The mass weighed without one of its pieces

The invariant that survives a boost found the quantity every observer agrees on: the energy and momentum of a system, combined as $E^2 - (pc)^2$, give $(mc^2)^2$ in every frame, and the mass of a particle that decayed can be found by adding the four-momenta of its products and taking that combination. Every unstable particle discovered in a collider since the 1960s has been found that way, as a bump in the invariant mass of its decay products. The cone a decay cannot leave followed the same decay into the laboratory and found the invariant mass staying put while every other kinematic quantity moved with the parent.

The method needs every product to be measured. The W boson, the carrier of the weak force, decays a third of the time into a charged lepton and a neutrino, and the neutrino interacts so weakly that it crosses every detector ever built without leaving a mark. Its energy and momentum are unknown, the invariant mass cannot be formed, and yet the W’s mass is known to a part in ten thousand. The way it is measured is a small masterpiece of relativistic kinematics: give up on the direction that cannot be measured and use the two that can.

What a hadron collider cannot know #

Ws are made in proton collisions, when a quark from one proton meets an antiquark from the other. Each carries an unknown share of its proton’s momentum, so the W they make moves along the beam at an unknown speed, different in every collision. That is the first thing lost: the W’s motion along the beam, and with it any quantity that depends on it. The second is the neutrino.

What remains is the plane across the beam. The colliding quarks had almost no momentum in that plane, so the total momentum of everything produced, summed in that plane, must be nearly zero. A detector that measures every other particle in the collision — the electron, and the spray of debris from the protons’ remnants — can therefore infer the neutrino’s momentum across the beam as whatever is needed to balance the sum. Its momentum along the beam stays unknown. So an experiment knows the electron completely, the neutrino’s transverse momentum by inference, and nothing about either particle’s motion along the beam that it can use.

A peak made by a sine #

In the W’s own frame the decay is simple. The two products fly apart back to back, each carrying exactly half the W’s rest energy, $M/2 = 40.19$ GeV, in a direction that varies from decay to decay. Seen from the laboratory, the electron’s momentum across the beam is that $M/2$ multiplied by the sine of the angle it makes with the beam, and that is where the first figure’s peak comes from. The sine of an angle rises quickly near zero and lingers near its maximum, changing little between seventy and ninety degrees. Electrons emitted over a wide range of angles near ninety degrees all have transverse momenta just below $M/2$, and they pile up there. In the figure’s dashed curve, for a W with no sideways motion and no spread of mass, the density rises without limit as the transverse momentum approaches $M/2$ and then stops dead.

The pile-up is called a Jacobian peak, after the factor $1/\sqrt{(M/2)^2 - p_T^2}$ that changing variables from angle to transverse momentum introduces. The neutrino energy an angle fixes found a peak of the same kind in the neutrinos from pion decays, gathered at one energy because a range of pion energies all give nearly the same neutrino energy at a fixed angle. In both, a smooth spread of one variable is folded into a spike in another by a function that turns over.

The spike is what makes the peak useful: its position is $M/2$, read off a distribution of electrons alone. Carlo Rubbia’s UA1 and the UA2 collaboration at CERN discovered the W in 1983 with a few dozen decays, and the electrons’ transverse momenta crowded towards forty GeV. But the solid and lighter curves show the peak’s weakness. The W is not at rest across the beam: it recoils against whatever else the collision made, by a few GeV on average, and a sideways boost adds to the electron’s transverse momentum at first order, smearing the peak and shifting it.

An edge that sideways motion barely moves #

The better quantity uses both particles. Form the invariant mass as though the momenta along the beam were zero — use only the transverse momenta of the electron and the neutrino and the angle between them in the transverse plane:

$$m_T = \sqrt{2\,p_T^{e}\,p_T^{\nu}\,(1-\cos\Delta\varphi)}.$$

Two things are true of it. It is unchanged by a boost along the beam, because a boost along the beam does not touch transverse momenta; so the unknown speed of the W along the beam does not matter. And it can never exceed the W’s mass: dropping the momenta along the beam can only reduce the invariant mass, so $m_T \le M$ in every decay, with equality only when both products leave at right angles to the beam. The distribution of $m_T$ therefore rises to an edge at the W’s mass and stops, as the second figure shows. The only events beyond it come from Ws that were themselves heavier than average: a W is not a fixed mass but a resonance, with the natural width of 2.1 GeV that the width that is a lifetime related to its lifetime of 3 × 10⁻²⁵ seconds.

