CMSSMP21008 ; CERNEP2023257  
Measurement of multidifferential cross sections for dijet production in protonproton collisions at $ \sqrt{s} = $ 13 TeV  
CMS Collaboration  
28 December 2023  
Submitted to Eur. Phys. J. C  
Abstract: A measurement of the dijet production cross section is reported based on protonproton collision data collected in 2016 at $ \sqrt{s}= $ 13 TeV by the CMS experiment at the CERN LHC, corresponding to an integrated luminosity of up to 36.3 fb$ ^{1} $. Jets are reconstructed with the anti$ k_{\mathrm{T}} $ algorithm for distance parameters of $ R= $ 0.4 and 0.8. Cross sections are measured doubledifferentially (2D) as a function of the largest absolute rapidity $ y_{\text{max}} $ of the two jets with the highest transverse momenta $ p_{\mathrm{T}} $ and their invariant mass $ m_{1,2} $, and tripledifferentially (3D) as a function of the rapidity separation $ y^{*} $, the total boost $ y_{\text{b}} $, and either $ m_{1,2} $ or the average $ p_{\mathrm{T}} $ of the two jets. The cross sections are unfolded to correct for detector effects and are compared with fixedorder calculations derived at nexttonexttoleading order in perturbative quantum chromodynamics. The impact of the measurements on the parton distribution functions and the strong coupling constant at the mass of the Z boson is investigated, yielding a value of $ \alpha_\mathrm{S}(m_{\mathrm{Z}})= $ 0.1179 $ \pm $ 0.0019.  
Links: eprint arXiv:2312.16669 [hepex] (PDF) ; CDS record ; inSPIRE record ; HepData record ; CADI line (restricted) ; 
Figures & Tables  Summary  Additional Figures  References  CMS Publications 

Figures  
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Figure 1:
Illustration of the dijet rapidity phase space, highlighting the relationship between the variables used for the 2D and 3D measurements. The colored triangles are suggestive of the orientation of the two jets in the different phase space regions in the laboratory frame, assuming that the beam line runs horizontally. 
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Figure 2:
Response matrix for the 2D measurement as a function of $ m_{1,2} $ using jets with $ R = $ 0.8. The entries represent the probability for a dijet event generated in the phase space region ($ m_{1,2}^\text{gen} $, $ y_{\text{max}}^\text{gen} $) indicated on the $ x $ axis to be reconstructed in the phase space region ($ m_{1,2}^\text{rec} $, $ y_{\text{max}}^\text{rec} $) indicated on the $ y $ axis. Response matrices for all other jet sizes and observables can be found in Appendix app:supplementarymaterial. 
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Figure 3:
Breakdown of the experimental uncertainty for the 2D measurements as a function of $ m_{1,2} $ using jets with $ R = $ 0.4 (left) and 0.8 (right). The individual components are discussed in Section 7, and the abbreviation "Unf." refers to the unfolding uncertainties. The shaded area represents the sum in quadrature of all statistical and systematic uncertainty components. 
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Figure 3a:
Breakdown of the experimental uncertainty for the 2D measurements as a function of $ m_{1,2} $ using jets with $ R = $ 0.4 (left) and 0.8 (right). The individual components are discussed in Section 7, and the abbreviation "Unf." refers to the unfolding uncertainties. The shaded area represents the sum in quadrature of all statistical and systematic uncertainty components. 
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Figure 3b:
Breakdown of the experimental uncertainty for the 2D measurements as a function of $ m_{1,2} $ using jets with $ R = $ 0.4 (left) and 0.8 (right). The individual components are discussed in Section 7, and the abbreviation "Unf." refers to the unfolding uncertainties. The shaded area represents the sum in quadrature of all statistical and systematic uncertainty components. 
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Figure 4:
Breakdown of the experimental uncertainty for the 3D measurement as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $\ using jets with $ R = $ 0.4. The individual components are discussed in Section 7. The shaded area represents the sum in quadrature of all statistical and systematic uncertainty components. Similar plots for all other jet sizes and observables can be found in Appendix app:supplementarymaterial. 
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Figure 4a:
Breakdown of the experimental uncertainty for the 3D measurement as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $\ using jets with $ R = $ 0.4. The individual components are discussed in Section 7. The shaded area represents the sum in quadrature of all statistical and systematic uncertainty components. Similar plots for all other jet sizes and observables can be found in Appendix app:supplementarymaterial. 
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Figure 4b:
Breakdown of the experimental uncertainty for the 3D measurement as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $\ using jets with $ R = $ 0.4. The individual components are discussed in Section 7. The shaded area represents the sum in quadrature of all statistical and systematic uncertainty components. Similar plots for all other jet sizes and observables can be found in Appendix app:supplementarymaterial. 
