Properties of the W boson

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In particle physics, the $W$ boson is one of the mediators of the weak interaction, the only example of parity-violating interactions observed in nature. The $W$ boson refers to a particle or its anti-particle, labelled $W^+$ and $W^-$, of charge $+e$ and $-e$ respectively. It has a spin of 1. Its mass, 80.4 GeV, is about 80 times the mass of the proton, and its lifetime is $3\times 10^{-25}$ s.

Its primary experimental manifestation at low energy is radioactivity, specifically $\beta$ decays of unstable nuclei. At high energy, it appears as a resonance in the scattering cross sections of leptons or quarks. It also plays an important role in flavour physics, as mediator of $CP$-violating interactions and neutrino oscillations.

The $W$ boson is intimately linked to the $Z$ and Higgs bosons. While the $W$ was postulated to account for directly observable experimental facts, the $Z$ and the Higgs boson were introduced only on theoretical grounds, as necessary consequences of the simplest possible gauge theory describing weak interactions. The $W$ and $Z$ were established as fundamental particles at the CERN $Sp\bar pS$ collider in 1983, and the Higgs boson was discovered at the CERN LHC in 2012.


Experimental motivation

The weak interaction plays a fundamental role in nature. It was initially introduced to explain the so-called "$\beta$ decays", involving the emission of an electron and a neutrino, of radioactive isotopes or low-mass mesons or hadrons. Examples of $\beta$ decays are:

  • pion decay via $\pi^\pm \to \mu^\pm \nu_\mu$ with a half-life of about 2.6$\times 10^{-8}$
  • muon decay via $\mu^\pm \to e^\pm \nu_e \nu_\mu$ with a half-life of about 2.2$\times 10^{-6}$ s;
  • neutron decay via $n \to p e^- \bar{\nu_e}$, with a half-life of about 11 min;
  • tritium decay via $^3_1 H \to ^3_2 He e^- \bar{\nu_e}$, with a half-life of 12.3 years;
  • carbon-14 decay via $^{14}_{6}C \to \, ^{14}_{7}N e^- \bar{\nu_e}$, with a half-life of 5725 years.

A prominent example is neutron decay, which in terms of elementary particles is described as the transition of a down quark into an up quark, with the emission of a $W^-$ boson which decays into an electron and an electron-neutrino. While the decay rates of the processes above differ by several orders of magnitude, they are much slower than decays mediated by electromagnetic or strong-interactions, with typical half-lives of $10^{-17}$ s and $10^{-24}$ s, respectively. This observed hierarchy was the primary motivation to introduce the weak interaction.

Weak interactions also manifest themselves in the form of low-energy nuclear fusion reactions. For example, the reaction $p p \to {^2}H + e^+ + \nu_e$ is the initial reaction of the "proton-proton chain" and the primary source of solar energy.

In contrast to the other forces of the Standard Model, the weak interaction is not only responsible for the transfer of momentum and energy, but also converts one type of elementary particle into another.

Theoretical description

The weak interaction was initially described by E. Fermi using a point-like, four-fermion interaction vertex ( Feynman:1958ty), with transition rates governed by a single, dimensionful parameter $G_F$. The most precise experimental determination of this parameter is obtained from the muon lifetime, which benefits from a very accurate theoretical prediction in terms of $G_F$ ( vanRitbergen:1999fi):

\begin{equation} \frac{1}{\tau_\mu} = \frac{G_\text{F}^2 m_\mu^5}{192 \pi^3} \left(1 - 8 \frac{m_e^2}{m_\mu^2} -12 \frac{m_e^4}{m_\mu^4} \ln\frac{m_e^2}{m_\mu^2} + \cdots \right) \tag{1} \end{equation}

where $\tau_\mu$ represents the muon lifetime, and $m_e$, $m_\mu$ are the electron and muon masses. The most precise measured value of $\tau_\mu$, $\tau_\mu=2 196 980.3(2.2)$ ps, leads to a value of the Fermi constant of $G_F =1.166 378 7(6)\times 10^{-5}$ GeV$^{-2}$. The same value successfully accounts for all measured weak decay lifetimes.

The Fermi constant $G_F$ being a dimensionful parameter, scattering amplitudes in the Fermi theory diverge at high-energy, and higher-order corrections are not renormalizable ( Feynman:1958ty), in contrast to Quantum Electrodynamics (QED) for which finite predictions, were established in the 1940's ( Gell-Mann:1954yli). It was therefore postulated, in analogy with QED, that the Fermi interaction actually describes the low-energy properties of an interaction involving exchanges of intermediate vector bosons. If these bosons are massive, the dimensionful $G_F$ is replaced by a new, dimensionless coupling constant, and the mass scale originates from the propagator of the exchanged boson. This discussion is summarized in Figure 1.

