Chapter 7 — Summary and Problems · Elementary Particle Physics

Thirty-two problems and two underlying facts — the interaction changes flavour, and its coupling carries units. Nearly everything here is one of those two asked again.

🎯 Why this matters

A problem set this repetitive says something about the chapter. The weak interaction is not conceptually deep so much as consequentially wide: one vertex, applied everywhere, is most of what there is to practise.

Thirty-two problems, and they split more cleanly than most chapters’ because the weak interaction has exactly two things to say and the problems say them repeatedly.

eleven are Cabibbo bookkeeping — 7.1, 7.2, 7.18, 7.19, 7.20, 7.23, 7.24, 7.25, 7.30, 7.31, 7.32. Every one reduces to reading two CKM factors off a diagram and squaring them;
eight are cross-sections and event rates — 7.6, 7.13, 7.14, 7.16, 7.17, 7.27, 7.28, and 7.7’s tail. These are the ones where the answer is a detector the size of a building;
four are the $m^5$ law — 7.5, 7.12, 7.21, 7.22, plus 7.32 where it fights the Cabibbo angle and wins;
four are helicity and parity — 7.9, 7.10, 7.15, 7.26;
three are kinematics — 7.4, 7.12, 7.26;
and 7.3, 7.8, 7.11, 7.29 are one-offs worth doing anyway: the units of $G_F$ , two decays that do not happen, why muon chambers sit outside the iron, and Pontecorvo’s test.

The book’s Solutions section covers none of them. All thirty-two are worked here, and every number is computed in a snippet rather than asserted.

What the chapter established

How chapter 7 argues — each box is forced by the one before it, not assumed

Wu, 1957P violated, maximallythe current is V − Aleft-chiral fields onlyπ → eν, 10⁻⁴the test it had to passuniversalityone G_F for e, μ, τ…but not for quarksK/π says the rate is wrongCabibbo rotationone coupling, two axesno flavour-changing neutral current⇒ GIM ⇒ charm must exist3 families ⇒ CKMone phase survives ⇒ CP→ chapter 8where that phase does its worktwo loose endsG_F has units → 293 GeVg_L, g_R unexplainedboth → chapter 9

Supplied. The chapter reads as a list of discoveries; it is actually a chain, and the arrows are the part worth holding on to. Nothing in it is postulated. Parity violation is measured and forces the structure; the structure predicts a suppression nobody would guess from phase space and the suppression is measured; universality is measured and then fails for quarks, which forces a rotation; the rotation predicts a neutral current that changes flavour, which is absent, and that absence forces a fourth quark.

The two dashed boxes are what the chapter cannot close — a coupling with dimensions, and two constants left as measured numbers — and both are exactly what chapter 9 exists to explain.

Each is a measurement, not a postulate — which is the chapter's method throughout.

what↕	how it was established↕	the number↕
The interaction is effective	$G_F$ carries dimensions	$1.166\times10^{-5}$ GeV⁻², naming 293 GeV
P is violated, maximally	Wu's polarized ⁶⁰Co	$\alpha \approx -1$
The current is V − A	$\pi\to e\nu$ suppressed by $10^{-4}$ against phase space	$m_e^2/m_\mu^2 = 2.34\times10^{-5}$
C is violated, maximally	the C-image of a real process does not exist	$h(\bar\nu) = +1$ always
The coupling is universal	the τ's two leptonic decays; μ and τ beta decay	$g_\mu/g_e = 1.001 \pm 0.002$
Quarks mix	the $K/\pi$ leptonic ratio	$\sin\theta_C = 0.2243$
A neutral current exists	Gargamelle: hadrons and no muon	1973