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Figure 5:
Theoretical predictions for the 2D (left) and 3D (right) cross sections, as a function of $ m_{1,2} $, illustrated here in the rapidity regions 1.0 $ < y_{\text{max}} < $ 1.5 and $ y_{\text{b}} < $ 0.5, $ y^{*} < $ 0.5, together with the corresponding sixpoint scale uncertainty for $ \mu_{\mathrm{R}}=\mu_{\mathrm{F}}=m_{1,2} $ using the CT18 NNLO PDF set. In the upper panels, the curves and symbols are slightly shifted for better visibility. The lower panels show the ratio to the respective prediction at LO\@. The fluctuations in the NNLO predictions are due to the limited statistical precision of the calculation. 
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Figure 5a:
Theoretical predictions for the 2D (left) and 3D (right) cross sections, as a function of $ m_{1,2} $, illustrated here in the rapidity regions 1.0 $ < y_{\text{max}} < $ 1.5 and $ y_{\text{b}} < $ 0.5, $ y^{*} < $ 0.5, together with the corresponding sixpoint scale uncertainty for $ \mu_{\mathrm{R}}=\mu_{\mathrm{F}}=m_{1,2} $ using the CT18 NNLO PDF set. In the upper panels, the curves and symbols are slightly shifted for better visibility. The lower panels show the ratio to the respective prediction at LO\@. The fluctuations in the NNLO predictions are due to the limited statistical precision of the calculation. 
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Figure 5b:
Theoretical predictions for the 2D (left) and 3D (right) cross sections, as a function of $ m_{1,2} $, illustrated here in the rapidity regions 1.0 $ < y_{\text{max}} < $ 1.5 and $ y_{\text{b}} < $ 0.5, $ y^{*} < $ 0.5, together with the corresponding sixpoint scale uncertainty for $ \mu_{\mathrm{R}}=\mu_{\mathrm{F}}=m_{1,2} $ using the CT18 NNLO PDF set. In the upper panels, the curves and symbols are slightly shifted for better visibility. The lower panels show the ratio to the respective prediction at LO\@. The fluctuations in the NNLO predictions are due to the limited statistical precision of the calculation. 
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Figure 6:
Nonperturbative correction factors obtained for jets with $ R = $ 0.4 (left) and 0.8 (right) as a function of $ m_{1,2} $, illustrated here in the rapidity region ($ y_{\text{b}} < $ 0.5, $ y^{*} < $ 0.5). Individual correction factors are first derived from simulation using eight different MC configurations. The largest and smallest value obtained in each observable bin is then used to define the final correction factor and its associated uncertainty. The correction values are larger for jets with $ R = $ 0.8, increasing to over 20% in the lowest $ m_{1,2} $ bin. 
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Figure 6a:
Nonperturbative correction factors obtained for jets with $ R = $ 0.4 (left) and 0.8 (right) as a function of $ m_{1,2} $, illustrated here in the rapidity region ($ y_{\text{b}} < $ 0.5, $ y^{*} < $ 0.5). Individual correction factors are first derived from simulation using eight different MC configurations. The largest and smallest value obtained in each observable bin is then used to define the final correction factor and its associated uncertainty. The correction values are larger for jets with $ R = $ 0.8, increasing to over 20% in the lowest $ m_{1,2} $ bin. 
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Figure 6b:
Nonperturbative correction factors obtained for jets with $ R = $ 0.4 (left) and 0.8 (right) as a function of $ m_{1,2} $, illustrated here in the rapidity region ($ y_{\text{b}} < $ 0.5, $ y^{*} < $ 0.5). Individual correction factors are first derived from simulation using eight different MC configurations. The largest and smallest value obtained in each observable bin is then used to define the final correction factor and its associated uncertainty. The correction values are larger for jets with $ R = $ 0.8, increasing to over 20% in the lowest $ m_{1,2} $ bin. 
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Figure 7:
Electroweak correction factors obtained for jets with $ R = $ 0.4 (left) and 0.8 (right) as a function of $ m_{1,2} $ in the five different $ y_{\text{max}} $ regions. The corrections depend strongly on the kinematic properties of the jets and are observed to be largest at central rapidities for $ m_{1,2} > $ 1 TeV. 
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Figure 7a:
Electroweak correction factors obtained for jets with $ R = $ 0.4 (left) and 0.8 (right) as a function of $ m_{1,2} $ in the five different $ y_{\text{max}} $ regions. The corrections depend strongly on the kinematic properties of the jets and are observed to be largest at central rapidities for $ m_{1,2} > $ 1 TeV. 
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Figure 7b:
Electroweak correction factors obtained for jets with $ R = $ 0.4 (left) and 0.8 (right) as a function of $ m_{1,2} $ in the five different $ y_{\text{max}} $ regions. The corrections depend strongly on the kinematic properties of the jets and are observed to be largest at central rapidities for $ m_{1,2} > $ 1 TeV. 
