1. Introduction

The weak interaction plays an absolutely fundamental role in nature. It is, for example, responsible for the spontaneous decay of radioactive isotopes, and it is the weak process $p+p\to {}^{2}H+e^{+}+\nu$ that is responsible for the burning of the sun. Written in terms of elementary particle interactions, the weak charged-current interaction is the conversion of a u-type quark into a d-type quark under the emission of an electron and an electron neutrino, that is, $u+d\to d+e+{\nu }_e$. In contrast to the other forces of the Standard Model, the weak interaction not only is responsible for the transfer of momentum and energy, but also allows the conversion of one type of elementary particle into another. One striking feature of the weak interaction is its slowness, or in other words its low rate of occurrence, caused by the large mass of the quanta of the weak-interaction fields, known as the $W$ and $Z$ bosons.

In 1933, Enrico Fermi published a first attempt toward a theoretical description of the weak interaction (Fermi, 1934), which could describe the beta decay of atomic nuclei. In his theory, the transition of neutrons to protons is described by a point-like four-particle interaction with a single coupling parameter, $G_F$, known today as the Fermi constant, defining the interaction strength. In the following years, it was shown that the Fermi theory can successfully describe not only a variety of nuclear processes but also the decay of the muon.

Originally, Fermi assumed that the weak interaction was a purely vectorial coupling between matter fields. In 1956, however, Lee and Yang (Lee & Yang, 1956) realized that also axial couplings are in principle allowed, which would lead to parity violation in weak decay processes. Parity violation was experimentally proven in the following year (Wu, Ambler, Hayward, Hoppes, & Hudson, 1957). The correct tensor structure of the interaction, vector minus axial vector, or $V-A$, was finally found in 1958, independently by Sudarshan and Marshak (Sudarshan & Marshak, 1958) and by Feynman and Gell-Mann (Feynman & Gell-Mann, 1958). This structure implied that the weak interaction acts only on the left-handed chiral components of fermion wave-functions. But while very successful in describing nuclear decay rates and asymmetries, it was rapidly realized that this description of the weak interaction as a four-fermion interaction yields diverging cross sections at energies in the order of $100\text{GeV}/c^2$, and could only be an effective theory valid at much lower energies.

While it was Oskar Klein who first suggested that the weak interaction could be mediated by massive charged fields, it was Julian Schwinger and his student Sheldon Glashow who tried as early as 1956 to formulate the weak interaction as a gauge theory and to unify it with the electromagnetic force. The starting point was an SU(2) theory in which the two corresponding charged current bosons describe the weak interaction, while the neutral current boson mitigates the electromagnetic interaction (Glashow, 1961). Steven Weinberg and Abdus Salam extended this approach in 1967 by postulating a neutral weak current and an additional U(1) gauge group (Salam, 1968; Weinberg, 1967). The gauge fields of this theory are initially massless, contradicting the low observed decay rates; the associated symmetry must thus be broken. This symmetry breaking is realized by the Higgs mechanism, and leads to massive vector bosons in the theory (Englert & Brout, 1964; Guralnik, Hagen, & Kibble, 1964; Higgs, 1964). Despite its attractive theoretical features, the initial formulation of this theory did not receive much attention in the first years after its publication, until Martinus Veltman and Gerardus ’t Hooft proved in 1971 (’t Hooft & Veltman, 1972) that the unified electroweak theory was actually renormalizable. The unified electroweak theory then appeared as a viable physical theory with consistent, testable predictions. This triggered a long-standing, worldwide research program in the experimental community.

In section 2, the theoretical aspects of the electroweak theory and its initial experimental verification are summarized. This is put in the context of high-precision measurements of the properties of the $W$ and $Z$ boson in later years, and the implications for our understanding of fundamental physics are presented.

2. Electroweak Unification and Its Experimental Verification

The symmetry group of the electroweak theory has an SU(2)xU(1) structure, where the associated generators are named the weak isopin $T$ and the weak hyper-charge $Y$. The corresponding coupling constants are $g$ and $g^{\prime }$, and the fields are labeled $W_1,W_2,W_3$ for SU(2) and $B_0$ for U(1), respectively. Left-handed fermions feel the weak interaction, and act as doublets of SU(2) with weak isospin $T=1/2$ and quantum numbers $T_3=\pm 1/2$ for the “up” and “down” elements of the doublet, respectively. Right-handed fermions are insensitive to this interaction and singlets of SU(2), so $T=T_3=0$. The U(1) quantum numbers, $Y$, are assigned such that the familiar electric charge is recovered for all fermions after electroweak symmetry breaking.

An overview of the weak isospin charges $T$ and $T_3$, as well as the hyper-charge $Y$, assigned to all elementary fermions (quarks, leptons, and neutrinos) is given in Table 1. The parity violating property of the weak interaction and its observed V-A structure are realized by assigning different charges to the left- and right-handed components of the fermion fields.

