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Fox-Wolfram moments

The Fox-Wolfram moments $H_l$, $l = 0, 1, 2, \ldots$, are defined by [Fox79]

\begin{displaymath}
H_l = \sum_{i,j} \frac{ \vert\mathbf{p}_i\vert \, \vert\mathbf{p}_j\vert }
{E_{\mathrm{vis}}^2} \, P_l (\cos \theta_{ij}) ~,
\end{displaymath} (291)

where $\theta_{ij}$ is the opening angle between hadrons $i$ and $j$ and $E_{\mathrm{vis}}$ the total visible energy of the event. Note that also autocorrelations, $i = j$, are included. The $P_l(x)$ are the Legendre polynomials,
$\displaystyle P_0(x)$ $\textstyle =$ $\displaystyle 1 ~,$  
$\displaystyle P_1(x)$ $\textstyle =$ $\displaystyle x ~,$  
$\displaystyle P_2(x)$ $\textstyle =$ $\displaystyle \frac{1}{2} \, (3x^2 -1) ~,$  
$\displaystyle P_3(x)$ $\textstyle =$ $\displaystyle \frac{1}{2} \, (5x^3 - 3x) ~,$  
$\displaystyle P_4(x)$ $\textstyle =$ $\displaystyle \frac{1}{8} \, (35x^4 - 30x^2 + 3) ~.$ (292)

To the extent that particle masses may be neglected, $H_0 \equiv 1$. It is customary to normalize the results to $H_0$, i.e. to give $H_{l0} = H_l / H_0$. If momentum is balanced then $H_1 \equiv 0$. 2-jet events tend to give $H_l \approx 1$ for $l$ even and $\approx 0$ for $l$ odd.


next up previous contents
Next: Jet masses Up: Event Shapes Previous: Thrust   Contents
Stephen Mrenna 2007-10-30