CMSPASSMP21008  
Multidifferential measurement of the dijet cross section in protonproton collisions at $ \sqrt{s}= $ 13 TeV  
CMS Collaboration  
7 December 2022  
Abstract: A measurement of the dijet production cross section is reported based on an integrated luminosity of 36.3 fb$ ^{1} $ of protonproton collision data collected in 2016 at $ \sqrt{s}= $ 13 TeV by the CMS detector at the CERN LHC. Jets are reconstructed with the anti$ k_{\text{T}} $ algorithm for distance parameters of $ R= $ 0.4 and $ R= $ 0.8 and differential cross sections are measured as a function of the kinematic properties of the two jets with largest transverse momenta. Doubledifferential (2D) measurements are presented as a function of the largest absolute rapidity $ y_{\text{max}} $ of the two jets and the dijet invariant mass $ m_{1,2} $. Tripledifferential (3D) measurements are presented as a function of the dijet rapidity separation $ y^{*} $, the total boost $ y_{\text{b}} $ of the dijet system, and either $ m_{1,2} $ or the average dijet transverse momentum $ \langle p_{\text{T}} \rangle_{1,2} $ as the third variable. The measured 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 2D and 3D measurements on determinations of the parton distribution functions and the strong coupling constant is investigated, with the inclusion of the 3D cross sections yielding the more precise value of $ \alpha_{\text{s}}(m_{\text{Z}})= $ 0.1201 $ \pm $ 0.0020.  
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These preliminary results are superseded in this paper, Submitted to EPJC. The superseded preliminary plots can be found here. 
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 matrices for the 2D measurement as a function of $ m_{1,2} $ using jets with $ R = $ 0.8 (top), and for the 3D measurement as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ using jets with $ R = $ 0.4 (bottom). The former includes all five $ y_{\text{max}} $ regions while the latter only shows a subset of the full 3D response matrix, covering the five rapidity regions with $ y_{\text{b}} < $ 0.5. The entries represent the probability for a dijet event generated in the phase space region indicated on the $ x $ axis to be reconstructed in the phase space region indicated on the $ y $ axis. 
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Figure 2a:
Response matrix for the 2D measurement as a function of $ m_{1,2} $ using jets with $ R = $ 0.8. It includes all five $ y_{\text{max}} $ regions while the latter only shows a subset of the full 3D response matrix, covering the five rapidity regions with $ y_{\text{b}} < $ 0.5. The entries represent the probability for a dijet event generated in the phase space region indicated on the $ x $ axis to be reconstructed in the phase space region indicated on the $ y $ axis. 
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Figure 2b:
Response matrix for the 3D measurement as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ using jets with $ R = $ 0.4. The entries represent the probability for a dijet event generated in the phase space region indicated on the $ x $ axis to be reconstructed in the phase space region indicated on the $ y $ axis. 
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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 $ R = $ 0.8 (right). The shaded area represents the sum in quadrature of the individual 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. The shaded area represents the sum in quadrature of the individual 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 = $ 0.8. The shaded area represents the sum in quadrature of the individual 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 shaded area represents the sum in quadrature of the individual uncertainty components. 
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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 shaded area represents the sum in quadrature of the individual uncertainty components. 
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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 shaded area represents the sum in quadrature of the individual uncertainty components. 
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Figure 5:
Theory 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. 
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Figure 5a:
Theory predictions for the 2D 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 panel, the curves and symbols are slightly shifted for better visibility. The lower panel shows the ratio to the respective prediction at LO. 
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Figure 5b:
Theory predictions for the 3D 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 panel, the curves and symbols are slightly shifted for better visibility. The lower panel shows the ratio to the respective prediction at LO. 
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Figure 6:
Nonperturbative correction factors obtained for jets with $ R = $ 0.4 (left) and $ R = $ 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 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. 
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Figure 6b:
Nonperturbative correction factors obtained for jets with $ R = $ 0.8 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 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 $ R = $ 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 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 = $ 0.8 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:
Overview of the 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. 
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Figure 8a:
Overview of the dijet cross sections, illustrated here for the 2D measurement as a function of $ m_{1,2} $ using jets with $ R = $ 0.8. 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. 
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Figure 8b:
Overview of the 3D measurement as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ using jets with $ R = $ 0.4. 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. 
