Exercises — Chapter 31: The Standard Model and Nuclear Physics

Section 31.1–31.2: The Quark Model and QCD Basics

Problem 31.1 — Quark Composition and Quantum Numbers (Warm-up)

(a) Verify the electric charge, baryon number, and strangeness of each of the following hadrons from their quark content:

Use the quark quantum numbers: $Q_u = +2/3$, $Q_d = Q_s = -1/3$, $B_q = 1/3$, $S_s = -1$ (strangeness carried by the $s$ quark only).

(b) The $\Omega^-$ baryon ($sss$) has $J^\pi = 3/2^+$. Explain how the Pauli exclusion principle is satisfied for three identical quarks in the same spatial state with all spins aligned, given the color quantum number.

(c) Write the fully antisymmetric color wavefunction for a baryon:

$$\chi_{\text{color}} = \frac{1}{\sqrt{6}} \sum_{\text{permutations}} \epsilon_{abc} |a\rangle_1 |b\rangle_2 |c\rangle_3$$

Verify that this is a color singlet (invariant under SU(3)$_{\text{color}}$ transformations).

Problem 31.2 — The Origin of Nucleon Mass

The proton mass is $m_p = 938.3$ MeV/$c^2$. The up quark mass is $m_u \approx 2.2$ MeV/$c^2$ and the down quark mass is $m_d \approx 4.7$ MeV/$c^2$ (at the $\overline{\text{MS}}$ scale of 2 GeV).

(a) Calculate the total current quark mass contribution to the proton ($uud$) and neutron ($udd$). What fraction of each hadron's mass comes from the quark masses?

(b) The proton-neutron mass difference is $m_n - m_p = 1.293$ MeV. The difference in quark mass content is $\Delta m_q = (m_d - m_u) = 2.5 \pm 0.5$ MeV. The electromagnetic self-energy contribution is estimated to be $\Delta m_{\text{EM}} \approx -0.76$ MeV (the proton's charge increases its mass relative to the neutron). Verify that $\Delta m_q + \Delta m_{\text{EM}} \approx m_n - m_p$ within uncertainties.

(c) Discuss the physical implications for nuclear physics and astrophysics if the quark mass difference were reversed ($m_u > m_d$), making the proton heavier than the neutron. Consider the stability of hydrogen and the consequences for stellar nucleosynthesis.

Problem 31.3 — Meson Quark Content and Isospin

(a) The three pions form an isospin triplet ($T = 1$, $T_z = +1, 0, -1$). Show that the quark content assignments $\pi^+ = u\bar{d}$, $\pi^- = d\bar{u}$, and $\pi^0 = (u\bar{u} - d\bar{d})/\sqrt{2}$ are consistent with these isospin quantum numbers, using $T_z(u) = +1/2$ and $T_z(d) = -1/2$.

(b) The $\rho$ mesons ($\rho^+$, $\rho^0$, $\rho^-$) have the same quark content as the pions but $J^\pi = 1^-$ instead of $0^-$. What is the difference in the quark spin configuration between a pion and a $\rho$ meson?

(c) The neutral kaon system consists of $K^0 = d\bar{s}$ and $\bar{K}^0 = s\bar{d}$. These are not mass eigenstates — the physical particles $K_S^0$ and $K_L^0$ are superpositions. Why can $K^0$ and $\bar{K}^0$ mix (while $\pi^+$ and $\pi^-$ cannot)?

Problem 31.4 — Running of $\alpha_s$

The leading-order running of the strong coupling constant is:

$$\alpha_s(Q^2) = \frac{12\pi}{(33 - 2n_f) \ln(Q^2/\Lambda_{\text{QCD}}^2)}$$

where $n_f$ is the number of active quark flavors at scale $Q$.

(a) Using $\alpha_s(M_Z = 91.2 \text{ GeV}) = 0.118$ and $n_f = 5$ (active flavors at $Q = M_Z$), solve for $\Lambda_{\text{QCD}}$.

(b) Calculate $\alpha_s$ at $Q = 10$ GeV, $Q = 2$ GeV, and $Q = 0.5$ GeV (using the same $\Lambda_{\text{QCD}}$ and appropriate $n_f$).

