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CMS-SMP-22-005 ; CERN-EP-2024-066
Measurement of multijet azimuthal correlations and determination of the strong coupling in proton-proton collisions at $ \sqrt{s}= $ 13 TeV
Accepted for publication in EPJC
Abstract: A measurement is presented of a ratio observable that provides a measure of the azimuthal correlations among jets with large transverse momentum $ p_{\mathrm{T}} $. This observable is measured in multijet events over the range of $ p_{\mathrm{T}} = $ 360-3170 GeV based on data collected by the CMS experiment in proton-proton collisions at a centre-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 134 fb$ ^{-1} $. The results are compared with predictions from Monte Carlo parton-shower event generator simulations, as well as with fixed-order perturbative quantum chromodynamics (pQCD) predictions at next-to-leading-order (NLO) accuracy obtained with different parton distribution functions (PDFs) and corrected for nonperturbative and electroweak effects. Data and theory agree within uncertainties. From the comparison of the measured observable with the pQCD prediction obtained with the NNPDF3.1 NLO PDFs, the strong coupling at the Z boson mass scale is $ \alpha_\mathrm{S}(m_{\mathrm{Z}})= $ 0.1177 $ \pm $ 0.0013 (exp) $ _{-0.0073}^{+0.0116} $ (theo) $ = $ 0.1177 $_{-0.0074}^{+0.0117} $, where the total uncertainty is dominated by the scale dependence of the fixed-order predictions. A test of the running of $ \alpha_\mathrm{S} $ in the TeV region shows no deviation from the expected NLO pQCD behaviour.
Figures & Tables Summary References CMS Publications
Figures

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Figure 1:
Example of the number of entries contributing to the numerator and denominator of the $ R_{\Delta\phi}(p_{\mathrm{T}}) $ ratio, Eq. (1), for 2-jet (left) and 3-jet (right) events, with all jets having $ p_{\mathrm{T}} > p_{\text{Tmin}}^{\text{nbr}}= $ 100 GeV. The 2-jet topology does not contribute (null numerator) to the $ R_{\Delta\phi}(p_{\mathrm{T}}) $ ratio when the azimuthal distance for neighbouring jets is fixed to 2 $ \pi/3 < \Delta\phi < 7\pi/ $ 8. In the 3-jet topology, each jet is considered as a reference, and its azimuthal separations ($ \Delta\phi{,} $ 1 and $ \Delta\phi{,} $ 2) to other neighbouring jets (with $ p_{\mathrm {T,1}}^{\text {nbr}} $ and $ p_{\mathrm{T,2}}^{\text {nbr}} $) are computed. Each neighbouring jet with $ \Delta\phi $ within the specified interval increments the entries of the numerator, whereas the denominator simply counts the number of jets in the event.

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Figure 2:
Probability matrix for the $ N(p_{\mathrm{T}},n) $ distribution built using PYTHIA8 simulated events. The horizontal axis corresponds to the generator-level jet $ p_{\mathrm{T}} $, and the vertical axis to the reconstructed-level jet $ p_{\mathrm{T}} $. The 4 $ \times $ 4 structure of the matrix corresponds to the bins of neighbouring jets $ n $ (labelled in the uppermost row and rightmost column), and indicates migrations among those bins. The horizontal and vertical axes of each cell correspond to the $ p_{\mathrm{T}} $ of the jets, and each cell indicates the migrations among the jet $ p_{\mathrm{T}} $ bins. The range of colours covers from 10$^{-6} $ to 1, and indicates the probability of migrations from a given (generator) particle-level bin to the corresponding (reconstructed) detector-level bin.

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Figure 3:
Bin-to-bin correlation matrix for the $ R_{\Delta\phi}(p_{\mathrm{T}}) $ distribution at the particle level, where the value 1 ($-$1) corresponds to fully (anti)correlated bins. For illustration purposes, only bins with (anti)correlations larger (smaller) than 0.05 ($-$0.05) are shown also as text.

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Figure 4:
The $ R_{\Delta\phi}(p_{\mathrm{T}}) $ observable as a function of $ p_{\mathrm{T}} $, compared with MC generator predictions at LO (left) and at NLO (right) accuracy. The LO predictions are obtained with PYTHIA8 tunes CUETP8M1 and CUETP8M2, and HERWIG++ tune UE-EE-5-CTEQ6L1 MC event generators. The NLO predictions are obtained with POWHEG interfaced with each of the aforementioned MC event generators. The experimental data are represented with black dots and the MC predictions with coloured lines. The lower panel of each plot shows the ratio between MC predictions and experimental data. The total experimental uncertainties are indicated by the vertical error bars (upper panels) and coloured band (lower panels) correspondingly.

