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Sphericity

The sphericity tensor is defined as [Bjo70]

\begin{displaymath}
S^{\alpha \beta} = \frac{\displaystyle \sum_i p^{\alpha}_{i}...
...eta}_{i} } {\displaystyle \sum_i \vert\mathbf{p}_i\vert^2 } ~,
\end{displaymath} (280)

where $\alpha, \beta = 1, 2, 3$ corresponds to the $x$, $y$ and $z$ components. By standard diagonalization of $S^{\alpha \beta}$ one may find three eigenvalues $\lambda_1 \geq \lambda_2 \geq \lambda_3$, with $\lambda_1 + \lambda_2 + \lambda_3 = 1$. The sphericity of the event is then defined as
\begin{displaymath}
S = \frac{3}{2} \, (\lambda_2 + \lambda_3) ~,
\end{displaymath} (281)

so that $0 \leq S \leq 1$. Sphericity is essentially a measure of the summed $p_{\perp}^2$ with respect to the event axis; a 2-jet event corresponds to $S \approx 0$ and an isotropic event to $S \approx 1$.

The aplanarity $A$, with definition $A = \frac{3}{2} \lambda_3$, is constrained to the range $0 \leq A \leq \frac{1}{2}$. It measures the transverse momentum component out of the event plane: a planar event has $A \approx 0$ and an isotropic one $A \approx \frac{1}{2}$.

Eigenvectors $\mathbf{v}_j$ can be found that correspond to the three eigenvalues $\lambda_j$ of the sphericity tensor. The $\mathbf{v}_1$ one is called the sphericity axis (or event axis, if it is clear from the context that sphericity has been used), while the sphericity event plane is spanned by $\mathbf{v}_1$ and $\mathbf{v}_2$.

The sphericity tensor is quadratic in particle momenta. This means that the sphericity value is changed if one particle is split up into two collinear ones which share the original momentum. Thus sphericity is not an infrared safe quantity in QCD perturbation theory. A useful generalization of the sphericity tensor is

\begin{displaymath}
S^{(r) \alpha \beta} = \frac{\displaystyle \sum_i \vert\math...
...eta}_{i} }{\displaystyle
\sum_i \vert\mathbf{p}_i\vert^r } ~,
\end{displaymath} (282)

where $r$ is the power of the momentum dependence. While $r = 2$ thus corresponds to sphericity, $r = 1$ corresponds to linear measures calculable in perturbation theory [Par78]:
\begin{displaymath}
S^{(1) \alpha \beta} = \frac{\displaystyle \sum_i \frac{\dis...
...}_i\vert} }
{\displaystyle \sum_i \vert\mathbf{p}_i\vert } ~.
\end{displaymath} (283)

Eigenvalues and eigenvectors may be defined exactly as before, and therefore also equivalents of $S$ and $A$. These have no standard names; we may call them linearized sphericity $S_{\mathrm{lin}}$ and linearized aplanarity $A_{\mathrm{lin}}$. Quantities derived from the linear matrix that are standard in the literature are instead the combinations [Ell81]

$\displaystyle C$ $\textstyle =$ $\displaystyle 3 ( \lambda_1 \lambda_2 + \lambda_1 \lambda_3 +
\lambda_2 \lambda_3 ) ~,$ (284)
$\displaystyle D$ $\textstyle =$ $\displaystyle 27 \lambda_1 \lambda_2 \lambda_3 ~.$ (285)

Each of these is constrained to be in the range between 0 and 1. Typically, $C$ is used to measure the 3-jet structure and $D$ the 4-jet one, since $C$ is vanishing for a perfect 2-jet event and $D$ is vanishing for a planar event. The $C$ measure is related to the second Fox-Wolfram moment (see below), $C = 1 - H_2$.

Noninteger $r$ values may also be used, and corresponding generalized sphericity and aplanarity measures calculated. While perturbative arguments favour $r = 1$, we know that the fragmentation `noise', e.g. from transverse momentum fluctuations, is proportionately larger for low-momentum particles, and so $r > 1$ should be better for experimental event axis determinations. The use of too large an $r$ value, on the other hand, puts all the emphasis on a few high-momentum particles, and therefore involves a loss of information. It should then come as no surprise that intermediate $r$ values, of around 1.5, gives the best performance for event axis determinations in 2-jet events, where the theoretical meaning of the event axis is well-defined. The gain in accuracy compared with the more conventional choices $r = 2$ or $r = 1$ is rather modest, however.


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