Measuring the W mass

Knowing the W boson exists is one thing; pinning its mass to a few parts in ten thousand is another, and it is one of the sharpest tests the Standard Model faces. The difficulty is peculiar to the W. The Z boson announces its mass by resonance — scan the $e^+e^-$ energy across $91$ GeV and the cross-section lineshape traces out a Breit–Wigner peak whose centre is $m_Z$ . The W refuses to cooperate: at LEP it is made in pairs, $e^+e^-\to W^+W^-$ , and once the beam energy clears the pair threshold $\sqrt s>2m_W\approx161$ GeV the process is not resonant, so there is no peak to sit on. Worse, the W lives only about $3\times10^{-25}$ s and is never seen directly — every measurement rebuilds it from its decay fermions, which for the neutrino channel includes a particle that leaves no trace at all. Two ideas rescue the measurement, and this article is about both. At LEP, the total four-momentum of the event is known exactly — it is $(\sqrt s,\vec 0)$ — so a kinematic fit can recover even an escaping neutrino and sharpen the reconstructed mass to tens of MeV. At a hadron collider like the Tevatron, the colliding partons carry unknown momentum fractions, so the W's longitudinal boost is unknown and only transverse momenta balance; the trick there is the transverse mass, an observable built entirely from transverse quantities that is blind to the missing boost and piles events against a sharp Jacobian edge sitting exactly at $m_W$ . The measured value, $m_W\approx80.4$ GeV known to better than $\pm20$ MeV, now sits at the centre of the electroweak precision programme — and, since 2022, at the centre of a genuine anomaly.

Key takeaways

Unlike the Z, W-pair production at LEP ( $\sqrt s>2m_W$ ) is not resonant, so $m_W$ cannot be read off a cross-section peak; direct reconstruction of the decay fermions is used instead.
$e^+e^-\to W^+W^-$ appears in three topologies — fully leptonic ( $\ell\nu\ell\nu$ , 10.6%), semileptonic ( $\ell\nu q\bar q'$ , 43.9%), and fully hadronic ( $q\bar q'q\bar q'$ , 45.4%) — set by $\mathrm{BR}(W\to q\bar q')=67.41\%$ .
At LEP the event's total four-momentum is fixed at $(\sqrt s,\vec0)$ ; a constrained kinematic fit uses $\sum E=\sqrt s,\ \sum\vec p=0$ to recover the unseen neutrino and sharpen the mass.
At a hadron collider the longitudinal boost is unknown, so $m_W$ is extracted from the transverse mass $m_T=\sqrt{2p_T^\ell p_T^\nu(1-\cos\phi_{\ell\nu})}$ , which has a Jacobian edge at $m_T=m_W$ .
The finite width appears because the reconstructed mass follows a Breit–Wigner Lorentzian of width $\Gamma_W\approx2.1$ GeV about $m_W$ , not a sharp delta.
Combined results: LEP $m_W=80.376\pm0.033$ GeV; the CDF II (2022) hadron-collider value $80.4335\pm0.0094$ GeV sits $\sim7\sigma$ above the Standard-Model fit.
Key idea: because each W decays to detectable fermions you can rebuild its four-momentum event by event; the finite width shows up as a Lorentzian spread of reconstructed masses, and clever kinematic constraints — total energy at LEP, transverse balance at a hadron collider — recover the unmeasured neutrino and locate $m_W$ .

The core insight

Because each W decays to detectable fermions, you can rebuild the W four-momentum from its decay products event by event; the finite width appears as the Lorentzian spread of reconstructed masses, and clever kinematic constraints — the fixed CM energy at LEP, transverse balance at a hadron collider — recover the unmeasured neutrino momentum and fix $m_W$ .

Where this lives today

The W mass is not a curiosity but a linchpin of the electroweak precision programme. In the Standard Model $m_W$ , the Z mass, the weak mixing angle and the Fermi constant are not independent: at tree level $m_W^2(1-m_W^2/m_Z^2)=\pi\alpha/\sqrt2\,G_F$ , and the small correction factor $\Delta r$ that fixes this relation at loop level depends on the top-quark and Higgs masses through virtual particles that never appear as real states. So a precise $m_W$ is a window onto physics running around inside loops — new heavy particles would shift the prediction. That is why the measurement is pushed so hard: the global fit predicts $m_W\approx80.357$ GeV to about $6$ MeV, and any measured value that disagrees is potentially a discovery. The 2022 CDF II result, $80.4335\pm0.0094$ GeV, does exactly that — it lies roughly $7\sigma$ above the fit and in tension with the ATLAS ( $\approx80.37$ GeV) and LHCb measurements, and its resolution, whether experimental (recoil modelling, parton distributions, lepton energy scale) or genuinely new physics, is one of the live questions in the field. Meanwhile the LHC experiments continue to refine $m_W$ in the harsh, high-pileup hadron environment where controlling the recoil and the parton distributions is the whole game, and a future $e^+e^-$ collider running a WW threshold scan — the FCC-ee programme — aims to measure $m_W$ to better than $1$ MeV, turning the modest LEP threshold method of this article into one of the most stringent tests the Standard Model will ever face.