(c) At what value of $Q$ does $\alpha_s = 1$ (approximately)? Compare this to typical nuclear momentum scales ($p_F \approx 250$ MeV for nucleons in a nucleus). What does this tell you about the applicability of perturbative QCD to nuclear physics?

(d) If nature had 20 quark flavors instead of 6, would QCD still be asymptotically free? What is the critical number of flavors?

Section 31.3–31.4: Confinement and Asymptotic Freedom

Problem 31.5 — The Cornell Potential

The quark-antiquark potential is often parametrized as the Cornell potential:

$$V(r) = -\frac{4}{3} \frac{\alpha_s}{r} + \sigma r + C$$

where $\alpha_s \approx 0.39$, $\sigma \approx 0.18$ GeV$^2$ (the string tension), and $C$ is a constant. Use $\hbar c = 0.197$ GeV$\cdot$fm.

(a) Convert the string tension to more intuitive units: GeV/fm and tonnes (using $1 \text{ GeV/fm} = 1.602 \times 10^5$ N $\approx 16.3$ tonnes).

(b) At what quark-antiquark separation $r_0$ does the potential energy equal zero (ignoring $C$)?

(c) At what separation $r_{\text{break}}$ does the potential energy equal the mass of a pion-pion pair ($2m_\pi c^2 \approx 280$ MeV), approximately the threshold for string breaking?

(d) Compare $r_0$ and $r_{\text{break}}$ to the proton charge radius ($\sim 0.84$ fm). Discuss the physical implications.

Problem 31.6 — The Uncertainty Principle and Nucleon Mass

Quarks are confined inside the nucleon, with a confinement radius of approximately $r \sim 0.5$ fm.

(a) Use the uncertainty principle $\Delta p \sim \hbar/r$ to estimate the typical momentum of a confined quark.

(b) Since the quark mass ($\sim 5$ MeV) is much less than $\Delta p \cdot c$, the quarks are ultrarelativistic. Estimate their kinetic energy using $E \approx pc$.

(c) With three quarks, the total kinetic energy is approximately $3 \times pc$. Compare this to the proton mass. (This is a crude estimate — the binding energy from gluon exchange must also be considered — but it gives the right order of magnitude.)

(d) Repeat the estimate for an electron confined to a proton-sized region. What kinetic energy would the electron have? Why does this rule out electrons as nuclear constituents (the problem that motivated the search for the neutron)?

Section 31.5–31.6: Lattice QCD and Chiral EFT

Problem 31.7 — Lattice QCD Parameters

A lattice QCD calculation uses a $48^3 \times 96$ lattice (48 spatial sites in each direction, 96 temporal sites) with lattice spacing $a = 0.09$ fm.

(a) What is the physical spatial extent $L$ of the lattice in fm? Is this large enough to contain a single nucleon (radius $\sim 0.84$ fm)? What about a deuteron (RMS radius $\sim 2.0$ fm)?

(b) What is the highest momentum (ultraviolet cutoff) that can be resolved on this lattice? Express in GeV.

(c) What is the lowest nonzero momentum (infrared cutoff)? Express in MeV.

(d) The pion Compton wavelength is $\lambda_\pi = \hbar/(m_\pi c) \approx 1.41$ fm. Is the lattice extent large enough that the pion field is not significantly affected by the finite volume?

(e) The computational cost of a lattice QCD calculation scales approximately as $(L/a)^3 \times (T/a) \times (1/m_q^2)$. If halving the lattice spacing (while keeping the physical volume fixed) increases the cost by a factor of $2^4 = 16$ (more sites in each dimension), estimate the factor by which the cost increases when going from $a = 0.12$ fm to $a = 0.06$ fm, keeping $L$ and $T$ fixed.

Problem 31.8 — Chiral Symmetry and the Pion Mass

The Gell-Mann–Oakes–Renner relation connects the pion mass to the quark masses:

$$m_\pi^2 f_\pi^2 = -(m_u + m_d) \langle \bar{q}q \rangle$$

where $f_\pi = 92.2$ MeV is the pion decay constant and $\langle \bar{q}q \rangle \approx -(250 \text{ MeV})^3$ is the quark condensate.

(a) Using $m_\pi = 135$ MeV (neutral pion), calculate $m_u + m_d$ from this relation. Compare to the PDG values ($m_u + m_d \approx 7$ MeV at 2 GeV).