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Figure 8:
Differential dijet cross sections, illustrated here for the 2D measurement as a function of $ m_{1,2} $ using jets with $ R = $ 0.8 (left), and the 3D measurement as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ using jets with $ R = $ 0.4 (right). The markers and lines indicate the measured unfolded cross sections and the corresponding NNLO predictions, respectively. For better visibility, the values are scaled by a factor depending on the rapidity region, as indicated in the legend. Analogous plots for all other jet sizes and observables can be found in Appendix app:supplementarymaterial. 
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Figure 8a:
Differential dijet cross sections, illustrated here for the 2D measurement as a function of $ m_{1,2} $ using jets with $ R = $ 0.8 (left), and the 3D measurement as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ using jets with $ R = $ 0.4 (right). The markers and lines indicate the measured unfolded cross sections and the corresponding NNLO predictions, respectively. For better visibility, the values are scaled by a factor depending on the rapidity region, as indicated in the legend. Analogous plots for all other jet sizes and observables can be found in Appendix app:supplementarymaterial. 
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Figure 8b:
Differential dijet cross sections, illustrated here for the 2D measurement as a function of $ m_{1,2} $ using jets with $ R = $ 0.8 (left), and the 3D measurement as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ using jets with $ R = $ 0.4 (right). The markers and lines indicate the measured unfolded cross sections and the corresponding NNLO predictions, respectively. For better visibility, the values are scaled by a factor depending on the rapidity region, as indicated in the legend. Analogous plots for all other jet sizes and observables can be found in Appendix app:supplementarymaterial. 
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Figure 9:
Comparison of the 2D dijet cross section as a function of $ m_{1,2} $ to fixedorder theoretical calculations at NNLO, using jets with $ R = $ 0.4 (left) and 0.8 (right). Shown are the ratios of the measured cross sections (markers) to the predictions obtained using the CT18 NNLO PDF set. The error bars and shaded yellow regions indicate the statistical and the total experimental uncertainties of the data, respectively, and the hatched teal band indicates the sum in quadrature of the PDF, NP, and scale uncertainties. Alternative theoretical predictions obtained using other global PDF sets are shown as colored lines. Similar plots for the individual rapidity regions can be found in Appendix app:supplementarymaterial. 
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Figure 9a:
Comparison of the 2D dijet cross section as a function of $ m_{1,2} $ to fixedorder theoretical calculations at NNLO, using jets with $ R = $ 0.4 (left) and 0.8 (right). Shown are the ratios of the measured cross sections (markers) to the predictions obtained using the CT18 NNLO PDF set. The error bars and shaded yellow regions indicate the statistical and the total experimental uncertainties of the data, respectively, and the hatched teal band indicates the sum in quadrature of the PDF, NP, and scale uncertainties. Alternative theoretical predictions obtained using other global PDF sets are shown as colored lines. Similar plots for the individual rapidity regions can be found in Appendix app:supplementarymaterial. 
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Figure 9b:
Comparison of the 2D dijet cross section as a function of $ m_{1,2} $ to fixedorder theoretical calculations at NNLO, using jets with $ R = $ 0.4 (left) and 0.8 (right). Shown are the ratios of the measured cross sections (markers) to the predictions obtained using the CT18 NNLO PDF set. The error bars and shaded yellow regions indicate the statistical and the total experimental uncertainties of the data, respectively, and the hatched teal band indicates the sum in quadrature of the PDF, NP, and scale uncertainties. Alternative theoretical predictions obtained using other global PDF sets are shown as colored lines. Similar plots for the individual rapidity regions can be found in Appendix app:supplementarymaterial. 
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Figure 10:
Comparison of the 3D dijet cross section for jets with $ R = $ 0.4 (left) and 0.8 (right) as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theoretical calculations at NNLO, shown here for three out of the total of 15 rapidity regions. The data points and predictions for alternative PDFs are analogous to those in Fig. 9. In addition, the separate contributions to the theory uncertainty due to the CT18 PDFs, NP corrections, and sixpoint scale variations are shown explicitly. Similar plots for all rapidity regions and observables can be found in Appendix app:supplementarymaterial. 
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Figure 10a:
Comparison of the 3D dijet cross section for jets with $ R = $ 0.4 (left) and 0.8 (right) as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theoretical calculations at NNLO, shown here for three out of the total of 15 rapidity regions. The data points and predictions for alternative PDFs are analogous to those in Fig. 9. In addition, the separate contributions to the theory uncertainty due to the CT18 PDFs, NP corrections, and sixpoint scale variations are shown explicitly. Similar plots for all rapidity regions and observables can be found in Appendix app:supplementarymaterial. 
