The Running Angle, and Four Million Z Decays · Elementary Particle Physics

Two machines were built for the same measurement and shaped differently, because a ring buys statistics and a linear collider buys polarization.

🎯 Why this matters

Two machines with different systematics are worth more than one machine twice as good. Once the dominant uncertainty stops being statistical, the only way to check a number is to measure it with something that fails differently.

§9.7 measured the bosons to a few per cent, which was enough to establish that they exist and are what the theory says. This section is about what happens when you improve that by a factor of a thousand — and the first thing that happens is that “the weak mixing angle” stops being a single number.

§9.8 Corrections, and what they are sensitive to

The tree-level relations of §9.3 came out about 4 % below the measured masses. The gap is loop corrections, and the two that matter are these:

Fig. 9.23 — the corrections that carry information

Click a vertex or an internal line.

Bettini Fig. 9.23, redrawn. Both are corrections to the W mass; both are small. What separates them is how they depend on the unknown mass in the loop — and that difference decided which particle got predicted and which did not.

💡 What this really says — quadratic versus logarithmic — why the top was predicted and the Higgs was not

This is the single most useful thing in §9.8 and the book states it in two sentences without drawing the consequence.

The top correction goes as $G_F m_t^2$ . The Higgs correction goes as $\ln M_H$ . Both are measured through the same quantity — a per-cent-level shift in $M_W$ — but they invert completely differently.

Suppose you measure the total correction to 10 %. Then:

for the top, $\delta m_t/m_t = \tfrac12 \times 10\,\% = 5\,\%$ . A measurement worth having. And indeed in 1993, before any top was seen, the fit gave $m_t = 166 \pm 27$ GeV — against the 172.7 eventually measured;
for the Higgs, $\delta M_H/M_H = 10\,\%$ of a logarithm, which means $M_H$ is determined only up to a multiplicative factor of $e^{0.1\times(\text{lever})}$ — in practice a factor of two or three. The pre-discovery fit gave $M_H = 94^{+29}_{-24}$ GeV, and the 95 % upper limit was 152 GeV: a band, not a value.

An engineer will recognise the shape immediately: a quadratic response gives you a measurement; a logarithmic response gives you an order of magnitude. It is the difference between a linear sensor and a decibel meter. If you want to locate something precisely, you need an observable that changes fast as it moves — and no amount of experimental effort converts a log into a power law.

That asymmetry structured two decades of particle physics. The top was found where the electroweak fit said it would be. The Higgs was hunted across a decade of mass, and the search had to be designed for the whole range from 114 GeV to 1 TeV (§9.13) precisely because the fit could not do better.

Because $g$ and $g'$ are couplings, they run — and so does their ratio. The evolution of $\sin^2\theta_W$ is slower and stranger than that of $\alpha$ (§5.8) or $\alpha_s$ (§6.5), because both the numerator and the denominator move:

Standard Model prediction (Jegerlehner 2017)
atomic parity violation, |Q| ~ 2.4 MeV
polarized Møller scattering, |Q| = 0.16 GeV
LEP1, on the Z pole

Fig. 9.24 redrawn schematically. Note the vertical scale: the whole variation across EIGHT orders of magnitude in Q is about 3 %. Note also the kink at M_W — the slope changes sign, and that is the section's real content.

⚠️ The slope inverts at M_W, and the reason is not the one you expect

In QCD the running of $\alpha_s$ has one sign because gluon loops (which anti-screen) beat quark loops (which screen) — §6.5. The electroweak gauge bosons carry weak charge too, so you would expect the same competition here, and you would expect it at all energies.

But the $W$ is massive. Below $Q \approx M_W$ the $W$ loops are suppressed and only fermion loops contribute; above it, they switch on.

Here is the part that is easy to get wrong. The $W$ loops do not simply reverse the running of the ratio by anti-screening. They contribute to the evolution of $g$ , the SU(2) coupling — and not to that of $g'$ , the U(1) coupling, because the $B$ field carries no weak isospin and the $W$ s carry no hypercharge. Since $\tan\theta_W = g'/g$ , changing only the denominator’s running is what inverts the slope of the angle.

So the kink at $M_W$ in Fig. 9.24 is a threshold, not a change in the nature of the interaction — and it is direct evidence that the $W$ is charged under the group it mediates.

