction to this distribution is currently unknown, we model it by supplementing the massive NLO prediction of the parton level Monte Carlo generator Zbb4 [44], with the NNLO coefficient of the massless fixed-order result.

We define the fraction of b-quark events as the ratio of the total b-quark production cross section divided by the total hadronic cross section,

$$\begin{aligned} r_b(Q) \equiv \frac{\sigma ^{\mathrm{}}_{\mathrm {massive}}(e^+e^- \rightarrow b{\bar{b}})}{\sigma ^{\mathrm{}}_{\mathrm {massive}}(e^+e^- \rightarrow \mathrm {hadrons})}. \end{aligned}$$

We evaluate the ratio of these cross sections at NNLO accuracy ($\mathcal O(\alpha _{\mathrm{S}}^{2})$ in the strong coupling) including the exact b-quark mass corrections at $\mathcal O(\alpha _{\mathrm{S}}^{})$ and the leading mass terms up to $(m_b^2/Q^2)^2$ at $\mathcal O(\alpha _{\mathrm{S}}^{2})$ [45]. We note that the electroweak coupling factors do not cancel in this ratio and the summation over quark flavours has to be carried out explicitly when computing $\sigma ^{\mathrm{}}_{\mathrm {massive}}(e^+e^- \rightarrow \mathrm {hadrons})$.

Distributions for $\Big [\frac{1}{\sigma _{\mathrm{t}}} \frac{{\mathrm{d}}\varSigma (\chi ,Q)}{{\mathrm{d}}\cos \chi }\Big ]_{\mathrm {massive}}$ were generated for each of the considered energies using a pole b-quark mass of $m_b = 4.75{\,\text {Ge}\text {V}}$, which is consistent with world average estimations of pole mass $4.78\pm 0.06{\,\text {Ge}\text {V}}$ [46].

In order to assess the uncertainty associated to the modelling of b-quark mass corrections, we have investigated two alternative approaches for including them in our predictions. In approach A Eq. (17) is modified to

$$\begin{aligned} \frac{1}{\sigma _{\mathrm{t}}} \frac{{\mathrm{d}}\varSigma (\chi ,Q)}{{\mathrm{d}}\cos \chi }= & {} \bigg [\frac{1}{\sigma _{\mathrm{t}}} \frac{{\mathrm{d}}\varSigma (\chi ,Q)}{{\mathrm{d}}\cos \chi }\bigg ]_{\mathrm {massless}} \\&+ r_b(Q)\bigg [\frac{1}{\sigma _{\mathrm{t}}} \frac{{\mathrm{d}}\varSigma (\chi ,Q)}{{\mathrm{d}}\cos \chi }\bigg ]^{NLO}_{\mathrm {massive}} \\&-r_b(Q)\bigg [\frac{1}{\sigma _{\mathrm{t}}} \frac{{\mathrm{d}}\varSigma (\chi ,Q)}{{\mathrm{d}}\cos \chi }\bigg ]^{NLO}_{\mathrm {massless}} , \end{aligned}$$

i.e., we simply subtract the massless fixed-order NLO prediction multiplied by the fraction $r_b(Q)$ of b-quark events and add back the corresponding massive NLO distribution. Approach B is defined in a way very similar to our baseline, Eq. (17), but we do not include any NNLO corrections to the massive distribution,

$$\begin{aligned} \frac{1}{\sigma _{\mathrm{t}}} \frac{{\mathrm{d}}\varSigma (\chi ,Q)}{{\mathrm{d}}\cos \chi }= & {} (1 - r_b(Q))\bigg [\frac{1}{\sigma _{\mathrm{t}}} \frac{{\mathrm{d}}\varSigma (\chi ,Q)}{{\mathrm{d}}\cos \chi }\bigg ]_{\mathrm {massless}} \\&+ r_b(Q)\bigg [\frac{1}{\sigma _{\mathrm{t}}} \frac{{\mathrm{d}}\varSigma (\chi ,Q)}{{\mathrm{d}}\cos \chi }\bigg ]^{NLO}_{\mathrm {massive}} . \end{aligned}$$

Hence, $\Big [\frac{1}{\sigma _{\mathrm{t}}} \frac{{\mathrm{d}}\varSigma (\chi ,Q)}{{\mathrm{d}}\cos \chi }\Big ]^{NLO}_{\mathrm {massive}}$ is simply the prediction obtained with Zbb4.

