Measurement Units; Collisions and Decays · Elementary Particle Physics

An experiment produces counts. Everything else on this page — cross-sections, widths, branching ratios — is what those counts become after dividing by something that had to be measured separately.

🎯 Why this matters

The denominator is where the work is. A cross-section is only ever as good as the luminosity it was divided by, so one systematic error in the beam measurement moves every number the experiment publishes.

So far everything has been geometry. This section turns it into numbers an experiment reports: how many events per second, out of how much beam, for how long.

§1.6 Natural units, in practice

The reference page established the convention. Here is what it costs you in day-to-day arithmetic. Keep the second as the unit of time, choose the unit of length so that $c = 1$ , and the unit of mass so that $\hbar = 1$ . Then

[L] = [T], \qquad [M] = [E] = [P] = [L^{-1}] = [T^{-1}],

and there is one dimension left in all of physics. Three numbers are worth memorising:

The three constants to know by heart (Eqs. 1.53–1.55), and the conversions they generate

Quantity	Value↕	What it is for
$\hbar$		turns a width into a lifetime and back
$c$	3 × 10²³ fm s⁻¹	turns a time into a distance — how far light goes in a nanosecond (30 cm)
$\hbar c$		turns an energy into a length; "200 MeV fm" is close enough for order-of-magnitude work
$1\ \text{MeV}$	1.52 × 10²¹ s⁻¹ · 1 MeV⁻¹ = 197 fm	the two directions of ħc, written out
$1\ \text{ps}^{-1}$	0.65 meV	a picosecond lifetime is a sub-meV width — which is why b-hadron widths are never quoted
$1\ \text{m}$	5.07 × 10⁶ eV⁻¹	laboratory scale, in natural units

⚖️ Natural-units converter

= 1 GeV

eV1.00000 × 10⁹
keV1.00000 × 10⁶
MeV1000
GeV1
TeV0.001
J1.60218 × 10⁻¹⁰
kg1.78266 × 10⁻²⁷
K1.16045 × 10¹³

ħ = c = 1 makes E, m and p one dimension. Back to SI: m = E/c², T = E/k.

The coupling that has no units

\htmlClass{t-a}{\alpha} = \frac{\htmlClass{t-q}{q_e^2}}{4\pi\htmlClass{t-e}{\varepsilon_0}\,\htmlClass{t-h}{\hbar c}} \approx \frac{1}{137}, \qquad \frac{q_e^2}{4\pi\varepsilon_0} = \alpha\hbar c \approx 2.3\times10^{-28}\ \text{J m}

(1.56–1.57)

The electron charge, made dimensionless by dividing out ħc. This is the number that measures how strongly light couples to matter.

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.

💡 What this really says — the strength of a force is a pure number or it is nothing

“How strong is electromagnetism?” has no answer in coulombs, because coulombs are a human choice. It has an answer as a pure number: about 1/137. Every QED amplitude comes with one factor of

\sqrt{\alpha}

per vertex, so a two-vertex process is suppressed by α, a four-vertex one by α², and the perturbation series converges beautifully. Chapter 6 meets the opposite situation:

\alpha_s \approx 0.12

at the Z mass and of order 1 at low energy, where the same series stops converging and the whole method fails.

🔢 Worked example — width and lifetime are one measurement (Example 1.7)

The

\pi^0

lives

\tau = 8.4\times10^{-17}

s. Its width:

\Gamma = \frac{\hbar}{\tau} = \frac{6.6\times10^{-16}\ \text{eV s}}{8.4\times10^{-17}\ \text{s}} = 7.8\ \text{eV}.

The $\eta$ has a measured width $\Gamma = 1.3$ keV. Its lifetime:

\tau = \frac{\hbar}{\Gamma} = \frac{6.6\times10^{-16}\ \text{eV s}}{1300\ \text{eV}} = 5.1\times10^{-19}\ \text{s}.

Notice which one was measured in each case. The π⁰’s 8 eV width is far below any spectrometer’s resolution, so you time it instead. The η’s 5×10⁻¹⁹ s is far below any clock, so you measure the width instead. One quantity, two instruments — you pick whichever side of $\Gamma\tau = \hbar$ your apparatus can reach.