However, the fields $W_1,W_2,W_3$ and $B_0$ are massless by construction and hence cannot represent our physical reality. This problem is solved in two steps. First, the known physical vector boson fields, that is, $W^{+},W^{-},Z$ as well as the photon field $\gamma$, are formulated as linear superpositions of the original fields, via the relations

where the mixing between the fields is described by a new parameter of the theory, namely the electroweak mixing angle ${\theta }_W$, which can be expressed in terms of the couplings constants $g$ and $g^{\prime }$ as

The mixing also defines the charges of all fermions with respect to to the physical fields, in particular the electromagnetic charge given by $Q=T_3+\frac{1}{2}Y$. The electromagnetic coupling constant, that is, the electric charge $e$, is related to $g$ and $g^{\prime }$ via $e=g^{\prime }\cdot \cos {\theta }_W=g\cdot \sin {\theta }_W$ and is by construction a vector-interaction. Similarly the coupling strength of fermions to $W^{\pm }$ fields is given by $(g\cdot |T_3|)/\sqrt{2}$.

The situation for the $Z$ boson is more complicated. While the corresponding charge is simply given by $g_Z=g/\cos {\theta }_W$, it has an additional pre-factor for the vectorial and the axial-vector components of the interaction, denoted

respectively. The second important step in the construction of the full electroweak theory is the introduction of spontaneous symmetry breaking. The Higgs mechanism gives mass to the $W^{\pm }$ and $Z$ bosons, while leaving the photon massless. This is technically realized by constructing the Higgs field such that it does not couple to the electromagnetic charge Q, but to all other nontrivial combinations of the weak isospin and the hyper-charge.

This common source of mass relates the $W$ and $Z$ boson masses through their couplings, which can be expressed in terms of ${\theta }_W$ as:

The $\rho$ parameter, defined as $\rho =M_W/(M_Z\cos {}^2{\theta }_W)$, is conventionally introduced to summarize this relation; in the SM, at tree level, $\rho \equiv 1$.

In analogy to the electromagnetic interaction, the weak interaction is propagated by the exchange of $W$ and $Z$ bosons, as illustrated in Figure 1. While the exchange of $W$ bosons always implies a change of flavor, that is, a change within the quark- or lepton-doublets, between the incoming and outgoing particles, interactions mediated by the $Z$ boson do not change the type of fermions involved. While the fundamental coupling constants within the electroweak theory are all of similar magnitude, interactions via the weak boson are suppressed at low energies. This is due to the appearance of the vector-boson masses in the propagator terms in the theoretical calculations of observable processes. Loosely speaking, the Fermi constant $G_F$ is replaced in the full electroweak theory by a momentum-dependent quantity, which involves the actual coupling of the interaction as well as the mass of the associated vector boson, that is, $G_F(q^2)\sim -g^2/(q^2-M_W^2)$, where $q^2$ is the energy transfer of the reaction. For small values of $q^2$, that is, $\left|q^2\right|\ll M_W^2$, this expression reduces to a direct relation between $M_W$ and $G_F$,

Here, the weak coupling parameter $g$ has been replaced by the weak mixing angle and the fine structure constant ${\alpha }_{em}=e^2/4\pi$. The Fermi theory therefore emerges from the full electroweak theory as the approximation in the low-energy limit. The fundamental relations of the electroweak theory are encoded in Equations 2.5 and 2.6, respectively. Once the values for the Fermi constant $G_F$, the fine-structure constant ${\alpha }_{em}$ and the weak mixing angle ${\theta }_W$ are experimentally determined, the masses of the $W$ and $Z$ bosons can be predicted. Since $G_F$ and ${\alpha }_{em}$ were already known in 1971, the immediate questions to be answered by experiments were clear: First of all, does a neutral weak current interaction exist and if so, what is the likelihood of its occurrence compared to the charged weak current processes? The answer to this question would deliver an estimate of the weak mixing angle, as it defines the relative strength between neutral and charged weak processes.

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Figure 1. Feynman diagrams of the allowed tree-level $W^{+}$ (left), $W^{-}$ (center), and $Z$ boson (right) exchanges, representing weak charged current and weak neutral current processes within the first generation of quarks and leptons.

Weak neutral and charged current interactions can be studied via collisions of neutrinos with atomic nuclei, leading to distinct signatures. In a charged current interaction, the incoming neutrino converts into a charged lepton; hence, one expects to observe an energetic lepton as well as remnants of the nucleus and other light hadrons in the final state. In contrast, one would not observe any lepton in a neutral current interaction, but just the final state products of the scattered nucleus. This experimental approach was adopted by the Gargamelle experiment at the European Center for Nuclear Research (CERN), operating from 1970 to 1979, which analyzed events produced in collisions of neutrinos with freon in a bubble chamber. The neutrinos were produced by a 26 GeV proton beam colliding with a beryllium target, yielding predominantly muon neutrinos, thus muons were expected in the final state of charged current interactions. The neutral current interactions were experimentally challenging to identify, as they could be mimicked by neutron collisions. A typical event display of a weak current interaction is shown in Figure 2. Finally, at a conference in Bonn in August 1973, the Gargamelle collaboration reported on the discovery of neutral current events (Hasert et al., 1973). Moreover, the ratios between neutral and charged current interactions $0.21\pm 0.03$ and $0.45\pm 0.09$ were presented for neutrino and antineutrino beams, respectively, consistent with the expectations of the unified electroweak theory. Even more important, a first estimate of the weak mixing angle was derived, implying a mass of the $W$ boson of $M_W\approx 70$ GeV.

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Figure 2. Photo of a weak current event, recorded by the Gargamelle experiment (Gargamelle Collaboration, 2017) (left) and schematic drawing of the corresponding reaction (right).

