In April this year the CDF collaboration shocked the particle physics world with a new measurement of the mass of the W boson particle , with a precision better than all previous results combined, and a value of the W mass higher than the prediction of the Standard Model by 7 times the uncertainty. If confirmed, this would be the clearest deviation from the Standard Model yet, to be added to a list of cracks that have appeared recently, such as the indications of lepton flavour violation reported by LHCb  and the deviation in the muon electromagnetic moment confirmed last year by the Muon g-2 collaboration .
The W and Z bosons play a central role in the theory of the Electroweak interactions, as the carriers of the weak nuclear force. Their properties are related to fundamental symmetries of the theory, and while the masses are not precisely fixed, their relations with other observables in the theory are fixed. This means that if enough other properties are measured, a precise prediction can be obtained, which can then be compared to a direct experimental measurement.
Measuring the masses of these bosons directly is therefore crucial to test the self-consistency of the theory we call the Standard Model of particle physics. In stark contrast to the measurement of the Z boson mass, the measurement of the W boson mass is a considerably bigger challenge. In its leptonic decay, the Z boson produces two muons or two electrons which can be measured very precisely. The W boson, on the other hand, having an electric charge, decays into only one electron or one muon, plus one neutrino which cannot be experimentally measured directly, but has to be inferred from momentum imbalance. The curious reader can read more  and . But let’s look briefly at the history of the measurement of the W boson mass.
The W and Z bosons have masses of around 80 GeV and 90 GeV, respectively. This means that they could not be produced directly until very powerful accelerators came about. The first accelerator with a center-of-mass energy large enough to directly produce them was the SppS at CERN, which operated from 1981 to 1991 in the same tunnel that is still in use for the SPS. Promptly in early 1983, the first discovery of the W boson was announced by the UA1 collaboration, with the UA2 collaboration following shortly after. Despite the larger mass, the Z boson discovery was announced only a short few months later. Looking more closely at the W boson, the discovery was announced with only 6 (!) events from UA1, a far cry from the hundreds of millions of W boson events that the LHC produced in only its Run-2 operation. In fact, the number of events in the first publication by UA1 was so small that they were given the letters A-F, as seen in Figure 1.
Figure 1: The transverse mass mT of the first six candidate W events recorded by UA1 , labeled A-F, with the horizontal error bar indicating the precision of the mT measurement for each event.
This publication from 1983 also quotes the first measurement of the mass of the W boson, at 81 ± 5 GeV, in very good agreement with the expectations at the time, and also with the current understanding, albeit with an uncertainty that is more than 500 times higher than today.
By the end of the 1980s, UA1 and UA2 had collected hundreds of W events and measured the W and Z boson masses to about 1% precision. The most precise W mass measurement, by UA2, yielded 80.79 ± 0.90 GeV. By this time the CDF experiment at the Tevatron collider at Fermilab also published its first W mass measurement based on 22 recorded W events: 80.0 ± 4.1 GeV.
Besides the impressive increase in precision in a very short amount of time, the importance of the discovery of these two massive bosons cannot be overstated. Having massive bosons in the Standard Model inherently requires a mechanism to give the masses to these bosons. This mechanism is known as the Higgs-Brout-Englert mechanism, which also predicts another (scalar) boson in the Standard Model, namely the Higgs boson which was discovered only some 30 years later by the ATLAS and CMS experiments at CERN. So in many ways, the success of the theoretical predictions of the 60’s and 70's, combined with the formidable performance of the accelerators of the 80’s are responsible for the LHC and its vast physics program in the 21st century.
Fig. 2 A summary of measurements of the W boson mass at the SppS, LEP, Tevatron, and LHC colliders since the first discovery. The upper panel shows the evolution of the precision over the years, the lower panel shows the gradual increase in the number of W boson events collected by the experiments and analysed for measuring the W boson mass (With data from here)
The next level of precision in the measurement of the W boson mass was achieved at the LEP collider at CERN, where four experiments ALEPH, DELHI, L3 and OPAL profited from the very clean e+e- environment. The first phase (LEP1) measured the Z boson properties with unprecedented precision before crossing the energy threshold for W pair production in 1996, allowing precise measurements of the W boson mass at and above threshold. The great advantage of e+e- collisions is that in most cases the events measured in the detector contain only the decay products of the W bosons and nothing else, and that the collision energy is known very precisely. This allows using the conservation of energy and momentum as a powerful constraint to further improve the statistical and systematic precision of the measurements.
