Virtual Particle Collider — energy budget & event yield

Concept: Head-on e⁺e⁻ collisions: the Lorentz-invariant energy √s = √((E₁+E₂)² − (p₁−p₂)²) is the creation budget — a particle pair is accessible only if √s ≥ 2m. Above threshold, the annihilation rate σ(e⁺e⁻→μ⁺μ⁻) = (4πα²/3s)·β(3−β²)/2·F_Z(s) rises from zero at √s = 2m_μ, then falls as 1/s through the photon — until the Z⁰ takes over: at √s = M_Z = 91.19 GeV the rate leaps ~200× (≈2 nb at Born level; LEP measured ≈1.5 nb) and then falls away again. The expected yield is N = σ·∫L dt·ε. The model is improved Born (γ/Z interference, no initial-state radiation). Threshold chips are labelled by the particles that emerge: BB̅ marks the open-flavour B⁺B⁻ threshold at 2m_B = 10.56 GeV; bound bb̅ already appears as the Υ(1S) resonance at 9.46 GeV, and tt̅ is a free-quark threshold because the top decays before it can hadronise.
Serves: Intro particle / modern physics units on colliders and cross-sections; works pre-formalism (thresholds unlock visually) and post-formalism via the ⚛︎ The Physics sheet. Numbers are engine-exact: √s = 10 GeV gives σ = 0.869 nb; 1 nb × 1 fb⁻¹ × ε = 0.9 gives 9.0×10⁵ events.
Inquiry: Open the muon channel at √s = 2 GeV → raise both beams to 10 GeV and watch reach grow while rate drops ×25 → compare 25+1 GeV vs 5+5 GeV (same √s = 10: asymmetry only buys a boost) → cross the W⁺W⁻ threshold at 161 GeV → plan a run: luminosity and efficiency set the count.
Misconception: “More energy ⇒ more events.” Wrong twice: (1) away from the Z⁰ pole this s-channel rate scales as 1/s — energy widens the menu of particles, luminosity buys statistics (the one exception, the Z⁰ resonance, is a narrow 2.5 GeV window at 91 GeV); (2) beam energies don’t simply add — unequal beams waste energy on center-of-mass motion, visible as E₁+E₂ − √s.
Decay prongs: Unstable products decay in flight: after the pair flies out, its track ends fan into short prongs tipped with dots — the secondaries of the decay. The sim draws 2 prongs for τ⁺τ⁻ and 3 for B⁺B⁻, W⁺W⁻, ZZ and tt̄; μ⁺μ⁻ tracks stay clean because muons cross the whole detector without decaying. The fan marks the decay vertex schematically — prong counts indicate “a few secondaries”, not the exact multiplicity of each real decay mode.

Principle	Where it appears
Predict before reveal (optional)	Predict card asks what doubling beam energy does to the μ⁺μ⁻ count; sim stays fully playable without answering.
Misconception encoded in a choice	“≈4× more” carries the intuition; picking it triggers the rebuttal (σ = 4πα²/3s), which is never shown as standing prose.
Invariant vs. varying	25+1 GeV and 5+5 GeV give the same √s = 10 GeV: √s is the invariant that matters; E₁+E₂ and the boost are not, shown as “lost to boost”.
Operational quantities	Every claim is a labelled readout — √s, σ in nb, expected events — computed with the exact engine formulas.
Formal layer on demand	Invariant mass, γ/Z cross-section with the Z⁰ factor, angular distribution and yield equations sit behind the top bar’s ⚛︎ The Physics revision sheet.

Virtual e⁺e⁻ Collider

Teach me Hide Text

Speed

Head-on collision

Guided Inquiry

1 · The creation budget

Let’s start with the budget. Two 1 GeV beams meet head-on, and √s = 2 GeV is the energy available to create new particles. Heavier pairs sit above that budget, at least for now.

2 · What does τ⁺τ⁻ cost?

Now let’s go looking for taus. The tau’s mass is 1.777 GeV, and e⁺e⁻ collisions create matter in particle–antiparticle pairs. So what do you think √s has to reach before taus can appear?

3 · Energy buys reach — and rate?

So energy buys reach — but does it buy rate too? Both beams are 5 GeV, so √s = 10 GeV and τ⁺τ⁻ is open. Now double each beam to 10 GeV. What do you think happens to the muon-pair count in the same run?

4 · Do beam energies add?

Let’s go back to 5 + 5 GeV, where √s = 10 GeV. Now spend 26 GeV unequally instead: 25 + 1 GeV. How much energy do you think is available to create new particles?

5 · Two separate accounts

So here’s the big idea. Energy sets what you can make. How many events you record is a separate account: the cross-section, the luminosity delivered, and the detector efficiency.

6 · Count events, not hopes

One last planning problem. At 5 + 5 GeV you record about 8×10⁵ muon pairs per fb⁻¹, and you need ten times more. Which change do you think delivers that?

