• White Tripla physics T-shirt, front view (flat-lay mockup), view 1

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Tripla

The Anomaly That Counts Colours Physics

\[\pi^{0}\rightarrow\gamma\gamma: \quad \Gamma = \frac{\alpha^{2} m_\pi^{3}}{64\pi^{3} f_\pi^{2}} \approx 7.8\ \mathrm{eV} \quad (f_\pi \approx 92\ \mathrm{MeV},\ 3 \text{ colours})\]
$29.99 USD
$29.99 USD
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Tripla, a pion-sprite with three coloured crown gems, bursts into two photon wings.

The Physics

For massless fermions the axial current is conserved classically, but quantum effects break it: a triangle loop with one axial and two vector vertices spoils the conservation. This is the ABJ (chiral) anomaly, and it gives the neutral pion its fast decay into two photons. At leading order, with three quark colours and the up and down quark charges built in, the predicted width is about 7.8 eV.

π⁰ → γγ: Γ = α² m_π³ / (64 π³ f_π²) ≈ 7.8 eV (f_π ≈ 92 MeV, 3 colours)

  • Γ: decay width of π⁰ → γγ (eV)
  • α: fine-structure constant ≈ 1/137.036 (dimensionless)
  • m_π: neutral pion mass, 134.977 MeV
  • f_π: pion decay constant in the ~92 MeV convention (MeV)

PrimEx-II measured Γ = 7.80 ± 0.12 eV in 2020, confirming the anomaly prediction at the percent level. The anomaly was found by Adler (Phys. Rev. 177, 2426, 1969) and by Bell and Jackiw (Nuovo Cimento A 60, 47, 1969).

Honest caveat: The match with three colours is strong evidence for three quark colours, not a proof. Small corrections beyond the anomaly are still studied, and f_π ≈ 92 MeV here is the convention that goes with this formula (others quote about 130 MeV).

On the shirt

Front: Tripla, the pixel-art character, with the title. Back: the anomaly formula, the PrimEx-II measurement and the 1969 papers. Printed on a unisex tee.

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Details

Size & fitS–5XL

Unisex heavy cotton (Gildan 5000), classic fit.

SizeWidthLengthSleeve
S182815.1
M202916.5
L223018
XL243119.5
2XL263221
3XL283322.4
4XL303423.7
5XL323525

Ter vergelijking: maat M is 20 in (50.80 cm) breed, ongeveer 3.143 × 1034 Plancklengtes (lP = 1.616255 × 10−35 m). Een meetlint is sneller.

Measurements in inches, ±1.5 in tolerance.

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Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

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The physics

I count to three. A neutral pion decays into two photons, and how fast it does that depends on how many colours quarks come in.

In 1969 Stephen Adler, and John Bell with Roman Jackiw, found that a symmetry that holds classically fails once quantum effects are included: the axial anomaly. It fixes the pion's two-photon decay width, Γ = α²m_π³/(64π³f_π²), with the quark colour count built into the prefactor. With three colours it gives about 7.8 eV.

In 2020 the PrimEx-II experiment at Jefferson Lab measured 7.80 ± 0.12 eV. That agreement is strong evidence that quarks come in three colours, though not a proof on its own.

Equations

\[\pi^{0}\rightarrow\gamma\gamma: \quad \Gamma = \frac{\alpha^{2} m_\pi^{3}}{64\pi^{3} f_\pi^{2}} \approx 7.8\ \mathrm{eV} \quad (f_\pi \approx 92\ \mathrm{MeV},\ 3 \text{ colours})\]
\[\text{Measured: } \Gamma = 7.80 \pm 0.12\ \mathrm{eV} \quad (\text{PrimEx-II, 2020})\]
SymbolMeaningUnit
\(\pi^{0}\) neutral pion \(\mathrm{1}\)
\(\gamma\) photon \(\mathrm{1}\)
\(\Gamma\) decay width of the neutral pion to two photons \(\mathrm{eV}\)
\(\alpha\) fine-structure constant ≈ 1/137.036 \(\mathrm{1}\)
\(m_\pi\) neutral-pion mass, ≈ 135 MeV/c² \(\mathrm{eV}\)
\(f_\pi\) pion decay constant (≈ 92 MeV convention) \(\mathrm{eV}\)

Sources

  1. Adler (1969) Axial-Vector Vertex in Spinor Electrodynamics, Phys. Rev. 177, 2426
  2. Bell & Jackiw (1969) A PCAC puzzle: π0→γγ in the σ-model, Nuovo Cim. A 60, 47
  3. Larin et al. (PrimEx-II) (2020) Precision measurement of the neutral pion lifetime, Science 368, 506

Field notes

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