§9.9 Why LEP was a circle, and SLC was not

For a circular collider, the power radiated by the beam goes as $E^4$ at fixed radius. That is the whole design argument, and it has a crossover:

Bettini pp. 381–382. The trade-off sits at 200–300 GeV, which is why LEP was the last of its kind at that energy and why every proposal above it is linear.

machine↕	what it spends energy on↕	the advantage↕
LEP, 27 km circular	synchrotron radiation, $\propto E^4$ — the particles are reused every turn but must be kept in orbit	enormous luminosity, because each bunch collides again on every revolution. Four experiments × 4 million $Z$ each
SLC, linear	accelerating each particle once, then discarding it	no orbit to maintain, so no $E^4$ penalty — and the beams can be polarized, which gives access to asymmetries LEP cannot reach

And the reason to build either is that an $e^+e^-$ collider is a precision instrument in a way a hadron collider structurally cannot be:

every event is interesting — not one in $10^9$ , as in §9.6;
the events are clean — there is no “rest of the event”, because the colliding objects are elementary;
and the initial state is a pure quantum state of known quantum numbers, $J^{PC} = 1^{--}$ , with the full centre-of-mass energy available and known.

That last point is what makes 20 ppm possible. Compare with §9.6, where the parton energies are unknown and everything had to be done transversely.

\sigma^{\text{peak}}(e^+e^-\to f\bar f) = \htmlClass{t-k}{\frac{12\pi}{M_Z^2}}\; \frac{\htmlClass{t-in}{\Gamma_e}\,\htmlClass{t-out}{\Gamma_f}}{\htmlClass{t-g}{\Gamma_Z^2}}

(9.78)

Bettini p. 383, the Breit–Wigner (9.77) evaluated at √s = M_Z. Every quantity on the right is separately measurable, which turns this from a formula into a constraint.

Every symbol, one at a time

Hover or tap a symbol above — it lights up in the equation and its meaning, units and type appear here.

the line shape, and what it counts

import numpy as np
MZ, GZ, Gl, Gh = 91.1875, 2.4952, 83.984e-3, 1744.4e-3
H = 389.379                                      # ub GeV^2
peak = lambda Gf, G: 12*np.pi*Gl*Gf/(MZ**2*G**2)*H*1e3     # nb

print("the peak cross-sections, from the measured widths")
print(f"  sigma(e+e- -> mu+ mu-) = 12 pi Gamma_l^2 /(M_Z^2 Gamma_Z^2) = {peak(Gl, GZ):5.2f} nb")
print(f"     book Example 9.4: 2.1 nb;  at L = 1e31 that is one Z per {1/(peak(Gl,GZ)*1e-33*1e31):.0f} s")
print(f"  sigma(e+e- -> hadrons) = 12 pi Gamma_l Gamma_h/(M_Z^2 Gamma_Z^2) = {peak(Gh, GZ):.2f} nb")
print( "     measured (9.81): 41.540 +- 0.037 nb -- it closes EXACTLY, because")
print( "     the quoted \"peak cross-section\" IS this formula, corrected to the pole")

Ginv = GZ - 3*Gl - Gh
print("\nnow the neutrino census")
print(f"  Gamma_inv = Gamma_Z - 3 Gamma_l - Gamma_h")
print(f"            = {GZ*1e3:.1f} - 3 x {Gl*1e3:.3f} - {Gh*1e3:.1f} = {Ginv*1e3:.2f} MeV")
print( "  the SM ratio Gamma_nu/Gamma_l = 1.991  (from c_Z alone: 0.2500/0.1256)")
print(f"  N_nu = (Gamma_inv/Gamma_l)/1.991 = {Ginv/Gl:.4f}/1.991 = {Ginv/Gl/1.991:.3f}")
print( "     book (9.88): 2.984 +- 0.008")

Gnu = Ginv/3
print("\nwhy the HEIGHT is the observable and not the width:")
print( "  sigma_peak goes as 1/Gamma_Z^2, so d(sigma)/sigma = -2 d(Gamma)/Gamma")
print(f"  one extra neutrino species adds Gamma_nu = {Gnu*1e3:.0f} MeV, i.e. +{Gnu/GZ*100:.1f}% on")
print(f"  Gamma_Z -- and therefore -{(1-1/(1+Gnu/GZ)**2)*100:.1f}% on the peak height.")
print("\n  peak hadronic cross-section vs the number of light neutrinos:")
for n in (2, 3, 4):
    G = GZ + (n - 3)*Gnu
    print(f"     N_nu = {n}  ->  Gamma_Z = {G:.3f} GeV,  sigma_peak = {peak(Gh, G):.1f} nb")
print("  those are 15% apart on a quantity measured to 0.1%.  no contest.")