Once the mass range of the $W$ and the $Z$ bosons was known, it opened the door for a dedicated experimental search, that is, direct production of the $W$ or $Z$ boson in particle collisions and the observation of their decay products. The $W^{+}$ and $W^{-}$ bosons can be directly produced in the annihilation processes $u+\overline{d}\to W^{+}$ and $d+\overline{u}\to W^{-}$, respectively, once the center-of-mass energy of the initial state quarks is close to the mass of the $W$ boson. Since quarks do not exist as free particles, direct quark-antiquark collisions cannot be realized experimentally. The best alternatives are collisions of protons and anti-protons, since the constituent quarks share a large fraction of the nucleon momenta and anti-quarks are dominantly present in the anti-proton. The center-of-mass energy in the quark-antiquark collision, $\sqrt{s_{q\overline{q}}}$, and the proton-antiproton collision, $\sqrt{S_{p\overline{p}}}$, can be related via

where $x_q$ and $x_{\overline{q}}$ denote the fractions of the nucleon momentum carried by the initial-state quarks (also referred to as partons). It was expected that the relevant momentum fractions are on the order of $x_q\approx x_{\overline{q}}\approx 0.25$, implying center of mass energies of a proton-antiproton collider on the order of $\sqrt{S_{p\overline{p}}}\approx 400-600$ GeV. The main experimental challenges in the construction of such a collider were the expected low beam densities and thus small integrated luminosities.

In 1976, the Super Proton Synchrotron (SPS) collider at CERN achieved a beam energy of 400 GeV during its commissioning run. In the same year, new ideas to upgrade and modify the SPS into a proton anti-proton collider, later known as the Super Proton–Antiproton Synchrotron $(Sp\overline{p}S)$, were presented (Rubbia, McIntyre, & Cline, 1978). The existing SPS was upgraded from a one-beam synchrotron to a two-beam collider, storing counter-rotating bunches of protons and antiprotons at an energy of 270 GeV per beam. This did not yet solve the initial problem of low densities of the anti-proton beam, which was produced by collisions of protons, accelerated by CERN’s proton synchroton to energies of 26 $\text{GeV}/c$, on a solid target. The decisive idea was to employ a special technique called stochastic cooling to compress the beam phase-space in a small anti-proton storage ring before the injection to the $Sp\overline{p}S$ (van der Meer, 1972). The $Sp\overline{p}S$ was operational from 1981 until 1991.

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Figure 3. Schematic illustration of the UA1 detector and its subsystems (Astbury et al., 1978; UA1 Collaboration, 1983a).

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Figure 4. Picture of the UA2 detector in the 1987–1990 configuration (Di Lella & Rubbia, 2015).

Given the short lifetimes of $W$ and $Z$ bosons of below $3\cdot {10}^{-25}\text{s}$, only their decay products can be observed experimentally. The most promising decay channels are those that contain at least one electron or one muon in the final state, that is, $W^{\pm }\to e{\nu }_e$, $W^{\pm }\to {\mu }^{\pm }{\nu }_{\mu }$, $Z\to e^{+}{e}^{-}$ and $Z\to {\mu }^{+}{\mu }^{-}$, associated with very small rates of background processes. The search for such signatures was conducted at the UA1 and UA2 experiments. The UA1 detector (UA1 Collaboration, 1983a), shown in Figure 3, was a general-purpose detector of cylindrical shape, 5.8 m long and 2.3 m in diameter. Tracks of charged particles were recorded in a drift chamber placed in a 0.7 Tesla magnetic field, allowing the measurement of their momenta as well as signs of their electric charge. The energy of each electron and photon was measured in an electromagnetic calorimeter and the energy of each long-lived hadron was measured in a hadronic calorimeter, both situated around the drift chamber. Muons were detected by eight planes of large drift chambers, enclosing the whole detector volume. The UA2 detector (UA2 Collaboration, 1983a), shown in Figure 4, had a more limited scope than UA1 and was designed specifically to detect electrons from the $W$ and $Z$ boson decays. Its major component was a highly granular calorimeter for precise energy measurements of electrons and photons, but it was also well adapted to the detection of hadronic jets. In contrast to UA1, UA2 had no muon detection system. The expected experimental signature of a $W$ boson decay in the detectors was therefore one highly energetic reconstructed charged lepton, as well as a momentum imbalance caused by the neutrino, which cannot be directly detected. The experimental signature of a leptonic $Z$ boson decay is even more striking, as two oppositely charged, highly energetic leptons are expected. An event display of a $Z$ boson candidate in the muon decay channel, recorded by the UA1 experiment, is shown in Figure 5. The first published invariant mass distribution of $Z$ boson candidates in the electron decay channel from the UA2 collaboration is shown in Figure 6.

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Figure 5. Event display of a $Z$ boson event candidate in the muon decay channel, recorded by the UA1 experiment (Di Lella & Rubbia, 2015).

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Figure 6. Invariant mass distribution of two oppositely charged, highly energetic electrons, recorded by the UA2 experiment (Di Lella & Rubbia, 2015).