Figure 1: Illustration of the neutron (left) and muon (middle) decay in the Fermi theory, with decay rates determined by a common coupling constant. The muon decay in the intermediate vector boson hypothesis is shown right, with a dimensionless coupling constant and a scale given by the mass of the exchanged boson.

A theory with massive intermediate vector bosons is, however, inconsistent with the requirement of local gauge invariance. The weak interaction could therefore not be interpreted as resulting from an internal symmetry, as is the case for Quantum Electrodynamics (QED) and, as consequence, would loose is renormalizability. The requirement of local gauge invariance could be satisfied by the identification of a suitable symmetry group that includes the photon and the charged $W$ bosons in its set of gauge fields, and the introduction of the Brout-Englert-Higgs mechanism ( Higgs:1964pj Englert:1964et) to generate mass terms in the theory through the spontaneous breaking of the gauge symmetry.

With the experimental evidence obtained by C.S. Wu ( PhysRev.105.1413) that the weak interaction violates parity, i.e. the invariance of an interaction when spatially inverting its coordinates around the origin, the Fermi interaction was extended to include a structure acting differently for left-handed and right-handed chiral components of the fermion wave-functions. The simplest description compatible with the experimental data is a vector -- axial-vector (V-A) coupling structure. While vector current interactions change sign under a parity transformation, axial-vector current interactions do not. The interference between those two terms creates parity violation.

These developments led to the formulation of the Standard Model in 1967 ( Weinberg:1967tq, Goldstone:1962es, Glashow:1961tr, Salam:1968rm). In the Standard Model, the weak interaction arises from the assumption that left-handed fermions are organized in doublets:

\begin{equation} \Psi^i_L(x) = \left( \begin{array}{c} \nu_L(x)\\ e_L(x)\\ \end{array} \right),\, \left( \begin{array}{c} u_L(x)\\ d_L(x)\\ \end{array} \right),\, \cdots \tag{2} \end{equation}

and that the theory is invariant under local gauge transformations of the type:

\begin{equation} \Psi^i_L(x) \to e^{i\vec{\tau}\vec{\theta}(x)}\Psi^i_L(x). \end{equation}

In Eq. 2, $e_L$, $\nu_L$ represent the electron and electron neutrino, and $u_L$ and $d_L$ represent the up and down quarks, together forming the first generation of fermions. The doublet structure of the theory is dictated by the observed transitions, and the simplest gauge group compatible with these requirements is $SU(2)$, whose generators are the Pauli matrices $\tau^{1,2,3}$. Imposing local gauge invariance requires the introduction of three corresponding gauge fields $W^{1,2,3}_\mu$. In addition to $SU(2)$, a $U(1)$ interaction, mediated by a vector field $B_\mu$, is required to account for the electromagnetic interaction after symmetry breaking. This leads to the following Lagrangian describing the interactions between the gauge fields $W^i_\mu$, $B_\mu$ and the fermion doublets $\Psi_L$: \begin{equation} {\cal L}_\text{int} = \sum_i \, g \, \bar{\Psi}^i_L \vec{\tau} \vec{W}_\mu \gamma^\mu \Psi^i_L \, + \, \sum_j \, g' \, y_j \bar{\Psi}^j B_\mu \gamma^\mu \Psi^j \end{equation}

where $g$ and $g'$ are the coupling constants associated to the $SU(2)$ and $U(1)$ gauge groups, respectively. A pair of charge-conjugate bosons, as was the primary goal of this construction, is defined as follows:

\begin{equation} W^+_\mu = \frac{W^1_\mu + i W^2_\mu}{\sqrt{2}} \,\,\,\,\,\,\, W^-_\mu = \frac{W^1_\mu - i W^2\mu}{\sqrt{2}} \end{equation}

The remaining component $W^3_\mu$ and the $U(1)$ gauge field $B_\mu$ mix to give, after symmetry breaking, a massless field $A_\mu$ and a massive field $Z_\mu$, respectively identified as the photon, and a new neutral vector boson:

\begin{equation} \left( \begin{array}{c} A_\mu \\ Z_\mu \\ \end{array} \right) \, = \left( \begin{array}{cc} \cos\theta_W & \sin\theta_W \\ -\sin\theta_W & \cos\theta_W \\ \end{array} \right)\, \left( \begin{array}{c} B_\mu \\ W^3_\mu \\ \end{array} \right) \end{equation}