(b) What would $m_\pi$ be if the quark masses were doubled? If they were halved?

(c) In the chiral limit ($m_u = m_d = 0$), the pions would be exactly massless Goldstone bosons. What would happen to the range of the nuclear force ($\sim \hbar/(m_\pi c)$) in this limit? Would nuclei still exist? (Hint: consider the balance between the infinite-range pion exchange and the short-range repulsion.)

(d) The kaon mass is $m_K \approx 496$ MeV. The kaon is a pseudo-Goldstone boson associated with the breaking of SU(3) flavor symmetry, with $m_K^2 \propto (m_u + m_s)$. Estimate the strange quark mass $m_s$ from the ratio $m_K^2/m_\pi^2$, assuming $m_s \gg m_u, m_d$.

Problem 31.9 — Chiral EFT Power Counting

In Weinberg's chiral power counting, the chiral order $\nu$ of an irreducible diagram contributing to the $A$-nucleon potential is:

$$\nu = -4 + 2A + 2L + \sum_i \Delta_i$$

where $L$ is the number of loops and $\Delta_i \geq 0$ is the vertex index of vertex $i$ (determined by the number of nucleon lines and pion lines and the number of derivatives/pion mass insertions at that vertex).

(a) Show that the leading-order ($\nu = 0$) two-nucleon ($A = 2$) potential has $L = 0$ and $\sum_i \Delta_i = 0$. What are the allowed topologies? (Answer: one-pion exchange and contact terms.)

(b) Show that the leading three-nucleon force ($A = 3$) first appears at $\nu = 3$ (N$^2$LO). With $L = 0$, what constraint does this place on $\sum_i \Delta_i$?

(c) The four-nucleon force first appears at $\nu = 4$ (N$^3$LO). Verify this using the power counting formula with $A = 4$ and $L = 0$.

(d) For typical nuclear momenta $Q \sim m_\pi = 140$ MeV and $\Lambda_\chi \sim 1$ GeV, estimate the size of corrections at NLO, N$^2$LO, and N$^3$LO relative to LO, using the ratio $(Q/\Lambda_\chi)^\nu$.

Section 31.7–31.9: Nucleon Structure and the Proton Radius

Problem 31.10 — Proton Form Factors and the Charge Radius

The Sachs electric form factor of the proton is approximately described by the dipole form:

$$G_E^p(Q^2) = \left(1 + \frac{Q^2}{0.71 \text{ GeV}^2}\right)^{-2}$$

(a) Calculate $dG_E^p/dQ^2$ at $Q^2 = 0$ and use the relation $\langle r^2 \rangle = -6 \, dG_E/dQ^2|_{Q^2=0}$ to extract the proton charge radius $r_p$. Express in fm.

(b) The dipole form factor is the Fourier transform of an exponential charge distribution $\rho(r) = \rho_0 e^{-r/a}$. Determine $a$ in fm and verify that $\langle r^2 \rangle = 12a^2$ gives the same radius.

(c) The muonic hydrogen measurement gives $r_p = 0.841$ fm. Determine the value of $\Lambda^2$ (the parameter replacing 0.71 GeV$^2$ in the dipole formula) that would reproduce this radius.

(d) At $Q^2 = 1$ GeV$^2$, calculate $G_E^p$ using both the standard dipole and your modified dipole from part (c). By what percentage do they differ? At $Q^2 = 0.01$ GeV$^2$?

Problem 31.11 — The Neutron Charge Distribution

The neutron is electrically neutral ($G_E^n(0) = 0$), but it has a nonzero charge radius: $\langle r^2 \rangle_n = -0.116$ fm$^2$.

(a) The negative value of $\langle r^2 \rangle_n$ means the outer part of the neutron carries net negative charge. In the quark model, the neutron is $udd$. Qualitatively, why would the outer region be negative? (Hint: Consider the pion cloud $n \to p + \pi^-$.)

(b) A simple model represents the neutron charge distribution as two concentric shells:

$$\rho(r) = \frac{Q_+}{4\pi R_+^2} \delta(r - R_+) - \frac{Q_+}{4\pi R_-^2} \delta(r - R_-)$$

with $R_+ < R_-$ (positive charge inside, negative charge outside), and equal and opposite total charges ($Q_+ = -Q_-$). Derive the expression for $\langle r^2 \rangle$ and show it is negative when $R_- > R_+$.

