The cone a decay cannot leave

A particle at rest that breaks into two sends them opposite ways with equal and opposite momenta, and the size of those momenta is fixed entirely by the three masses. There is nothing else it can do: momentum conservation forces the directions and energy conservation forces the magnitude.

Set the parent moving and none of that is visible. The two products arrive at a detector at angles that depend on where they were going in the rest frame, with energies that depend on the same thing, and the symmetric picture has been folded forward into something that looks nothing like it.

What survives the folding is more useful than what is lost. Everything below is done by writing the products’ four-momenta in the parent’s rest frame, where the answer is short, and boosting them — rather than by writing a formula in the laboratory frame and evaluating it. The two routes give the same numbers and only the first makes it obvious which of the results are kinematics and which are assumptions about the decay.

The rest frame, which is all masses #

Before anything is boosted, the rest-frame picture has to be got right, and it is shorter than it looks.

A parent of mass $M$ decaying into masses $m_1$ and $m_2$ gives each product a momentum

$$p^* = \frac{\sqrt{[M^2 - (m_1+m_2)^2][M^2 - (m_1-m_2)^2]}}{2M}$$

and that is the whole of it: no interaction, no coupling, no matrix element. Conservation of energy and momentum in the rest frame is two equations in two unknowns, and this is the solution.

Two features of that expression carry most of the practical consequences. It vanishes when $M = m_1 + m_2$, which is the threshold: a parent barely heavier than its products gives them almost no momentum, so they emerge nearly at rest and nearly together. And it grows as $M$ increases past that, so a decay with a large mass difference throws its products apart hard.

A decay near threshold is therefore hard to see and a decay far above it is easy, for reasons that have nothing to do with how often it happens. The products of a near-threshold decay travel almost parallel and share the parent’s velocity almost exactly, which makes them look like a single particle; the products of a decay far above threshold arrive at a wide angle carrying a large invariant mass.

What the boost does to the angles #

The rest frame is uniform: with no spin to orient it, a parent decays equally in every direction. Boosting compresses that.

The edge is worth understanding rather than remembering. In the lab, the product’s momentum is the vector sum of its rest-frame momentum and the parent’s carrying motion. If the parent’s contribution is larger, no combination of the two can point backwards, and the possible directions sweep out a cone. If it is smaller, some combinations do point backwards, and every angle remains available.

The comparison is between the parent’s speed and the product’s own speed in the rest frame, not between the parent’s speed and anything absolute. A slow parent emitting a fast product has no cone; a fast parent emitting a fast product may still have none.

At large boosts the cone’s half-angle is the rest-frame momentum divided by the parent’s — the same $1/\gamma$ scaling that turns an isotropic emitter into a searchlight in the sky that crowds into a cone. It is why a high-energy experiment sees narrow jets rather than isotropic sprays, and why the detector elements nearest the beam have to be the finest: everything from one decay arrives within a few degrees, and separating the products becomes a question of angular resolution.

The spectrum that is a rectangle #

The energies do something so clean that it is worth stating as a result rather than as a step.

The lab energy is $\gamma(E^* + \beta p^* \cos\theta^*)$, which is linear in $\cos\theta^*$. An isotropic distribution is uniform in $\cos\theta^*$. A linear function of a uniformly distributed variable is uniformly distributed. So the spectrum is a rectangle, with sharp edges at $\gamma(E^* \pm \beta p^*)$ and nothing between them but a constant.

This is one of the few exact results in a subject full of approximations, and its usefulness is that the two edge positions carry everything. Their sum gives $2\gamma E^*$ and their difference gives $2\gamma\beta p^*$, so measuring where the spectrum starts and stops determines the parent’s speed and the rest-frame momentum — and therefore, given the daughter masses, the parent’s mass. A parent that lives too briefly to detect can be weighed by the shape of a histogram.

The result is also a warning. A flat spectrum with two edges says “two-body decay” and says nothing whatever about the interaction that caused it, because no dynamics survives into the shape. Everything the dynamics can do lives in whether the rest-frame distribution is isotropic in the first place.

The same rectangle, with the parent unseen #

The flat spectrum is worth one worked case, because it is how a whole class of measurements is made.

