Chapter 9 — Summary and Problems · Elementary Particle Physics

Nothing here can be answered from memory. Every question descends from the same short structure, so a problem you cannot finish identifies exactly which part of it you have not understood.

🎯 Why this matters

A chapter that predicts can be examined in a way a chapter that catalogues cannot. Every one of these has an answer the theory fixes, which is why the set is both the longest and the most uniform in the book.

Thirty-six problems, the largest set in the book, and unusually well distributed:

eight are the electroweak bookkeeping — 9.5 to 9.8, 9.28, 9.29, 9.32, 9.33 — where you assign isospin and hypercharge and then use them as a checksum on processes that charge conservation cannot decide;
nine are widths and rates — 9.9 to 9.16, 9.23 — all of which reduce to the two master constants of §9.5 and the $c_Z$ table;
six are experiment design — 9.1 to 9.3, 9.22, 9.24, 9.30 — asking what a beam, a detector or a machine has to deliver;
five are error propagation and counting — 9.18, 9.19, 9.21, 9.25, 9.31;
and the rest are the ones worth doing even if you skip the others: 9.20 on why the weak force is weak, 9.27 on a dark-matter line, 9.34 on flavour blindness, and 9.35–9.36, which arrive at the fact that a nucleus’s weak charge is essentially its neutron number.

The book supplies no solutions. All thirty-six are worked here.

What the chapter established

Every row traces to one angle, θ_W ≈ 29° — except the last two, which trace to a field value, υ = 246 GeV.

what↕	how↕	the number↕
The weak and electromagnetic interactions share a description	one rotation takes $(W_3, B)$ to $(\gamma, Z)$	21 couplings → 2 constants
…and the angle is the same everywhere	ν–e scattering, atomic parity violation, Møller, LEP asymmetries, $M_W/M_Z$	$\sin^2\theta_W = 0.232$ over 8 decades in $Q$
The mediators exist, at the predicted masses	a $p\bar p$ collider, and a Jacobian edge	$M_W = 80.377\pm0.012$ , $M_Z = 91.1875\pm0.0021$
They carry weak charge themselves	$e^+e^-\to W^+W^-$ diverges without the $WWZ$ vertex	measured to 209 GeV
There are three families and no more	the height of the Z resonance	$N_\nu = 2.984 \pm 0.008$
The masses come from a broken symmetry	a scalar doublet with a degenerate vacuum	$\upsilon = 246$ GeV; 4 d.o.f. → 3 eaten + 1 H
…and the scalar exists, with the right couplings	$\gamma\gamma$ and $ZZ^*\to4\ell$ , then couplings vs mass	$M_H = 125.25$ GeV, $\mu = 1.002\pm0.057$

“21 couplings → 2 constants”, drawn: every fermion’s pair of Z couplings, from one angle

g_Lg_R−0.5−0.25+0.25+0.5+0.25−0.25ν(+0.500, 0)e μ τ(−0.268, +0.232)u c t(+0.345, −0.155)d s b(−0.423, +0.077)g_L = T₃ − Q sin²θ_W g_R = − Q sin²θ_WTwo integers you already know, and ONE measured number. Nothing else is free.Turn sin²θ_W and all four points slide along fixed lines — the neutrino cannot move at all, since Q = 0.

Supplied — the chapter’s first row claims “21 couplings → 2 constants” and the book leaves it as a sentence. Each fermion type couples to the Z with two independent strengths, left and right, and a pre-electroweak theory would have had to measure every one of them. Here all eight numbers on the plot come from $g_L = T_3 - Q\sin^2\theta_W$ and $g_R = -Q\sin^2\theta_W$ with $\sin^2\theta_W = 0.232$ — the weak isospin and charge are known integers, so one measured angle fixes the lot.

That is what unification buys, and it is why measuring $\sin^2\theta_W$ in eight different places over eight decades in $Q$ (row 2) is a test rather than a repetition. The neutrino sits on the $g_R = 0$ axis for a reason no measurement could change: it has no electric charge, so the $Q\sin^2\theta_W$ term vanishes and a right-handed neutrino would couple to nothing at all.

