For ( , , ), label the as particle 1 and the two as particles 2 and 3, and use the…

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

Symmetry and Flavour, Ch Four-vectors at work: boosts, invariants and the shape of a final state

Problem

For

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

(

M = 1869.66\,\mathrm{MeV}

m_K = 493.68\,\mathrm{MeV}

m_\pi = 139.57\,\mathrm{MeV}

), label the

K^-

as particle 1 and the two

\pi^+

as particles 2 and 3, and use the Dalitz variables

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

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

. ((a)) Starting from the energies

E_2^*

and

E_3^*

of particles 2 and 3 in the rest frame of the

12

pair, derive the general expressions for the maximum and minimum of

m_{23}^2

at fixed

m_{12}^2

. ((b)) Evaluate these limits at

m_{12} = 1.000\,\mathrm{GeV}

and give the corresponding range of

m_{23}

(in

\mathrm{GeV}

, to three significant figures).

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