Build-a-Hadron — quark combination lab

Concept: Hadrons are colour-neutral quark bundles: baryons (qqq) and mesons (qq̅). Charge, isospin, strangeness, charm and bottomness are simple sums over constituents, tied together by the Gell-Mann–Nishijima relation Q = I₃ + ½(B+S+C+B̃+T).
Serves: Introductory particle physics / Standard Model units. Works before the formalism (build and discover) and after it (open The Physics to verify every build).
Inquiry: Which combinations are missing? Multi-heavy baryons (Ξcc⁺ — its partner Ξcc⁺⁺ was found by LHCb in 2017 — Ωcc⁺, Ωccc⁺⁺, Ξbb, Ωbbb…) are predicted but unseen: their cards estimate the 2mc² pair-production cost and what experiments must deliver (LHC luminosity, HL-LHC statistics). Top-quark hadrons can never form — the top decays (~5×10⁻²⁵ s) before hadronization (~10⁻²³ s).
Misconception: "Three identical quarks (uuu) would violate Pauli exclusion." Colour charge supplies the hidden quantum number: Δ⁺⁺ exists, and its existence is precisely what forced physicists to invent colour.

Principle	Where it appears
Predict, then test	The uuu prediction chips commit a guess, then auto-build Δ⁺⁺ as evidence.
Misconception targeted structurally	The "Pauli forbids it" chip encodes the classic error; the rebuttal (colour charge) appears only on engagement.
Invariant made visible	Q = I₃ + ½(B+S+C+B̃+T) is checked live for every hadron built; baryon number and colour neutrality always hold.
Operational quantities	Mass, spin, charge, lifetime, decay mode and discovery year are read off each card, never narrated vaguely.
Frontier inquiry	Undiscovered heavies show predicted mass, 2mc² energy cost and the experimental bottleneck; top combos show why "impossible" differs from "not yet".

Build-a-Hadron

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Guided Inquiry

1 · The hadron factory

Let’s start with the basics. A hadron is a bundle of quarks held together by the strong force. Each quark carries an electric charge — u, c and t at +⅔, d, s and b at −⅓ — so a hadron’s charge is just the sum over its constituents. Here’s your first build on screen: u u d — the proton, Q = +1.

2 · Three of a kind?

Now let’s push that a step further. The proton already holds two identical u quarks. Could a baryon hold three of them — u u u — when quarks are fermions and no two may occupy the same state? What do you think?

3 · White with only two?

So far, white has taken all three colours — yet two-quark hadrons exist too. How do you think a pair can be white?

4 · The strangeness ladder

Now for a little history. On screen: Ξ⁻ (d s s), Strange S = −2 — each s lowers S by one. By 1962, only S = −3 (s s s) was still unfilled. So can a gap in a pattern predict a particle? What do you think?

5 · Not yet ≠ never

One last puzzle. c c c breaks no rule: three colours, charge +2. Yet it’s never been seen, while a top quark can never form a hadron at all. What do you think separates the two cases?

Here’s something to leave you with: twelve of this catalogue’s ground-state baryons are predicted but still unseen, each priced by the 2mc² cost of making its quarks. Why not go and build a few?

Listen

Mode

Heavy quarks c, b, t

Total spin

same quarks, different spin — different hadron

Explore

Current build

Charge Q	0
Isospin I₃	0
Strange S	0
Charm C	0
Bottom B̃	0

colour: incomplete (0/3)

Check: complete the combo first.

Only colour-neutral states escape confinement — r g b, or colour + anticolour.

The Physics

Build-a-Hadron · what this sim shows, the physics behind it, and every control

At a glance

Quarks combine into hadrons: three make a baryon, such as the proton, and a quark with an antiquark makes a meson, such as the pion. This sim adds up the labels of the quarks you choose, then looks the combination up in a catalogue of hadrons.

What you'll learn

How a hadron's charge and its other labels follow from the quarks inside it.
What colour is, and how it limits which quarks can bind.
How one set of quarks can make two different particles.
How a hadron that has been observed differs from one that is only predicted, and from one that cannot form.

The sim's layout: pick quarks, fill the slots, read the card.