3 Extraction procedure

To extract the strong coupling the predictions described above were confronted with the available data sets. Namely, the data obtained in SLD [47], L3 [48], DELPHI [49], OPAL [50, 51], TOPAZ [52], TASSO [53], JADE [54], MAC [55], MARKII [56], CELLO [57] and PLUTO [58] experiments were included. The information on used data is summarised in Tab. 1.

Table 1 Data used in the extraction procedure. The average of $\sqrt{s}$ is given in the brackets

The criteria to include the data were high precision of differential distributions obtained with charged and neutral final state particles in the full $\chi$ range, presence of corrections for detector effects, correction for initial state photon radiation and sufficient amount of supplementary information. Therefore, data sets without supplementary information [59], with large uncertainties [60], superseded datasets [61, 62] and measurements unfolded only to charged particles in the final state [63] are not included in the analysis.

The data sets selected for the extraction procedure have high precision and the measurements from different experiments performed at close energy points are consistent.⁷ This justifies their use in the extraction procedure in a wide centre-of-mass energy interval, similarly to studies of thrust [43] and C parameter [64] and allows us to target the highest precision of $\alpha _{\mathrm{S}}$ determination with available theoretical predictions.

3.1 Monte Carlo generation setup

In a previous study [23] the non-perturbative effects for the EEC distribution were modelled with an analytic approach. In this paper the non-perturbative effects in the $e^{+}e^{-}\rightarrow hadrons$ process are modelled using state-of-the-art particle-level Monte Carlo (MC) generators. The non-perturbative corrections of the energy–energy correlation distributions were extracted as ratios of energy–energy correlation distributions at hadron and parton level in the simulated samples.

In this study the MC generators SHERPA2.2.4 [65]⁸ and Herwig7.1.1, [66,67,68] were used.

The $e^{+}e^{-}\rightarrow hadrons$ MC samples were generated at centre-of-mass energies $\sqrt{s}=14.0$, 22.0, 29.0, 34.0, 43.5, 53.3, 59.5 and $91.2{\,\text {Ge}\text {V}}$. In all cases, the simulation of initial state radiation was disabled and generator settings were defaults if the opposite is not stated explicitly. The value of the strong coupling used for the hard process was set to $\alpha _{\mathrm{S}}(M_{Z})=0.1181$ [46].

The SHERPA2.2.4 samples were generated with the MENLOPS method using the matrix element generators AMEGIC [69], COMIX [70] and the GoSam [71] one-loop library to produce matrix elements for $e^{+}e^{-}\rightarrow Z/\gamma \rightarrow \mathrm{2,3,4,5\ partons}$ processes. The $2-$parton final state processes had NLO accuracy in perturbative QCD. The QCD matrix elements were calculated assuming massive b-quarks. The merging parameter $Y_{\mathrm {cut}}$ was set to $10^{-2.75} \simeq 1.778\times 10^{-3}$.

To test the fragmentation and hadronization model dependence, the events generated with SHERPA2.2.4 were hadronized using the Lund string fragmentation model [72] or the cluster fragmentation model [73]. The first setup is labelled below as $S^{L}$ and the second as $S^{C}$.

To assure proper fragmentation of heavy quarks and heavy hadron decays the cluster fragmentation model was adjusted. The value of SPLIT_LEADEXPONENT parameter was set to 1.0, the parameter M_DIQUARK_OFFSET was set to 0.55, the production of charm and beauty baryons was enhanced by factors 0.8 and 1.7.

