$W$-Boson Mass, Electroweak Precision Tests and SMEFT

The discovery of the Higgs boson Aad et al. (2012); Chatrchyan et al. (2012) marks a great success of the Standard Model (SM) in particle physics. It opens the door to explore mass origin of the massive elementary particles and the underlying physics for electroweak symmetry breaking (EWSB), and thus has far-reaching impacts for the development of particle physics. The measurements of the $W$ boson can also play a significant role in this direction. As a direct consequence of EWSB, the $W$-boson mass ($m_{W}$) could be determined by the $Z$-boson mass ($m_{Z}$) and the Weinberg angle ($\theta_{W}$). Thanks to the high precision achieved in electroweak precision tests (EWPTs), $m_{W}$ has been well-constrained in the SM. We denote this measured value as $m_{W}^{\rm EW}$. Then any deviation from $m_{W}^{\rm EW}$ in the direct measurements of $m_{W}$ may serve as a signal of physics beyond the SM (BSM).

Recently, the Collider-Detector-at-Fermilab (CDF) collaboration at Tevatron reported its updated measurement of $m_{W}$ Aaltonen et al. (2022a). With $\sim 4.24\times 10^{6}$ semileptonic $W$-decay events, this measurement yields $m_{W}=80.4335\pm 0.0094$ GeV Aaltonen et al. (2022a). Here a combination of the almost quadrupled statistics and the improvements in systematic effects such as parton distribution function and lepton resolution leads to a greatly reduced uncertainty of $\sim 9.4$ MeV. This precision is approximately halved compared to that of the previous CDF measurement Aaltonen et al. (2012) and also exceeds the current experimental average ($\sim 12$ MeV) Zyla et al. (2020). Such a $W$-boson mass results in a tension of $\sim 7\sigma$ with the latest EW global fit of $m_{W}^{\rm EW}=80.3545\pm 0.0059$ GeV de Blas et al. (2021). Caution should be taken when we attend to make an interpretation of this tension. There is a significant tension between the new CDF result and the direct measurements performed by the D0 collaboration at the Tevatron Abazov et al. (2012) and the ATLAS/LHCb collaborations at the Large Hadron Collider (LHC) Aaboud et al. (2018); Aaij et al. (2022), while the latter ones are in good agreement with $m_{W}^{\rm EW}$. This indicates that a better understanding of the uncertainties in these measurements is needed, to ensure a solid interpretation of the data. Notably, dedicated (HL-)LHC runs with a low instantaneous luminosity and upgraded detectors may also measure $m_{W}$ with a precision $\lesssim 10$ MeV ATL (2018a); Azzi et al. (2019), falsifying or strengthening the new CDF result.

With these subtleties in mind, we would explore the potential origin of this discrepancy from the BSM physics. For this purpose, we will simply assume that the direct measurements of $m_{W}$ will later converge to a value close to the CDF one Aaltonen et al. (2022a). It is already known that the BSM scenarios exist where the $m_{W}$ value can deviate from the SM prediction without heavily disturbing other electroweak precision observables (EWPOs) Berthier and Trott (2015); Bjørn and Trott (2016); Corbett et al. (2021). The examples include the singlet López-Val and Robens (2014) or doublet Lopez-Val and Sola (2013) extensions of the SM Higgs sector where the corrections to $m_{W}$ enter at one-loop level. A significant correction to $m_{W}$ can be also achieved in supersymmetry with the sfermion-loop contributions Heinemeyer et al. (2013); Diessner and Weiglein (2019); Bagnaschi et al. (2022). Moreover, a positive correction to $m_{W}$ is possible via a tree-level mixing between the SM $Z$ boson and the BSM “dark photon” Curtin et al. (2015); Algueró et al. (2022). In composite Higgs theories, $m_{W}$ can be shifted away from its SM prediction due to the UV-scale suppressed interactions Bellazzini et al. (2014).

In this paper, we will try to explore this discrepancy in the framework of the SM effective field theory (SMEFT) and make a prediction on its impacts for the ongoing and upcoming measurements at colliders. In the SMEFT, we parametrize the leading effects of the BSM physics above the EW scale with a set of dimension-six (6D) operators

where $\mathcal{L}_{\text{SM}}$ defines the SM, and $c_{i}$ and $\Lambda$ denote dimensionless Wilson coefficients and the cutoff scale of the BSM physics, respectively. Instead of presenting a comprehensive study on the whole set of 6D operators, we will focus on the key ones which are closely related to the $W$-boson physics. We would view this as a first step for more refined analyses later.

