Reconstructing the rest frame of systems

Reconstructing the rest frame of τ+ τ- systems

P e t e r R o s e n d a h lT h o m a s B u r g e s s

B j a r n e S t u g u

I F T U n i v e r s i t y o f B e r g e n

Peter Rosendahl, University of Bergen Spaatind - 05 jan 2012

Motivation

We want to study resonances decaying to a pair of taus.

Unfortunately, some observables are best defined the rest frame of τ pairs, especially variables for studying the τ polarisation

At LHC, Z and Higgs bosons are produced with high boost

Due to neutrinos in the τ decays, no trivial way exists to reconstruct the resonance rest frame

2


Idea

For resonances at the Z mass or above, the τ-jets are collinear.

In resonance rest frame, the visible τ-jets are nearly back-to-back.

Proposed method:

Reconstruct rest frame by finding boost that makes τ-jets back-to-back

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τ-jet

τ-jet


z-Boost Reconstruction

We define acollinarity, α, as the deviation from being back-to-back

Z and Higgs bosons at LHC are produced with a predominantly longitudinal boost

Search for the boost along the z-axis, βz, that minimises α between the τ-jets

zβ

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8

α0.0

0.5

1.0

1.5

2.0

2.5

3.0

|<0.1zβ|

<0.1tβ|<0.1,

zβ|

>0.3tβ|<0.1,

zβ|

<0.7zβ0.5<

<0.1tβ<0.7,

zβ0.5<

>0.3tβ<0.7,

zβ0.5<

>0.9zβ

<0.1tβ>0.9,

zβ

>0.3tβ>0.9,

zβ

α as function of βz

Easy to minimise since α always has single minimum - can be found with a

simple binary search

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Full Boost Reconstruction

Method can easily be extended to search for the full boost - not only along the z-axis.

First find βz

Estimate transverse direction by summing up τ-jets pT and ETmiss

Finally, search for minimum along transverse direction to find βT

[rad.]RFα0.0 0.2 0.4 0.6 0.8 1.0

rate

(Nor

mal

ised

)0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035-τ +τ → Z

Gen. RF

Reco.(Z) RF

Reco.(XYZ) RF

Lab. Frame

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Z →τ+τ- Pythia8 + Tauola simulation

In many cases reconstructing βz will be sufficient


Performance

Performance studied on Z→ τ- τ+ Pythia8+Tauola simulations

Rest frame is nicely found for all generated boost values

Better performance in full boost method,

However needs accurate transverse direction as input

|Recoβ - Genβ|0.0 0.1 0.2 0.3 0.4 0.5 0.6

rate

(Nor

mal

ised

)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

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0.16

0.18

Z-method

XYZ-method

MeansZ-method 0.149XYZ-method 0.076

Note: Log scale

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Reconstructed vs generated boost


Applications

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Polarisation studyDecays of τ-leptons carry information about polarisation

Effect strongest in τ→πν

Looking at E(π-)/MZ

Effects washed out in detector frame (▲)

Almost fully recovered in reconstructed frame (△)

Z/ Mvis2 E0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

rate

(Nor

mal

ised

)0.00

0.01

0.02

0.03

0.04

0.05

0.06 Gen. h= 1Gen. h=-1Reco. h= 1Reco. h=-1Lab. h= 1Lab. h=-1

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τ→πν decays from Pythia8 + Tauola simulation


Spin Reconstruction

Polarisation configuration depends on the boson spin

Spin information can be extracted by studying the polarisation correlation of τ+ vs. τ-

rate

(Nor

mal

ised

)

1

2

3

4

5

6

7

-610×

H / M- 2 E0 0.2 0.4 0.6 0.8 1

H /

M+

2 E

0

0.2

0.4

0.6

0.8

1

rate

(Nor

mal

ised

)

1

2

3

4

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7

8

-610×

Z / M- 2 E0 0.2 0.4 0.6 0.8 1

Z /

M+

2 E

0

0.2

0.4

0.6

0.8

1Z (spin 1) H (spin 0)

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In detector frame In recon. frame

0.23 0.47

Sensitivity to spinFrom these distributions a likelihood function can be created

and a sensitivity to spin defined as

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logL-0.4 -0.2 0.0 0.2 0.4 0.6

Ent

ries/

0.01

0

2

4

6

8

10

12

14

16

Z

Mean 1.47e-01RMS 1.33e-02

HiggsMean -1.64e-01RMS 1.60e-02

Sensitivity 0.47

The Evis distributions for single ⇡ decay modes shown in figure 7 for the generated

and reconstructed heavy boson RFs show that the true correlations are partly recovered

in the reconstructed RF using the XYZ-method proposed in section 2. Furthermore, it is

shown that these correlations are much weakened in the laboratory frame.

4.1 Analysis of the sensitivity to spin

Conservation of angular momentum leads to distinctly di↵erent helicity configurations when

comparing the final states of Z0 and H decays in their rest frames. While the spin compo-

nents along the direction of flight of the ⌧+ must add up to ±1 for the Z0, the sum must

be 0 for the H decays. Thus the spin e↵ects can be studied by looking at the correlations

in Evis of the two ⌧ -leptons as shown in figure 8.

rate

(Nor

mal

ised

)

1

2

3

4

5

6

7

-610×

H / M- 2 E0 0.2 0.4 0.6 0.8 1

H /

M+

2 E

0

0.2

0.4

0.6

0.8

1

(a) Spin 0 (H sample).

rate

(Nor

mal

ised

)

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2

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8

-610×

Z / M- 2 E0 0.2 0.4 0.6 0.8 1

Z /

M+

2 E

0

0.2

0.4

0.6

0.8

1

(b) Spin 1 (Z0sample).