The numbers, all at once

every numeric answer in the chapter's problems

import numpy as np
GF, s2, MZ, MW, me, u = 1.1663788e-5, 0.232, 91.0, 80.0, 0.5109989e-3, 1.66053907e-27
cz = lambda i3, q: i3 - s2*q
K = GF*MZ**3/(3*np.sqrt(2)*np.pi); Ge = GF*MW**3/(6*np.pi*np.sqrt(2))
Gn = K*.25; Gl = K*(cz(-.5,-1)**2+cz(0,-1)**2)
Gu = 3*K*(cz(.5,2/3)**2+cz(0,2/3)**2); Gd = 3*K*(cz(-.5,-1/3)**2+cz(0,-1/3)**2)
Gh = 2*Gu+3*Gd; G3 = 3*Gn+3*Gl+Gh
GW = 3*Ge + 3*Ge*(.974**2+.99**2+.224**2+.22**2); H = 389.379

Ne = 547e3/u/2; sg = 1.7e-45*24; phi = 1/(4*sg*Ne)
print(f"9.1   duty cycle {2*6e-3/14.4:.1e}; fluence {phi:.1e} /m^2 per pulse; I = {phi*3.2**2:.1e} nu/pulse")
print(f"9.2   theta_max = {np.sqrt(2*me/20)*1e3:.1f} mrad at 20 GeV; need ~0.7 mrad; iron excluded")
mpi, mmu = .13957039, .1056584; Es = (mpi**2-mmu**2)/(2*mpi); g = 200/mpi; b = np.sqrt(1-1/g**2)
print(f"9.3   E* = {Es*1e3:.1f} MeV; E_max = {g*Es*(1+b):.1f} GeV = {g*Es*(1+b)/200:.3f} E_pi; E_min = {g*Es*(1-b)*1e6:.0f} keV")
print(f"      theta*=50 mrad -> theta_lab = {np.arctan(np.sin(.05)/(g*(np.cos(.05)+b)))*1e6:.1f} urad")
print( "9.4   sigma(nu)/sigma(nubar) = 3 EXACTLY, from angular momentum alone")

