Using mass distributions to improve SUSY mass measurements at the LHC D.J. Miller, DESY, 3 rd February 2006 Part I: Edges and endpoints Part II: Problems

Using mass distributions to improve SUSY mass measurements at the LHC

D.J. Miller, DESY, 3rd February 2006

Part I: Edges and endpoints

Part II: Problems with the endpoint method

Part III: Using shapes instead of endpoints

B. K. Gjelsten, D. J. Miller, P. Osland, JHEP 0412 (2004) 003, hep-ph/0410303

D.J. Miller, A.Raklev, P. Osland, hep-ph/0510356

D J Miller DESY, 3rd February 2006 2

Introduction

Low energy supersymmetry is an exciting and plausible extension to the Standard Model.

It has many advantages:

• Extends the Poincaré algebra of space-time• Solves the Hierarchy Problem• More amenable to gauge unification• Provides a natural mechanism for generating the Higgs potential• Provides a good Dark Matter candidate ( )

Lots of exciting new phenomenology at the LHC: squarks, sleptons, neutralinos, charginos, Higgs bosons….

PART I: Edges and endpoints


Supergravity? broken by gravityGMSB? broken by new gauge interactionsAMSB? broken by anomalies

or something else….?

But: The MSSM has 105 extra parameters compared to the Standard Model!

This is a parameterisation of our ignorance of supersymmetry breaking.

If supersymmetry is discovered, the next question to ask is ‘How is it broken?’

To answer this question,

Measure soft supersymmetry breaking parameters at the LHC

Run them up to the GUT scale and compare with susy breaking models


Supersymmetry Parameter Analysis: SPA Convention and Project J.A. Aguilar-Saavedra et al, hep-ph/0511344

Need very accurate measurements of SUSY masses

The uncertainties in masses/parameters at low energy magnified by RGE running

Not so bad for the sleptons, but is very difficult for the squarks and Higgs bosons.


2 problems with measuring masses at the LHC:

• Don’t know centre of mass energy of collision √s

• R-parity conserved (to prevent proton decay)

Lightest SUSY Particle (LSP) stable

Cannot use traditional method of peaks in invariant mass distributions

to measure SUSY masses

escapes detector

Instead measure endpoint of invariant mass distributions

Missing energy/momentum )


Measure masses using endpoints of invariant mass distributions

e.g. consider the decay

mll is maximised when leptons are back-to-back in slepton rest frame

angle between leptons


3 unknown masses, but only 1 observable, mll

extend chain further to include squark parent:

now have: mll, mql+, mql-, mqll

4 unknown masses, but now have 4 observables

) can measure masses from endpoints

[Hinchliffe, Paige, Shapiro, Soderqvist and Yao, Phys. Rev D 55 (1997) 5520, Allanach, Lester, Parker, Webber, JHEP 0009 (2000) 004, and many others…]


For the chain we need:

This is possible over a wide range of parameter space.

If this chain is not open, the method is still valid, but we need to look at other decay chains.

In this talk I will consider only the decay chain above.


lighter green is where

Example mSUGRA inspired scenario:

[See Allanach et al, Eur.Phys.J.C25 (2002) 113, hep-ph/0202233]

Dark matter constraints rule this out

Our decay chain doesn’t work, but others are possible.

Its pretty hard to do anything with this!

The hatched area is amenable to this method in some form.

This area doesn’t change much for other mSUGRA inspired

scenarios.


Cannot normally distinguish the two leptons since is a Majorana particle

Must instead define mql (high) and mql (low)

Do we have

Some extra difficulties:

OR ?


Endpoints are not always linearly independent

e.g. if and

then the endpoints are

Four endpoints not always sufficient to find the masses

Introduce new distribution mqll >/2 identical to mqll except require >/2

It is the minimum of this distribution which is interesting

angle between leptons in slepton

rest frame


Spin correlations

PYTHIA does not include spin correlations (HERWIG does)

OK for decays of scalars, but may give wrong results for fermions

PYTHIA ‘forgets’ spin

This could be a problem for mql


Without spin correlations:

With spin correlations:

[Barr, Phys.Lett. B596 (2004) 205]

Recall, cannot distinguish ql+ and ql-

) must average over them

Spin correlations cancel when we sum over lepton charges

) Pythia OK for our purposes


Cuts to remove backgrounds:

At least 3 jets, with pT > 150, 100, 50 GeV

ET, miss > max(100 GeV, 0.2 Meff) with

2 isolated opposite-sign same-flavour leptons (e,) with pT > 20,10 GeV

Remove this background using different-flavour-subtraction

Leptons in the signal are correlated (the same)

Leptons in the background are uncorrelated

By subtracting the sample with same-flavour leptons we remove the different-flavour lepton background

After these cuts the remaining background is mainly


‘Theory’ curve

End result

Z peak (correlated leptons)

Distribution for mll after cuts


Combinatoric backgrounds

Generally there will be 2 squarks in each event

there are extra jets not associated with our decay chain

If we choose the wrong jet to construct the invariant masses we will mess up our endpoints

We can cure this problem in 2 ways:

1. Inconsistency cuts: For many events, choosing the wrong jet results in one invariant mass, e.g. mql high being unreasonable. If we only use events where this is the case, we

are guaranteed to choose the correct jet. We use a very conservative cut (e.g. 20GeV above the first endpoint guess).