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Figure 9:
Comparison of the 2D dijet cross section as a function of $ m_{1,2} $ to fixedorder theory calculations at NNLO, using jets with $ R = $ 0.4 (left) and $ R = $ 0.8 (right). Shown are the ratios of the measured cross sections (markers) to 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 total theory uncertainty. Alternative theory predictions obtained using other global PDF sets are shown as colored lines. 
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Figure 9a:
Comparison of the 2D dijet cross section as a function of $ m_{1,2} $ to fixedorder theory calculations at NNLO, using jets with $ R = $ 0.4. Shown are the ratios of the measured cross sections (markers) to 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 total theory uncertainty. Alternative theory predictions obtained using other global PDF sets are shown as colored lines. 
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Figure 9b:
Comparison of the 2D dijet cross section as a function of $ m_{1,2} $ to fixedorder theory calculations at NNLO, using jets with $ R = $ 0.8. Shown are the ratios of the measured cross sections (markers) to 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 total theory uncertainty. Alternative theory predictions obtained using other global PDF sets are shown as colored lines. 
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Figure 10:
Comparison of the 3D dijet cross section for jets with $ R = $ 0.4 (left) and $ R = $ 0.8 (right) as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theory 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 continuation. In addition, the separate contributions to the theory uncertainty due to the CT18 PDFs, NP corrections, and sixpoint scale variations are shown explicitly. 
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Figure 10a:
Comparison of the 3D dijet cross section for jets with $ R = $ 0.4 as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theory calculations at NNLO, shown here for one of the total of 15 rapidity regions. The data points and predictions for alternative PDFs are analogous to those in continuation. In addition, the separate contributions to the theory uncertainty due to the CT18 PDFs, NP corrections, and sixpoint scale variations are shown explicitly. 
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Figure 10b:
Comparison of the 3D dijet cross section for jets with $ R = $ 0.8 as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theory calculations at NNLO, shown here for one out of the total of 15 rapidity regions. The data points and predictions for alternative PDFs are analogous to those in continuation. In addition, the separate contributions to the theory uncertainty due to the CT18 PDFs, NP corrections, and sixpoint scale variations are shown explicitly. 
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Figure 10c:
Comparison of the 3D dijet cross section for jets with $ R = $ 0.4 as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theory calculations at NNLO, shown here for one out of the total of 15 rapidity regions. The data points and predictions for alternative PDFs are analogous to those in continuation. In addition, the separate contributions to the theory uncertainty due to the CT18 PDFs, NP corrections, and sixpoint scale variations are shown explicitly. 
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Figure 10d:
Comparison of the 3D dijet cross section for jets with $ R = $ 0.8 as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theory calculations at NNLO, shown here for one out of the total of 15 rapidity regions. The data points and predictions for alternative PDFs are analogous to those in continuation. In addition, the separate contributions to the theory uncertainty due to the CT18 PDFs, NP corrections, and sixpoint scale variations are shown explicitly. 
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Figure 10e:
Comparison of the 3D dijet cross section for jets with $ R = $ 0.4 as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theory calculations at NNLO, shown here for one out of the total of 15 rapidity regions. The data points and predictions for alternative PDFs are analogous to those in continuation. In addition, the separate contributions to the theory uncertainty due to the CT18 PDFs, NP corrections, and sixpoint scale variations are shown explicitly. 
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Figure 10f:
Comparison of the 3D dijet cross section for jets with $ R = $ 0.8 as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theory calculations at NNLO, shown here for one out of the total of 15 rapidity regions. The data points and predictions for alternative PDFs are analogous to those in continuation. In addition, the separate contributions to the theory uncertainty due to the CT18 PDFs, NP corrections, and sixpoint scale variations are shown explicitly. 
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Figure 11:
PDFs obtained in a fit to HERA DIS together with CMS 2D dijet data. Shown are the PDFs of the up and down valence quarks (top row), of the gluon (bottom left), and of the total sea quarks (bottom right) as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The colored bands correspond to the individual contributions to the total PDF uncertainty. 
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Figure 11a:
PDFs obtained in a fit to HERA DIS together with CMS 2D dijet data. Shown are the PDFs of the up valence quark as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The colored bands correspond to the individual contributions to the total PDF uncertainty. 
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Figure 11b:
PDFs obtained in a fit to HERA DIS together with CMS 2D dijet data. Shown are the PDFs of the down valence quark as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The colored bands correspond to the individual contributions to the total PDF uncertainty. 