The physical parameters $e$ and $\sin\theta_W$ are expressed in terms of the initial gauge coupling constants following $e = g \sin\theta_W = g' \cos\theta_W$. In terms of physical fields and parameters, the interactions are

\begin{eqnarray} {\cal L}_\text{int} &=& {\cal L}_\text{CC} + {\cal L}_\text{NC} + {\cal L}_\text{QED} \nonumber\\ {\cal L}_\text{CC} &=& \frac{e}{\sqrt{2}\sin\theta_W} \left\{ W^-_\mu \, [\bar{u}_L \gamma^\mu d_L \, + \, \bar{\nu}_e \gamma^\mu e^-] + W^+_\mu \, [u_L \gamma^\mu \bar{d}_L \, + \, \nu_e \gamma^\mu e^+] \right\} \nonumber\\ {\cal L}_\text{NC} &=& \frac{e}{2\sin\theta_W\cos\theta_W} Z_\mu \sum_f \bar{f} \gamma^\mu (v_f - a_f \gamma^5) f \nonumber\\ {\cal L}_\text{QED} &=& e A_\mu \sum_f Q_f \bar{f} \gamma^\mu f \end{eqnarray}

where ${\cal L}_{CC} $ and ${\cal L}_{NC} $ describe the interactions of the fermions with $W$ and $Z$ bosons, respectively, and ${\cal L}_{QED} $ describes electromagnetism. Fermions and anti-fermions are denoted $f$ and $\bar f$, while the electric charge of a fermion is denoted $Q_f$ and $\theta _W$ is the Weinberg electroweak mixing angle. Diagrams illustrating the interactions between the gauge bosons and the fermions are given in Figure 2.

Figure 2: Coupling between a W boson and a lepton-neutrino pair (left), and an up-down quark doublet (middle), and between a photon or Z boson and a fermion-antifermion pair.

A distinctive feature of the $SU(2)$ gauge group is that it is non-Abelian ( Abers:1973qs, Gross:1973id), which implies interactions among the weak gauge bosons. Interactions between $W$ bosons and either a photon or a $Z$ boson are predicted following the Lagrangian below:

\begin{eqnarray} {\cal L}_\text{3V} = -ig \, \Bigl[ \, & (W^+_{\mu\nu}W^{-\mu} - W^{+\mu}W^{-}_{\mu\nu})(A^\nu\sin\theta_W-Z^\nu\cos\theta_W) \nonumber \\ &+ W^+_\nu W^-_\mu (A^{\mu\nu}\sin\theta_W - Z^{\mu\nu}\cos\theta_W) \, \Bigr]. \end{eqnarray}

Four-gauge-boson interaction vertices also occur in the theory :

\begin{align} {\cal L}_\text{4V} = &-\frac{g^2}{4} \,\Biggl\{\,\Bigl[\, 2\,W^{+}_\mu\,W^{-\mu} + (\,A_\mu\,\sin \theta_\text{W} - Z_\mu\,\cos \theta_\text{W} \,)^2 \,\Bigr]^2 \\ &- \Bigl[\, W_\mu^{+}\, W_\nu^{-} + W^{+}_\nu \, W^{-}_\mu + \left(\, A_\mu\,\sin \theta_\text{W} - Z_\mu\,\cos \theta_\text{W} \,\right)\left(\, A_\nu\,\sin \theta_\text{W} - Z_\nu\,\cos \theta_\text{W} \,\right)\, \Bigr]^2\,\Biggr\} . \end{align}

The corresponding so-called triple and quartic gauge interactions are depicted in Figure 3.

Figure 3: Coupling between a W-boson pair and a Z boson as well as between four $W^-$ bosons, and between a W-boson pair, a Z boson, and a photon. The vertex $ZZ \to \gamma$ does not exist since the Z boson carries no electric charge.

Finally, $W$ bosons interact with the Higgs boson, the scalar field introduced to generate gauge-invariant mass terms for the $W$ and $Z$ bosons. Interactions include trilinear and quartic couplings, according to

\begin{align} {\cal L}_\text{HV} =\left(\,g\,m_\text{H} H + \frac{\,g^2\,}{4}\;H^2\,\right)\left(\,W^{+}_\mu\,W^{-\mu} + \frac{1}{\,2\,\cos^2\,\theta_\text{W}\,}\;Z_\mu\,Z^\mu\,\right) , \end{align}

and are illustrated, for $W$ bosons, in Figure 4.

Figure 4: Coupling between a W-boson pair and one (left) or two (right) Higgs bosons.

