Chapter 5 — Summary and Problems · Elementary Particle Physics

The problems are markedly less theoretical than the chapter that produced them, and that is the examination: a theory chapter is tested by whether you can put numbers to it.

🎯 Why this matters

Order-of-magnitude work is what the rest of the book assumes without saying so. An exact calculation is what a computer is for; deciding which terms may be dropped before starting is not.

Chapter 5 built the first quantum field theory the book treats properly, and its nineteen problems are noticeably less about QED than the chapter was. Half of them are order-of-magnitude estimates and kinematics — the skills a chapter this theoretical is at risk of letting atrophy.

They fall into four families:

five are estimates — 5.1, 5.2, 5.3, 5.13, 5.14. Every one is “turn a size into a momentum, or a mass into a length”, and none needs a field theory;
four are Breit–Wigner arithmetic — 5.7, 5.8, 5.9, and 5.10’s threshold. These reach back to §4.1 and §4.9, and 5.9 is the peak-area method doing real work;
three are the running of α — 5.11 and 5.12 directly, and 5.4 obliquely, since it asks where a virtual particle actually is;
six are selection rules and diagram-drawing — 5.5, 5.6, 5.15, 5.16, 5.17 and 5.18, the last of them an isospin measurement;
and 5.19 is a collider-design estimate, which turns out to be the one that reaches furthest outside the chapter.

The book supplies worked solutions for only four of them — 5.2, 5.6, 5.14 and 5.17. The other fifteen are worked here, and every number below is computed in a snippet rather than asserted.

What the chapter established

The chapter summary (p. 224), with where each item was built

You should now have	What that means in practice	§
the QED Lagrangian, the prototype of all the others	Three terms: a free fermion, a free field, and one interaction term that is a current dotted into a potential. Every interaction in the rest of the book is that same shape with a different current and a different gauge group.	5.1
the Lamb–Retherford experiment, which created quantum field theory	A 1057.8 MHz splitting that the Dirac equation says is exactly zero. Measuring something the theory forbids is worth more than measuring something it predicts.	5.2–5.3
local gauge invariance generating the interaction itself	The single deepest idea in the book. Demand that a phase be choosable independently at every point, and the photon and its coupling are forced — not added. Chapter 6 does it with SU(3) and gets eight gluons that carry charge; Chapter 9 does it with SU(2)×U(1) and needs the Higgs to survive.	5.1
Lagrangians as scalar products of fermion currents, vector or axial	The vector current ψ̄γ^μψ is the object at every QED vertex. Adding an axial piece ψ̄γ^μγ⁵ψ is the entire difference between electromagnetism and the weak interaction — one term, and parity is gone.	5.1, 5.6
the basic structure of Feynman diagrams	√α per vertex, a propagator per internal line, sum over diagrams, square at the end. A netlist, evaluated — not a picture of an event, as §5.4 insists and problem 5.4 proves with a number.	5.4–5.5
the process e⁺e⁻ → μ⁺μ⁻	One diagram, and two results that come from structure rather than algebra: σ ∝ 1/s by dimensions alone, and 1 + cos²θ from helicity conservation. The control curve against which the hadronic cross-section is read.	5.7
the running of the electromagnetic coupling	1/α is linear in ln Q² with a kink at every threshold. The constant does not exist; 1/137 is one point on a curve, and 1/128.9 is another. Chapter 6 runs the same machinery and gets the opposite sign.	5.8
the magnetic moments as stringent tests of the SM	Thirteen digits measured and thirteen calculated. The electron's became the definition of α; the muon's became a search, and it currently sits 1.5σ from theory with 95% of that error bar coming from one lattice-QCD number.	5.9a–5.9b

The third line is the one that outlives the chapter. Everything else here is QED; <strong>“a local gauge invariance generates the interaction itself”</strong> is the organising principle of every remaining chapter, and it is stated once, in passing, in a summary bullet.

