Using mass distributions to improve SUSY mass measurements

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 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, 3 rd February 2006

PART I: Edges and endpoints 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…. D J Miller DESY, 3 rd February 2006 2

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? ’ Supergravity? GMSB? AMSB? broken by gravity broken by new gauge interactions broken by anomalies or something else…. ? 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 D J Miller DESY, 3 rd February 2006 3

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. Need very accurate measurements of SUSY masses Supersymmetry Parameter Analysis: SPA Convention and Project J. A. Aguilar-Saavedra et al, hep-ph/0511344 D J Miller DESY, 3 rd February 2006 4

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 Missing energy/momentum ) escapes detector Cannot use traditional method of peaks in invariant mass distributions to measure SUSY masses Instead measure endpoint of invariant mass distributions D J Miller DESY, 3 rd February 2006 5

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 D J Miller DESY, 3 rd February 2006 6

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 [Hinchliffe, Paige, Shapiro, Soderqvist and Yao, Phys. Rev D 55 (1997) 5520, Allanach, Lester, Parker, Webber, JHEP 0009 (2000) 004, and many others…] D J Miller DESY, 3 rd February 2006 7

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. D J Miller DESY, 3 rd February 2006 8

Example m. SUGRA inspired scenario: [See Allanach et al, Eur. Phys. J. C 25 (2002) 113, hep-ph/0202233] Our. Dark decay chain doesn’t work, matter constraints others areout possible. rule this Thebut hatched area is amenable to this method in some form. Its pretty hard to do lighter green is where This area doesn’t change much anything with this! for other m. SUGRA inspired scenarios. D J Miller DESY, 3 rd February 2006 9

Some extra difficulties: Cannot normally distinguish the two leptons since is a Majorana particle Do we have OR ? Must instead define mql (high) and mql (low) D J Miller DESY, 3 rd February 2006 10

Endpoints are not always linearly independent e. g. if and then the endpoints are Four endpoints not always sufficient to find the masses angle between leptons in slepton rest frame Introduce new distribution mqll > /2 identical to mqll except require > /2 It is the minimum of this distribution which is interesting D J Miller DESY, 3 rd February 2006 11

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 D J Miller DESY, 3 rd February 2006 12

Without spin correlations: With spin correlations: Recall, cannot distinguish ql+ and ql) must average over them Spin correlations cancel when we sum over lepton charges ) Pythia OK for our purposes [Barr, Phys. Lett. B 596 (2004) 205] D J Miller DESY, 3 rd February 2006 13

Cuts to remove backgrounds: " At least 3 jets, with p. T > 150, 100, 50 Ge. V " ET, miss > max(100 Ge. V, 0. 2 Meff) with " 2 isolated opposite-sign same-flavour leptons (e, ) with p. T > 20, 10 Ge. V After these cuts the remaining background is mainly 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 D J Miller DESY, 3 rd February 2006 14

Distribution for mll after cuts End result ‘Theory’ curve Z peak (correlated leptons) D J Miller DESY, 3 rd February 2006 15

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. 20 Ge. V 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. 3. Both these methods use only data (no theory input). D J Miller DESY, 3 rd February 2006 16

Inconsistency cut: Mixed events: The final result has been rescaled to allow comparison with theory curve. About ¼ of the events survive. D J Miller This seems to work much better. Notice that beyond the kinematic maximum, the background is very well predicted. DESY, 3 rd February 2006 17

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] D J Miller DESY, 3 rd February 2006 18

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 Measured Error (Ge. V) mass (Ge. V) D J Miller 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 DESY, 3 rd February 2006 19

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 all the mass measurements improve. D J Miller DESY, 3 rd February 2006 precisely (e. g. 50 Me. V) then 20

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 D J Miller DESY, 3 rd February 2006 I will explain these blue curves later 21

PART II: Problems with the endpoint method 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-SPS 1 a points. linear fit lower part of plot obscured by background endpoint mismeasured For SPS 1 a, this isn’t such a problem because the edges are almost linear and the backgrounds are not large compared to the signal. D J Miller DESY, 3 rd February 2006 22

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 Ge. V! D J Miller DESY, 3 rd February 2006 Here, there is a sudden drop to zero 23

Quantify this by asking how large the final ‘feature’ is compared to the total height of the distribution. Many parameter scenarios have dangerous “feet” or “drops”. D J Miller DESY, 3 rd February 2006 e. g. b a r=a/b 24

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 hierarchy e. g. This splits the mass-space into different regions, each of which may contain a mass solution which fits the measured endpoint. D J Miller DESY, 3 rd February 2006 25

For example, in SPS 1 a, 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 Ge. V. region boundary true mass false mass In this case, the false mass is far enough away that this should not be a problem. D J Miller DESY, 3 rd February 2006 26

Region boundaries: 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. Endpoints with errors Nominal endpoints model point D J Miller region boundary DESY, 3 rd February 2006 27

second mass solutions - at SPS 1 a this is caused by D J Miller DESY, 3 rd February 2006 28

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. D J Miller DESY, 3 rd February 2006 29

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 D J Miller DESY, 3 rd February 2006 30

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 D J Miller DESY, 3 rd February 2006 31

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 D J Miller , v. DESY, 3 rd February 2006 32

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. for D J Miller with DESY, 3 rd February 2006 33

Finally The multi-function form of this is coming from the question “can maximum opening angle or not”? reach its 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. D J Miller DESY, 3 rd February 2006 34

In this derivation we have completely ignored the widths of the particles In principle, in every event, each particle has a definite p 2 which plays the role of m 2 in our derivation. So our derivation is OK is we now smear p 2 around m 2. derived distribution with no widths new distribution D J Miller DESY, 3 rd February 2006 35

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. Problem 1 solved We have an analytic expression for “r”, the quantity that tells us when we have a foot or a drop. 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. Problem 2 solved D J Miller DESY, 3 rd February 2006 36

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. Problem 3 solved We can now use the data from (almost) the entire distribution, not just the edge, so statistical error will get better too. 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. D J Miller DESY, 3 rd February 2006 37

Step 1: compare our analytic results with the parton level of PYTHIA, with no other effects. (SPS 1 a) Works very well – only deviations are statistical D J Miller DESY, 3 rd February 2006 38

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. D J Miller DESY, 3 rd February 2006 39

Its fairly obvious why this is: Only the cut on lepton PT is dangerous, but low lepton PT means low invariant masses D J Miller DESY, 3 rd February 2006 40

Step 3: compare with PYTHIA with cuts and FSR causes a slight shift of the entire distribution to lower mass. D J Miller DESY, 3 rd February 2006 41

Step 4: compare with detector level We used Acer. Det with (a simplified version of ATLFAST) [E. Richter-Was, hep-ph/0207355] 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 D J Miller Some combinatoric background remains because we were very conservative with our inconsistency cut DESY, 3 rd February 2006 42

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] D J Miller DESY, 3 rd February 2006 43

Things to do 1. So far, we have only simulated with Acer. Det 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? D J Miller DESY, 3 rd February 2006 44

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: non-linear edges feet and drops multiple solutions 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! D J Miller DESY, 3 rd February 2006 45