Precision measurement of the $W$-boson mass: Theoretical contributions and uncertainties

We perform a comprehensive analysis of electroweak, QED, and mixed QCD-electroweak corrections underlying the precise measurement of the $W$-boson mass $M_W$ at hadron colliders. By applying a template fitting technique, we detail the impact on $M_W$ of next-to-leading order electroweak and QCD corrections, multiple photon emission, lepton pair radiation, and factorizable QCD-electroweak contributions. As a by-product, we provide an up-to-date estimate of the main theoretical uncertainties of perturbative nature. Our results can serve as a guideline for the assessment of the theoretical systematics at the Tevatron and LHC and allow a more robust precision measurement of the $W$-boson mass at hadron colliders.

Figure 1

Relative effect due to lepton-pair corrections on the $W$ transverse mass distribution, for $W\to \mu \nu$ (left plot) and $W\to e\nu$ (right plot) decays at the Tevatron ($\sqrt{s}=1.96\mathrm{TeV}$). The plots show the relative difference between the Horace-3.1 predictions for multiple FSR with and without pair emission.

Figure 2

Relative difference, for the ${\mu }^{+}\nu$ invariant mass distribution, normalized to the prediction of Powheg-v2 with NLO QCD corrections interfaced to the Pythia QCD PS, of two different implementations of EW corrections: predictions of Powheg-v2 two-rad with NLO $\mathrm{QCD}+\mathrm{EW}$ corrections (red dots), and of the old version of Powheg-v2 with NLO $\mathrm{QCD}+\mathrm{EW}$ corrections (blue dots). Both codes are interfaced to the Pythia QCD PS and to Photos. For reference, the EW corrections normalized to the LO are reported in the insets.

Figure 3

Same as in Fig. 2 for the lepton-pair transverse mass (left plot) and for the lepton transverse momentum (right plot).

Figure 4

Fixed-order predictions for the transverse mass distribution, in the case of $W^{+}$ production with muons in the final state at LHC 14 TeV and acceptance cuts as in Table 7. We show different perturbative approximations, including only NLO QCD, only NLO EW, and the sum of the two sets of corrections. In the left plot we show the shape of the distributions and in the right plot the relative effect of the radiative corrections, normalized to the LO prediction.

Figure 5

Relative contribution of higher-order FSR and higher-order FSR plus pair radiation normalized to one-photon emission, for $W^{+}$ decay into muons (left) and into bare electrons (right). Predictions from Horace-3.1 at Tevatron energy, with the acceptance cuts of Table 3.

Figure 6

Shape of the distribution of the relative lepton-photon transverse momentum $p_T^{\ell \gamma }$ for the decays $W^{+}\to {\mu }^{+}\nu$ (left plots) and $W^{+}\to e^{+}\nu$ (right plots) at the LHC 14 TeV, with acceptance cuts as in Table 12. In the upper plots we show the comparison of the results obtained interfacing Powheg-v2 NLO QCD results with Pythia-qed and Photos. In the lower plots we show the comparison of the results obtained with Horace including QED FSR effects and those obtained interfacing Horace LO with Photos, in both cases without QCD corrections.

Figure 7

Upper plots: lepton-pair transverse mass (left plots) and lepton transverse momentum (right plots) distributions in different approximations: without QCD corrections (Horace LO and Horace with QED FSR PS) and with QCD corrections (Powheg-v2 NLO $\mathrm{QCD}+\mathrm{QCD}$ PS and Powheg-v2 NLO $\mathrm{QCD}+\mathrm{QCD}$ PS interfaced to Photos) for the decay $W^{+}\to {\mu }^{+}\nu$ at the LHC 14 TeV, with acceptance cuts as in Table 12. Lower plots: relative contribution of QED FSR normalized to the LO predictions and of QED $\mathrm{FSR}+\text{mixed}$ QCD-QED corrections normalized to the Powheg-v2 NLO $\mathrm{QCD}+\mathrm{QCD}$ PS predictions.

Figure 8

Upper plots: lepton-pair transverse mass (left plots) and lepton transverse momentum (right plots) distributions, for the decay $W^{+}\to {\mu }^{+}\nu$ at the LHC 14 TeV, with acceptance cuts as in Table 12, according to three different approximations: Powheg-v2 with only QCD corrections, Powheg-v2 with only QCD corrections interfaced to Photos and Powheg-v2 two-rad with NLO ($\mathrm{QCD}+\text{EW}$) corrections matched to Photos. Lower plots: relative contribution of QED $\mathrm{FSR}+\text{mixed}$ QCD-QED effects of the ${\mathrm{QCD}}_{\mathrm{NLOPS}}\times {\mathrm{QED}}_{\mathrm{PS}}$ approximation and of QED $\mathrm{FSR}+\text{mixed}$ QCD-EW corrections of the ${\mathrm{QCD}}_{\mathrm{NLOPS}}\times {\mathrm{EW}}_{\mathrm{NLOPS}}$ approximation, both normalized to the Powheg-v2 with only QCD corrections results.

Figure 9

Comparison of different approximations of the ${\mu }^{+}\nu$ invariant mass distribution simulated at the LHC 14 TeV with acceptance cuts as in Table 12. Relative difference of the predictions obtained with Photos compared to the ones with Pythia-qed as tools to simulate QED FSR effects. The comparison is based on results computed with Powheg-v2 with only QCD corrections interfaced to Photos or Pythia-qed (black dots) and on results computed with Powheg-v2 two-rad with NLO ($\mathrm{QCD}+\mathrm{EW}$) corrections, matched to Photos or Pythia-qed (blue dots).

Figure 10

Same as in Fig. 9 for the lepton-pair transverse mass (left plot) and for the lepton transverse momentum (right plot) distributions.