ARGUS distribution
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Probability density function c = 1. | |||
Cumulative distribution function c = 1. | |||
| Parameters |
c
>
0
{\displaystyle c>0}
cut-off (real) χ > 0 {\displaystyle \chi >0} curvature (real) | ||
|---|---|---|---|
| Support | x ∈ ( 0 , c ) {\displaystyle x\in (0,c)\!} | ||
| see text | |||
| CDF | see text | ||
| Mean |
μ
=
c
π
/
8
χ
e
−
χ
2
4
I
1
(
χ
2
4
)
Ψ
(
χ
)
{\displaystyle \mu =c{\sqrt {\pi /8}}\;{\frac {\chi e^{-{\frac {\chi ^{2}}{4}}}I_{1}({\tfrac {\chi ^{2}}{4}})}{\Psi (\chi )}}}
where I1 is the Modified Bessel function of the first kind of order 1, and Ψ ( x ) {\displaystyle \Psi (x)} is given in the text. | ||
| Mode | c 2 χ ( χ 2 − 2 ) + χ 4 + 4 {\displaystyle {\frac {c}{{\sqrt {2}}\chi }}{\sqrt {(\chi ^{2}-2)+{\sqrt {\chi ^{4}+4}}}}} | ||
| Variance | c 2 ( 1 − 3 χ 2 + χ ϕ ( χ ) Ψ ( χ ) ) − μ 2 {\displaystyle c^{2}\!\left(1-{\frac {3}{\chi ^{2}}}+{\frac {\chi \phi (\chi )}{\Psi (\chi )}}\right)-\mu ^{2}} | ||
cut-off (
curvature (
is given in the text.
In physics, the ARGUS distribution, named after the particle physics experiment ARGUS, is the probability distribution of the reconstructed invariant mass of a decayed particle candidate in continuum background.
Definition
The probability density function (pdf) of the ARGUS distribution is:
f ( x ; χ , c ) = χ 3 2 π Ψ ( χ ) ⋅ x c 2 1 − x 2 c 2 exp { − 1 2 χ 2 ( 1 − x 2 c 2 ) } , {\displaystyle f(x;\chi ,c)={\frac {\chi ^{3}}{{\sqrt {2\pi }}\,\Psi (\chi )}}\cdot {\frac {x}{c^{2}}}{\sqrt {1-{\frac {x^{2}}{c^{2}}}}}\exp {\bigg \{}-{\frac {1}{2}}\chi ^{2}{\Big (}1-{\frac {x^{2}}{c^{2}}}{\Big )}{\bigg \}},}
for 0 ≤ x < c {\displaystyle 0\leq x<c} . Here χ {\displaystyle \chi } and c {\displaystyle c} are parameters of the distribution and
Ψ ( χ ) = Φ ( χ ) − χ ϕ ( χ ) − 1 2 , {\displaystyle \Psi (\chi )=\Phi (\chi )-\chi \phi (\chi )-{\tfrac {1}{2}},}
. Here
χ
{\displaystyle \chi }
and
c
{\displaystyle c}
are parameters of the distribution and
where Φ ( x ) {\displaystyle \Phi (x)} and ϕ ( x ) {\displaystyle \phi (x)} are the cumulative distribution and probability density functions of the standard normal distribution, respectively.
and
ϕ
(
x
)
{\displaystyle \phi (x)}
are the Cumulative distribution function
The cumulative distribution function (cdf) of the ARGUS distribution is
F ( x ) = 1 − Ψ ( χ 1 − x 2 / c 2 ) Ψ ( χ ) {\displaystyle F(x)=1-{\frac {\Psi \left(\chi {\sqrt {1-x^{2}/c^{2}}}\right)}{\Psi (\chi )}}} .
.
Parameter estimation
Parameter c is assumed to be known (the kinematic limit of the invariant mass distribution), whereas χ can be estimated from the sample X1, …, Xn using the maximum likelihood approach. The estimator is a function of sample second moment, and is given as a solution to the non-linear equation
1 − 3 χ 2 + χ ϕ ( χ ) Ψ ( χ ) = 1 n ∑ i = 1 n x i 2 c 2 {\displaystyle 1-{\frac {3}{\chi ^{2}}}+{\frac {\chi \phi (\chi )}{\Psi (\chi )}}={\frac {1}{n}}\sum _{i=1}^{n}{\frac {x_{i}^{2}}{c^{2}}}} .
