Impact parameter formula

[Anon]

I recently learned of the definition of the 2D impact parameter Dxy in ParticlePropagator.cc. I understand that the formula is supposed to be (angular momentum)/pt, yielding a signed Dxy. However, I am worried that it may give incorrect results for very highly-displaced particles, where the pt vector evaluated at a particle's production vertex may be rotated relative to its pt vector at closest (extrapolated) approach to the beam. The problem is that the "angular momentum" is currently computed with \vec{x} and \vec{p} evaluated at different points on a particle's trajectory. The returned number will always give an underestimate of the true magnitude of Dxy.

Probably this would only represent a problem in practice in only a very small subset of applications, but it seems trivial to fix.

Many thanks.

[Michele Selvaggi]

Hi,

let me rephrase what I understand you are saying:

"the present implementation is inconstistent because the momentum is computed at vertex instead of being computed at the Xd,Yd coordinates of closest approach"

So one should either compute both the momentum and position at vertex or both at (Xd,Yd). The two approaches should give exactly the same result so I might as well take both P and X at vertex, so no calculation is needed.

If you confirm, I can easily fix it.

Thanks for spotting this,
Michele

[Anon]

Since this is not a central-force problem, angular momentum around the center of the detector is not conserved. Only the magnitude of pt is conserved. If you want to use an angular momentum approach, I think you would need to have the pt vector evaluated at (xd,yd). Regardless, at least the magnitude of Dxy is certainly sqrt(xd*xd+yd*yd), and is trivial to compute.

Actually, on further study, my understanding of the signing of Dxy (at least in b-tagging and in some displaced NP searches) is that it is not based on angular momentum, but rather on taking the dot product of (xd,yd) with the direction of the jet containing the track. If you want a definition track-by-track, independent of how it is clustered, I'm not sure whether there is a unique sensible prescription.

[Michele Selvaggi]

Hi,

actually when tracks are used for b-tagging the sign of dxy is defined precisely as you say. You can have a look at the relevant module:

​https://github.com/delphes/delphes/blob/master/modules/TrackCountingBTagging.cc

I have updated formula there so that dxy is computed as sqrt(xd*xd+yd*yd) instead of using angular momentum formula.