In 1983, a sufficient number of candidate events had been collected by the UA1 and UA2 collaborations, led by Carlo Rubbia and Pierre Darriulat, to announce the discovery of the $W$ and $Z$ bosons as mediators of the electroweak interaction (Arnison et al., 1983; Banner et al., 1983; UA1 Collaboration, 1983a, 1983b). The initial mass measurements yielded values of $80\pm 5\text{GeV}/c^2$ and ${91.9}_{-1.4}^{+1.3}$ $\text{GeV}/c^2$ for the $W$ and $Z$ boson, respectively. Just one year after the discovery of the electroweak gauge bosons, the Nobel Prize was awarded to Carlo Rubbia and Simon van der Meer “for their decisive contributions to the large project, which led to the discovery of the field particles $W$ and $Z$.”

3. Importance of the $W$ and $Z$ Boson Properties for the Standard Model

Beyond being interesting in their own right, the properties of the $W$ and $Z$ boson reflect the fundamental relations of the Standard Model, and their measurement tests these predictions as summarized in the following. The relations 2.5 and 2.6 can be used to predict two observables, when inserting the three others. A common choice of the observables to be used for the predictions are those with the smallest experimental uncertainties, that is, the fine structure constant ${\alpha }_{em}$, the $Z$ boson mass $M_Z$, and the Fermi constant $G_F$. Knowing these, the other observables of the electroweak sector, in particular the $W$ boson mass $M_W$ and $\sin {}^2{\theta }_W$, can be predicted and confronted with experimental results.

This approach allows the mass of the $W$ boson to be expressed at tree level as

If the measured values of $M_Z$, $G_F$, and ${\alpha }_{em}$ are inserted in Equation 3.1, a value of $M_W=79827\pm 5\text{MeV}$ is predicted. Comparing this prediction to the current world average value of $M_W=80379\pm 13\text{MeV}$ (Erler & Schott, 2019), a significant discrepancy becomes apparent. The reason for this discrepancy is the quantum nature of the electroweak theory: quantum fluctuations alter the simple relations in 2.5 and 2.6. For example, the $W$ boson can emit and reabsorb a Higgs boson or split up into a top-quark and a bottom-quark loop, as illustrated in Figure 7.

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Figure 7. Feynman diagrams of important first-order loop corrections to the $W$ boson propagator.

When calculating the impact of these quantum corrections to the predicted $W$ boson mass, it turns out that $M_W$ exhibits a logarithmic dependence on $M_H$, a dependence on the quark masses that is dominated by a quadratic dependence on the mass of the heaviest quark $m_t$, and an approximately linear dependence on $M_Z$, ${\alpha }_{em}$ and the strong coupling constant ${\alpha }_S$.¹ An approximate expression for $M_W$, valid for $M_H>M_W$, is (ALEPH, DELPHI, L3, OPAL, SLD, LEP Electroweak Working Group, SLD Electroweak Group, & SLD Heavy Flavour Group, 2006)

where $\Delta r$ includes all radiative corrections to $M_W$, $\Delta {\alpha }_{em}$ is the difference between the electromagnetic coupling constant evaluated at $q^2=0$ and $q^2=M_Z{}^2$, and $\Delta \rho$ is the quantum correction to the tree-level expression for $\rho$, now defined as $\rho =1+\Delta \rho$.

Similarly, the vector and axial-vector couplings between the $Z$ boson and the fermions become

where

Hence, precise measurements of all relevant observables of the electroweak sector plus the top-quark mass, $m_t$, and the strong coupling constant ${\alpha }_S$, allow a test of the consistency of the Standard Model by fitting all model parameters to all measured observables, while respecting their relations.

Historically, even more important have been the predictions and the constraints on previously unknown parameters that enter equations 3.2–3.7. After the discovery of the $W$ and $Z$ bosons by the UA1 and UA2 experiments and the subsequent precision measurements of their masses and the electroweak mixing angle by the LEP collaborations, as discussed in more detail in the following, the top-quark mass was predicted to be $m_t=174\pm 21$ GeV (Pietrzyk, 1994), a prediction that was confirmed by the discovery of this particle at the Tevatron proton-antiproton collider (CDF Collaboration, 1995; D0 Collaboration, 1995). In the experiments at this collider, top quarks are produced in pairs, through the process

As in $W$-boson final states, hadronic decays of this particle, $t\to Wb\to q{\overline{q}}^{\prime }b$, allow the reconstruction of the invariant mass of the decaying particle. The observed invariant-mass distribution led to an initial measurement of the top-quark mass of $m_t=176\pm 13$ GeV (CDF Collaboration, 1995), in agreement with the Standard Model prediction. This confirmed prediction remains today one of the most striking successes of physics. The observed top-quark invariant-mass distribution from CDF Collaboration (1995), and the historical evolution of the indirect determination of $m_t$ before and after the top-quark discovery are illustrated in Figure 8.

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Figure 8. Left: reconstructed invariant mass distribution in top-quark candidate events, from the CDF top-quark discovery publication (CDF Collaboration, 1995). Right: prediction of the mass of the top quark, as a function of the year when it was derived. The bands show the 68% confidence level obtained by the LEP EW WG (light blue), the PDG (grey), and the Gfitter group (green). The direct measurements are shown as dots. Also shown are fit results taking direct searches for the SM Higgs boson into account (green). Taken from Gfitter Collaboration (2018).

