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:

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

where

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

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

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

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

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

with the error function erf.

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

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