Illustrated Physics
physics, drawn
Concept

Kinematics — where it appears

What conservation of energy and momentum permit, regardless of the forces responsible. A decay's angles and energy spectrum are kinematics; how often it happens is not — which is why edges, thresholds and endpoints are the features an experiment is built around.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

How fast something looks as it moves across the sky. The apparent transverse speed of a source, in units of the speed of light, against the angle between its motion and the line of sight, at β = 0.8, β = 0.95, β = 0.99. Every curve rises above one over a range of angles, reaching 1.33 at 36.8°, 3.04 at 18.4°, 7.02 at 8.1° — and those maxima are γβ at arccos β, found by searching the drawn curves rather than put into them. Nothing is moving faster than light. What has happened is that the source has come closer between the two observations, so the second flash had less far to travel and arrived sooner than it would have done; dividing the transverse distance by the interval between arrivals therefore gives too large a speed. Below β = 1/√2 no angle produces the illusion at all, so seeing it is a measurement: it puts a floor under the speed and a ceiling on the angle at once.

The motion that measures faster than light

Take two photographs of a jet a year apart, measure how far a blob moved across the sky, divide by a year, and the answer can be seven times the speed of light. Nothing has broken. The blob came closer between the two pictures, so the second flash had less far to travel and arrived early, and the interval between arrivals is not the interval between departures.

relativity · Velocity addition The one number the boost leaves alone. The mass of a lambda into a proton and a pion, reconstructed from the two products' laboratory energies and momenta alone, against how fast the parent was moving — for five different rest-frame emission angles. Every curve is the same horizontal line. The lab energies vary by more than a factor of ten across this range and the angles between the products vary from almost 180° to a few degrees; the combination E² − p² of the pair does not vary at all, to 3.2e-15, which the figure requires before drawing anything. The light curves are the energies of the two products, on the same axis and to a different scale, drawn to show how much is moving while the invariant does not. This is the whole method of particle physics. A parent that lives for 10⁻²³ seconds is never detected; what is detected is two tracks, and their invariant mass is computed and histogrammed over millions of events. A parent that exists shows up as a peak at its own mass, at the same place whatever the beam energy, which is what makes the peak believable.

The cone a decay cannot leave

A particle at rest breaks into two and they go opposite ways. Set the parent moving and the whole pattern folds forward — into a cone with a hard edge, beyond which nothing is emitted at any rest-frame angle at all. The energy spectrum that comes out is exactly rectangular, and the one number the boost leaves alone is how the parent is identified at all.

relativity · Relativistic dynamics The rectangle a spin tilts. The laboratory energy of the pion from a tau into a pion and a neutrino moving at 0.8 of the speed of light, for samples of 60,000 decays whose parents have αP = +1, 0, −1 along their line of flight. Every sample fills the same interval, 313 to 2667 MeV: the edges are set by the masses and the speed, and they do not move. Inside that interval an unpolarised sample is flat and a polarised one is tilted, its height at each energy 1 + αP times the cosine of the rest-frame angle that energy corresponds to. Reading the slope back from each sample gives +0.993, −0.010, −1.006, each with a standard error near 0.006. A parent spinning along its flight throws the pion forwards in its own frame, and the boost turns forwards into more energetic, so the tilt of an energy spectrum measures a polarisation without any angle being measured at all.

The slope a spin leaves in a spectrum

An unpolarised parent decaying in flight gives its products a rectangle of energies. Give the parent a spin along its line of flight and the rectangle tilts, while its two edges stay exactly where they were. The tilt is the polarisation, it can be read without ever seeing which way the parent was going — and which variable it is read from decides how many decays the reading costs.

relativity · Relativistic dynamics Resonances drawn as bands. 60,000 decays of a D⁰ decaying to K⁻π⁺π⁰, accepted from 1,287,365 flat ones in proportion to the square of an amplitude built from 3 short-lived intermediate states, each decaying to two of the three products. A ρ⁺(770) in m²(π⁺π⁰), 67.0 per cent of the rate on its own; a K⁻(892) in m²(K⁻π⁰), 25.5 per cent of the rate on its own; a K⁰(892) in m²(K⁻π⁺), 33.2 per cent of the rate on its own. Each appears as a band at its own mass squared — vertical, horizontal or diagonal according to which pair it decays to — holding 51 per cent, 19 per cent, 23 per cent of the decays within one width of its mass, where phase space alone would put 34, 10, 9. The separate fractions add to 126 per cent, not 100, because the amplitudes interfere where the bands overlap. The magnitudes and phases are a model chosen to make all three visible, not a fit to data.

The plane in which three bodies are flat

A particle breaking into two gives each product a fixed energy; one breaking into three gives none of them one. What it gives instead is a plane of two invariant masses in which a decay with no forces spreads perfectly evenly inside a curved boundary — so every band, dark stripe and bright crossing a real decay draws there is a force, its spin, or a phase between two routes to the same three particles.

relativity · Relativistic dynamics

Named alongside it

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

Angular distributionBeamingDecayFour-momentumMeasurementInvariant massThe Lorentz transformationParitySpinApparent speedCentre of massConservation

All concepts