def W(x):
    c = lambda i3, q: i3 - x*q
    return (K*(c(-.5,-1)**2+c(0,-1)**2), 3*K*(c(.5,2/3)**2+c(0,2/3)**2), 3*K*(c(-.5,-1/3)**2+c(0,-1/3)**2))
for tag, x in (("9.9   sin^2 = 3/8: ", 3/8), ("9.10  sin^2 = 1/4: ", .25)):
    l, uu, dd = W(x); h = 2*uu+3*dd; t = 3*Gn+3*l+h
    print(f"{tag} Gl = {l*1e3:3.0f}, Gu = {uu*1e3:3.0f}, Gd = {dd*1e3:3.0f} MeV, Gh = {h:.2f} GeV")
    print(f"      Gamma_Z = {t:.2f} GeV,  Gnu/Gh = {Gn/h:.3f}" + ("   (vs 0.099 measured -> dead)" if x > .3 else ""))
print(f"9.11  BR(W -> e nu) = 1/9 = 11.1% by COUNTING; {Ge/GW*100:.1f}% from the widths")
print(f"9.12  g_Zee^2/g_Wenu^2 = {2*(cz(-.5,-1)**2+cz(0,-1)**2)/(1-s2):.3f};  rate ratio {Gl/Ge:.2f}")
print(f"9.13  g_Zuu^2/g_Wud^2 = {2*(cz(.5,2/3)**2+cz(0,2/3)**2)/(1-s2):.3f};  rate ratio {Gu/(3*.974**2*Ge):.2f}")
print(f"9.14  10 nb -> {10*(GW-3*Ge)/Ge:.0f} nb into hadrons ({10*GW/Ge:.0f} nb into everything)")
brs = [Gl/(N*Gn+3*Gl+Gh)*100 for N in (3,4,5)]
rat = [(G3/(N*Gn+3*Gl+Gh))**2 for N in (3,4,5)]
print(f"9.15  N=3,4,5: BR(mumu) = {brs[0]:.2f}, {brs[1]:.2f}, {brs[2]:.2f}%;  peak sigma ratios 1, .{rat[1]*1000:.0f}, .{rat[2]*1000:.0f}")
print(f"9.16  {12*np.pi*Gl*Gl/(MZ**2*G3**2)*H*1e3:.1f} nb at the Z peak;  {(4*np.pi/3)/MW**2*(3*.974**2*Ge)*Ge/GW**2*H*1e3:.1f} nb at the W peak")
E = np.sqrt(MZ**2+140**2)
print(f"9.17  BOTH electrons at {E/MZ*MZ/2:.1f} GeV, each {np.degrees(np.arctan(1/((E/MZ)*(140/E)))):.0f} deg from the beam")
E1, E2, th, dth = 60., 40., np.radians(140), np.radians(1)
m = np.sqrt(2*E1*E2*(1-np.cos(th)))
r = .5*np.sqrt((.15/np.sqrt(E1))**2 + (.15/np.sqrt(E2))**2 + (dth/np.tan(th/2))**2)
print(f"9.18  m = {m:.1f} GeV, dm = {r*m:.1f} GeV -- the ENERGY term dominates 4:1")
print(f"9.19  20% on sin^2 -> {80*.1:.0f} GeV on M_W;  1% on M_Z/M_W -> {2*(MW/MZ)**2*.01:.3f} on sin^2")
q2 = (.197)**2
print(f"9.20  the weak force is {1/(7.4**2*q2/(q2+MW**2)):.1e} times weaker at 1 fm -- ALL propagator")
EZ = np.sqrt(MZ**2+100**2)
print(f"9.21  x = {MZ/2000:.3f} each at rest;  {(EZ+100)/2000:.3f} and {(EZ-100)/2000:.3f} at p_L = 100 GeV")
print(f"9.22  {17e-36*1e32*86400:.0f} W pairs per day at LEP2")
G4 = 4*Gn+3*Gl+Gh; d = 1-(G3/G4)**2
print(f"9.23  one extra neutrino: +{(G4/G3-1)*100:.1f}% on Gamma_Z, -{d*100:.1f}% on the peak height")
print(f"9.24  {(5/d)**2:.0f} events for a 5 sigma exclusion -- a few hours of LEP")
Rs = [3.1*(Ge/GW)/(Gl/(N*Gn+3*Gl+Gh)) for N in (3,4,5)]
print(f"9.25  R = {Rs[0]:.1f}, {Rs[1]:.1f}, {Rs[2]:.1f} for N = 3, 4, 5;  R < 10.1 excludes N >= 4")
uL, uR, dL, dR = cz(.5,2/3), cz(0,2/3), cz(-.5,-1/3), cz(0,-1/3)
sn = (uL**2+dL**2) + (uR**2+dR**2)/3; sb = (uL**2+dL**2)/3 + (uR**2+dR**2)
print(f"9.26  CC ratio 3 exactly;  NC ratio {sn/sb:.1f}")
mz = 91.1876; mc = (136 + np.sqrt(136**2 + mz**2))/2
print(f"9.27  m_chi = 136 GeV (gamma gamma) or {mc:.1f} GeV (Z gamma)")
print(f"9.30  Glashow needed E_nu = M_W^2/2m_e = {MW**2/(2*me):.1e} GeV = 6 PeV")
print(f"9.31  lambda-bar = {197/16:.1f} fm >> R_A = 4 fm -> COHERENT; T_max = {2*16**2/70.7e3*1e3:.1f} keV")
print(f"9.32  c_Z^2 = {uL**2:.3f}, {uR**2:.3f}, {dL**2:.3f}, {dR**2:.3f};  isoscalar combination {sn:.3f}")
print(f"9.33  isoscalar combination {sb:.3f};  ratio {sn/sb:.1f}")
print( "9.34  exactly 1 -- the neutral current is flavour-blind")
print(f"9.35  c_A = +-1/2 ALWAYS;  c_V(e) = {cz(-.5,-1)+cz(0,-1):.3f}, nearly zero")
cVu, cVd = uL+uR, dL+dR
print(f"9.36  proton c_V = {2*cVu+cVd:+.2f}, neutron c_V = {cVu+2*cVd:+.2f} -> Q_W = -N, {abs((cVu+2*cVd)/(2*cVu+cVd)):.0f}:1")
print("\n(this snippet uses the book's sin^2 = 0.232 throughout; problems 9.32-9.36")
print(" state sin^2 = 0.23, which moves the last digit of c_V and the 9.36 ratio.")
print(" both are the book's own values, used where the book uses them.)")

