A general method for determining the masses of semi-invisibly decaying particles at hadron colliders
aa r X i v : . [ h e p - ph ] D ec A general method for determining the massesof semi-invisibly decaying particles at hadron colliders
Konstantin T. Matchev and Myeonghun Park
Physics Department, University of Florida, Gainesville, FL 32611, USA (Dated: 27 December, 2010)We present a general solution to the long standing problem of determining the masses of pair-produced, semi-invisibly decaying particles at hadron colliders. We define two new transverse kine-matic variables, M CT ⊥ and M CT k , which are suitable one-dimensional projections of the contrans-verse mass M CT . We derive analytical formulas for the boundaries of the kinematically allowedregions in the ( M CT ⊥ , M CT k ) and ( M CT ⊥ , M CT ) parameter planes, and introduce suitable variables D CT k and D CT to measure the distance to those boundaries on an event per event basis. We showthat the masses can be reliably extracted from the endpoint measurements of M maxCT ⊥ and D minCT (or D minCT k ). We illustrate our method with dilepton t ¯ t events at the LHC. PACS numbers: 14.80.Ly,12.60.Jv,11.80.Cr
The ongoing run of the Large Hadron Collider (LHC)at CERN will finally provide the first glimpse of physicsat the TeV scale. In large part, the excitement surround-ing the LHC is fueled by the anticipation of the unknown:no one knows for sure where or how the first signal of newphysics beyond the standard model (BSM) will show up.Yet, complementary and independent arguments fromparticle physics and astrophysics suggest that the bestplace to look for new physics is a channel with missingtransverse energy /E T , caused by unseen new particlescontributing to the dark matter of the Universe.Unfortunately, the study of missing energy signaturesposes a tremendous challenge at hadron colliders like theLHC. The first fundamental difficulty is related to thevery nature of hadron colliders, where in each event thepartonic center-of-mass energy √ ˆ s and longitudinal mo-mentum p z of the initial state are unknown. To makematters worse, the lifetime of the dark matter particleis typically protected by a new parity symmetry, whichguarantees that in every event the missing particles comein pairs, thus proliferating the number of unknown pa-rameters describing the final state event kinematics. p (¯ p ) p (¯ p ) U : ~U T V : ( m , ~p T ) V : ( m , ~p T ) PP CCM p ↑↓ M c ↑↓ FIG. 1: The generic event topology under consideration. Allparticles visible in the detector are clustered into three groups:upstream objects U with total transverse momentum ~U T , andtwo composite visible particles V i ( i = 1 , m i and total transverse momentum ~p iT . The generic topology of a “new physics” /E T event issketched in Fig. 1. Consider the inclusive productionof an identical pair of new “parent” particles P . Eachparent P decays semi-invisibly to a set V i ( i = 1 ,
2) ofstandard model (SM) particles, which are visible in thedetector, and a dark matter particle C (from now onreferred to as the “child”) which escapes detection. Ingeneral, the parent pair is accompanied by a number ofadditional “upstream” objects U (typically jets) with to-tal transverse momentum ~U T . They may originate fromvarious sources such as initial state radiation or decays ofeven heavier particles. We shall not be interested in theexact details of the physics responsible for U , adopting afully inclusive approach to the production of the parents P . Given this general setup, the goal is to determine