For a two-to-two process the centre-of-mass differential cross section is d d ^* = |…

PHYSICS · Multi-part · Fermi University Physics · Particle Physics & Quantum Fields · Particles, interactions and experiment · Particles, interactions and experiment

Symmetry and Flavour, Ch From amplitude to rate: Lorentz-invariant phase space, widths and cross sections

Problem

For a two-to-two process the centre-of-mass differential cross section is dσdΩ∗=∣M∣2‾ pf∗64π2s pi∗.\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega^*}=\frac{\overline{|\mathcal M|^2}\,p_f^*}{64\pi^2 s\,p_i^*}. For elastic scattering (the same two particles, masses m1,m2m_1,m_2, in the initial and final state) pi∗=pf∗≡p∗p_i^*=p_f^*\equiv p^* and E1∗=E3∗E_1^*=E_3^*. ((a)) Using the Mandelstam invariant t=(p1−p3)2t=(p_1-p_3)^2, show that for elastic scattering t=−2p∗2(1−cos⁡θ∗)t=-2p^{*2}(1-\cos\theta^*), and use this to derive dσdt=∣M∣2‾64πs p∗2.\frac{\mathrm{d}\sigma}{\mathrm{d}t}=\frac{\overline{|\mathcal M|^2}}{64\pi s\,p^{*2}}. ((b)) A beam of π−\pi^- mesons scatters elastically off protons at rest at s=1.90 GeV\sqrt s=1.90\,\mathrm{GeV}, where the centre-of-mass momentum is p∗=710 MeVp^*=710\,\mathrm{MeV}. Taking a constant, angle-independent squared amplitude ∣M∣2‾=40\overline{|\mathcal M|^2}=40 (dimensionless, as appropriate for a 2→22\to2 process) as a stand-in for the true πp\pi p amplitude, evaluate dσ/dt\mathrm{d}\sigma/\mathrm{d}t at t=0t=0 in mb GeV−2\mathrm{mb\,GeV^{-2}}, using (ℏc)2=0.3894 mb GeV2(\hbar c)^2=0.3894\,\mathrm{mb\,GeV^2}.

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