Quiz — Chapter 31: The Standard Model and Nuclear Physics

Instructions: Select the best answer for each question. Detailed solutions follow at the end.

Question 1. The quark content of the proton is $uud$. The up quark has charge $+2/3\,e$ and the down quark has charge $-1/3\,e$. What fraction of the proton's mass ($938$ MeV/$c^2$) is accounted for by the current quark masses ($m_u \approx 2$ MeV, $m_d \approx 5$ MeV)?

(a) About 1% (b) About 10% (c) About 30% (d) About 50%

Question 2. The strong coupling constant $\alpha_s$ decreases at higher energies (shorter distances). This property is called:

(a) Confinement (b) Chiral symmetry breaking (c) Asymptotic freedom (d) Color screening

Question 3. In QCD, gluons differ from photons in QED because gluons:

(a) Are massive particles with spin 0 (b) Carry color charge and interact with each other (c) Are electrically charged (d) Mediate only attractive forces

Question 4. The nuclear force between protons and neutrons is best described as:

(a) A fundamental force between elementary particles (b) A direct consequence of one-gluon exchange between nucleons (c) A residual effect of the strong (color) force, analogous to the van der Waals force (d) A force mediated by W and Z bosons

Question 5. Confinement in QCD means that:

(a) Quarks always move at the speed of light inside hadrons (b) Only color-neutral combinations of quarks and gluons can exist as free particles (c) The strong coupling constant is always greater than 1 (d) Gluons cannot propagate between quarks

Question 6. The pions are much lighter than other hadrons because they are:

(a) Made of the lightest quarks (b) Held together by the weakest color force (c) Pseudo-Goldstone bosons of spontaneously broken chiral symmetry (d) Bound states of gluons only

Question 7. In chiral effective field theory, three-nucleon forces first appear at which order?

(a) Leading order (LO) (b) Next-to-leading order (NLO) (c) Next-to-next-to-leading order (N$^2$LO) (d) N$^3$LO

Question 8. Lattice QCD calculations have demonstrated that the proton mass is approximately $938$ MeV even when the $u$ and $d$ quark masses are set to zero. This shows that:

(a) Quarks are not real constituents of the proton (b) The Higgs mechanism generates most of the proton mass (c) The proton mass arises primarily from the energy of the confined gluon field and quark kinetic energy (d) The quark model is incorrect

Question 9. Robert Hofstadter received the Nobel Prize in 1961 for demonstrating through electron scattering that:

(a) Quarks exist inside the proton (b) The proton has a finite size with a smooth charge distribution (c) The neutron has zero magnetic moment (d) Nuclear forces obey the Yukawa potential

Question 10. The proton spin puzzle refers to the finding that:

(a) The proton spin is not exactly $1/2$ (b) Quarks carry only about $30$--$40\%$ of the proton's spin, with the rest from gluon spin and orbital angular momentum (c) The proton's magnetic moment disagrees with the Dirac prediction (d) The proton's spin is entirely due to gluon angular momentum

Question 11. The proton radius puzzle arose from a discrepancy between measurements using:

(a) Electron scattering vs. neutron scattering (b) Muonic hydrogen spectroscopy vs. electron scattering and electronic hydrogen spectroscopy (c) X-ray diffraction vs. electron microscopy (d) Compton scattering vs. pair production

Question 12. The EMC effect, discovered in 1983, showed that:

(a) Muons and electrons scatter identically from protons (b) The quark structure functions of nucleons are modified when nucleons are bound inside a nucleus (c) The proton contains exactly three quarks (d) Deep inelastic scattering violates QCD predictions

Question 13. The QCD scale parameter $\Lambda_{\text{QCD}} \approx 200$--$300$ MeV represents:

(a) The mass of the lightest gluon (b) The energy at which perturbative QCD ceases to be reliable and non-perturbative effects dominate (c) The binding energy of the proton (d) The mass of the pion

Question 14. At very high temperatures ($T > 155$ MeV), nuclear matter undergoes a transition to:

(a) A neutron gas (b) A quark-gluon plasma in which quarks and gluons are deconfined (c) A Bose-Einstein condensate of pions (d) A crystal lattice of nucleons

Question 15. Which experimental facility, currently under construction, is designed to make definitive measurements of the gluon contribution to the proton spin?