The physical picture

Start at LEP, where the W was pinned most cleanly. Above the pair threshold the beam makes two W bosons back-to-back, each of which decays within $10^{-25}$ s into a fermion–antifermion pair. Whether both decay to leptons, one to leptons and one to quarks, or both to quarks sorts the events into three cleanly separated topologies, and their relative rates simply count the branching ratios. The decisive advantage of an $e^+e^-$ machine is that the initial state is known exactly: the two W bosons together must carry the full beam four-momentum $(\sqrt s,\vec0)$ . That single fact turns the messy job of measuring four decay fermions into a constrained fit that can even solve for an escaping neutrino. The first figure lays out the three topologies and the constraint that makes them powerful.

Figure

At LEP2 above the pair threshold √s > 2m_W ≈ 161 GeV, e+e- → W+W- yields three topologies — fully leptonic 10.6%, semileptonic 43.9%, and fully hadronic 45.4% — and the fixed total four-momentum Σ E = √s, Σ p = 0 lets a kinematic fit sharpen the reconstructed mass toward m_W = 80.4 GeV

A hadron collider throws away exactly the advantage LEP enjoys. In $p\bar p\to WX$ the W is made by a single quark–antiquark pair, $u\bar d\to W^+$ , but the quark and antiquark carry unknown momentum fractions $x_1,x_2$ of their parent hadrons, so the W is born with an unknown longitudinal boost along the beam. The total longitudinal momentum of the event is therefore useless, and the beam remnants spray down the pipe unmeasured. Only in the transverse plane does momentum balance, and only there can the neutrino be reconstructed — as the missing transverse momentum recoiling against everything visible. The variable that makes a virtue of this restriction is the transverse mass, and its distribution carries the mass measurement in a single sharp feature. The second figure is the money plot.

Figure

The transverse-mass spectrum dN/dm_T for W → μν piles up against a Jacobian edge at m_T = m_W: the endpoint fixes m_W = 80.4 GeV, while the finite width Γ_W = 2.1 GeV and detector resolution smear it. Because m_T = √(2 p_T^ℓ p_T^ν (1 − cos φ)) uses only transverse quantities, it is immune to the unknown longitudinal boost

The core equations

The finite lifetime of the W means its reconstructed mass is never a single number. Because production and decay proceed through the W propagator $1/(q^2-m_W^2+im_W\Gamma_W)$ , recalled from aq-pp-w-boson-propagator, the invariant mass built from the decay fermions is distributed as a relativistic Breit–Wigner about $m_W$ — the same lineshape derived in aq-pp-resonances-breit-wigner.

\frac{dN}{dm}\ \propto\ \frac{m_W^2\,\Gamma_W^2}{\left(m^2-m_W^2\right)^2+m_W^2\,\Gamma_W^2} \qquad (1)

where: $m$ — invariant mass reconstructed from the decay fermions; $m_W$ — pole mass (the quantity we want); $\Gamma_W\approx2.1$ GeV — total width, the full width at half maximum of the peak. Reading: the "mass" of any one event scatters about $m_W$ with a spread set by $\Gamma_W$ , so a measurement is a fit of this shape to many events, not a reading of one. At LEP the mass is not read from a resonance but from a threshold. Just above pair production the cross section is governed by the velocity $\beta$ of the outgoing W bosons, because the dominant $t$ -channel neutrino-exchange amplitude produces the pair in an S-wave — so $\sigma$ rises steeply from zero, and its value near $161$ GeV is acutely sensitive to $m_W$ .

\sigma(e^+e^-\to W^+W^-)\ \propto\ \beta\ ,\qquad \beta=\sqrt{1-\frac{4m_W^2}{s}} \qquad (2)

where: $\beta$ — speed of each W in the CM frame; $s=(\sqrt s)^2$ — squared beam energy. Reading: a small shift in $m_W$ moves the threshold and changes $\beta$ sharply, so measuring $\sigma$ at one well-chosen energy measures $m_W$ — the LEP "threshold method". The reason an $e^+e^-$ collider can afford the direct-reconstruction method well above threshold is the exact initial state. Every measured and unmeasured particle in the event must sum to the beam four-momentum, giving four constraints (five if the two W masses are set equal).