The numbers, all at once

Fifteen answers, computed rather than quoted. Each is derived in full in the problem set below.

every numeric answer in the chapter's problems

import numpy as np
hbarc = 197.3269804                       # MeV fm
me, mmu, mp, mpi, mK = 0.51099895, 105.6583755, 938.27208816, 139.57039, 493.677
alpha = 1/137.035999166
GeV2_ub = 389.3793721                     # GeV^-2 -> microbarn

print("5.1  atomic electron beta = alpha            = %.4f" % alpha)
print("     proton in a nucleus (R ~ 4 fm)          = %.3f" % ((hbarc/4)/mp))
print("     quark in a nucleon (R ~ 0.8 fm)         ~ 1  (p = %.0f MeV >> m_q)"
      % (hbarc/0.8))
print("5.2  Bohr radius hbar c/(m_e alpha)          = %.1f pm" % (hbarc/(me*alpha)/1e3))
for n in (2, 3):
    print("5.3  P3/2-P1/2 splitting at n = %d           = %.1f ueV"
          % (n, 13.6*alpha**2/(2*n**3)*1e6))
print("5.4  vertex separation, electron rest frame  = %.0f fm  (any sqrt s)"
      % (hbarc/(2*me)))
print("5.5  Compton tree diagrams                   = 2  (s and u; no t)")
print("5.6  next-to-tree Compton diagrams           = 17")
print("5.7  sigma(mu mu) off resonance, 86.8 nb/s:")
for m, n in [(0.77526,'rho'), (3.0969,'J/psi'), (9.4603,'Upsilon'), (91.1876,'Z')]:
    print("       %-8s %8.3f nb" % (n, 86.8/m**2))
print("     fraction beyond 90 deg                  = 1/2 exactly")
m, Bee, Bh = 3.0969, 0.059, 0.877
pref = 12*np.pi/m**2*GeV2_ub
print("5.8  J/psi peak: mu mu %.2f ub, hadrons %.0f ub, ratio to point-like %.0f"
      % (pref*Bee*Bee, pref*Bee*Bh, pref*Bee*Bee/(86.8/m**2/1000)))
G, r = 54e-6, 310/8
Gll = G/(3 + r)
print("5.9  Upsilon: Gamma_ll = %.2f keV, Gamma_h = %.1f keV  (PDG Gamma_ee = 1.34)"
      % (Gll*1e6, r*Gll*1e6))
print("5.10 threshold E1 for E2 = 10 eV             = %.1f GeV"
      % ((me*1e6)**2/10/1e9))
print("     if E1 = 2E2: E2 = %.0f keV, beta_cm = %.3f"
      % (me*1e6/np.sqrt(2)/1e3, 1/3))
run = lambda inv, q0, q1, zf: inv - zf/(3*np.pi)*np.log(q1**2/q0**2)
mt2 = 2*172.69
print("5.11 alpha^-1(1 TeV^2) = %.1f  (%.1f with the top threshold)"
      % (run(129, 91.0, 1000.0, 6.667), run(run(129, 91.0, mt2, 6.667), mt2, 1000.0, 8.0)))
print("5.12 quark/lepton slope ratio, 4 < Q^2 < 10  = %.0f"
      % ((3*(2/3)**2 + 2*3*(1/3)**2) / 2))
print("5.13 pair thresholds, 2m + 2m^2/M:")
for M, mm, lbl in [(15.9949*931.494, me, 'e+e- on oxygen'), (me, me, 'e+e- on an electron'),
                   (mp, mmu, 'mu+mu- on a proton')]:
    print("       %-22s %9.3f MeV" % (lbl, 2*mm + 2*mm*mm/M))
print("5.14 exotic-hydrogen Bohr radii:")
a0 = hbarc/(me*alpha)
for m_, lbl in [(mmu,'mu- p'), (mpi,'pi- p'), (mK,'K- p'), (mp,'pbar p')]:
    mR = m_*mp/(m_+mp)
    print("       %-7s reduced mass %6.1f MeV -> a = %6.1f fm" % (lbl, mR, a0*me/mR))
print("     electron capture releases                = 13.6 eV")
print("5.15 omega -> pi+pi-: strong NO (G-parity), EM YES (~1.5% observed)")
print("5.16 diagram counts: (c)(d) t+u, (e) s+t, (f) s, (g) t")
print("5.17 (a) C=-1 l=1 I=1  (b) C=+1 l=0 I=0  (c) C=+1 l=0 I=0 or 2")
print("5.18 rho0 pi0 / all rho pi measured %.3f; I=0 predicts 1/3, I=1 predicts 0,"
      % (0.56/1.69))
print("     I=2 predicts 2/3  ->  I(J/psi) = 0")
Ee, Eg, rho, sig, Ne = 100e9, 0.25e-3, 3e8, 7.9e-30, 1.6e12
lam = 1/(rho*sig); t = lam/2.99792458e8; g = Ee/(me*1e6)
print("5.19 E_gamma,FT = %.0f eV; lambda = %.1e m, t = %.0f yr, rate = %.1f /s;"
      % (Eg*Ee/(me*1e6), lam, t/3.156e7, Ne/t))
print("     backscattered photon = %.0f MeV" % (4*g*g*Eg/1e6))