In summary, a consistent description of weak interactions in terms of a gauge theory requires the introduction, in addition to the experimentally motivated $W^+$ and $W^-$ fields, of

  • one additional, neutral vector boson, the $Z$, ensuring the gauge invariance of the interaction under $SU$(2) and mediating a new, neutral-current interaction among neutrinos;
  • a neutral, scalar particle, the Higgs boson $H$, required to maintain gauge invariance in presence of massive fields.

While the theoretical description of the interactions between $W$ bosons and fermions was adjusted to reproduce the experimental facts, the $Z$ and Higgs bosons and their interactions are genuine predictions o the theory.

In the Standard Model, the Fermi constant $G_F$ is re-interpreted in terms of the physical parameters $e$ and $\sin\theta_W$ as

\begin{equation} G_F = \frac{\sqrt{2}}{8}\left(\frac{e}{\sin\theta_W m_W}\right)^2. \end{equation}

Before the discovery of the $W$ boson, neutrino scattering experiments indicated $\sin\theta_W\sim 0.2$, yielding $m_W\sim 80$ GeV. In this context, the weakness of the weak interaction thus appears as a consequence of the large mass of the mediating particle, rather than the smallness of the coupling constant itself. This early estimation of $m_W$ guided the experimental program that led to the discovery of the $W$.

Search, discovery and available data

The apparent weakness of the weak interaction and the fact that low-energy weak transitions can be described by a point-like interaction indicate that the masses of the $W$ and the $Z$ must be large, and that the electroweak bosons can only be observed as resonances in high energy particle collision. The dominant production mechanism is the annihilation process of one quark and one anti-quark, e.g. $u \bar d \rightarrow W^+$ or $d \bar u \rightarrow W^-$, where the energies of the initial quarks must match the mass of the $W$ boson.

The Super Proton Anti-Proton Synchrotron ($Sp\bar p S$) collider at CERN, built in the 1970's, was designed to collide protons and anti-protons at a centre of mass energy of up to 540 GeV, providing a rich sample of quark and anti-quark collisions, which where recorded by the UA1 and UA2 experiments. Given their short lifetime, the existence of $W$ bosons has to be inferred by their decay products. Leptonic $W$-boson decays, in particular $W\to e\nu_e$ and $W\to \mu\nu_\mu$, have small experimental background contributions and can be identified in particle detectors by the reconstruction of one high energetic electron or muon and an imbalance in the total transverse momentum of all particles measured in an event. The latter quantity is known as missing transverse momentum, and occurs since neutrinos leave the detector without any interaction.

On 24 February 1983 the UA1 collaboration published a paper describing the discovery of the $W$ boson, based on six collision events with high energetic electrons and missing transverse momentum, shown in Figure 8. The discovery of the W boson was announced by the UA1 and UA2 experiments in a joint press conference at CERN the following day. Carlo Rubbia and Simon van der Meer were awarded the Nobel prize in 1984 for this achievement (Figure 5).

Figure 5: Left: Distributions of the lepton and neutrino transverse momenta for the first W boson candidate events recorded by the UA1 experiment. Right: Picture of the UA2 detector at CERN. Credit: UA1 and UA2 Collaborations, Luigi Di Lella1 and Carlo Rubbia, CERN

The discovery of the $W$ boson was followed by a long-term, and still ongoing experimental program aiming for measurements of its properties with ever increasing precision. An important step in this program was achieved at the electron-positron collider LEP, between 1996 and 2000, after it reached the energy threshold for the $W$-boson pair-production process $e^+ e^- \rightarrow W^+W^-$. An advantage of lepton colliders is the possibility to study also the hadronic decay modes of the W bosons, thanks to manageable backgrounds. About 100,000 W bosons have been recorded and analysed at LEP by the ALEPH, DELPHI, L3 and OPAL experiments.

The CDF and D0 experiment at the Tevatron collider (1983-2011) studied W boson again in proton anti-proton collisions, at significantly higher centre of mass energies of 1.96 TeV. About 4 million $W$-boson decays have been detected in the electron and muon decay channels. With the start of the Large hadron collider at CERN, $W$ bosons are studied in proton-proton collisions for the first time. By the end of 2018, more than 300 million leptonic $W$-boson decays have been recorded by the ATLAS, CMS and LHCb experiments.