\sum_i E_i=\sqrt s,\qquad \sum_i \vec p_i=\vec 0,\qquad \big[\,m(W^+)=m(W^-)\,\big] \qquad (3)

where: the sum runs over all final-state fermions (jets and leptons); the bracketed equal-mass condition is an optional fifth constraint. Reading: four conservation equations are enough to solve for the three unknown components of an escaping neutrino, which is what makes the semileptonic channel the "golden" one. At a hadron collider only the transverse plane is closed, so the neutrino is not solved but inferred: its transverse momentum is minus the vector sum of everything else the detector sees.

\vec p_T^{\,\nu}=-\,\vec p_T^{\,\ell}-\vec p_T^{\,\text{recoil}}=-\!\!\sum_{\text{visible}}\vec p_T \qquad (4)

where: $\vec p_T^{\,\ell}$ — measured charged-lepton transverse momentum; $\vec p_T^{\,\text{recoil}}$ — transverse momentum of the hadronic recoil (initial-state radiation and beam remnants). Reading: the neutrino is the event's missing transverse momentum; its longitudinal momentum stays forever unknown, which is exactly why a longitudinally-blind observable is needed. That observable is the transverse mass. Built only from transverse energies and momenta, for massless leptons it collapses to a compact form in the lepton–neutrino opening angle.

m_T^2=\left(E_T^\ell+E_T^\nu\right)^2-\left|\vec p_T^{\,\ell}+\vec p_T^{\,\nu}\right|^2\ \xrightarrow{\,m_\ell,m_\nu\to0\,}\ 2\,p_T^\ell\,p_T^\nu\left(1-\cos\phi_{\ell\nu}\right) \qquad (5)

where: $E_T=\sqrt{p_T^2+m^2}\to p_T$ for massless particles; $\phi_{\ell\nu}$ — azimuthal angle between lepton and neutrino. Reading: no longitudinal quantity appears, so $m_T$ is invariant under the unknown boost along the beam — the whole point of the construction. The measurement lives in the shape of the $m_T$ (or lepton- $p_T$ ) distribution, which has a Jacobian singularity. For a W at rest the lepton transverse momentum is $p_T^\ell=\tfrac12 m_W\sin\theta^*$ , and the density piles up where $dp_T^\ell/d\theta^*\to0$ .

\frac{dN}{dp_T^\ell}\ \propto\ \frac{1}{\sqrt{\,1-\left(2p_T^\ell/m_W\right)^2\,}}\ ,\qquad m_T\le m_W \qquad (6)

where: $\theta^*$ — decay angle in the W rest frame; the density diverges at $p_T^\ell=m_W/2$ and $m_T=m_W$ . Reading: the endpoint of the transverse mass and the peak of the lepton $p_T$ both sit at a value fixed by $m_W$ — the edge is the measurement.

Derivation AThe transverse mass and its Jacobian edge

The goal is to show from scratch why an observable built out of nothing but transverse quantities can measure a mass at all, and why its distribution ends in a sharp edge sitting exactly at $m_W$ . We define the transverse mass, reduce it to the compact opening-angle form for massless decay products, prove the bound $m_T\le m_W$ by evaluating it for a W at rest, and finally show that the event density diverges at the endpoint because the map from decay angle to transverse mass has zero slope there. The result is that the location of the edge — not the value of any single event — carries the mass, and that this edge is untouched by the unknown longitudinal boost that plagues a hadron collider.

Step 1. Define $m_T^2=(E_T^\ell+E_T^\nu)^2-(\vec p_T^{\,\ell}+\vec p_T^{\,\nu})^2$ with transverse energy $E_T=\sqrt{p_T^2+m^2}$ . For massless leptons $E_T\to p_T$ , and expanding the square gives $m_T^2=2p_T^\ell p_T^\nu-2\,\vec p_T^{\,\ell}\!\cdot\vec p_T^{\,\nu}=2p_T^\ell p_T^\nu(1-\cos\phi_{\ell\nu})$ . Why: only transverse components enter, so $m_T$ cannot depend on the boost along the beam — it is the same in every frame related by a longitudinal Lorentz transformation, which is precisely the unknown at a hadron collider.

Step 2. Evaluate $m_T$ for a W at rest. In the W frame each fermion has energy $m_W/2$ and is emitted at polar angle $\theta^*$ , so $p_T^\ell=p_T^\nu=\tfrac12 m_W\sin\theta^*$ , and the two are back-to-back in azimuth, $\phi_{\ell\nu}=\pi$ , giving $1-\cos\phi=2$ . Then $m_T^2=2\cdot(\tfrac12 m_W\sin\theta^*)^2\cdot2=m_W^2\sin^2\theta^*$ , so $m_T=m_W\sin\theta^*$ . Why: this is the key inequality — since $\sin\theta^*\le1$ , every event has $m_T\le m_W$ , with equality only for a decay transverse to the beam; the true mass is the maximum the transverse mass can reach.