Fig. 1

Data and Monte Carlo predictions obtained with the $S^{L}$ and $S^{C}$ setups at parton and hadron level. Reweighting was applied

For the cross-check of SHERPA2.2.4 samples, the Herwig7.1.1 generator was used. The Herwig7.1.1 samples were generated with improved unitarized merging [74] using the MadGraph5 [75] matrix element generator and the GoSam [71] one-loop library to produce matrix elements of the $e^{+}e^{-}\rightarrow Z/\gamma \rightarrow \mathrm{2,3,4,5\ partons}$ processes. The $2-$parton final state processes again had NLO accuracy in perturbative QCD and the matrix elements were calculated assuming massive b-quarks. The merging parameter was set to $\sqrt{s}\times 10^{-1.25} \simeq \sqrt{s}\times 5.623\times 10^{-2}$. For the modelling of the hadronization process the default implementation of the cluster fragmentation model [76] was used. To improve the modelling of beauty production at the lowest energies, the b-quark nominal mass was changed from the default value of $5.3{\,\text {Ge}\text {V}}$ to $5.1{\,\text {Ge}\text {V}}$. This setup is labelled below as $H^{M}$.

Fig. 2

Data and Monte Carlo predictions obtained with the $H^{M}$ setup at parton and hadron level. Reweighting was applied

3.2 Estimation of hadronization effects from MC models

Estimation of hadronization corrections is an integral part of comparing the parton-level QCD predictions to the data measured on hadron (particle) level. Despite the fact that under certain conditions the local parton-hadron duality leads to close values of quantities on parton and hadron level, the difference between them is not negligible and should be taken into account in precise analyses. One way to do so is to apply correction factors estimated from MC simulations to the perturbative QCD prediction. The factors, called hadronization corrections H / P, are defined as ratios of the corresponding quantities at parton level to the same quantities at hadron level at every point of the considered distribution.

To obtain the EEC distributions, the generated MC samples were processed in the same way as data (see e.g. Ref. [50]), using partons before hadronization for parton-level calculations and undecayed/stable particles for hadron-level calculations. For the parton-level calculations the parton energies were used as provided by the MC generators.

The predictions obtained with all setups describe the data well for all ranges of $\chi$ with the exception of the regions near $\chi =0^{\circ }$ and $\chi =90^{\circ }$, for all values of $\sqrt{s}$. For $\sqrt{s}<29{\,\text {Ge}\text {V}}$ the $H^{M}$ setup is sensitive to the b-quark mass and the corresponding predictions are not reliable.

However, to assure an even better description of data, a reweighting procedure was applied to the simulated samples. The samples were reweighted at hadron level on an event-by-event basis to describe the data and the corresponding event weights were propagated to the parton level. The resulting distributions are shown in Fig. 1 for the SHERPA2.2.4 setups and in Fig. 2 for the Herwig7.1.1 setup.

As the SHERPA2.2.4 setups give the most stable and physically reliable predictions these are used in the analysis for reference hadronization corrections ($S^{L}$) and for systematic studies ($S^{C}$). The corresponding hadronization corrections together with parametrizations are shown in Fig. 3 for reweighted samples.

Fig. 3

Hadronization corrections obtained with different setups of Monte Carlo event simulations for every energy point and corresponding parametrizations. Event-by-event reweighting was applied. The used fit range is indicated with a thick line

3.3 Estimation of statistical correlations between measurements from MC models

To perform an accurate extraction procedure, the available data and uncertainties were examined and for every measured set of data a covariance matrix was built. The procedure consisted of multiple steps.

In the first step the systematic uncertainties were recalculated and separated from statistical uncertainties when this was possible. For the measurements with the uncertainties rounded to one significant digit [50, 53, 54] the uncertainties were expanded assuming maximal uncertainty before rounding. The measurements of TASSO [53] were converted from the $d\varSigma /d\cos \chi$ form to $d\varSigma /d\chi$ using values of $\cos \chi$ on the bin edges.

