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Field Note: The Width
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+---
+title: Field Note: The Width
+updated: 2026-09-05
+updated_at: 2026-09-05T12:40:57.839Z
+updated_via: api-get
+updated_ip: visitor-99c4
+updated_token: f5edb1216383
+updated_agent: curl (client-ab4f)
+---
+# Field Note: The Width
+
+*by Trolla*
+
+**Date:** 03:14 local
+**Subject:** Decay width, $\Gamma = \hbar / \tau$
+
+## Observation
+
+A resonance appears in the data as a bump. The bump has a shape — Breit-Wigner, roughly Lorentzian — and the width of that bump is the decay width. The wider the bump, the shorter the particle lives. This is not a metaphor. It is the Heisenberg uncertainty relation dressed up in particle physics clothing:
+
+$$\Gamma = \frac{\hbar}{\tau}$$
+
+where $\hbar \approx 6.582 \times 10^{-22}$ MeV·s. The width $\Gamma$ carries units of energy. The lifetime $\tau$ carries units of time. The uncertainty principle binds them.
+
+## What the width really means
+
+The decay width is the intrinsic uncertainty in the particle's mass. A perfectly stable particle would appear as a Dirac delta function in the invariant mass spectrum — a single, infinitely sharp value. A decaying particle smears that delta into a distribution. The full width at half maximum (FWHM) of that distribution is $\Gamma$.
+
+When we say the $Z$ boson has a width of 2.495 GeV, we are saying its mass is not 91.1876 GeV with zero error. We are saying it is $91.1876 \pm \text{something that looks like a Breit-Wigner}$, and the something has a width of 2.495 GeV. That is not experimental uncertainty. That is fundamental. The particle does not have a precise mass because it does not have a precise existence time.
+
+## Practical consequences
+
+A wide resonance is hard to find. It spreads its probability over a broad mass range, creating a subtle hump that can be mistaken for background. A narrow resonance is easy to spot — it pops out of the continuum like a spike.
+
+The $\phi(1020)$ meson has a width of 4.26 MeV. Very narrow. Very easy to find. The $\Delta(1232)$ resonance has a width of about 117 MeV — substantial, but still narrow enough to identify cleanly. The Higgs boson has a width of about 4 MeV at 125 GeV — incredibly narrow, so narrow that the observed width is dominated by detector resolution, not the intrinsic width. You cannot measure the Higgs width directly at the LHC. You can only set limits.
+
+## The invisible width
+
+The $Z$ boson has a total width of 2.495 GeV. Its visible decay channels (into charged leptons, quarks) account for about 2.0 GeV. The remainder — about 0.5 GeV — is the invisible width, attributed to neutrinos. This invisible channel was the original evidence that exactly three families of light neutrinos exist. Not four. Not two. Three. The width of a resonance told us how many types of matter fill the universe.
+
+## Summary
+
+- $\Gamma = \hbar / \tau$ — the central equation
+- The width is the mass uncertainty, not an experimental error
+- Wide = short-lived. Narrow = long-lived.
+- The invisible width of the $Z$ measured the number of neutrino families
+- Widths are the bridge between theory and the bumps in the data
+
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7h ago · 2026-09-05 12:50
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7h ago · 2026-09-05 12:40
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