Figure 8. Energy correlations in the reconstructed RF using the XYZ-method of a ⌧+⌧� pair withboth ⌧ -leptons decaying to h± ⌫⌧ . To avoid e↵ects coming from the mass di↵erences all energiesare scaled with the mass of the decaying boson.

To quantify the observed di↵erence between spin 0 and spin 1 particles, the Z0 and H

samples were split into training and test samples. Probability density functions for both

spins, P0

(E⌧� , E⌧+) and P1

(E⌧� , E⌧+), were constructed as the fraction of events in a bin

of (E⌧� , E⌧+) in the corresponding training sample. From this a likelihood, L, is created

for N events from both training samples by

logL =NX

i=0

�log(P i

1

)� log(P i0

)�/N (4.1)

The test samples of Z0 and H events are divided in sub-samples of N events and the

likelihood is calculated for each sub-sample. The distribution of logL when N = 1000 and

when both ⌧ -lepton decays to h± ⌫⌧ is shown in figure 9.

Inspired by [15], a sensitivity, S, is defined such that the number of standard deviations

between the means of the two distributions, n�, is given by

n� = SpN (4.2)

– 10 –

The Evis distributions for single ⇡ decay modes shown in figure 7 for the generated

and reconstructed heavy boson RFs show that the true correlations are partly recovered

in the reconstructed RF using the XYZ-method proposed in section 2. Furthermore, it is

shown that these correlations are much weakened in the laboratory frame.

4.1 Analysis of the sensitivity to spin

Conservation of angular momentum leads to distinctly di↵erent helicity configurations when

comparing the final states of Z0 and H decays in their rest frames. While the spin compo-

nents along the direction of flight of the ⌧+ must add up to ±1 for the Z0, the sum must

be 0 for the H decays. Thus the spin e↵ects can be studied by looking at the correlations

in Evis of the two ⌧ -leptons as shown in figure 8.

rate

(Nor

mal

ised

)

1

2

3

4

5

6

7

-610×

H / M- 2 E0 0.2 0.4 0.6 0.8 1

H /

M+

2 E

0

0.2

0.4

0.6

0.8

1

(a) Spin 0 (H sample).

rate

(Nor

mal

ised

)

1

2

3

4

5

6

7

8

-610×

Z / M- 2 E0 0.2 0.4 0.6 0.8 1

Z /

M+

2 E

0

0.2

0.4

0.6

0.8

1

(b) Spin 1 (Z0sample).

Figure 8. Energy correlations in the reconstructed RF using the XYZ-method of a ⌧+⌧� pair withboth ⌧ -leptons decaying to h± ⌫⌧ . To avoid e↵ects coming from the mass di↵erences all energiesare scaled with the mass of the decaying boson.

To quantify the observed di↵erence between spin 0 and spin 1 particles, the Z0 and H

samples were split into training and test samples. Probability density functions for both

spins, P0

(E⌧� , E⌧+) and P1

(E⌧� , E⌧+), were constructed as the fraction of events in a bin

of (E⌧� , E⌧+) in the corresponding training sample. From this a likelihood, L, is created

for N events from both training samples by

logL =NX

i=0

�log(P i

1

)� log(P i0

)�/N (4.1)

The test samples of Z0 and H events are divided in sub-samples of N events and the

likelihood is calculated for each sub-sample. The distribution of logL when N = 1000 and

when both ⌧ -lepton decays to h± ⌫⌧ is shown in figure 9.

Inspired by [15], a sensitivity, S, is defined such that the number of standard deviations

between the means of the two distributions, n�, is given by

n� = SpN (4.2)

– 10 –

Sensitivity

Likelihoods for Z and Higgs


[GeV]0 20 40 60 80 100 120 140 160 180 200

rate

(Nor

mal

ised

)0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045hτhτ → 0 Z

° > 170jetsφΔ

Z-method

Alternative Mass Estimation

Neutrinos in the τ decay makes reconstruction the mass of a resonance non-trivial

In the resonance rest frame, the leading τ-jet energy will accumulate towards half the resonance mass

As a simple approximation the kinematic edge can be found at the steepest slope in the leading τ-energy distribution

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Mass can be extrapolated similarly to W mass measurements

Z →τ+τ- Pythia8 + Tauola simulation

2E1


Comparison to other methods

Both methods can be applied on all events

Methods currently used at the LHC, only works for a fraction of suitable event topologies

In the region where the Coll. Approx. breaks down, this method can be applied without using ETmiss

Method is extremely fast and simple

more 1000x faster than likelihood method currently used by CMS and ATLAS

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Summary and Outlook

A way of reconstructing rest frame of τ pair resonances and its applications have been presented

Method and its applications are published atarxiv:1105.6003 (recently accepted in JHEP)

Suitable to all τ decay channels and not necessarily specific to τ pairs

No detector effects have been included in plots shown

However, we are investigating use of the method in ATLAS

C++ code available on request

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Peter Rosendahl, University of Bergen Spaatind - 05 jan 2012 14

Thank you for your attention

Reconstructing the rest frame of systems

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