For the data from TOPAZ [52] the systematic uncertainty was calculated from an estimated relative systematic uncertainty of $\pm 4\%$ [52].

Taking into account the uncertainties of $\alpha _{\mathrm{S}}$ extraction analysis [58] the same was done for PLUTO [58] data. The systematic uncertainty estimation of $\pm 5\%$ for TASSO [53] is based on the upper limit of $10\%$ for the total uncertainty mentioned in the paper [53]. The systematic uncertainties from DELPHI [49] and SLD [47] were used as provided in the original papers.

For all remaining data sets the published combined uncertainty was treated as statistical.

The measurements of $\varSigma$ are provided in the original publications without correlations between the individual points. The correlation matrix was estimated from the Monte Carlo samples in terms of Fisher correlation coefficients [77, 78]. Some of the obtained correlation matrices are shown in Fig. 4.

Fig. 4

A graphical representation of statistical correlation matrices for $\varSigma (\chi )$ obtained from Monte Carlo simulated events with the $S^{L}$ setup for OPAL [51] data at $\sqrt{s}=91.2{\,\text {Ge}\text {V}}$ (left), TOPAZ [52] data at $\sqrt{s}=53.3{\,\text {Ge}\text {V}}$ (centre), and JADE [54] data at $\sqrt{s}=22{\,\text {Ge}\text {V}}$ (right). The bottom left corner for each figure corresponds to $\chi =0^{\circ }$ and the bottom right to $\chi =180^{\circ }$

The obtained correlation coefficients are sizeable, up to 0.5 for the closest points, which highlights the importance of properly taking into account the correlations between measured points in the fits. The obtained correlation matrix together with statistical uncertainties was used to build a statistical covariance matrix for every data set.

To construct the systematic covariance matrix, the systematic uncertainties from the original publications were used with an assumption that these are positively correlated with correlation coefficient $\rho =0.5$ between closest points. The correlations between the uncertainties of data from different experiments or different beam energies were neglected. The final covariance matrix used in the fit for every data set was a sum of statistical and systematic covariance matrices.

3.4 Fit procedure and estimation of uncertainties

The strong coupling extraction procedure is based on the comparison of data to the perturbative QCD prediction combined with non-perturbative (hadronization) corrections. The perturbative part of the predictions was calculated in every bin as described in previous sections. To tame the statistical fluctuations present in the obtained binned hadronization correction distributions, these were parametrized with analytic functions, expressed as a sum of polynomials of $\chi -90^{\circ }$. The value of the fitted function at the bin centre was used as the correction factor.

To find the optimal value of $\alpha _{\mathrm{S}}$, the MINUIT2 [79, 80] program was used to minimize

$$\begin{aligned} \chi ^2(\alpha _{\mathrm{S}})=\sum _\mathrm{data\ sets}\chi ^2(\alpha _{\mathrm{S}})_\mathrm{data\ set}, \end{aligned}$$

where the $\chi ^2(\alpha _{\mathrm{S}})$ value was calculated for each data set as

$$\begin{aligned} \chi ^2(\alpha _{\mathrm{S}})=(\varvec{D}-\varvec{P}(\alpha _{\mathrm{S}}))V^{-1}(\varvec{D}-\varvec{P}(\alpha _{\mathrm{S}}))^{T}, \end{aligned}$$

with $\varvec{D}$ standing for the vector of data points, $\varvec{P}(\alpha _{\mathrm{S}})$ for the vector of calculated predictions and V for the covariance matrix for $\varvec{D}$. The default scale used in the fit procedure was $\mu =Q=\sqrt{s}$.