print("\nExample 9.6 -- why the vertex detector must resolve 10 um:")
for nm, m, tau in (('D0', 1864.84, 0.4103), ('B0', 5279.66, 1.519)):
    g = 50000/m; bg = np.sqrt(g*g - 1)
    print(f"  {nm} at 50 GeV: beta.gamma = {bg:4.1f}, flight = {bg*tau*0.299792458:.2f} mm")
print("  both millimetres, so tagging c and b at LEP is comfortable --")
print("  contrast the 28 um of an unboosted B at the Upsilon(4S), Sec. 8.6")

prints

the peak cross-sections, from the measured widths
sigma(e+e- -> mu+ mu-) = 12 pi Gamma_l^2 /(M_Z^2 Gamma_Z^2) =  2.00 nb
   book Example 9.4: 2.1 nb;  at L = 1e31 that is one Z per 50 s
sigma(e+e- -> hadrons) = 12 pi Gamma_l Gamma_h/(M_Z^2 Gamma_Z^2) = 41.54 nb
   measured (9.81): 41.540 +- 0.037 nb -- it closes EXACTLY, because
   the quoted "peak cross-section" IS this formula, corrected to the pole

now the neutrino census
Gamma_inv = Gamma_Z - 3 Gamma_l - Gamma_h
          = 2495.2 - 3 x 83.984 - 1744.4 = 498.85 MeV
the SM ratio Gamma_nu/Gamma_l = 1.991  (from c_Z alone: 0.2500/0.1256)
N_nu = (Gamma_inv/Gamma_l)/1.991 = 5.9398/1.991 = 2.983
   book (9.88): 2.984 +- 0.008

why the HEIGHT is the observable and not the width:
sigma_peak goes as 1/Gamma_Z^2, so d(sigma)/sigma = -2 d(Gamma)/Gamma
one extra neutrino species adds Gamma_nu = 166 MeV, i.e. +6.7% on
Gamma_Z -- and therefore -12.1% on the peak height.

peak hadronic cross-section vs the number of light neutrinos:
   N_nu = 2  ->  Gamma_Z = 2.329 GeV,  sigma_peak = 47.7 nb
   N_nu = 3  ->  Gamma_Z = 2.495 GeV,  sigma_peak = 41.5 nb
   N_nu = 4  ->  Gamma_Z = 2.661 GeV,  sigma_peak = 36.5 nb
those are 15% apart on a quantity measured to 0.1%.  no contest.

Example 9.6 -- why the vertex detector must resolve 10 um:
D0 at 50 GeV: beta.gamma = 26.8, flight = 3.30 mm
B0 at 50 GeV: beta.gamma =  9.4, flight = 4.29 mm
both millimetres, so tagging c and b at LEP is comfortable --
contrast the 28 um of an unboosted B at the Upsilon(4S), Sec. 8.6

N_ν = 2
N_ν = 3 — the data
N_ν = 4
LEP, error bars ×10

Fig. 9.30 redrawn. Three curves differing only in Γ_Z, through one extra or one fewer neutrino species. The peak heights are 47.7, 41.5 and 36.5 nb — 15 % apart, on a quantity measured to 0.1 %. Note this is the HEIGHT doing the work, not the width.

⚙️ Engineer’s bridge — measure the quantity your apparatus is good at, not the one the formula names

The neutrino count is a width measurement that was not done by measuring a width, and the reasoning generalises.

You want $\Gamma_Z$ . The obvious route is the FWHM of the resonance. But a width in energy fights the beam energy spread — every imperfection in the machine broadens your peak the same way the physics does, and disentangling them is exactly the systematic you cannot control.

The peak height, on the other hand, is a cross-section. It is measured by counting events against a luminosity, and the luminosity at LEP was known to better than a per mille from small-angle Bhabha scattering — a pure QED process you can compute. So the height is limited by a calculable normalisation rather than by machine imperfections.

And the sensitivity is better as well as cleaner. Since $\sigma_{\text{peak}} \propto 1/\Gamma_Z^2$ , a 1 % change in the width produces a 2 % change in the height. You get a factor of two in leverage for free, on top of the systematic advantage.

The general move — reformulate the measurement in terms of the observable your apparatus measures best, even when the formula is written in terms of another one — recurs throughout this book. It is §9.4’s ratio $R$ , which cancels the detector. It is §9.7’s Jacobian edge, which sidesteps the unmeasurable longitudinal momentum. It is §8.8’s double ratio. Here it converts “measure a width” into “count events”, and the answer is $N_\nu = 2.984 \pm 0.008$ .