prints

5.1  atomic electron beta = alpha            = 0.0073
   proton in a nucleus (R ~ 4 fm)          = 0.053
   quark in a nucleon (R ~ 0.8 fm)         ~ 1  (p = 247 MeV >> m_q)
5.2  Bohr radius hbar c/(m_e alpha)          = 52.9 pm
5.3  P3/2-P1/2 splitting at n = 2           = 45.3 ueV
5.3  P3/2-P1/2 splitting at n = 3           = 13.4 ueV
5.4  vertex separation, electron rest frame  = 193 fm  (any sqrt s)
5.5  Compton tree diagrams                   = 2  (s and u; no t)
5.6  next-to-tree Compton diagrams           = 17
5.7  sigma(mu mu) off resonance, 86.8 nb/s:
     rho       144.419 nb
     J/psi       9.050 nb
     Upsilon     0.970 nb
     Z           0.010 nb
   fraction beyond 90 deg                  = 1/2 exactly
5.8  J/psi peak: mu mu 5.33 ub, hadrons 79 ub, ratio to point-like 589
5.9  Upsilon: Gamma_ll = 1.29 keV, Gamma_h = 50.1 keV  (PDG Gamma_ee = 1.34)
5.10 threshold E1 for E2 = 10 eV             = 26.1 GeV
   if E1 = 2E2: E2 = 361 keV, beta_cm = 0.333
5.11 alpha^-1(1 TeV^2) = 125.6  (125.3 with the top threshold)
5.12 quark/lepton slope ratio, 4 < Q^2 < 10  = 1
5.13 pair thresholds, 2m + 2m^2/M:
     e+e- on oxygen             1.022 MeV
     e+e- on an electron        2.044 MeV
     mu+mu- on a proton       235.113 MeV
5.14 exotic-hydrogen Bohr radii:
     mu- p   reduced mass   95.0 MeV -> a =  284.7 fm
     pi- p   reduced mass  121.5 MeV -> a =  222.6 fm
     K- p    reduced mass  323.5 MeV -> a =   83.6 fm
     pbar p  reduced mass  469.1 MeV -> a =   57.6 fm
   electron capture releases                = 13.6 eV
5.15 omega -> pi+pi-: strong NO (G-parity), EM YES (~1.5% observed)
5.16 diagram counts: (c)(d) t+u, (e) s+t, (f) s, (g) t
5.17 (a) C=-1 l=1 I=1  (b) C=+1 l=0 I=0  (c) C=+1 l=0 I=0 or 2
5.18 rho0 pi0 / all rho pi measured 0.331; I=0 predicts 1/3, I=1 predicts 0,
   I=2 predicts 2/3  ->  I(J/psi) = 0
5.19 E_gamma,FT = 49 eV; lambda = 4.2e+20 m, t = 44596 yr, rate = 1.1 /s;
   backscattered photon = 38 MeV

Why the muon, when the electron is measured three million times better

10⁻¹⁶10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁹absolute uncertainty on the anomaly (log scale)the walla_e measured — 1.3 × 10⁻¹⁶one electron in a trap, thirteen digitsa_e predicted — 7.2 × 10⁻¹⁰limited by the input value of α, not by QEDthe measurement is 5 500 000× better than the comparison it feedsa_μ measured — 4.1 × 10⁻¹⁰a_μ predicted — 5.8 × 10⁻¹⁰

Both comparisons are limited at ≈ 7 × 10⁻¹⁰ — to within 1 %. So the muon’s 43 000× sensitivity is not divided by anything.

Supplied, and it corrects a comparison that is easy to get backwards. All four numbers are the chapter’s own: Eq. (5.69) for the predicted $a_e$ and its α-dominated uncertainty, Eqs. (5.76) and (5.78) for the muon.

