A beam of mesons with kinetic energy strikes protons at rest

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Quantum Mechanics · Particle Physics & Quantum Fields

Symmetry and Flavour, Ch Isospin as a group: SU(2), Clebsch-Gordan ratios and G-parity

Problem

A beam of

\pi^+

mesons with kinetic energy

T_\pi=195\,\mathrm{MeV}

strikes protons at rest, forming the

\Delta(1232)

resonance (

J=\tfrac32

). Take

m_p=938.3\,\mathrm{MeV}

m_\pi=139.6\,\mathrm{MeV}

. ((a)) Using the relativistic invariant

s=m_p^2+m_\pi^2+2m_pE_\pi

and the two-body centre-of-mass momentum formula, find the pion centre-of-mass momentum

k^*

at this beam energy. ((b)) Using the Breit-Wigner peak cross-section formula

\sigma_{\max}=\dfrac{2J+1}{(2s_\pi+1)(2s_N+1)}\dfrac{4\pi}{k^{*2}}\,B_{\rm in}B_{\rm out}

and the fact that

\pi^+p

is a pure isospin-

\tfrac32

state (

B_{\rm in}=B_{\rm out}=1

), predict

\sigma_{\max}(\pi^+p\to\pi^+p)

. ((c)) Using the isospin Clebsch-Gordan decomposition of

\pi^-p

and

\pi^0n

into

I=\tfrac32

and

I=\tfrac12

, predict

\sigma_{\max}(\pi^-p\to\pi^0n)

at the same beam energy.