# 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

_______________________

Let's establish a few quantities:

To that end, we utilize a guess, σ'

Substituing this, we find...

Dropping terms of <E

We can substitute for σ

And solving for σ

This is an estimator of σ

_______

-Gene

-Gene

_______________________

Let's establish a few quantities:

- E
_{intr}: error on the measurement by the element in question

- σ
_{intr}^{2}= <E_{intr}^{2}> : intrinsic resolution of the element, and its relation to an ensemble of errors in measurement

- σ
- E
_{proj}: error on the track projection to that element (excluding the element from the track fit)- σ
_{proj}^{2}= <E_{proj}^{2}> : resolution of the track projection to an element, and its relation to an ensemble of errors in track projections

- σ
- 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
_{incl}^{2}= <(E_{intr}- E_{track})^{2}> : resolution from the inclusive residuals

- σ
- R
_{excl}= E_{intr}- E_{proj}: residual difference between the measurement and the exclusive track fit

- σ
_{excl}^{2}= <(E_{intr}- E_{proj})^{2}> : resolution from the exclusive residuals

- σ

_{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

_{excl}^{2}= <E_{intr}^{2}> + <E_{proj}^{2}> = σ_{intr}^{2}+ σ_{proj}^{2}**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_{proj}^{2}, and w_{intr}= 1/σ'_{intr}^{2}:E

= [(E

= [(σ

_{track}= [(w_{intr}E_{intr}) + (w_{proj}E_{proj})] / (w_{intr}+ w_{proj})= [(E

_{intr}/ σ'_{intr}^{2}) + (E_{proj}/ σ_{proj}^{2})] / [(1/σ'_{intr}^{2}) + (1/σ_{proj}^{2})]= [(σ

_{proj}^{2}E_{intr}) + (σ'_{intr}^{2}E_{proj})] / (σ'_{intr}^{2}+ σ_{proj}^{2})Substituing this, we find...

σ

= <E

= σ

_{incl}^{2}= <(E_{intr}- E_{track})2>= <E

_{intr}^{2}> - 2 <E_{intr}E_{track}> + <E_{track}^{2}>= σ

_{intr}^{2}- 2 <E_{intr}{[(σ_{proj}^{2}E_{intr}) + (σ'_{intr}^{2}E_{proj})] / (σ'_{intr}^{2}+ σ_{proj}^{2})}> + <{[(σ_{proj}^{2}E_{intr}) + (σ'_{intr}^{2}E_{proj})] / (σ'_{intr}^{2}+ σ_{proj}^{2})}^{2}}>Dropping terms of <E

_{intr}E_{proj}>, replacing terms of <E_{proj}^{2}> and <E_{intr}^{2}> with σ_{proj}^{2}and σ_{intr}^{2}respectively, and multiplying through such that all terms on the right-hand-side of the equation have the denominator (σ'_{intr}^{2}+ σ_{proj}^{2})^{2}, we findσ

= (σ

= σ'

_{incl}^{2}= [(σ_{intr}^{2}σ'_{intr}^{4}) + (2 σ_{intr}^{2}σ'_{intr}^{2}σ_{proj}^{2}) + (σ_{intr}^{2}σ_{proj}^{4}) - (2 σ_{intr}^{2}σ'_{intr}^{2}σ_{proj}^{2}) - (2 σ'_{intr}^{2}σ_{proj}^{4}) + (σ'_{intr}^{4}σ_{proj}^{2}) + (σ_{intr}^{2}σ_{proj}^{4})] / (σ'_{intr}^{2}+ σ_{proj}^{2})^{2}= (σ

_{intr}^{2}σ'_{intr}^{4}+ σ'_{intr}^{4}σ_{proj}^{2}) / (σ'_{intr}^{2}+ σ_{proj}^{2})^{2}= σ'

_{intr}^{4}(σ_{intr}^{2}+ σ_{proj}^{2}) / (σ'_{intr}^{2}+ σ_{proj}^{2})^{2}We can substitute for σ

_{proj}^{2}using σ_{excl}^{2}= σ_{intr}^{2}+ σ_{proj}^{2}:σ

σ

_{incl}^{2}= σ'_{intr}^{4}σ_{excl}^{2}/ (σ'_{intr}^{2}+ σ_{excl}^{2}- σ_{intr}^{2})^{2}σ

_{incl}= σ'_{intr}^{2}σ_{excl}/ (σ'_{intr}^{2}+ σ_{excl}^{2}- σ_{intr}^{2})And solving for σ

_{intr}^{2}we find:σ

_{intr}^{2}= σ'_{intr}^{2}+ σ_{excl}^{2}- (σ'_{intr}^{2}σ_{excl}/ σ_{incl})**σ**_{intr}= √{ σ_{excl}^{2}- σ'_{intr}^{2}[(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._______

-Gene

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