Step 3. Find the density. With $m_T=m_W\sin\theta^*$ we have $m_T^2=m_W^2(1-\cos^2\theta^*)$ , so $2m_T\,dm_T=-2m_W^2\cos\theta^*\,d\cos\theta^*$ and $dN/dm_T=(dN/d\cos\theta^*)\,|d\cos\theta^*/dm_T|=(dN/d\cos\theta^*)\,\dfrac{m_T}{m_W^2|\cos\theta^*|}$ . As $\theta^*\to\pi/2$ , $\cos\theta^*\to0$ and the density diverges — an integrable Jacobian singularity right at $m_T=m_W$ . Why: the pile-up is not a coincidence of dynamics but pure kinematics; the transverse mass spends a disproportionate number of events near its endpoint, which is what makes the edge a sharp, statistically powerful marker of $m_W$ .

Step 4. Restore reality. The finite width smears the sharp $m_T=m_W$ endpoint into the Breit–Wigner of equation (1), and detector resolution on $p_T^\ell$ and on the recoil widens it further; a genuine transverse recoil $p_T^W\ne0$ shifts individual events but leaves the edge position far more stable than the lepton- $p_T$ peak. Why: this is why experiments fit the full templated $m_T$ shape rather than read the endpoint by eye, and why $m_T$ is preferred over the lepton $p_T$ spectrum — the edge is first-order insensitive to the W's own transverse motion.

Go deeper · Derivation B · 2 minWhy the WW threshold turn-on pins the mass

Derivation BWhy the WW threshold turn-on pins the mass

The goal is to explain the LEP threshold method: how running the collider at a single, carefully chosen energy just above $2m_W$ turns a cross-section measurement into a mass measurement, and to show that the sensitivity is greatest right at the turn-on. We establish the near-threshold behaviour $\sigma\propto\beta$ from the S-wave character of the dominant amplitude, differentiate the velocity with respect to $m_W$ , and show that the resulting sensitivity diverges as the threshold is approached — so that a modest cross-section measurement translates into a competitive mass, subject to the tradeoff that the rate itself vanishes exactly at threshold.

Step 1. Near threshold the pair is produced almost at rest, and the dominant $t$ -channel neutrino-exchange diagram of $e^+e^-\to W^+W^-$ yields an S-wave final state, whose phase space and matrix element combine to give $\sigma\propto\beta$ with $\beta=\sqrt{1-4m_W^2/s}$ . Why: an S-wave two-body cross section rises linearly in the outgoing speed, so the entire steepness of the turn-on is controlled by how $\beta$ depends on $m_W$ — the mass is encoded in the shape of the rise, not in a peak.

Step 2. Differentiate the velocity at fixed beam energy. From $\beta^2=1-4m_W^2/s$ , $2\beta\,d\beta=-(8m_W/s)\,dm_W$ , so $\dfrac{d\beta}{dm_W}=-\dfrac{4m_W}{s\,\beta}$ . Since $\sigma\propto\beta$ , the fractional sensitivity is $\dfrac{1}{\sigma}\dfrac{d\sigma}{dm_W}=\dfrac{1}{\beta}\dfrac{d\beta}{dm_W}=-\dfrac{4m_W}{s\,\beta^2}$ . Why: the factor $1/\beta^2$ blows up as $\beta\to0$ , so a fixed shift in $m_W$ produces an ever-larger fractional change in the cross section the closer the beam sits to threshold — maximum leverage lives at the turn-on.

Step 3. Choose the running point. Exactly at threshold $\beta=0$ and $\sigma=0$ , so there are no events to count; the optimum is a compromise a little above threshold — LEP took a dedicated run at $\sqrt s\approx161$ GeV — where $\beta$ is small enough for high sensitivity but the rate is large enough for a measurement. Why: the threshold method trades statistics against sensitivity, and the sweet spot delivers a per-experiment precision of a few hundred MeV from a single energy point, a valuable cross-check on direct reconstruction.

Step 4. Read the number. At $\sqrt s=161$ GeV a shift $\Delta m_W=-0.1$ GeV moves $\beta$ from $0.0498$ to $0.0705$ , a $41\%$ rise in $\sigma$ — an enormous response to a $0.1\%$ change in the mass. Why: this outsized fractional response is the whole content of the method; even a coarse cross-section measurement at one energy resolves $m_W$ to the hundred-MeV level, and combining many experiments and the direct-reconstruction data brings LEP to $\pm33$ MeV.