Every small number in chapter 7, on one axis — and what makes each one small

10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹1dimensionless ratio (log scale)2.3 × 10⁻⁵ — (m_e/m_μ)², the helicity factor aloneV − A couples chirality; a massless lepton could not be produced at all1.2 × 10⁻⁴ — the measured Γ(π→eν)/Γ(π→μν)5.5× above the mass factor: phase space pushes back, and loses5.3 × 10⁻² — tan²θ_C, every |ΔS| = 1 rate against ΔS = 0not a weaker force: the same coupling, resolved onto a different axis0.95 = cos²θ_C — and 1.000 for the leptonsuniversality holds exactly; only the quarks pay the rotation

Supplied. The chapter’s results are a set of small numbers scattered across forty pages; put on one axis they sort into exactly two families. The two on the left are chirality — the interaction refuses to touch a right-chiral field, so a decay to the lighter lepton is the more forbidden one, which is the opposite of what phase space wants and is why the 1.2 × 10⁻⁴ is the chapter’s sharpest measurement.

The two on the right are a rotation — nothing is suppressed at all, a unit vector is being resolved onto two axes, and cos²θ_C + sin²θ_C = 1 says no probability is lost. Values verified in the block below; the phase-space factor is 5.487, giving a predicted 1.28 × 10⁻⁴ against the measured 1.23 × 10⁻⁴, the remainder being radiative corrections.

The numbers, all at once

every numeric answer in the chapter's problems

import numpy as np
GF, hbar, c = 1.1663788e-5, 6.582119569e-25, 2.99792458e8
GeV2_m2, u = 3.893793721e-32, 1.66053907e-27
me, mmu, mtau = 0.51099895e-3, 0.1056583755, 1.77686
mpi, mp, mn, mL, mSm = 0.13957039, 0.93827209, 0.93956542, 1.115683, 1.197449
alpha, s_C, c_C = 1/137.035999, 0.2243, 0.97373

hbarc = 197.3269804e-3*1e-15 * 1.602176634e-10
print(f"7.3   G_F = {1.17e-5/(1.602176634e-10)**2*hbarc**3:.3e} J m^3     -- energy x volume, the SI face of GeV^-2")

g = (29/2)/mtau; bg = np.sqrt(g*g-1)
print(f"7.4   tau flight at PEP: beta.gamma = {bg:.2f}, d = {bg*c*290.3e-15*1e3:.2f} mm")
t16, t18 = [2.2e-6*b*(mmu/mtau)**5*1e12 for b in (0.16, 0.1782)]
print(f"7.5   tau_tau = {t16:.3f} ps (book's BR 16%); {t18:.3f} ps with the measured 17.82%")
print( "        -- against 0.290 ps observed, so the m^5 law is exact and the input was stale")

s10, s100 = [4*np.pi*alpha**2/(3*s)*GeV2_m2 for s in (100., 10000.)]
print(f"7.6   sigma(e+e- -> mu mu) = {s10*1e37:.3f} nb at 10 GeV, {s100*1e40:.2f} pb at 100 GeV   (~ 1/s)")

pk = 3.15*8.617333e-5*2
print(f"7.7   relic neutrinos: <p> = 3.15 kT = {pk:.1e} eV, v = {pk/50e-3:.4f} c = {pk/50e-3*c/1e3:.0f} km/s")
print(f"7.12  muon decay electron: p_min = 0, p_max = {(mmu**2-me**2)/(2*mmu)*1e3:.2f} MeV = m_mu/2")

P13 = (5e3*1.2e7/1.6726219e-27)*7e-43
print(f"7.13  1e6 neutrinos across the Earth: P = {P13:.1e}, so {1e6*P13:.0f} interact")
print(f"7.14  sigma(nu e)/sigma(nu N) = {me/(0.2*mp):.2e}, INDEPENDENT of energy")
print(f"        sigma/E = {2*GF**2*me/np.pi*GeV2_m2:.2e} (electron), "
      f"{0.2*2*GF**2*mp/np.pi*GeV2_m2:.2e} m^2/GeV (nucleon)")