(c) Using the measured $\langle r^2 \rangle_n = -0.116$ fm$^2$ and assuming $R_+ = 0.3$ fm (the core), estimate $R_-$ if $Q_+ = +e/3$ (a rough estimate of the $u$ quark contribution). Compare to the nucleon size.

Problem 31.12 — The Proton Spin Budget

The proton spin sum rule is:

$$\frac{1}{2} = \frac{1}{2}\Delta\Sigma + \Delta G + L_q + L_g$$

Modern experimental determinations give approximately: - $\Delta\Sigma \approx 0.33 \pm 0.05$ (quark spin, from inclusive DIS) - $\Delta G \approx 0.25 \pm 0.10$ (gluon spin, from RHIC $pp$ collisions, for $x > 0.05$)

(a) Calculate the quark spin contribution $\frac{1}{2}\Delta\Sigma$. What percentage of the proton spin does this represent?

(b) Assuming the gluon spin contribution is $\Delta G = 0.25$, what is the combined orbital angular momentum $L_q + L_g$ required to satisfy the sum rule?

(c) The naive quark model predicts $\Delta\Sigma = 1$ ($\Delta u = 4/3$, $\Delta d = -1/3$, $\Delta s = 0$). The experimental value $\Delta\Sigma \approx 0.33$ is much smaller. One contribution to the suppression is the strange sea: $\Delta s \approx -0.05$. Calculate what $\Delta u + \Delta d$ must be if $\Delta s = -0.05$ and $\Delta\Sigma = 0.33$.

(d) The Electron-Ion Collider (EIC) will measure $\Delta G$ with much better precision and will access $L_q$ through generalized parton distributions. If the EIC determines $\Delta G = 0.30 \pm 0.03$, what constraint does this place on $L_q + L_g$?

Problem 31.13 — Muonic vs. Electronic Hydrogen

In hydrogen-like atoms, the energy shift due to the finite nuclear size (the proton radius) is:

$$\Delta E_{\text{size}} = \frac{2}{3} \frac{Z \alpha}{\hbar c} \left(\frac{Ze^2}{4\pi\epsilon_0}\right) |\psi_{nS}(0)|^2 \langle r_p^2 \rangle$$

where $|\psi_{nS}(0)|^2 = (Z/n a_0)^3/\pi$ for the $S$-states of a hydrogen-like atom with Bohr radius $a_0 = \hbar/(Z\alpha m_\ell c)$ ($m_\ell$ being the lepton mass).

(a) Calculate the ratio of $|\psi_{1S}(0)|^2$ for muonic hydrogen to that for ordinary hydrogen. (Use $m_\mu/m_e = 206.8$.)

(b) The $2S$--$2P$ Lamb shift in electronic hydrogen is approximately $1058$ MHz ($\approx 4.4 \times 10^{-6}$ eV). The proton-size contribution is approximately $1.2\%$ of this. What is the absolute size of the proton-radius correction in electronic hydrogen?

(c) In muonic hydrogen, the $2S$--$2P$ splitting is much larger ($\approx 206$ meV), and the proton-size contribution is about $2\%$ of this splitting. Calculate the proton-size correction in muonic hydrogen. By what factor is it enhanced relative to electronic hydrogen?

(d) A change of $\delta r_p = 0.04$ fm in the proton radius (the original puzzle discrepancy) changes $\langle r_p^2 \rangle$ by approximately $2 r_p \delta r_p$. Estimate the corresponding energy shift in muonic hydrogen using your result from part (c). Compare this to the experimental precision achieved by the CREMA collaboration ($\sim 0.005$ meV).

