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%  C4 formulae and data - RCED 12/3/05
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\def\lambdabar{{\mathchar'26\mskip-10mu\lambda}}

\vskip0.5cm

\leftline{\bf C4 Particle Physics formulae and data}
\medskip

Unless otherwise indicated, the questions on this paper use natural
units with $\hbar=c=1$. The energy unit is GeV. 
\[
\begin{tabular}{ll}
Cross sections & $1\GeV^{-2} = 0.3894\mb$ \\
Length & $1\GeV^{-1} = 0.1973\fm$ \\
Time & $1\GeV^{-1} = 6.582\times10^{-25}\s$ \\
Fermi constant & $G_{\rm F} = 1.166\times10^{-5}\GeV^{-2}$
\end{tabular}
\]

\vskip0.5cm

\leftline{\bf Dirac (Dirac-Pauli representation) and Pauli matrices}
\medskip

\[
\gamma^0=\left(\begin{array}{rr} I & 0 \\ 0 & -I \end{array}\right),~~
\vgamma=\left(\begin{array}{rr}
0 & \vsigma \\ -\vsigma & 0 \end{array}\right),~~
\gamma^5=\left(\begin{array}{rr} 0 & I \\ I & 0 \end{array}\right)
\]
\[
\sigma_1=\left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right),~~
\sigma_2=\left(\begin{array}{rr} 0 & -\rm i \\ \rm i & 0 \end{array}\right),~~
\sigma_3=\left(\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right)
\]

\vskip0.5cm

\leftline{{\bf Rotation matrices}~~~
$\displaystyle
\langle j,m^\prime|\e^{-\i J_y\theta}|j,m\rangle=d^j_{m^\prime m}(\theta)$}

\[
d^{1/2}_{++}=d^{1/2}_{--}=\cos(\theta/2)\,;~~~
d^{1/2}_{-+}=-d^{1/2}_{+-}=\sin(\theta/2).
\]
\[
d^1_{11} = d^1_{-1-1}=(1+\cos\theta)/2\,;~~~
d^1_{1-1} = d^1_{-11}=(1-\cos\theta)/2\,; 
\]
\[ 
d^1_{00} = \cos\theta\,;~~~
d^1_{01} = -d^1_{10}=-d^1_{0-1}=d^1_{-10}=\sin\theta/\sqrt{2}\,.
\]

\vskip0.5cm

\leftline{{\bf Spherical harmonics}~~~
$\displaystyle Y^m_l(\theta,\phi)$}

\[
Y^0_0=\sqrt{1\over 4\pi}\, ;~~~Y^0_1=\sqrt{3\over 4\pi}\cos\theta\, ;~~
Y^{\pm 1}_1=\mp\sqrt{3\over 8\pi}\sin\theta\,\e^{\pm \i\phi}\,.
\]
\[
Y^0_2=\sqrt{5\over 16\pi}(3\cos^2\theta-1)\,;~~
Y^{\pm 1}_2=\mp\sqrt{15\over 8\pi}\sin\theta\cos\theta\,\e^{\pm \i\phi}\,;~~
Y^{\pm 2}_2=\sqrt{15\over 32\pi}\sin^2\theta\,\e^{\pm 2\i\phi}\,.
\]

\vskip0.5cm

\leftline{\bf CKM quark mixing matrix}

The mixing of the charge $-e/3$ quark mass eigenstates (d, s, b) is
expressed in a $3\times 3$ unitary matrix $V$:
\[
\left(\matrix{\dq^\prime \cr \sq^\prime \cr \bq^\prime\cr}\right)=
\left(\matrix{V_{\uq\dq} & V_{\uq\sq} & V_{\uq\bq} \cr
              V_{\cq\dq} & V_{\cq\sq} & V_{\cq\bq} \cr
              V_{\tq\dq} & V_{\tq\sq} & V_{\tq\bq} \cr}\right)
\left(\matrix{\dq\cr \sq\cr \bq\cr}\right)\,.
\]
The magnitudes of the elements, derived from the Particle Data Group 2004 
tables, are given below. The number in brackets gives an estimate of the
uncertainty in the last digit. Note that these values may not give an exactly
unitary matrix, but this has no significance.
\[
V=\left(\matrix{0.975(0) & 0.224(3) & 0.004(1) \cr
                0.224(3) & 0.974(1) & 0.042(2) \cr
                0.009(5) & 0.040(3) & 0.999(0) \cr}\right)
\]