We organize this paper in the following way. We will introduce the analysis formalism in Section II, which was originally developed in Chiu et al. (2018). The analysis results of the SMEFT and the predictions for the ongoing and future tests will be presented in Section III. We will conclude in Section IV.

The observables to be considered in this paper are summarized in Tab. 1, which include $m_{W}$, the EWPOs and one Higgs observable. Below is the list of the EWPOs:

Here $\Gamma_{f}=\Gamma(Z\rightarrow f\bar{f})$, $\Gamma_{\rm had}=\Gamma(Z\to\text{hadrons})$ and $\Gamma_{Z}$ are the partial and total decay widths of the $Z$ boson. $g_{V(A)}^{f}$ is the (axial-)vector couplings of fermion $f$. $\Gamma_{W}$ and $\text{BR}_{W\to{\rm had}}$ denote the total decay width and branching fraction of hadronic decays of $W$ boson. Note that we take the experimental average at the LHC and Tevatron as the value of $\sin^{2}\theta_{\text{eff}}^{\text{lep}}$, which is thus uncorrelated with the asymmetry observables ($A$’s) measured at LEP.

Totally six CP-even 6D operators will be considered in this study, which include

Here we work in the Warsaw basis Grzadkowski et al. (2010).¹¹1Our $\mathcal{O}_{T}$ is slightly different from $\mathcal{O}_{HD}=(H^{\dagger}D_{\mu}H)^{*}(H^{\dagger}D_{\mu}H)$ in the original Warsaw basis. They are related as $\mathcal{O}_{T}=-2\mathcal{O}_{HD}-\frac{1}{2}\partial_{\mu}\left(H^{\dagger}H\right)\partial^{\mu}\left(H^{\dagger}H\right)$. These operators can affect the EW physics by renormalizing wave functions, shifting the definition of EW parameters and the SM couplings, and introducing new vertices. We will follow the linear formalism developed in Chiu et al. (2018) to analyze these effects. Strictly speaking, there are additional CP-even 6D operators that could affect EWPTs such as Higgs coupling to quarks Ellis et al. (2021). Yet our choice already suffices for the leading-order discussions on the potential BSM physics to explain the new CDF result, and explore other possible measurable consequences.

To derive the relative shifts caused by these operators in the observables listed Eq. (II), we need three basic input parameters: the $Z$-boson mass $m_{Z}$, the Fermi constant $G_{F}$, and the fine-structure constant $\alpha^{-1}(m_{Z})$. Their experimental values and uncertainties are presented in Tab. 1. For the other input parameters that contribute to the EW predictions in the SM at one or higher-loop levels, namely $m_{h}$, $m_{t}$, $\alpha_{s}(m_{Z})$, their values are taken to be the same as those in de Blas et al. (2021). Following the procedures described in Fan et al. (2015); d’Enterria and Srebre (2016); Dubovyk et al. (2018) with the updated parameter values, we calculate the SM values for the EWPOs which agree with de Blas et al. (2021) decently, with a difference $\lesssim 1\sigma$. As for theoretical uncertainties, only the one for $m_{W}$ is relevant. The overall uncertainty of $\sim 5.9$ MeV for the $m_{W}$ prediction is evenly contributed by the uncertainty of $m_{t}$ and the high-order corrections. By taking this effect into consideration, the predicted uncertainties for these observables also match well with de Blas et al. (2021).

To show how the 6D operators affect these EWPOs, let us consider $m_{W}$ as an example. The $W$-boson mass receives the contributions from these operators via the EW-parameter shifts only, which yields

Here $\theta_{W}$ and $g_{Z}$ are defined as

where $g$ is the SM $SU(2)$ coupling. Then we have

The parameter shifts depend on the Wilson coefficients of the 6D operators as

where $v=246$GeV is the SM Higgs VEV. $\delta Z_{Z}$ and $\delta Z_{A}$ are field renormalization factors induced by $\mathcal{O}_{WB}$. They are given by

where $g^{\prime}$ is the SM $U(1)_{Y}$ coupling. Combining these derivations, one will find the dependence of $m_{W}$ on the Wilson coefficients of these 6D operators at linear level. Generalizing this discussion, we have

for all EWPOs. The coefficients $C_{\mathcal{O}}$’s are summarized in Tab. 2.

