Chapter 32, Problem 31P

To determine

$\left(a\right)$

Show that $B^{-}$ meson and $b{u}^{-}$ quark combination is for all quantum numbers,

Answer to Problem 31P

Solution:

$B^{-}$ And $b{u}^{-}$ quark combination is consistent for all quantum numbers.

Explanation of Solution

Given info:

$B^{-}$ Meson $=b{u}^{-}$ quark.

Formula used:

In a reaction charge, Baryon number, lepton number and strangeness number are conserved.

Baryon number: proton and neutrons both have baryons.

$\begin{array}{l}\text{Lepton number Le}\\ {\overline{V}}_e-1\\ e^{+}-1\\ {\overline{V}}_{\mu }-1\\ {\mu }^{+}-1\\ {\overline{V}}_{\Sigma }-1\\ {\Sigma }^{+}-1\\ V_e\text{ 1}\\ {\text{e}}^{-}\text{ 1}\\ V_{\mu }\text{ 1}\\ {\mu }^{-}1\\ V_{\Sigma }1\\ {\Sigma }^{-}1\end{array}$

Strangeness number:

Strangeness number of quarks up, down and strange.

$\begin{array}{l}\text{ strangeness}\\ \text{U 0}\\ \text{d 0}\\ \text{s }-\text{1}\end{array}$

Calculation:

Charge conservation $:\text{-}e=\frac{1}{3}e-\left(\frac{2}{3}e\right)$

$-e=-e$

Baryon conservation:$\frac{1}{3}-\frac{1}{3}=0$

Strangeness conservation:$0+0=0$

Topness:$0+0=$ 0

Spin:

$\begin{array}{l}0=\frac{1}{2}-\frac{1}{2}\\ 0=0\end{array}$

Bottomness:

$\begin{array}{l}-1=-1+0\\ -1=-1\end{array}$

To determine

$\left(b\right)$

What are the quark combinations for $B^{+},B^0,{\overline{B}}^0?$

Answer to Problem 31P

Solution:

$B^{+}=\overline{b}u,B^0=b\overline{d},{\overline{B}}^0=\overline{b}d$ Are the quark combination for $B^{+}$, $B^0,{\overline{B}}^0$

Explanation of Solution

Given info:

$B^{+},B^0,{\overline{B}}^0$

Formula used:

$B^{+}$ Is the antiparticle of $B^{-}$

$B^0$ Is neutral and must have bottom quark.

${\overline{B}}^0$ Is the antiparticle of $B^0.$

Calculation:

$B^{+}$ Is the antiparticle of $B^{-}.$

Therefore, $B^{+}$$=\overline{b}u$

$B^0$ Is aneutral entity and must have a bottom quark.

$B^0=b\overline{d}$

$B^0$ Is the antiparticle to $B^0$

$B^0=\overline{b}d.$