Go deeper · Derivation C · 2 minRecovering the neutrino with the LEP kinematic fit

Derivation CRecovering the neutrino with the LEP kinematic fit

The goal is to show precisely how the fixed initial state at an $e^+e^-$ collider lets a semileptonic $\ell\nu q\bar q'$ event be fully reconstructed despite the invisible neutrino, and how the surplus of constraints beyond the bare count sharpens the mass resolution. We tally the unknowns, count the conservation equations that energy–momentum balance supplies, show that the neutrino is exactly determined, and then explain how imposing the additional equal-mass condition over-constrains the fit and pulls the reconstructed mass toward its true value. The payoff is a resolution far finer than the raw jet-energy measurement would suggest.

Step 1. Count the unknowns in $\ell\nu q\bar q'$ . The charged lepton and the two jets are measured; the only unknowns are the three components of the neutrino momentum (its energy follows from masslessness, $E_\nu=|\vec p_\nu|$ ). So there are three unknowns. Why: at a hadron collider these three would be irreducible, but at LEP the closed initial state supplies enough equations to solve for all of them, which is the structural reason the semileptonic channel is "golden".

Step 2. Count the constraints. Total energy–momentum conservation, equation (3), gives four equations: $\sum E=\sqrt s$ and $\sum\vec p=\vec0$ . Three unknowns against four constraints leaves the neutrino fully determined and one relation to spare — a "4C" fit. Why: solving for $\vec p_\nu$ turns an event with an invisible particle into a completely reconstructed one, so the leptonic W's invariant mass can be built just as if the neutrino had been seen.

Step 3. Spend the spare constraint. Requiring the two reconstructed W masses to be equal, $m(W^+)=m(W^-)$ , adds a fifth equation — a "5C" fit — that trades the residual freedom for a tighter mass estimate, since jet-energy mismeasurements are corrected against the beam-energy constraint. Why: the dominant uncertainty in the raw reconstruction is jet-energy resolution; the beam-energy constraint effectively recalibrates each event's jets, shrinking the per-event mass error from several GeV to roughly $2$ – $3$ GeV.

Step 4. Fit the ensemble. The width $\Gamma_W$ and residual resolution spread the per-event masses into a Breit–Wigner-plus-Gaussian shape; fitting this template over thousands of events across the semileptonic and hadronic channels extracts the peak position $m_W$ to tens of MeV, the LEP combined result being $80.376\pm0.033$ GeV. Why: the mass is a property of the ensemble's peak, so the finite per-event resolution and the intrinsic width both enter the template but neither limits the final precision, which is statistics- and systematics-driven.

Limiting cases

Case	Result	Meaning
Exactly at threshold $\sqrt s=2m_W$	$\beta\to0$ , $\sigma\to0$	no events to count; run just above threshold where $d\sigma/dm_W$ is largest
Well above threshold $\sqrt s\gg2m_W$	WW non-resonant; use direct reconstruction	mass from the invariant mass of decay fermions, not a lineshape peak
W produced at rest (hadron collider)	lepton $p_T$ peaks at $m_W/2$ , $m_T$ edge at $m_W$	pure Jacobian singularity fixes the mass
W with transverse recoil $p_T^W\ne0$	lepton- $p_T$ peak smears; $m_T$ edge stable	why $m_T$ is preferred over the lepton $p_T$ spectrum
Zero width $\Gamma_W\to0$	reconstructed mass $\to\delta(m-m_W)$ ; edge infinitely sharp	the finite width Lorentzian-broadens the peak and the edge
Narrow-width limit $\Gamma_W/m_W\approx2.7\%\ll1$	four-fermion process factorises into two on-shell W's	justifies the two-real-W picture used for mass extraction

Common trapThe transverse mass of a single event equals the W mass

It does not. Equation (5) and Derivation A show that $m_T\le m_W$ for every event, with equality reached only in the special case of a decay exactly transverse to the beam. A single event's $m_T$ is therefore a lower bound on $m_W$ , scattered below it by the decay angle, the finite width, and detector resolution — reading one event as "the mass" would systematically underestimate $m_W$ . The mass lives in the shape of the whole distribution: it is the location of the Jacobian edge where the events pile up, extracted by fitting a template of the full $m_T$ spectrum. The same caution applies to the lepton $p_T$ spectrum, whose peak sits at $m_W/2$ only for a W with no transverse recoil. And underneath all of this is a subtler point: because the reconstructed mass follows the Breit–Wigner of equation (1), the WW final state is really a four-fermion process through two virtual W propagators, not two textbook on-shell particles — the two-real-W picture is an approximation good to order $\Gamma_W/m_W$ , not an exact statement.

