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ARGUS distribution

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ARGUS distribution

In physics, the ARGUS distribution, named after the particle physics experiment ARGUS,[1] 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; \chi, c ) = \frac{\chi^3}{\sqrt{2\pi}\,\Psi(\chi)} \cdot \frac{x}{c^2} \sqrt{1-\frac{x^2}{c^2}} \exp\bigg\{ -\frac12 \chi^2\Big(1-\frac{x^2}{c^2}\Big) \bigg\},

for 0 ≤ x < c. Here χ, and c are parameters of the distribution and

\Psi(\chi) = \Phi(\chi)- \chi \phi( \chi ) - \tfrac{1}{2} ,

and Φ(·), ϕ(·) are the cumulative distribution and probability density functions of the standard normal distribution, respectively.

Differential equation

The pdf of the ARGUS distribution is a solution of the following differential equation:

\left\{\begin{array}{l} c^2 x (c-x) (c+x) f'(x)+f(x) \left(-c^4-c^2 \left(\chi ^2-2\right) x^2+\chi ^2 x^4\right)=0, \\[10pt] f(1)=-\frac{\sqrt{2-\frac{2}{c^2}} \chi ^3 e^{\frac{\chi ^2}{2 c^2}}}{c^2 \left(\sqrt{2} \chi -\sqrt{\pi} e^{\frac{\chi ^2}{2}} \operatorname{erf}\left(\frac{\chi }{\sqrt{2}}\right)\right)} \end{array}\right\}

Cumulative distribution function

The cumulative distribution function (cdf) of the ARGUS distribution is

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 speed of light), 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 - \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 \scriptstyle\hat\chi is consistent and asymptotically normal.

Generalized ARGUS distribution

Sometimes a more general form is used to describe a more peaking-like distribution:

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\{ -\frac12 \chi^2\left(1-\frac{x^2}{c^2}\right) \right\}, \qquad 0 \leq x \leq c,

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:

\frac{c}{\sqrt2\chi}\sqrt{(\chi^2-2p-1)+\sqrt{\chi^2(\chi^2-4p+2)+(1+2p)^2}}

p = 0.5 gives a regular ARGUS, listed above.

References

  1. ^ (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.

Further reading

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