in-dependently the mass M p of the parent and the mass M c of the child in terms of U , V and V .In the past, several approaches to this problem havebeen proposed, but each has its own limitations. Forexample, the classic method of invariant mass endpoints[1, 2] only applies when the visible SM particles in V i arisefrom a sufficiently long decay chain. Attempts at directreconstruction [3] of the children momenta are again lim-ited to long decay chains only. In this letter, we shall con-sider the extreme, most challenging example where eachvisible set V i consists of a single SM particle of fixed mass m i . A perfect testing ground for this scenario is providedby dilepton t ¯ t events (already observed at the LHC [4])and we shall use that example in our numerical illustra-tions below. The role of the parent P (child C ) will beplayed by the SM W -boson (SM neutrino), each V i is aSM lepton ( e or µ ), while U is composed of the two b -jetsfrom the top quark decays, plus any additional QCD jetsfrom initial state radiation (ISR).For such extremely short decay chains, the only viablealternative at the moment is provided by the methodsbased on the M T variable [5]. There, at least in princi-ple, the individual masses M p and M c can be determinedby observing a “kink” feature in the M T endpoint as afunction of a hypothesized trial mass M c for C [6], or by T k T ⊥ ~U T ~p T ~p T ~/P T = − ~p T − ~p T − ~U T ~p T ⊥ ~p T k ~p T ⊥ ~p T k FIG. 2: Decomposition of the observed transverse momentumvectors from Fig. 1 in the transverse plane. exploring the U T dependence of the M T endpoint [7].Compared to those M T approaches, our method herehas two advantages. First, it is simpler – it uses only theobserved objects U , V and V in the event and makesno reference to the missing particle kinematics (or mass).Second, it is more precise, since it utilizes the whole kine-matic boundary of the relevant two-dimensional distri-bution and not just the kinematic endpoint of its one-dimensional projection. We proceed in three easy steps. Step I. Orthogonal decomposition of the observed trans-verse momenta with respect to the ~U T direction. TheTevatron and LHC collaborations currently use fixed axescoordinate systems to describe their data. Instead, wepropose to rotate the coordinate system from one eventto another, so that the transverse axes are always alignedwith the direction T k selected by the measured upstreamtransverse momentum vector ~U T and the direction T ⊥ orthogonal to it (see Fig. 2). The visible transverse mo-mentum vectors from Fig. 1 are then decomposed as ~p iT k ≡ U T (cid:16) ~p iT · ~U T (cid:17) ~U T , (1) ~p iT ⊥ ≡ ~p iT − ~p iT k = 1 U T ~U T × (cid:16) ~p iT × ~U T (cid:17) . (2) Step II. Constructing the transverse and longitudinalcontransverse masses M CT ⊥ and M CT k . Our startingpoint is the original contransverse mass variable [8] M CT = q m + m + 2 ( e T e T + ~p T · ~p T ) , (3)where e iT is the “transverse energy” of V i e iT = q m i + | ~p iT | . (4)For events with U T = 0, M CT has an upper endpointwhich is insensitive to the unknown √ ˆ s , providing onerelation among M p and M c [8, 9] M maxCT ( U T = 0) = q m + m + 2 m m cosh ( ζ + ζ ) , (5)where sinh ζ i ≡ λ ( M p , M c , m i )2 M p m i , (6) λ ( x, y, z ) ≡ x + y + z − xy − xz − yz . (7) Unfortunately, the U T = 0 limit is not particularlyinteresting at hadron colliders (especially for inclusivestudies), since a significant amount of upstream U T istypically generated by ISR (and other) jets. One pos-sible fix is to use all events, but modify