Listen

Beams

e⁻ energy E₁5.00 GeV

e⁺ energy E₂5.00 GeV

link beams (symmetric)

Show σ(√s) plot

Experimental readouts

∫L dt1.0 fb⁻¹

efficiency ε0.90

μ⁺μ⁻ events7.82×10⁵

Readouts

√s	10.00 GeV
σ(μ⁺μ⁻)	0.869 nb
lost to boost	0.00 GeV

energy budget√s = √((E₁+E₂)² − (p₁−p₂)²) = 2E_beam for equal beams

γ/Z rateσ = 4πα²/(3s) · β(3−β²)/2 · F_Z(s), β = √(1−4m_μ²/s)

Z⁰ factorF_Z = 1 + 2g_v²Re χ + (g_v²+g_a²)²|χ|², χ = (G_F M_Z²/2√2πα)·s/(s−M_Z²+iM_ZΓ_Z)

limitsF_Z → 1 for √s ≪ M_Z (σ ∝ 1/s); σ_peak ≈ 12πΓ_eeΓ_μμ/(M_Z²Γ_Z²) ≈ 2 nb Born (LEP: ≈1.5 nb after ISR)

angular shapedσ/dΩ = α²/(4s)(1+cos²θ)

yieldN = σ · ∫L dt · ε

The Physics

The particle collider · what this sim shows, the physics behind it, and every control

At a glance

This sim collides an electron beam with a positron beam head-on and shows what the collision can create and how many events you would record. Every collider is judged by what it can make, how often, and how many events it records.

What you'll learn

The energy available for making new particles, from the two beam energies.
Which particle pairs a collision can create.
How the muon-pair rate depends on that energy.
How a count of recorded events is built up.

The beams meet at the centre of a detector.

What's on screen

Beams: e⁻ blue from the left, e⁺ pink from the right, each labelled with its energy. √s is the headline.
Products: two tracks leaving back to back, usually μ⁻ (green) and μ⁺ (purple). A heavier pair, when affordable, replaces every third event.
Threshold chips and σ(√s) plot: a chip fills in once its pair can be made; the dot marks the current √s.
Readouts: √s, σ(μ⁺μ⁻), "lost to boost" (section 04 explains it), and the event count.

See itFire one collision at the opening settings, 5 + 5 GeV. Watch for: the two beam bunches meeting at the centre, then a pair of tracks leaving in opposite directions.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
e⁻ energy E₁	Energy of the electron beam.	0.1–105 GeV opens at 5 GeV
e⁺ energy E₂	Energy of the positron beam.	0.1–105 GeV opens at 5 GeV
link beams	While ticked, moving one beam moves the other too. Untick it to give the beams different energies.	on / off opens on
Show σ(√s) plot	Shows or hides the cross-section plot and threshold chips. Hidden, the collision gets the full height.	on / off opens on
∫L dt	The integrated luminosity: how much collision data the run gathers.	0.01–100 fb⁻¹ opens at 1.0
efficiency ε	The fraction of muon pairs the detector records.	0–1 opens at 0.90
μ⁺μ⁻ events	The number of muon pairs recorded. A readout, fed by ∫L dt and ε.	readout
Readouts	√s, the muon-pair cross-section σ, and the energy lost to boost. Click the header to open the panel.	panel
⏸ Pause / ▶ Play	Freezes the collision cycle or restarts it. Moving a beam, ∫L dt or ε pauses the run; on release the sim replays one collision and holds.	top bar
Speed	How fast the collision is drawn. The numbers do not change.	0.25×–4× opens at 1×
↻ Reset	Puts the beams, ∫L dt and ε back to their opening values (while a Guided Inquiry card is open, to that card's setup).	top bar

Fixed: head-on e⁺e⁻ beams only, and only μ⁺μ⁻ pairs are counted. Each event's direction is drawn at random, so no two collisions look alike.

Each pair opens at 2m

The electron and positron annihilate, and the energy comes back out as matter. Charge and lepton number must balance, so what appears is a particle and its antiparticle: a lone μ⁻ cannot be made. A pair of mass m therefore costs at least 2m.

A pair channel opens when √s reaches 2m.

The six chips in the sim, with the sim's masses. The dashed line is the opening √s = 10 GeV.

Below 2m the rate is exactly zero, not just small. For muons the σ readout shows "0, closed" and the beams pass straight through.
Only μ⁺μ⁻ is counted. The other pairs show which channels are open; they never enter σ or the event count.
The beams stop at 105 GeV each, so √s ≤ 210 GeV and the tt̄ chip (345.4 GeV) never fills.

See itRaise both beams from 1.7 to 1.8 GeV. Watch for: the τ⁺τ⁻ chip changing from dashed to filled, and tau tracks with decay prongs at their ends.

Unequal beams spend energy on motion

The energy available for creation is √s. Momentum cannot be destroyed: whatever net momentum p₁ − p₂ the beams bring in, the products must carry out, and that motion cannot go into mass.

Same √s, different cost. The sim shows the difference as "lost to boost".