What's on screen

Quark tray (top): the six flavours u, d, s, c, b, t, each with its electric charge.
Slots (centre): three for a baryon, two for a meson. Wavy lines are gluons.
Identity card (right): name, mass, spin, charge and lifetime once the slots are full. Purple means predicted only; red means no hadron.
Library (bottom): every hadron you build is kept as a chip.
Current build (sidebar): the sums Q, I₃, S, C, B̃ and the colour line.

See itBuild a proton from two u quarks and one d quark. Watch for: the three quarks fusing into one particle, and Q = +1 in the readout.

The controls

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

Control	What it changes	Range
Mode	Baryon qqq has three slots. Meson has a quark slot and an antiquark slot. Switching mode clears the slots.	Baryon / Meson opens on Baryon
Heavy quarks c, b, t	Unlocks the charm, bottom and top quarks. While it is off they are dashed in the tray and cannot be placed.	on / off opens on
Quark tray	Click a quark to put it in the next empty slot. In Meson mode the top row is the quark and the bottom row the antiquark. Click a filled slot to empty it. When all slots are full, the next click starts a new build.	u d s c b t
Total spin	Chooses the spin state of the same quarks: Lowest ↑↓ (spin ½ for a baryon, 0 for a meson) or Raised ↑↑ (3⁄2, or 1). The note under it says when no partner state exists. Every new build starts on Lowest.	Lowest / Raised opens on Lowest
Frontier combo ▶	Builds the next of the 12 baryons marked predicted only, and turns Heavy quarks on.	12, in turn
Clear	Empties the slots and keeps the mode.	button
Library	Each new hadron you build is kept as a chip with its discovery year. Click a chip to reload it. The counter shows how many of the 105 entries you have built.	0 – 105
Current build	The sums Q, I₃, S, C and B̃ over the filled slots, and the colour line.	panel
▶ Play / ⏸ Pause	Starts or freezes the animation: the turning ring of quarks and the gluon lines.	top bar
Speed	Runs the same animation faster or slower.	0.25×–4× opens at 1×
↻ Reset	Returns to the opening proton, with Heavy quarks on and the library back to that one entry.	top bar

B (⅓ per quark) and topness T are summed too but have no readout row; T is 0 in every hadron that can form.

Every label is a sum over the quarks

The six flavours at their electric charge, each with the extra label it carries.

A hadron's Q, I₃, S, C and B̃ are each the sum over its quarks. An antiquark counts with the opposite sign: u has Q = +⅔, so ū has −⅔.
Only u and d carry isospin (I₃ = ±½). Each heavier flavour carries one label of its own.
The signs are conventions: the s quark has S = −1 and the b quark has B̃ = −1, while c has C = +1 and t has T = +1.
Baryon number B is ⅓ per quark: 1 for a baryon, 0 for a meson.
Mass, spin and lifetime are not sums. The sim looks them up in its catalogue of hadrons.

See itBuild the neutron (u d d), then swap one d for an s. Watch for: I₃ and S changing in the readout while Q stays 0.

Only colour-neutral bundles are seen

The sim's slot colours. Left: u u d. Right: u d̄. Wavy lines are gluons.

Quarks carry a strong-force charge called colour: red, green or blue. Antiquarks carry anticolours. The names have nothing to do with visible colour.
No single quark has ever been seen alone. Only colour-neutral bundles come out: this is confinement.
Two ways to be neutral: one quark of each colour (a baryon), or a colour with its anticolour (a meson). That is the whole Mode switch.
Colour also lets three identical quarks share a hadron, as in u u u: the three differ in colour.
The colour line in the readout says "r g b → white ✓" only once every slot is full.

See itFill all three baryon slots with u. Watch for: three identical quarks, one in each colour slot, and "r g b → white ✓".

See itSwitch to Meson mode and build a u quark with a d̄ antiquark. Watch for: two slots, red and cyan, and "r + r̄ → white ✓".

The charge follows from the other labels

The Gell-Mann–Nishijima relation. Hypercharge Y collects every label except isospin.

It holds for each quark on its own, so it holds for any sum of quarks. The sim checks every build against it. I₃ and Y are also the two axes on which hadron families are drawn: for the u, d, s baryons Y = S + 1.

Mode

Spin

Quarks

Tray Heavy quarks c, b, t

These controls move the sim too.