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Figure 5:
Theoretical predictions for the cross sections corresponding to the numerator (left) and denominator (right) of the $ R_{\Delta\phi}(p_{\mathrm{T}}) $ ratio, Eq. (1), obtained using the NNPDF3.1 NLO PDF set. The coloured bands represent the LO and NLO scale uncertainties derived with a six-point variation of $ \mu_\mathrm{R} $ and $ \mu_\mathrm{F} $ from the central reference value. The lower panels show the ratios to the respective LO predictions.

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Figure 6:
Nonperturbative correction factors for the numerator (upper left) and denominator (upper right) of the $ R_{\Delta\phi}(p_{\mathrm{T}}) $ ratio, Eq. (1), using PYTHIA8 with tunes CUETP8M1 and CUETP8M2, HERWIG++ with tune UE-EE-5-CTEQ6L1, and POWHEG interfaced with each of them. The lower plot shows the NP correction factors (blue line) for $ R_{\Delta\phi}(p_{\mathrm{T}}) $ and their uncertainties.

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Figure 7:
Electroweak corrections for the numerator (blue) and denominator (green) of Eq. (1), and for the $ R_{\Delta\phi}(p_{\mathrm{T}}) $ ratio itself (red). The solid lines correspond to the additive combination of NLO EW corrections to the QCD process (NLO QCD$ \,+\, $EW), and the markers represent the multiplicative combination (NLO QCD$ \,\times\, $EW).

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Figure 8:
The $ R_{\Delta\phi}(p_{\mathrm{T}}) $ observable as a function of $ p_{\mathrm{T}} $, compared with fixed-order theoretical calculations at NLO accuracy using the ABMP16 (upper left), CT18 (upper right), MSHT20 (lower left), and NNPDF3.1 (lower right) NLO PDF sets. The experimental data are indicated with blue dots (with error bars representing the total experimental uncertainty), the theoretical prediction for the default $ \alpha_\mathrm{S}(m_{\mathrm{Z}}) $ for each PDF set with black solid lines, the scale uncertainties with red bands, and the PDF uncertainties with green bands. The lower panel of each plot shows the ratio between experimental data and theoretical predictions.

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Figure 9:
Sensitivity of the $ R_{\Delta\phi}(p_{\mathrm{T}}) $ ratio to the strong coupling constant $ \alpha_\mathrm{S}(m_{\mathrm{Z}}) $. The data are indicated with blue dots with error bars representing the total experimental uncertainty. In each plot, the lines represent fixed-order NLO theoretical calculations obtained with ABMP16 (upper left), CT18 (upper right), MSHT20 (lower left) and NNPDF3.1 (lower right) NLO PDF sets. Solid green (red) lines indicate maximum (minimum) values, and dotted black lines intermediate values of $ \alpha_\mathrm{S}(m_{\mathrm{Z}}) $ for each PDF set.

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Figure 10:
Minimization of the $ \chi^2 $ between experimental measurements and theoretical predictions for the $ R_{\Delta\phi}(p_{\mathrm{T}}) $ ratio, with respect to $ \alpha_\mathrm{S}(m_{\mathrm{Z}}) $ for the ABMP16, CT18, MSHT20, and NNPDF3.1 NLO PDF sets. In this plot, only experimental uncertainties are included in the covariance matrix. The minimum value of $ \alpha_\mathrm{S}(m_{\mathrm{Z}}) $ found for each PDF set is indicated with a dashed line and corresponds to the central result. The experimental uncertainty is estimated from the $ \alpha_\mathrm{S}(m_{\mathrm{Z}}) $ values for which the $ \chi^2 $ is increased by one unit with respect to the minimum value.

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Figure 11:
Determination of $ \alpha_\mathrm{S}(m_{\mathrm{Z}}) $ from the $ R_{\Delta\phi}(p_{\mathrm{T}}) $ ratio with the NNPDF3.1 PDF set (red), in comparison with previous NLO determinations of $ \alpha_\mathrm{S}(m_{\mathrm{Z}}) $ from inclusive jet (magenta), dijet (green), and multijet (blue) measurements. The horizontal error bars indicate the total uncertainty (experimental and theoretical). The world-average $ \alpha_\mathrm{S}(m_{\mathrm{Z}}) $ value is represented by the vertical dashed black line and its uncertainty by the yellow band.

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Figure 12:
Running of the strong coupling $ \alpha_\mathrm{S}(Q) $ (dashed line) evolved using the current world-average value $ \alpha_\mathrm{S}(m_{\mathrm{Z}}) = $ 0.1180 $ \pm $ 0.0009 [5] together with its associated total uncertainty (yellow band). The four new extractions from the present analysis (Table 5) are shown as filled red circles, compared with results from the H1[93,94,90], ZEUS [95], D0 [11,12], CMS [14],17,18,22], and ATLAS [24,21] experiments. The vertical error bars indicate the total uncertainty (experimental and theoretical). All the experimental results shown in this figure are based on fixed-order predictions at NLO accuracy in pQCD.
Tables

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Table 1:
The different HLT $ p_{\mathrm{T}} $ thresholds used in the measurement and the corresponding integrated luminosities for each data-taking year.