prints

9.1   duty cycle 8.3e-04; fluence 3.7e+10 /m^2 per pulse; I = 3.8e+11 nu/pulse
9.2   theta_max = 7.1 mrad at 20 GeV; need ~0.7 mrad; iron excluded
9.3   E* = 29.8 MeV; E_max = 85.4 GeV = 0.427 E_pi; E_min = 10 keV
    theta*=50 mrad -> theta_lab = 17.4 urad
9.4   sigma(nu)/sigma(nubar) = 3 EXACTLY, from angular momentum alone
9.9   sin^2 = 3/8:  Gl = 103, Gu = 247, Gd = 309 MeV, Gh = 1.42 GeV
    Gamma_Z = 2.23 GeV,  Gnu/Gh = 0.116   (vs 0.099 measured -> dead)
9.10  sin^2 = 1/4:  Gl =  82, Gu = 275, Gd = 357 MeV, Gh = 1.62 GeV
    Gamma_Z = 2.36 GeV,  Gnu/Gh = 0.102
9.11  BR(W -> e nu) = 1/9 = 11.1% by COUNTING; 11.0% from the widths
9.12  g_Zee^2/g_Wenu^2 = 0.327;  rate ratio 0.37
9.13  g_Zuu^2/g_Wud^2 = 0.373;  rate ratio 0.44
9.14  10 nb -> 61 nb into hadrons (91 nb into everything)
9.15  N=3,4,5: BR(mumu) = 3.44, 3.22, 3.03%;  peak sigma ratios 1, .876, .773
9.16  2.1 nb at the Z peak;  8.8 nb at the W peak
9.17  BOTH electrons at 83.5 GeV, each 33 deg from the beam
9.18  m = 92.1 GeV, dm = 1.4 GeV -- the ENERGY term dominates 4:1
9.19  20% on sin^2 -> 8 GeV on M_W;  1% on M_Z/M_W -> 0.015 on sin^2
9.20  the weak force is 3.0e+03 times weaker at 1 fm -- ALL propagator
9.21  x = 0.045 each at rest;  0.118 and 0.018 at p_L = 100 GeV
9.22  147 W pairs per day at LEP2
9.23  one extra neutrino: +6.9% on Gamma_Z, -12.4% on the peak height
9.24  1622 events for a 5 sigma exclusion -- a few hours of LEP
9.25  R = 9.9, 10.6, 11.3 for N = 3, 4, 5;  R < 10.1 excludes N >= 4
9.26  CC ratio 3 exactly;  NC ratio 2.4
9.27  m_chi = 136 GeV (gamma gamma) or 149.9 GeV (Z gamma)
9.30  Glashow needed E_nu = M_W^2/2m_e = 6.3e+06 GeV = 6 PeV
9.31  lambda-bar = 12.3 fm >> R_A = 4 fm -> COHERENT; T_max = 7.2 keV
9.32  c_Z^2 = 0.119, 0.024, 0.179, 0.006;  isoscalar combination 0.308
9.33  isoscalar combination 0.129;  ratio 2.4
9.34  exactly 1 -- the neutral current is flavour-blind
9.35  c_A = +-1/2 ALWAYS;  c_V(e) = -0.036, nearly zero
9.36  proton c_V = +0.04, neutron c_V = -0.50 -> Q_W = -N, 14:1

(this snippet uses the book's sin^2 = 0.232 throughout; problems 9.32-9.36
state sin^2 = 0.23, which moves the last digit of c_V and the 9.36 ratio.
both are the book's own values, used where the book uses them.)

Three of those repay a second look.

9.20 is the chapter in one line. The weak coupling is larger than the electric one, and at 1 fm the weak force is three thousand times weaker. Every bit of that is the propagator: $q^2/(q^2+M_W^2)$ against $1/q^2$ . “Weak” was never a statement about a charge.

9.35 finds a near-zero that is not an accident. The electron’s vector coupling to the Z is $c_V = -\tfrac12 + 2\sin^2\theta_W = -0.04$ , vanishing exactly at $\sin^2\theta_W = 1/4$ — the same special point that makes §9.4’s ratio $R$ equal 1. Nature sits a few per cent away from it, and half the precision electroweak programme consists of measuring how far.

9.36 ends somewhere surprising. Because the proton’s vector weak charge is $+0.04$ and the neutron’s is $-0.50$ , a nucleus’s weak charge is essentially $-N$ — it counts neutrons, by more than an order of magnitude. And because the effect adds coherently over them, a part-per-million parity violation becomes measurable in a heavy atom. That is why atomic parity-violation experiments use caesium and thallium, and it is a tabletop measurement of the same angle the LHC measures.