The edge’s virtue is its indifference to the W’s sideways motion. A small transverse boost raises one product’s transverse momentum and lowers the other’s, and to first order the changes cancel in $m_T$; only their squares remain. The lighter curve, for Ws moving sideways at six GeV on average, has its edge in the same place, a little softened.

Why the edge cannot be passed #

The bound $m_T \le M$ has a short proof that says exactly what is being thrown away. Describe each massless lepton by its transverse momentum, its direction φ in the transverse plane and its rapidity y along the beam. The invariant mass of the pair is then

$$M^2 = 2\,p_T^{e}\,p_T^{\nu}\,\big(\cosh\Delta y - \cos\Delta\varphi\big),$$

and the transverse mass is the same expression with $\cosh\Delta y$ replaced by 1. Since $\cosh$ of anything is at least 1, the transverse mass is always the smaller, and the two agree only when the leptons have the same rapidity. Everything the measurement gives up is in that one difference of rapidities, the one quantity the neutrino’s invisibility along the beam hides; and because only the difference appears, the bound holds however fast the W was moving along the beam.

The Z as a ruler #

An edge is only as good as the scale it is read on. A one-part-in-ten-thousand measurement of the W’s mass needs the electron’s momentum measured to better than that, and the neutrino’s inferred momentum understood almost as well, and no detector is built that accurately. The calibration comes from the W’s neutral partner. The Z boson decays to an electron and a positron, or a muon pair, with both products seen, so its invariant mass can be formed directly, and its mass is known from the LEP colliders to two parts in a hundred thousand. Measuring the Z’s mass in the same detector, from the same kind of leptons, calibrates the momentum scale: the measured Z peak is made to fall where LEP put it.

The Z also stands in for the neutrino. Remove one of its two leptons from the analysis by hand, pretend it was not seen, and the event looks like a W decay: one lepton, an imbalance in the transverse plane, and the same debris from the colliding protons. Comparing the inferred momentum of the removed lepton with its actual, measured momentum tests the recoil model that the W analysis relies on. The precision of the W mass is, to a large extent, the precision with which the Z can be made to imitate it.

What a push along the beam changes #

The third figure makes the indifference to the unknown motion along the beam explicit. The electron’s energy depends on that motion: it is the box-shaped spectrum the cone a decay cannot leave found for a decay in flight, a single line at forty GeV for a W at rest, stretching to a box from five to nearly three hundred GeV for a W at a rapidity of two. Since the experiment cannot know the W’s rapidity, the electron’s energy is useless for weighing it. Its transverse momentum is the same distribution at every rapidity, the three curves lying exactly on one another. The angle a collider counts in logarithms introduced rapidity as the variable in which boosts along the beam simply add; the transverse momentum and the transverse mass are the quantities those boosts do not reach at all.

Peak against edge #

The fourth figure measures the difference. As the W’s mean sideways motion grows from nothing to fifteen GeV, the falling edge of the electron’s transverse momentum, doubled to put it in units of the mass, moves outward by several GeV; the edge of the transverse mass hardly moves. A mass fitted from the electron alone needs the W’s sideways motion modelled to an accuracy of a few per cent of itself to reach a precision of ten MeV in the mass; one fitted from the transverse mass needs it far less.

The transverse mass has its own weakness, and it is the neutrino. Its transverse momentum is inferred from everything else the detector sees, and the debris of the protons’ remnants is measured far less precisely than the electron. Real measurements fit both distributions — and the muon equivalents, since a W decays to a muon and a neutrino as often as to an electron — and compare the masses they give, and the agreement or disagreement between them is one of the checks on how well the sideways motion and the detector’s response are understood.

The W’s mass, measured #

The W’s mass matters because it is predicted. Once the masses of the Z boson, the top quark and the Higgs boson are known, along with the strength of the electromagnetic and weak forces, the Standard Model fixes the W’s mass through quantum corrections — the virtual particles that, in the vacuum around a W, slightly change its mass from the value the simplest form of the theory would give. A particle not in the model, if it couples to the W, would change those corrections and move the W’s mass. So a precise measurement is a search for the unknown, conducted by weighing something known.