N16 = (1/86400)/(6e14*1e-48*0.40)
print(f"7.16  GALLEX: {N16:.1e} nuclei = {N16*71*u/1e3:.1f} t of Ga-71 = {N16*71*u/0.4/1e3:.1f} t of natural Ga")
lam = 1/((7.7e3/(56*u)*26)*1.7e-45)
print(f"7.17  iron for one interaction: {lam:.1e} m = {lam/2.99e11:.0f}x the Earth's orbit, {lam/c/86400:.1f} days")
print(f"7.21  BR(b -> c e nu) = 1/(3 + 2x3) = {1/9:.3f}    measured 0.108")
print(f"7.22  Gamma(D+)/Gamma(mu) measured {7e10*2.2e-6:.2e}; |V_cs|^2 (m_c/m_mu)^5 = {0.975**2*(1.27/mmu)**5:.2e}")

t2, t4 = (s_C/c_C)**2, (s_C/c_C)**4
print(f"7.23  D+ ratios  CF : SCS : DCS = 1 : {t2:.4f} : {t4:.5f}")
Vcb, Vub, Vud, Vus = 40.8e-3, 3.82e-3, c_C, s_C
b = [Vcb**2*Vud**2, Vcb**2*Vus**2, Vub**2*Vud**2, Vub**2*Vus**2]
print(f"7.25  B0 ratios  (1):(2):(4):(3) = 1 : {b[1]/b[0]:.5f} : {b[2]/b[0]:.5f} : {b[3]/b[0]:.5f}")

ps = (mpi**2-mmu**2)/(2*mpi); Es = np.sqrt(ps**2+mmu**2)
Epi = np.sqrt(0.25+mpi**2); gp, bp = Epi/mpi, 0.5/Epi
print(f"7.26  pi+ at 500 MeV: p_mu from {gp*(bp*Es-ps)*1e3:.1f} to {gp*(ps+bp*Es)*1e3:.1f} MeV -- all forward")
Ne = (10/86400)/(1e10*1e-47*0.5)
print(f"7.27  water Cherenkov for 10 events/day: {Ne:.1e} electrons = {Ne/10*18e-3/6.02214076e23/1e6:.1f} kton")
lam28 = 1/((1e17/(56*u))*3e-46)
print(f"7.28  supernova core: lambda = {lam28:.0f} m = {lam28/1e3:.1f} km -- SMALLER than the core")
print(f"7.29  threshold for nubar p -> e+ n: {(mn+me-mp)*1e3:.3f} MeV; the flux reaches {mmu/2*1e3:.1f} MeV")

Qn, QL = (mSm-mn-me)*1e3, (mSm-mL-me)*1e3
print(f"7.32  phase space (Q_n/Q_L)^5 = {(Qn/QL)**5:.0f} beats Cabibbo 1/tan^2 = {1/t2:.1f}")
print(f"        net {(Qn/QL)**5*t2:.1f} against a measured {1.017e-3/5.73e-5:.1f} -- agreement "
      f"{abs((Qn/QL)**5*t2/(1.017e-3/5.73e-5)-1)*100:.0f}%")