Where it breaks: the detector cancels only for the channels that share it, and the invisible width does not. It is obtained by subtraction — total minus everything visible — so it inherits every uncertainty in the hadronic and leptonic widths instead of carrying its own, and it is a statement about what was not seen. That also means the number counts anything light, neutral and non-interacting that the Z can reach, not neutrinos specifically: a fourth-generation neutrino heavier than $M_Z/2$ would not appear in it at all. The measurement bounds a width, and reading it as “there are three families” imports an assumption it never made.

🔬 Experiment card — ALEPH, DELPHI, L3 and OPAL at LEP, CERN, 1989–2000

Apparatus

A 27 km

e^+e^-

storage ring, with four independent detectors on it — each a central tracking chamber in a solenoidal field, electromagnetic and hadronic calorimetry, large muon chambers, and silicon micro-strip vertex detectors of 10 μm resolution close to the beam pipe for tagging charm and beauty. The Stanford Linear Collider ran in parallel with far lower luminosity but polarized beams, giving asymmetries LEP could not reach.

What is measured

The shape of the Z resonance — the hadronic cross-section scanned against machine energy, since it is the largest channel. Then the partial widths from the individual channels, and the forward–backward and polarization asymmetries. The beam energy itself was calibrated by resonant depolarization, reaching Δ√s = ±2 MeV, sensitive enough that the fitted energy tracked the tides in Lake Geneva and the passage of the TGV to Paris.

The result

Four million

Z

decays per experiment, giving

$M_Z = 91.1875 \pm 0.0021\;\text{GeV}\;(23\;\text{ppm}), \qquad \Gamma_Z = 2.4952 \pm 0.0023\;\text{GeV}$

$\sigma^0_{\text{had}} = 41.540 \pm 0.037\;\text{nb}, \qquad \Gamma_\ell = 83.984 \pm 0.086\;\text{MeV}$

and, from the invisible width, $N_\nu = 2.984 \pm 0.008$ .

What it proved

That there are three light neutrino species and no more — and hence, if families are universal, three families. That the

Z

couples identically to the three charged leptons, to 2 ‰: universality for the neutral current, where §7.8 had tested it only for the charged one.

And it turned the electroweak theory into a precision instrument. Every number above became an input to §9.20’s global fit, and the loop corrections they exposed predicted the top mass before it was found.

What LEP actually measured

Bettini Eqs. (9.79)–(9.88), from the combined LEP and SLD fit at decommissioning. Compare the last column with §9.5's tree-level predictions and §9.7's 1983 values.

quantity↕	measured↕	precision↕	compare↕
$M_Z$	91.1875 ± 0.0021 GeV	23 ppm	UA1 got 93 ± 3 — a factor 1400 better
$\Gamma_Z$	2.4952 ± 0.0023 GeV	0.09 %	tree level predicted 2.42 — 3 % low, and the gap is the corrections
$\sigma^0_{\text{had}}$	41.540 ± 0.037 nb	0.09 %	this is $12\pi\Gamma_e\Gamma_h/M_Z^2\Gamma_Z^2$ by definition, corrected to the pole
$\Gamma_\ell$	83.984 ± 0.086 MeV	0.1 %	tree level 83. The three leptonic widths agree to 2 ‰ — universality, now for the neutral current
$\Gamma_h$	1744.4 ± 2.0 MeV	0.1 %	tree level 1670, 4 % low
$R_c = \Gamma_c/\Gamma_h$	0.1721 ± 0.0030	1.7 %	separable only because charm has a measurable flight distance — Example 9.6
$R_b = \Gamma_b/\Gamma_h$	0.21629 ± 0.00036	0.17 %	the same, and better, because beauty lives longer
$N_\nu$	2.984 ± 0.008	0.3 %	three, and no more — for any neutrino below $M_Z/2$

The invisible width row is the one that counts particles: subtract the visible widths from the total and divide by a predicted number, and you have the number of light neutrino species .

Two other rows are worth a second look.

The leptonic widths agree to 2 ‰. §7.8 tested lepton universality in the charged current and §9.7 repeated it at 7 %. This is the same statement for the neutral current, at a precision two orders of magnitude better, and it is a separate fact: nothing forces the $Z$ to treat the families alike merely because the $W$ does.

