Questions on quarks and the hadrons built from them

Question 1

State the electric charge of the up quark and of the down quark, as fractions of the elementary charge e.

Accepted Answer

Up quark: +2/3 e (1). Down quark: −1/3 e (1).

Question 2

State the quark composition of a proton and of a neutron.

Accepted Answer

Proton: up, up, down (uud) (1). Neutron: up, down, down (udd) (1).

Question 3

State the baryon number of a quark and of an antiquark.

Accepted Answer

A quark has baryon number +1/3 (1); an antiquark has −1/3 (1).

Question 4

State the strangeness of the strange quark and of the anti-strange quark.

Accepted Answer

Strange quark: strangeness −1 (1). Anti-strange quark: strangeness +1 (1).

Question 5

The π⁻ is the antiparticle of the π⁺ (up, anti-down). State the quark composition of the π⁻ and show that its charge is −1e.

Accepted Answer

The π⁻ is a down quark and an anti-up quark (1). Charge = (−1/3) + (−2/3) = −1e (1).

Question 6

State two properties of a quark that take the opposite sign for its antiquark.

Accepted Answer

Charge (1); baryon number or strangeness (1). (Any two of the three.)

Question 7

In beta-minus decay a down quark changes into an up quark, and in beta-plus decay an up quark changes into a down quark. State the charge of the W boson emitted in each case.

Accepted Answer

Beta-minus: the W boson carries charge −1, so it is a W⁻ (1). Beta-plus: the W boson carries charge +1, so it is a W⁺ (1).

Question 8

Name the only fundamental interaction that can change the flavour of a quark, and state what follows for the strangeness of the particles in a strong interaction.

Accepted Answer

The weak interaction (1). Because no other interaction changes quark flavour, a strong interaction cannot change the number of strange quarks present, so strangeness is conserved exactly in it (1).

Question 9

Show, using the quark charges, that a proton (uud) has a charge of +1e.

Accepted Answer

Charge = (+2/3) + (+2/3) + (−1/3) (1) = +3/3 = +1e (1)

Question 10

Show, using the quark charges, that a neutron (udd) has a charge of 0.

Accepted Answer

Charge = (+2/3) + (−1/3) + (−1/3) (1) = 0 (1)

Question 11

A positive pion is made of an up quark and an anti-down quark. Calculate its charge and its baryon number, showing your working.

Accepted Answer

Charge = (+2/3) + (+1/3) (1) = +1e (the anti-down quark has charge +1/3) (1) Baryon number = (+1/3) + (−1/3) (1) = 0, confirming it is a meson (1)

Question 12

State the quark composition of an antiproton and show that its charge is −1e.

Accepted Answer

An antiproton is anti-up, anti-up, anti-down (1) Charge = (−2/3) + (−2/3) + (+1/3) (1) = −3/3 = −1e (1)

Question 13

The Δ⁺⁺ baryon has charge +2e, baryon number +1 and strangeness 0. Deduce its quark composition, showing your reasoning. Quark charges: up +2/3 e, down −1/3 e.

Accepted Answer

Baryon number +1 means three quarks (1). With strangeness 0, only up (+2/3) and down (−1/3) are available, and the only trio whose charges sum to +2 is three up quarks (1). Check: 3 × (+2/3) = +2e, so the composition is uuu (1).

Question 14

In beta-minus decay a down quark changes into an up quark, emitting a W⁻ boson. Show that charge is conserved at this vertex.

Accepted Answer

Before: the down quark has charge −1/3. After: up (+2/3) + W⁻ (−1) (1). (+2/3) + (−1) = −1/3, equal to the charge before, so charge is conserved (1).

Question 15

Explain, in terms of quark content, why kaons have strangeness +1 or −1 but pions always have strangeness 0.

Accepted Answer

Each kaon contains exactly one strange or one anti-strange quark (1): an anti-strange quark gives the kaon strangeness +1 and a strange quark gives strangeness −1 (1). Pions are built only from up and down quarks and their antiquarks, all of which have strangeness 0 (1).

Question 16

The beta-minus decay of a neutron is written at quark level as d → u + e⁻ + anti-ν_e. Show that baryon number and electron lepton number are both conserved at this vertex.

Accepted Answer

Baryon number before = +1/3, the baryon number of one quark (1)
After = +1/3 for the up quark, and 0 for the electron and the antineutrino, giving +1/3, so baryon number is conserved (1)
Electron lepton number: 0 before; after = (+1) for the electron + (−1) for the electron antineutrino = 0, so it is conserved as well (1)

Question 17

A Ξ⁻ baryon has quark content dss. It decays into a Λ⁰ (uds) and a π⁻ (a down quark and an anti-up quark). The strange quark has charge −1/3 e and strangeness −1; the up quark has charge +2/3 e. Determine the quark that changes flavour, the charge of the W boson emitted, and the interaction responsible.

Accepted Answer

One strange quark becomes an up quark, s → u; the other strange quark and the down quark are spectators (1)
Charge at the vertex: −1/3 → +2/3 plus the W, so the W carries −1 and is a W⁻ (1)
Strangeness runs −2 → −1, and only the weak interaction changes quark flavour, so the decay is weak (1)

Question 18

A student writes the quark change in beta-plus decay as u → d + W⁻. Explain, using the quark charges, why the boson emitted must be a W⁺ instead, and state what that boson becomes.

