Weak decays of bottom hadrons

Some exclusive branching ratios now are known for $\mathrm{B}$ decays. In this version, the $\mathrm{B}^0$, $\mathrm{B}^+$, $\mathrm{B}_{\mathrm{s}}^0$ and $\Lambda_{\b }^0$ therefore appear in a similar vein to the one outlined above for $\mathrm{D}_{\mathrm{s}}^+$ and $\Lambda_{\c }^+$ above. That is, all leptonic channels and all hadronic two-body decay channels are explicitly listed, while hadronic channels with three or more particles are only given in terms of a quark content. The $\mathrm{B}_{\c }$ is exceptional, in that either the bottom or the charm quark may decay first, and in that annihilation graphs may be non-negligible. Leptonic and semileptonic channels are here given in full, while hadronic channels are only listed in terms of a quark content, with a relative composition as given in [Lus91]. No separate branching ratios are set for any of the other weakly decaying bottom hadrons, but instead a pure `spectator quark' model is assumed, where the decay of the $\b$ quark is the same in all hadrons and the only difference in final flavour content comes from the spectator quark. Compared to the charm decays, the weak decay matrix elements are given somewhat larger importance in the hadronic decay channels.

In semileptonic decays $\b\to \c\ell^- \overline{\nu}_{\ell}$ the $\c$ quark is combined with the spectator antiquark or diquark to form one single hadron. This hadron may be either a pseudoscalar, a vector or a higher resonance (tensor etc.). The relative fraction of the higher resonances has been picked to be about 30%, in order to give a leptonic spectrum in reasonable experiment with data. (This only applies to the main particles $\mathrm{B}^0$, $\mathrm{B}^+$, $\mathrm{B}_{\mathrm{s}}^0$ and $\Lambda_{\b }^0$; for the rest the choice is according to the standard composition in the fragmentation.) The overall process is therefore $H \to h \ell^- \overline{\nu}_{\ell}$, where $H$ is a bottom antimeson or a bottom baryon (remember that $\overline{\mathrm{B}}$ is the one that contains a $\b$ quark), and the matrix element used to distribute momenta is

$$\vert{\cal M}\vert^2 = (p_H p_{\nu}) (p_{\ell} p_h) ~.$$ (278)

Again decay product masses have been neglected in the matrix element, but in the branching ratios the $\tau^- \overline{\nu}_{\tau}$ channel has been reduced in rate, compared with $\mathrm{e}^- \overline{\nu}_{\mathrm{e}}$ and $\mu^- \overline{\nu}_{\mu}$ ones, according to the expected mass effects. No CKM-suppressed decays $\b\to \u\ell^- \overline{\nu}_{\ell}$ are currently included.

In most multi-body hadronic decays, e.g. $\b\to \c\d\overline{\mathrm{u}}$, the $\c$ quark is again combined with the spectator flavour to form one single hadron, and thereafter the hadron and the two quark momenta are distributed according to the same matrix element as above, with $\ell^- \leftrightarrow \d$ and $\overline{\nu}_{\ell} \leftrightarrow \overline{\mathrm{u}}$. The invariant mass of the two quarks is calculated next. If this mass is so low that two hadrons cannot be formed from the system, the two quarks are combined into one single hadron. Else the same kind of approach as in hadronic charm decays is adopted, wherein a multiplicity is selected, a number of hadrons are formed and thereafter momenta are distributed according to phase space. The difference is that here the charm decay product is distributed according to the $V-A$ matrix element, and only the rest of the system is assumed isotropic in its rest frame, while in charm decays all hadrons are distributed isotropically.

Note that the $\c$ quark and the spectator are assumed to form one colour singlet and the $\d\overline{\mathrm{u}}$ another, separate one. It is thus assumed that the original colour assignments of the basic hard process are better retained than in charm decays. However, sometimes this will not be true, and with about 20% probability the colour assignment is flipped around so that $\c\overline{\mathrm{u}}$ forms one singlet. (In the program, this is achieved by changing the order in which decay products are given.) In particular, the decay $\b\to \c\mathrm{s}\overline{\mathrm{c}}$ is allowed to give a $\c\overline{\mathrm{c}}$ colour-singlet state part of the time, and this state may collapse to a single $\mathrm{J}/\psi$. Two-body decays of this type are explicitly listed for $\mathrm{B}^0$, $\mathrm{B}^+$, $\mathrm{B}_{\mathrm{s}}^0$ and $\Lambda_{\b }^0$; while other $\mathrm{J}/\psi$ production channels appear from the flavour content specification.

The $\mathrm{B}^0$- $\overline{\mathrm{B}}^0$ and $\mathrm{B}_{\mathrm{s}}^0$- $\overline{\mathrm{B}}_{\mathrm{s}}^0$ systems mix before decay. This is optionally included. With a probability

$${\cal P}_{\mathrm{flip}} = \sin^2 \left( \frac{x \, \tau} { 2\, \langle \tau \rangle} \right)$$ (279)

a $\mathrm{B}$ is therefore allowed to decay like a $\overline{\mathrm{B}}$, and vice versa. The mixing parameters are by default $x_{\d } = 0.7$ in the $\mathrm{B}^0$- $\overline{\mathrm{B}}^0$ system and $x_{\mathrm{s}} = 10$ in the $\mathrm{B}_{\mathrm{s}}^0$- $\overline{\mathrm{B}}_{\mathrm{s}}^0$ one.

In the past, the generic $\mathrm{B}$ meson and baryon decay properties were stored for `particle' 85, now obsolete but not yet removed. This particle contains a description of the free $\b$ quark decay, with an instruction to find the spectator flavour according to the particle code of the actual decaying hadron.