.
The solution exists and is unique, provided that the right-hand side is greater than 0.4; the resulting estimator χ ^ {\displaystyle \scriptstyle {\hat {\chi }}} is consistent and asymptotically normal.
is Generalized ARGUS distribution
Sometimes a more general form is used to describe a more peaking-like distribution:
f ( x ) = 2 − p χ 2 ( p + 1 ) Γ ( p + 1 ) − Γ ( p + 1 , 1 2 χ 2 ) ⋅ x c 2 ( 1 − x 2 c 2 ) p exp { − 1 2 χ 2 ( 1 − x 2 c 2 ) } , 0 ≤ x ≤ c , c > 0 , χ > 0 , p > − 1 {\displaystyle f(x)={\frac {2^{-p}\chi ^{2(p+1)}}{\Gamma (p+1)-\Gamma (p+1,\,{\tfrac {1}{2}}\chi ^{2})}}\cdot {\frac {x}{c^{2}}}\left(1-{\frac {x^{2}}{c^{2}}}\right)^{p}\exp \left\{-{\frac {1}{2}}\chi ^{2}\left(1-{\frac {x^{2}}{c^{2}}}\right)\right\},\qquad 0\leq x\leq c,\qquad c>0,\,\chi >0,\,p>-1} F ( x ) = Γ ( p + 1 , 1 2 χ 2 ( 1 − x 2 c 2 ) ) − Γ ( p + 1 , 1 2 χ 2 ) Γ ( p + 1 ) − Γ ( p + 1 , 1 2 χ 2 ) , 0 ≤ x ≤ c , c > 0 , χ > 0 , p > − 1 {\displaystyle F(x)={\frac {\Gamma \left(p+1,\,{\tfrac {1}{2}}\chi ^{2}\left(1-{\frac {x^{2}}{c^{2}}}\right)\right)-\Gamma (p+1,\,{\tfrac {1}{2}}\chi ^{2})}{\Gamma (p+1)-\Gamma (p+1,\,{\tfrac {1}{2}}\chi ^{2})}},\qquad 0\leq x\leq c,\qquad c>0,\,\chi >0,\,p>-1}
F
(
x
)
=
Γ
(
p
+
1
,
1
2
χ
2
(
1
−
x
2
c
2
)
)
−
Γ
(
p
+
1
,
1
2
χ
2
)
Γ
(
p
+
1
)
−
Γ
(
p
+
1
,
1
2
χ
2
)
,
0
≤
x
≤
c
,
c
>
0
,
χ
>
0
,
p
>
−
1
{\displaystyle F(x)={\frac {\Gamma \left(p+1,\,{\tfrac {1}{2}}\chi ^{2}\left(1-{\frac {x^{2}}{c^{2}}}\right)\right)-\Gamma (p+1,\,{\tfrac {1}{2}}\chi ^{2})}{\Gamma (p+1)-\Gamma (p+1,\,{\tfrac {1}{2}}\chi ^{2})}},\qquad 0\leq x\leq c,\qquad c>0,\,\chi >0,\,p>-1}
where Γ(·) is the gamma function, and Γ(·,·) is the upper incomplete gamma function.
Here parameters c, χ, p represent the cutoff, curvature, and power respectively.
