Intrinsic resolution in a tracking element

Foreword: This has probably been worked out in a textbook somewhere, but I wanted to write it down for my own sake. This is a re-write (hopefully more clear, with slightly better notation) of Appendix A of my PhD thesis (I don't think it was well-written there)...

-Gene

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Let's establish a few quantities:

E_intr : error on the measurement by the element in question
- σ_intr² = <E_intr²> : intrinsic resolution of the element, and its relation to an ensemble of errors in measurement
E_proj : error on the track projection to that element (excluding the element from the track fit)
- σ_proj² = <E_proj²> : resolution of the track projection to an element, and its relation to an ensemble of errors in track projections
E_track : error on track fit at an element including the element in the fit
R_incl = E_intr - E_track : residual difference between the measurement and the inclusive track fit
- σ_incl² = <(E_intr - E_track)²> : resolution from the inclusive residuals
R_excl = E_intr - E_proj : residual difference between the measurement and the exclusive track fit
- σ_excl² = <(E_intr - E_proj)²> : resolution from the exclusive residuals

Let us further assume that the projection from the track fit excluding the element E_proj is uncorrelated with the intrinsic error of the measurement from the element: <E_proj E_intr> = 0. This implies that we can write:

σ_excl² = <E_intr²> + <E_proj²> = σ_intr² + σ_proj²

Our goal is to determine σ_intr given that we can only observe σ_incl and σ_excl.

To that end, we utilize a guess, σ'_intr, and write down a reasonable estimation of E_track using a weighted average of E_intr and E_proj, where the weights are w_proj = 1/σ_proj², and w_intr = 1/σ'_intr²:

E_track = [(w_intr E_intr) + (w_proj E_proj)] / (w_intr + w_proj)
= [(E_intr / σ'_intr²) + (E_proj / σ_proj²)] / [(1/σ'_intr²) + (1/σ_proj²)]
= [(σ_proj² E_intr) + (σ'_intr² E_proj)] / (σ'_intr² + σ_proj²)

Substituing this, we find...

σ_incl² = <(E_intr - E_track)2>
= <E_intr²> - 2 <E_intr E_track> + <E_track²>
= σ_intr² - 2 <E_intr {[(σ_proj² E_intr) + (σ'_intr² E_proj)] / (σ'_intr² + σ_proj²)}> + <{[(σ_proj² E_intr) + (σ'_intr² E_proj)] / (σ'_intr² + σ_proj²)}²}>

Dropping terms of <E_intr E_proj>, replacing terms of <E_proj²> and <E_intr²> with σ_proj² and σ_intr² respectively, and multiplying through such that all terms on the right-hand-side of the equation have the denominator (σ'_intr² + σ_proj²)², we find

σ_incl² = [(σ_intr² σ'_intr⁴) + (2 σ_intr² σ'_intr² σ_proj²) + (σ_intr² σ_proj⁴) - (2 σ_intr² σ'_intr² σ_proj²) - (2 σ'_intr² σ_proj⁴) + (σ'_intr⁴ σ_proj²) + (σ_intr² σ_proj⁴)] / (σ'_intr² + σ_proj²)²
= (σ_intr² σ'_intr⁴ + σ'_intr⁴ σ_proj²) / (σ'_intr² + σ_proj²)²
= σ'_intr⁴ (σ_intr² + σ_proj²) / (σ'_intr² + σ_proj²)²

We can substitute for σ_proj² using σ_excl² = σ_intr² + σ_proj²:

σ_incl² = σ'_intr⁴ σ_excl² / (σ'_intr² + σ_excl² - σ_intr²)²
σ_incl = σ'_intr² σ_excl / (σ'_intr² + σ_excl² - σ_intr²)

And solving for σ_intr² we find:

σ_intr² = σ'_intr² + σ_excl² - (σ'_intr² σ_excl / σ_incl)
σ_intr = √{ σ_excl² - σ'_intr² [(σ_excl / σ_incl) - 1] }

This is an estimator of σ_intr. Ideally, σ_intr and σ'_intr should be the same. One can iterate a few times starting with a good guess for σ'_intr and then replacing it in later iterations with the σ_intr found from the previous iteration until the two are approximately equal.

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-Gene