The last figure shows where the measurements stand. The LEP colliders weighed Ws produced in pairs from electron–positron collisions, where the energy is known; the hadron-collider experiments at Fermilab and CERN used the transverse mass and the lepton’s transverse momentum. In 2022 the CDF collaboration, analysing its full Tevatron data, reported 80,433.5 MeV with an uncertainty of 9.4, seven standard deviations above the prediction of 80,357. A discrepancy of seventy-six MeV in eighty thousand would be a sign of new physics. The CMS experiment’s measurement of 2024, of comparable precision, agrees with the prediction instead, as did earlier measurements from ATLAS, LHCb and the Tevatron’s D0. The disagreement is unresolved, and it turns on the things this essay’s figures are about: how well a few GeV of the W’s sideways motion are modelled, how well the momentum scale of the detector is calibrated, and how much the shape of the edge depends on the colliding quarks’ share of their protons’ momenta.

Weighing what is never seen #

The transverse mass has outgrown the W. Any heavy particle that decays into something visible and something that escapes leaves the same signature, an imbalance of momentum in the transverse plane, and the same trick weighs it. Searches for new particles at the Large Hadron Collider are, to a large extent, searches for an imbalance in the transverse plane that the known invisible particle, the neutrino, cannot account for — the signature expected of the particles that might make up the dark matter, which would leave the detector as silently as a neutrino does. When such particles are produced in pairs, each decaying to something visible and something invisible, a generalisation of the transverse mass that minimises over the unknown way the invisible momenta are shared between the two decays gives an edge at the parent’s mass in the same way, and it is the variable those searches are built on. In each case the measurement concedes what cannot be known along the beam and extracts a mass from what is left across it.

Where the kinematics holds #

The figures use the simplest form of the decay. A real W is polarised, because the weak force couples only to left-handed particles, so its decay products are not emitted isotropically in its frame but with an angular distribution that depends on the W’s charge and the direction of the colliding quarks; that changes the shape of the Jacobian peak and must be modelled. The figures take the leptons to be massless, which for an electron is exact to a part in a hundred thousand. They draw the W’s sideways motion from an assumed distribution with a six-GeV mean, where the real distribution comes from the physics of the colliding quarks’ emissions and is one of the largest uncertainties in the measurement. And they omit the detector entirely: real distributions are smeared by the resolution of the electron’s measurement, the much coarser resolution of the inferred neutrino momentum, and the photons the electron radiates as it leaves the decay.

What the figures cannot show #

The figures show distributions of simulated decays, not the measurement. A real analysis fits a measured distribution with templates simulated for many trial masses and finds the mass whose template matches best, and its uncertainty is dominated not by the number of Ws, which runs to millions, but by how well every ingredient of the simulation is known. Nor can the figures show the neutrino’s absence, which is the whole of the problem. The energy that did not all arrive found the neutrino first as missing energy in beta decay, inferred from a spectrum that could not be explained without it. Seventy years on, the same absence is still measured by what it leaves unbalanced.

Still open: whether the W is heavier than predicted #

The CDF result and the measurements that disagree with it cannot all be right unless something about the analyses is wrong, and identifying what has not been possible from the published information alone. New measurements from the large LHC data sets, with better-understood detectors and improved theory of the W’s production, are expected to sharpen the comparison, and combinations that account for the correlated uncertainties between experiments — the shared model of the protons’ internal structure, above all — are under way. If CDF is confirmed, the Standard Model’s account of the vacuum around a W is incomplete; if not, the episode will have taught the field how fragile a one-part-in-ten-thousand measurement of an edge can be.

The habit worth carrying away is to measure in the directions that are known. When a W decays and its neutrino escapes, the invariant mass cannot be formed, but across the beam the electron’s momentum piles up in a Jacobian peak at M/2 = 40.19 GeV and the transverse mass $\sqrt{2p_T^e p_T^\nu(1-\cos\Delta\varphi)}$ rises to an edge at the W’s mass that no decay can exceed — blind to the W’s unknown motion along the beam, and moved by its sideways motion only at second order, 0.01 GeV for 6 GeV where the electron’s edge moves 2.44. The mass is weighed from the shadow it casts across the beam.