The electron’s measurement cannot be spent at its own accuracy — testing it against theory needs a value of α from a different experiment entirely, and that input is five and a half million times coarser than the trap. Comparing the two experiments would say the electron wins by 3 × 10⁶ and that the muon should lose despite its 43 000× sensitivity; comparing what each test is actually limited by says the two are level, and the 43 000 survives intact. That is why one of these experiments makes headlines and the other has become the world’s best determination of α.

Two answers worth pulling out

86.8 nb / s — the point-like law (5.47)
problem 5.7 — the four resonance energies
problem 5.8 — the J/ψ peak height, 5300 nb

Problems 5.7 and 5.8 on one axis. The line falls four orders of magnitude from the ρ to the Z purely as 1/s; the single orange point is how far above it a narrow resonance stands — a factor of 590 at the J/ψ. This is Fig. 5.27 reconstructed from two problems, and it is why that figure's vertical axis has to be logarithmic.

💡 What this really says — Problem 5.4 is the most important one here, and it looks like the least

It asks for a distance and gets two wildly different answers from the same diagram: about 0.02 fm in the centre of mass, and 193 fm in the electron’s rest frame — the second one independent of the collider energy.

Both are correct, and that is the point. If the vertices were events in a history, their separation would be a fact about the world and every observer would have to agree on the kind of thing it is. They do not. The internal line carries $q^2 = s > 0$ , so the separation is time-like, and a boost trades that time for distance without limit.

A Feynman diagram is a term in a perturbation series with a definite value. It is not a picture of what happened, and problem 5.4 is the cheapest available proof — which is presumably why the book set it.

⚙️ Engineer’s bridge — problem 5.9, and why you compute the ratio first

Problem 5.9 hands you two measured areas and a total width, and asks for two partial widths. The tempting route is to plug each area into the Breit–Wigner formula, carry $6\pi^2/m^2$ , convert GeV⁻¹ to nb MeV, and solve. It works, and it is the wrong first move.

Take the ratio of the two areas instead. Everything cancels — the $6\pi^2$ , the $m^2$ , the total width, the unit conversion, and the unknown $\Gamma_{\ell\ell}$ that appears in both — leaving $\Gamma_h/\Gamma_{\ell\ell} = 310/8 = 38.75$ in one step, with no constants and therefore no chance of a units error. One linear equation then closes the problem.

This is the same instinct as §5.9a’s ratiometric measurements and §5.9b’s $\omega_a/\omega_c$ : form the dimensionless combination as early as possible. It is worth doing even when the absolute calculation is easy, because the ratio is where the systematic errors cancel — and, in a hand calculation, where the factors of $(\hbar c)^2$ cannot bite you. Substituting back at the end to reproduce the given 8 nb MeV is then a genuine check rather than a tautology.

Where it breaks: working in natural units and restoring $\hbar c$ at the end protects you from dimensional slips and from nothing else. It will not catch a factor of 2, a missing colour factor, or a wrong power of the coupling — the errors that actually dominate — and it introduces one of its own: the final substitution is only a check if you knew the target number independently, which on a novel calculation you do not. Dimensional discipline is a cheap filter, not a verification.