Once the top-quark mass was experimentally known, the mass of the Higgs boson could be predicted as the last remaining free parameter of the electroweak sector. The compatibility of the Standard Model with the available precision measurements for different hypothetical mass values of the Higgs boson, encoded as a ${\chi }^2$ distribution, is shown in Figure 9. The historical development of the indirect determination of the Higgs boson mass is also shown. By the end of the operation of the LEP collider in 2001, the indirect prediction of the Higgs boson mass via the global electroweak fit resulted in a value between 50 and 160 $\text{GeV}$ (ALEPH, DELPHI, L3, OPAL, SLD, LEP Electroweak Working Group et al., 2006). Hence, it was known at that time already that the Large Hadron Collider (LHC), which was constructed at CERN from 1998 onward and started operation in 2008, had to find the Higgs boson, if it existed.

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Figure 9. Left: the “blue-band” plot from 2005, illustrating the estimate of the Higgs boson mass in the Standard Model from electroweak precision measurements, provided by the LEP Electroweak Working Group (ALEPH et al., 2006; LEP Electroweak Working Group, n.d.). The excluded area by direct searches is shown in yellow, while the blue band illustrates the ${\chi }^2$ distribution of the global electroweak fit. Right: prediction of the mass of the Higgs boson, as a function of the year when it was derived. The bands show the 68% confidence level obtained by the LEP EW WG (light blue), the PDG (grey), and by the Gfitter group (dark blue). The direct measurements are shown as dots. Also shown are fit results taking direct searches for the SM Higgs boson into account (green). Taken from Gfitter Collaboration (2018).

With the discovery of the Higgs boson with a mass of 125 $\text{GeV}$ by the ATLAS and CMS collaborations in 2012 (ATLAS Collaboration, 2012; CMS Collaboration, 2012), the last free parameter of the Standard Model was experimentally fixed. This opened the door to searches for new fundamental particles that could alter the relations of the electroweak theory via quantum fluctuations. Such tests rely fundamentally on precision measurements of all electroweak observables, in particular the properties of the $W$ and $Z$ bosons. Since these indirect tests are largely model independent, they provide useful guidance for theory developments and for the design of future experiments.

4. Measurements of the Properties of the Electroweak Gauge Bosons

In the year of the $W$ and $Z$ boson discovery, an initial measurement of the $W$ boson mass was performed by comparing selected event distributions in data with Monte Carlo predictions using different mass hypotheses. The two distributions most sensitive to $M_W$ are the transverse momentum spectrum of the charged decay lepton, $p_T^l$, and the transverse mass of the system, given by $m_T=\sqrt{2\cdot p_T^l\cdot p_T^{\nu }\cdot (1-\cos \varphi )}$, where $p_T^{\nu }$ denotes the reconstructed missing transverse energy in the event caused by the neutrino and $\varphi$ denotes the opening angle between the reconstructed decay lepton and the direction of missing transverse energy. The plane transverse to the beam axis is of special importance for hadron colliders, as the momentum in the beam direction of the interacting initial-state partons is not known and thus only the momentum balance before and after the collision in the transverse plane can be imposed. The charged lepton $p_T^l$ spectrum is expected to peak at $M_W/2$, while the $m_T$ distribution should end at $M_W$, as it can be interpreted as a projection of the invariant mass in the transverse plane. The distributions observed in the electron decay channel of the $W$ boson at the UA1 experiment are shown in Figure 10. While the $p_T^l$ distribution is affected by the modeling of the $W$-boson transverse momentum in hadronic collisions, which depends on a detailed understanding of perturbative quantum chromodynamics (QCD) as well as non-perturbative QCD models, these effects are significantly reduced for the $m_T$ distribution. The measurement of the Z-boson mass in hadronic collisions is significantly easier, as the 4-momenta of both decay leptons can be directly reconstructed and an invariant mass can be formed (Figure 6). The invariant mass only mildly depends on the modeling of the underlying proton–anti-proton collision and is mainly affected by the knowledge of the energy and momentum response of the detector. This basic methodology for the mass measurements of the electroweak vector bosons at hadron colliders has not changed up to the present day.

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Figure 10. The distribution of the electron transverse momentum (left) and the transverse mass derived from the measured electron and neutrino vectors (right). The two curves show the results of a fit to the hypotheses $W\to e\nu$ and $W\to e\nu \nu$ (UA1 Collaboration, 1985).

After the first measurements at the experiments where they were discovered, the masses of $W$ and the $Z$ bosons were most precisely measured at LEP in the 1990s. The $Z$ boson appears as a resonance in the reaction

where $f$ denotes a fermion (electron, muon, $\tau$ lepton, or quark). The measurement thus relies on a measurement of the cross section (or counting rate) of the process as a function of the center-of-mass energy $\sqrt{s}$; the position of the peak reflects the pole of the resonance and hence the mass of the particle. This procedure is represented in Figure 11, where the data points represent the cross sections measured at seven values of $\sqrt{s}$ ranging between 88 and 94 GeV [27]. The line shows the theoretical prediction, in which the $Z$-boson mass, its width, ${\Gamma }_Z$, and the peak cross section, ${\sigma }_0$, are free parameters that are adjusted to the data.

