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 Crystall Ball function is given by:

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

where

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

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

$N$ is a normalization factor and $\alpha$, $n$, ${\bar {x}}$ and $\sigma$ are parameters which are fitted with the data.

External link

• J. E. Gaiser, Appendix-F Charmonium Spectroscopy from Radiative Decays of the J/Psi and Psi-Prime, Ph.D. Thesis, SLAC-R-255, (1982). (This is a 205 page document in .pdf form – the function is defined on p. 178.)