Problem 31.13a — The EMC Effect (Extended)

The EMC effect is the observation that the structure function per nucleon in a nucleus differs from that of a free nucleon:

$$R(x) = \frac{F_2^A(x, Q^2)}{A \cdot F_2^N(x, Q^2)}$$

where $x$ is the Bjorken scaling variable. Experimentally, $R(x)$ shows four distinct regions:

(a) In a simple Fermi gas model of the nucleus, nucleons have momenta up to the Fermi momentum $p_F \approx 250$ MeV/$c$. A nucleon at rest in the lab frame has momentum fraction $x = Q^2/(2M_N \nu)$ from 0 to 1 in deep inelastic scattering. If the nucleon has additional momentum $p$ from nuclear motion, the effective $x$ range extends to $x > 1$. Show that Fermi motion can qualitatively explain region IV: the nuclear structure function is enhanced at $x > 0.7$ because nuclear motion smears the nucleon structure function to larger $x$ values.

(b) The EMC effect in region III (a depletion of $\sim 10$--$15\%$ for heavy nuclei like iron) was initially surprising because it suggests the quark-gluon structure of nucleons is modified in the nuclear medium. List at least three proposed explanations for the EMC effect and briefly describe the physical mechanism of each.

(c) The size of the EMC effect scales approximately linearly with the nuclear density $\rho_A$. For iron ($A = 56$, $\rho \approx 0.16$ fm$^{-3}$), the depletion at $x = 0.5$ is about $13\%$. Estimate the depletion for deuterium ($A = 2$, average density much lower). Why is measuring the EMC effect in deuterium experimentally challenging?

(d) The EMC effect has been recently correlated with short-range nucleon-nucleon correlations (SRC) — pairs of nucleons at very short distances that have high relative momentum. Explain qualitatively why nucleons in SRC pairs might have modified quark distributions compared to isolated nucleons.

Problem 31.13b — The Running Coupling and the QCD Phase Diagram

(a) The QCD coupling $\alpha_s(Q)$ becomes strong ($\alpha_s > 1$) at momentum transfers below $Q \sim 1$ GeV. Sketch (qualitatively) a plot of $\alpha_s$ versus $Q$ from $Q = 0.1$ GeV to $Q = 100$ GeV on a logarithmic scale. Mark the regions where perturbative QCD is applicable and where it is not.

(b) The QCD phase diagram plots temperature $T$ versus baryon chemical potential $\mu_B$. At $T = 0$ and $\mu_B = 0$, the ground state is the vacuum. At $T = 0$ and $\mu_B \approx 923$ MeV (the nucleon mass minus the binding energy per nucleon), we have cold nuclear matter at saturation density. At high $T$ and low $\mu_B$, lattice QCD predicts a crossover transition to the quark-gluon plasma at $T_c \approx 155$ MeV. Sketch the QCD phase diagram and label these regions.

(c) At high $\mu_B$ and low $T$, theory predicts a first-order phase transition line ending at a critical point. Why can lattice QCD not directly compute the phase diagram at finite baryon chemical potential? (Hint: the fermion determinant becomes complex — the "sign problem.")

(d) The Beam Energy Scan program at RHIC systematically varies the collision energy of gold-gold collisions to map the QCD phase diagram. What experimental signatures would indicate the location of the critical point? Discuss at least two observables.

Synthesis Problems

Problem 31.14 — The Van der Waals Analogy

(a) For two neutral hydrogen atoms separated by a distance $R \gg a_0$ (the Bohr radius), the Van der Waals potential is $V(R) = -C_6/R^6$ with $C_6 \approx 6.5 \, E_h a_0^6$ ($E_h = 27.2$ eV is the Hartree energy). Calculate the Van der Waals potential energy at $R = 5a_0$ in eV. Compare this to the hydrogen ionization energy (13.6 eV).

(b) The nuclear analogue: at $r = 2$ fm, the one-pion exchange potential between two nucleons has magnitude roughly $V_\pi \sim 10$--$20$ MeV. Compare this to the nucleon mass (939 MeV) — the analogue of the ionization energy.

(c) In both cases, the residual force is much weaker than the fundamental binding energy. Estimate the ratio $V_{\text{residual}}/E_{\text{binding}}$ for each case and compare.

(d) Despite the analogy, there is a qualitative difference: the Van der Waals potential can be calculated perturbatively from QED ($\alpha \approx 1/137$), while the nuclear force cannot be calculated perturbatively from QCD ($\alpha_s \sim 1$). Explain why this difference in coupling strength is the central challenge for deriving nuclear forces from first principles.

Problem 31.15 — From Quarks to the Deuteron (Comprehensive)

This problem traces the chain from QCD to the simplest nucleus.