⚠️ Two errata on this page of the book

p. 17 lists the seven fixed SI constants. Two are misprinted:

the elementary charge is given as $1.626\,176\,634\times10^{-19}$ C; the defined value is $1.\mathbf{602}\,176\,634\times10^{-19}$ C — as the book’s own Appendix 2 prints, and as Example 1.10 on p. 32 uses. The wrong digits look copied from the Planck constant on the line above;
the luminous efficacy $K_\text{cd}$ is given as 633 lm/W; the SI defines 683 lm/W.

Neither affects any physics in the book — the candela never appears again — but they are in a list of defined constants, where a typo is a contradiction rather than an imprecision.

§1.7 What an experiment actually counts

Two kinds of process, and both are computed the same way. A collision $a+b \to$ anything, where each distinct final state is a channel with its own partial cross-section, and the sum over channels is the total. And a decay, where each channel has a partial width $\Gamma_i$ , the sum is the total width $\Gamma$ , and

\tau = \frac{1}{\Gamma}, \qquad \text{BR}_i = \frac{\Gamma_i}{\Gamma}.

⚙️ Engineer’s bridge — branching ratios are a probability distribution

The branching ratios of a particle sum to exactly 1, because the particle must do something. So the decay table of any unstable particle is a normalised probability distribution over outcomes, and the total width is a rate constant — the same object as the λ in an exponential decay law, or 1/RC.

That is not an analogy: $N(t) = N_0e^{-\Gamma t}$ is literally the same equation, and $\Gamma$ is literally the reciprocal time constant. The only unfamiliar part is that particle physicists quote the rate constant in MeV, because $\Gamma\tau = \hbar$ lets them.

Where it breaks: the reciprocal-time-constant picture assumes a clean exponential, and that assumption is a property of an isolated, narrow unstable state rather than a general fact. A broad resonance is not well described by “decays with time constant τ” at all — the ρ of §4.1 has a width comparable to its own mass, so it is a line shape rather than a lifetime, and it never propagates far enough for the exponential to be a meaningful description of anything. $\Gamma\tau = \hbar$ is a conversion; it stops being a description once Γ is not small compared with the mass.

Cross-section: the effective area

\htmlClass{t-R}{R_i} = \htmlClass{t-s}{\sigma}\,\htmlClass{t-N}{N_t}\,\htmlClass{t-F}{\Phi_b}

(1.59)

The definition of the cross-section: it is whatever number makes this equation true.

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.

⚙️ Engineer’s bridge — σ is a hit probability, ℒσ is a throughput

Picture each target centre as a disc of area σ. The chance a beam particle crossing a thin slab hits one is (number of discs) × (area each) ÷ (total area) — so σ is a hit probability per unit areal density of targets. That is why it has units of area even though nothing is geometrically that size: the neutrino cross-section of

10^{-47}

m² corresponds to no physical object.

Combine the target and beam factors into one number, the luminosity $\mathcal{L}$ , and you get the equation an accelerator physicist lives by:

\text{rate} = \mathcal{L}\,\sigma .

Throughput = offered load × probability of success. ℒ is the machine’s job; σ is nature’s; the product is your data rate. Everything about how an experiment is designed — target size, running time, trigger thresholds — follows from where its σ sits on the scale below.

Where it breaks: the effective-area picture is a geometric intuition and the quantity is not geometric. A cross-section can be far larger than the target — on a resonance it is set by $\lambda^2$ , the probe’s wavelength squared, which is why the neutron capture cross-sections of some nuclei exceed their physical area by orders of magnitude — and it depends on energy, so you cannot paint a fixed patch on the target and be done. Read σ as a rate coefficient with the units of area; read it as an area and every resonance becomes a paradox.

luminosity ℒ1e38 m⁻²s⁻¹ = 1e34 cm⁻²s⁻¹cross-section σ8e-30 m² = 0.08 b

σ = 0.08 b = 80 mb
rate = ℒσ = 8e8 events per second
= 2.525e16 per year of continuous running

Far more than any detector can record — this is why a trigger exists (§9.14).

rate at the current luminosity

Twenty-four decades of cross-section, from the total pp cross-section down to a neutrino's. Everything an experiment does — target size, running time, trigger design — is a response to where on this line its process sits.

🔢 Worked example — the luminosity of a fixed-target run (Example 1.9)

A beam of intensity

I = 10^{13}

s⁻¹ hits a liquid-hydrogen target (

\rho = 60

kg m⁻³) of length

l = 10

cm. Hydrogen has

A = 1

, so the number of target protons per cubic metre is

n_t = \rho N_A \times 10^{3} = 60 \times 6.022\times10^{23} \times 10^{3} = 3.61\times10^{28}\ \text{m}^{-3},

(the $10^3$ converts kilograms to grams, because $N_A$ counts per gram-mole) and

\mathcal{L} = I\,n_t\,l = 10^{13} \times 3.61\times10^{28} \times 0.1 = 3.6\times10^{40}\ \text{m}^{-2}\text{s}^{-1}.