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Figure 11c:
PDFs obtained in a fit to HERA DIS together with CMS 2D dijet data. Shown are the PDFs of the gluon as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The colored bands correspond to the individual contributions to the total PDF uncertainty. 
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Figure 11d:
PDFs obtained in a fit to HERA DIS together with CMS 2D dijet data. Shown are the PDFs of the total sea quarks as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The colored bands correspond to the individual contributions to the total PDF uncertainty. 
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Figure 12:
PDFs obtained in a fit to HERA DIS together with CMS 3D dijet data. Shown are the PDFs of the up and down valence quarks (top row), of the gluon (bottom left), and of the total sea quarks (bottom right) as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The colored bands correspond to the individual contributions to the total PDF uncertainty. 
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Figure 12a:
PDFs obtained in a fit to HERA DIS together with CMS 3D dijet data. Shown are the PDFs of the up valence quark as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The colored bands correspond to the individual contributions to the total PDF uncertainty. 
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Figure 12b:
PDFs obtained in a fit to HERA DIS together with CMS 3D dijet data. Shown are the PDFs of the down valence quark as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The colored bands correspond to the individual contributions to the total PDF uncertainty. 
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Figure 12c:
PDFs obtained in a fit to HERA DIS together with CMS 3D dijet data. Shown are the PDFs of the gluon as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The colored bands correspond to the individual contributions to the total PDF uncertainty. 
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Figure 12d:
PDFs obtained in a fit to HERA DIS together with CMS 3D dijet data. Shown are the PDFs of the total sea quarks as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The colored bands correspond to the individual contributions to the total PDF uncertainty. 
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Figure 13:
PDFs obtained in a fit to HERA DIS together with CMS 2D dijet data compared to a fit to HERA DIS data alone. Shown are the PDFs of the up and down valence quarks (top row), of the gluon (bottom left), and of the total sea quarks (bottom right) as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The hatched bands indicate the Hessian fit uncertainty and are shown in the lower panels as a relative uncertainty with respect to the corresponding central values. The solid black line shows the ratio between the fitted central values. 
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Figure 13a:
PDFs obtained in a fit to HERA DIS together with CMS 2D dijet data compared to a fit to HERA DIS data alone. Shown are the PDFs of the up valence quark as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The hatched bands indicate the Hessian fit uncertainty and are shown in the lower panels as a relative uncertainty with respect to the corresponding central values. The solid black line shows the ratio between the fitted central values. 
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Figure 13b:
PDFs obtained in a fit to HERA DIS together with CMS 2D dijet data compared to a fit to HERA DIS data alone. Shown are the PDFs of the down valence quark as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The hatched bands indicate the Hessian fit uncertainty and are shown in the lower panels as a relative uncertainty with respect to the corresponding central values. The solid black line shows the ratio between the fitted central values. 
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Figure 13c:
PDFs obtained in a fit to HERA DIS together with CMS 2D dijet data compared to a fit to HERA DIS data alone. Shown are the PDFs of the gluon as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The hatched bands indicate the Hessian fit uncertainty and are shown in the lower panels as a relative uncertainty with respect to the corresponding central values. The solid black line shows the ratio between the fitted central values. 
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Figure 13d:
PDFs obtained in a fit to HERA DIS together with CMS 2D dijet data compared to a fit to HERA DIS data alone. Shown are the PDFs of the total sea quarks as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The hatched bands indicate the Hessian fit uncertainty and are shown in the lower panels as a relative uncertainty with respect to the corresponding central values. The solid black line shows the ratio between the fitted central values. 
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Figure 14:
PDFs obtained in a fit to HERA DIS together with CMS 3D dijet data compared to a fit to HERA DIS data alone. Shown are the PDFs of the up and down valence quarks (top row), of the gluon (bottom left), and of the total sea quarks (bottom right) as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The hatched bands indicate the Hessian fit uncertainty and are shown in the lower panels as a relative uncertainty with respect to the corresponding central values. The solid black line shows the ratio between the fitted central values. 
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Figure 14a:
PDFs obtained in a fit to HERA DIS together with CMS 3D dijet data compared to a fit to HERA DIS data alone. Shown are the PDFs of the up valence quark as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The hatched bands indicate the Hessian fit uncertainty and are shown in the lower panels as a relative uncertainty with respect to the corresponding central values. The solid black line shows the ratio between the fitted central values. 
