Illustrated Physics
physics, drawn
Concept

Jacobian peak — where it appears

A sharp edge in the distribution of a measured quantity at the point where it stops changing with the hidden variable that generates it. In a two-body decay the transverse momentum piles up at half the parent's mass, which lets a mass be read from particles that are only partly seen.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as two-body decay — the same set of essays touches all of them, so they are one junction rather than several.

The neutrino's energy against the pion's, at four angles. The energy of the neutrino from a pion's decay against the pion's energy, for neutrinos going exactly forward and at 1°, 2.5° and 4° to the pion's line of flight. Straight ahead the neutrino takes a fixed share, 0.427, of the pion's energy, so pions of every energy give neutrinos of every energy. Off axis the curve rises, flattens and falls: at 2.5° it peaks at 0.683 GeV for a pion of 3.2 GeV, and every pion between 2 and 10 GeV gives a neutrino between 0.61 and 0.68 GeV. The flat top is where γθ = 1. Its height, p*/θ, depends on the angle alone — the rest-frame momentum of the decay, 29.8 MeV/c, divided by the angle in radians.

The neutrino energy an angle fixes

A pion breaks into a muon and a neutrino, and the neutrino carries off a fixed 29.8 MeV in the pion's own frame. In the laboratory, straight along the pion's path, it carries a fixed share of whatever energy the pion had, so a beam of pions of every energy makes neutrinos of every energy. Two and a half degrees to one side, something else happens. The neutrino's energy, as a function of the pion's, rises to a maximum and falls again, and around the maximum pions from two to ten GeV all give neutrinos within a tenth of a GeV of each other. An experiment in Japan builds its detector deliberately off the axis of its own beam for that reason — trading most of its neutrinos for the ones at the energy it needs.

relativity · Relativistic dynamics A peak at half the mass, made by a sine. The momentum across the beam of the electron from 200,000 simulated W decays, as a probability density per GeV, for a W of mass 80.37 GeV. For a W exactly at rest across the beam, of exactly its mass (dashed), the electron carries M/2 = 40.19 GeV in the W's frame and shows a fraction sin θ of it across the beam, so the distribution climbs to a sharp peak at M/2 and stops: the Jacobian peak, piled up because sin θ hardly changes near θ = 90°. The W's natural width of 2.1 GeV rounds the peak (solid, peak at 39.9 GeV); its motion across the beam, a few GeV on average, smears it further and pushes a tail past M/2 (lighter). The position of the peak measures the mass, and is shifted by anything that moves the W.

The mass weighed without one of its pieces

A W boson decays into an electron and a neutrino, and the neutrino leaves every detector without a trace. Its mass cannot be found the usual way, by adding up the energy and momentum of what it decayed into, because half of what it decayed into is missing. What survives is the part of the motion across the beam, where the neutrino's momentum can be inferred from the imbalance it leaves. There the electron's momentum piles up at half the W's mass, and a quantity built from both particles ends at a sharp edge at the full mass. The world's best measurements of that edge disagree by seventy-five parts in eighty thousand.

relativity · Mass-energy

Named alongside it

The objects these essays reach for when they reach for this one.

Two-body decayDoppler factorInvariant massLorentz boostLorentz invarianceNeutrinoNeutrino beamNeutrino oscillationPionRapidityRelativistic kinematicsTransverse mass

All concepts