§7.12–7.13Measuring the Cabibbo Angle, and the Neutral Current

§7.11 listed the CKM elements as measured numbers. This section does two of them properly, and the pair makes a point that the table could not: the theoretically cleanest measurement is almost never the most precise one, and choosing between them is the real work.

Then §7.13 closes the chapter with the discovery that the neutral current exists — made, remarkably, by noticing that something was missing.

|V_ud|: three routes, and the cleanest is the hardest

$|V_{ud}| \approx \cos\theta_C$ is measured three ways: superallowed $0^+ \to 0^+$ nuclear beta decays (the most precise), the neutron lifetime, and the pion’s $\pi_{e3}$ decay, Eq. (7.99),

$\pi_{e3}$ is the theorist’s choice by a wide margin. There is no nucleus, so no nuclear-structure corrections; it is a pure vector transition, so the matrix element is simple; and its form factor $f_+(0)$ is fixed by isospin symmetry, because $\pi^+$ and $\pi^0$ sit in the same multiplet. Almost nothing has to be calculated.

It is also, experimentally, close to impossible. The branching ratio is $10^{-8}$, and the reason is a single small number:

import numpy as np
GF, hbar = 1.1663788e-5, 6.582119569e-25          # GeV^-2, GeV s
mpip, mpi0, Vud, tau_pi = 0.13957039, 0.1349768, 0.97373, 2.6033e-8

D = mpip - mpi0
print(f"(7.100) m_pi+ - m_pi0 = {D*1e3:.4f} MeV        book: 4.5936 +- 0.0005")
print(f"        that is only {D/mpip*100:.2f}% of the pion mass, and it enters at the FIFTH power")

G    = GF**2 * D**5 / (30*np.pi**3) * Vud**2 * (1 - D/(2*mpip))**3
Gtot = hbar/tau_pi
print(f"\n(7.102) evaluated with |V_ud| = {Vud}:")
print(f"  Gamma(pi_e3)  = {G:.4e} GeV")
print(f"  Gamma_total   = hbar/tau_pi = {Gtot:.4e} GeV")
print(f"  BR            = {G/Gtot:.4e}        measured (7.103): 1.036e-08")
print(f"  agreement: {abs(G/Gtot/1.036e-8-1)*100:.1f}%  -- and the piece deliberately left out of this")
print( "  arithmetic is f(eps)(1+delta_EM), which the book calls 'a few per cent'.")

print("\nwhy the branching ratio is 10^-8:")
print( "  had the whole pion mass been available, Delta^5 would be")
print(f"  (m_pi/Delta)^5 = {(mpip/D)**5:.1e} times larger.")
print("\nwhat that costs the experimenter:")
print( "  a 1e-3 relative uncertainty needs ~1e6 pi_e3 events,")
print(f"  hence {1e6/1.036e-8:.1e} pion decays in total     book: 1e14")

(7.100) m_pi+ - m_pi0 = 4.5936 MeV        book: 4.5936 +- 0.0005
      that is only 3.29% of the pion mass, and it enters at the FIFTH power

(7.102) evaluated with |V_ud| = 0.97373:
Gamma(pi_e3)  = 2.6985e-25 GeV
Gamma_total   = hbar/tau_pi = 2.5284e-17 GeV
BR            = 1.0673e-08        measured (7.103): 1.036e-08
agreement: 3.0%  -- and the piece deliberately left out of this
arithmetic is f(eps)(1+delta_EM), which the book calls 'a few per cent'.

why the branching ratio is 10^-8:
had the whole pion mass been available, Delta^5 would be
(m_pi/Delta)^5 = 2.6e+07 times larger.

what that costs the experimenter:
a 1e-3 relative uncertainty needs ~1e6 pi_e3 events,
hence 9.7e+13 pion decays in total     book: 1e14

|V_us|: tagging a kaon you never see

The KLOE measurement of $|V_{us}|$ uses a trick worth understanding on its own, because it is a piece of quantum mechanics used as an experimental instrument.

DAΦNE is a $\phi$-factory: an $e^+e^-$ collider parked at $\sqrt{s} = m_\phi = 1.020$ GeV. About 34 % of the time the $\phi$ decays to two neutral kaons. Because the pair is produced in an $L = 1$ state, its wave function is antisymmetric under exchange — so the two kaons can never be in the same state. If one decays as a $K_S$, the other must be a $K_L$.

