Impact of the parton distribution function uncertainties on the measurement of the $W$ boson mass at the Tevatron and the LHC

We study at a quantitative level the impact of the uncertainties on the value of the $W$ boson mass measured at hadron colliders due to: i) the proton parton distribution functions (PDFs), ii) the value of the strong coupling constant ${\alpha }_s$ and iii) the value of the charm mass used in the PDF determination. The value of the $W$ boson mass is extracted, by means of a template fit technique, from the lepton-pair transverse mass distribution measured in the charged current Drell-Yan process. We study the determination of $m_W$ at the Tevatron and at the LHC with 7 and 14 TeV of center-of-mass energy in a realistic experimental setup. The analysis has been done at the Born level using the event generator HORACE and at NLO-QCD using the event generators DYNNLO and ResBos. We consider the three global PDF sets, CTEQ6.6, MSTW2008, and NNPDF2.1. We estimate that the total PDF uncertainty on $m_W$ is below 10 MeV both at the Tevatron and at the LHC for all energies and final states. We conclude that PDF uncertainties do not challenge a measurement of the $W$ boson mass at the level of 10 MeV accuracy.

Figure 1

Left plot: the transverse mass distributions at the Tevatron at Born level obtained with two values of $m_W$ which differ by 10 MeV. Right plot: the bin by bin ratio of these two distributions.

Figure 2

Flowchart that summarizes the procedure used to determine the shift in $m_W$ induced by any given PDF set. More details are provided in the text.

Figure 3

${\chi }^2$ distributions obtained fitting Born level pseudodata with Born level templates for the same fixed PDF set at the Tevatron kinematics. The different curves correspond to different pseudodata samples each with different statistics. The $\Delta {\chi }^2=1$ rule indicates the resolution, at 68% C.L., on the $W$ mass.

Figure 4

Relative PDF uncertainties in the Born level transverse mass distributions, computed with respect the respective central PDF set. From top to bottom: LHC 7 TeV $W^{+}$ and $W^{-}$ and LHC 14 TeV $W^{+}$ and $W^{-}$. Both the PDF uncertainties on the standard distribution, Eq. (4), and the normalized distribution, Eq. (5), are shown.

Figure 5

Same as Fig. 4 for the Tevatron.

Figure 6

Relative difference between the distributions obtained with the central PDF set of CTEQ6.6, MSTW08 and NNPDF2.1, normalized to the CTEQ6.6 result. We show the results both for the normalized and for the standard distributions for LHC 7 TeV (upper plots) and 14 TeV (lower plots).

Figure 7

Same as Fig. 6 for the Tevatron.

Figure 8

Same as Fig. 4 but now the transverse mass distributions have been computed at NLO-QCD using the event generator DYNNLO.

Figure 9

Same as Fig. 5 but now the transverse mass distributions have been computed at NLO-QCD using the event generator DYNNLO.

Figure 10

The relative PDF uncertainty in the standard transverse mass distributions for NNPDF2.1 for different colliders, energy, and final states.

Figure 11

Comparison of the PDF–only uncertainty and the combined $\mathrm{PDF}+{\alpha }_s$ uncertainty of the transverse mass distribution for NNPDF2.1. For simplicity we show only the distributions at the LHC 7 TeV, the distributions for Tevatron and LHC 14 TeV are quantitatively very similar.

Figure 12

For NNPDF2.1 we show the dependence on $m_c$ of the transverse mass distribution, expressed as relative deviation from the central NNPDF2.1 set with $m_c^2=2{\mathrm{GeV}}^2$. We show results both for the normalized and for the standard transverse mass distributions. We consider only the LHC 14 TeV case, where charm mass effects are known to be more important.

Figure 13

Graphical representation of the results of Table  for the various colliders and final states considered. In each case we draw the envelope of the results from the PDF sets to define the total PDF uncertainty ($\pm {\delta }_{\mathrm{pdf}}^{\mathrm{tot}}$) as a thick solid line. The dashed line marks the position of the nominal value $m_W^0=80.398\mathrm{GeV}$ used to generate the pseudodata.

Figure 14

The distribution of the 100 values of $m_W$ obtained from each of the 100 replicas of the NNPDF2.1 analysis at the LHC 7 TeV, compared to the results of reweighting NNPDF2.1 with the recent ATLAS and CMS data on $W$ lepton asymmetries (left plot) and by the reweighting of $W$ lepton asymmetries pseudodata at the same kinematics than the published LHC data but assuming with a 1% total experimental uncertainty (right plot). In each case, the spread of the distributions represents the PDF uncertainty. See text for more details.