The mode is:
c 2 χ ( χ 2 − 2 p − 1 ) + χ 2 ( χ 2 − 4 p + 2 ) + ( 1 + 2 p ) 2 {\displaystyle {\frac {c}{{\sqrt {2}}\chi }}{\sqrt {(\chi ^{2}-2p-1)+{\sqrt {\chi ^{2}(\chi ^{2}-4p+2)+(1+2p)^{2}}}}}}
The mean is:
μ = c p π Γ ( p ) Γ ( 5 2 + p ) χ 2 p + 2 2 p + 2 M ( p + 1 , 5 2 + p , − χ 2 2 ) Γ ( p + 1 ) − Γ ( p + 1 , 1 2 χ 2 ) {\displaystyle \mu =c\,p\,{\sqrt {\pi }}{\frac {\Gamma (p)}{\Gamma ({\tfrac {5}{2}}+p)}}{\frac {\chi ^{2p+2}}{2^{p+2}}}{\frac {M\left(p+1,{\tfrac {5}{2}}+p,-{\tfrac {\chi ^{2}}{2}}\right)}{\Gamma (p+1)-\Gamma (p+1,\,{\tfrac {1}{2}}\chi ^{2})}}}
where M(·,·,·) is the Kummer's confluent hypergeometric function.
The variance is:
σ 2 = c 2 ( χ 2 ) p + 1 χ p + 3 e − χ 2 2 + ( χ 2 − 2 ( p + 1 ) ) { Γ ( p + 2 ) − Γ ( p + 2 , 1 2 χ 2 ) } χ 2 ( p + 1 ) ( Γ ( p + 1 ) − Γ ( p + 1 , 1 2 χ 2 ) ) − μ 2 {\displaystyle \sigma ^{2}=c^{2}{\frac {\left({\frac {\chi }{2}}\right)^{p+1}\chi ^{p+3}e^{-{\tfrac {\chi ^{2}}{2}}}+\left(\chi ^{2}-2(p+1)\right)\left\{\Gamma (p+2)-\Gamma (p+2,\,{\tfrac {1}{2}}\chi ^{2})\right\}}{\chi ^{2}(p+1)\left(\Gamma (p+1)-\Gamma (p+1,\,{\tfrac {1}{2}}\chi ^{2})\right)}}-\mu ^{2}}
p = 0.5 gives a regular ARGUS, listed above.
References
- ^ Albrecht, H. (1990). "Search for hadronic b→u decays". Physics Letters B. 241 (2): 278–282. Bibcode:1990PhLB..241..278A. doi:10.1016/0370-2693(90)91293-K. (More formally by the ARGUS Collaboration, H. Albrecht et al.) In this paper, the function has been defined with parameter c representing the beam energy and parameter p set to 0.5. The normalization and the parameter χ have been obtained from data.
- ^ Confluent hypergeometric function
Further reading
- Albrecht, H. (1994). "Measurement of the polarization in the decay B → J/ψK*". Physics Letters B. 340 (3): 217–220. Bibcode:1994PhLB..340..217A. doi:10.1016/0370-2693(94)01302-0.
- Pedlar, T.; Cronin-Hennessy, D.; Hietala, J.; Dobbs, S.; Metreveli, Z.; Seth, K.; Tomaradze, A.; Xiao, T.; Martin, L. (2011). "Observation of the hc(1P) Using e+e− Collisions above the DD Threshold". Physical Review Letters. 107 (4): 041803. arXiv:1104.2025. Bibcode:2011PhRvL.107d1803P. doi:10.1103/PhysRevLett.107.041803. PMID 21866994. S2CID 33751212.
- Lees, J. P.; Poireau, V.; Prencipe, E.; Tisserand, V.; Garra Tico, J.; Grauges, E.; Martinelli, M.; Palano, A.; Pappagallo, M.; Eigen, G.; Stugu, B.; Sun, L.; Battaglia, M.; Brown, D. N.; Hooberman, B.; Kerth, L. T.; Kolomensky, Y. G.; Lynch, G.; Osipenkov, I. L.; Tanabe, T.; Hawkes, C. M.; Soni, N.; Watson, A. T.; Koch, H.; Schroeder, T.; Asgeirsson, D. J.; Hearty, C.; Mattison, T. S.; McKenna, J. A.; et al. (2010). "Search for Charged Lepton Flavor Violation in Narrow Υ Decays". Physical Review Letters. 104 (15): 151802. arXiv:1001.1883. Bibcode:2010PhRvL.104o1802L. doi:10.1103/PhysRevLett.104.151802. PMID 20481982. S2CID 14992286.