A beam of charged pions decaying in flight produces muons, and the muon’s lab energy spectrum is a rectangle whose edges depend on the pion’s momentum. Nothing in the apparatus sees a pion decay; what is seen is a muon, and the pion’s momentum is inferred from where the rectangle starts and stops.

The same construction with the pion replaced by a kaon gives a different rectangle, with different edges, because the masses differ. A beam containing both produces two overlapping rectangles, and the four edges say what the mixture is — a composition measurement made without identifying a single particle individually.

A spectrum with structure that is purely kinematic is a more reliable measurement than one with structure that is dynamical, because kinematics is known exactly and dynamics is not. That is the reason edges, thresholds and endpoints are the features experiments are built around, and the reason a smooth bump is always the harder claim.

The number that does not move #

The most important consequence is the one the first figure draws.

Combine the two products’ four-momenta and take $E^2 - p^2$ of the sum. That quantity is a Lorentz invariant, so it is the same in the lab as in the parent’s rest frame, and in the rest frame it is simply the parent’s mass. It therefore does not depend on the boost, on the emission angle, on the beam energy, or on anything else about how the decay happened to be produced.

The figure computes it from lab quantities across the whole range of parent speeds and five different emission angles, and requires it to be constant to a part in $10^9$ before drawing anything. That is not a decoration: an error in the boost would show up as a drift, and a drift is exactly what a mistake in this calculation looks like.

This is the entire method by which unstable particles are found. Nothing detects a particle that lives for $10^{-23}$ seconds. What is detected is two tracks; their invariant mass is computed; the calculation is repeated over millions of events and histogrammed. A parent that exists appears as a peak at its own mass, in the same place whatever the beam energy — and it is that last property, more than the peak’s height, that makes it believable.

The same invariance is what the invariant that survives a boost sets out in general, and the same combination is what makes a collider worth building rather than a fixed target, since the collision that wastes most of the energy is the statement that the invariant mass of a beam and a stationary target grows only as the square root of the beam energy.

Where the edges go blunt #

The rectangle’s edges are the measurement, so it is worth knowing what softens them, because in practice none of them is sharp.

The parent’s own speed is never a single number. A beam has a momentum spread, and a parent produced in a collision has whatever momentum the production process gave it — a distribution rather than a value. Each parent speed produces its own rectangle, and what is observed is the sum of rectangles of different widths, which has sloping sides rather than vertical ones.

The parent’s mass is not a single number either, if it is short-lived: a particle with a lifetime $\tau$ has a mass spread of order $\hbar/\tau$, which is the width that is a lifetime applied to a particle rather than to a resonance in a circuit. For a broad resonance that spread dominates everything else.

And the detector contributes its own resolution, which is usually the largest of the three at low energy and the smallest at high.

Disentangling those is the whole of the analysis, and the way it is done is to note that they scale differently: the beam spread is a fixed fraction of the momentum, the natural width is a fixed mass, and the detector’s resolution grows with energy in a way that has been calibrated separately. Measuring the same peak at several beam energies separates them.

Two photons, which have no rest frame #

A decay into two massless products has no threshold speed to be above, because a photon travels at $c$ in every frame and can never be slower than its parent. There is no forward cone with an edge. There is something else instead.

The relation is short: $\sin(\theta_{\min}/2) = m/E$. At rest the photons go exactly opposite ways; at high energy they close in.

That number decides how finely a calorimeter must be divided. Below an opening angle comparable with the detector’s granularity, the two photons land in the same cell and the pair is indistinguishable from a single photon of twice the energy — which is the commonest failure mode of this measurement, and it fails systematically rather than randomly: every high-energy pion is mis-measured the same way, so the effect is a bias rather than a widening.

Reading a lifetime off a length #

There is a second measurement that the same bookkeeping supports, and it is the one that turns a decay into a clock.

A particle of proper lifetime $\tau$ moving with Lorentz factor $\gamma$ travels a mean distance $\gamma\beta c\tau$ before decaying. For a lambda, whose proper lifetime is about 260 picoseconds, that is about eight centimetres per unit of $\gamma\beta$ — a distance a tracking detector resolves easily. So the decay point is visible as a vertex displaced from where the particle was made, and the displacement measured over many events gives the lifetime.

