CMSSMP22015 ; CERNEP2024010  
Measurement of energy correlators inside jets and determination of the strong coupling $ \alpha_\mathrm{S} (m_\mathrm{Z}) $  
CMS Collaboration  
21 February 2024  
Accepted for publication in Phys. Rev. Lett.  
Abstract: Energy correlators that describe energyweighted distances between two or three particles in a jet are measured using an event sample of $ \sqrt{s} = $ 13 TeV protonproton collisions collected by the CMS experiment and corresponding to an integrated luminosity of 36.3 fb$ ^{1} $. The measured distributions reveal two key features of the strong interaction: confinement and asymptotic freedom. By comparing the ratio of the two measured distributions with theoretical calculations that resum collinear emissions at approximate nexttonexttoleading logarithmic accuracy matched to a nexttoleading order calculation, the strong coupling is determined at the Z boson mass: $ \alpha_\mathrm{S} (m_\mathrm{Z}) = $ 0.1229 $ ^{+0.0040}_{0.0050} $, the most precise $ \alpha_\mathrm{S} (m_\mathrm{Z}) $ value obtained using jet substructure observables.  
Links: eprint arXiv:2402.13864 [hepex] (PDF) ; CDS record ; inSPIRE record ; HepData record ; Physics Briefing ; CADI line (restricted) ; 
Figures  
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Figure 1:
Measured (unfolded) E2C distributions, compared with three MC predictions in the jet $ p_{\mathrm{T}} $ regions mentioned in the legends. The lower panels show the ratios to the PYTHIA8 CP5 ($ p_{\mathrm{T}} $ordered showers) reference. The data statistical and systematic uncertainties are shown by vertical bars and hatched boxes, respectively; the PYTHIA8 uncertainty is shown by the blue band. The three $ x_\text{L} $ regions are described in the text. 
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Figure 2:
Measured E3C/E2C ratio (left) and their ratio to predictions (right) in the perturbative $ x_\text{L} $ region, in the jet $ p_{\mathrm{T}} $ regions mentioned in the legends. The $ \text{NLO}+\text{NNLL}_\text{approx} $ theoretical predictions [19] are corrected to hadronlevel and normalized to the measured data. The statistical and experimental systematic uncertainties are shown with error bars and boxes, respectively. 
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Figure 3:
Fitted slopes of the measured E3C/E2C ratios, in the eight jet $ p_{\mathrm{T}} $ regions, compared to the corresponding theoretical predictions for three different $ \alpha_\mathrm{S} $ values. 
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Figure A1:
Measured (unfolded) E2C distributions compared with the PYTHIA8 CP5 prediction (upper) and all the MC predictions (lower) in the eight $ p_{\mathrm{T}} $ regions. Statistical and experimental systematic uncertainties are shown with error bars and hatched bands, respectively. The theory uncertainty in the PYTHIA8 CP5 prediction is shown with the blue bands. 
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Figure A1a:
Measured (unfolded) E2C distributions compared with the PYTHIA8 CP5 prediction (upper) and all the MC predictions (lower) in the eight $ p_{\mathrm{T}} $ regions. Statistical and experimental systematic uncertainties are shown with error bars and hatched bands, respectively. The theory uncertainty in the PYTHIA8 CP5 prediction is shown with the blue bands. 
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Figure A1b:
Measured (unfolded) E2C distributions compared with the PYTHIA8 CP5 prediction (upper) and all the MC predictions (lower) in the eight $ p_{\mathrm{T}} $ regions. Statistical and experimental systematic uncertainties are shown with error bars and hatched bands, respectively. The theory uncertainty in the PYTHIA8 CP5 prediction is shown with the blue bands. 
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Figure A2:
Measured (unfolded) E3C distributions compared with the PYTHIA8 CP5 prediction (upper) and all the MC predictions (lower) in the eight $ p_{\mathrm{T}} $ regions. Statistical and experimental systematic uncertainties are shown with error bars and hatched bands, respectively. The theory uncertainty in the PYTHIA8 CP5 prediction is shown with the blue bands. 
png pdf 
Figure A2a:
Measured (unfolded) E3C distributions compared with the PYTHIA8 CP5 prediction (upper) and all the MC predictions (lower) in the eight $ p_{\mathrm{T}} $ regions. Statistical and experimental systematic uncertainties are shown with error bars and hatched bands, respectively. The theory uncertainty in the PYTHIA8 CP5 prediction is shown with the blue bands. 
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Figure A2b:
Measured (unfolded) E3C distributions compared with the PYTHIA8 CP5 prediction (upper) and all the MC predictions (lower) in the eight $ p_{\mathrm{T}} $ regions. Statistical and experimental systematic uncertainties are shown with error bars and hatched bands, respectively. The theory uncertainty in the PYTHIA8 CP5 prediction is shown with the blue bands. 
