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,\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) , with the error function erf.…

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,\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) , 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 \bar 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.

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