Reference

Constants

Every number the engine computes with, and where each one comes from. These values are read live from the same module the cross sections use, so what you see here is what the calculations ran on — not a transcription of it. They are grouped by what kind of claim each one is: measured, fixed by definition, or mathematical.

Physical

Measured. Each carries an uncertainty and an adjustment year, and cites the evaluation it was taken from — a better experiment could move any of them.

$\alpha$ Fine-structure constant 1/α = 137.035999178(21)

0.0072973525643 dimensionless

THE COUPLING IS NOT A CONSTANT. α is a function of the energy at which you probe the charge — vacuum polarization screens the bare charge, so a harder collision sees a larger effective coupling (α ≈ 1/128 at the Z mass). The value here is the low-energy Thomson limit α(0), which is the right one at the leading order this course works to: the running is itself a loop effect. Part 3 derives it and turns α into α(μ).

P. Mohr, D. Newell, B. Taylor and E. Tiesinga (CODATA), "CODATA Recommended Values of the Fundamental Physical Constants: 2022," published via the NIST Reference on Constants, Units and Uncertainty. https://physics.nist.gov/cgi-bin/cuu/Value?alph

Why this source, and how it was checked

Look for

The "fine-structure constant" entry: 7.297 352 5643(11) × 10⁻³.

Why this source

CODATA is the evaluated world adjustment the PDG itself defers to for this quantity, and the value sits on a single citable page rather than inside a review.

How it was checked

Read directly from the NIST page for this quantity on 2026-07-27, which states its adjustment year as 2022. Every digit stored in constants.ts was compared against it. This caught a real error: the stored value had been 7.2973525693e-3 — the CODATA 2018 value (α⁻¹ = 137.035999084) — while the comment beside it claimed 2022. Corrected to the 2022 value, a shift of 7×10⁻¹⁰ relative that moves no quoted number.
$m_e$ Electron mass 0.510 998 950 69(16) MeV

0.00051099895069 GeV

Carried exactly through every cross section in this course rather than dropped. Its square is around 10⁻¹¹ of s at √s = 10 GeV, so it is negligible at collider energies but not near threshold — and keeping it means the massless limit is something the engine can be checked against rather than something it assumes.

P. Mohr, D. Newell, B. Taylor and E. Tiesinga (CODATA), "CODATA Recommended Values of the Fundamental Physical Constants: 2022," published via the NIST Reference on Constants, Units and Uncertainty. https://physics.nist.gov/cgi-bin/cuu/Value?mec2mev

Why this source, and how it was checked

Look for

The "electron mass energy equivalent in MeV" entry: 0.510 998 950 69(16) MeV.

Why this source

The PDG Lepton Summary Table quotes the same value, taken from CODATA — so this is the primary source rather than a restatement of one.

How it was checked

Read directly from the NIST page for this quantity on 2026-07-27, which states its adjustment year as 2022. Every digit stored in constants.ts was compared against it. The stored value had been truncated to 0.51099895 MeV; it now carries all of CODATA's digits, so the number on this page is the number the engine uses.
$m_\mu$ Muon mass 105.658 3755(23) MeV

0.1056583755 GeV

Sets the threshold of the first cross section this course computes: e⁺e⁻ → μ⁺μ⁻ cannot proceed below twice this mass, √s ≈ 211.3 MeV, and the cross section is exactly zero beneath it.

P. Mohr, D. Newell, B. Taylor and E. Tiesinga (CODATA), "CODATA Recommended Values of the Fundamental Physical Constants: 2022," published via the NIST Reference on Constants, Units and Uncertainty. https://physics.nist.gov/cgi-bin/cuu/Value?mmuc2mev

Why this source, and how it was checked

Look for

The "muon mass energy equivalent in MeV" entry: 105.658 3755(23) MeV.

Why this source

As for the electron mass — the PDG lepton listing reproduces the CODATA value.

How it was checked

Read directly from the NIST page for this quantity on 2026-07-27, which states its adjustment year as 2022. Every digit stored in constants.ts was compared against it. The stored value already matched to every digit.

Conversions

Exact, but not mathematics. Fixed by the SI definitions rather than by measurement, so no experiment will ever change them — though a redefinition of the units would.

$(\hbar c)^2$ Cross-section conversion 0.389 379 3722 GeV²·mbarn

389379.3722 nb·GeV²

EXACT, not measured. Since the 2019 SI redefinition, the Planck constant, the speed of light and the electronvolt are all fixed by definition, so ℏc = 197.326 980 459… MeV·fm carries no uncertainty and neither does its square. This is a unit conversion, not an experimental result — the only number on this page with no error bar that is still about the world rather than about mathematics.

P. Mohr, D. Newell, B. Taylor and E. Tiesinga (CODATA), "CODATA Recommended Values of the Fundamental Physical Constants: 2022," published via the NIST Reference on Constants, Units and Uncertainty. https://physics.nist.gov/cgi-bin/cuu/Value?hbcmevf

Why this source, and how it was checked

Look for

The "reduced Planck constant times c" entry: 197.326 980 4… MeV·fm, marked exact.

Why this source

Because ℏc is exact, this constant is better computed than quoted — and it is, in constants.test.ts, from the defining SI values of h, c and the electronvolt.

How it was checked

Recomputed from h = 6.626 070 15×10⁻³⁴ J·s, c = 299 792 458 m/s and 1 eV = 1.602 176 634×10⁻¹⁹ J, all exact by definition, giving 389379.3722. The stored value had been 389379.3721 — wrong in its last digit, which for an exactly computable quantity is simply a typo. The test recomputes it, so it cannot drift again.

Mathematical

Defined. Nothing could ever change them and there is no source to cite, so each is given by its definition and recomputed from it in the test suite.

$\gamma$ Euler–Mascheroni constant

0.5772156649015329 dimensionless

The constant of the ε-expansion. Γ(ε) = 1/ε − γ + O(ε), so γ appears beside the pole of every one-loop integral in Part 3 — and whether it is carried explicitly or absorbed into the scale μ̄ is precisely what distinguishes the MS and MS-bar schemes.

$\gamma = \lim_{n \to \infty}\left(\sum_{k=1}^{n} \frac{1}{k} - \ln n\right)$
$\zeta(3)$ Apéry's constant

1.2020569031595942 dimensionless

Reaches loop results through the third derivative of lnΓ: ψ″(1) = −2ζ(3), so it enters as soon as an ε-expansion is carried to third order. Irrational, as Apéry proved in 1978; whether ζ(5) is remains open.

$\zeta(3) = \sum_{k=1}^{\infty} \frac{1}{k^3}$
$\sqrt{\pi}$ Root pi

1.7724538509055159 dimensionless

The scale of every half-integer value of the gamma function, and so of the surface area of a sphere in d dimensions, which is where it enters a loop integral. Computed from Math.PI rather than transcribed, so there are no digits here to get wrong.

$\sqrt{\pi} = \Gamma(\tfrac{1}{2})$
$\ln 4\pi$ Log four pi

2.5310242469692907 dimensionless

Arrives with γ in the combination −γ + ln4π that dimensional regularization produces from (4πμ²/Δ)^ε. Named because that pairing is not a coincidence: absorbing the two together is what defines the MS-bar scheme. Computed, not transcribed.

$\ln 4\pi$