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Figure 10b:
Comparison of the 3D dijet cross section for jets with $ R = $ 0.4 (left) and 0.8 (right) as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theoretical calculations at NNLO, shown here for three out of the total of 15 rapidity regions. The data points and predictions for alternative PDFs are analogous to those in Fig. 9. In addition, the separate contributions to the theory uncertainty due to the CT18 PDFs, NP corrections, and sixpoint scale variations are shown explicitly. Similar plots for all rapidity regions and observables can be found in Appendix app:supplementarymaterial. 
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Figure 10c:
Comparison of the 3D dijet cross section for jets with $ R = $ 0.4 (left) and 0.8 (right) as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theoretical calculations at NNLO, shown here for three out of the total of 15 rapidity regions. The data points and predictions for alternative PDFs are analogous to those in Fig. 9. In addition, the separate contributions to the theory uncertainty due to the CT18 PDFs, NP corrections, and sixpoint scale variations are shown explicitly. Similar plots for all rapidity regions and observables can be found in Appendix app:supplementarymaterial. 
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Figure 10d:
Comparison of the 3D dijet cross section for jets with $ R = $ 0.4 (left) and 0.8 (right) as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theoretical calculations at NNLO, shown here for three out of the total of 15 rapidity regions. The data points and predictions for alternative PDFs are analogous to those in Fig. 9. In addition, the separate contributions to the theory uncertainty due to the CT18 PDFs, NP corrections, and sixpoint scale variations are shown explicitly. Similar plots for all rapidity regions and observables can be found in Appendix app:supplementarymaterial. 
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Figure 10e:
Comparison of the 3D dijet cross section for jets with $ R = $ 0.4 (left) and 0.8 (right) as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theoretical calculations at NNLO, shown here for three out of the total of 15 rapidity regions. The data points and predictions for alternative PDFs are analogous to those in Fig. 9. In addition, the separate contributions to the theory uncertainty due to the CT18 PDFs, NP corrections, and sixpoint scale variations are shown explicitly. Similar plots for all rapidity regions and observables can be found in Appendix app:supplementarymaterial. 
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Figure 10f:
Comparison of the 3D dijet cross section for jets with $ R = $ 0.4 (left) and 0.8 (right) as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theoretical calculations at NNLO, shown here for three out of the total of 15 rapidity regions. The data points and predictions for alternative PDFs are analogous to those in Fig. 9. In addition, the separate contributions to the theory uncertainty due to the CT18 PDFs, NP corrections, and sixpoint scale variations are shown explicitly. Similar plots for all rapidity regions and observables can be found in Appendix app:supplementarymaterial. 
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Figure 11:
Parton distributions obtained in a fit to HERA DIS data together with the CMS 2D or 3D dijet measurements. The top panels show the PDFs of the up and down valence quarks (upper row), of the gluon (lower left), and of the total sea quarks (lower right) as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The middle (lower) panels show the relative uncertainty contributions obtained for the 2D (3D) fit, as well as the ratios of the fitted central values. 
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Figure 11a:
Parton distributions obtained in a fit to HERA DIS data together with the CMS 2D or 3D dijet measurements. The top panels show the PDFs of the up and down valence quarks (upper row), of the gluon (lower left), and of the total sea quarks (lower right) as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The middle (lower) panels show the relative uncertainty contributions obtained for the 2D (3D) fit, as well as the ratios of the fitted central values. 
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Figure 11b:
Parton distributions obtained in a fit to HERA DIS data together with the CMS 2D or 3D dijet measurements. The top panels show the PDFs of the up and down valence quarks (upper row), of the gluon (lower left), and of the total sea quarks (lower right) as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The middle (lower) panels show the relative uncertainty contributions obtained for the 2D (3D) fit, as well as the ratios of the fitted central values. 
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Figure 11c:
Parton distributions obtained in a fit to HERA DIS data together with the CMS 2D or 3D dijet measurements. The top panels show the PDFs of the up and down valence quarks (upper row), of the gluon (lower left), and of the total sea quarks (lower right) as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The middle (lower) panels show the relative uncertainty contributions obtained for the 2D (3D) fit, as well as the ratios of the fitted central values. 
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Figure 11d:
Parton distributions obtained in a fit to HERA DIS data together with the CMS 2D or 3D dijet measurements. The top panels show the PDFs of the up and down valence quarks (upper row), of the gluon (lower left), and of the total sea quarks (lower right) as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The middle (lower) panels show the relative uncertainty contributions obtained for the 2D (3D) fit, as well as the ratios of the fitted central values. 
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Figure 12:
Parton distributions obtained in a fit to HERA DIS data together with the CMS dijet data, compared to a fit to HERA DIS data alone. Shown are the PDFs of the up and down valence quarks (upper row), of the gluon (lower left), and of the total sea quarks (lower right) as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The bands indicate the fit uncertainty and are shown in the lower panels as a relative uncertainty with respect to the corresponding central values. The lines in the lower panels show the ratios between the fitted central values. 