prints

7.3   G_F = 1.440e-62 J m^3     -- energy x volume, the SI face of GeV^-2
7.4   tau flight at PEP: beta.gamma = 8.10, d = 0.70 mm
7.5   tau_tau = 0.262 ps (book's BR 16%); 0.291 ps with the measured 17.82%
      -- against 0.290 ps observed, so the m^5 law is exact and the input was stale
7.6   sigma(e+e- -> mu mu) = 0.869 nb at 10 GeV, 8.69 pb at 100 GeV   (~ 1/s)
7.7   relic neutrinos: <p> = 3.15 kT = 5.4e-04 eV, v = 0.0109 c = 3255 km/s
7.12  muon decay electron: p_min = 0, p_max = 52.83 MeV = m_mu/2
7.13  1e6 neutrinos across the Earth: P = 2.5e-05, so 25 interact
7.14  sigma(nu e)/sigma(nu N) = 2.72e-03, INDEPENDENT of energy
      sigma/E = 1.72e-45 (electron), 6.33e-43 m^2/GeV (nucleon)
7.16  GALLEX: 4.8e+28 nuclei = 5.7 t of Ga-71 = 14.2 t of natural Ga
7.17  iron for one interaction: 2.7e+14 m = 914x the Earth's orbit, 10.5 days
7.21  BR(b -> c e nu) = 1/(3 + 2x3) = 0.111    measured 0.108
7.22  Gamma(D+)/Gamma(mu) measured 1.54e+05; |V_cs|^2 (m_c/m_mu)^5 = 2.39e+05
7.23  D+ ratios  CF : SCS : DCS = 1 : 0.0531 : 0.00282
7.25  B0 ratios  (1):(2):(4):(3) = 1 : 0.05306 : 0.00877 : 0.00047
7.26  pi+ at 500 MeV: p_mu from 282.5 to 504.1 MeV -- all forward
7.27  water Cherenkov for 10 events/day: 2.3e+33 electrons = 6.9 kton
7.28  supernova core: lambda = 3100 m = 3.1 km -- SMALLER than the core
7.29  threshold for nubar p -> e+ n: 1.804 MeV; the flux reaches 52.8 MeV
7.32  phase space (Q_n/Q_L)^5 = 319 beats Cabibbo 1/tan^2 = 18.8
      net 16.9 against a measured 17.7 -- agreement 5%

Two of those deserve a second look, because they are the chapter’s method in miniature.

7.32 puts two large effects against each other and both are needed. The $\Sigma^-\to n$ channel is Cabibbo-suppressed relative to $\Sigma^-\to\Lambda$ — it should lose by a factor 19 — and it wins by 18, because $Q^5$ favours it by 319. Net prediction 17, measured 17.7. Neither effect alone gets within an order of magnitude of the answer.

7.21 gets a branching ratio right from pure counting. The $W^-$ has three leptonic channels and two hadronic ones, and the hadronic ones come in three colours, so $\mathrm{BR}(b\to ce\bar\nu) = 1/9 = 11\,\%$ against a measured 10.8 %. Drop the colour factor and you get $1/5 = 20\,\%$ — wrong by a factor of two. Chapter 6’s colour is doing load-bearing work in a Chapter 7 problem.