That is 10³⁶ cm⁻²s⁻¹ — a hundred times more than the LHC. Fixed targets win on luminosity by a wide margin and lose on energy by a wider one, which is the trade-off §1.4–1.5 quantified.

The beam gets used up

Each slab of target removes a constant fraction of the surviving beam, so the intensity falls exponentially:

I(z) = I_0\,e^{-n_t\sigma_\text{tot} z}, \qquad L_\text{abs} = \frac{1}{n_t\sigma_\text{tot}} .

⚙️ Engineer’s bridge

This is the same differential equation as an RC discharge, and as Beer–Lambert absorption in optics, for the same structural reason: a constant removal probability per unit of the independent variable.

L_\text{abs}

is the space-domain time constant. Later in this chapter two more lengths appear with identical mathematics and different physics — the radiation length

X_0

for electrons and the collision length

\lambda_0

for hadrons (§1.11). Same exponential, three different removal mechanisms.

Where it breaks: the shared exponential hides that the three lengths are defined by different removal mechanisms, and they are not interchangeable. A collision length removes a hadron from the beam by any interaction; a radiation length is an energy-loss scale for an electron that survives; an interaction length in a calorimeter is about starting a shower. Quoting “the attenuation length” without saying which process removes what is the commonest way to be wrong by a factor of a few in a shielding estimate.

target density ρ60 kg m⁻³mass number A1σ per centre40 mb

n_t = ρN_A×10³/A = 3.613e28 m⁻³
L_abs = 1/(n_t σ) = 6.919 m
a 10 cm target removes 1.43 % of the beam

I(z)/I₀ = exp(−z/L_abs)

The same exponential you know from RC discharge and from Beer–Lambert absorption — and for the same reason: a constant removal probability per unit length. L_abs is its time constant.

Where cross-sections come from

\htmlClass{t-W}{W} = 2\pi\,\htmlClass{t-M}{\left|M_{fi}\right|^2}\,\htmlClass{t-r}{\rho(E)}, \qquad \htmlClass{t-M}{M_{fi}} = \langle f\,|\,H_\text{int}\,|\,i\rangle

(1.58, 1.68)

Fermi's golden rule. Every rate in this book is one of these.

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.

💡 What this really says — every rate is dynamics × kinematics

The golden rule factorises the problem into two independent pieces:

$|M|^2$ : how strongly the interaction couples them. Hard to compute, requires a theory.
$\rho(E)$ : how much room the final state has. Easy to compute, requires only masses and conservation laws.

An enormous amount of physics is deciding which of the two is doing the work. A decay can be slow because the coupling is feeble (weak decays) or because the phase space is tiny (the neutron’s 878 s lifetime, where only 1.3 MeV is released — see the mass-defect discussion in §1.4–1.5). Distinguishing those two explanations is a recurring move in Chapters 2, 4 and 7.

For the important special case of a two-body decay, the phase-space integral can be done once and for all, giving

\Gamma(a \to c\,d) = \frac{p_f}{8\pi m^2}\,\left\langle\left|M_{a,cd}\right|^2\right\rangle ,

where $p_f$ is the momentum of either product in the parent’s rest frame and the angle brackets denote the average of $|M|^2$ over directions. The whole kinematic dependence of a two-body decay rate is that single factor $p_f$ — which is why a decay switches on so sharply just above threshold, and why Chapter 4’s resonances have the shapes they do.

