Crystal Ball function

Examples of the Crystal Ball function.

The Crystal Ball function, named after the Crystal Ball Collaboration (hence the capitalized initial letters), is a probability density function (PDF) commonly used to model various lossy processes in high-energy physics such as Bremsstrahlung by electrons. 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:

$f(x;\alpha ,n,\overline{x},\sigma )=N\cdot \{\begin{array}{ll}\exp (-\frac{(x-\overline{x}{)}^2}{2{\sigma }^2}),& \text{for }\frac{x-\overline{x}}{\sigma }>-\alpha \\ A\cdot (B-\frac{x-\overline{x}}{\sigma }{)}^{-n},& \text{for }\frac{x-\overline{x}}{\sigma }⩽-\alpha \end{array},$

where

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

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

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

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

$D=\sqrt{\frac{\pi }{2}}(1+\mathrm{erf}\left(\frac{\left|\alpha \right|}{\sqrt{2}}\right))$,

with the error function erf.

The parameters of the function (that are usually determined by a fit) are:

• $N$ is a normalization factor (Skwarnicki 1986)
• $\alpha >0$ defines the point where the PDF changes from a power-law to a Gaussian distribution
• $n>1$ is the power of the power-law tail
• $\overline{x}$ and $\sigma$ are the mean and the standard deviation of the Gaussian

External links

• 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.)
• M. J. Oreglia, A Study of the Reactions psi prime --> gamma gamma psi, Ph.D. Thesis, SLAC-R-236 (1980), Appendix D.
• T. Skwarnicki, A study of the radiative CASCADE transitions between the Upsilon-Prime and Upsilon resonances, Ph.D Thesis, DESY F31-86-02(1986), Appendix E.