Past exam of the particle physics course of the University of Oxford / 2017 / 2 / d / Solution

Use the explicit charged channels $D^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}$ and $K^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}$, labeling the two positive pions as 1,2 and the negative one as 3. With $m_{\pi }=0.139\mathrm{GeV}$, the opposite-charge pair mass squares have the rangesTheir exact correlated boundaries follow by taking the pair rest-frame angle to its endpoints. For general daughter masses, define $\Lambda (a,b,c)=a^2+b^2+c^2-2ab-2ac-2bc$. At fixed $s=s_{13}$,The charged-pion thresholds give the following numerical comparison:

The area is about $3150$ times smaller for the kaon. Including the $M_0^{-2}$ phase-space prefactor, its integrated labelled three-body phase space is about $220$ times smaller; neither comparison predicts a ratio of weak decay widths without the matrix elements.

A ${\rho }^0\to {\pi }^{+}{\pi }^{-}$ intermediate state produces horizontal or vertical bands near $s_{13}$ or $s_{23}=m_{\rho }^2\simeq (0.775\mathrm{GeV}{)}^2$ in the charm decay. Scalar pion-pair amplitudes, including broad low-mass $S$-wave contributions and accessible $f_0(980)$, can also populate the plot. The kaon pair mass cannot exceed $0.355\mathrm{GeV}$, so it cannot contain an on-shell ${\rho }^0$ or $f_0(980)$ band; smooth nonresonant and rescattering effects remain possible. Channels containing neutral pions use their supplied $0.135\mathrm{GeV}$ mass and shift these boundaries slightly without changing the conclusion. The figure shows kinematic domains and representative accessible bands, not an assumed measured intensity.