In the rest frame of a parent of mass decaying into three bodies labelled with

PHYSICS · Subjective · 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

In the rest frame of a parent of mass

M

decaying into three bodies labelled

1,2,3

with

E_1+E_2+E_3=M

, the Dalitz variables are

m_{12}^2=(p_1+p_2)^2

and

m_{23}^2=(p_2+p_3)^2

. Show that

m_{12}^2

depends only on

E_3

and that

m_{23}^2

depends only on

E_1

, derive the constant Jacobian relating

\mathrm{d}E_1\,\mathrm{d}E_3

\mathrm{d}m_{12}^2\,\mathrm{d}m_{23}^2

, and evaluate it numerically in

\mathrm{MeV^2}

for

K^+\to\pi^+\pi^+\pi^-

, with

M_{K^+}=493.68\,\mathrm{MeV}

(label the two identical pions

1

and

2

, and the

\pi^-

as particle

3

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