At $e^{+}{e}^{-}$ colliders, $W$ bosons are produced in pairs, in the process

The $W$ bosons decay into lepton-neutrino $(W\to \mathcal{l}\nu )$ or quark-antiquark pairs $(W\to q{\overline{q}}^{\prime })$. While neutrinos escape detection, quarks give rise to jets of hadronic particles in the detectors. The invariant mass of the decaying boson can thus be reconstructed in this final state, as illustrated in Figure 12. The invariant mass distribution observed in the data is compared to the corresponding prediction obtained from a simulation of the experiment (DELPHI Collaboration, 2008). Again, the $W$-boson mass is a free parameter in this procedure, and is adjusted to optimize the agreement with the data.

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Figure 11. The $Z$ resonance, measured from $e^{+}{e}^{-}\to q\overline{q}$ cross sections as a function of $\sqrt{s}$ (ALEPH, DELPHI, L3, OPAL, SLD, LEP Electroweak Working Group et al., 2006). The data points represent the measured cross sections, and the green line is the theoretical prediction for the preferred values of $m_Z$ and ${\Gamma }_Z$. The pink curve represents the corresponding Breit-Wigner distribution, unfolded for QED radiative corrections.

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Figure 12. $W$-boson invariant mass distributions in $e^{+}{e}^{-}\to W^{+}{W}^{-}$ events, in the $WW\to \mu \nu q\overline{q}$ channel. The points are the detector-level data, and the histograms are the simulated distributions obtained for $M_W=80.40\text{GeV}$ (DELPHI Collaboration, 2008).

The final measurements of the $W$ and $Z$ boson masses at LEP achieved a precision of $\delta m_Z\sim 2\text{MeV}$ and $\delta M_W\sim 30\text{MeV}$, respectively. Besides the extremely precise theoretical predictions allowing an interpretation of the data to such precision, the most stringent experimental requirement was an exquisite calibration of the beam energy (Working Group on LEP Energy, 1995).

At proton colliders, $W$ and $Z$ bosons are produced by quark-antiquark fusion, and experimentally observed through their leptonic decays:

The initial-state quarks carry an unknown fraction of the proton or anti-proton beam energy. The collision energy (cf. Eq. 2.7) is thus unknown $a$$priori$, and can only be inferred from the measured final-state particles. The techniques employed to measure the $Z$-boson mass at LEP can thus not be employed. Instead, the measurement procedure relies on detailed simulations of the production and decay processes, and of the interactions between the final state particles and the detectors. In this context, the precisely known masses of other particles are used to adjust the response of the calorimeters and tracking detectors.

Since its precise measurement at LEP, the $Z$-boson mass has often been used as the main reference for detector calibration at hadron colliders. This was notably the case for measurements of $M_W$ performed at UA2 (UA2 Collaboration, 1992), D0 (D0 Collaboration, 2014), and ATLAS (ATLAS Collaboration, 2018). The mass of the $J/\psi$ meson, a particle also known to very high precision but much lighter than the $W$ boson, was used as a calibration reference at CDF (CDF Collaboration, 2014). Figure 13 illustrates the observed $Z$-boson signal in ATLAS together with the calibrated MC prediction (ATLAS Collaboration, 2016).

In $W\to \mathcal{l}\nu$ decays, each final-state particle carries on average half of the energy of the decaying boson. The energy of a $W$ boson at rest is just its mass; hence, the final-state momentum distributions can be expected to contain information about $M_W$. The $p_{\text{T}}^{\mathcal{l}}$ distribution observed in ATLAS is shown in Figure 14. Thanks to the large accumulated $W$ and $Z$ boson samples, experiments at the Tevatron and LHC produced measurements of $M_W$ with a precision of about 20 MeV per measurement. An estimate of the average of all current experimental values for $M_W$ is (Gfitter Collaboration, 2018):

It should be noted that all measurements of $M_W$ at hadron colliders are limited by modeling uncertainties such as the parton density functions of the proton or the transverse momentum description of the $W$ boson in hadron collisions. In fact, several of these model uncertainties are currently larger at the LHC compared to the Tevatron, which is due to the higher center-of-mass energies but also due to the differences between the proton-proton and the proton–anti-proton initial state. A further experimental advantage at the Tevatron is the relatively small number of simultaneous multiple-particle collisions compared to the LHC, leading to a significantly better resolution of the transverse-mass observable, which is less impacted by modeling uncertainties. Given that the statistical uncertainties are subdominant in this measurement, future improvements in precision rely mainly on a better understanding of the underlying physics processes.

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Figure 13. Observed muon-pair invariant mass distribution in the $pp\to {\mu }^{+}{\mu }^{-}+X$ channel. The $Z$ boson resonance observed in data is used as a reference to calibrate the prediction of the simulation (ATLAS Collaboration, 2016).

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Figure 14. Observed $p_{\text{T}}^{\mathcal{l}}$ distribution in $pp\to \mu \nu +X$ events. The position of the peak in data reflects the value of the $W$-boson mass. The prediction of the simulation includes calibration corrections, and the assumed value of $M_W$ is adjusted to optimize the agreement with data (ATLAS Collaboration, 2018).

