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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

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D J MillerDESY, 3rd February 20062 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

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D J MillerDESY, 3rd February 20063 Supergravity? broken by gravity GMSB?broken by new gauge interactions AMSB?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

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D J MillerDESY, 3rd February 20064 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.

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D J MillerDESY, 3rd February 20065 2 problems with measuring masses at the LHC: Dont 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 )

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D J MillerDESY, 3rd February 20066 Measure masses using endpoints of invariant mass distributions e.g. consider the decay m ll is maximised when leptons are back-to-back in slepton rest frame angle between leptons

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D J MillerDESY, 3rd February 20067 3 unknown masses, but only 1 observable, m ll extend chain further to include squark parent: now have: m ll, m ql +, m ql -, m qll 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…]

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D J MillerDESY, 3rd February 20068 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.

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D J MillerDESY, 3rd February 20069 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 doesnt 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 doesnt change much for other mSUGRA inspired scenarios.

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D J MillerDESY, 3rd February 200610 Cannot normally distinguish the two leptons since is a Majorana particle Must instead define m ql (high) and m ql (low) Do we have Some extra difficulties: OR ?

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D J MillerDESY, 3rd February 200611 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 m qll > /2 identical to m qll except require > /2 It is the minimum of this distribution which is interesting angle between leptons in slepton rest frame

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D J MillerDESY, 3rd February 200612 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 m ql

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D J MillerDESY, 3rd February 200613 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

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D J MillerDESY, 3rd February 200614 Cuts to remove backgrounds: At least 3 jets, with p T > 150, 100, 50 GeV E T, miss > max(100 GeV, 0.2 M eff ) with 2 isolated opposite-sign same-flavour leptons (e, ) with p T > 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

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D J MillerDESY, 3rd February 200615 Theory curve End result Z peak (correlated leptons) Distribution for m ll after cuts

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D J MillerDESY, 3rd February 200616 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. m ql 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).

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D J MillerDESY, 3rd February 200617 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.

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D J MillerDESY, 3rd February 200618 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 E exp 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]

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D J MillerDESY, 3rd February 200619 We can do this blind (i.e. input masses into the Monte Carlo only and dont look at them again until we are done) and see what we get Input mass (GeV) Measured mass (GeV) Error (GeV) 96.196.33.8 143.0143.23.8 176.8177.03.7 537.2537.56.1 491.9492.413.4

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D J MillerDESY, 3rd February 200620 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.

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D J MillerDESY, 3rd February 200621 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

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D J MillerDESY, 3rd February 200622 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 isnt 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

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D J MillerDESY, 3rd February 200623 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

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D J MillerDESY, 3rd February 200624 Quantify this by asking how large the final feature is compared to the total height of the distribution. e.g. a b r=a/b Many parameter scenarios have dangerous feet or drops.

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D J MillerDESY, 3rd February 200625 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.

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D J MillerDESY, 3rd February 200626 For example, in SPS 1a, using values of m qll, m ql high, m ql low and m ll with no errors, fitting to the LSP mass returns a second solution at around 80 GeV. true mass false mass region boundary In this case, the false mass is far enough away that this should not be a problem.

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D J MillerDESY, 3rd February 200627 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 point region boundary Nominal endpoints Endpoints with errors Region boundaries:

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D J MillerDESY, 3rd February 200628 second mass solutions - at SPS 1a this is caused by

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D J MillerDESY, 3rd February 200629 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.

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D J MillerDESY, 3rd February 200630 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

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D J MillerDESY, 3rd February 200631 An example invariant mass distribution Consider This invariant mass is not easily measurable since we cannot tell which lepton is l f, 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 l n in the rest frame of angle between q and l f in the rest frame of

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D J MillerDESY, 3rd February 200632 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.

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D J MillerDESY, 3rd February 200633 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.

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D J MillerDESY, 3rd February 200634 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.

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D J MillerDESY, 3rd February 200635 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

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D J MillerDESY, 3rd February 200636 Problem 1 solved We calculate m qll, m ql high, m ql low and m ll in this way (but not m qll > /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.

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D J MillerDESY, 3rd February 200637 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.

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D J MillerDESY, 3rd February 200638 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)

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D J MillerDESY, 3rd February 200639 Step 2: Compare with parton level with cuts (previously defined) Cuts cause a decrease in events for low invariant mass, but dont affect the high invariant mass edge.

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D J MillerDESY, 3rd February 200640 Its fairly obvious why this is: Only the cut on lepton P T is dangerous, but low lepton P T means low invariant masses

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D J MillerDESY, 3rd February 200641 Step 3: compare with PYTHIA with cuts and FSR FSR causes a slight shift of the entire distribution to lower mass.

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D J MillerDESY, 3rd February 200642 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

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D J MillerDESY, 3rd February 200643 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]

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D J MillerDESY, 3rd February 200644 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 m qll > /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?

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D J MillerDESY, 3rd February 200645 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

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