Q71P - University Physics with Modern Physics

Question

Determine the electric charge, baryon number, strangeness quantum number, and charm quantum number for the following quark combinations: \[\left( \mathbf{a} \right)\underline{\phantom{xxx}}\mathbf{uus},\underline{\phantom{xxx}}\left( \mathbf{b} \right)\underline{\phantom{xxx}}\mathbf{c\bar{s}},\underline{\phantom{xxx}}\left( \mathbf{c} \right)\underline{\phantom{xxx}}\mathbf{\bar{d}\bar{d}\bar{u}},\] and \[\left( \mathbf{d} \right)\underline{\phantom{xxx}}\mathbf{\bar{c}b}.\]

Step-by-Step Solution

Answer

1. The electric charge, baryon number, strangeness quantum number, and charm quantum number for $uus$ are $+1,\underline{\phantom{xxx}}+\text{1},\underline{\phantom{xxx}}-\text{1}$ and $0$ respectively. 

2. The electric charge, baryon number, strangeness quantum number, and charm quantum number for  $c\bar{s}$ are $+1,\underline{\phantom{xxx}}0,\underline{\phantom{xxx}}+1$ and $+1$ respectively

3. The electric charge, baryon number, strangeness quantum number, and charm quantum number for $\bar{d}\bar{d}\bar{u}$ are $+1,\underline{\phantom{xxx}}-1,\text{ 0}$ and $0$ respectively. 

4. The electric charge, baryon number, strangeness quantum number, and charm quantum number for   $\bar{c}b$ are $-1,\underline{\phantom{xxx}}0,\underline{\phantom{xxx}}0$ and $-1$ respectively. 

Step 1: Define Quark

Hadrons are composites of constituents with spin $-\frac{1}{2}$ fermions called quarks

Step 2: Given information

The table shows the properties of the original quarks

Symbol	$Q/e$	Baryon number$\left( B \right)$	Strangeness $\left( S \right)$	Charm
$u$	$\frac{2}{3}$	$\frac{1}{3}$	$0$	$0$
$d$	-\frac{1}{3}	$\frac{1}{3}$	$0$	$0$
$s$	-\frac{1}{3}	$\frac{1}{3}$	$-1$	$0$
$c$	$\frac{2}{3}$	$\frac{1}{3}$	$0$	$1$
$\bar{s}$	$\frac{1}{3}$	$-\frac{1}{3}$	$1$	$0$
$\bar{d}$	$\frac{1}{3}$	$-\frac{1}{3}$	$0$	$0$
$\bar{u}$	$-\frac{2}{3}$	$-\frac{1}{3}$	$0$	$0$
$\bar{c}$	$-\frac{2}{3}$	$-\frac{1}{3}$	$0$	$-1$
$b$	$-\frac{1}{3}$	$\frac{1}{3}$	$0$	$0$

Step 3: Determine $Q,\text{ B, S }\!\!\And\!\!\text{ C}$

a)

 For the quarks: $uus$

$\begin{align}

  & Q=\frac{2}{3}+\frac{2}{3}+\left( -\frac{1}{3} \right) \\ 

 & =+1 \\ 

 & B=\frac{1}{3}+\frac{1}{3}+\frac{1}{3} \\ 

 & =+1   

\end{align}$

$\begin{align}

  & S=0+0-1 \\ 

 & =-1 \\ 

 & C=0+0+0 \\ 

 & =0   

\end{align}$

Hence, the electric charge, baryon number, strangeness quantum number, and charm quantum number for $uus$ are $+1,\underline{\phantom{xxx}}+\text{1},\underline{\phantom{xxx}}-\text{1}$ and $0$ respectively.

 

b)

For the quarks: $c\bar{s}$

$\begin{align}

  & Q=\frac{2}{3}+\frac{1}{3} \\ 

 & =+1 \\ 

 & B=\frac{1}{3}-\frac{1}{3} \\ 

 & =0   

\end{align}$

$\begin{align}

  & S=0+1 \\ 

 & =+1 \\ 

& C=1+0 \\ 

 & =+1   

\end{align}$

 

Hence, the electric charge, baryon number, strangeness quantum number, and charm quantum number for  $c\bar{s}$ are $+1,\underline{\phantom{xxx}}0,\underline{\phantom{xxx}}+1$ and $+1$ respectively

Step 4: Determine $Q,\text{ B, S }\!\!\And\!\!\text{ C}$.

c)

For the quarks: $\bar{d}\bar{d}\bar{u}$

$\begin{align}

  & Q=\frac{1}{3}+\frac{1}{3}+\left( -\frac{2}{3} \right) \\ 

 & =0 \\ 

 & B=\left( -\frac{1}{3} \right)+\left[ -\frac{1}{3} \right]+\left[ -\frac{1}{3} \right] \\ 

 & =-1   

\end{align}$

$\begin{align}

  & S=0+0+0 \\ 

 & =0 \\ 

 & C=0+0+0 \\ 

 & =0   

\end{align}$

Hence, the electric charge, baryon number, strangeness quantum number, and charm quantum number for $\bar{d}\bar{d}\bar{u}$ are $+1,\underline{\phantom{xxx}}-1,\text{ 0}$ and $0$ respectively.

 

d)

 

For the quarks: $\bar{c}b$

$\begin{align}

  & Q=-\frac{2}{3}+\frac{1}{3} \\ 

& =-1 \\ 

 & B=\left( -\frac{1}{3} \right)+\frac{1}{3} \\ 

 & =0   

\end{align}$

$\begin{align}

  & S=0+0 \\ 

 & =0 \\ 

 & C=-1+0 \\ 

 & =-1   

\end{align}$

Hence, the electric charge, baryon number, strangeness quantum number, and charm quantum number for  $\bar{c}b$ are $-1,\underline{\phantom{xxx}}0,\underline{\phantom{xxx}}0$ and $-1$ respectively.