Energy loss improvements
and tracking
Niels van Eldik, Peter Kluit, Alan Poppleton, Andi Salzburger,
Sharka Todorova
Common Tracking Meeting 4 July 2013
1
Back tracking and
the impact of the Landau
If one has a Landau distribution for the Eloss one can calculate
the distribution at the IP – so after backtracking- by the following
convolution:
N(p’) = Integral Landau(E,EMOP,σL)*Gaus(p’-E, σp) dE
Note that also the impact of the track variation in the Barrel calorimeter
can be written in this form where σp = EMOPdL/L.
The point is that the MOP value of the distribution N(p’) is not
EMOP anymore. If σL is very small the shift will be very small.
However for larger values of σL the shift will depend on σp .
This means that in the backtracking the new EMOP value after the
convolution should be used.
The proposal is to calculate the correction analytically using per
track the σp.
2
Numerical issues with the Landau
distribution and convolutions
The second issue is that one has to use numerical integration
and FFT to get the Landau convoluted with a Gaussian
distribution. It is far from easy to reach the required numerical
precision.
To circumvent these problems I decided to move to a set of
analytical functions that allow convolution and fast robust fitting.
The Landau will be approximated by:
Landau(x) = x/(a2+x2) 2 for x>0
x = E + E0
a = 3.59524 σL
EMOP = E0 + a/√3
Gaussian(σreso) ≈ Breit-Wigner 2 ~ 1/(b2+x2) 2
b ≈ 0.87388 σreso
These functions can be convoluted analytically and fitted to the
data.
3
Approximation to the Landau distribution
In red the chosen approximation to the Landau function
Landau(x) = x/(a2+x2) 2 for x>0 compared to the root
implementation
4
Data fitted to the model
Here a fit to some extreme cases.
First a Barrel fit where the distribution is mostly gaussian.
p0 = normalization
p1 = mean
p2 = a ~ Landau width
p3 = b = BW2 width
So b ~ sigma resolution
5
Data fitted to the model
• The numerical analytical functions are in place. They are
able to describe the data. For quite extreme cases from
the Gaussian to the Landau and the Resolution
dominated. More examples and fits are in the backup
Next slides:
• Work out the formulae to give corrections to EMOP
after convolution with a Gaussian. Give the mathematics
for Landau distributions and 68% (95%) CL intervals.
• Finally some implications for MS and CB track fitting are
discussed.
6
Modeling the shift in EMOP
• Problem: suppose we have a MOP value and we have a
Landau error σL and a gaussian sigma σreso what is the
shift Δ EMOP?
• Here we use the parameters
• a = 3.59524 σL
• b = 0.87388 σreso
Δ EMOP = function (a,b)
• It has a special form Δ EMOP = a f(b/a)
• f(x=b/a) is shown on the next page
• It has an asymptotic behaviour where f->1 for x->inf
• The function can be approximated by
f(x) = (p0*x*x+p1*x*x*x)/(1+p2*x+p3*x*x+p4*x*x*x)
p0
p2
p4
1.75479e+00
4.28703e+00
6.05171e-01
p1
p3
5.88545e-01
4.36873e+00
7
Modeling the shift in EMOP
8
Typical example in the Calorimeter:
the shift in EMOP
Barrel a = 200 MeV
EC
a = 450 MeV
Assume a dp/p of 2%
This gives the curves on
the left*.
So at high momenta one
gets the full shift of a. At
low momenta the shift is
smaller.
That is why the Jpsi pT
> 5 GeV is very different
from the Z pT > 20 GeV
where the MOP shift
starts to rise.
* For simplicity it is assumed that a= 3.595 σL does not change with p
9
Typical example in the Muon
Spectrometer: the shift in EMOP
Barrel a = 20 MeV
EC
a = 45 MeV
Assume a dp/p of 2%
This gives the curves on
the left.
So at high momenta one
gets the full shift of a. At
3 GeV (EC 6 GeV) the
shift is already 8 (20)
MeV.
The dependence is
rather small but should
be accounted for…
10
Convoluting Landau distributions and
the track fit
• Problem: suppose we have several EMOP values and
errors. As discussed in the Muon-PUB-2008-002 note,
the formula to combine these values is:
*
This has very special features:
1.The MOP value shifts depending on the MOP errors (7)
2.The errors do not add up quadratically but linearly (8)
This has implications for track fitting.
It means that Eloss has to be combined using these formula.