📝 Chapter 5 — all 19 problems

0/19 solved

5.1orders of magnitudetheory
Estimate the speeds of an atomic electron, a proton in a nucleus and a quark in a nucleon.
- $\hbar c = 197.3$ MeV fm
- $\alpha = 1/137$ ; $m_e = 0.511$ MeV, $m_p = 938$ MeV, $m_{u,d} \sim$ a few MeV
- nuclear radius a few fm; nucleon radius $\approx 0.8$ fm
5.2the Bohr radius from uncertainty alonetheory
Evaluate the order of magnitude of the radius of the hydrogen atom.
- $\Delta p\,\Delta r \approx \hbar$
- $E = p^2/2m_e - q_e^2/4\pi\varepsilon_0 r$
5.3fine structuretheory
Calculate the energy difference due to the spin–orbit coupling between the levels $P_{3/2}$ and $P_{1/2}$ for $n = 2$ and $n = 3$ for the hydrogen atom ( $Rhc = 13.6$ eV).
- $E_{fs} = -\dfrac{Rhc\,\alpha^2}{n^3}\left[\dfrac{1}{j+1/2} - \dfrac{3}{4n}\right]$
- $\alpha^2 = 5.325\times10^{-5}$
5.4where a virtual particle livestheory
Consider the process $e^+e^- \to \mu^+\mu^-$ at energies much larger than the masses. Evaluate the spatial distance between the two vertices of the diagram Fig. 5.19 (s channel) in the CM reference frame and in the reference frame in which the electron is at rest.
- the virtual photon carries $q^2 = s > 0$
- $\hbar c = 197.3$ MeV fm
5.5Compton scattering at tree leveltheory
Draw the tree-level diagrams for the Compton scattering $\gamma + e^- \to \gamma + e^-$ .
- QED vertices only; every vertex joins two fermion lines and one photon
5.6counting diagramstheory
Draw the diagrams at the next-to-tree-level order for the Compton scattering (17 in total).
- one order higher means two more vertices, i.e. one extra internal photon or one loop
5.7the point-like cross-sectiontheory
Give the values that the cross-section of $e^+e^- \to \mu^+\mu^-$ would have in the absence of resonance at the ρ, the ψ, the ϒ and the Z. What is the fraction of the angular cross-section $\theta > 90°$ ?
- $\sigma = \dfrac{4\pi\alpha^2}{3s} = \dfrac{86.8\ \text{nb}}{s\,[\text{GeV}^2]}$ , Eq. (5.47)
- $\mathrm{d}\sigma/\mathrm{d}\Omega \propto 1 + \cos^2\theta$ , Eq. (5.45)
- $m_\rho = 0.775$ , $m_{J/\psi} = 3.097$ , $m_\Upsilon = 9.460$ , $m_Z = 91.19$ GeV
5.8resonance peak cross-sectionstheory
Calculate the cross-sections of the processes $e^+e^- \to \mu^+\mu^-$ and $e^+e^- \to$ hadrons at the J/ψ peak ( $m_\psi = 3.097$ GeV) and the ratio of the former to its value in the absence of resonance. Neglect the masses and use the Breit–Wigner approximation. [ $\Gamma_e/\Gamma = 5.9\%$ , $\Gamma_h/\Gamma = 87.7\%$ ]
- $\sigma_{\text{peak}} = \dfrac{4\pi}{k^2}\,g\,B_iB_f$ with $g = \dfrac{2J+1}{(2s_1+1)(2s_2+1)}$
- $J(J/\psi) = 1$ , so $g = 3/4$ ; massless leptons give $k = m/2$
- $(\hbar c)^2 = 389.4$ GeV² μb
5.9the peak-area methodtheory
Consider the narrow resonance ϒ ( $m = 9.460$ GeV) observed at $e^+e^-$ colliders in $e^+e^-\to\mu^+\mu^-$ and $e^+e^-\to$ hadrons. Its width is $\Gamma = 54$ keV. The measured peak areas are $\int\sigma_{\mu\mu}\mathrm{d}E = 8$ nb MeV and $\int\sigma_h\mathrm{d}E = 310$ nb MeV. In the Breit–Wigner approximation calculate the partial widths $\Gamma_\mu$ and $\Gamma_h$ . Assume all the leptonic widths equal.
- $\displaystyle\int\sigma\,\mathrm{d}E = \frac{6\pi^2}{m^2}\frac{\Gamma_i\Gamma_f}{\Gamma}$ for $e^+e^-$ onto a $J = 1$ resonance
- $\Gamma = 3\Gamma_{\ell\ell} + \Gamma_h$ if the only channels are three lepton flavours and hadrons
- $(\hbar c)^2 = 389.4$ GeV² μb
5.10photon–photon pair productiontheory
Two photons flying in opposite directions collide, with energies $E_1$ and $E_2$ . (1) Find the minimum $E_1$ for $\gamma_1\gamma_2 \to e^+e^-$ if $E_2 = 10$ eV. (2) Answer the same if $E_1 = 2E_2$ . (3) Find the centre-of-mass speed in the latter case. (4) Draw the lowest-order Feynman diagram.
- for two head-on massless particles, $s = 4E_1E_2$
- threshold: $s \geq (2m_e)^2$
- $\beta_{CM} = |\mathbf{p}_{tot}|/E_{tot}$