That is a tagging scheme with no inefficiency in the logic. And it is usable because the two lifetimes differ enormously:

import numpy as np
mphi, mK0 = 1.019461, 0.497611                     # GeV

p  = np.sqrt((mphi/2)**2 - mK0**2)                 # each kaon, in the phi rest frame
bg = p/mK0
print(f"sqrt(s) = m_phi = {mphi:.3f} GeV                book: 1.020 GeV")
print(f"each kaon: p* = {p*1e3:.1f} MeV,  beta.gamma = {bg:.4f}\n")

lam = {}
for nm, tau, book in (('K_S', 0.8954e-10, '0.6'), ('K_L', 5.116e-8, '350')):
    lam[nm] = bg * 2.99792458e10 * tau             # cm
    print(f"  lambda({nm}) = beta.gamma c tau = {lam[nm]:8.2f} cm     book: {book} cm")

print(f"\na factor {lam['K_L']/lam['K_S']:.0f} between them.  that is why the tag works: a decay")
print( "vertex metres from the interaction point can only be a K_L, and")
print( "a K_L on one side guarantees a K_S on the other.")
print("\nnote how little goes into this -- the phi mass, the kaon mass,")
print( "and the two lifetimes.  no model of anything.")

sqrt(s) = m_phi = 1.019 GeV                book: 1.020 GeV
each kaon: p* = 110.5 MeV,  beta.gamma = 0.2220

lambda(K_S) = beta.gamma c tau =     0.60 cm     book: 0.6 cm
lambda(K_L) = beta.gamma c tau =   340.56 cm     book: 350 cm

a factor 571 between them.  that is why the tag works: a decay
vertex metres from the interaction point can only be a K_L, and
a K_L on one side guarantees a K_S on the other.

note how little goes into this -- the phi mass, the kaon mass,
and the two lifetimes.  no model of anything.

KLOE collected about 400 million $K_SK_L$ pairs, isolated some 13 000 semileptonic $K_S$ decays, and measured, in Eq. (7.107),

which with the lattice form factor $f_+(0) = 0.9698 \pm 0.0017$ gives $|V_{us}| = 0.2240 \pm 0.0024$.

The Cabibbo hierarchy, seen directly

Example 7.1 is the cleanest demonstration in the chapter that $\theta_C$ is doing real work. Take one particle, the $D^0$, and four decays that differ only in which CKM elements sit at the two vertices:

s, c = 0.2243, 0.97373                    # sin, cos of the Cabibbo angle
t2, t4 = (s/c)**2, (s/c)**4
print(f"tan^2(theta_C) = {t2:.4f}       book: 0.05")
print(f"tan^4(theta_C) = {t4:.5f}      book: 0.0025\n")

print("                        predicted   measured    meas/pred")
for nm, pred, meas in (('K+K-  / pi+K-  (SCS)', t2, 0.1030),
                       ('pi+pi-/ pi+K-  (SCS)', t2, 0.0369),
                       ('K+pi- / pi+K-  (DCS)', t4, 0.00344)):
    print(f"  {nm}  {pred:9.4f}  {meas:9.4f}   {meas/pred:8.2f}")

print("\nthe DCS ratio, suppressed by FOUR powers of the Cabibbo angle,")
print(f"comes out within {abs(0.00344/t4-1)*100:.0f}% of a one-line estimate.  that is the hierarchy")
print(f"confirmed over a range of {0.1030/0.00344:.0f} in rate.")
print("\nbut look at the two SCS rows: at this level of approximation they")
print(f"should be EQUAL, and they differ by a factor {0.1030/0.0369:.1f}.  the book's own")
print("caveat covers it -- 'the colour field effects (difficult)' -- and")
print("it is worth seeing that an estimate this crude is right about the")
print("ORDERING and wrong about the individual numbers by up to 2x.")