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Figure 14b:
PDFs obtained in a fit to HERA DIS together with CMS 3D dijet data compared to a fit to HERA DIS data alone. Shown are the PDFs of the down valence quark as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The hatched bands indicate the Hessian fit uncertainty and are shown in the lower panels as a relative uncertainty with respect to the corresponding central values. The solid black line shows the ratio between the fitted central values. 
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Figure 14c:
PDFs obtained in a fit to HERA DIS together with CMS 3D dijet data compared to a fit to HERA DIS data alone. Shown are the PDFs of the gluon as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The hatched bands indicate the Hessian fit uncertainty and are shown in the lower panels as a relative uncertainty with respect to the corresponding central values. The solid black line shows the ratio between the fitted central values. 
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Figure 14d:
PDFs obtained in a fit to HERA DIS together with CMS 3D dijet data compared to a fit to HERA DIS data alone. Shown are the PDFs of the total sea quarks as a function of the fractional parton momentum $ x $ at a factorization scale equal to the top quark mass. The hatched bands indicate the Hessian fit uncertainty and are shown in the lower panels as a relative uncertainty with respect to the corresponding central values. The solid black line shows the ratio between the fitted central values. 
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Figure 15:
The response matrices for the 2D measurements as a function of $ y_{\text{max}} $ and $ m_{1,2} $ for jets with $ R = $ 0.4 (left) and $ R = $ 0.8 (right). The figure description corresponds to that of Fig. 2. 
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Figure 15a:
The response matrices for the 2D measurements as a function of $ y_{\text{max}} $ and $ m_{1,2} $ for jets with $ R = $ 0.4. The figure description corresponds to that of Fig. 2. 
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Figure 15b:
The response matrices for the 2D measurements as a function of $ y_{\text{max}} $ and $ m_{1,2} $ for jets with $ R = $ 0.8. The figure description corresponds to that of Fig. 2. 
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Figure 16:
Partial response matrices for the 3D measurements as a function of $ m_{1,2} $ using jets with $ R = $ 0.4 (left) and $ R = $ 0.8 (right), shown here for the five rapidity regions with $ y_{\text{b}} < $ 0.5. The figure description corresponds to that of Fig. 2. 
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Figure 16a:
Partial response matrices for the 3D measurements as a function of $ m_{1,2} $ using jets with $ R = $ 0.4, shown here for the five rapidity regions with $ y_{\text{b}} < $ 0.5. The figure description corresponds to that of Fig. 2. 
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Figure 16b:
Partial response matrices for the 3D measurements as a function of $ m_{1,2} $ using jets with $ R = $ 0.8, shown here for the five rapidity regions with $ y_{\text{b}} < $ 0.5. The figure description corresponds to that of Fig. 2. 
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Figure 17:
Partial response matrices for the 3D measurements as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ using jets with $ R = $ 0.4 (left) and $ R = $ 0.8 (right), shown here for the five rapidity regions with $ y_{\text{b}} < $ 0.5. The figure description corresponds to that of Fig. 2. 
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Figure 17a:
Partial response matrices for the 3D measurements as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ using jets with $ R = $ 0.4, shown here for the five rapidity regions with $ y_{\text{b}} < $ 0.5. The figure description corresponds to that of Fig. 2. 
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Figure 17b:
Partial response matrices for the 3D measurements as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ using jets with $ R = $ 0.8, shown here for the five rapidity regions with $ y_{\text{b}} < $ 0.5. The figure description corresponds to that of Fig. 2. 
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Figure 18:
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 (left) and $ R = $ 0.8 (right). The figure description corresponds to that of Fig. 8. 
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Figure 18a:
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 figure description corresponds to that of Fig. 8. 
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Figure 18b:
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 = $ 0.8. The figure description corresponds to that of Fig. 8. 
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Figure 19:
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 $ R = $ 0.8 (right). The figure description corresponds to that of Fig. 8. 
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Figure 19a:
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. The figure description corresponds to that of Fig. 8. 
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Figure 19b:
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 = $ 0.8. The figure description corresponds to that of Fig. 8. 
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Figure 20:
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 = $ 0.4 (left) and $ R = $ 0.8 (right). The figure description corresponds to that of Fig. 8. 
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Figure 20a:
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 = $ 0.4. The figure description corresponds to that of Fig. 8. 
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Figure 20b:
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 figure description corresponds to that of Fig. 8. 