5.11running αtheory
Calculate the reciprocal of the fine-structure constant at $Q^2 = 1$ TeV², knowing that $\alpha^{-1}(M_Z^2) = 129$ and $M_Z = 91$ GeV. Assume no particles beyond the known ones exist.
- $\alpha^{-1}(Q^2) = \alpha^{-1}(\mu^2) - \dfrac{z_f}{3\pi}\ln\dfrac{|Q|^2}{\mu^2}$ , Eq. (5.52)
- $z_f = 6.67$ for $10 < |Q| < 100$ GeV, $z_f = 8$ above the top
5.12what sets the slopetheory
If no threshold is crossed, $\alpha^{-1}(Q^2)$ is a linear function of $\ln(|Q|^2/\mu^2)$ . What is the ratio between the quark and lepton contributions to the slope of this linear dependence for $4 < Q^2 < 10$ GeV²?
- slope $= -z_f/3\pi$ , with $z_f = \sum$ (colour) × (charge)²
- a fermion contributes once $m < Q/2$
5.13pair-production thresholdstheory
Calculate the energy threshold $E_\gamma$ for the conversion of a photon into an $e^+e^-$ pair in the electric field of (1) an oxygen nucleus, (2) an electron, and (3) for the production of a $\mu^+\mu^-$ pair in the field of a proton. In which configuration is $m_{ee}$ a minimum?
- $s = M^2 + 2E_\gamma M$ for a photon on a target of mass $M$ at rest
- threshold: $\sqrt s \geq M + 2m$
5.14exotic atomstheory
Weakly decaying negative particles may come to rest in matter and be captured by a nucleus. Consider capture by a proton. (a) Evaluate the Bohr radius for the $\mu^-p$ , $\pi^-p$ , $K^-p$ and $\bar pp$ systems. (b) Calculate the energy released in the capture of an electron at rest into the ground state by a proton.
- $a = \hbar c/(m_R c^2\alpha)$ and $E_1 = 13.6\ \text{eV}\times m_R/m_e$
- $m_R = mM/(m+M)$ with $M = m_p$
5.15G-parity and isospin selection rulestheory
Is the decay $\omega \to \pi^+\pi^-$ allowed by strong interactions? Is it allowed by electromagnetic interactions?
- $\omega$ : $I^G(J^{PC}) = 0^-(1^{--})$
- $G(n\pi) = (-1)^n$
- strong: ΔI = 0 and G conserved. EM: ΔI = 0 or 1, G not conserved
5.16drawing diagramstheory
Draw the lowest-order Feynman diagrams for (a) $\pi^0\to\gamma\gamma$ , (b) $\Sigma^0\to\Lambda\gamma$ , (c) $e^-e^-\to e^-e^-$ , (d) $e^+e^+\to e^+e^+$ , (e) $e^+e^-\to e^+e^-$ , (f) $e^+e^-\to\mu^+\mu^-$ , (g) $e^+\mu^+\to e^+\mu^+$ .
- one QED vertex = two fermion lines + one photon
- identical particles in the final state ⇒ an extra exchange diagram
5.17C, ℓ and I of a two-pion systemtheory
Determine the charge conjugation, the lowest orbital momentum and the isospin of the 2π systems in (a) $\eta\to\pi^+\pi^-\gamma$ , (b) $\omega\to\pi^+\pi^-\gamma$ , (c) $\rho^0\to\pi^+\pi^-\gamma$ , and state ΔI. (d) Are $\eta\to\pi^0\pi^0\gamma$ , $\omega\to\pi^0\pi^0\gamma$ and $\rho^0\to\pi^0\pi^0\gamma$ allowed?
- $C(\gamma) = -1$ ; $C(\eta) = +1$ , $C(\omega) = -1$ , $C(\rho^0) = -1$
- $C(\pi^+\pi^-) = (-1)^l$ ; the total 2π wave function must be symmetric
- EM allows ΔI = 0 or 1
5.18isospin from branching ratiostheory
Given $\mathrm{BR}(J/\psi\to\rho\pi) = (1.69\pm0.15)\times10^{-2}$ and $\mathrm{BR}(J/\psi\to\rho^0\pi^0) = (0.56\pm0.07)\times10^{-2}$ , determine the isospin of the J/ψ.
- $\rho$ and $\pi$ both have $I = 1$
- $1\otimes1 = 0\oplus1\oplus2$
- $|\langle I,0|1,0;1,0\rangle|^2 = 1/3, 0, 2/3$ for $I = 0, 1, 2$
5.19inverse Compton scattering on the CMBtheory
LEP stores $e^\pm$ at $E_e = 100$ GeV. Take CMB photons of $E_\gamma = 0.25$ meV at density $\rho = 3\times10^8$ m⁻³. (a) What photon energy on an electron at rest gives the same CM energy? (b) With $\sigma = (\alpha\hbar/m_ec)^2 = 7.9\times10^{-30}$ m², find the interaction length and time, and the event rate for $N_e = 1.6\times10^{12}$ stored electrons (take $\langle\cos\theta\rangle = 0$ ). (c) What is the energy of photons backscattered at 180° after a head-on collision?
- $s = m_e^2 + 2E_\gamma E_e(1-\cos\theta)$
- $\lambda = 1/\rho\sigma$
- backscatter: $E'_\gamma = 4\gamma^2E_\gamma/(1 + 4\gamma E_\gamma/m_e)$