For equal head-on beams, p₁ = p₂ and nothing is reserved: √s = 2E.
5 + 5 GeV and 25 + 1 GeV both give √s = 10 GeV. The second brings in 26 GeV, of which 16 GeV is lost to the boost.

See itFire 25 GeV against 1 GeV. Watch for: √s still reading 10.00 GeV although the beams add to 26 GeV, with lost to boost reading 16.00 GeV.

Below the Z⁰, σ falls as 1/s

The cross-section σ is the rate per unit of data. For μ⁺μ⁻ the sim uses three factors:

The first is the photon term. The second switches the channel on over a few tenths of a GeV above 2m_μ; by √s = 1 GeV it is 0.999. The third, F_Z, stays within 3% of 1 below 50 GeV; section 07 covers where it does not. In that range σ ∝ 1/s: doubling √s quadruples s and cuts σ to a quarter.

σ(μ⁺μ⁻) against √s, both axes logarithmic. Solid: the sim's curve. Dashed: the photon term alone.

See itFire 5 + 5 GeV, then 10 + 10 GeV. Watch for: σ in the card falling to a quarter of its first value when both beams double.

The event count is σ · ∫L dt · ε

Three separate quantities multiply into the number of events you record:

σ is the physics, ∫L dt is the machine, ε is the detector. In the sim's units, 1 nb × 1 fb⁻¹ = 10⁶ events.

E₁ E₂ ∫L dt ε

These sliders move the sim too. With link beams ticked, moving one beam moves both.

Beam energy enters only through σ. ∫L dt and ε just scale the count.
ε is a fraction: from 0.90, even a perfect detector adds only 11%. Ten times the data gives ten times the events.

See itTake 5 + 5 GeV. Raise ∫L dt tenfold, then raise ε to 1. Watch for: the event count in the card growing tenfold for the data, and by about 11% for the detector.

The Z⁰ peak breaks the 1/s rule

F_Z contains the Z⁰ propagator χ, the spike in section 05's plot. Its denominator never reaches zero, because the Z⁰ has a finite width Γ_Z.

Below about 50 GeV, F_Z is within 3% of 1 and the photon 1/s rule holds. Above the pole F_Z falls only slowly: 5.9 at 100 GeV, 1.8 at 120 GeV, 1.2 at 210 GeV.
At √s = M_Z = 91.1876 GeV, F_Z ≈ 190 and σ jumps from 0.0104 nb to 1.98 nb.
The peak is narrow: σ halves about 1.25 GeV either side of the pole in √s, and 5 GeV away it is down to about 6–7% of the peak.
A resonance is not a threshold. Making a pair of Z bosons is a separate chip, ZZ, at 2M_Z = 182.4 GeV.

See itStep √s through 80, 91.19 and 100 GeV. Watch for: σ jumping about 40 to 60 times higher at the pole than at 80 and 100 GeV, and a "Z⁰ resonance" label above the collision.

Where it matters

LEP. From 1989 to 1995 LEP ran on the Z⁰ peak; its four experiments recorded about 17 million Z decays, 3.4% of them to μ⁺μ⁻. In 1996 LEP passed 2m_W ≈ 161 GeV to make W pairs, and it reached 209 GeV by 2000.
B factories. KEKB and PEP-II ran at about √s = 10.58 GeV, just above the B⁺B⁻ chip. Their beams were deliberately unequal (8 + 3.5 GeV at KEKB): the boost sends the B mesons far enough to measure their decay times.
Colliders against fixed targets. To make μ⁺μ⁻ by hitting an electron at rest you need a beam of about 44 GeV. Head-on beams need only about 0.11 GeV each.

Go deeper: what the sim leaves out

What the sim evaluates: the Born-level formula, with no initial-state radiation.

A beam particle can radiate a photon before it collides, lowering the √s actually used. This initial-state radiation is not in the sim.
Including it lowers the measured peak by roughly a quarter compared with the sim's 1.98 nb.

Summary

In one line. The beams set √s, which decides which pairs can be made; how many you record depends on σ, the data and the detector.

Key points

√s = √((E₁+E₂)² − (p₁−p₂)²), or 2E for equal beams.
A pair of mass m opens at √s = 2m; below that the rate is zero.
Below about 50 GeV, σ ∝ 1/s; at the Z⁰ pole it is about 190 times the photon-only value.
N = σ · ∫L dt · ε: more data means a bigger count.

Common mix-ups

"More beam energy gives more events." Below the Z⁰ the rate falls as 1/s. Energy widens the menu; ∫L dt raises the count.
"The beam energies just add." Only for equal head-on beams; otherwise √s is smaller than E₁ + E₂.
"Below threshold the pair is just rare." It is impossible: the rate is zero.

Things to try on your own

Unlink the beams and find a new pair of energies that gives √s = 10.00 GeV.
Find the lowest equal beam energy at which the muon channel opens.
Choose ∫L dt so that 20 + 20 GeV records as many muon pairs as 5 + 5 GeV at 1 fb⁻¹.