the definition(3) to approximately compensate for the transverse ~U T boost [9]. One then recovers a distribution whose end-point is still given by (5). Alternatively, one could stickto the original M CT variable, and simply account for the U T dependence of its endpoint as M maxCT ( U T ) = q m + m + 2 m m cosh (2 η + ζ + ζ )(8)where ζ i were already defined in (6), andsinh η ≡ U T M p , cosh η ≡ s U T M p . (9)Our approach here is to utilize the one-dimensionalprojections from eqs. (1,2) and construct one-dimensionalanalogues of the M CT variable M CT ⊥ ≡ q m + m + 2 ( e T ⊥ e T ⊥ + ~p T ⊥ · ~p T ⊥ ) , (10) M CT k ≡ q m + m + 2 (cid:0) e T k e T k + ~p T k · ~p T k (cid:1) , (11)where the corresponding “transverse energies” are e iT ⊥ ≡ q m i + | ~p iT ⊥ | , e iT k ≡ q m i + | ~p iT k | . (12)The benefit of the decomposition (10,11) is that one gets“two for the price of one”, i.e. two independent and com-plementary variables instead of the single variable (3).The variable M CT ⊥ in particular is very useful for ourpurposes. To illustrate the basic idea, it is sufficient toconsider the most common case, where V i is approxi-mately massless ( m i = 0), as the leptons in our t ¯ t ex-ample. A crucial property of M CT ⊥ is that its endpointis independent of U T : M maxCT ⊥ = M p − M c M p , ∀ U T . (13)In fact the whole M CT ⊥ distribution is insensitive to U T :d N d M CT ⊥ = N ⊥ δ ( M CT ⊥ ) + ( N tot − N ⊥ ) d ¯ N d M CT ⊥ , (14)where N ⊥ is the number of events in the zero bin M CT ⊥ = 0. Using phase space kinematics, we find thatthe shape of the remaining (unit-normalized) zero-bin-subtracted distribution is simply given byd ¯ N d ˆ M CT ⊥ ≡ − M CT ⊥ ln ˆ M CT ⊥ (15)in terms of the unit-normalized M CT ⊥ variableˆ M CT ⊥ ≡ M CT ⊥ M maxCT ⊥ . (16) FIG. 3: Zero-bin subtracted M CT ⊥ distribution after cuts, for t ¯ t dilepton events. The yellow (lower) portion is our signal,while the blue (upper) portion shows t ¯ t combinatorial back-ground with isolated leptons arising from τ or b decays. The observable M CT ⊥ distribution for our t ¯ t exampleis shown in Fig. 3, for 10 fb − of LHC data at 7 TeV.Events were generated with PYTHIA [10] and processedwith the PGS detector simulator [11]. We apply stan-dard background rejection cuts as follows [4]: we requiretwo isolated, opposite sign leptons with p iT >
20 GeV, m ℓ + ℓ − >
12 GeV, and passing a Z -veto | m ℓ + ℓ − − M Z | >
15 GeV; at least two central jets with p T >
30 GeV and | η | < .
4; and a /E T cut of /E T >
30 GeV ( /E T >
20 GeV)for events with same flavor (opposite flavor) leptons. Wealso demand at least two b -tagged jets, assuming a flat b -tagging efficiency of 60%. With those cuts, the SMbackground from other processes is negligible [4].Fig. 3 demonstrates that the M CT ⊥ endpoint can bemeasured quite well. Since the theoretically predictedshape (15) is distorted by the cuts, we use a linear slopewith Gaussian smearing, and fit for the endpoint and theresolution parameter. We find M maxCT ⊥ = 80 . M maxCT ⊥ = 80 . M p and M c . At this point,a second, independent constraint can in principle be ob-tained from an analogous measurement of the M maxCT end-point (8) at a fixed value of U T (resulting in loss in statis-tics), after which the two masses can be found from M p = U T M maxCT ( U T ) M maxCT ⊥ ( M maxCT ( U T )) − ( M maxCT ⊥ ) , (17) M c = q M p (cid:0) M p − M maxCT ⊥ (cid:1) . (18)However, the orthogonal decomposition (10,11) offers an-other approach, which we pursue in the last step. Step III. Fitting to kinematic boundary lines.
