Crystal Ball function



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:


 * $$f(x;\alpha,n,\bar x,\sigma) = N \cdot \begin{cases} \exp(- \frac{(x - \bar x)^2}{2 \sigma^2}), & \mbox{for }\frac{x - \bar x}{\sigma} > -\alpha \\

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

where


 * $$A = \left(\frac{n}{\left| \alpha \right|}\right)^n \cdot \exp\left(- \frac {\left| \alpha \right|^2}{2}\right)$$,
 * $$B = \frac{n}{\left| \alpha \right|} - \left| \alpha \right|$$,
 * $$N = \frac{1}{\sigma (C + D)}$$,
 * $$C = \frac{n}{\left| \alpha \right|} \cdot \frac{1}{n-1} \cdot \exp\left(- \frac {\left| \alpha \right|^2}{2}\right)$$,
 * $$D = \sqrt{\frac{\pi}{2}} \left(1 + \operatorname{erf}\left(\frac{\left| \alpha \right|}{\sqrt 2}\right)\right)$$.

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