Figure 2 shows the evolution of the precision on the W boson mass from the different accelerators and experiments in the upper panel. The lower panel shows how this increase in precision was achieved, namely by a very large increase in the number of W bosons produced by the accelerators. The LEP experiments achieved competitive precision with considerably fewer events due to the relatively favourable experimental conditions for precision measurements in e+e- collisions.
LEP yielded a rich harvest of measurements with extraordinary precision  and firmly established the consistency of the Standard Model, at a level of precision not seen before: quantum loop corrections had to be included in the theoretical predictions to match the experimental observations. With a final combined W boson mass from the four LEP experiments of 80.376 ± 0.033 GeV, it was possible to probe minor effects from top quarks and the hypothetical Higgs boson by their invisible interactions with the W bosons in the vacuum, even before these particles had been observed directly. After the top quark had been observed in 1995 at Fermilab, and its mass constrained with direct measurements, the mass of the elusive Higgs boson could be predicted to fall between 115 and 285 GeV with a confidence level of 95%.
The next leap in our precise understanding of the electroweak sector came with the discovery of the Higgs boson in the predicted mass range at around 125 GeV, by the ATLAS and CMS experiments at the LHC. Immediately the mass of the discovered boson was measured with good precision, very much reducing the remaining uncertainty in the Standard Model predictions, constraining the theoretical prediction of the W boson mass to better than 0.1 per mille: 80’357 ± 6 MeV.
Thanks to the higher collision energy and luminosity at the LHC beams, the LHC experiments quickly collected samples of W bosons comparable or even larger than recorded at the Tevatron. This allowed ATLAS to publish the first W boson mass measurement at the LHC in 2018 : 80’370 ± 7 (stat) ± 17 (syst) MeV, followed by LHCb in 2022 : 80’354 ± 23 (stat) ± 22 (syst) MeV. Both measurements are in good agreement with the Standard Model prediction, as shown in Fig 2. However! These measurements did not come easy.
The difficulty of the measurement of the W boson mass comes from many different factors, both from experimental sources, and from the underlying theory. It is important to note here that an uncertainty of 9 MeV in a measurement of around 80 GeV corresponds to a precision of 1 part in 10’000.
First of all, as outlined above, in the most sensitive leptonic decay channel of the W boson, a neutrino is emitted, which cannot be measured experimentally. This means that the mass of the W boson is not directly experimentally accessible, and has to be estimated from other observables in the detector. Especially at hadron colliders like the Tevatron or the LHC, inferring the W boson mass is usually done by either looking only at the momentum of the charged lepton (electron or muon) from the W boson decay, or via a variable called the transverse mass (mT), which incorporates the missing transverse energy which is proportional to the momentum of the not-measured neutrino. While the experimental precision in the measurement of the charged lepton momenta is high, getting to values of 1 part in 10’000 is still a formidable challenge and takes years for experimentalists to achieve. Furthermore, if the missing transverse momentum is used, the scale and resolution in the missing transverse momentum has to be known to a high degree of accuracy as well. This aspect is comparatively easier at the Tevatron than the LHC, largely due to the design of the detectors and also due to the fact that the LHC has a larger number of collisions per bunch crossing than any accelerator before.
Even after all these experimental ingredients are known to the needed precision, the difficulties in the measurement are not over. Any measurement of the W boson mass relies to some degree on underlying calculations from the theory. This reliance on theoretical calculations stem, for instance, from the fact that protons (or protons plus anti-protons in the case of the Tevatron) are not fundamental particles, but are made of quarks and gluons called partons. In a given proton-(anti-)proton collision, one parton from each of these (anti-)protons collide to produce the W boson, as illustrated in Fig. 3. In the case of the Tevatron, the types of partons that can collide to produce a W boson are more readily available and their momentum distributions are better understood than at the LHC. The exact energy fraction that these partons carry is not precisely known, which complicates the underlying calculations considerably.