Reproduce it

import numpy as np
NA, hbar = 6.02214076e23, 6.582119569e-16      # hbar in eV s

print(f"pi0: Gamma = hbar/tau = {hbar/8.4e-17:.2f} eV        (book: 8 eV)")
print(f"eta: tau = hbar/Gamma = {hbar/1300:.2e} s     (book: 5e-19 s)")
print(f"proton Compton wavelength = {2*np.pi/938.272*197.3269804:.4f} fm")

I, rho, l = 1e13, 60.0, 0.10                   # Example 1.9, liquid hydrogen
nt = rho * 1e3 * NA / 1.0                      # A = 1;  1e3 converts kg -> g
print(f"Example 1.9: n_t = {nt:.3e} m^-3, L = {I*nt*l:.3e} m^-2 s^-1   (book: 3.6e40)")

sig = 40e-31                                   # 40 mb, in m^2
print(f"L_abs in liquid H2 (sigma = 40 mb) = {1/(nt*sig):.3f} m; "
      f"10 cm removes {(1-np.exp(-0.1*nt*sig))*100:.2f}%")

Llhc = 1e38                                    # 1e34 cm^-2 s^-1
print(f"LHC: rate = {Llhc*80e-31:.3e} /s -> {Llhc*80e-31/4e7:.1f} interactions "
      f"per 40 MHz crossing")
print(f"LHC Higgs: {Llhc*50e-40:.3f} /s = {Llhc*50e-40*3.156e7:.3e} per year of "
      f"continuous running")
print(f"alpha*hbar*c = {197.3269804e-9*1.602176634e-19/137.035999166:.3e} J m   "
      f"(book: 2.3e-28)")

prints

pi0: Gamma = hbar/tau = 7.84 eV        (book: 8 eV)
eta: tau = hbar/Gamma = 5.06e-19 s     (book: 5e-19 s)
proton Compton wavelength = 1.3214 fm
Example 1.9: n_t = 3.613e+28 m^-3, L = 3.613e+40 m^-2 s^-1   (book: 3.6e40)
L_abs in liquid H2 (sigma = 40 mb) = 6.919 m; 10 cm removes 1.43%
LHC: rate = 8.000e+08 /s -> 20.0 interactions per 40 MHz crossing
LHC Higgs: 0.500 /s = 1.578e+07 per year of continuous running
alpha*hbar*c = 2.307e-28 J m   (book: 2.3e-28)

Erratum — two of the seven defining constants on p. 17

The list of the seven constants the 2019 SI fixes by definition contains two wrong values. In a list of defined constants a typo is not an inaccuracy, it is a contradiction.

The elementary charge is printed as $e = 1.\mathbf{626}\,176\,634\times10^{-19}$ C. It is $1.\mathbf{602}\,176\,634\times10^{-19}$ C — as the book’s own Appendix 2 (p. 495) prints and Example 1.10 (p. 32) uses. The slip is visible on the page: the line directly above gives the Planck constant as $6.\mathbf{626}\,070\,15\times10^{-34}$ J s, and the “626” has been carried down one line.
The luminous efficacy is printed as $K_{cd} = 633$ lm/W. It is 683 lm/W. This one never matters again — no candela appears anywhere else in the book — but it is in the same list.

Everything downstream in this book uses the correct charge, so no result is affected. Take the values from Appendix 2, not from this page.

🔑 If you remember only three things

Natural units are a statement, not a shortcut. Setting ħ = c = 1 asserts that these are conversion factors between human units rather than facts about the world waiting to be measured.
A width and a lifetime are one measurement. Γ = ħ/τ means a resonance’s shape and a clock’s reading are the same number, reached from two directions.
A lifetime alone does not measure a coupling. The golden rule’s two factors can pull opposite ways, so phase space must be divided out before a rate says anything about strength.

Where this goes next

§1.8 computes an actual cross-section — Rutherford’s — and shows that scattering is a Fourier transform.
§1.11 reuses the exponential-attenuation mathematics twice more, for radiation length and collision length.
§1.12 builds the machines that deliver the luminosity, and §9.14 meets the consequence: 40 MHz of crossings against a recordable few hundred hertz is the trigger problem, and it is set entirely by the numbers in the widget above.

✅ Check yourself — units, cross-sections and rates

0/5 answered · 0 correct

1.The π⁰ width is quoted as a lifetime (8.4×10⁻¹⁷ s) but the η's is quoted as a width (1.3 keV). Why the different conventions for two similar mesons?
2.Why is the fine-structure constant, and not the electron charge, the honest measure of how strong electromagnetism is?
3.The LHC runs at $\mathcal{L} = 10^{34}$ cm⁻²s⁻¹ with an inelastic pp cross-section of about 80 mb, and crossings happen at 40 MHz. What follows?
4.Fermi's golden rule splits a rate into $|M_{fi}|^2$ and $\rho(E)$ . What is the practical value of that split?
5.A 10 cm liquid-hydrogen target removes only about 1.4 % of the beam. What is the absorption length telling you, and what other quantities in this chapter share the mathematics?