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Figure 21:
Detailed breakdown of the experimental uncertainty for the 3D measurements as a function of $ m_{1,2} $ using jets with $ R = $ 0.4 (left) and $ R = $ 0.8 (right), in six out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The figure description corresponds to that of Fig. 4. 
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Figure 21a:
Detailed breakdown of the experimental uncertainty for the 3D measurements as a function of $ m_{1,2} $ using jets with $ R = $ 0.4, in six out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The figure description corresponds to that of Fig. 4. 
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Figure 21b:
Detailed breakdown of the experimental uncertainty for the 3D measurements as a function of $ m_{1,2} $ using jets with $ R = $ 0.8, in six out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The figure description corresponds to that of Fig. 4. 
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Figure 22:
(continuation of Fig. 21) Detailed breakdown of the experimental uncertainty for the 3D measurements as a function of $ m_{1,2} $ using jets with $ R = $ 0.4 (left) and $ R = $ 0.8 (right), in the remaining nine out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The figure description corresponds to that of Fig. 4. 
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Figure 22a:
(continuation of Fig. 21) Detailed breakdown of the experimental uncertainty for the 3D measurements as a function of $ m_{1,2} $ using jets with $ R = $ 0.4, in the remaining nine out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The figure description corresponds to that of Fig. 4. 
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Figure 22b:
(continuation of Fig. 21) Detailed breakdown of the experimental uncertainty for the 3D measurements as a function of $ m_{1,2} $ using jets with $ R = $ 0.8, in the remaining nine out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The figure description corresponds to that of Fig. 4. 
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Figure 23:
Detailed breakdown of the experimental uncertainty for the 3D measurements as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ using jets with $ R = $ 0.4 (left) and $ R = $ 0.8 (right), in six out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The figure description corresponds to that of Fig. 4. 
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Figure 23a:
Detailed breakdown of the experimental uncertainty for the 3D measurements as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ using jets with $ R = $ 0.4, in six out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The figure description corresponds to that of Fig. 4. 
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Figure 23b:
Detailed 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 figure description corresponds to that of Fig. 4. 
png pdf 
Figure 24:
(continuation of Fig. 23) Detailed breakdown of the experimental uncertainty for the 3D measurements as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ using jets with $ R = $ 0.4 (left) and $ R = $ 0.8 (right), in the remaining nine out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The figure description corresponds to that of Fig. 4. 
png pdf 
Figure 24a:
(continuation of Fig. 23) Detailed breakdown of the experimental uncertainty for the 3D measurements as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ using jets with $ R = $ 0.4, in the remaining nine out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The figure description corresponds to that of Fig. 4. 
png pdf 
Figure 24b:
(continuation of Fig. 23) Detailed 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 figure description corresponds to that of Fig. 4. 
png pdf 
Figure 25:
Comparison of the 2D dijet cross section for jets with $ R = $ 0.4 (left) and $ R = $ 0.8 (right) as a function of $ m_{1,2} $ to fixedorder theory calculations at NNLO, shown here for three inner $ y_{\text{max}} $ regions. The figure description corresponds to that of Fig. 9. 
png pdf 
Figure 25a:
Comparison of the 2D dijet cross section for jets with $ R = $ 0.4, as a function of $ m_{1,2} $ to fixedorder theory calculations at NNLO, shown here for three inner $ y_{\text{max}} $ regions. The figure description corresponds to that of Fig. 9. 
png pdf 
Figure 25b:
Comparison of the 2D dijet cross section for jets with $ R = $ 0.8, as a function of $ m_{1,2} $ to fixedorder theory calculations at NNLO, shown here for three inner $ y_{\text{max}} $ regions. The figure description corresponds to that of Fig. 9. 