Another, even more demanding theoretical uncertainty comes from the momentum of the W boson itself. As explained before, given that experimentalists don’t have direct access to the W boson, they have to rely on the calculations of the underlying W boson momentum from theorists. These calculations are very difficult in the low W boson momentum area where most of the Ws are produced, and can crucially not be easily cross-checked in distributions from the data and can therefore carry large uncertainties. So the experimentalists’ choice in how to deal with this uncertainty is a very critical aspect of a W boson mass measurement. All the most recent measurements, by the ATLAS, LHCb, and CDF collaborations have made very particular choices on how to deal with this uncertainty. Many of these choices are heavily debated in the community, and are usually the most scrutinized aspects of these types of measurements.
All in all, getting to one part in 10’000 in overall uncertainty at any accelerator and experiment is a formidable challenge, and involves years of study by many physicists, both experimental and theoretical ones.
The recent CDF measurement represents the culmination of decades of work by the CDF collaboration to understand their detector, data, analysis techniques and associated systematic uncertainties to an extreme level of detail for a hadron collider experiment. This, combined with their chosen theoretical model, incorporating knowledge of the distribution of quarks and gluons within the proton and anti-proton, and Standard Model calculations of how they interact to produce a W boson, its resulting momentum, and the direction of the lepton and neutrino decay products, combine to produce a measured value of the W mass in significant tension with the Standard Model. Assuming no large mistakes, or (severe) under-estimation of systematic uncertainties, the most tantalizing interpretation is that the Electroweak theory underpinning the Standard Model is incorrect, inconsistent or incomplete in an observable way, giving hints of new physics beyond the Standard Model which might be just around the corner. One possible alternative is that critical aspects of the (anti)-proton structure or QCD are insufficiently understood. Aside from the 7 sigma disagreement with the indirect Standard Model prediction, the measurement is also in significant tension with previous measurements from LHCb, ATLAS, and D0. A misunderstanding of the proton structure or of QCD could manifest differently depending on the center-of-mass energy, proton-proton vs proton-antiproton collisions, and different choices for the corresponding theoretical models or analysis techniques.
A complete understanding will require further discussion and interpretation of the CDF result, continued theoretical progress on the production of W bosons at hadron colliders, and further measurements from the LHC experiments, and eventually at future colliders. The LHC Electroweak Working Group serves as a cross-experiment and theory forum where a complete set of experimental and theoretical issues can be discussed. In addition, the dedicated LHC-Tevatron mW combination working group is charged with producing combinations of mW measurements from both the LHC and Tevatron experiments, including an accurate treatment of correlations of systematic uncertainties, and potentially incorporating updated knowledge of the theoretical aspects or more coherent treatments across measurements. Theoretical understanding of W production at hadron colliders continues to improve as well, with improvements to the methodology used to model the proton structure, as well as the incorporation of new measurements from the LHC, and ever more accurate QCD calculations of W and Z production, with a special focus on the challenging low momentum region dominated by soft gluon emission. The existing ATLAS and LHCb measurements incorporate only a small fraction of the data recorded so far, with even more to come in Run 3 and at HL-LHC, including dedicated “low-pileup” runs with reduced numbers of simultaneous proton collisions to allow the measurement of the W properties with increased precision or reduced systematic uncertainties. The eagerly awaited first measurement of mW from CMS, as well as future measurements with even better precision from ATLAS, CMS, and LHCb, in conjunction with continuing improvements to the theoretical understanding will eventually shed light on whether the value of mW represents a breaking point for the Standard Model. In the longer term, future e+e- colliders could provide the opportunity to measure mW to a precision of less than 1 MeV, about 1 part in 100,000, and confirm the validity of the Standard Model, or confirm the existence of a larger discrepancy which we might be seeing the first hints of today.
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