Worked examples

1 · The transverse mass of a $W\to\mu\nu$ event (fully worked)

A candidate $W^+\to\mu^+\nu$ event at a hadron collider has a muon with $p_T^\ell=38.0$ GeV and missing transverse momentum $p_T^\nu=42.0$ GeV, separated in azimuth by $\phi_{\ell\nu}=168^\circ$ . Compute the transverse mass, compare it with $m_W=80.4$ GeV, and say what the event contributes to the measurement.

Step 1. The neutrino is not seen directly; its transverse momentum is the missing transverse momentum of equation (4), here $42.0$ GeV, pointing opposite the vector sum of the muon and hadronic recoil. Its longitudinal momentum is unknown and, crucially, not needed.

Step 2. Apply the massless transverse-mass formula (5): $m_T=\sqrt{2\,p_T^\ell\,p_T^\nu\,(1-\cos\phi_{\ell\nu})}$ . Evaluate the angular factor: $\cos168^\circ=-0.9781$ , so $1-\cos\phi=1.9781$ .

Step 3. Insert the numbers: $2\times38.0\times42.0=3192\ \mathrm{GeV}^2$ , times $1.9781$ gives $6314\ \mathrm{GeV}^2$ , so $m_T=\sqrt{6314}=\mathbf{79.5\ GeV}$ .

Step 4. Interpret. The result sits just below $m_W=80.4$ GeV, exactly as the bound $m_T\le m_W$ demands — this muon–neutrino pair was emitted a little away from the transverse direction ( $\sin\theta^*\approx79.5/80.4=0.989$ , i.e. $\theta^*\approx81^\circ$ ). One event like this is a single entry near the top of the spectrum; thousands of them trace out the Jacobian edge whose position is fitted to yield $m_W$ . m_T = 79.5 GeV, below and consistent with m_W = 80.4 GeV; the event populates the edge that carries the mass.

2 · Sensitivity of the LEP threshold method

LEP ran a dedicated point at $\sqrt s=161$ GeV to measure $m_W$ from the WW cross-section turn-on. Using $\sigma\propto\beta$ , quantify how sharply the cross section responds to a $0.1$ -GeV change in $m_W$ .

Step 1. Compute $\beta$ for $m_W=80.40$ GeV: $\beta=\sqrt{1-4(80.40)^2/(161)^2}=\sqrt{1-25856.6/25921}=\sqrt{0.002483}=\mathbf{0.0498}$ . The W bosons crawl away at $5\%$ of $c$ — the pair is nearly at rest.

Step 2. Recompute for $m_W=80.30$ GeV: $4(80.30)^2/(161)^2=25792.4/25921=0.99504$ , so $\beta=\sqrt{0.004963}=\mathbf{0.0705}$ .

Step 3. Take the ratio. Since $\sigma\propto\beta$ , the cross section for the lighter mass is larger by $0.0705/0.0498=\mathbf{1.41}$ — a $\mathbf{41\%}$ increase in $\sigma$ for a mere $0.1\%$ decrease in $m_W$ .

Step 4. Read off the physics. This outsized fractional response — captured by the $1/\beta^2$ sensitivity of Derivation B — is why a single-energy cross-section measurement resolves the mass at all. The tradeoff is statistics: sitting closer to threshold sharpens the leverage but starves the event rate, so LEP chose $161$ GeV as the compromise. A 0.1-GeV mass shift moves the near-threshold cross section by ≈ 41%.

3 · Why the semileptonic channel is golden

From $\mathrm{BR}(W\to q\bar q')=0.6741$ , work out the fraction of $e^+e^-\to W^+W^-$ events in each topology and explain which is best for measuring $m_W$ .

Step 1. Let $h=0.6741$ be the hadronic branching ratio and $\ell=1-h=0.3259$ the leptonic one. The two W decays are independent, so the topology fractions are products.

Step 2. Fully hadronic ( $q\bar q'q\bar q'$ ): $h^2=0.6741^2=\mathbf{45.4\%}$ . Semileptonic ( $\ell\nu q\bar q'$ ): $2h\ell=2(0.6741)(0.3259)=\mathbf{43.9\%}$ . Fully leptonic ( $\ell\nu\ell\nu$ ): $\ell^2=0.3259^2=\mathbf{10.6\%}$ . The three sum to $100\%$ .

Step 3. Rank them for the mass. The fully leptonic channel has two neutrinos and only four constraints — under-determined, so it is not used for $m_W$ . The fully hadronic channel is fully visible but suffers colour reconnection and Bose–Einstein correlations between the two overlapping hadronic systems (the W decay length $c\tau\approx0.09$ fm is far shorter than the $\sim1$ fm hadronisation scale), a systematic of tens of MeV.