Notably, the main EWPOs are sensitive to four linear combinations of the six Wilson coefficients only: $\xi_{0}=-1.1c_{WB}+2c_{T}-4c^{(3)l}_{L}+4c^{(3)l}_{LL}$, $\xi_{\pm}=c_{L}^{(3)l}\pm c^{l}_{L}$ and $c_{R}^{e}$. Explicitly, $\sigma_{\rm had}$ is sensitive to all of them; $A_{b}$ and $R_{b}$ depend on $\xi_{0}$ only; $A_{\text{FB}}^{b,\ell}$, $A_{\ell}$, and $\sin^{2}\theta_{\text{eff}}^{\text{lep}}$ have the same dependence on $\xi_{0}$, $\xi_{+}$ and $c_{R}^{e}$; and $R_{\ell}$ depends on $\xi_{0}$, $\xi_{+}$ and $c_{R}^{e}$ uniquely. This necessarily leaves two (approximately) runaway directions. These directions could be partly lifted by $\Gamma_{Z}$ and the three $W$ observables, which have different dependences on the variables beyond $\xi_{0,\pm}$ and $c_{R}^{e}$. In addition, we will include the relative Higgs diphoton signal strength in the gluon fusion channel, namely $\mu_{ggh}^{\gamma\gamma}/\mu_{\rm SM}$, to enhance the constraint on $\mathcal{O}_{WB}$. Only the ATLAS measurements Rossi (2022) will be considered here. The CMS results are similar CMS (2020) and combining them will not qualitatively change the conclusion. With this set of observables, all runaway directions can be lifted, which we will discuss more in next section.

In this paper, we plan to study the potential to further test the relevant SMEFT at future circular $e^{-}e^{+}$ colliders such as FCC-ee Abada et al. (2019) and CEPC Dong et al. (2018) also. These machines, with extraordinary integrated luminosities and state-of-the-art detector technologies, would provide unprecedented precisions for measuring $m_{W}$ and other EWPOs. For concreteness, we consider the ones at FCC-ee Abada et al. (2019) and list them in Table 1 also. These projected precisions receive contributions from both the systematic and statistical uncertainties. The projected $m_{W}$ precision goes below $0.5$ MeV, improved by more than one order of magnitude compared to the CDF measurement, thanks to the dedicated beam energy scanning around the $W^{+}W^{-}$ threshold. The other EWPTs would also be improved substantially at future $e^{-}e^{+}$ colliders. Notably, we will use BR($h\to\gamma\gamma$)/BR($h\to\gamma\gamma$)_SM to replace $\mu_{ggh}^{\gamma\gamma}/\mu_{\rm SM}$ here as they are mutually equivalent given the six SMEFT operators in Eq. (II) while the latter is not directly measurable at $e^{-}e^{+}$ colliders. Such a choice also allows us to combine the HL-LHC and future $e^{-}e^{+}$ collider data for a further precision improvement Fan et al. (2015).

At a future $e^{-}e^{+}$ collider, the theoretical uncertainties for the considered EWPOs become more relevant. Although not playing a significant role at the current stage, these theoretical uncertainties may become comparable to the experimental ones as the latter are expected to be improved more by the time of running a future $e^{-}e^{+}$ collider. For most of the relevant EWPOs, the major factors determining their theoretical uncertainties by that time, namely the expected input parameter precisions and the intrinsic uncertainties in relation to higher-order corrections, have been discussed in d’Enterria and Srebre (2016); Freitas et al. (2019). The exceptions are $\sigma_{\rm had}^{0}$, $\Gamma_{W}$ and BR_W→had. For these observables, no future projections for their intrinsic uncertainties from the higher-order corrections are available. So we adopt a conservative assumption that their future values will be reduced by half compared to current ones. The projected theoretical uncertainties are summarized in Table 1.

Our paper explores the possible new physics origin of the recent CDF $m_{W}$ measurement, focusing on alleviating the tension between the CDF result and EWPTs. We carry out a model-independent analysis using a subset of 6D operators which EWPOs are most sensitive to, in the SMEFT framework. We implement three different fits and show that to accommodate the CDF result, new physics has to generate $\mathcal{O}_{T}=\frac{1}{2}(H^{\dagger}\overset{\text{$\leftrightarrow$}}{D}_{\mu}H)^{2}$ with a coefficient $c_{T}({\rm TeV}/\Lambda)^{2}\gtrsim 0.01$. This suggests a new energy scale of multiple TeV for tree-level effects and sub TeV for loop-level effects, which could be searched for either through direct searches or other indirect probes.

While the CDF result clearly needs to be cross-checked with the upcoming LHC measurements, it certainly urges the particle physics community to prepare better for understanding the potential physical cause of the anomalies from indirect precision measurements. This could be crucial for the planning of future collider projects. As we have shown, a future $e^{-}e^{+}$ collider covering the $Z$-pole, $WW$ threshold, Higgs factory, and $t\bar{t}$ threshold modes could be highly relevant or even decisive in this regard.