For the MS track fit I would propose to collect the total
material between the measurements and use the formulae
above to aggregate the Eloss.
* I prefer the form: (σ1+σ2) ln (σ1+σ2) - σ1 ln σ1- σ2 ln σ2
11
Landau distributions and CL intervals
We want to derive analytic expression for the different
CL intervals. For a pure Landau distribution we define σ+
and σ- as containing 68% of the events on the upper
Big!
lower side of the MOP Eloss value.
68% CL
σ- = 1.02 σL
σ+ = 4.65 σL
95% CL
2σ- = 1.82 σL
2σ+ = 21.86 σL
After convolution with a Gaussian with σReso the formulae
become (checked using the Landau-BW approximations):
σ- = √ (1.02 σL )2 + (σReso) 2
σ+ = √ (1.82 σL )2 + (σReso) 2
2σ- = √ (4.65 σL )2 + (2 σReso) 2
2σ+ = √ (21.86 σL )2 + (2 σReso) 2
These expressions we need for Calorimeter object
in the CB trackfit.
12
Treatment of the Calorimeter Eloss
• Proposal: from the Calorimeter Tracking Geometry we
obtain:
1)The EMOP (from ID to MS)
2)The σL on the Eloss
3)The dL/L: uncertainty on track length in Calorimeter
NB in the calculation of 1) and 2) the Landau formalism of
slide 11(adding errors linearly) should be used.
From this we calculate:
EMOP corrected using the formalism on slide 7 (dL/L gives a
smearing and therefore shifts the mean).
We also need σ+ and σ- from slide 12.
This allows to make the comparison of the measured
Eloss in the Calorimeter with EMOP corrected + 2 σ.
13
Treatment of the Calorimeter Eloss
• Next step is to determine the Eloss and errors that should
be used in case no Calorimeter measurement is used.
Now we have to calculate:
EMOP corrected using dp/p inside MS and dL/L using the
formalism on slide 7. We also need σ+ and σ- from slide 12.
The value of the EMOP corrected will depend on whether
we looked at the measured Eloss or not. NB if we did look the
Landau tail is reduced. Currently I work on the mathematics
for this case.
This Eloss object can be used for two purposes:
•Back extrapolation of the MS track to the IP
-> SA parameters
•Perform the Combined fit
-> CB parameters
14
Treatment of the MuonSpectrometer
Eloss
To do a perfect job inside the Muon Spectrometer
• One should firstly aggregate according to the formalism
for Landau distributions
• Secondly one should reiterate using the uncertainty on
the momentum to update EMOP corrected and then refit. This is
depicted on slide 10.
• Question: what is done in the Step Propagator? Is the
MOP value returned? How is the aggregation of the Eloss
done (is the Landau formalism used)?
• One could use the Landau formalism in the track fit.
• However one needs a re-iteration propagating the error
on the momentum into the EMOP value.
• For the calorimeter Eloss object one could make the
object taking into account the momentum error…
15
Backup slides
16
Introduction
Currently the Energy loss description for muons passing the
calorimeter has a precision of about Et 100 MeV.
For mass measurements using the combined and or muon
(standalone) measurements it is important to improve
significantly the precision of the E loss description.
The target that we want to put here is a description that is
accurate at e.g. the 10 MeV level.
This we want to achieve on a track-track basis.
This project will need improvements in several areas:
1)the Tracking Calorimeter description: E loss MOP and sigma
modeling
2) (analytic) computing and reconstruction
17
Introduction: Physics
Concerning the physics aspects:
Firstly one can get rid of part of the Landau tail. In particular for
isolated muons. This is done by the Muid Combined algorithm.
Secondly, we cannot fully get rid of the Landau tails; but we can
give the right (well-centered) MOP value. This means that the
MOP of di-muon resonances like Jpsi and Z will be (about) well
centered.
Thirdly, we cannot get the “mean” of the momentum or mass
distribution well-centered. This has to do with the fact that a
Landau distribution does NOT have a mean. So depending on
the cuts, the “mean” will change. The MOP value is however well
defined.
18
Calorimeter Tracking geometry
The current situation is the following.
There is a layer based description of the calorimeter to describe
the E loss and its error.
This description is not used in Muid because it is not precise
enough. The Muid description is basically an eta-phi-p map that
gives MOP Eloss and its error.
