Standard Model and Beyond · Problem 2.8

Lie Groups, Lie Algebras, and Symmetries

Physics — Particle Physics · massive-scalar-electrodynamics-scattering

Problem

Consider π+(p⃗1)π−(p⃗2)→π+(p⃗3)π−(p⃗4)\pi^+(\vec{p}_1)\pi^-(\vec{p}_2) \to \pi^+(\vec{p}_3)\pi^-(\vec{p}_4) in massive scalar electrodynamics, i.e., with VI=gϕ†ϕA,V_I = g\phi^\dagger\phi A, where AA, the analog of the electromagnetic field, is a Hermitian spin-0 field with mass μ≠0\mu \neq 0. The analog of the charge is gg, which now has dimensions of mass. (a) Find expressions for the differential and total cross sections in the CM to lowest nontrivial order, in terms of ss, mm, μ\mu, gg, and cos⁡θ\cos\theta. (b) Define the dimensionless variable x=s/m2≥4x = s/m^2 \geq 4, and specialize to the values m=g=1m = g = 1 GeV, μ=0.5\mu = 0.5 GeV. Plot dσ/dcos⁡θd\sigma/d\cos\theta vs cos⁡θ\cos\theta in units of 1 fm2=10−26 cm21\,\text{fm}^2 = 10^{-26}\,\text{cm}^2 for x=4,4.2x = 4, 4.2, and 4.44.4. Use any plotting program. (c) Plot σ\sigma in units of fm2\text{fm}^2 versus xx for the same parameter values and the range 4≤x≤54 \leq x \leq 5.

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