Fox-Wolfram moments

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

$$H_l = \sum_{i,j} \frac{ \vert\mathbf{p}_i\vert \, \vert\mathbf{p}_j\vert } {E_{\mathrm{vis}}^2} \, P_l (\cos \theta_{ij}) ~,$$ (288)

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,

$$P_0(x) = 1 ~,$$

$$P_1(x) = x ~,$$

$$P_2(x) = \frac{1}{2} \, (3x^2 -1) ~,$$

$$P_3(x) = \frac{1}{2} \, (5x^3 - 3x) ~,$$

$$P_4(x) = \frac{1}{8} \, (35x^4 - 30x^2 + 3) ~.$$ (289)

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.