The fit ranges were chosen to avoid regions where resummed predictions or hadronization correction calculations are not reliable. The selected fit ranges were $117^{\circ }{-}165^{\circ }$, $60^{\circ }{-}165^{\circ }$ and $60^{\circ }{-}160^{\circ }$. The uncertainty on the fit result was estimated with the $\chi ^2+1$ criterion as implemented in the MINUIT2 program. The results of the fits are given in Tab. 2 for each fit range. In order to assess the impact of the NNLO corrections, in Tab. 2 we also present the results obtained using NLO+NNLL predictions. The NNLO corrections affect the fit in a moderate but non-negligible way. The obtained values of $\chi ^2$ divided by the number of degrees of freedom in the fit are of order unity for all cases. The corresponding distributions obtained from the fits for different $\sqrt{s}$ points are shown in Figs. 5, 6, 7 and 8.

Table 2 Results of the fits of the matched predictions at NLO+NNLL and NNLO+NNLL accuracy to experimental data. The given uncertainty is fit uncertainty scaled by $\sqrt{\chi ^{2}/ndof}$

Fig. 5

Fits of theory predictions to the data at $\sqrt{s}=59.5-91.2{\,\text {Ge}\text {V}}$. The used fit range is shown with thick line. For the ratio plot only the uncertainties of the data are taken into account

Fig. 6

Fits of theory predictions to the data for $\sqrt{s}=34-53.3{\,\text {Ge}\text {V}}$. The used fit range is shown with thick line. For the ratio plot only the uncertainties of the data are taken into account

Fig. 7

Fits of theory predictions to the data for $\sqrt{s}=22-29{\,\text {Ge}\text {V}}$. The used fit range is shown with thick line. For the ratio plot only the uncertainties of the data are taken into account

Fig. 8

Fits of theory predictions to the data for $\sqrt{s}=14{\,\text {Ge}\text {V}}$. The used fit range is shown with thick line. For the ratio plot only the uncertainties of the data are taken into account

Fig. 9

Dependence of fit results on the renormalization scale (upper left), resummation scale (upper right), non-perturbative simulation model (bottom left) choice and b mass corrections (bottom right). The fit range for $S^{L}$, $S^{C}$ and $H^{M}$ setups is $60^{\circ }{-}160^{\circ }$. The fit range for the $An.^{DMW}$ setup is $117^{\circ }{-}165^{\circ }$

Fig. 10

Dependence of fit results on the used data sets. The fit range for $S^{L}$, $S^{C}$ and $H^{M}$ setups is $60^{\circ }{-}160^{\circ }$

The systematic uncertainties of the obtained results were estimated with procedures used in previous studies [81]. To estimate the bias of the obtained result caused by the absence of higher-order terms in the perturbative predictions, the renormalization scale variation procedure was performed. In this procedure the fits were repeated, with variation of the renormalization scale in the range between $x_{R}=1/2$ and $x_{R}=2$.

The bias of hadronization model selection is studied using the $S^{L}$ and $S^{C}$ setups of hadronization corrections, see results in Fig. 9. The bias related to the ambiguity of resummation scale choice was estimated by varying $x_{L}$ in the range between $x_{L}=1/2$ and $x_{L}=2$. To estimate the bias related to the ambiguity of our prescription implementing the unitarity constraint in the resummed calculation (see Eq. (16)), two values of p were used: $p=1$ and $p=2$. The difference between results obtained with two options is negligible. In all cases above the sizes of the biases were estimated numerically as half of the difference between the maximal and minimal $\alpha _{\mathrm{S}}$ value obtained in the corresponding set of fits. To estimate the potential bias of the result caused by imperfections of specific hadronization model and parton shower model, the fits were repeated with hadronization corrections obtained with all setups described in previous subsections. The numerical value of the bias was obtained as half of the difference between the $\alpha _{\mathrm{S}}$ values obtained using non-perturbative corrections from Lund and cluster hadronization models implemented in SHERPA2.2.4. From Fig. 9 it is seen that the estimated biases are relatively independent and, therefore combined in the final result as such.

