In natural units ( ) the quoted leading formula for muon-pair production in…

PHYSICS · Subjective · Fermi University Physics · Particle Physics & Quantum Fields · Particles, interactions and experiment · Particles, interactions and experiment

Particle Physics, Ch Decay rates, cross sections and phase space

Problem

In natural units (ℏ=c=1\hbar = c = 1) the quoted leading formula for muon-pair production in electron-positron collisions, well above the muon-pair threshold, is σ(e+e−→μ+μ−)=4πα23s,\sigma(e^+e^-\to\mu^+\mu^-) = \frac{4\pi\alpha^2}{3s}, where α=1/137.0\alpha = 1/137.0 is the fine-structure constant (Chapter 8) and s\sqrt{s} is the collision energy. (a) Show that the formula has the dimensions of an area. (b) Evaluate σ(e+e−→μ+μ−)\sigma(e^+e^-\to\mu^+\mu^-) in nb\mathrm{nb} at s=3.0 GeV\sqrt{s} = 3.0\,\mathrm{GeV}, using (ℏc)2=0.3894 GeV2 mb(\hbar c)^2 = 0.3894\,\mathrm{GeV^2\,mb}. (c) Treating the quarks as free once made, each in three colours, predict σ(e+e−→hadrons)\sigma(e^+e^-\to\text{hadrons}) at s=3.0 GeV\sqrt{s} = 3.0\,\mathrm{GeV}, where only uu, dd and ss pairs can be made, and at s=6.0 GeV\sqrt{s} = 6.0\,\mathrm{GeV}, where cc pairs can be made too but bb pairs cannot.

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