It isknown that two-dimensional correlation plots reveal alot more information than one-dimensional projected his-tograms [2, 12]. To this end, consider the scatter plot of M CT ⊥ vs M CT k in Fig. 4(a), where for illustration we FIG. 4: Scatter plots of (a) M CT ⊥ versus M CT k and (b) M CT ⊥ versus M CT , for a fixed representative value U T = 75 GeV.The solid lines show the corresponding boundaries definedin (20) and (23), for the correct value of M maxCT ⊥ and severaldifferent values of M p as shown. used 10,000 events at the parton level. For a given valueof M CT ⊥ , the allowed values of M CT k are bounded by M ( lo ) CT k ( M CT ⊥ ) ≤ M CT k ≤ M ( hi ) CT k ( M CT ⊥ ) , (19)where M ( lo ) CT k ( M CT ⊥ ) = 0 and M ( hi ) CT k ( M CT ⊥ ) = M maxCT ⊥ (cid:18)q − ˆ M CT ⊥ cosh η + sinh η (cid:19) . (20)Fig. 4(a) reveals that the endpoint M maxCT k of the one-dimensional M CT k distribution is obtained at M CT ⊥ = 0 M maxCT k = M ( hi ) CT k (0) = M maxCT ⊥ (cosh η + sinh η )= 12 (cid:18) − M c M p (cid:19) (cid:16)q M p + U T + U T (cid:17) . (21)Notice that events in the zero bins M CT ⊥ = 0 and M CT k = 0 fall on one of the axes and cannot be dis-tinguished on the plot.Now consider the scatter plot of M CT ⊥ vs M CT shownin Fig. 4(b). M CT is similarly bounded by M ( lo ) CT ( M CT ⊥ ) ≤ M CT ≤ M ( hi ) CT ( M CT ⊥ ) , (22)where this time M ( lo ) CT ( M CT ⊥ ) = M CT ⊥ and M ( hi ) CT ( M CT ⊥ ) = M maxCT ⊥ (cid:18) cosh η + q − ˆ M CT ⊥ sinh η (cid:19) . (23)We see that the endpoint M maxCT of the one-dimensional M CT distribution is also obtained for M CT ⊥ = 0: M maxCT = M ( hi ) CT (0) = M maxCT ⊥ (cosh η + sinh η ) = M maxCT k . (24) FIG. 5: D CT distributions for four different values of M p (and M c given from (18)). The yellow (light shaded) histogramsuse only events in the zero bin M CT ⊥ = 0. The red solid linesshow linear binned maximum likelihood fits. (GeV) p M
40 60 80 100 120 140 160 180 200 ( G e V ) m i n C T D -25-20-15-10-50510 w = M p M =0 minCT D FIG. 6: Fitted values of D minCT as a function of M p . Fig. 4 reveals a conceptual problem with one- dimensional projections. While all points in the vicin-ity of the boundary lines (20) and (23) are sensitiveto the masses, the M maxCT ⊥ endpoint is extracted mostlyfrom events with M CT ⊥ ∼ M maxCT ⊥ , while the M maxCT k and M maxCT endpoints are extracted mostly from the eventswith M CT ⊥ ∼
0. The events near the boundary, but with intermediate values of M CT ⊥ , will not enter efficiently ei-ther one of these endpoint determinations.So how can one do better, given the knowledge of theboundary line (23)? In the spirit of [13], we define thesigned distance to the corresponding boundary, e.g. D CT ( M p , M c ) ≡ M ( hi ) CT ( M CT ⊥ , U T , M p , M c ) − M CT and similarly for D CT k . The key property of this variableis that for the correct values of M p and M c , its lowerendpoint D minCT is exactly zero (see Fig. 5(b)): D minCT ( M p , M c ) = 0 . (25)In that case the boundary line provides a perfectly snugfit to the scatter plot — notice the green boundary linemarked “80” in Fig. 4(b). While in general eq. (25) rep-resents a two-dimensional fit to M p and M c , in practiceone can already use the M maxCT ⊥ measurement to reducethe problem to a single degree of freedom, e.g. the par-ent mass M p , as presented in Figs. 4 and 5. We see thatthe correct parent mass M p = 80 GeV provides a perfectenvelope, for which D minCT = 0. If, on the other hand, M p is too low, a gap develops between the outlying pointsin the scatter plot and their expected boundary, whichresults in D minCT >
0. Conversely, if M p is too high, someof the outlying points from the scatter plot fall outsidethe boundary and have D CT <
0, leading to D minCT < D minCT as afunction of M p from our PGS data sample is shown inFig. 6, which suggests that a W mass measurement atthe level of a few percent might be viable. Acknowledgments.
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