As discussed in Section 2, the structure of the Standard Model theory implies relations between the $W$- and $Z$-boson masses, and their couplings to fermions. In particular, the leading-order expressions for the vector and axial couplings between the $Z$-boson and fermions are

where $T_3$ and $Q$ are listed in Table 1 for all fermions, and ${{\displaystyle \sin }}^2{\theta }_W$ is at leading order defined by $M_W/M_Z$. This coupling structure is parity violating: in $W$- and $Z$-boson production, final-state fermions are preferably oriented along the direction of the incoming fermions, and anti-fermions tend to follow the direction of the initial anti-fermions. This is different for QED, where such asymmetries are absent. Experimentally, this implies the existence of so-called forward-backward asymmetries, defined as

where $f$ and $f^{\prime }$ denote the type of the initial- and final-state fermions, respectively, and ${\sigma }_{\text{F,B}}^{f,f^{\prime }}$ denotes the $f\overline{f}\to f^{\prime }{\overline{f}}^{\prime }$ cross sections in the forward and backward directions. On the $Z$ resonance, this quantity can be related to the theory via the expressions:

and $c_v^f$ and $c_A^f$ are defined in Equation 4.6. Measuring $A_{\text{FB}}$ at $\sqrt{s}\sim M_Z$ is thus equivalent to determining the ratio of the vector- to axial-coupling strengths, which is ultimately a function of ${{\displaystyle \sin }}^2{\theta }_{\text{W}}$.

For pure photon exchange, for example, for $\sqrt{s}\ll M_Z$ where the influence of the $Z$-boson exchange is negligible, the asymmetry is 0. At the $Z$ peak, considering the process $e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-}$ for simplicity, using the couplings defined in Table 1 and assuming $M_W\sim 80$ GeV and $M_Z\sim 91$ GeV, the leading-order expressions yield to $A_{\text{FB}}^{e,\mu }\sim 0.02$. In the intermediate mass range, or above the $Z$ peak, the interference between photon and $Z$-boson exchange yields much larger values of $A_{\text{FB}}$. This behavior is illustrated in Figure 15 in the vicinity of the $Z$ resonance (ALEPH, DELPHI, L3, OPAL, SLD, LEP Electroweak Working Group et al., 2006).

At hadron colliders, the picture is obscured by the fact that the directions of the initial fermion and anti-fermion are $a$$priori$ unknown. However, the proton constituents at large momentum fraction ($x_q\sim {10}^{-1}$ and above, the “valence” region) are mostly quarks, rather than anti-quarks; at low momentum fractions, in the “sea,” equal amounts of quarks and anti-quarks are expected. Therefore, $Z$ bosons with a large longitudinal momentum in the direction of a given beam are likely to be following the direction of the initial quark.

A practical quantity relating the longitudinal momentum of the $Z$ boson and the initial quark momentum fractions is the rapidity, $y$, which can be measured in the laboratory frame from the final-state leptons as $y=\frac{1}{2}ln\frac{E+p_z}{E-p_z}$, and is related to the momentum fractions of the initial quarks by $y=\frac{1}{2}\ln \frac{x_1}{{\overline{x}}_2}$. High values of $y$ correspond to $x_1\gg x_2$, and indicate events where $x_1$ is most likely the quark, and $x_2$ the antiquark. The $z$ axis can then be oriented along the direction of the quark, and the asymmetry can be measured in this frame. At low values of rapidity, $x_1\sim x_2$, the quark and antiquark directions cannot be identified; the physical asymmetry is diluted by this indetermination, and the oberved asymmetry is close to 0. This is illustrated in Figure 16, where the invariant-mass dependence of the asymmetry is studied in intervals of increasing rapidity (CMS Collaboration, 2018a). As can be seen, for $\sqrt{s_{q\overline{q}}}\sim m_Z$, the asymmetry is just above zero, as was the case at LEP.

The measured asymmetries are interpreted in terms of ${{\displaystyle \sin }}^2{\theta }_{\text{eff}}$, following Eqs. 4.8–4.9 and 4.6. To this date, the most precise measurements are still from the lepton colliders, thanks to the cleaner initial state; due to improving analysis techniques and a better knowledge of the proton structure, LHC measurements are becoming competitive. The current world average for this quantity is (Erler & Schott, 2019)

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Figure 15. Observed forward-backward asymmetry as a function of $\sqrt{s}$, in $e^{+}{e}^{-}\to q\overline{q}$ events at LEP (ALEPH, DELPHI, L3, OPAL, SLD, LEP Electroweak Working Group et al., 2006) . The points represent the data, and the green line is the theoretical prediction for the preferred value of ${{\displaystyle \sin }}^2{\theta }_{\text{W}}$. The pink curve represents the theoretical prediction unfolded for QED radiative corrections.

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Figure 16. Observed forward-backward asymmetry in $pp\to {\mu }^{+}{\mu }^{-}+X$ at the LHC, as a function of $m_{\mu \mu }$ and $y_{\mu \mu }$ (CMS Collaboration, 2018a). The points represent the data, and the red line is the prediction of the simulation for the preferred value of ${{\displaystyle \sin }}^2{\theta }_{\text{eff}}$.

One of the remaining mysteries of physics is the existence of three generations of quarks and leptons. Why more than one generation? And why just three, and not more? Indeed, the theory of the Standard Model was initially written for the lightest generation only, and later just extended to the higher generations $assuming$ that the fermion couplings are just replicated from one generation to the next.

