A first determination of the strong coupling α S at approximate N 3 LO order in a global PDF fit

Abstract

We present the first determination of the value of the strong coupling via a simultaneous global fit of the proton parton distribution functions (PDFs) at approximate ${\text{N}}^3$ LO (a ${\text{N}}^3$ LO) order in QCD. This makes use of the MSHT global PDF fitting framework, and in particular the recent theoretical advances t l;;mhat allow a PDF fit to now be performed at this order. The value of the strong coupling is found to be ${\alpha }_S\left(M_Z^2\right)\left({\mathrm{aN}}^3\mathrm{LO}\right)=0.1170\pm 0.0016$ . This is in excellent agreement with the NNLO value of ${\alpha }_S\left(M_Z^2\right)\left(\text{NNLO}\right)=0.1171\pm 0.0014$ , indicating that good perturbative convergence has been found. The resulting uncertainties, calculated using the MSHT dynamic tolerance procedure, are somewhat larger, but more accurate, at a ${\text{N}}^3$ LO, due to the missing higher order theoretical uncertainties that are included at this order, but not at NNLO. We in addition present a detailed breakdown of the individual dataset sensitivity to the value of the strong coupling, with special focus on the impact of fitting dijet rather than inclusive jet data. This choice is found to have a non-negligible impact, but with overall good consistency found, especially at a ${\text{N}}^3$ LO.

Fig. 1

The ${\chi }^2$ profile for the NNLO (left) and aN$_{}^3$LO (right) PDF fits as ${\alpha }_S(M_Z^2)$ is scanned from 0.112 to 0.122. The best fit ${\alpha }_S(M_Z^2)$ value, best fit ${\chi }^2$ and number of datapoints are given on the figures

Fig. 2

The individual datasets $\mathrm{\Delta }{\chi }^2={\chi }^2-{\chi }_0^2$ for different values of ${\alpha }_S(M_Z^2)$, within the global PDF fits at NNLO (red) and aN$_{}^3$LO (green). This figure provides the profiles with ${\alpha }_S(M_Z^2)$ for a small selection of the fixed target deep inelastic scattering datasets included

Fig. 3

The individual datasets $\mathrm{\Delta }{\chi }^2={\chi }^2-{\chi }_0^2$ for different values of ${\alpha }_S(M_Z^2)$, within the global PDF fits at NNLO (red) and aN$_{}^3$LO (green). This figure provides the profiles with ${\alpha }_S(M_Z^2)$ for a small selection of the collider Drell–Yan datasets included

Fig. 4

The individual datasets $\mathrm{\Delta }{\chi }^2={\chi }^2-{\chi }_0^2$ for different values of ${\alpha }_S(M_Z^2)$, within the global PDF fits at NNLO (red) and aN$_{}^3$LO (green). This figure provides the profiles with ${\alpha }_S(M_Z^2)$ for a small selection of the collider inclusive jet datasets included

Fig. 5

The individual datasets $\mathrm{\Delta }{\chi }^2={\chi }^2-{\chi }_0^2$ for different values of ${\alpha }_S(M_Z^2)$, within the global PDF fits at NNLO (red) and aN$_{}^3$LO (green). This figure provides the profiles with ${\alpha }_S(M_Z^2)$ for a small selection of the collider top production datasets included

Fig. 6

The $\mathrm{\Delta }{\chi }^2={\chi }^2-{\chi }_0^2$ for different values of ${\alpha }_S(M_Z^2)$, within the global PDF fit at aN$_{}^3$LO for the ATLAS 8 TeV Z $p_T$ dataset. In addition the $\mathrm{\Delta }{\chi }^2=1,T^2$ limits are marked, the latter correspond to the bounds these data provide on ${\alpha }_S(M_Z^2)$ in the MSHT global fit approach with a dynamic tolerance

Fig. 7

The overall and dataset by dataset best fit value and upper and lower bounds on ${\alpha }_S(M_Z^2)$, for a selection of the datasets in the global fit. The overall upper and lower bounds are given by the horizontal dashed lines, whilst the coloured vertical solid lines show the individual dataset bounds. The upper plot is the NNLO fit and the lower is the aN$_{}^3$LO fit

Fig. 8

Posterior variations of the qg and gg splitting functions at the best fit value of ${\alpha }_S(M_Z^2)=0.117$, as well as for ${\alpha }_S(M_Z^2)=0.116$, 0.118 indicated by the dashed red curves. In the most visible part of the plots, i.e. below $x\approx {10}^{-2}$ the lower (upper) curves correspond to the values of 0.116 (0.118). Also shown are the prior and lower order results, for comparison

Fig. 9

The ${\chi }^2$ profile for the LHC jet (left) or dijet (right) data only as ${\alpha }_S(M_Z^2)$ is scanned from 0.112 to 0.122, comparing the PDF fits at NNLO, aN$_{}^3$LO and aN$_{}^3$LO with the K-factors fixed at the values corresponding to the global best fit for ${\alpha }_S(M_Z^2)$

Fig. 10

The ${\chi }^2$ profile for the NNLO (left) and aN$_{}^3$LO (right) PDF fits as ${\alpha }_S(M_Z^2)$ is scanned from 0.112 to 0.122, comparing the global total ${\chi }^2$ profiles for the LHC inclusive jets fit with that including instead the dijets data. The aN$_{}^3$LO plot also shows the effect of fixing the fitted aN$_{}^3$LO k-factors to the values corresponding to the global best fit for ${\alpha }_S(M_Z^2)$

Fig. 11

The impact of varying ${\alpha }_S(M_Z^2)$ in the aN$_{}^3$LO PDF fit on (left) the extracted gluon and (right) total singlet PDFs. This demonstrates the correlations between the PDFs and ${\alpha }_S(M_Z^2)$

Fig. 12

Cross section uncertainties for gluon fusion Higgs, Z, and $W^{\pm }$ production at $\sqrt{s}=14$ TeV at (left) NNLO with the MSHT20nnlo PDFs and (right) N$_{}^3$LO with the MSHT20aN3LO PDFs. The blue bars are the PDF uncertainties, the green represent the ${\alpha }_S$ uncertainty, and the red bars are the combined PDF+${\alpha }_S$ uncertainty, added in quadrature