Couplings to fermions

The weak isospin charge is fixed (in contrast with QED), up to a global coupling constant $g$. The couplings between $W$ bosons and fermion doublets are therefore universal. The $W$-boson decay widths into lepton and quark doublets are respectively given by

\begin{equation} \Gamma_{\ell\nu} = \frac{G_F M_W^3}{6\pi \sqrt{2}}(1+\delta_\text{EW}), \,\,\,\,\,\,\, \Gamma_{q\bar{q'}} = \frac{N_c |V_{q\bar{q'}}|^2G_F M_W^3}{6\pi \sqrt{2}}(1+\delta_\text{EW}+\delta_\text{S}), \tag{3} \end{equation}

where $N_c=3$ is the number of quark colours, $V_{q\bar{q'}}$ is the Cabibbo-Kobayashi-Maskawa (CKM) matrix element, introduced below, corresponding for the considered quark doublet, and $\delta_{EW} \sim -0.4$% and $\delta_S \sim +4.1$% reflect electroweak and strong radiative corrections, as illustrated in Figure 6.

Figure 6: Representative diagrams for radiative corrections in $W$-boson decays to leptons (left) and quarks (middle and right).

These calculations are combined to derive the $W$ branching fractions, giving $B(W\to\ell\nu)$=10.86% and $B(W \to hadrons) = $67.42%. The leptonic branching fractions are given by flavour, since electron, muon and $\tau$ decays can be measured separately; the hadronic branching fraction is summed over all hadronic final states as first and second generation quark decays can not be distinguished with sufficient precision.

The $W$-boson decay fractions have been measured at LEP ( ParticleDataGroup:2022pth), the Tevatron ( ParticleDataGroup:2022pth) and most recently at the LHC ( ATLAS:2020xea, CMS:2022mhs). The LEP measurements rely on $e^+ e^- \to W^+ W^-$ events, while hadron collider measurement exploit both direct $W$-boson production, $pp\to W + X$, and $W$-bosons from top-quark decays, $pp\to t\bar{t}+X$ followed by $t\to Wb$.

A summary of recent measurements is given in Figures 7 and 8. While most measurements are in agreement with the Standard Model predictions, the LEP measurement of $B(W\to\tau\nu)$ has long been in significant disagreement. Recent LHC measurements now match or exceed LEP in precision, confirming the SM also for the $\tau$ decay mode.

Figure 7: Summary of W-boson branching fraction measurements and lepton universality tests by the ATLAS Collaboration. Credit: ATLAS Collaboration, Nature Phys. 17 (2021) 7, 813-818, e-Print: 2007.14040
Figure 8: Summary of W-boson branching fraction measurements and lepton universality tests by the CMS Collaboration. Credit: CMS Collaboration, Phys.Rev.D 105 (2022) 7, 072008, e-Print: 2201.07861

The electroweak interaction couples $W$ bosons to fermions in "interaction eigenstates", different from the "mass eigenstates" which propagate in free space and can be detected. Mass and flavour eigenstates are related by the CKM matrix $V_{CKM} $ ( Cabibbo:1963yz, Kobayashi:1973fv) in the case of quarks, and by the PMNS matrix $U_{PMNS} $ ( Maki:1962mu) in the case of leptons. By convention, the quark flavours ($u$, $d$, $s$, $c$, $b$, $t$) are defined by the mass eigenstates, whereas the lepton flavours ($e$, $\nu_e$, $\mu$, $\nu_\mu$, $\tau$, $\nu_\tau$) are defined by their interaction eigenstates. Transition amplitudes involving $W$-boson couplings to quarks thus need to include the relevant CKM matrix elements. Again by convention, these matrices are defined as applying to the down-type fermions, following

\begin{equation} \left( \begin{array}{c} d'\\ s'\\ b'\\ \end{array} \right) = \left( \begin{array}{ccc} V_{ud} & V_{us} & V_{ub}\\ V_{cd} & V_{cs} & V_{cb}\\ V_{td} & V_{ts} & V_{tb}\\ \end{array} \right)\, \left( \begin{array}{c} d\\ s\\ b\\ \end{array} \right),\,\,\,\,\,\,\,\,\,\,\, \left( \begin{array}{c} \nu_e\\ \nu_\mu\\ \nu_\tau\\ \end{array} \right) = \left( \begin{array}{ccc} U_{e1} & U_{e2} & U_{e3}\\ U_{\mu 1} & U_{\mu 2} & U_{\tau 3}\\ U_{\tau 1} & U_{\tau 2} & U_{\tau 3}\\ \end{array} \right)\, \left( \begin{array}{c} \nu_1\\ \nu_2\\ \nu_3\\ \end{array} \right) \end{equation}

where the quark interaction eigenstates are primed, and $\nu_{1,2,3}$ refer to the neutrino mass eigenstates. The Standard Model does not predict the values of the CKM and PMNS matrix elements. An average of recent experimental data (ParticleDataGroup:2022pth) gives, with a confidence level of 68\%,