0/36 solved

9.1neutrino beamstheory
CHARM2 studied $\nu_\mu e \to \nu_\mu e$ with a narrow-band beam of mean energy 24 GeV. The SPS gave two pulses 6 ms apart on a 14.4 s cycle. The useful target mass was 547 t with a square section 3.2 m on a side, containing equal numbers of protons and neutrons. Find the duty cycle. For one interaction every four pulses on average, what neutrino fluence and what intensity are needed?
- $\sigma/E = 1.7\times10^{-45}$ m²/GeV
- M = 547 t, l = 3.2 m
- Z/A = 1/2
9.2kinematicstheory
An electron of 20 GeV is detected by CHARM2, produced by elastic $\nu e$ scattering. How large can its scattering angle be? What accuracy is needed in the direction measurement to verify it? Could the calorimeter be built of iron?
- $E_e\theta_e^2 \le 2m_e$
- $m_e = 0.511$ MeV
9.3beam designtheory
A narrow-band $\nu_\mu$ beam is made from a monochromatic $\pi^+$ beam of 200 GeV decaying as $\pi^+\to\mu^+\nu_\mu$ . Find the neutrino energy in the pion rest frame, the maximum and minimum laboratory energies, and the laboratory angle of a neutrino emitted at $\theta^* = 50$ mrad.
- $E_\pi = 200$ GeV
- $m_\pi = 139.57$ , $m_\mu = 105.66$ MeV
9.4CC cross-sectionstheory
Consider the charged-current cross-sections for neutrinos and antineutrinos on nuclei with equal numbers of protons and neutrons, neglecting all masses. Show that the neutrino total cross-section is three times the antineutrino one.
- V−A: neutrinos are left-handed, antineutrinos right-handed
- an isoscalar target
9.5quantum numberstheory
Give the weak isospin, its third component and the weak hypercharge for $e^-_L$ , $\nu_L$ , $u_L$ , $d_R$ , and for their antiparticles.
- $Y_W = 2(Q - I_{Wz})$
- left-chirality fermions in doublets, right-chirality charged fermions in singlets
9.6quantum numberstheory
Give the weak isospin, its third component and the weak hypercharge for $\mu^-_L$ , $\tau^-_R$ , $t_L$ , $b_R$ , and for their antiparticles.
- $Y_W = 2(Q - I_{Wz})$
- left-chirality fermions in doublets, right-chirality charged fermions in singlets
9.7conservation lawstheory
Which of these are allowed and which forbidden, and why? $W^-\to d_L\bar u_L$ , $W^-\to u_L\bar u_L$ , $Z\to W^+W^-$ , $W^-\to e^-_R\bar\nu_R$ .
- all gauge bosons have $Y_W = 0$
- Table 9.1
9.8conservation lawstheory
Which of these are allowed and which forbidden? $d\to W^-u_L$ , $Z\to e^-_L\bar e_L$ , $W^-\to ZW^-$ , $W^-\to e^-_L\bar\nu_L$ .
- all gauge bosons have $Y_W = 0$
- Table 9.1
9.9Z widthstheory
A grand-unification theory based on SU(5) predicted $\sin^2\theta_W = 3/8$ , which experiment falsified. Find the partial widths of the Z in that hypothesis, and the ratio $\Gamma_\nu/\Gamma_h$ .
- $G_F M_Z^3/(3\sqrt2\pi) = 660$ MeV
- $c_Z = I_{Wz} - Q\sin^2\theta_W$
- $M_Z = 91$ GeV
9.10Z widthstheory
Repeat for $\sin^2\theta_W = 1/4$ : the partial and total widths of the Z, and $\Gamma_\nu/\Gamma_h$ .
- $M_Z = 91$ GeV
- $G_F M_Z^3/(3\sqrt2\pi) = 660$ MeV
9.11branching ratiostheory
Evaluate the branching ratio for $W\to e^+\nu_e$ .
- colour factor 3 for quarks
- $M_W < m_t$ , so the top channels are closed
9.12couplingstheory
Evaluate the ratio $g_{Zee}^2/g_{We\nu}^2$ and the decay-rate ratio $\Gamma(Z\to e^+e^-)/\Gamma(W\to e\nu)$ .
- $g_Z = (g/\cos\theta_W)c_Z$
- $g_{We\nu} = g/\sqrt2$
- $\sin^2\theta_W = 0.232$
9.13couplingstheory
Evaluate $g_{Zuu}^2/g_{Wud'}^2$ and $\Gamma(Z\to u\bar u)/\Gamma(W\to \bar d u)$ .
- $c_Z(u_L) = 1/2 - (2/3)s^2$ , $c_Z(u_R) = -(2/3)s^2$
9.14resonancestheory
Assume $\sigma(\bar u d \to e^-\bar\nu_e) = 10$ nb at the W resonance. Evaluate the total $\sigma(\bar u d \to q\bar q)$ at resonance.
- Breit–Wigner: $\sigma_{\max}\propto \Gamma_{\text{in}}\Gamma_{\text{out}}/\Gamma^2$
9.15neutrino countingtheory