The measurement needs the boost, which comes from the same reconstruction that gives the mass: the total momentum of the two products is the parent’s momentum, and the parent’s mass is the invariant. Divide one by the other and $\gamma\beta$ is known for that event. Nothing about the parent has been observed except through its products, and yet its speed, its mass and its lifetime are all recoverable.

This is the sense in which relativity is used rather than tested in particle physics. The transformations are the arithmetic every measurement is expressed in, and an error in them would show up not as a failed test but as an incoherent set of results — masses that depended on beam energy, lifetimes that depended on where the detector was.

Where the model stops #

The decay is treated as isotropic in the rest frame. A parent with spin, produced in a way that polarises it, decays anisotropically, and the flat spectrum acquires a slope. That slope is a measurement of the polarisation and is one of the standard ways spin is determined — so the deviation from the result here is itself an instrument.

Only two bodies. A three-body decay has no fixed rest-frame momentum, because the energy is shared among three, so the spectrum is not a rectangle and the invariant mass of any two of the three is not the parent’s. Beta decay is the historical example: its continuous spectrum was the anomaly that required the neutrino, and it is a direct consequence of there being three products rather than two.

The parent is assumed to have a definite momentum. In a real event it does not: what is known is the momenta of the products, and the parent’s momentum is their sum. The distinction matters for the angular figures, which are drawn as functions of a parent speed nobody measures independently.

Detector effects are absent. Every real invariant-mass peak is broadened by the resolution of the momentum measurement, often by far more than the particle’s own width, and separating the two is the central problem in measuring a lifetime this way.

And the tracks are assumed to be identified. Computing an invariant mass requires knowing each product’s mass to get its energy from its momentum, and assigning the wrong mass to a track puts the event in the wrong place. Half of a detector’s design is about telling a pion from a kaon so that this step can be taken.

Why the endpoint is where a mass is looked for #

The sharpest use of all this is the search for something that is not detected at all.

If a decay produces a particle that leaves no trace — a neutrino, or anything else that passes through the apparatus — the visible products do not carry the full energy, and their spectrum has an endpoint whose position depends on the invisible particle’s mass. A heavier invisible particle takes more of the budget, so the visible endpoint moves down.

That is how the neutrino’s mass is bounded from tritium decay: the electron’s spectrum ends slightly below where it would if the neutrino were massless, and the shift is the mass. The whole difficulty is that the shift is a tiny displacement of an endpoint where the count rate is already falling to zero, so almost no events are in the region that carries the information.

The measurement is not of a peak but of a shape near a boundary, and the boundary’s position is pure kinematics of exactly the kind computed here. What makes it a hard experiment is the statistics; what makes it a possible one is that the kinematics carries no unknowns beyond the one being measured.

What the pictures cannot show #

The invariant-mass figure draws five curves that coincide exactly, which is the result and is also a poor picture of what an experiment sees. A real histogram is a broad peak on a large smooth background, most of which comes from combining two products that were never from the same decay — the combinatorial background, which grows as the square of the number of tracks and is what limits every search of this kind.

The cone figures plot angle against rest-frame angle, which no detector measures. What a detector records is a distribution of laboratory angles, and recovering the rest-frame distribution from it requires knowing the parent’s momentum, which usually requires having reconstructed the decay already. The inference runs in the opposite direction from the figure.

Where the ladder goes next #

The relativistic-dynamics ladder began with the push that does not point where the body goes, where force and acceleration part company, went on to the collision that wastes most of the energy and the threshold arithmetic that justifies colliders, then to nothing is allowed to be rigid and to the centre that is not a place. This rung runs the same four-momentum bookkeeping backwards: one particle becoming two, and what the boost does to the pattern.

The rung after it is the decay of something produced with a known polarisation, where the rest-frame distribution is no longer isotropic and its shape becomes the measurement. The habit worth carrying is the one this rung is built on: when a boost scrambles everything observable, look for the combination it cannot touch — and then build the experiment around measuring that.