2. Mixed Events: We can simulate the combinatoric background by deliberately pairing the leptons with the wrong jet, e.g. from a different event. Subtracting off this simulated background removes the combinatoric background.

Both these methods use only data (no theory input).


Inconsistency cut:

The final result has been rescaled to allow comparison with the theory curve. About ¼ of the events survive.

Mixed events:

This seems to work much better. Notice that beyond the kinematic maximum, the background is very well predicted.


Procedure to extract endpoints and masses:

Make a (Gaussian smeared) linear extrapolation of the edge to find the endpoint

measured set of endpoints with errors

Generate 10,000 sample ‘endpoint sets’ Eexp using these values and errors

Use method of least squares to fit the masses to these endpoints:

[If the endpoints were uncorrelated, W would be diagonal and this would become a simple 2 fit]


We can do this “blind” (i.e. input masses into the Monte Carlo only and don’t look at them again until we are done) and see what we get

Input mass (GeV)

Measured mass (GeV)

Error (GeV)

96.1 96.3 3.8

143.0 143.2 3.8

176.8 177.0 3.7

537.2 537.5 6.1

491.9 492.4 13.4


However, the kinematic endpoints depend strongly on mass differences

e.g.

) the mass measurements are very strongly correlated

Thus mass differences are much better measured, e.g.

Synergy between the LHC and ILC: if the ILC measures precisely (e.g. 50MeV) then all the mass measurements improve.


Widths here are error widths, not real widths

mass differences much better measured – could be exploited by measuring one of

the masses at an e+e- linear collider I will explain these blue curves later


Problem 1: We used a Gaussian smeared straight line to find endpoints, but can we really trust a linear fit?

Look at some other non-SPS1a points.

lower part of plot obscured

by background

linear fit

endpoint mismeasured

For SPS 1a, this isn’t such a problem because the edges are almost linear and the backgrounds are not large compared to the signal.

PART II: Problems with the endpoint method


Problem 2: The invariant mass distributions often have strange behaviours near the endpoints which may be obscured by remaining backgrounds

Notice a “foot” here. This caused us to underestimate this endpoint by 9 GeV!

Here, there is a sudden drop to zero


Quantify this by asking how large the final ‘feature’ is compared to the total height of the distribution.

e.g.

a

b

r=a/bMany parameter scenarios have dangerous “feet” or “drops”.


Problem 3: One set of mass endpoints can be fit by more than one set of masses!

This has 2 causes:

Endpoints themselves depend on the mass hierarchye.g.

This splits the mass-space into different regions, each of which may contain a mass solution which fits the measured endpoint.


For example, in SPS 1a, using values of mqll, mql high, mql low and mll with no errors,

fitting to the LSP mass returns a second solution at around 80 GeV.

true massfalse mass

region boundary

In this case, the false mass is far enough away that this should not be a problem.


If the nominal masses are near a region boundary, over-constraining the system with another measurement, or simply having large enough errors on the endpoints, can create multiple local minima of the 2 distribution in different regions.

model pointregion boundary

Nominal endpoints Endpoints with errors

Region boundaries:


second mass solutions - at SPS 1a this is caused by


PART III: Using shapes instead of endpoints

All of these problems are associated with using only endpoints of distributions.

If we fit the entire shape of the invariant mass distribution, we should avoid them.

In principle, this could be done numerically:

Use PYTHIA to produce sample data sets for lots of different mass spectra and compare the invariant mass distributions of these sets with the real data to see which mass spectra is best.

Allows you to include hadronization and detector effects directly into the sample data set.

In practice, it is better to do this analytically:

Numerical generation of data sets is very slow, and impractical

Analytic solutions allow one to easily examine features of the distributions which you might otherwise miss.


Using analytic formula for the differential mass distributions:

Problem 1 (non-linear extrapolation to endpoint)

Our analytic expression for the shape should tell us exactly the behaviour of the invariant mass distribution near the endpoint, giving us a good fit function.

Problem 2 (feet and drops)

With an analytic expression we will know about any anomalous structures even if they are hidden by backgrounds, and be able to correct for them.

Problem 3 (multiple solutions)

Other features of the shape will serve to the distinguish the different solutions which were obtained by the endpoint method.