Take the number of light neutrinos to be 3, 4 or 5 in turn, changing nothing else. Evaluate for each the Z branching ratio into $\mu^+\mu^-$ and the ratio $\Gamma_\nu/\Gamma_Z$ , and the ratio of the peak hadronic cross-sections.
- $\Gamma_\nu = 165$ , $\Gamma_\ell = 83$ MeV, $\Gamma_h = 1.66$ GeV
- $\sigma_{\text{peak}}\propto 1/\Gamma_Z^2$
9.16resonancestheory
Calculate $\sigma(e^+e^-\to\mu^+\mu^-)$ at the Z peak and $\sigma(\bar u d\to e^-\bar\nu_e)$ at the W peak.
- $\sigma_{\text{peak}}(e^+e^-) = 12\pi\Gamma_e\Gamma_\mu/(M_Z^2\Gamma_Z^2)$
- $\sigma_{\text{peak}}(\bar ud) = \tfrac{4\pi}{3}\tfrac{1}{M_W^2}\tfrac{\Gamma_{ud}\Gamma_{e\nu}}{\Gamma_W^2}$
- $\hbar^2c^2 = 389.4\;\mu$ b GeV²
9.17kinematicstheory
A Z is produced at a $p\bar p$ collider at $\sqrt s = 540$ GeV, moving along the beams with $p_Z = 140$ GeV. It decays as $Z\to e^+e^-$ with the electrons at 90° to the beams in the Z rest frame. Calculate the two electron energies in the laboratory.
- $M_Z = 91$ GeV
9.18error propagationtheory
A $Z\to e^+e^-$ decay has calorimeter energies $E_1 = 60$ GeV and $E_2 = 40$ GeV, with resolution $\sigma_E/E = 0.15/\sqrt E$ . The measured angle between the tracks is $140°\pm1°$ . Find the error on $m_Z$ .
- $M = \sqrt{2E_1E_2(1-\cos\theta)}$
- $\frac{\delta M}{M} = \frac12\sqrt{\left(\frac{\delta E_1}{E_1}\right)^2+\left(\frac{\delta E_2}{E_2}\right)^2+\left(\frac{\delta\theta}{\tan(\theta/2)}\right)^2}$
9.19error propagationtheory
Before their discovery the predictions were $M_Z = 90$ and $M_W = 80$ GeV. If $\sin^2\theta_W = 0.23$ with a 20 % uncertainty, what is the uncertainty on $M_W$ ? If instead $M_Z/M_W$ is measured to 1 %, what is the uncertainty on $\sin^2\theta_W$ ?
- $M_W = 37.3\,\text{GeV}/\sin\theta_W$
- $\sin^2\theta_W = 1 - (M_W/M_Z)^2$
9.20propagatorstheory
At 1 GeV the weak charge $g$ is larger than the electric charge by $\sqrt{4\pi}/\sin\theta_W \approx 7.4$ . Why then is the electrostatic force between two electrons at 1 fm so much larger than the weak force?
- $\hbar c = 197$ MeV fm
- $M_W = 80$ GeV
9.21parton kinematicstheory
At the Tevatron ( $p\bar p$ , $\sqrt s = 2$ TeV), what are the approximate momentum fractions of the annihilating quark and antiquark for a Z produced at rest? In what fraction are they sea quarks? And if the Z has longitudinal momentum 100 GeV?
- $\hat s = x_q x_{\bar q} s$
- $M_Z = 91$ GeV
9.22luminositytheory
LEP2 was designed to study $e^+e^-\to W^+W^-$ . If the cross-section at $\sqrt s = 200$ GeV is 17 pb and the luminosity is $10^{32}$ cm⁻²s⁻¹, how many events are produced per day?
- 1 pb = $10^{-36}$ cm²
- 1 day = 86 400 s
9.23neutrino countingtheory
What is the percentage change in the Z total width for one additional neutrino type? What is the change in the peak hadronic cross-section?
- $\Gamma_\nu = 165$ MeV
- $\Gamma_Z = 2.40$ GeV
- $\sigma_{\text{peak}}\propto1/\Gamma_Z^2$
9.24statisticstheory
Working at the Z with an $e^+e^-$ collider and assuming statistical uncertainties only, how many events are needed to exclude a fourth neutrino at five standard deviations?
- one extra neutrino changes the peak cross-section by 12.4 %
9.25neutrino countingtheory
UA1 and UA2 measured $R = \dfrac{\sigma_W\,\mathrm{BR}(W\to e\nu)}{\sigma_Z\,\mathrm{BR}(Z\to e^+e^-)}$ from event counts alone, so that systematics cancel. Writing $R = \dfrac{\sigma_W}{\sigma_Z}\cdot\dfrac{\Gamma(W\to\ell\nu)}{\Gamma_W}\cdot\dfrac{\Gamma_Z}{\Gamma(Z\to\ell\ell)}$ , evaluate R for 3, 4 and 5 neutrinos and set an upper limit on $N_\nu$ from $R < 10.1$ at 90 % CL.
- $\sigma_W/\sigma_Z = 3.1$
- $\Gamma(W\to\ell\nu) = 224$ , $\Gamma_W = 2035$ MeV
- $\Gamma(Z\to\ell\ell) = 83$ MeV, $\Gamma_h = 1662$ MeV
9.26cross-section ratiostheory
Calculate the ratio of the CC cross-sections of neutrinos and antineutrinos on nuclei with equal numbers of protons and neutrons, considering only valence quarks. Repeat for the neutral-current interactions.
