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
;
α
,
n
,
x
¯
,
σ
)
=
N
⋅
{
exp
(
−
(
x
−
x
¯
)
2
2
σ
2
)
,
for
x
−
x
¯
σ
>
−
α
A
⋅
(
B
−
x
−
x
¯
σ
)
−
n
,
for
x
−
x
¯
σ
⩽
−
α
where
A
=
(
n
|
α
|
)
n
⋅
exp
(
−
|
α
|
2
2
)
,
B
=
n
|
α
|
−
|
α
|
,
N
=
1
σ
(
C
+
D
)
,
C
=
n
|
α
|
⋅
1
n
−
1
⋅
exp
(
−
|
α
|
2
2
)
,
D
=
π
2
(
1
+
erf
(
|
α
|
2
)
)
.
N
(Skwarnicki 1986) is a normalization factor and
α
,
n
,
x
¯
and
σ
are parameters which are fitted with the data. erf is the error function.
