Strong and electromagnetic decays

The decays of hadrons containing the `ordinary' $\u$, $\d$ and $\mathrm{s}$ quarks into two or three particles are known, and branching ratios may be found in [PDG96]. (At least for the lowest-lying states; the four $L = 1$ meson multiplets are considerably less well known.) We normally assume that the momentum distributions are given by phase space. There are a few exceptions, where the phase space is weighted by a matrix-element expression, as follows.

In $\omega$ and $\phi$ decays to $\pi^+ \pi^- \pi^0$, a matrix element of the form

$$\vert{\cal M}\vert^2 \propto \vert \mathbf{p}_{\pi^+} \times \mathbf{p}_{\pi^-} \vert^2$$ (258)

is used, with the $\mathbf{p}_{\pi}$ the pion momenta in the rest frame of the decay. (Actually, what is coded is the somewhat more lengthy Lorentz invariant form of the expression above.)

Consider the decay chain $P_0 \to P_1 + V \to P_1 + P_2 + P_3$, with $P$ representing pseudoscalar mesons and $V$ a vector one. Here the decay angular distribution of the $V$ in its rest frame is

$$\vert{\cal M}\vert^2 \propto \cos^2 \theta_{02} ~,$$ (259)

where $\theta_{02}$ is the angle between $P_0$ and $P_2$. The classical example is $\mathrm{D}\to \mathrm{K}^* \pi \to \mathrm{K}\pi \pi$. If the $P_1$ is replaced by a $\gamma$, the angular distribution in the $V$ decay is instead $\propto \sin^2 \theta_{02}$.

In Dalitz decays, $\pi^0$ or $\eta \to \mathrm{e}^+\mathrm{e}^- \gamma$, the mass $m^*$ of the $\mathrm{e}^+\mathrm{e}^-$ pair is selected according to

$${\cal P}(m^{*2}) \, \d m^{*2} \propto \frac{\d m^{*2}}{m^{*2... ...{1}{ (m_{\rho}^2 - m^{*2})^2 + m_{\rho}^2 \Gamma_{\rho}^2 } ~.$$ (260)

The last factor, the VMD-inspired $\rho^0$ propagator, is negligible for $\pi^0$ decay. Once the $m^*$ has been selected, the angular distribution of the $\mathrm{e}^+\mathrm{e}^-$ pair is given by

$$\vert{\cal M}\vert^2 \propto (m^{*2} - 2 m_{\mathrm{e}}^2) \... ...\mathrm{e}^+})^2 + (p_{\gamma} p_{\mathrm{e}^-})^2 \right\} ~.$$ (261)

Also a number of simple decays involving resonances of heavier hadrons, e.g. $\Sigma_{\c }^0 \to \Lambda_{\c }^+ \pi^-$ or $\mathrm{B}^{*-} \to \mathrm{B}^- \gamma$ are treated in the same way as the other two-particle decays.