\begin{equation} V_\text{CKM} = \left( \begin{array}{ccccccc} 0.97401 \pm 0.00011 & & & 0.22650 \pm 0.00048 & & & 0.00361 \pm 0.00010\\ 0.22636 \pm 0.00048 & & & 0.97320 \pm 0.00011 & & & 0.04053 \pm 0.00072\\ 0.00854 \pm 0.00019 & & & 0.03978 \pm 0.00071 & & & 0.999172 \pm 0.000029\\ \end{array} \right), \end{equation} for the quark mixing matrix. Three-standard-deviation confidence intervals are given for the neutrino mixing matrix ( ParticleDataGroup:2022pth):

\begin{equation} U_\text{PMNS} = \left( \begin{array}{ccccccc} 0.801 - 0.845 & & & 0.513 - 0.579 & & & 0.143 - 0.155 \\ 0.234 - 0.500 & & & 0.471 - 0.689 & & & 0.637 - 0.776 \\ 0.271 - 0.525 & & & 0.477 - 0.694 & & & 0.613 - 0.756 \\ \end{array} \right). \end{equation}

The experimental status of the CKM matrix is discussed in detail here.

Mass and Width

In the Standard Model, the mass of the $W$ boson, $W$, appears in the propagator term of the $W$ boson and is expressed at tree level as

\begin{equation} \tag{4} M_W^2 = \frac{M_Z^2}{2} \cdot\left(1+\sqrt{1-\frac{\sqrt{8}\cdot \pi \cdot \alpha_{em}}{G_F\cdot M_Z^2}}\right). \end{equation}

Using the measured values of $M_{Z}$, $G_F$ and $\alpha_{em}$ in Equation~(4) yields $M_{W} = 79827 \pm 5\,MeV$, in strong disagreement with the experimental value. Virtual corrections, induced by the evolution of $\alpha_{em}$ as a function of energy and by the heavy particles of the spectrum, predominantly the Higgs Boson and the top quark, however modify this relation to:

\begin{eqnarray} \tag{5} M_W^2 &=& \frac{M_Z^2}{2} \cdot\left(1+\sqrt{1-\frac{\sqrt{8}\cdot \pi \cdot \alpha_{em}}{G_F\cdot M_Z^2}\frac{1}{1-\Delta r}}\right), \text{ where}\\ \Delta r &=& \Delta\alpha_{em} -\frac{\cos^{2}\theta}{\sin^{2}\theta} \Delta\rho, \text{ and}\\ \Delta\rho &=& \frac{3 G_F M_W^2}{8\sqrt{2}\pi^2}\left[\frac{m_t^2}{M_W^2} -\frac{\sin^{2}\theta}{\cos^{2}\theta}\left( \ln{\frac{M_H^2}{M_W^2} - \frac{5}{6}}\right) + \cdots \right]. \end{eqnarray}

In the above relations, $\Delta\alpha_{em}$ is the difference between the electromagnetic coupling constant evaluated at $q^2=0$ and $q^2=M_{Z}^2$, and $\Delta\rho$ accounts for the influence of the top quark and the Higgs boson. These corrections imply a quadratic dependence of $M_W$ on the mass of the top quark $M_t$, and logarithmic dependence on the mass of the Higgs boson $M_H$. The measured masses of $M_H$ ( ATLAS:2015yey) and $M_t$ ( ATLAS:2014wva) thus yield a precise prediction for the $W$-boson mass, which can be tested by comparing the predicted value of $M_W$ with measurement (Figure 9).

Extensions of the standard model modify the expression of $\Delta r$. In supersymmetric models, the new particles of the spectrum enter the $W$ boson mass relations and shift it towards higher masses ( Heinemeyer:2013dia). Precision measurements of the W boson mass are thus a tool to search for physics beyond the SM. Predictions for $m_W$ in the SM and in the minimal supersymmetric model (MSSM) are illustrated in Figure 10.

Figure 9: Prediction for the W boson mass as a function of the top quark mass in the SM, at leading order, accounting for the evolution of the fine-structure constant, and including for complete higher-order corrections with an experimental value of the top quark mass of 172.5 GeV. Credit: Maarten Boonekamp
Figure 10: Prediction for the W boson mass as a function of top quark mass in the MSSM, for various configurations of the supersymmetric parameter space. Credit: S. Heinemeyer, W. Hollik, G. Weiglein, L. Zeune, JHEP 12 (2013) 084. e-Print: 1311.1663.

