We introduce a variable useful for measuring masses of particles pair
produced at hadron colliders, where each particle decays to one particle that
is directly observable and another particle whose existence can only be
inferred from missing transverse momenta. This variable is closely related to
the transverse mass variable commonly used for measuring the $W$ mass at hadron
colliders, and like the transverse mass our variable extracts masses in a
reasonably model independent way. Without considering either backgrounds or
measurement errors we consider how our variable would perform measuring the
mass of selectrons in a mSUGRA SUSY model at the LHC

Lester, CG

Summers, DJ

Physics Letters B

arXiv

C.G Lester

D.J Summers

Crossref

Measuring masses of semi-invisibly decaying particle pairs produced at hadron colliders

Apollo (Cambridge)

arXiv:hep-ph/9906349v1  12 Jun 1999Cavendish–HEP–99/07June 1999Measuring masses of semi-invisibly decayingparticles pair produced at hadron collidersC. G. Lester1 and D. J. Summers2High Energy Physics Group, Cavendish Laboratory,Madingley Road, Cambridge CB3 0HE, EnglandAbstractWe introduce a variable useful for measuring masses of particles pair produced athadron colliders, where each particle decays to one particle that is directly observableand another particle whose existence can only be inferred from missing transversemomenta. This variable is closely related to the transverse mass variable commonlyused for measuring the W mass at hadron colliders, and like the transverse mass ourvariable extracts masses in a reasonably model independent way. Without consideringeither backgrounds or measurement errors we consider how our variable would performmeasuring the mass of selectrons in a mSUGRA SUSY model at the LHC.1email: lester@hep.phy.cam.ac.uk2email: summers@hep.phy.cam.ac.ukBefore 1983, e+e− colliders dominated the search for new particles, their clean final statesmaking identification of new particles far easier than at hadron machines. However in 1983this changed with the observation of the W and Z bosons by the UA1[1, 3] and UA2[2, 4]collaborations at the CERN pp̄ collider. Despite the more difficult conditions where theinitial state protons and anti-protons gave rise to an underlying event which complicatesmeasurement of the final state, the higher energies available to a pp̄ machine meant theydiscovered the W and Z bosons before other machines had enough energy to produce them.This trend continued in 1994/5 when CDF and DØ discovered the t quark at the Tevatronpp̄ collider [5]. To date the only direct experimental evidence that we have for the t quarkis from the Tevatron.Clearly, the greater energies available to pp̄ machines gives them an advantage in discov-ering fundamental particles, and we can hope that this trend will be continued at the LHC.Upon discovery of a new particle the first question that is asked is what is the mass of thisparticle. There are two fundamentally different ways of measuring the mass of a particle.Usually one tries to extract the mass of the particle directly from the observed events. Forexample UA1 and UA2 detected the Z boson in its decay to lepton pairs e+e− or µ+µ−. Foreach event that contains a pair of leptons one can construct the mass of the particle thatproduced the leptons as,m2 = (pl+ + pl−)2 (1)and on an event by event basis obtain a direct estimate the mass of the particle. Alternativelyone could make an indirect model dependent measure of the Z mass by measuring the“Z event” cross-section, and using a model that predicts the Z cross-section in terms of theZ mass, perform a fit to the measured cross-section. For measuring the Z mass the directmeasurement is superior, if only for its lack of model dependence, however when measuringthe mass of the W boson the case is less clear. At the CERN pp̄ collider the W bosonwas detected in its decay to a charged lepton and a neutrino, however the neutrino goesunobserved and so only gives rise to missing momentum. This means that one can’t directlyobserve the W mass from the lepton and neutrino momenta. UA1 [1] formed the transversemass variable,m2T = 2(EeT /ET − peT · /pT ), (2)where /E2T ≡ /p2T . This variable has the property that,m2T ≤ m2W (3)with equality possible for events where the lepton and neutrino are produced with the samerapidity. Thus although one can’t obtain the W mass from a single event, one can obtaina lower limit on the W mass. In addition if one obtains the lower mass bound from manyevents, this can approach the W mass, and so one can extract the W mass in a modelindependent way. UA1 [1] along with UA2 [2] also performed a model depenendent fit tothe pT spectrum for the lepton, extracting the W mass from that model.In practice, though, measuring the W mass with mT does have some small model depen-dence, as the precise fraction of events which occur with mT close to mW is still dependent1pppT/p2p1Figure 1: Diagram of the generic process that we consider. A hadronic collision that leads toa pair of particles being produced, which each decay into one particle that is observed withmomenta p1 and p2 respectively; and one particle (shown as a wavy lines) that is not directlydetected, and whose presence can only be inferred from the missing transverse momentum,/pT .on the physics processes which produce the W boson, and how the W boson decays. Inaddition, the missing transverse momentum is poorly measured experimentally comparedwith pT (l), so the theoretical model dependence of the measurement of mW from the pT (l)spectrum is balanced by the experimental error on extracting mW from the edge of the mTspectrum.In this paper we wish to introduce a variable which measures particle masses, which liketransverse mass has little dependence on exactly how such massive particles are produced.The variable is used for the generic process shown in figure 1, where a hadronic collision pairproduces a massive particle whose dominant decay is into one observed and one unobservedparticle. This unobserved particle can only be detected from the missing momentum that itcarries away, and that the massive particle is pair produced means that we can only measurethe missing momentum of the pair of invisible particles. Although this may sound like anunusual process to look for new particles, it naturally occurs in any theory where there is an(approximately) conserved charge, and the lightest particle with that charge is only weaklyinteracting. Two examples of where such a situation can occur are SUSY models and a4th lepton generation. In R-parity conserving SUSY models, sparticles are pair produced,and cascade decay to the lightest sparticle, which must be stable and is expected to not bedirectly detectable. Slepton production and decay can often follow this route:pp → X + l̃+R l̃−R → X + l+l−χ̃01χ̃01. (4)In such an event the pair of lightest SUSY particles, χ̃, go unobserved and only leave theirsignature as missing transverse momentum.For a 4th generation lepton the charged lepton would be pair produced in a Drell-Yantype process, decaying to a neutrino and a W boson,pp → X + l+4 l−4 → X + ν̄l4W+νl4W− (5)2the (probably massive) νl4 going unobserved, while the W bosons could be detected in theirdecays to either lν or to jets.We now look specifically at the process given in equation 4, although the variable whichwe define would work identically in any process where a particle is pair produced and decaysto one visible and one invisible particle.The variable that we wish to introduce is closely related to mT , however the standarddefinition of mT , given in equation 2, assumes that the unobserved particle is massless, sowe return to the derivation of this variable. For the decay,l̃ → lχ̃ (6)for arbitrary momenta we can write,m2l̃= m2l + m2χ̃ + 2(ET lET χ̃ cosh(∆η) − pT l · pT χ̃) (7)where ET =√p2T + m2 and ∆η is the difference in rapidity, η = 12ln[(E + pz)/(E − pz)],between between the l and χ̃.Now as cosh η ≥ 1 we have,m2l̃≥ m2T (pT l,pT χ̃) ≡ m2l + m2χ̃ + 2(ET lET χ̃ − pT l · pT χ̃). (8)This gives a version of transverse mass valid for arbitrary masses, with equality when the land χ̃ are produced with the same rapidity. Notice that ET l and ET χ̃ depend on m2l and m2χ̃respectively.The transverse mass can’t be formed directly from the process in equation (4), as boththe