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Figure 12a:
Parton distributions obtained in a fit to HERA DIS data together with the CMS dijet data, compared to a fit to HERA DIS data alone. Shown are the PDFs of the up and down valence quarks (upper row), of the gluon (lower left), and of the total sea quarks (lower right) as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The bands indicate the fit uncertainty and are shown in the lower panels as a relative uncertainty with respect to the corresponding central values. The lines in the lower panels show the ratios between the fitted central values. 
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Figure 12b:
Parton distributions obtained in a fit to HERA DIS data together with the CMS dijet data, compared to a fit to HERA DIS data alone. Shown are the PDFs of the up and down valence quarks (upper row), of the gluon (lower left), and of the total sea quarks (lower right) as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The bands indicate the fit uncertainty and are shown in the lower panels as a relative uncertainty with respect to the corresponding central values. The lines in the lower panels show the ratios between the fitted central values. 
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Figure 12c:
Parton distributions obtained in a fit to HERA DIS data together with the CMS dijet data, compared to a fit to HERA DIS data alone. Shown are the PDFs of the up and down valence quarks (upper row), of the gluon (lower left), and of the total sea quarks (lower right) as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The bands indicate the fit uncertainty and are shown in the lower panels as a relative uncertainty with respect to the corresponding central values. The lines in the lower panels show the ratios between the fitted central values. 
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Figure 12d:
Parton distributions obtained in a fit to HERA DIS data together with the CMS dijet data, compared to a fit to HERA DIS data alone. Shown are the PDFs of the up and down valence quarks (upper row), of the gluon (lower left), and of the total sea quarks (lower right) as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The bands indicate the fit uncertainty and are shown in the lower panels as a relative uncertainty with respect to the corresponding central values. The lines in the lower panels show the ratios between the fitted central values. 
Tables  
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Table 1:
Overview of the singlejet (dijet) triggers deployed for the different $ p_{\mathrm{T}} $ ($ \langle p_{\mathrm{T}} \rangle $) thresholds at the HLT, and the corresponding integrated luminosities. 
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Table 2:
Nominal values and variations of parameters used to determine the PDF model uncertainty. Variations marked with an asterisk are in conflict with the requirement $ \mu_{\mathrm{F},\,0} < m_\mathrm{c} $ and thus cannot be used directly for the uncertainty estimation. Following Ref. [63], the results obtained for the opposite variation are symmetrized in these cases. 
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Table 3:
Goodnessoffit values for the fits to the HERA DIS data alone, and together with the CMS dijet measurements, using the PDF parametrization given in Eq. 8. The table shows the partial $ \chi^2 $ values divided by the number of data points for the HERA DIS datasets and each of the dijet rapidity regions. The total $ \chi^2 $ value, divided by the number of degrees of freedom, is given at the bottom of the table. 
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Table 4:
Goodnessoffit values for the fits to the HERA DIS data alone, and together with the CMS dijet measurements, including all rapidity regions. The table shows the partial $ \chi^2 $ values divided by the number of data points for the HERA DIS datasets and each of the dijet rapidity regions. The total $ \chi^2 $ value, divided by the number of degrees of freedom, is given at the bottom of the table. 
Summary 
The dijet production cross section is measured based on pp collision data recorded by the CMS detector in 2016 at $ \sqrt{s} = $ 13 TeV, corresponding to an integrated luminosity of up to 36.3 fb$ ^{1} $. The measurements are performed doubledifferentially (2D) as a function of the dijet invariant mass $ m_{1,2} $ in five regions of the maximal absolute rapidity $ y_{\text{max}} $\ of the two jets with the largest transverse momenta, and tripledifferentially (3D) as a function of either $ m_{1,2} $ or the average transverse momentum $ \langle p_{\mathrm{T}} \rangle_{1,2} $ in 15 bins of the rapidity variables $ y^{*} $\ and $ y_{\text{b}} $. The latter two variables correspond to the rapidity separation of the two jets, and the total boost of the dijet system, respectively. All measurements are performed for jets clustered using the anti$ k_{\mathrm{T}} $ jet algorithm with distance parameters $ R= $ 0.4 and 0.8, and the cross sections are unfolded in all measurement dimensions simultaneously to correct for detector effects. This is the first time that such a large set of multidifferential dijet measurements for two observables, $ \langle p_{\mathrm{T}} \rangle_{1,2} $ and $ m_{1,2} $, and two jet distance parameters, $ R = $ 0.4 and 0.8, is made available for comparison to theory and use in fits of the parton distribution functions (PDFs) of the proton. Predictions at nexttonexttoleading order (NNLO) in perturbative quantum chromodynamics, supplemented with electroweak and nonperturbative corrections are observed to describe the data better for $ R = $ 0.8. Using the measurement of $ m_{1,2} $ for $ R = $ 0.8, the PDFs of the proton are determined simultaneously in fits to the dijet measurements together with deepinelastic scattering data from the HERA experiments following the approach described in earlier HERAPDF analyses [1,2,63]. The results obtained from the double and tripledifferential measurements are compatible within the estimated uncertainties. The inclusion of either of the dijet measurements leads to an improved determination of the PDFs compared to fits to HERA data alone. In particular, the uncertainty in the gluon distribution at fractional proton momenta $ x > $ 0.1 is reduced, with the 3D dijet data providing tighter constraints at higher values of $ x $ compared to the 2D data. The strong coupling constant at the Z boson mass is determined simultaneously with the PDFs, yielding consistent results between the 2D and 3D dijet measurements, with the former resulting in the slightly more precise value of $ \alpha_\mathrm{S}(m_{\mathrm{Z}}) = $ 0.1179 $ \pm $ 0.0019 at NNLO. The impact of subleadingcolor contributions to the leadingcolor NNLO calculation used here is not yet known [42]. Apart from being useful as inputs to PDF fits or studies of jet size dependence, the present 2D and 3D measurements for two jet size parameters, $ R= $ 0.4 and 0.8, and for the two dijet observables $ m_{1,2} $ and $ \langle p_{\mathrm{T}} \rangle_{1,2} $, provide an ideal testing ground for further investigations. 