This assumption has been heavily tested in past and present experiments. The most direct tests are provided by measurements and comparisons of the vector-boson branching fractions, and have been performed using the large event samples available at LEP, the Tevatron, and the LHC. Two examples are given in Figure 17, which illustrates leptonic branching fractions of the $W$ and the $Z$ boson. If the couplings between the bosons and the leptons of all generations are equal, then equal branching fractions (up to small mass-dependent corrections) into all leptonic final states should be observed. Branching fractions into fermion pairs of a given type are measured by counting the number of events featuring the corresponding final state as well as the total width of the $Z$ boson, ${\Gamma }_Z$.

The $W$ boson leptonic branching fractions correspond to the decays $W\to e{\nu }_e$, $W\to \mu {\nu }_{\mu }$, and $W\to \tau {\nu }_{\tau }$, and are selected from $e^{+}{e}^{-}\to W^{+}{W}^{-}$ events at LEP. The experiments at the Tevatron and the LHC are mostly sensitive to electron and muon decays in single $W$-boson production. Leptonic $Z$-boson decays are selected from $e^{+}{e}^{-}\to Z$ and $pp,p\overline{p}\to Z$ events. As can be seen from Figure 17, the universality of leptonic couplings is tested at the percent level for $W$ bosons, and at the per mill level for $Z$ bosons. Perfect agreement is found for electrons and muons, both for $W$ and $Z$ boson decays. The $W\to \tau {\nu }_{\tau }$ branching fraction differs from the other measurements at the level of about three standard deviations. This possible hint of a violation of the Standard Model prediction will be confirmed or invalidated at the LHC. A summary of the $W$ and $Z$ boson properties is given in Table 2.

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Figure 17. Left: measured $W\to \mathcal{l}\nu$ branching fractions at LEP (ALEPH et al., 2013). Right: Measured ratios of $W\to e\nu$ to $W\to \mu \nu$, and $Z\to ee$ to $Z\to \mu \mu$ cross sections at the LHC, compared to the Standard Model predictions (ATLAS Collaboration, 2017).

5. Outlook

The previous sections gave a partial, introductory overview of precision measurements of the $W$ and $Z$ resonance properties and their implications as tests of the validity of the Standard Model.

The precision of past, present, and future measurements of the relevant parameters in the electroweak sector of the Standard Model are summarized in Table 3. The first column summarizes the status after LEP and with finalized top-quark mass measurements at the Tevatron, and before the discovery of the Higgs boson. The second column represents the current situation at the LHC. It accounts for the now precisely known Higgs boson mass and otherwise reflects improvements in the precision of the top-quark mass (ATLAS Collaboration, 2019; CMS Collaboration, 2018b) and $W$-boson mass determinations. By the end of the High-Luminosity LHC program, further improvements in $M_W$, $m_t$ and ${{\displaystyle \sin }}^2{\theta }_W$ by factors of two to three are expected (ATLAS Collaboration, 2018, October), due to increased statistics and projected improvements in the perturbative calculations and the proton parton-density functions (Abdul Khalek, Bailey, Gao, Harland-Lang, & Rojo, 2019). The LHC measurements should then drive the overall precision of these quantities. Correspondingly, the precision in the indirect determination of $m_H$ improves from 30 to about 9 GeV, increasingly tightening the comparison with the measured value. Any significant deviation between the predicted and measured value could directly hint to physics beyond the Standard Model.

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Figure 18. Comparisons of ${\chi }^2$ distributions for scanning different Higgs boson masses (left), top quark masses (middle) and W boson masses (right) using the current and future experimental uncertainties via the Gfitter program (Gfitter Collaboration, 2014). The theoretical uncertainties are indicated by the filled areas.

Figure 18 illustrates the comparison between the direct and indirect determinations of $m_H$, $m_t$, and $M_W$ for present and projected uncertainties. The narrower $\Delta {\chi }^2$ curves show the improvements in the indirect measurements. Their intersection with the dashed line at $\Delta {\chi }^2=1$ defines the one-standard-deviation uncertainty, which should ultimately reach 9 GeV for $m_H$ as previously discussed, and 1 GeV for $m_t$.

It should also be noted that if the uncertainties improve but the central values remain at their current values, the differences between the direct and indirect measurements can reach three standard deviations, potentially excluding the Standard Model predictions or, equivalently, indicating the presence of physics beyond the SM. Future $e^{+}{e}^{-}$ colliders (CEPC Collaboration, 2018; FCC Collaboration, 2019) have the potential to bring the indirect uncertainties down to $\delta m_H\sim 2$ GeV and $\delta M_W<1$ MeV, which constitutes a giant leap in the power of this test.

This brief discussion shows how the $W$ boson and the $Z$ boson, whose discoveries signaled the birth of the Standard Model, slowly evolved from fuzzy silhouettes to precisely drawn characters. The precise measurements of their properties is a continuing experimental program, already spanning about forty years and five high-energy colliders. Measurements performed at one experiment provided guidance in the design of the next, leading to the discovery of the previously unconfirmed top quark and Higgs boson. An order of magnitude improvement in precision can still be expected from ongoing and future programs. The analysis of the properties of the $W$ and $Z$ bosons, the top quark, and the Higgs boson provide stringent tests of the theoretical relations of the Standard Model and could ultimately decide its fate.