\newpage

\leftline{\bf Clebsch--Gordan coefficients}
\medskip
\begin{table}[hb]
\begin{tabular}{cccccccccc}
1& $\times$ &${1\over 2}$& &&&&&& \\
&&&&&&&&& \\[-6pt]
\hline
&&&&&&&&& \\[-6pt]
&&& $J$ & $~{3\over 2}$ & $~{3\over 2}$ & $~{1\over 2}$ & $~{3\over 2}$ &
$~{1\over 2}$ & $~{3\over 2}$   \\[6pt]
$m_1$ && $m_2$ & $M$ & $+{3\over 2}$ & $+{1\over 2}$ & $+{1\over 2}$ & 
$-{1\over 2}$ & $-{1\over 2}$ & $-{3\over 2}$ \\[2pt]
\hline
$+1$ && $+{1\over 2}$ && 1 &&&&& \\[3pt]
$+1$ && $-{1\over 2}$ &&& $\sqrt{1/3}$& $\sqrt{2/3}$ &&& \\[3pt]
$~0$ && $+{1\over 2}$ &&& $\sqrt{2/3}$& $-\sqrt{1/3}$ &&& \\[3pt]
$~0$ && $-{1\over 2}$ &&&&& $\sqrt{2/3}$& $\sqrt{1/3}$ &  \\[3pt]
$-1$ && $+{1\over 2}$ &&&&& $\sqrt{1/3}$& $-\sqrt{2/3}$ &  \\[3pt]
$-1$ && $-{1\over 2}$ &&&&&&& 1 \\[3pt]
\hline
\end{tabular}
\end{table}

\begin{table}[hb]
\begin{tabular}{ccccccccccccc}
1& $\times$ & 1& &&&&&&&&& \\
&&&&&&&&&&&& \\[-6pt]
\hline
&&&&&&&&&&&& \\[-6pt]
&&& $J$ & $~2$ & $~2$ & $~1$ & $~2$ &$~1$ & $~0$& $~2$& $~1$& $~2$ \\[6pt]
$m_1$ && $m_2$ & $M$ & $+2$ & $+1$ & $+1$ & 
$~0$ & $~0$ & $~0$& $-1$ & $-1$ & $-2$ \\[2pt]
\hline
$+1$ && $+1$ && 1 &&&&&&&& \\[3pt]
$+1$ && $~0$ &&& $\sqrt{1/2}$& $\sqrt{1/2}$ &&&&&& \\[3pt]
$~0$ && $+1$ &&& $\sqrt{1/2}$& $-\sqrt{1/2}$ &&&&&& \\[3pt]
$+1$ && $-1$ &&&&& $\sqrt{1/6}$& $\sqrt{1/2}$ & $\sqrt{1/3}$ &&&  \\[3pt]
$~0$ && $~0$ &&&&& $\sqrt{2/3}$& $0$ & $-\sqrt{1/3}$ &&& \\[3pt]
$-1$ && $+1$ &&&&& $\sqrt{1/6}$& $-\sqrt{1/2}$ & $\sqrt{1/3}$ &&&  \\[3pt]
$~0$ && $-1$ &&&&&&&& $\sqrt{1/2}$& $\sqrt{1/2}$ & \\[3pt]
$-1$ && $~0$ &&&&&&&& $\sqrt{1/2}$& $-\sqrt{1/2}$ & \\[3pt]
$-1$ && $-1$ &&&&&&&&&& 1 \\[3pt]
\hline
\end{tabular}
\end{table}

\vskip0.5cm

\leftline{\bf Breit-Wigner resonance formula}
\medskip

The formula represents the energy dependence of the total cross-section 
$\sigma(i\to f)$ for unpolarised scattering between a two-body initial 
state $i$ to a final state $f$, in the vicinity of a resonance of rest-mass
energy $M$, spin $J$ and total width $\Gamma$. 
\[
\sigma(i\to f)= \pi \lambdabar^2 g 
{\Gamma_i\Gamma_f\over [(E-M)^2+\Gamma^2/4]},
\]
where $\displaystyle \lambdabar={\hbar c\over pc}$,  
$\displaystyle g = {2J+1\over (2s_a+1)(2s_b+1)}$, $p$ is the magnitude of the 
centre-of-mass momentum of the initial state particles, $s_a$, $s_b$ are their
spins and $\Gamma_i,~\Gamma_f$ the initial and final state partial widths. 
\newpage

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% end of C4 data file
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