Additional Figures  
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Additional Figure 1:
Response matrices for the 2D measurements as a function of $ y_{\text{max}} $ and $ m_{1,2} $ for jets with $ R = $ 0.4. The details correspond to those of Fig. 2. 
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Additional Figure 2:
Partial response matrices for the 3D measurements as a function of $ m_{1,2} $ using jets with $ R = $ 0.4 (upper) and 0.8 (lower), shown here for the five rapidity regions with $ y_{\text{b}} < $ 0.5. The details correspond to those of Fig. 2. 
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Additional Figure 2a:
Partial response matrices for the 3D measurements as a function of $ m_{1,2} $ using jets with $ R = $ 0.4 (upper) and 0.8 (lower), shown here for the five rapidity regions with $ y_{\text{b}} < $ 0.5. The details correspond to those of Fig. 2. 
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Additional Figure 2b:
Partial response matrices for the 3D measurements as a function of $ m_{1,2} $ using jets with $ R = $ 0.4 (upper) and 0.8 (lower), shown here for the five rapidity regions with $ y_{\text{b}} < $ 0.5. The details correspond to those of Fig. 2. 
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Additional Figure 3:
Partial response matrices for the 3D measurements as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ using jets with $ R = $ 0.4 (upper) and 0.8 (lower), shown here for the five rapidity regions with $ y_{\text{b}} < $ 0.5. The details correspond to those of Fig. 2. 
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Additional Figure 3a:
Partial response matrices for the 3D measurements as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ using jets with $ R = $ 0.4 (upper) and 0.8 (lower), shown here for the five rapidity regions with $ y_{\text{b}} < $ 0.5. The details correspond to those of Fig. 2. 
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Additional Figure 3b:
Partial response matrices for the 3D measurements as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ using jets with $ R = $ 0.4 (upper) and 0.8 (lower), shown here for the five rapidity regions with $ y_{\text{b}} < $ 0.5. The details correspond to those of Fig. 2. 
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Additional Figure 4:
Overview of the 2D dijet cross section as a function of $ m_{1,2} $ in all 5 $ y_{\text{max}} $ regions, using jets with $ R = $ 0.4. The details correspond to those of Fig. 8. 
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Additional Figure 5:
Overview of the 3D dijet cross section as a function of $ m_{1,2} $ in all 15 $ {(y^{*}\!, y_{\text{b}})} $ regions, using jets with $ R = $ 0.4 (left) and 0.8 (right). The details correspond to those of Fig. 8. 
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Additional Figure 5a:
Overview of the 3D dijet cross section as a function of $ m_{1,2} $ in all 15 $ {(y^{*}\!, y_{\text{b}})} $ regions, using jets with $ R = $ 0.4 (left) and 0.8 (right). The details correspond to those of Fig. 8. 
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Additional Figure 5b:
Overview of the 3D dijet cross section as a function of $ m_{1,2} $ in all 15 $ {(y^{*}\!, y_{\text{b}})} $ regions, using jets with $ R = $ 0.4 (left) and 0.8 (right). The details correspond to those of Fig. 8. 
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Additional Figure 6:
Overview of the 3D dijet cross section as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ in all 15 $ {(y^{*}\!, y_{\text{b}})} $ regions, using jets with $ R = $ 0.8. The details correspond to those of Fig. 8. 