0/32 solved

7.1which interaction is it?theory
Draw the Feynman quark diagrams of the following decays: $K^{*+}\to K^0+\pi^+$ ; $n\to p+e^-+\bar\nu_e$ ; $\pi^+\to\mu^++\nu_\mu$ .
- a W is exchanged only in a weak decay
- the strong interaction conserves every flavour
7.2four more, same questiontheory
Draw the Feynman quark diagrams of: $\pi^+\to\pi^0+e^++\nu_e$ ; $\rho^+\to\pi^0+\pi^+$ ; $K^0\to\pi^-+\pi^+$ ; $\Lambda\to p+e^-+\bar\nu_e$ .
- $\pi^+ = u\bar d$ , $\pi^0 = (u\bar u - d\bar d)/\sqrt2$ , $\rho^+ = u\bar d$ , $K^0 = d\bar s$ , $\Lambda = uds$
7.3G_F in SI unitstheory
Find the value of the Fermi constant $G_F$ in SI units, knowing that $G_F/(\hbar c)^3 = 1.17\times10^{-5}$ GeV $^{-2}$ .
- $\hbar c = 197.327$ MeV fm
- 1 GeV $= 1.602177\times10^{-10}$ J
7.4how far a tau fliestheory
PEP was an $e^+e^-$ collider running at $\sqrt s = 29$ GeV. For $e^+e^-\to\tau^+\tau^-$ , find the average distance a $\tau$ flies before decaying.
- $m_\tau = 1.777$ GeV, $\tau_\tau = 290.3$ fs
- $d = \beta\gamma\, c\,\tau$
7.5the tau lifetime from the muon'stheory
For $\mu^+\to e^+\nu_e\bar\nu_\mu$ (BR 100%) and $\tau^+\to e^+\nu_e\bar\nu_\tau$ (BR 16%), with $\tau_\mu = 2.2$ μs, calculate the $\tau$ lifetime.
- $\Gamma \propto G_F^2 m^5$ — Eq. (7.13)
- $m_\mu = 105.658$ MeV, $m_\tau = 1776.86$ MeV
7.6the QED point cross-sectiontheory
Neglecting masses, calculate $\sigma(e^+e^-\to\mu^+\mu^-)$ at $\sqrt s = 10$ GeV and at $\sqrt s = 100$ GeV.
- $\sigma = 4\pi\alpha^2/3s$
- $1\,\text{GeV}^{-2} = 3.894\times10^{-32}\,\text{m}^2 = 0.3894$ mb
7.7what a neutrino istheory
What are the differences between a neutrino and an antineutrino? What is conserved in neutrino scattering? Complete $\nu_e + n \to\;?$ . If neutrinos are massless, what is the direction of their spin, and of antineutrinos'? The Universe is full of neutrinos at about 2 K — what is their average speed if the mass is 50 meV?
- lepton number $L = +1$ for $\nu$ , $-1$ for $\bar\nu$
- $\langle p\rangle \approx 3.15\,kT$ for a relativistic Fermi gas
- $k = 8.617\times10^{-5}$ eV/K
7.8single-pion production, and two decays that do not happentheory
Write the reaction(s) by which a $\nu_\mu$ can produce a single pion hitting (a) a proton and (b) a neutron. Does $\mu^-\to e^-\gamma$ exist? Does $\mu^-\to e^-e^+e^-$ exist? Give reasons.
- a $\nu_\mu$ charged-current vertex gives $\nu_\mu\to\mu^-$ , raising the hadronic charge by one
- the FCNC limits of Eq. (7.110)–(7.111)
7.9testing parity in hyperon productiontheory
A $\pi^-$ beam hits a target and $\Lambda$ hyperons are produced inclusively. We measure the momentum $\mathbf{p}_\Lambda$ and the polarization $\mathbf{P}$ of the hyperon. How can we check whether parity is conserved in these reactions? What do you expect?
- $\mathbf{P}$ is an axial vector; momenta are polar vectors
- the production is a strong process
7.10seeing parity violation in a hyperon decaytheory
How can you observe parity violation in the decay $\Lambda\to p\pi^-$ ?
- $\Lambda$ has $J^P = \tfrac12^+$ ; the proton is $\tfrac12^+$ and the pion $0^-$
- a polarized sample gives access to $\mathbf{J}\cdot\mathbf{p}$
7.11why muons get through irontheory
Muons have the same interactions as electrons. Why does a $\mu$ of a few GeV pass through an iron slab while an electron of the same energy does not?
- radiative energy loss $\propto 1/m^2$
- iron: $X_0 = 1.76$ cm, $dE/dx|_{\min} = 1.45$ MeV cm²/g, $\rho = 7.87$ g/cm³
7.12the muon decay electron spectrum, at its endstheory
What are the minimum and maximum momenta of the electron from the decay of a $\mu$ at rest?