Besides the estimations, several cross-checks of the obtained results were performed. First, the datasets were grouped according to their energies and fitted separately for each energy. The results are shown in Fig. 10. There is no visible trend for the fitted value of $\alpha _{\mathrm{S}}$ with energy in the $S^{L}$ and $S^{C}$ setups. For the $H^{M}$ setup, the results of the fits are not reliable below $\sqrt{s} < 29{\,\text {Ge}\text {V}}$ due to the sensitivity of this setup to the b-quark mass. In addition to the MC hadronization models the fits were also performed with the analytic hadronization model of Dokshitzer, Marchesini and Webber (DMW) [82]. In this setup, non-perturbative effects in EEC were accounted for by multiplying the Sudakov form factor by a correction of the form

$$\begin{aligned} S_{\mathrm {NP}} = e^{-\frac{1}{2} a_1 b^2} (1 - 2 a_2 b). \end{aligned}$$

Here $a_1$ and $a_2$ are non-perturbative parameters that can be related in the dispersive approach to certain moments ${\bar{\alpha }}_{q,p}$ of the strong coupling $\alpha _{\mathrm{S}}$ [82]. These moments are the fit parameters of the analytic model. The results obtained from the fits with this setup are listed in Tab. 2. They show a high degree of dependence on the selected fit range, but are close to results obtained with the Monte Carlo based hadronization corrections in the range $117^{\circ }{-}165^{\circ }$, see Tab. 2. Hence we conclude that away from the back-to-back region, the analytic model cannot fully account for hadronization effects.

4 Results and discussions

In this paper we presented the first combined analysis and extraction of $\alpha _{\mathrm{S}}$ at NNLO+NNLL precision from energy–energy correlation in electron-positron annihilation. Moreover, our analysis is the first extraction of the strong coupling based on Monte Carlo hadronization corrections obtained from NLO Monte Carlo setups at NNLO+NNLL precision. For the central value of the final result we quote the results obtained from the fits with the $S^{L}$ hadronization model in the range $60^{\circ }{-}160^{\circ }$ with uncertainties and estimations of biases obtained as described above.

At NNLO+NNLL accuracy we obtain the best fit value of

$$\begin{aligned} \alpha _{\mathrm{S}}(M_{Z})= & {} 0.11750\pm 0.00018 {( exp.)}\pm 0.00102{(hadr.)}\\&\pm 0.00257{(ren.)}\pm 0.00078{(res.)}. \end{aligned}$$

In order to appreciate the impact of NNLO corrections, we also quote the result of the fit at NLO+NNLL accuracy

$$\begin{aligned} \alpha _{\mathrm{S}}(M_{Z})= & {} 0.12200\pm 0.00023 {( exp.)}\pm 0.00113{(hadr.)}\\&\pm 0.00433{(ren.)}\pm 0.00293{(res.)}. \end{aligned}$$

We see that the inclusion of the NNLO corrections has a moderate but non-negligible effect on the extracted value of $\alpha _{\mathrm{S}}$.

It has been explicitly checked that there are no correlations between estimated biases, therefore, the combined values with combined estimations of bias at NNLO+NNLL accuracy are:

$$\begin{aligned} \alpha _{\mathrm{S}}(M_{Z})= 0.11750\pm 0.00287{\text{( }comb.)}\end{aligned}$$

while in comparison, for NLO+NNLL accuracy we obtain:

$$\begin{aligned} \alpha _{\mathrm{S}}(M_{Z})= 0.12200\pm 0.00535{\text{( }comb.)}. \end{aligned}$$

The value obtained from the analysis in NNLO+NNLL approximation is in agreement with the world average as of 2017 [83], however it is visibly lower than the results from measurements performed for other $e^{+}e^{-}$ observables using NNLO perturbative QCD predictions and MC hadronization models [83]. The estimated uncertainties are dominated by the uncertainty on the theoretical predictions. The results obtained in this study can be compared to those described in the original publications with NLO+NLL precision as well as the results obtained with analytic hadronization model in the sister paper [23].