Step 4. The winner. The semileptonic channel has exactly one neutrino, recoverable by the kinematic fit of Derivation C, and no cross-talk between W's — a clean, fully reconstructed event. Fractions 45.4% / 43.9% / 10.6%; the 43.9% semileptonic channel is the golden one for m_W.

Physics of the portrait

The mass of the W is measured by a variable, not a person — an observable clever enough to see through the one thing a hadron collider cannot measure. Its story is the story of learning to live without the longitudinal momentum.

Origins · 1 mintransverse mass and how this was discovered

transverse mass

the boost-invariant observable that pins m_W where the longitudinal momentum is unknown · precision W-mass measurement at hadron colliders

When the W and Z were discovered at CERN's $Sp\bar pS$ collider in 1983, the experimenters faced a problem with no clean solution: in $p\bar p\to W\to e\nu$ the neutrino escapes, and because the colliding partons carry unknown momentum fractions the W's longitudinal boost is unknown too, so the invariant mass of the decay pair simply cannot be formed. The response was to build a mass out of the one plane in which momentum is conserved. The transverse mass $m_T=\sqrt{2p_T^\ell p_T^\nu(1-\cos\phi_{\ell\nu})}$ , sharpened in 1983 by theorists including Barger, Phillips, Smith, Van Neerven and Vermaseren as a refinement of the earlier Jacobian-peak method, uses only transverse quantities and is therefore blind to the missing boost — and, as a pure matter of kinematics, its distribution ends in a Jacobian edge sitting exactly at $m_W$ . UA1 and UA2 read the first W masses from the Jacobian peak of the electron $p_T$ spectrum; the Tevatron experiments CDF and D0 turned $m_T$ into a precision tool, controlling the lepton energy scale against the known Z mass and the recoil model against $Z\to\ell\ell$ data until they reached $\sim15$ MeV. LEP approached the same number from the opposite direction — a fixed, fully-known initial state and a kinematic fit — and the two methods agree. The transverse mass is now the workhorse of every hadron-collider W-mass measurement, and it is the variable at the heart of the 2022 CDF II result of $80.4335\pm0.0094$ GeV that stands, unexplained, some seven standard deviations above the Standard-Model expectation.

History

1983

— UA1 and UA2 read the first W mass from the Jacobian peak of the lepton spectrum at CERN's Spp̄S; the transverse-mass variable is introduced

1989

— LEP and the SLC begin precision Z measurements that sharpen the Standard-Model prediction for m_W

1996

— LEP2 crosses the W-pair threshold at √s = 161 GeV, measuring m_W from the cross-section turn-on

2012

— the Tevatron experiments CDF and D0 reach ≈ 15–20 MeV precision on m_W using the transverse mass

2022

— CDF II reports m_W = 80.4335 ± 0.0094 GeV, about 7σ above the Standard-Model electroweak fit

Going further

Texts. Thomson, Modern Particle Physics, §16.3 (properties of the W boson), §16.3.1 (measurement of the W mass and width at LEP), §16.3.2 (measurement of the W mass at the Tevatron) · Halzen & Martin, Quarks and Leptons, Ch. 15 (weak interactions at high energy) · Griffiths, Introduction to Elementary Particles, 2nd ed., Ch. 9–10 · Peskin & Schroeder, An Introduction to QFT, §21.3 (electroweak precision) · Schwartz, QFT and the Standard Model, Ch. 29–31. Key checks: $m_T=\sqrt{2p_T^\ell p_T^\nu(1-\cos\phi)}\le m_W$ with the Jacobian singularity at the endpoint; the threshold sensitivity $\tfrac1\sigma\,d\sigma/dm_W=-4m_W/s\beta^2$ and the numerical $41\%$ response at $161$ GeV; the topology fractions $45.4/43.9/10.6\%$ from $\mathrm{BR}(W\to q\bar q')=0.6741$ ; and the LEP result $m_W=80.376\pm0.033$ GeV and $\Gamma_W=2.195\pm0.083$ GeV.

Prerequisites. Properties of the W Bosons · Breit–Wigner Resonances · The W-Boson Propagator · The Z Resonance at LEP. Leads to. Electroweak Radiative Corrections · The Top Quark · The Higgs Discovery.

Ask the tutor: Want to see why the transverse-mass edge stays fixed at $m_W$ even when the W is kicked sideways by QCD recoil, while the lepton $p_T$ peak drifts — the exact reason $m_T$ won?

Why can the Z mass be measured from a peak but the W mass cannot?