png pdf 
Figure 25c:
Comparison of the 2D dijet cross section for jets with $ R = $ 0.4 (left) and $ R = $ 0.8 (right) as a function of $ m_{1,2} $ to fixedorder theory calculations at NNLO, shown here for three inner $ y_{\text{max}} $ regions. The figure description corresponds to that of Fig. 9. c 
png pdf 
Figure 25d:
Comparison of the 2D dijet cross section for jets with $ R = $ 0.4 (left) and $ R = $ 0.8 (right) as a function of $ m_{1,2} $ to fixedorder theory calculations at NNLO, shown here for three inner $ y_{\text{max}} $ regions. The figure description corresponds to that of Fig. 9. d 
png pdf 
Figure 25e:
Comparison of the 2D dijet cross section for jets with $ R = $ 0.4 (left) and $ R = $ 0.8 (right) as a function of $ m_{1,2} $ to fixedorder theory calculations at NNLO, shown here for three inner $ y_{\text{max}} $ regions. The figure description corresponds to that of Fig. 9. e 
png pdf 
Figure 25f:
Comparison of the 2D dijet cross section for jets with $ R = $ 0.4 (left) and $ R = $ 0.8 (right) as a function of $ m_{1,2} $ to fixedorder theory calculations at NNLO, shown here for three inner $ y_{\text{max}} $ regions. The figure description corresponds to that of Fig. 9. f 
png pdf 
Figure 26:
(continuation of Fig. 25) Comparison of the 2D dijet cross section for jets with $ R = $ 0.4 (left) and $ R = $ 0.8 (right) as a function of $ m_{1,2} $ to fixedorder theory calculations at NNLO, shown here for two outermost $ y_{\text{max}} $ regions. The figure description corresponds to that of Fig. 9. 
png pdf 
Figure 26a:
(continuation of Fig. 25) Comparison of the 2D dijet cross section for jets with $ R = $ 0.4, as a function of $ m_{1,2} $ to fixedorder theory calculations at NNLO, shown here for two outermost $ y_{\text{max}} $ regions. The figure description corresponds to that of Fig. 9. 
png pdf 
Figure 26b:
(continuation of Fig. 25) Comparison of the 2D dijet cross section for jets with $ R = $ 0.8, as a function of $ m_{1,2} $ to fixedorder theory calculations at NNLO, shown here for two outermost $ y_{\text{max}} $ regions. The figure description corresponds to that of Fig. 9. 
png pdf 
Figure 26c:
(continuation of Fig. 25) Comparison of the 2D dijet cross section for jets with $ R = $ 0.4 (left) and $ R = $ 0.8 (right) as a function of $ m_{1,2} $ to fixedorder theory calculations at NNLO, shown here for two outermost $ y_{\text{max}} $ regions. The figure description corresponds to that of Fig. 9. c 
png pdf 
Figure 26d:
(continuation of Fig. 25) Comparison of the 2D dijet cross section for jets with $ R = $ 0.4 (left) and $ R = $ 0.8 (right) as a function of $ m_{1,2} $ to fixedorder theory calculations at NNLO, shown here for two outermost $ y_{\text{max}} $ regions. The figure description corresponds to that of Fig. 9. d 
png pdf 
Figure 27:
Comparison of the 3D dijet cross section as a function of $ m_{1,2} $ to fixedorder theory calculations at NNLO, using jets with $ R = $ 0.4 (left) and $ R = $ 0.8 (right), in six out of the total 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The figure description corresponds to that of Fig. 10. 
png pdf 
Figure 27a:
Comparison of the 3D dijet cross section as a function of $ m_{1,2} $ to fixedorder theory calculations at NNLO, using jets with $ R = $ 0.4, in six out of the total 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The figure description corresponds to that of Fig. 10. 
png pdf 
Figure 27b:
Comparison of the 3D dijet cross section as a function of $ m_{1,2} $ to fixedorder theory calculations at NNLO, using jets with $ R = $ 0.8, in six out of the total 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The figure description corresponds to that of Fig. 10. 
png pdf 
Figure 28:
(continuation of Fig. 27) Comparison of the 3D dijet cross section as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theory calculations at NNLO, using jets with $ R = $ 0.4 (left) and $ R = $ 0.8 (right), in the remaining nine out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The figure description corresponds to that of Fig. 10. 
png pdf 
Figure 28a:
(continuation of Fig. 27) Comparison of the 3D dijet cross section as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theory calculations at NNLO, using jets with $ R = $ 0.4, in the remaining nine out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The figure description corresponds to that of Fig. 10. 
png pdf 
Figure 28b:
(continuation of Fig. 27) Comparison of the 3D dijet cross section as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theory calculations at NNLO, using jets with $ R = $ 0.8, in the remaining nine out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The figure description corresponds to that of Fig. 10. 
png pdf 
Figure 29:
Comparison of the 3D dijet cross section as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theory calculations at NNLO, using jets with $ R = $ 0.4 (left) and $ R = $ 0.8 (right), in six out of the total 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The figure description corresponds to that of Fig. 10. 