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Table 2:
Values of the $ R_{\Delta\phi}(p_{\mathrm{T}}) $ observable in different $ p_{\mathrm{T}} $ intervals, and associated experimental uncertainties.

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Table 3:
Default and range of $ \alpha_\mathrm{S}(m_{\mathrm{Z}}) $ values used in the different NLO PDF sets.

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Table 4:
Results for $ \alpha_\mathrm{S}(m_{\mathrm{Z}}) $, associated uncertainties, and goodness-of-fit per degree of freedom ($ \chi^2/n_\text{dof} $), obtained from the measured $ R_{\Delta\phi}(p_{\mathrm{T}}) $ distribution compared with theoretical predictions using different NLO PDF sets.

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Table 5:
Values of $ \alpha_\mathrm{S}(m_{\mathrm{Z}}) $ and $ \alpha_\mathrm{S}(Q) $ determined in four different jet $ p_{\mathrm{T}} $ fitting subregions corresponding to an average scale $ \langle Q \rangle $ over each $ p_{\mathrm{T}} $ interval.
Summary
A measurement of the $ R_{\Delta\phi}(p_{\mathrm{T}}) $ ratio, sensitive to azimuthal correlations in multijet events, has been presented using proton-proton collision data collected by the CMS experiment at a centre-of-mass energy of 13 TeV and corresponding to an integrated luminosity of 134 fb$ ^{-1} $. The experimental data are compared with predictions from Monte Carlo (MC) event generators, PYTHIA8 with tunes CUETP8M1 and CUETP8M2, HERWIG++ with tune UE-EE-5-CTEQ6L1, and POWHEG interfaced with each one of them. Deviations between data and MC predictions are observed in all cases, except for PYTHIA8 tune CUETP8M2, which gives a good overall description of the measurement. The measurement is also compared with fixed-order perturbative quantum chromodynamics (pQCD) predictions at next-to-leading-order (NLO) accuracy using the NLOJET++ package within the FASTNLO framework. Those predictions are extracted for four different NLO parton distribution function (PDF) sets, ABMP16, CT18, MSHT20, and NNPDF3.1. Corrections for nonperturbative (NP) effects are evaluated using all the aforementioned MC event generators, and are applied to the fixed-order predictions. The predictions are additionally corrected for electroweak (EW) effects that become important at large jet transverse momenta. Generally, the fixed-order predictions are in agreement with the experimental data in the phase space of this analysis, and they provide a good description of the measured $ R_{\Delta\phi}(p_{\mathrm{T}}) $ distribution for all PDF sets. {\tolerance=2400 Based on a comparison of the measured $ R_{\Delta\phi}(p_{\mathrm{T}}) $ distribution and the theoretical predictions, the strong coupling at the scale of the Z boson mass is: $ \alpha_\mathrm{S}(m_{\mathrm{Z}})= $ 0.1177 $_{-0.0068}^{+0.0114} $ (scale) $ \pm $ 0.0013 (exp) $ \pm $ 0.0011 (NP) $ \pm $ 0.0010 (PDF) $ \pm $ 0.0003 (EW) $ \pm $ 0.0020 (PDF choice) $=$ 0.1177 $_{-0.0074}^{+0.0117} $, using calculations based on the NNPDF3.1 NLO PDF set. Alternative $ \alpha_\mathrm{S}(m_{\mathrm{Z}}) $ results obtained with other PDF sets are compatible among each other, as well as with the central result of this work, and with the current world average, $ \alpha_\mathrm{S}(m_{\mathrm{Z}}) = $ 0.1180 $ \pm $ 0.0009. The spread of the $ \alpha_\mathrm{S}(m_{\mathrm{Z}}) $ values obtained from different PDF sets is used for the assignment of the ``PDF choice'' uncertainty quoted in the final strong coupling constant derived here. The dominant uncertainty in this measurement originates from the scale dependence of the NLO pQCD predictions, and is expected to be significantly reduced with the future inclusion of fixed-order predictions at next-to-NLO accuracy. The evolution of the strong coupling as a function of the energy scale, $ \alpha_\mathrm{S}(Q) $, has been tested up to $ Q\approx $ 2 TeV, a higher scale than that probed in previous H1, ZEUS, D0, CMS, and ATLAS measurements. This test has been performed by choosing as energy scale $ Q $ the average jet transverse momentum in the different intervals considered, and no deviation from the expected NLO pQCD running of the strong coupling is observed.
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