The Z is produced singly and directly in $e^+e^-\to Z$ , so scanning the beam energy across $91$ GeV traces the resonance and its lineshape peak is $m_Z$ . The W is charged, so it cannot be made singly in $e^+e^-$ ; it comes in pairs, and once the beam clears the pair threshold $\sqrt s>2m_W$ the process $e^+e^-\to W^+W^-$ is a continuum, not a resonance — there is no peak to scan. The mass must instead be reconstructed from the decay fermions of each W event by event, or read from the threshold turn-on of the pair cross section. It is a direct consequence of the W being charged and the Z neutral.

What exactly is the "missing transverse momentum" and why only transverse?

When a neutrino escapes a detector it carries away momentum unseen, so the visible particles do not balance. In the transverse plane the initial state has essentially zero net momentum (the beams are collinear), so momentum conservation forces the visible transverse momenta plus the neutrino's to sum to zero — the neutrino's $p_T$ is minus the vector sum of everything measured. Along the beam this argument fails: the colliding partons carry unknown longitudinal momentum fractions, and the beam remnants vanish down the pipe unmeasured, so there is no longitudinal balance to exploit. That asymmetry — closed transverse plane, open longitudinal axis — is the entire reason the transverse mass exists.

Why is the transverse mass preferred over the lepton

p_T

spectrum?

Both have a Jacobian singularity fixed by $m_W$ — the lepton $p_T$ peaks at $m_W/2$ and the transverse mass ends at $m_W$ . The difference is robustness. The lepton $p_T$ peak is smeared and shifted whenever the W itself has transverse momentum from initial-state QCD radiation, and modelling that recoil precisely is hard. The transverse mass, because it combines the lepton and neutrino together, is first-order insensitive to the W's transverse boost, so its edge stays put even when the recoil is imperfectly known. Experiments use both as cross-checks, but $m_T$ carries the smaller systematic and usually drives the result.

Why does the finite width matter for a mass measurement?

Because the W is not a stable particle with a single sharp mass. Its propagator $1/(q^2-m_W^2+im_W\Gamma_W)$ makes the invariant mass built from the decay fermions scatter about $m_W$ as a Breit–Wigner of width $\Gamma_W\approx2.1$ GeV — the "mass" of one event is drawn from a distribution, not fixed. A measurement therefore fits this Lorentzian shape (convolved with detector resolution) to many events and extracts the pole position $m_W$ from the peak. The width also smears the Jacobian edge of the transverse mass, so $\Gamma_W$ enters every template; conveniently, the same data can fit both the peak position ( $m_W$ ) and its width ( $\Gamma_W$ ) at once.

What are colour reconnection and Bose–Einstein correlations, and why do they only bite the four-jet channel?

In the fully hadronic $W^+W^-\to q\bar q'q\bar q'$ channel the two W bosons decay almost on top of each other: the W decay length $c\tau=\hbar c/\Gamma_W\approx0.09$ fm is far shorter than the $\sim1$ fm scale over which quarks hadronise. So the two hadronic systems overlap in space-time and can exchange soft gluons (colour reconnection) or have their identical bosons correlated (Bose–Einstein), blurring which hadrons came from which W. That cross-talk shifts the reconstructed mass by tens of MeV and is a leading systematic. The semileptonic channel has only one hadronic W, so there is nothing to reconnect to — another reason it is golden.

Why is the CDF II 2022 W-mass result such a big deal?

Because $m_W$ is over-constrained in the Standard Model. Given the Z mass, the Fermi constant and the fine-structure constant, the theory predicts $m_W\approx80.357$ GeV to about $6$ MeV, with the small loop correction $\Delta r$ depending on the top and Higgs masses. Any precise disagreement points to new virtual particles. The CDF II measurement, $80.4335\pm0.0094$ GeV, sits roughly $7\sigma$ above that prediction and also in tension with the ATLAS and LHCb values, which agree with the Standard Model. Either an experimental effect (recoil modelling, parton distributions, energy calibration) is being missed, or there is genuinely new physics — and disentangling the two is one of the sharpest open questions in precision electroweak physics.

Measuring the W mass

Where this lives today

The physical picture

The core equations

Limiting cases

Worked examples

1 · The transverse mass of a $W\to\mu\nu$ event (fully worked)

2 · Sensitivity of the LEP threshold method

3 · Why the semileptonic channel is golden

Physics of the portrait

Going further

Before this

Where this leads

Measuring the W mass

Where this lives today

The physical picture

The core equations

Limiting cases

Worked examples

1 · The transverse mass of a W→μνW\to\mu\nuW→μν event (fully worked)

2 · Sensitivity of the LEP threshold method

3 · Why the semileptonic channel is golden

Physics of the portrait

Going further

Before this

Where this leads

1 · The transverse mass of a $W\to\mu\nu$ event (fully worked)