png pdf 
Figure 29a:
Comparison of the 3D dijet cross section as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theory calculations at NNLO, using jets with $ R = $ 0.4, in six out of the total 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The figure description corresponds to that of Fig. 10. 
png pdf 
Figure 29b:
Comparison of the 3D dijet cross section as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theory calculations at NNLO, using jets with $ R = $ 0.8, in six out of the total 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The figure description corresponds to that of Fig. 10. 
png pdf 
Figure 30:
(continuation of Fig. 29) Comparison of the 3D dijet cross section as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theory calculations at NNLO, using jets with $ R = $ 0.4 (left) and $ R = $ 0.8 (right), in the remaining nine out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The figure description corresponds to that of Fig. 10. 
png pdf 
Figure 30a:
(continuation of Fig. 29) Comparison of the 3D dijet cross section as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theory calculations at NNLO, using jets with $ R = $ 0.4, in the remaining nine out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The figure description corresponds to that of Fig. 10. 
png pdf 
Figure 30b:
(continuation of Fig. 29) Comparison of the 3D dijet cross section as a function of $ \langle p_{\mathrm{T}} \rangle_{1,2} $ to fixedorder theory calculations at NNLO, using jets with $ R = $ 0.8, in the remaining nine out of 15 $ {(y^{*}\!, y_{\text{b}})} $ bins. The figure description corresponds to that of Fig. 10. 
Tables  
png pdf 
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. 
png pdf 
Table 2:
PDF parametrizations obtained after the $ \chi^2 $ scan for the fits of the HERA DIS data together with the CMS 2D or 3D dijet measurements. 
png pdf 
Table 3:
Goodnessoffit values for the fits to the HERA DIS data alone and together with the CMS 2D dijet measurement, using the final PDF parametrization for the 2D fits. 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. 
png pdf 
Table 4:
Goodnessoffit values for the fits to the HERA DIS data alone and together with the CMS 3D dijet measurement, using the final PDF parametrization for the 3D fits. 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. 
png pdf 
Table 5:
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_\text{c} $ and thus cannot be used directly for the uncertainty estimation. Following Ref. [56], the results obtained for the opposite variation are symmetrized in these cases. 
Summary 
Dijet production cross sections are presented doubledifferentially as a function of the dijet invariant mass $ m_{1,2} $ in five regions of the maximal absolute rapidity $ y_{\text{max}} $\ of the two leading$ p_{\mathrm{T}} $ jets, and tripledifferentially 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}} $. The latter correspond to the rapidity separation of the two jets, and the total boost of the dijet system, respectively. All measurements are performed for jets clustered using the anti$ k_{\mathrm{T}} $ jet algorithm with distance parameters $ R= $ 0.4 and $ R= $ 0.8 and the cross sections are unfolded in all measurement dimensions simultaneously to correct for detector effects. The data are compared to fixedorder calculations at nexttonexttoleading order in perturbative QCD, supplemented with electroweak and nonperturbative corrections. The theory is observed to describe the data better for the larger jet radius $ R = $ 0.8. Focusing on the larger jet radius and the dijet invariant mass $ m_{1,2} $ as an observable, the parton distribution functions (PDFs) of the proton are determined in fits to the dijet measurements together with deepinelastic scattering data from the HERA experiments following the approach described in earlier HERAPDF analyses [54,55,56]. The inclusion of the presented measurements leads to an improved determination of the PDFs compared to fits to HERA data alone. In particular, the uncertainty in the gluon PDF at medium to large fractional momenta $ x $ of the proton is reduced. The results obtained from the double and tripledifferential measurements are shown to be compatible within the estimated uncertainties. Fits including the strong coupling constant $ \alpha_{\mathrm{s}}(m_{\mathrm{Z}}) $ as a free parameter yield consistent results between the double and tripledifferential dijet measurements, where the latter gives the slightly more precise value of $ \alpha_{\mathrm{s}}(m_{\mathrm{Z}}) = $ 0.1201 $ \pm $ 0.0020 at NNLO\@. The impact of subleadingcolor contributions to the leadingcolor NNLO calculation used here is not known yet [34]. Apart from being useful input to PDF fits or studies of jet size dependence, the presented double and tripledifferential measurements for two jet size parameters $ R= $ 0.4 and $ R= $ 0.8, and for the two dijet observables invariant mass $ m_{1,2} $ and average transverse momentum $ \langle p_{\mathrm{T}} \rangle_{1,2} $ provide an ideal testing ground for further investigations. 
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