- $\mu^-\to e^-\bar\nu_e\nu_\mu$ , a three-body decay
- $m_\mu = 105.658$ MeV, $m_e = 0.511$ MeV
7.13neutrinos through the Earththeory
Cosmic rays produce $\nu_\mu$ and $\nu_e$ in the atmosphere. In a sample of $N = 10^6$ $\nu_\mu$ of 1 GeV, how many interact crossing the Earth along a diameter? ( $\sigma = 7$ fb, $\rho = 5\times10^3$ kg m $^{-3}$ , $R \approx 6000$ km.)
- $P = n\sigma$ with $n$ the column density of nucleons
- nucleon mass $1.673\times10^{-27}$ kg
- 1 fb $= 10^{-43}$ m²
7.14why neutrinos scatter off nucleons, not electronstheory
With $\sigma(\nu_\mu e\to\mu\nu_e) = G_F^2s/\pi$ and $\sigma(\nu N\to\mu h) = 0.2\,G_F^2s/\pi$ (the 0.2 from the quark distribution), calculate their ratio at $E_\nu = 50$ GeV. How does the ratio depend on energy? Calculate $\sigma/E$ for both.
- for a fixed target, $s \approx 2m_{\text{target}}E_\nu$
- $m_e = 0.511$ MeV, $m_N = 939$ MeV
7.15one diagram or two?theory
Draw the tree-level Feynman diagrams for elastic $\nu_ee^-$ scattering. What is different for $\nu_\mu e^-$ ?
- the W changes flavour at each vertex; the Z⁰ does not
- $\nu_e$ and $e^-$ belong to the same lepton family
7.16how much gallium GALLEX neededtheory
GALLEX measured the solar $\nu_e$ flux via $\nu_e + {}^{71}\mathrm{Ga}\to{}^{71}\mathrm{Ge}+e^-$ . Taking $\Phi = 6\times10^{14}$ m $^{-2}$ s $^{-1}$ , $\sigma = 10^{-48}$ m², efficiency $\varepsilon = 40\%$ : how many $^{71}$ Ga nuclei give one interaction per day? What $^{71}$ Ga mass is that? What natural gallium mass, if the abundance is 40 %?
- rate per nucleus $=\Phi\sigma$
- 1 day $= 86400$ s; $u = 1.661\times10^{-27}$ kg
7.17iron for one neutrino interactiontheory
How many metres of Fe must a 1 GeV $\nu_\mu$ traverse to interact, on average, once? How long does that take? Compare with the diameter of the Earth's orbit. ( $\sigma = 0.017$ fb, $\rho = 7.7\times10^3$ kg m $^{-3}$ , $Z = 26$ , $A = 56$ .)
- $\lambda = 1/(n\sigma)$
- $Z = 26$ electrons per atom; 1 AU $= 1.496\times10^{11}$ m
7.18Cabibbo bookkeeping for charmtheory
Write a Cabibbo-favoured (CF) and a singly Cabibbo-suppressed (SCS) semileptonic decay of the $c$ quark. Write three CF and three SCS decays of the $D^+$ .
- $V_{cs}\approx\cos\theta_C$ (large), $V_{cd}\approx\sin\theta_C$ (small)
- $D^+ = c\bar d$
7.19the b quark's favoured decaytheory
Draw the Feynman diagram for anti-bottom quark decay favoured by the mixing. Write three favoured decay modes of the $B^+$ .
- $|V_{cb}| = 40.8\times10^{-3}$ , $|V_{ub}| = 3.82\times10^{-3}$
- $B^+ = u\bar b$
7.20top decaytheory
Draw the principal Feynman diagrams for top quark decay.
- $|V_{tb}| = 1.014$ , $|V_{ts}| = 41.5\times10^{-3}$ , $|V_{td}| = 8.6\times10^{-3}$
- $m_t = 173$ GeV, $M_W = 80.4$ GeV
7.21counting the W's optionstheory
Draw the Feynman diagrams for bottom and charm decays. Estimate the ratio $\Gamma(b\to c+e^-+\bar\nu_e)/\Gamma(b\to c)$ .
- the W couples with the same strength to every fermion pair (universality, §7.8)
- quarks come in three colours
7.22why a charmed meson decays 10⁵ times faster than a muontheory
Given $\Gamma(D^+\to\bar K^0e^+\nu_e) = (7\pm1)\times10^{10}$ s $^{-1}$ and $\Gamma(\mu^+\to e^+\nu_e\bar\nu_\mu) = 1/(2.2\,\mu\text{s})$ , justify the ratio of the two.
- $\Gamma \propto G_F^2m^5$ — the same dimensional argument as Eq. (7.13)
- $m_c \approx 1.27$ GeV, $m_\mu = 0.1057$ GeV, $|V_{cs}|\approx0.975$
7.23favoured, suppressed, doubly suppressedtheory