neutralinos give rise to missing momentum, however we can experimentally measure thesum of their transverse momenta as the missing transverse momenta in the event,/pT = pT χ̃a + pT χ̃b. (9)If pT χ̃a and pT χ̃b were obtainable, then one could form two transverse masses, and using therelationship (8) obtain,m2l̃≥ max{m2T (pT l−,pT χ̃a), m2T (pT l+,pT χ̃b)} (10)However, not knowing the form of the splitting (9), the best we can say is that:m2l̃≥ M2T2 ≡ min/p1+/p2=/pT[max {m2T (pT l−, /p1), m2T (pT l+ , /p2)}](11)With the minimization over all possible 2-momenta, /p1,2, such that their sum gives theobserved missing transverse momentum, /pT . This is the variable, called MT2, that we wishto introduce. This bound we can obtain directly from experimentally measured parameters.Although not totally transparent, for particular momenta, MT2 can be equal to ml̃; therequirement being that for both slepton decays the lepton and neutralino are produced at3the same rapidity (although the sleptons themselves can be at differing rapidities), and inaddition,(pT l−ET l−− pT χ̃aET χ̃a)∝(pT l+ET l+− pT χ̃bET χ̃b). (12)We have not managed to derive a general analytic expression for the minimization oversplittings of /pT ; largely because an experimental measurement of /pT only measures themissing transverse momentum, and neither the missing energy nor the missing longitudinalmomenta. This means that /pT is not a 4 vector, which means that MT2 can not be calculatedin a manifestly Lorentz invariant manner. The complication that this introduces to theminimization is enough to making an analytic solution non trivial. However if we take oneof the parameters that we minimize over to be m2 = /p2T (i.e. that /ET =√m2 + /p2T ), andin addition note that longitudinal momenta play no part in the definition of MT2, then theremaining minimization becomes Lorentz invariant, and this means that it can be solvedanalytically. This leaves one minimization over m2 that we performed numerically. Theform of this minimization is not particularly illuminating, and so we do not give it here;however a computer code for evaluating M2T2 is available from the authors.Of course, the variable MT2 is only good at extracting particle masses from processeshaving many events close to the maximum allowed value, and this depends on the physicsof the process being measured. Hence, a priori, we can not say that MT2 is useful formeasuring particle masses in all processes. However we expect typical physics processes tohave reasonable numbers of events close to the maximum value allowed for MT2 and henceexpect it to be a useful variable. To illustrate the variable in use, we consider a modelwhich allows the process shown in equation (4), at the expected LHC centre of mass energy√s = 14TeV. Our SUSY model is the fifth minimal supergravity model (mSUGRA) [6]point selected by the LHC Committee in 1996 for detailed study by the ATLAS and CMScollaborations [7]. This model is characterized by,{tanβ = 2.1, m1/2 = 300 GeV, m0 = 100 GeV, A0 = 300 GeV, µ > 0}. (13)We generate events using the SUSY version of the Herwig generator [8]. For our purposesthe only important features of the model that we use to produce our events are,ml̃R = 157.1 GeV mχ̃01 = 121.5 GeV. (14)The angular distributions for the sleptons are given by the Drell-Yan process that producesspin-0 particles, while the decay of the sleptons is isotropic due again to their being spinless.Herwig serves to build up a reasonably realistic underlying event, i.e. that /pT 6= pT l+ +pT l−.Defining the missing momenta as,/pT = pT χ̃a + pT χ̃b, (15)we generate 1105 events, which corresponds to an integrated luminosity of about 30 fb−1,which the LHC should collect in approximately one year of low luminosity running.4Figure 2: MT2 distribution for the process pp → X+ l̃+R l̃−R → X+l+l−χ̃01χ̃01 at the LHC. Withml̃ = 157.1 GeV and mχ̃ = 121.5 GeV, assuming the actual value for mχ̃ when calculatingMT2. The data with error bars are 1105 events, that the LHC would collect in approximately1 year of running at low luminosity, i.e. L ≃ 30 fb−1. The histogram represents L ≃ 500 fb−1to show the shape of the distribution