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Figure A3:
Ratio of the unfolded E3C and E2C distributions compared with the PYTHIA8 CP5 prediction (upper) and all the MC predictions (lower) in the eight $ p_{\mathrm{T}} $ regions. Statistical and experimental systematic uncertainties are shown with error bars and hatched bands, respectively. The theory uncertainty in the PYTHIA8 CP5 prediction is shown with the blue bands. 
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Figure A3a:
Ratio of the unfolded E3C and E2C distributions compared with the PYTHIA8 CP5 prediction (upper) and all the MC predictions (lower) in the eight $ p_{\mathrm{T}} $ regions. Statistical and experimental systematic uncertainties are shown with error bars and hatched bands, respectively. The theory uncertainty in the PYTHIA8 CP5 prediction is shown with the blue bands. 
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Figure A3b:
Ratio of the unfolded E3C and E2C distributions compared with the PYTHIA8 CP5 prediction (upper) and all the MC predictions (lower) in the eight $ p_{\mathrm{T}} $ regions. Statistical and experimental systematic uncertainties are shown with error bars and hatched bands, respectively. The theory uncertainty in the PYTHIA8 CP5 prediction is shown with the blue bands. 
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Figure A4:
Ratio of the unfolded E3C and E2C distributions in the eight $ p_{\mathrm{T}} $ regions. The $ \text{NLO}+\text{NNLL}_\text{approx} $ theoretical predictions [19] and the corresponding uncertainties are corrected to hadronlevel and normalized to the measured data. The statistical and experimental systematic uncertainties are shown with error bars and boxes, respectively. 
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Figure A5:
The $ \chi^2 $ values determined from the fit of the measured data, as a function of $ \alpha_\mathrm{S} (m_\mathrm{Z}) $. 
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Figure A6:
The statistical correlation matrix of the bins of the unfolded E2C distribution. The bin number $ n $ in this figure is defined by $ n=22 i_{p_{\mathrm{T}}} + i_{x_\text{L}} $, where $ i_{p_{\mathrm{T}}} $ and $ i_{x_\text{L}} $ are the indices of the $ p_{\mathrm{T}} $ and $ x_\text{L} $ bins. In total there are ten $ p_{\mathrm{T}} $ bins, the eight presented in the analysis plus the overflow and underflow bins. The number of $ x_\text{L} $ bins per $ p_{\mathrm{T}} $ region is 22. 
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Figure A7:
The statistical correlation matrix of the bins of the unfolded E3C distribution. The bin number $ n $ in this figure is defined by $ n=22 i_{p_{\mathrm{T}}} + i_{x_\text{L}} $, where $ i_{p_{\mathrm{T}}} $ and $ i_{x_\text{L}} $ are the indices of the $ p_{\mathrm{T}} $ and $ x_\text{L} $ bins. In total there are ten $ p_{\mathrm{T}} $ bins, the eight presented in the analysis plus the overflow and underflow bins. The number of $ x_\text{L} $ bins per $ p_{\mathrm{T}} $ region is 22. 
png pdf 
Figure A8:
The statistical correlation matrix of the bins of the unfolded E3C/E2C distribution. The bin number $ n $ in this figure is defined by $ n=22 i_{p_{\mathrm{T}}} + i_{x_\text{L}} $, where $ i_{p_{\mathrm{T}}} $ and $ i_{x_\text{L}} $ are the indices of the $ p_{\mathrm{T}} $ and $ x_\text{L} $ bins. In total there are ten $ p_{\mathrm{T}} $ bins, the eight presented in the analysis plus the overflow and underflow bins. The number of $ x_\text{L} $ bins per $ p_{\mathrm{T}} $ region is 22. 
Summary 
In summary, the E2C and E3C (two and threeparticle energy correlators) jet substructure observables, have been measured using a sample of protonproton collisions at $ \sqrt{s} = $ 13 TeV, collected by the CMS experiment and corresponding to an integrated luminosity of 36.3 fb$ ^{1} $. A multidimensional unfolding has been performed, of the jet $ p_{\mathrm{T}} $, of the (largest) distance between particles in a pair or a triplet, $ x_\text{L} $, and of the product of their energy weights, to compare the data with samples of events simulated with different parton showering algorithms and hadronization models. The results provide a highprecision measurement of jet properties described by QCD and can help validate future higherorder corrections in parton shower algorithms. The strong coupling at the Z boson mass, $ \alpha_\mathrm{S} (m_\mathrm{Z}) $, is extracted by comparing the measured E3C/E2C ratio with calculations at approximate nexttonexttoleading logarithmic accuracy matched to a nexttoleading order: $ \alpha_\mathrm{S} (m_\mathrm{Z}) = $ 0.1229 $ ^{+0.0040}_{0.0050} $. This is the most precise determination of $ \alpha_\mathrm{S} $ using jet substructure techniques. The result benefits greatly from the development of novel jet substructure observables, which reduce the sensitivity to the quarkgluon composition, and from the availability of highprecision theoretical calculations. 
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