For (1) $D^+\to\bar K^0+\pi^+$ ; (2) $D^+\to K^++\bar K^0$ ; (3) $D^+\to K^++\pi^0$ : find the valence quark composition and establish whether each is favoured, suppressed or doubly suppressed.
- $D^+ = c\bar d$ , $K^+ = u\bar s$ , $\bar K^0 = s\bar d$ , $\pi^+ = u\bar d$
- $V_{cs}, V_{ud}$ large; $V_{cd}, V_{us}$ small
7.24the ΔS = ΔQ ruletheory
Given $\Gamma(\Sigma^-\to ne^-\bar\nu_e)/\Gamma_{\text{tot}}\approx10^{-3}$ and $\Gamma(\Sigma^+\to ne^+\nu_e)/\Gamma_{\text{tot}} < 5\times10^{-6}$ , give the reason for such a difference.
- $\Sigma^- = dds$ , $\Sigma^+ = uus$ , $n = udd$
- a single W vertex changes one quark
7.25sorting four B⁰ decaystheory
For (1) $B^0\to D^-+\pi^+$ ; (2) $B^0\to D^-+K^+$ ; (3) $B^0\to\pi^-+K^+$ ; (4) $B^0\to\pi^-+\pi^+$ : find the valence quark composition, establish the dependence of the partial rates on the mixing matrix elements, and sort them in decreasing order.
- $B^0 = d\bar b$ , $D^- = d\bar c$ , $\pi^+ = u\bar d$ , $\pi^- = d\bar u$ , $K^+ = u\bar s$
- $|V_{cb}| = 0.0408$ , $|V_{ub}| = 0.0038$ , $|V_{ud}| = 0.974$ , $|V_{us}| = 0.224$
7.26a decay in flighttheory
A pion of momentum $p_\pi = 500$ MeV decays as $\pi^+\to\mu^++\nu$ . Find the minimum and maximum $\mu$ momentum. What are the flavour and chirality of the neutrino?
- $m_\pi = 139.57$ MeV, $m_\mu = 105.66$ MeV
- boost: $p_{\parallel} = \gamma(p^*\cos\theta^* + \beta E^*)$
7.27sizing a water Cherenkov detectortheory
A large water Cherenkov detector observes solar neutrinos via $\nu_e+e^-\to\nu_e+e^-$ . With $\sigma = 10^{-47}$ m², flux $\Phi = 10^{10}$ m $^{-2}$ s $^{-1}$ and efficiency $\varepsilon = 50\%$ , what water mass gives 10 events a day?
- water has 10 electrons per molecule; $M(\mathrm{H_2O}) = 18$ g/mol
- $N_A = 6.022\times10^{23}$
7.28neutrinos trapped in a supernovatheory
A collapsing iron core reaches $\rho = 100\,000$ t mm $^{-3}$ . For $E_\nu = 10$ MeV and $\sigma \approx 3\times10^{-46}$ m² on iron, find the neutrino mean free path. ( $A_{Fe} = 56$ .)
- $\lambda = 1/(n\sigma)$ , with $n$ the number of iron nuclei per m³
- a neutron star core is $\sim10$ km across
7.29Pontecorvo's testtheory
In 1959 Pontecorvo proposed testing whether $\bar\nu_e$ and $\bar\nu_\mu$ are different particles. A low-energy $\pi^+$ beam is stopped in a target; the $\mu^+$ from their decays also stop and then decay. (1) What is the lowest-threshold reaction permitted if $\bar\nu_e=\bar\nu_\mu$ but forbidden if not? (2) What is its threshold? (3) Does the process provide any $\bar\nu_\mu$ above threshold?
- $\pi^+\to\mu^+\nu_\mu$ , then $\mu^+\to e^+\nu_e\bar\nu_\mu$
- $m_n = 939.565$ , $m_p = 938.272$ , $m_e = 0.511$ MeV
7.30charmed hyperons cascading downtheory
Give a cascade of Cabibbo-favoured decays through flavoured hyperons for $\Sigma_c^{++}(uuc)$ , $\Xi_c^+(usc)$ and $\Omega_c^0(ssc)$ .
- the favoured charm transition is $c\to s$ with $W^+\to u\bar d$
- $\Lambda = uds$ , $\Xi^0 = uss$ , $\Omega^- = sss$
7.31beauty hyperons cascading downtheory
Give a cascade of Cabibbo-favoured decays through flavoured hyperons for $\Sigma_b^+(uub)$ , $\Xi_b^-(dsb)$ and $\Lambda_b^0(udb)$ .
- the favoured $b$ transition is $b\to c$ ( $|V_{cb}|$ ), then $c\to s$
- $\Lambda_c^+ = udc$ , $\Xi_c^0 = dsc$
7.32phase space against the Cabibbo angletheory
The $\Sigma^-$ has two beta decays: $\Sigma^-\to ne^-\bar\nu_e$ with BR $(1.017\pm0.034)\times10^{-3}$ , and $\Sigma^-\to\Lambda e^-\bar\nu_e$ with BR $(5.73\pm0.27)\times10^{-5}$ . State the reason for the difference.
- $\Sigma^- = dds$ , $n = udd$ , $\Lambda = uds$
- $\Gamma\propto Q^5$
- $m_{\Sigma^-} = 1197.4$ , $m_n = 939.6$ , $m_\Lambda = 1115.7$ MeV