Errata found in the problems and solutions

where	as printed	should be
Problem 5.14(a)	the μ⁻p system is called <strong>muonium</strong>	muonic hydrogen. <strong>Muonium</strong> is μ⁺e⁻ — a genuinely different system, and the one whose spectroscopy tests QED without any nuclear structure at all. μ⁻p is what produced the proton-radius puzzle.
Solution to 5.14(b)	"It is 13.6 eV for an electron … hence 13.6 × 121/0.5 = 3.3 keV"	both numbers are right but the second answers a different question. 121 MeV is the <em>pion</em>–proton reduced mass, so 3.3 keV is the ground state of <strong>pionic</strong> hydrogen; part (b) asks about the electron, whose answer is the 13.6 eV already given.
Problem 5.19(b)	σ = (αħ/m_ec)² = 7.9 × 10⁻³⁰ m², called the Thomson cross-section	that is r_e², the classical electron radius squared. The Thomson cross-section is (8π/3)r_e² = 6.65 × 10⁻²⁹ m² — <strong>8.4 times larger</strong>. Working with the stated value is fine as an exercise, but the real LEP rate is nearer 10 s⁻¹ than 1 s⁻¹.

None of these changes a method, and two of them are labelling. They are recorded because a reader checking their own arithmetic against the book would otherwise conclude they had made a mistake.

🔑 If you remember only three things

One diagram gives two distances four orders of magnitude apart. Which frame you ask in decides what “how far apart” even means, and the problem that shows this looks like the easiest on the page.
A ratio is safer than either number inside it. Shared factors cancel, including the ones you would otherwise have had to look up and could have got wrong.
Every number in the chapter appears once more, together. Only in one table do the scales become comparable, which no individual section was able to do.

Where this goes next

Chapter 6 is the same construction with a different group. Every tool built here is reused within twenty pages: the Lagrangian becomes SU(3), the running coupling changes sign, the point-like cross-section of §5.7 becomes the denominator of R, and the hadronic term that limits the muon anomaly is computed on a lattice.

§6.1 — hadron production at $e^+e^-$ colliders, i.e. the numerator of problem 5.7’s ratio.
§6.5 — the evolution of $\alpha_s$ : problem 5.11 with the sign flipped.
§6.10 — lattice QCD, and the number §5.9b showed to be 95% of the muon’s theoretical error bar.

✅ Check yourself — Chapter 5 review

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1.Problem 5.1 finds an atomic electron at β = 0.007, a nucleon at β = 0.05 and a quark at β ≈ 1. What is actually responsible for the spread?
2.Problem 5.4 gives the separation of the two vertices as 0.02 fm in the CM frame and 193 fm in the electron rest frame, the latter independent of √s. How can both be right?
3.In problem 5.9 the first move is to divide the two measured peak areas. Why is that better than plugging each into the Breit–Wigner formula?
4.Problem 5.12 finds that quarks and leptons contribute equally to the slope of α⁻¹ in the window 4 < Q² < 10 GeV². What makes them equal?
5.Problem 5.19 finds an interaction time of 45 000 years and a rate of about one event per second. Is that a contradiction?