$R_b$ and $R_c$ exist at all only because charm and beauty hadrons fly far enough to leave a displaced vertex. Example 9.6 makes the point: at 50 GeV a $D^0$ travels 3.3 mm and a $B^0$ 4.3 mm, and a silicon micro-strip detector resolves 10 μm. Compare §8.6, where an unboosted $B$ at the $\Upsilon(4S)$ travels 28 μm and two entire accelerator complexes were built asymmetric to fix it.

Erratum — Example 9.4’s conversion constant, and Example 9.5’s answer

Two arithmetic problems on p. 383, in adjacent worked examples.

Example 9.4 ends with

5.3\times10^{-6}\;\text{GeV}^{-2}\times \mathbf{339}\;[\mu\text{b}\cdot\text{GeV}^{2}] = 2.1\;\text{nb}

The conversion constant is $\hbar^2c^2 = \mathbf{389.4}$ μb·GeV², not 339 — and the stated answer requires the correct value: $5.3\times10^{-6}\times389.4 = 2.06$ nb, while 339 would give 1.80 nb. The chapter contradicts itself, since Eq. (9.58) on p. 369 uses 388 for the same quantity. The units are also printed as μb·GeV $^{-2}$ , which cannot be right dimensionally.

Example 9.5 states 40.2 nb, but its own printed inputs give something else:

\frac{12\pi}{91^2}\,\frac{84\times1690}{2450^2} = 1.077\times10^{-4}\;\text{GeV}^{-2} \;\Rightarrow\; \mathbf{41.9}\;\text{nb}

with the correct 389.4. The stated 40.2 is what you get using $\Gamma_Z = 2495$ MeV — the measured value — rather than the 2450 printed in the same expression and used correctly in Example 9.4 next door.

The irony is worth noting: the book’s slip makes its own estimate look worse than it is. The correct 41.9 nb sits within 1 % of the measured $\sigma^0 = 41.540$ nb, while the printed 40.2 is 3 % away.

All confirmed on the renders of PDF pp. 387 and 401.

Aside — “Δ√s = ±2 MeV, which is 40 ppm” needs a reading

Taken at face value the arithmetic does not work: $2/91188 = 22$ ppm, not 40.

Two readings give 40. Either the ±2 MeV is a full range of 4 MeV, giving $4/91188 = 44$ ppm; or — more likely, given how LEP’s energy calibration actually worked — the 2 MeV refers to the beam energy of 45.6 GeV, of which it is $2/45594 = 44$ ppm, and $\Delta\sqrt{s}$ is then 4 MeV.

The LEP beam energy was calibrated by resonant depolarization, which measures the spin precession frequency of the stored beam and reaches about 1 MeV on $E_{\text{beam}}$ . It is one of the finest measurements in accelerator physics — sensitive enough that the fitted energy tracked the tides in Lake Geneva and the passage of the TGV to Paris. Either way the site quotes the figure that follows from the numbers: 20–25 ppm on $M_Z$ itself, which is Eq. (9.79).

🔑 If you remember only three things

Sensitivity to a heavy particle depends on how its mass enters. A quadratic dependence let the top be predicted; a logarithmic one left the Higgs almost unconstrained.
A factor of a thousand changes what is being tested. At a few per cent you are testing existence; at twenty parts per million you are testing loops.
The Z’s total width is a headcount. Everything that couples to it contributes, including whatever no apparatus can see.

Where this goes next

§9.10 takes LEP above the $Z$ to $\sqrt s = 209$ GeV and watches $e^+e^-\to W^+W^-$ , a cross-section that diverges unless the $Z$ couples directly to the $W$ s — the last untested structural feature of the theory. Then §9.11 measures $M_W$ and $m_t$ at the Tevatron precisely enough to make the loop corrections of §9.8 into a prediction about the Higgs mass, which §9.12 finally explains.

✅ Check yourself — running, and the Z line shape

0/6 answered · 0 correct

1.The W-mass correction goes as G_F m_t² but only as ln M_H. What did that asymmetry decide?
2.Why does the slope of sin²θ_W(Q) change sign at Q ≈ M_W?
3.N_ν is extracted from the peak HEIGHT of the resonance rather than from its width. Why is that the better measurement?
4.σ⁰_had = 41.540 ± 0.037 nb, and 12πΓ_eΓ_h/(M_Z²Γ_Z²) with the measured widths gives 41.54 nb. Is that a striking confirmation?
5.Example 9.4 ends '5.3×10⁻⁶ GeV⁻² × 339 μb·GeV⁻² = 2.1 nb'. What is wrong?
6.At LEP a D⁰ of 50 GeV flies 3.3 mm and a B⁰ 4.3 mm. Why does that matter for this section's results?