- the L+R configuration carries a factor 1/3
- $c_Z = I_{Wz}-Q\sin^2\theta_W$ , $\sin^2\theta_W = 0.232$
9.27kinematicstheory
Dark matter might consist of 'neutralinos' χ, identical to their antiparticles, annihilating as $\chi\chi\to\gamma\gamma$ and $\chi\chi\to Z^0\gamma$ . A gamma telescope observes a monochromatic line at $E_\gamma = 136$ GeV. Find $m_\chi$ in both hypotheses, assuming negligible initial kinetic energy.
- $M_Z = 91.19$ GeV
- the pair annihilates essentially at rest, so $\sqrt s = 2m_\chi$
9.28colour and chiralitytheory
(1) For a pair of quarks with colours R and B and a third quark G, is the force attractive or repulsive for the combinations RB + BR and RB − BR? (2) Is $W^-\to b_L\bar c_R$ allowed? (3) Is $Z\to\tau^-_R\bar\tau_R$ allowed?
- $3\otimes3 = 6\oplus\bar3$
- Table 9.1
9.29colour and chiralitytheory
Establish whether each is allowed, where the left superscript is colour and the right subscript chirality: (a) $W^-\to{}^Bs_L\,{}^B\bar u_R$ , (b) $W^-\to{}^Bd_R\,{}^B\bar u_L$ , (c) $W^-\to{}^Rd_L\,{}^B\bar u_R$ , (d) $Z\to{}^Gu_L\,{}^G\bar u_R$ , (e) $Z\to{}^Gu_R\,{}^G\bar u_R$ , (f) $Z\to{}^Gu_R\,{}^G\bar c_L$ , (g) $Z\to{}^Gt_L\,{}^G\bar t_R$ . And is $\tfrac{1}{\sqrt6}({}^Rq\,{}^Bq\,{}^Gq - \ldots)$ bound?
- gauge bosons carry no colour and no hypercharge
9.30historytheory
In 1960 Glashow suggested establishing charged weak bosons through the resonant reaction $\nu_e e^-\to W^-\to\ldots$ . Why did the idea not work in practice?
- $M_W = 80$ GeV
- $m_e = 0.511$ MeV
9.31neutrino detectiontheory
The 'boron neutrinos' from $^8\mathrm{B}\to2\alpha\,e^+\nu_e$ in the Sun reach $E_{\nu,\max} = 16$ MeV. Consider elastic coherent scattering on a $^{76}$ Ge nucleus of radius $R_A = 4$ fm. Is the resolving power enough to resolve the nuclear structure? What is the maximum recoil energy? How does the interaction probability depend on flavour?
- $\hbar c = 197$ MeV fm
- $m(^{76}\text{Ge}) \approx 70.7$ GeV
9.32neutral currentstheory
For neutral-current scattering of $\nu_\mu$ on the valence quarks of a nucleus, write the squared Z-charge factors of the quarks and their values at $\sin^2\theta_W = 0.23$ . Write the combination the total cross-section is proportional to, and evaluate it for the valence quarks and for a nucleus with equal numbers of u and d.
- $c_Z = I_{Wz}-Q\sin^2\theta_W$
- L+R carries a factor 1/3
9.33neutral currentstheory
Repeat problem 9.32 for antineutrinos.
- the same four $c_Z$ values
- for an antineutrino the 1/3 moves to the LEFT-handed quarks
9.34neutral currentstheory
Compare the neutral-current scattering of $\nu_e$ and $\nu_\mu$ of the same energy on the valence quarks of a nucleus. Consider the ratio of the left- and right-quark contributions in the two cases. How large is the ratio between these ratios?
- the Z couples identically to all neutrino flavours
9.35neutral currentstheory
Find the relative amplitudes of the vector and axial couplings to the Z for $\nu_e$ , $e^-$ , $u$ and $d$ , at $\sin^2\theta_W = 0.23$ .
- $c_V = c_L + c_R$ , $c_A = c_L - c_R$
9.36atomic parity violationtheory
Atomic electrons interact with the nucleus by photon exchange and, far more weakly, by Z exchange; the interference of the two gives observable parity violation. The main contribution comes from the product of the axial ELECTRON current and the vector QUARK currents. (a) Evaluate the axial Z-charges of the proton and the neutron. (b) Evaluate them for a nucleus with Z protons and N neutrons — is the larger contribution from protons or neutrons? (c) Explain why the product of the axial quark current and the vector electron current is less important.
- $c_A(q) = I_{Wz}$
- $c_V(u) = +0.193$ , $c_V(d) = -0.347$
- consider valence quarks only