Accepted Answer

The up quark has charge +2/3 and the down quark −1/3 (1)
The quark gives up (+2/3) − (−1/3) = +1 of charge, so the boson must carry +1, which is a W⁺ and not a W⁻ (1)
The W⁺ becomes a positron and an electron neutrino (1)

Question 19

Describe the beta-minus decay of a neutron in terms of its quarks, naming what the changed quark becomes and the particles emitted.

Accepted Answer

Inside the neutron (udd) one down quark changes into an up quark (1), turning it into a proton (uud) (1). The change emits a W⁻ boson (1), which becomes an electron and an electron antineutrino (1).

Question 20

A positive kaon is made of an up quark and an anti-strange quark. The anti-strange quark has charge +1/3 e and strangeness +1. Determine the charge and the strangeness of the K⁺, showing your working.

Accepted Answer

Charge = (+2/3) + (+1/3) (1) Charge = +1e (1) Strangeness = 0 (up) + (+1) (anti-strange) = +1 (1)

Question 21

Explain why the quark model assigns the neutron the composition udd rather than uud.

Accepted Answer

The composition must give the neutron's zero charge (1). uud gives +1e (the proton) (1), whereas udd gives (+2/3) + (−1/3) + (−1/3) = 0, matching the neutral neutron (1).

Question 22

A baryon has charge 0 and strangeness −1. Deduce its quark composition, using only up, down and strange quarks. Quark charges: up +2/3 e, down −1/3 e, strange −1/3 e; the strange quark has strangeness −1.

Accepted Answer

A baryon is three quarks, with no antiquarks (baryon number +1) (1). Strangeness −1 means exactly one strange quark, contributing charge −1/3 (1). The other two quarks must supply +1/3 between them, which one up (+2/3) and one down (−1/3) do (1). Composition: uds, with charge (+2/3) + (−1/3) + (−1/3) = 0 ✓ (1).

Question 23

A student claims to have detected a baryon with charge +1 and strangeness +1. Deduce whether such a particle can be built from up, down and strange quarks and their antiquarks. The strange quark has strangeness −1 and baryon number +1/3; an antiquark has baryon number −1/3.

Accepted Answer

A baryon has baryon number +1, so it must contain three quarks and no antiquarks, since each antiquark contributes −1/3 (1). Among the up, down and strange quarks, only the strange quark carries strangeness, and its value is −1; the anti-strange quark (+1) is not available inside a baryon (1). A baryon's strangeness can therefore only be 0, −1, −2 or −3, never +1 (1). The claim is impossible: no such particle exists in the quark model (1).

Question 24

State the maximum possible charge of a baryon built from up, down and strange quarks, and explain your reasoning. Quark charges: up +2/3 e, down −1/3 e, strange −1/3 e.

Accepted Answer

The up quark has the largest charge, +2/3 e (1). A baryon is three quarks, so the maximum charge is that of uuu: 3 × (+2/3) = +2e (1). Replacing any up with a down or a strange quark (both −1/3 e) lowers the total, so +2e is the maximum (1).

Question 25

A proton bound in a nucleus decays by beta-plus emission. Describe this decay in terms of the proton's quarks, naming what the changed quark becomes, the boson emitted and the particles that boson becomes.

Accepted Answer

Inside the proton (uud) one up quark changes into a down quark (1), turning it into a neutron (udd) (1). The change emits a W⁺ boson (1), which becomes a positron and an electron neutrino (1).

Question 26

A Λ⁰ baryon (uds) decays into a proton (uud) and a π⁻ (a down quark and an anti-up quark). Explain why this decay cannot proceed by the strong interaction, and state what this implies about the lifetime of the Λ⁰ compared with that of a particle which decays by the strong interaction.

Accepted Answer

Writing the quark content under both sides, the up and the down quark spectate and the strange quark must become an up quark, s → u, which is a change of flavour (1)
The strong interaction conserves quark flavour, and so conserves strangeness exactly, and cannot bring that change about (1)
Strangeness runs −1 → 0, a change of 1, which only the weak interaction allows (1)
The weak interaction is far less probable, so the Λ⁰ is comparatively long-lived beside a particle that decays by the strong interaction (1)

Question 27

A K⁰ meson is a down quark with an anti-strange quark, and it decays into a π⁺ (an up quark with an anti-down quark) and a π⁻ (a down quark with an anti-up quark). The anti-strange quark has charge +1/3 e and strangeness +1; the anti-up quark has charge −2/3 e. Deduce which quark or antiquark changes flavour and what it becomes, the charge of the W boson emitted and what that boson becomes, and the change in strangeness.

Accepted Answer

The K⁰'s down quark is a spectator: it appears again in the π⁻ (1)
So it is the anti-strange quark that changes flavour, becoming the anti-up quark that pairs with it to make the π⁻ (1)
Charge: +1/3 → −2/3 plus the W, so the W carries +1 and is a W⁺ (1)
That W⁺ becomes the up quark and the anti-down quark of the π⁺ (1)
Strangeness runs +1 → 0, a change of 1, so this is a weak decay (1)