(a) The Large Hadron Collider (LHC) (b) The Facility for Rare Isotope Beams (FRIB) (c) The Electron-Ion Collider (EIC) (d) The International Thermonuclear Experimental Reactor (ITER)

Solutions

1. (a) Three quarks contribute $2 \times 2.2 + 4.7 = 9.1$ MeV to the proton mass of 938 MeV — about 1%. The remaining 99% comes from the energy of the strong force (gluon field energy and quark kinetic energy from confinement).

2. (c) Asymptotic freedom is the property, discovered by Gross, Wilczek, and Politzer in 1973, that the QCD coupling constant decreases at higher energies (shorter distances). This is a consequence of gluon self-interaction and is opposite to the behavior of QED.

3. (b) Unlike photons, which are electrically neutral and do not self-interact, gluons carry color charge (each gluon carries one color and one anti-color). This means gluons interact with other gluons, leading to confinement and asymptotic freedom — the defining features of QCD that distinguish it qualitatively from QED.

4. (c) The nuclear force is a residual strong force. Just as the van der Waals force between electrically neutral atoms arises from residual electromagnetic interactions between their charged constituents, the nuclear force between color-neutral nucleons arises from residual color interactions between their constituent quarks and gluons.

5. (b) Confinement means quarks and gluons cannot be isolated — they are permanently bound inside color-neutral hadrons (baryons, mesons, and exotic states). Attempting to separate a quark-antiquark pair produces new hadrons rather than free quarks.

6. (c) The pions are pseudo-Goldstone bosons arising from the spontaneous breaking of chiral symmetry in QCD. They would be exactly massless if the $u$ and $d$ quark masses were zero. Their small mass ($\sim 140$ MeV, much less than other hadrons like the $\rho$ at 775 MeV) reflects the small quark masses.

7. (c) In Weinberg's chiral power counting, three-nucleon forces first appear at N$^2$LO ($\nu = 3$). The leading 3NF includes the Fujita-Miyazawa two-pion exchange mechanism, a one-pion exchange plus contact term, and a pure three-nucleon contact term.

8. (c) Lattice QCD calculations show that the proton mass in the chiral limit ($m_u = m_d = 0$) is approximately 860 MeV — barely different from 938 MeV. This demonstrates that the proton's mass comes primarily from the energy of the confined gluon field and the kinetic energy of quarks (via the uncertainty principle), not from the Higgs-generated quark masses.

9. (b) Hofstadter's electron scattering experiments at Stanford showed that the proton is not a point particle but has a finite size (charge radius $\sim 0.87$ fm) with a smooth charge distribution. This was one of the first indications that the proton has internal structure.

10. (b) The proton spin puzzle is the finding that quark spins account for only about 30--40% of the proton's total spin of $1/2$. The remainder comes from the gluon spin contribution ($\sim 40$--$60\%$) and the orbital angular momentum of quarks and gluons.

11. (b) The proton radius puzzle was a discrepancy between the proton charge radius measured by muonic hydrogen spectroscopy ($r_p = 0.841$ fm, CREMA 2010) and the earlier values from electron-proton scattering and electronic hydrogen spectroscopy ($r_p = 0.877$ fm). The puzzle is now largely resolved in favor of the smaller value.

12. (b) The European Muon Collaboration found that the structure function $F_2$ per nucleon is different for nucleons bound in a nucleus compared to free nucleons. This "EMC effect" — a $\sim 10$--$15\%$ suppression at intermediate $x$ — shows that the quark-gluon structure of nucleons is modified by the nuclear environment.

13. (b) $\Lambda_{\text{QCD}}$ is the scale at which the running coupling $\alpha_s$ formally diverges in the leading-order formula, signaling the breakdown of perturbative QCD. Below this scale, non-perturbative effects (confinement, hadron formation) dominate.

14. (b) Above the QCD transition temperature ($T_c \approx 155$ MeV), hadrons dissolve into a quark-gluon plasma (QGP) — a state of deconfined quarks and gluons. The QGP has been created at RHIC and the LHC and existed in the early universe during the first $\sim 10$ microseconds.

15. (c) The Electron-Ion Collider (EIC), under construction at Brookhaven National Laboratory (expected operations $\sim 2032$), will collide polarized electrons with polarized protons and ions. Its primary physics goals include mapping the gluon spin contribution to the proton spin and determining the role of orbital angular momentum.