Crystal Ball function

Image:CrystalBallFunction.svg

The Crystal Ball function, named after the Crystal Ball Collaboration (hence the capitalized initial letters), is a probability density function commonly used to model various lossy processes in high-energy physics. It consists of a Gaussian core portion and a power-law low-end tail, below a certain threshold. The function itself and its first derivative are both continuous.

The Crystal Ball function is given by:

:

A \cdot (B - \frac{x - \bar x}{\sigma})^{-n}, & \mbox{for }\frac{x - \bar x}{\sigma} \leqslant -\alpha \end{cases}

where

:$A\; =\; \backslash left(\backslash frac\{n\}\{\backslash left|\; \backslash alpha\; \backslash right|\}\backslash right)^n\; \backslash cdot\; \backslash exp\backslash left(-\; \backslash frac\; \{\backslash left|\; \backslash alpha\; \backslash right|^2\}\{2\}\backslash right)$,

:$B\; =\; \backslash frac\{n\}\{\backslash left|\; \backslash alpha\; \backslash right|\}\; -\; \backslash left|\; \backslash alpha\; \backslash right|$,

:$N\; =\; \backslash frac\{1\}\{\backslash sigma\; (C\; +\; D)\}$,

:$C\; =\; \backslash frac\{n\}\{\backslash left|\; \backslash alpha\; \backslash right|\}\; \backslash cdot\; \backslash frac\{1\}\{n-1\}\; \backslash cdot\; \backslash exp\backslash left(-\; \backslash frac\; \{\backslash left|\; \backslash alpha\; \backslash right|^2\}\{2\}\backslash right)$,

:$D\; =\; \backslash sqrt\{\backslash frac\{\backslash pi\}\{2\}\}\; \backslash left(1\; +\; \backslash operatorname\{erf\}\backslash left(\backslash frac\{\backslash left|\; \backslash alpha\; \backslash right|\}\{\backslash sqrt\; 2\}\backslash right)\backslash right)$.

$N$ (Skwarnicki 1986) is a normalization factor and $\backslash alpha$, $n$, $\backslash bar\; x$ and $\backslash sigma$ are parameters which are fitted with the data. erf is the error function.