There are two possible roads that improve the situation:
- Improve the E loss description of the Calorimeter
- Improve the precision of the Muid Eloss map
It is hard to conceive that a map with a precision of 10 MeV can
be made. We think however it is possible to achieve a very
precise description of the E loss in the calorimeter.
This would require a volume based TG description of the
calorimeter.
19
Calorimeter Tracking geometry
In the Muon system a volume based TG description gives a very
accurate description of the material traversed.
For the Muon system this is the best we can achieve.
A similar description of the calorimeter would give on a track-bytrack basis the E loss. And this is - in my opinion - the ‘ultimate’
description one wants to have.
Both Sharka and Andi are willing to work on this topic.
Requirements
Eloss: for a given track one wants to know the MOP Eloss value
and its error. Secondly one needs the track length in the
Calorimeter and its error.
Identification: crossed cells and the MOP E loss per cell and its
error (for use in Muon Calorimeter Identification algorithms)
20
Energy loss modeling
Energy loss modeling is described by a Landau distribution that
is characterized by a MOP value and a width or sigma value.
The MOP energy and the sigma depend on the momentum of
the muon and consists of a linear and a logarithmic term. The
tracking geometry takes uses dependencies. To obtain a precise
description Geant4 simulations and TG should agree on these
underlying parametrisations and the material characteristics
(such as X0).
A study of the Eloss shape in the MC in different regions of the
detector was performed.
It was found out that the Eloss shape in the Barrel does NOT
have a Landau shape. In the Endcap it can be described by a
Landau distribution.
21
Energy loss modeling
Fit: frac*Landau + (1-frac) Gaussian
Puzzle
Barrel has 83% Gaussian!
Only 17% Landau
Endcap only 4%
as expected 96% Landau
22
Energy loss EC plots for
one eta bin and a 2 GeV p range
Here eta = 2.0-2.1; p 12-14; 14-16 and 16-18 GeV
Enough stats in Endcap from 14 GeV onwards:
observe a large Landau tail (as expected)
23
Energy loss Barrel plots for
one eta bin and a 2 GeV p range
Barrel eta 0-0.1 same momentum range observe a
larger Gaussian component
24
Energy loss Barrel plots for
one eta bin and a 2 GeV p range
Barrel eta 0-0.1 to lower momenta: one observes an
increase in the Gaussian component
25
Barrel: Energy loss modeling
Barrel has more Gaussian shape. Why is this?
First observed by Alan and Kostas. Their explanation:
The distribution has two components: one of Landau
component and another track length component.
If the track length varies due to multiple scattering the mean of
the Landau will be smeared with a Gaussian.
Here a model
that indeed
does the job
for 10% track
length
smearing
26
Energy loss modeling
Conclude:
Energy loss modeling is described by a Landau distribution that
is characterized by a MOP value and a width or sigma value.
Need also to know the track length variations dL/L in the
Calorimeter to account for the effect observed in the Barrel.
That is why on slide this is added as a requirement to the TG
description.
Reconstruction and computing issues:
- Back tracking and the impact of the Landau tail.
- Numerical precision and the Landau distribution, convolutions.
- Approximation to the Landau distribution.
27
Numerical issues with the Landau
distribution
The first approach was to use the formula on slide 10 and
compute it.
This turns out to be not that trivial: the Landau distribution itself
using e.g. the root implementation has a problem. The MOP
value you put in is not the maximum of the distribution…
It is shifted by about 40 MeV.
MOP value of 3 GeV input
The maximum of the curve
is shifted by 40 MeV
The point is that we need a
precision of << 10 MeV on the
MOP value
28
Back tracking and
the impact of the Landau
Suppose we have a track measured in the MS its q/p distribution
will be rather Gaussian.
The momentum resolution dp/p in the Muon System is given
below:
29
Can the data be fitted
with the current model?
Here a fit to some extreme cases.
An Endcap fit where the distribution is mostly Landau
with small 0.02 dp/p smearing.
p0 = normalization
p1 = mean
p2 = a ~ Landau width
p3 = b = BW2 width
So b ~ sigma resolution
30
Can the data be fitted
with the current model?
Here a fit to some extreme cases.
An Endcap fit where the distribution is mostly Landau
with large 0.10 dp/p smearing.
p0 = normalization
p1 = mean
p2 = a ~ Landau width
p3 = b = BW2 width
So b ~ sigma resolution
Note the large b value
As expected the MOP
value increases wrt the
small smearing case
31