See itBuild u u s and read its labels. Watch for: I₃ = +1 and S = −1 in the readout, with Q = +1.

The same quarks can make two particles

What the Total-spin selector chooses between.

u u d with the spins opposed (total spin ½) is the proton: 938.3 MeV, stable. With all three aligned (3⁄2) it is the Δ⁺: 1232 MeV, gone in 5.6×10⁻²⁴ s.
Q, I₃ and S are identical. Only the spin state differs, and it shifts the mass by 293.7 MeV.
The letter S means two things: spin in the Total-spin note, strangeness in the readout.
For three identical quarks only spin 3⁄2 exists. The whole state must change sign when two quarks swap. Colour already does, the lowest state has no orbital motion, and identical flavours are symmetric, so the spin part must be symmetric, and only 3⁄2 is.

Quarks are fermions: swapping any two flips the sign of the whole state.

See itBuild the proton, then raise the spins. Watch for: the card changing from p to Δ⁺, with the mass jumping to 1232 MeV while Q stays +1.

See itBuild d d d, then ask for Raised. Watch for: the note under Total spin saying that symmetry forces 3⁄2.

Observed, predicted and impossible hadrons

The ten spin-3⁄2 u, d, s baryons in the catalogue at their I₃ and S. Right: each row's mean mass.

In 1962 the bottom rung was empty. The rows above it were about 150 MeV apart, so Gell-Mann predicted a particle with S = −3 and Q = −1 near 1.68 GeV. Brookhaven found the Ω⁻ in 1964, at 1672.5 MeV.
Predicted only (✧). 12 of the 35 baryon combinations in the catalogue, such as c c c, have not been seen. Baryon number is conserved, so a collision that brings none must make the baryon with an antibaryon. The card's Pair cost is that threshold, 2mc², with m the hadron's mass: about 9.6 GeV for c c c. It is a rough guide, not a law.
Impossible (⊘). The top quark decays in about 5×10⁻²⁵ s, while the sim puts the time to bind at about 10⁻²³ s. The top decays first, so no top hadron forms.

See itBuild s s s, then c c c, then a top. Watch for: the card changing from found, to "not yet observed" in purple, to "impossible" in red.

Where it matters

Strange particles come in pairs. The strong interaction conserves strangeness, so π⁻ + p → K⁰ + Λ⁰ makes one particle with S = +1 and one with S = −1. You can build both in the sim: d s̄ and u d s.
Finding new hadrons. LHCb found the doubly charmed Ξcc⁺⁺ (c c u) in 2017, at about 3621.6 MeV. The sim lists it as found.
Beyond three or two. Three quarks plus a quark–antiquark pair is also colour-neutral. LHCb has reported such five-quark states, called pentaquarks. The sim covers only baryons and mesons.

Go deeper: why the mass is not a sum

The masses of the u, u and d quarks add to about 9 MeV, under 1% of the proton's 938.3 MeV. Most of the rest is energy of the strong-force field and of the quarks' motion.
That is why the sim sums labels but reads masses from a table.

Summary

In one line. A hadron's labels are sums over its quarks, and only colour-neutral bundles are seen: three quarks, or a quark and an antiquark.

Key points

Q, I₃, S, C and B̃ add up over the quarks; an antiquark counts with the opposite sign.
Q = I₃ + ½(B + S + C + B̃ + T) holds for every hadron.
The same quarks with a different spin state are a different particle.
A combination is observed, predicted only, or impossible (the top quark).

Common mix-ups

"Three identical quarks would break the Pauli principle." Pauli forbids identical states, not identical flavours. The three u quarks differ in colour.
"Any two quarks make a meson." Red with green is still coloured. A meson needs an anticolour, so it is always a quark with an antiquark.
"Not yet observed means forbidden." Only the top quark is ruled out. c c c breaks no rule and is only costly.

Things to try on your own

Build d d s, u d s and u u s. Record I₃ and S for each, and check that Q = I₃ + ½(1 + S).
In Meson mode build u d̄ and then d ū. Compare Q and I₃, and the two cards.
Press Frontier combo until it comes back to the start. Compare the Pair costs of the baryons with c quarks and those with b quarks.