that would be obtained with huge statistics, with thenormalization modified to be the same as L ≃ 30 fb−1.Figure 3: Values of ml̃ that would be obtained from that largest MT2 value observed, wherediffering values of mχ̃ are used in the calculation of MT2, for the 1105 events shown in figure2. All events have been generated with ml̃ = 157.1 GeV and mχ̃ = 121.5 GeV.5In figure 2 we show the MT2 distribution for this model. Quite clearly the maximumMT2 value allowed corresponds well to the mass of the selectron, with the MT2 distributiontending smoothly to zero at that point. While such an edge is not as easy to measure as avertical drop, we still expect that such an edge would be detectable experimentally, and hencea viable means to measure the selectron mass. The lower bound on MT2 is mχ̃, however thisdoes not give a means to measure the LSP mass, the lower bound on MT2 follows directlyfrom equation (8), and this is a parameter that goes into the calculation of MT2. Indeed ifthe value of mχ̃ is unknown in addition to ml̃ one can only obtain a relationship betweenthe two. In figure 3 we show the value of ml̃ that would have been obtained from the sameevents shown in figure 2, using differing input values for mχ̃ in the calculation of MT2. Onecan see that slepton mass that one would extract from the events behaves approximatly as,ml̃ ≃ mχ̃ + constant, (16)and so any uncertainty in the LSP mass will be directly reflected as an error in the extractedselectron mass. It should also be noted that the edge of the MT2 distribution becomes hardto fit as the input χ̃ mass deviates from the physical χ̃ mass as the MT2 distributions developsa tail at the largest MT2 values.To conclude, in this paper we have introduced a new variable for measuring massesof particles produced at hadron colliders, where the longitudinal momentum of the hardscattering is typically unmeasured. It may be used when particles are pair produced, witheach decaying to one particle that is directly observed and one particle that is not directlyobserved. This variable is analogous to the transverse mass variable, mT , commonly usedfor measuring the W mass in its decay to lν at pp̄ colliders, except that it works wherethe particle being measured is pair produced and where the unseen particle is massive. Weexpect that the masses extracted using our variable, like those from the transverse mass,will be largely independent of the physics processes which produce the particles, and hencegive a viable means of extracting masses in a model independent way. As an illustration ofthis variable in action we consider measuring the mass of selectrons in SUSY at the LHC.The results look promising however as we have not considered the effects of any backgroundprocesses or experimental mis-measurement errors further study is required[9].AcknowledgementsWe would like to thank Andy Parker for helpful conversations. CGL wishes to thank hisfunding body, the Partice Physics and Astronomy Research Council (PPARC), for financialsupport.6References[1] G. Arnison et al., UA1 Collaboration, Phys. Lett. 122B (1983) 103, Phys. Lett. 129B(1983) 273.[2] G. Banner et al., UA2 Collaboration, Phys. Lett. 122B (1983) 476.[3] G. Arnison et al., UA1 Collaboration, Phys. Lett. 126B (1983) 398.[4] P. Bagnaia et al., UA2 Collaboration, Phys. Lett. 129B (1983) 130.[5] F. Abe et al., CDF Collaboration, Phys. Rev. D50, 2996 (1994), Phys. Rev. Lett. 73,225 (1994), Phys. Rev. Lett. 74, 2626 (1995);S. Abachi et al., DØ Collaboration, Phys.Rev.Lett. 74, 2632 (1995)[6] “The minimal supersymmetric standard model: Group Summary Report” Report ofthe MSSM Working Group at the GDR - Supersymétrie workshop, Montpellier, France,15-17 Apr 1998. PM-98-45, hep-ph/9901246, Dec 1998.[7] http://www.cern.ch/Committees/LHCC/SUSY96.htmlhttp://atlasinfo.cern.ch/Atlas/GROUPS/PHYSICS/SUSY/susy.html[8] Herwig 6.1. Cavendish-HEP-99/03. Paper in preparation.[9] Cambridge SUSY working group, work in progress.7

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Measuring masses of semi-invisibly decaying particle pairs produced at hadron colliders

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