(a) The QCD Lagrangian has six quark flavors and one coupling constant $g_s$ (plus quark masses). How many free parameters does QCD have in total?

(b) At low energies, the relevant degrees of freedom change from quarks and gluons to nucleons and pions. The leading-order chiral EFT Lagrangian for the $NN$ system has two contact terms (coupling constants $C_S$ and $C_T$) plus the pion-nucleon coupling $g_A$ and $f_\pi$. How do the number of parameters compare to the fundamental QCD level?

(c) The deuteron binding energy is $B_d = 2.225$ MeV. Explain why this cannot currently be computed directly from the QCD Lagrangian with controlled precision, despite QCD being the fundamental theory.

(d) In chiral EFT at N$^3$LO, the $NN$ potential has approximately 29 low-energy constants fitted to scattering data. With this potential, the deuteron binding energy is reproduced to within a few percent. Is this a "prediction from QCD" or a "fit to data"? Discuss the philosophical status of chiral EFT as a bridge between QCD and nuclear physics.

Problem 31.16 — The Quark-Gluon Plasma

At the Relativistic Heavy Ion Collider (RHIC), gold nuclei ($^{197}$Au) collide at center-of-mass energies up to $\sqrt{s_{NN}} = 200$ GeV per nucleon pair, creating a quark-gluon plasma (QGP).

(a) In a central (head-on) Au+Au collision, approximately how many nucleons participate? (Recall $A = 197$ for gold.) How many quarks and gluons are liberated, roughly, if each nucleon contains 3 valence quarks plus a sea of gluons and $q\bar{q}$ pairs?

(b) The initial temperature of the QGP produced at RHIC is estimated to be $T \approx 300$--$400$ MeV. Compare this to the QCD transition temperature $T_c \approx 155$ MeV and to the temperature of the Sun's core ($T_\odot \approx 1.3$ keV). By what factor does the QGP temperature exceed the solar core temperature?

(c) The QGP at RHIC behaves as a nearly perfect liquid with a viscosity-to-entropy-density ratio near the conjectured quantum lower bound $\eta/s = \hbar/(4\pi k_B)$. This was surprising — the naive expectation was a weakly interacting gas. Explain qualitatively why $\alpha_s$ at $T \sim 300$ MeV (corresponding to $Q \sim T$) is still large enough for strong interactions.

(d) The $J/\psi$ meson ($c\bar{c}$ bound state) is expected to be suppressed in the QGP because the Debye screening of the color force in the plasma prevents the $c$ and $\bar{c}$ quarks from binding. The screening radius $r_D$ decreases with increasing temperature. If the $J/\psi$ binding radius is approximately $0.25$ fm, estimate the temperature at which $r_D \sim 0.25$ fm, given that $r_D \approx 1/(g_s T)$ with $g_s \sim 2$ at the relevant scale.

Problem 31.17 — Isospin Symmetry and the Nuclear Force (Conceptual)

(a) In the quark model, the proton ($uud$) and neutron ($udd$) differ only by the interchange of one $u$ quark for one $d$ quark. If $m_u = m_d$ exactly, the strong interaction between nucleons would be exactly charge-independent ($V_{pp} = V_{nn} = V_{np}$ in the same isospin state). Explain why isospin symmetry is only approximate in nature and list two sources of isospin-symmetry breaking.

(b) The mass difference $m_d - m_u \approx 2.5$ MeV generates isospin-breaking effects in the nuclear force at the level of a few percent. One observable consequence is the Nolen-Schiffer anomaly: the predicted Coulomb energy difference between mirror nuclei (like $^{41}$Ca and $^{41}$Sc) does not fully account for the observed binding energy difference. Additional charge-symmetry-breaking nuclear force contributions are needed. Estimate the size of the nuclear CSB effect ($\sim (m_d - m_u)/\Lambda_\chi$) relative to the Coulomb correction.

(c) In chiral EFT, the leading isospin-breaking nuclear force comes from the pion mass difference ($m_{\pi^\pm} - m_{\pi^0} = 4.6$ MeV) and from a charge-symmetry-breaking contact term. At what chiral order do these corrections first appear? (Hint: the pion mass difference introduces a correction at leading order in isospin breaking but at NLO in the chiral expansion.)