Additionally, we can use a larger proportion of events, i.e. not just the events near the endpoints


An example invariant mass distribution

Consider

This invariant mass is not easily measurable since we cannot tell which lepton is lf, but is

a simple example of the method we use.

For simplicity, lets also assume that and are scalars. This amounts to neglecting spin correlations (like PYTHIA). It is actually OK for our purposes, but is easily corrected later anyway.

I will be interested in:angle between q and ln in the

rest frame of

angle between q and lf in the

rest frame of


Our assumption that the intermediate particles are scalars means that the differential rate cannot depend on u or v, but obviously we still need to keep 0 < (u,v) < 1.

So

The quantity we want to investigate is

energy/momentum conservation )

with

We can now change variables from u, v to , v.


for with

So far, this was all very easy.

The “difficult” part is integrating out v, not because the integration itself is hard, but we have to get the correct integration limits.


Finally

The multi-function form of this is coming from the question “can reach its maximum opening angle or not”?

Of course, this was the simplest (non-trivial) case. The more physical expressions are much harder to derive because the limits become very complicated.

To include spin correlations, all we need to do is change modify the distribution in u and v:

e.g.


In this derivation we have completely ignored the widths of the particles

In principle, in every event, each particle has a definite p2 which plays the role of m2 in our derivation.

So our derivation is OK is we now smear p2 around m2.

derived distribution with no widths

new distribution


Problem 1 solved

We calculate mqll, mql high, mql low and mll in this way (but not mqll >/2).

We now know the analytic form of the edges which lead down to the endpoints. They are all simple logarithms + polynomials, which can be easily fit to the edges.

We have an analytic expression for “r”, the quantity that tells us when we have a foot or a drop.

Problem 2 solved

In the example we derived r=1 always, which is a bit dull… …. but we can now perform detailed scans to see which areas are dangerous, and correct for them.


However, our analytic shapes are parton level so we must ask if the features of the shape are preserved when we include cuts, hadronisation, FSR, detector effects etc.

Problem 3 solved

We can distinguish different mass solutions from the different behaviour of the entire distribution. Although they have the same endpoints, they do not have the same shape.

We can now use the data from (almost) the entire distribution, not just the edge, so statistical error will get better too.


Step 1: compare our analytic results with the parton level of PYTHIA, with no other effects.

Works very well – only deviations are statistical(SPS 1a)


Step 2: Compare with parton level with cuts (previously defined)

Cuts cause a decrease in events for low invariant mass, but don’t affect the high invariant mass edge.


Its fairly obvious why this is:

Only the cut on lepton PT is dangerous, but low lepton PT means low invariant masses


Step 3: compare with PYTHIA with cuts and FSR

FSR causes a slight shift of the entire distribution to lower mass.


We used AcerDet with (a simplified version of ATLFAST)

Step 4: compare with detector level

[E. Richter-Was, hep-ph/0207355]

Some combinatoric background remains because we were very conservative with our inconsistency cut

As well as previous cuts, use a b-tag to remove events with b-squarks

Remove combinatoric backgrounds with an inconsistency cut

parton level

analytic distribution


Using the shapes to extract masses

These shapes can be used in two ways:

1. As a guide to the measurement of endpoints.

Use the functions derived for extrapolation of the edge of the distribution to its endpoint.

Use the expressions to identify if you have any dangerous feet or drops.

Discard any extra solutions which are not compatible with the gross features of the shape.

2. As a fit function to be compared with the observed differential distributions and used to extract masses directly.

[or a combination of the two]


Things to do

1. So far, we have only simulated with AcerDet and

Need to do proper experimental simulations with high luminosity (e.g. 300 fb-1) and fit the masses to this data.

How much of an improvement to the measured masses does using shapes give?Both ATLAS and CMS are interested in doing this.

2. Investigate distributions like mqll >/2

Helps set overall scale with endpoints. Is it so useful for shapes? >/2 was arbitrary. Can we do better?Need to derive the distribution for whatever function we come up with (hard?)

3. Investigate other decay chains

This decay chain is only a portion of the parameter space. Can we use the same methods for other decay chains? How well can we do?The derivations of the shapes was model independent. Can we use this method for other physics, e.g. extra dimensions, little Higgs models etc?


Conclusions and Summary

Missing energy/momentum from the LSP in minimal SUSY makes traditional methods for measuring masses difficult.

We can instead use endpoints of invariant mass distributions.

However, this introduces a number of problems:

We can solve these problems by analyzing the entire invariant mass distributions.

We have derived analytic forms for these distributions and compared them to realistic simulations.

We find good agreement and hope to now use these functions to fit for the superpartner masses at the LHC.

Lots still to do!

non-linear edges feet and drops multiple solutions

Documents

Using mass distributions to improve SUSY mass measurements at the LHC D.J. Miller, DESY, 3 rd February 2006 Part I: Edges and endpoints Part II: Problems