The most precise measurements of the W boson mass have been performed at hadron colliders, i.e. the Tevatron and the LHC. The measurements are performed in the electron and muon decay channels which a selection of $W$ boson candidates with small backgrounds. These final states however imply an experimental challenge, as the neutrino in the final state escapes detection and the $W$-boson decay kinematics are only partly reconstructed. The determination of $m_W$ relies on the reconstructed transverse momentum of the charged lepton, $p_{T}^\ell$, the missing transverse momentum, $p_{T}^{\nu}$, and the transverse mass, $m_{T}$, defined as $m_{T}=\sqrt{p_{T}^{l}p_{T}^{n} (1-\cos\phi)}$, where $\phi$ is the opening angle between the decay lepton and the missing momentum in the plane transverse to the beams.

The extraction of $M_W$ relies on the measurement and interpretation of the kinematic distribution of its decay products. If the $W$ boson is produced at rest, the peak of the final-state lepton transverse momentum distribution is $at M_W/2$, while the peak of the transverse mass distribution is located at $M_W$. The situation is complicated by several effects. First, a measurement with a relative precision of ${\cal O}(10^{-4})$ requires the energy and momentum calibration of the detectors to be understood at the corresponding level. Second, the $W$ boson is not produced at rest and the theoretical modelling of $W$-boson production and decay has to be understood in great detail. Figure 11 shows the reconstructed transverse momentum spectrum of leptons in the ATLAS experiment for different assumed values of the W boson mass, illustrating the required precision for the measurement. Figure 12 shows the measured transverse momentum spectrum of leptons in ATLAS, compared to a simulation assuming the best fit value of the W boson mass.

Figure 11: Lepton transverse momentum distribution in simulated events for a reference value of the W boson mass and its variations of $\pm 50$~MeV. Credit: ATLAS Collaboration, Eur.Phys.J.C 78 (2018) 2, 110, e-Print: 1701.07240.
Figure 12: Comparison between data and simulation for this distribution, using the measured value of the W boson mass. Credit: ATLAS Collaboration, Eur.Phys.J.C 78 (2018) 2, 110, e-Print: 1701.07240

Measurements at the LHC have been performed by the ATLAS collaboration, yielding $M_W = 80370 \pm 19$ MeV, and by the LHCb collaboration with a result of $M_W = 80354 \pm 32$ MeV. The D0 collaboration at the Tevatron measured a value of $M_W=80375 \pm 23$ MeV. Combining these hadron-collider measurements with the LEP collider average yields a value of $M_W = 80366.9 \pm 13.3$ MeV, with good compatibility. This average is also consistent with the prediction of $M_W$ in the Standard Model.

In 2022, the CDF collaboration published a new measurement using its full data-set, and obtained a value of $m_W=80433.5 \pm 9.4$ MeV, a precision better than that of all other experiments combined. This measurement disagrees with the Standard Model expectation at the level of 7 standard deviations, but also with the combined value of all other experiments by about four standard deviations. The origin of this discrepancy is presently under investigation.

Measurements of $M_W$ have also been performed separately for positive and negative $W$ bosons, showing compatible mass values and thus confirming the SM prediction. The most precise value was that obtained by ATLAS ( ATLAS:2017rzl), giving $M_W^+ - M_W^- = -29 \pm 28$ MeV.

The total $W$ boson decay width is calculated in the Standard Model from the sum of the partial widths given in Eq. 3, giving $\Gamma_W^{SM} = 2091 \pm 1$ MeV. While it is less sensitive to new physics than $m_W$ ( Rosner:1993rj, Denner:1990cpz), it still provides a useful test of the Standard Model, and a means of probing radiative corrections $\delta_{EW}$ and $\delta_{S}$. In particular, $\delta_{S} $ is a direct function of the strong coupling constant and a measurement of $\Gamma_W$ thus constrains this fundamental parameter.

Measurements of $\Gamma_W$ proceed similarly to those of $M_W$, namely through an analysis of the distributions of the decay products. The distribution most sensitive to $\Gamma_W$ is the transverse mass, especially at high values where it most directly reflects the tail of the $W$-boson Breit-Wigner distribution. The most precise measurements are provided by the CDF and D0 Collaborations, with a combined value of $\Gamma_W=2085 \pm 42$ MeV, in good agreement with the Standard Model prediction.

A summary of available measurements of $M_W$ and $\Gamma_W$ is shown in Figure 13.

Figure 13: Summary of available measurements of the W boson mass (left) and its width (right). Credit: Particle Data Group, PTEP 2022 (2022) 083C01,

Gauge interactions at high energy

As mentioned in Section 1, a distinctive feature of the electroweak interaction is that the $W$ bosons do not couple only to fermions, but also to the other vector bosons of the theory. This important prediction follows from the structure of the theory, and can be tested experimentally.

