Crystal Ball function explained

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 ⋅ \begin{cases} \exp( - ((x - \bar x) ^{2})/(2 \sigma ^{2}) ), & for (x - \bar x)/(\sigma) > - \alpha \\ A ⋅ (B - (x - \bar x)/(\sigma) ) ^{-n} , & for (x - \bar x)/(\sigma) \leqslant - \alpha \end{cases}

where

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

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.