Standard Model and Beyond · Problem 2.24

Lie Groups, Lie Algebras, and Symmetries

Physics — Particle Physics · mott-scattering-coulomb

Problem

(a) Consider the Mott scattering process in which an electron of momentum p=βEp = \beta E scatters from a static Coulomb potential of charge ZeZe, Aμ(x)=Ze4π∣x⃗∣(1,0,0,0).A^\mu(x) = \frac{Ze}{4\pi|\vec{x}|}(1,0,0,0). Show that the unpolarized (Mott) cross section for scattering angle θ\theta is dσˉdcos⁡θ=(Zα)2π(1−β2sin⁡2θ2)2β2p2sin⁡4θ2→β≪1Z2α2π2β2p2sin⁡4θ2.\frac{d\bar{\sigma}}{d\cos\theta} = \frac{(Z\alpha)^2\pi(1-\beta^2\sin^2\frac{\theta}{2})}{2\beta^2 p^2\sin^4\frac{\theta}{2}} \xrightarrow{\beta\ll 1} \frac{Z^2\alpha^2\pi}{2\beta^2 p^2\sin^4\frac{\theta}{2}}. The last formula is the Rutherford cross section. (b) Suppose the Coulomb potential for an electron in a nuclear field transformed as a scalar rather than as the time component of a four-vector, i.e., HI=−eψˉ(x)ψ(x)ϕ(x),ϕ(x)=Ze4π∣x⃗∣.\mathcal{H}_I = -e\bar{\psi}(x)\psi(x)\phi(x), \qquad \phi(x) = \frac{Ze}{4\pi|\vec{x}|}. Calculate the unpolarized differential cross section, and compare it with the Mott formula.

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