The first clear experimental evidence for gauge-boson self interactions was given at LEP, using the process $e^+e^-\to W^+W^-$. At leading order, this process receives contributions from neutrino exchange in the $t$ channel, and from photon and $Z$-boson exchange in the $s$ channel. The three contributions are illustrated in Figure 14.

Figure 14: Contributions to W-boson pair production in electron positron collisions.

The expected contributions to $W^+W^-$ production cross section are illustrated in Figure 15 and compared to the data from the LEP experiments ( ALEPH:2006bhb). The latter are in excellent agreement with the calculation including all diagrams of Figure 14. Calculations including only neutrino exchange, or removing the diagram involving the $Z$ boson, diverge at high energy.

Figure 15: Energy dependence of the WW production cross section at LEP originally produced. Credit: ALEPH, DELPHI, L3, OPAL, LEP Electroweak Collaborations, Phys.Rept. 532 (2013) 119-244, e-Print: 1302.3415.

Gauge-boson interactions most often involve $W$ bosons, and their study has been generalized to all combinations of known electroweak mediators. This involves final states with photon pairs ($\gamma\gamma$), a photon and a $W$ or $Z$ boson ($W\gamma$, $Z\gamma$), and weak boson pairs ($WW$, $WZ$, $ZZ$). While initiated in $e^+e^-$ collisions at LEP ($\sqrt{s}=161-210$ GeV), measurements were pursed in $p\bar{p}$ collisions at the Tevatron ($\sqrt{s}=1.8-1.96$ TeV) (e.g. CDF:2004tuf, D0:2004fqq), and $pp$ collisions at the LHC ($\sqrt{s}=7-13.6$ TeV) (e.g. CMS:2020mxy, CMS:2022woe, CMS:2021icx, ATLAS:2017nei, ATLAS:2023avk). This variety of initial states, final states and centre-of-mass energies allows a precise decomposition of the production cross sections into elementary interactions, testing the triple gauge couplings (TGC) of Figure 6 in the Standard Model and probing possible new interactions. A recent status of gauge-boson pair-production studies at the LHC is given in Figure 21, summarizing measurements performed at CMS. Measurements with the largest samples reach a precision at the level of a few percent, and good agreement with the Standard Model prediction is found. Similar results are obtained by ATLAS. The so-called "vector-boson fusion" (VBF) processes constitute a particular class of interactions. Vector-boson fusion events are selected by imposing the presence of two highly energetic forward jets, which signal the emission of two vector bosons from the incoming protons in the $t$ channel. These vector bosons then interact, or "fuse", producing final states of varying complexity.

VBF processes are initiated by $W$ and $Z$ bosons in a proportion of approximately 10:1. The simplest and dominant VBF processes are $WW\to Z$ and $WZ\to W$, followed by the decay of the $W$ and $Z$ to fermions. Leptonic decays are used in the vast majority of cases, as they allow a clean selection with low backgrounds. VBF processes, with vector bosons in the initial state and fermions in the final state, can be seen as "mirrors" of gauge-boson pair-production processes discussed above, and again provide tests of the TGC vertices of the Standard Model.

In Run 2 and Run 3 of the LHC (2015-present), particular attention is given to the vector-boson scattering (VBS) processes, a sub-class of VBF events containing pairs of vector bosons in the final state (for example, $WW\to WW$ events can be seen as $W$ bosons scattering off each other). VBS events become important at scattering energies in excess of a few TeV, and probe the quartic gauge couplings of the SM.

VBS cross sections receive contributions from TGCs and QGCs (Figure 3), but also from diagrams involving Higgs bosons (Figure 4). Representative TGC, QGC and Higgs contributions to the $WW\to WW$ scattering process are illustrated in Figure 16. VBS cross section calculations result in infinities, when diagrams involving the Higgs boson are ignored. Measurements of VBS processes thus test the Higgs sector of the electroweak theory, in a complementary way to Higgs boson property measurements themselves.

Figure 16: Representative TGC (left), QCG (middle) and Higgs (right) contributions to the WW to WW scattering process.

Measurements of VBF processes ($qqW$, $qqZ$, where $qq$ indicates the forward jets signing these events) achieve a typical precision of about 5 percent, and provide tests of TGC's complementary to those obtained from VBS processes, as argued above. VBS measurements ($qqW\gamma$, $qqZ\gamma$, $qqWW$, $qqWZ$, $qqZZ$) are still in their infancy, and achieve typical precisions of 30-50 percent, down to 10 percent for the most precise channel. Good agreement with the SM predictions is observed so far in all cases. Precision measurements of VBS processes is one of the major goals of the high-luminosity phase of the LHC ( Apollinari:2017lan}.


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