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Additional Figure 7:
Breakdown of the experimental uncertainty for the 3D measurements as a function of $ m_{1,2} $ using jets with $ R = $ 0.4 (left) and 0.8 (right), in six out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The details correspond to those of Fig. 4. 
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Additional Figure 7a:
Breakdown of the experimental uncertainty for the 3D measurements as a function of $ m_{1,2} $ using jets with $ R = $ 0.4 (left) and 0.8 (right), in six out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The details correspond to those of Fig. 4. 
png pdf 
Additional Figure 7b:
Breakdown of the experimental uncertainty for the 3D measurements as a function of $ m_{1,2} $ using jets with $ R = $ 0.4 (left) and 0.8 (right), in six out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The details correspond to those of Fig. 4. 
png pdf 
Additional Figure 8:
(continuation of Fig. 7) Breakdown of the experimental uncertainty for the 3D measurements as a function of $ m_{1,2} $ using jets with $ R = $ 0.4 (left) and 0.8 (right), in the remaining nine out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The details correspond to those of Fig. 4. 
png pdf 
Additional Figure 8a:
(continuation of Fig. 7) Breakdown of the experimental uncertainty for the 3D measurements as a function of $ m_{1,2} $ using jets with $ R = $ 0.4 (left) and 0.8 (right), in the remaining nine out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The details correspond to those of Fig. 4. 
png pdf 
Additional Figure 8b:
(continuation of Fig. 7) Breakdown of the experimental uncertainty for the 3D measurements as a function of $ m_{1,2} $ using jets with $ R = $ 0.4 (left) and 0.8 (right), in the remaining nine out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The details correspond to those of Fig. 4. 
png pdf 
Additional Figure 9:
Breakdown of the experimental uncertainty for the 3D measurements as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ using jets with $ R = $ 0.8, in six out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The details correspond to those of Fig. 4. 
png pdf 
Additional Figure 10:
(continuation of Fig. 9) Breakdown of the experimental uncertainty for the 3D measurements as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ using jets with $ R = $ 0.8, in the remaining nine out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The details correspond to those of Fig. 4. 
png pdf 
Additional Figure 11:
Comparison of the 2D dijet cross section for jets with $ R = $ 0.4 (left) and 0.8 (right) as a function of $ m_{1,2} $ to fixedorder theoretical calculations at NNLO, shown here for three inner $ y_{\text{max}} $ regions. The details correspond to those of Fig. 9. 
png pdf 
Additional Figure 11a:
Comparison of the 2D dijet cross section for jets with $ R = $ 0.4 (left) and 0.8 (right) as a function of $ m_{1,2} $ to fixedorder theoretical calculations at NNLO, shown here for three inner $ y_{\text{max}} $ regions. The details correspond to those of Fig. 9. 
png pdf 
Additional Figure 11b:
Comparison of the 2D dijet cross section for jets with $ R = $ 0.4 (left) and 0.8 (right) as a function of $ m_{1,2} $ to fixedorder theoretical calculations at NNLO, shown here for three inner $ y_{\text{max}} $ regions. The details correspond to those of Fig. 9. 
png pdf 
Additional Figure 11c:
Comparison of the 2D dijet cross section for jets with $ R = $ 0.4 (left) and 0.8 (right) as a function of $ m_{1,2} $ to fixedorder theoretical calculations at NNLO, shown here for three inner $ y_{\text{max}} $ regions. The details correspond to those of Fig. 9. 
png pdf 
Additional Figure 11d:
Comparison of the 2D dijet cross section for jets with $ R = $ 0.4 (left) and 0.8 (right) as a function of $ m_{1,2} $ to fixedorder theoretical calculations at NNLO, shown here for three inner $ y_{\text{max}} $ regions. The details correspond to those of Fig. 9. 
png pdf 
Additional Figure 11e:
Comparison of the 2D dijet cross section for jets with $ R = $ 0.4 (left) and 0.8 (right) as a function of $ m_{1,2} $ to fixedorder theoretical calculations at NNLO, shown here for three inner $ y_{\text{max}} $ regions. The details correspond to those of Fig. 9. 
png pdf 
Additional Figure 11f:
Comparison of the 2D dijet cross section for jets with $ R = $ 0.4 (left) and 0.8 (right) as a function of $ m_{1,2} $ to fixedorder theoretical calculations at NNLO, shown here for three inner $ y_{\text{max}} $ regions. The details correspond to those of Fig. 9. 
png pdf 
Additional Figure 12:
(continuation of Fig. 11) Comparison of the 2D dijet cross section for jets with $ R = $ 0.4 (left) and 0.8 (right) as a function of $ m_{1,2} $ to fixedorder theoretical calculations at NNLO, shown here for two outermost $ y_{\text{max}} $ regions. The details correspond to those of Fig. 9. 