Everything in chapter 9 descends from two numbers — and the three that do not are chapter 12’s problem

sin²θ_W = 0.232 (θ_W ≈ 28.8°)one rotation of (W₃, B) into (γ, Z)υ = 246 GeVthe field value in the vacuumg_L, g_Revery fermionM_W/M_Z= cos θ_WasymmetriesA_FB, A_LR, APVM_W, M_Zthe SCALE, not the ratio3 eaten d.o.f.→ W±, Z get massN_ν = 2.984from the Z width…and three things that need a THIRD input the model does not supply:M_H = √(2λ)·υ needs λ · every fermion mass needs its own Yukawa · neither is predicted — → ch12’s parameter count

Supplied. The summary table above says it in words — “every row traces to one angle, except the last two, which trace to a field value” — and the shape of the claim is worth seeing. θ_W fixes ratios and υ fixes the scale: the angle alone can never tell you how heavy a W is, and the vacuum value alone can never tell you how a fermion’s left and right couplings differ.

It also makes visible where the chapter stops. The Higgs mass and every fermion mass sit below the dashed line because each needs a coupling nobody can predict — 125.25 GeV is measured, not derived, and the same is true of all nine charged-fermion masses. That is precisely the bill §12.7 presents.

🔑 If you remember only three things

The arithmetic is the same each time. Charge assignments in, a coupling out, a rate from the coupling — thirty-six variations on three steps.
A wrong answer localises the error. The step that failed is identifiable rather than lost somewhere in the algebra, which is rare enough in a physics problem set to be worth exploiting.
The set is uniform because the theory is. Earlier chapters needed many kinds of problem because their physics was a catalogue; this one needs thirty-six of one kind.

Where this goes next

Chapter 10 is the hole this chapter left. §9.12 showed that the BEH mechanism gives neutrinos no mass at all, because the recipe needs a right-chirality partner that has never been observed. And neutrinos oscillate, so they have mass. Chapter 8’s formalism runs again — the same $P(\alpha\to\beta)$ , the same interference between mass eigenstates — but with mixing angles that are large rather than small, and with an origin of mass that is outside everything Chapters 7–9 built.

Chapter 11 turns to gravitational waves, and Chapter 12 collects what is left: the strong-CP problem, baryogenesis — where §8.11’s CP violation was shown to be far too small — dark matter, and the fact that §9.20’s 19 free parameters are 19 things nobody can explain.

✅ Check yourself — Chapter 9 in review

0/6 answered · 0 correct

1.Problem 9.20: the weak coupling g is 7.4 times the electric charge, yet at 1 fm the weak force is thousands of times weaker. Where does the suppression come from?
2.Problems 9.4 and 9.26 both give exactly 3 for the CHARGED-current ν/ν̄ ratio, but the neutral-current ratio is 2.4. Why the difference?
3.Problem 9.24 finds that 1600 events suffice to exclude a fourth neutrino at 5σ, yet each LEP experiment collected four million. What does that tell you?
4.Problem 9.30 asks why Glashow's 1960 proposal to make a W in ν_e e → W failed. What is the answer?
5.Problem 9.35 finds c_V(e) = −0.04, nearly zero, while c_A(e) = −1/2 exactly. What is behind each?
6.Problem 9.36 ends with the nuclear weak charge being essentially −N, the neutron number. Why do the protons contribute so little?