🔑 If you remember only three things

Changing flavour is what makes it weak, not the strength. The name is an accident of rates; the defining property is that nothing else in the book does it at all.
A coupling with dimensions turns every estimate into a question about scale. You cannot ask how big without also asking at what energy.
The two facts outlive the chapter. Chapters 8, 9 and 10 are applications of exactly these, which is why the repetition here is an investment rather than a drill.

Where this goes next

The chapter leaves three things unexplained on purpose, and each becomes a chapter.

The phase. §7.11 found that a 3 × 3 mixing matrix keeps one irremovable complex phase, and that with two families it could not. Chapter 8 makes that phase observable — in neutral-meson mixing, where the states that propagate are not the states that are produced, and in CP violation, which is the same mismatch seen as an asymmetry between matter and antimatter.

The couplings. §7.13 wrote the neutral current with two constants $g_L^i$ and $g_R^i$ and refused to say what they are. Chapter 9 derives them from the fermions’ electric charges, which is the moment the weak and electromagnetic interactions stop being two things. It also explains why $M_W$ has the value that makes $G_F$ what it is.

The other mixing. §7.9 repaired quark universality by rotating the down-type states. Chapter 10 finds the same phenomenon among neutrinos — with two differences that matter: the angles are large rather than small, and it requires neutrino mass, which the Standard Model as built through Chapter 9 does not provide.

✅ Check yourself — Chapter 7 in review

0/6 answered · 0 correct

1.Across the whole chapter, which single structural fact does the most explanatory work?
2.Which of these are consequences of the m⁵ law, Γ ∝ G_F² m⁵? (Select all that apply.)
3.Problem 7.21 gets BR(b → c e ν̄) = 1/9 from counting W decay channels. What makes the counting work?
4.Several problems (7.13, 7.17, 7.28) compute how far a neutrino travels before interacting. What distinguishes 7.28?
5.Problems 7.23, 7.25, 7.30 and 7.31 are all the same exercise. What is it?
6.Problem 7.5 gives τ_τ = 0.262 ps using the book's BR = 16 %, against 0.290 ps measured. What should you conclude?