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:

where

1. ,

2. ,

3. ,

4. ,

5. ,

with the error function erf.

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

• is a normalization factor (Skwarnicki 1986)

• defines the point where the PDF changes from a power-law to a Gaussian distribution

• is the power of the power-law tail

• 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.