Charmed hadron lifetimes

The lifetime of any particle is related to its mass by the Heisenberg uncertainty relation which states that

$$\Delta E \Delta t \ge \hbar \, .$$ (1)

This means that as the lifetime, $\tau$, of a particle becomes shorter, its energy (or mass) is less well determined. As particle physicists, we measure this uncertainty in mass as $\Gamma$, or the ``natural width,'' of a state which is proportional to $1/\tau$. For weakly decaying particles (which have relatively long lifetimes), $\Gamma$ is experimentally unmeasurable. However, for short-lived, strongly or electromagnetically decaying particles, the natural width is a physically observable effect. This is the case with the $J/\psi$ and also for the states as shown in sc_width. For each available decay mode or mechanism, the lifetime of a particle is decreased.

In the spectator quark model of weak decays, it is assumed that the non-decaying, or spectator, quarks do not interact with products of the virtual decay. If this were the case, the lifetimes of all the charmed hadrons would be identical. However, large differences between the charm lifetimes are seen. In order to explain these differences, additional effects which depend on the quark content must be considered. For Cabibbo favored charmed baryon decays there are three major decay diagrams which may contribute to lifetimes. These three contributions and a process that can only occur for charmed mesons are shown in intro:baryon_life.

 

Figure 1.4: Feynman diagrams for charm lifetime contributions. Each diagram represents a possible decay mode, depending on the initial quark content.

The spectator process in which there is no interaction between the spectator quarks and the produced quarks in the decay is shown schematically in intro:had_spec. This is commonly called an ``external'' decay diagram. It is also this mechanism at work in semi-leptonic decays where the virtual decays to a lepton-neutrino pair. This type of decay is available to all charmed particles. We denote its contribution to the lifetime as $\Gamma_{\text{ext}}$.

In intro:color_sup we show the diagram for an internal decay. In this case if there are two quarks of the same type in the final state, this can give rise to constructive or destructive interference due to Pauli suppression and the color charge (the produced quarks must have compatible colors with the spectator quarks). When the decay products of the virtual match the spectator quarks, the interference is destructive (denoted by $\Gamma^-_{\text{int}}$) and serves to increase the lifetime of the decaying particle. When there are spectator quarks, the quark arising from a quark decay interferes constructively with the spectator quarks, which is denoted by $\Gamma^+_{\text{int}}$ [19].

Exchange of a is possible in charmed baryons containing a quark and in charmed mesons containing a quark. This decay diagram is shown in intro:w_exc. We denote its contribution to the lifetime as $\Gamma_{\text{exc}}$. The presence of the exchange mechanism is believed to be the primary reason the charmed baryon lifetimes are typically shorter than the charmed meson lifetimes.

Finally, in the case of the , there is a Cabibbo favored mechanism by which the quark and quark can annihilate into a . This process in shown in intro:mes_ann and is denoted by $\Gamma_{\text{ann}}$. (For the , this process is Cabibbo suppressed.)

Summarizing the most important contributions for the charmed hadrons, we obtain

$$\begin{split} \Gamma(\dplus) & = \Gamma_{\text{ext}} + \Gamma... ..._{\text{ext}} + \tfrac{10}{3}\Gamma^+_{\text{int}} \end{split}$$ (2)

where the $\frac{10}{3}$ factor for the $\Omega_c^0$ arises since the final state contains three quarks which constructively interfere. A very good discussion of these contributions in the baryon case is given in theory:gublt. While this exercise shows conceptually why the lifetime is shorter than any of the meson lifetimes, an accurate calculation requires the inclusion of Cabibbo and doubly Cabibbo suppressed processes as well. intro:baryon_lifetimes shows a comparison of the charm lifetimes.

 

Figure 1.5: Lifetimes of the charmed hadrons [20].

intro:baryon_lifetimes