png pdf 
Additional Figure 12a:
(continuation of Fig. 11) Comparison of the 2D dijet cross section for jets with $ R = $ 0.4 (left) and 0.8 (right) as a function of $ m_{1,2} $ to fixedorder theoretical calculations at NNLO, shown here for two outermost $ y_{\text{max}} $ regions. The details correspond to those of Fig. 9. 
png pdf 
Additional Figure 12b:
(continuation of Fig. 11) Comparison of the 2D dijet cross section for jets with $ R = $ 0.4 (left) and 0.8 (right) as a function of $ m_{1,2} $ to fixedorder theoretical calculations at NNLO, shown here for two outermost $ y_{\text{max}} $ regions. The details correspond to those of Fig. 9. 
png pdf 
Additional Figure 12c:
(continuation of Fig. 11) Comparison of the 2D dijet cross section for jets with $ R = $ 0.4 (left) and 0.8 (right) as a function of $ m_{1,2} $ to fixedorder theoretical calculations at NNLO, shown here for two outermost $ y_{\text{max}} $ regions. The details correspond to those of Fig. 9. 
png pdf 
Additional Figure 12d:
(continuation of Fig. 11) Comparison of the 2D dijet cross section for jets with $ R = $ 0.4 (left) and 0.8 (right) as a function of $ m_{1,2} $ to fixedorder theoretical calculations at NNLO, shown here for two outermost $ y_{\text{max}} $ regions. The details correspond to those of Fig. 9. 
png pdf 
Additional Figure 13:
Comparison of the 3D dijet cross section as a function of $ m_{1,2} $ to fixedorder theoretical calculations at NNLO, using jets with $ R = $ 0.4 (left) and 0.8 (right), in six out of the total 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The details correspond to those of Fig. 10. 
png pdf 
Additional Figure 13a:
Comparison of the 3D dijet cross section as a function of $ m_{1,2} $ to fixedorder theoretical calculations at NNLO, using jets with $ R = $ 0.4 (left) and 0.8 (right), in six out of the total 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The details correspond to those of Fig. 10. 
png pdf 
Additional Figure 13b:
Comparison of the 3D dijet cross section as a function of $ m_{1,2} $ to fixedorder theoretical calculations at NNLO, using jets with $ R = $ 0.4 (left) and 0.8 (right), in six out of the total 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The details correspond to those of Fig. 10. 
png pdf 
Additional Figure 14:
(continuation of Fig. 13) Comparison of the 3D dijet cross section as a function of $ m_{1,2} $ to fixedorder theoretical calculations at NNLO, using jets with $ R = $ 0.4 (left) and 0.8 (right), in the remaining nine out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The details correspond to those of Fig. 10. 
png pdf 
Additional Figure 14a:
(continuation of Fig. 13) Comparison of the 3D dijet cross section as a function of $ m_{1,2} $ to fixedorder theoretical calculations at NNLO, using jets with $ R = $ 0.4 (left) and 0.8 (right), in the remaining nine out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The details correspond to those of Fig. 10. 
png pdf 
Additional Figure 14b:
(continuation of Fig. 13) Comparison of the 3D dijet cross section as a function of $ m_{1,2} $ to fixedorder theoretical calculations at NNLO, using jets with $ R = $ 0.4 (left) and 0.8 (right), in the remaining nine out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The details correspond to those of Fig. 10. 
png pdf 
Additional Figure 15:
Comparison of the 3D dijet cross section as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theoretical calculations at NNLO, using jets with $ R = $ 0.4 (left) and 0.8 (right), in six out of the total 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The details correspond to those of Fig. 10. 
png pdf 
Additional Figure 15a:
Comparison of the 3D dijet cross section as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theoretical calculations at NNLO, using jets with $ R = $ 0.4 (left) and 0.8 (right), in six out of the total 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The details correspond to those of Fig. 10. 
png pdf 
Additional Figure 15b:
Comparison of the 3D dijet cross section as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theoretical calculations at NNLO, using jets with $ R = $ 0.4 (left) and 0.8 (right), in six out of the total 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The details correspond to those of Fig. 10. 
png pdf 
Additional Figure 16:
(continuation of Fig. 15) Comparison of the 3D dijet cross section as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theoretical calculations at NNLO, using jets with $ R = $ 0.4 (left) and 0.8 (right), in the remaining nine out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The details correspond to those of Fig. 10. 
png pdf 
Additional Figure 16a:
(continuation of Fig. 15) Comparison of the 3D dijet cross section as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theoretical calculations at NNLO, using jets with $ R = $ 0.4 (left) and 0.8 (right), in the remaining nine out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The details correspond to those of Fig. 10. 
png pdf 
Additional Figure 16b:
(continuation of Fig. 15) Comparison of the 3D dijet cross section as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theoretical calculations at NNLO, using jets with $ R = $ 0.4 (left) and 0.8 (right), in the remaining nine out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The details correspond to those of Fig. 10. 
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