tan^2(theta_C) = 0.0531       book: 0.05
tan^4(theta_C) = 0.00282      book: 0.0025

                      predicted   measured    meas/pred
K+K-  / pi+K-  (SCS)     0.0531     0.1030       1.94
pi+pi-/ pi+K-  (SCS)     0.0531     0.0369       0.70
K+pi- / pi+K-  (DCS)     0.0028     0.0034       1.22

the DCS ratio, suppressed by FOUR powers of the Cabibbo angle,
comes out within 22% of a one-line estimate.  that is the hierarchy
confirmed over a range of 30 in rate.

but look at the two SCS rows: at this level of approximation they
should be EQUAL, and they differ by a factor 2.8.  the book's own
caveat covers it -- 'the colour field effects (difficult)' -- and
it is worth seeing that an estimate this crude is right about the
ORDERING and wrong about the individual numbers by up to 2x.

Example 7.2 runs the same argument one generation up, and it is a cleaner test because the suppressed and unsuppressed decays differ in only the light quark the W makes:

Both decays are $\bar b\to\bar c$ at the first vertex, so $|V_{cb}|$ cancels exactly and the entire ratio is one CKM factor. The measured value is $\approx0.05$ — agreement at the level of the estimate, and better than the $D^0$ ratios of Example 7.1 because there is no second effect competing.

That contrast is the lesson of the two examples read together. In the $D^0$ the two singly-suppressed modes should be equal and differ by a factor 2.8, because the final states are different enough for strong-interaction effects to matter. In the $B^-$ the two final states differ by one strange quark and nothing else, so those effects largely cancel and the naive estimate survives.

§7.13 Discovering something by its absence

Flavour-changing neutral currents are suppressed to invisibility (§7.10). Flavour-conserving ones — the ordinary neutral current neutral current (NC) a weak interaction mediated by Z⁰, which changes neither charge nor flavour. Predicted by the electroweak theory and found at Gargamelle in 1973 by the absence of a muon. defined in §7.12-7.13 — open in glossary , mediated by the Z⁰ — were predicted to exist, and finding them took until 1973.

The instrument was Gargamelle, a bubble chamber named after the mother of Rabelais’s giant, built under A. Lagarrigue from 1965 and filled with 15 tonnes of the freon CF₃Br. That choice of liquid is the whole experiment: heavy enough that neutrinos interact at an appreciable rate, and with a radiation length of only 11 cm, so photons convert inside the chamber and can be seen.

Every charged-current event has a muon in it: $\nu_\mu N \to \mu^- + \text{hadrons}$. If neutral currents exist, then Eq. (7.109),

should also happen — and it looks like a hadronic spray with no muon and no incoming visible track. The signature is a negation.

What makes the neutral current different

The chapter closes by naming two properties that separate NC from CC, and both matter later:

It is diagonal. The neutral current transforms a particle into itself — no flavour change, exactly as §7.10’s Eq. (7.84) required. Searches for the off-diagonal version have never found anything, and the limits are extraordinary:

It couples to both chiralities. This is the sharpest break with everything the chapter has established. The charged current is $\bar\psi_L\gamma^\mu\psi_L$ and nothing else — that was §7.3’s whole result. The neutral current has both:

with the left-handed sum running over all twelve fermions and the right-handed one over the charged fermions only — because a right-chirality neutrino, if it exists at all, couples to nothing.

Those constants $g_L^i$ and $g_R^i$ are left completely unexplained here. That is not an omission: they are the output of the electroweak theory, and Chapter 9 derives them from the fermions’ electric charges. The fact that a weak-interaction coupling turns out to be a function of electric charge is the single clearest signal that the two interactions are one interaction.

Where this goes next

Chapter 7 is done. It established the Lorentz structure of the charged current (V − A, maximal P and C violation), its universality across leptons, the quark mixing that rescues universality, and — in this section — the existence of a neutral current whose couplings the chapter cannot explain.

Three threads run out of it. Chapter 8 takes the CKM phase found in §7.11 and makes it observable, in neutral-meson mixing and CP violation. Chapter 9 explains where the W and Z came from, why $M_W$ has the value that makes $G_F$ what it is, and what $g_L$ and $g_R$ are. Chapter 10 repeats §7.9’s mixing story for neutrinos, where the two bases differing is the entire phenomenon — and requires neutrino mass, which nothing in this chapter provides.