Measuring Invisible Particle Masses Using a Single Short Decay Chain
aa r X i v : . [ h e p - ph ] S e p Prepared for submission to JHEP
Measuring Invisible Particle Masses Using aSingle Short Decay Chain
Hsin-Chia Cheng and Jiayin Gu
Department of Physics, University of California, Davis, CA 95616, U.S.A.
Abstract:
We consider the mass measurement at hadron colliders for a decay chain oftwo steps, which ends with a missing particle. Such a topology appears as a subprocessof signal events of many new physics models which contain a dark matter candidate.From the two visible particles coming from the decay chain, only one invariant masscombination can be formed and hence it is na¨ıvely expected that the masses of the threeinvisible particles in the decay chain cannot be determined from a single end point of theinvariant mass distribution. We show that the event distribution in the log( E T /E T )vs. invariant mass-squared plane, where E T , E T are the transverse energies of the twovisible particles, contains the information of all three invisible particle masses and allowsthem to be extracted individually. The experimental smearing and combinatorial issuespose challenges to the mass measurements. However, in many cases the three invisibleparticle masses in the decay chain can be determined with reasonable accuracies. ontents B.1 The mother particle Y is polarized and decays anisotropically 36B.2 Soft visible particles and off-shell effects 36 Two major questions that particle physics are facing today are the origin of the elec-troweak symmetry breaking and the identity of the dark matter in the universe. Theanswers to both questions may lie on the physics at the TeV scale. For the first questionthe TeV scale is the scale of the electroweak symmetry breaking. It is widely expectedthat there will be new states at the TeV scale to stabilize the electroweak scale, solv-ing the hierarchy problem of the Higgs sector. The second question is related to theTeV scale because the most promising dark matter candidate is a new stable weaklyinteracting massive particle (WIMP) at the weak scale. The thermal relic left from theBig Bang for such a particle provides the right amount of dark matter in the currentuniverse. It is quite possible that these two questions are related and have their com-mon origin from the TeV scale physics. The Large Hadron Collider (LHC) is currentlyrunning and searching for new physics at the TeV scale intensively. Discoveries may bemade at any time to provide us important clues to these fundamental questions.Even though we do not know the exact new theory at the TeV scale, a typicalcollider signature for new physics containing the dark matter is the missing transverse– 1 –nergy ( E T ) in an event. To ensure the stability of the dark matter particle, a newsymmetry is often needed such that the dark matter particle is the lightest particlecharged under this new symmetry while all standard model (SM) particles are neutral.There can be other new particles at the TeV scale which are charged under this newsymmetry, as it is often the case in a more complete theory, including the most popularsupersymmetric standard model (SSM) where the R -parity plays the role of the thenew symmetry [1–4]. There are many other models which share the same feature, e.g.,universal extra dimensions (UEDs) [5–7] and little Higgs theories with T -parity [8–10]. These new particles are necessarily pair-produced at the collider due to the newsymmetry. After being produced, they decay down to the dark matter particles whichescape the detector, yielding missing energies, together with SM particles coming fromthe decays. The standard signatures for this class of models are therefore jets and/orleptons with E T .Missing transverse energy is an important channel to search for new physics at thecolliders. Events with large missing transverse energies are easy to identify. However,once such signals beyond the SM backgrounds are discovered, it is a non-trivial taskto identify what these new physics events are and to reconstruct their kinematics.With at least two missing particles in each event, in general one cannot reconstructthe full kinematics on an event-by-event basis without additional information. Somemore sophisticated techniques need to be developed if one wants to determine theproperties of the invisible particles including the dark matter particle which appear inthe process. One of the first things that we would like to know about the new particlesare their masses. Once the masses are determined, they can be used to reconstructthe kinematics of the events, and hence to help determining other properties of thesenew particles. In the past decade, there have been many techniques developed todetermine the masses of the new particles in the decay chains which end up withmissing particles. (See Ref. [11] for a review.) The starting point is to isolate thesignal events that predominantly come from a particular topology and then find theappropriate kinematic variables or constraints which depend on the masses of the newparticles. Some variables, e.g., the invariant mass end point [13, 14] and the stransverse M T [15–17]) can be used in a wide variety of event topologies, but in general the bestmass determination may require different techniques for different event topologies.In recent years many studies have been focussed on symmetric decay chains [18–31]. The advantage of considering symmetric decay chains is that there are additionalkinematic constraints coming from mass equalities between the particles in the two The event topology can be tested by looking at distributions of various kinematic variables suchas the invariant masses [12]. – 2 – (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)
Y X N2 1
Figure 1 . Decay chain decay chains and the total missing transverse momentum, which allowed accurate massdetermination even for relatively short decay chains. On the other hand, in a modelwhere many different decay patterns can happen, there are often many more eventswith asymmetric decay chains. There have been attempts to generalize the techniquesdeveloped for symmetric decay chains to asymmetric chains, but so far only to differentmother or daughter particle masses between the two chains, and some prior knowledgeabout the asymmetry of the events is required [32, 33]. To be complete general, it isworthwhile to find kinematic techniques or variables which can be applied to a singledecay chain. Then they can be used even in a collection of different types of signalevents as long as they contain one identical decay chain. In the case of limited statistics,they may give the first estimate of the masses before there are enough symmetric-chainevents for analysis.In fact, the early attempts of the mass determination in SUSY-like events werebased on single decay chains [34–40]. However, to obtain enough kinematic constraints,long decay chains with at least 3 steps are required. There are many invariant masscombinations which can be formed with 3 or more visible particles, which provideenough mass relations among the invisible particles in the decay chain. So far there is notechnique for extracting the masses in a shorter decay chain with only two or less visibleparticles. Na¨ıvely one may imagine that this task is impossible. For example, considera decay chain starting with a mother particle Y which decays through an intermediateon-shell state X and ends up with the missing particle N as shown in Fig. 1. There isone visible particle from each step of the decays and the visible particles are labeled as1 and 2 in the figure. There are three invisible particles ( Y, X, N ) in the decay chainbut there is only one invariant mass that can be formed from the two visible particles.One invariant end point certainly cannot determine three masses, but only providesone relation among them. It is also well known that the shape of the invariant massdistribution depends on the spin of the intermediate particle X and the chiralness of itscouplings, but not the masses [41–53]. One needs to find two more independent massrelations in order to solve for the three masses. In this work we make the first attempt– 3 –n a model-independent way to measure the invisible particle masses of the short singledecay chain of Fig. 1. We show that in certain cases, it is possible to determine allthree masses in the decay chain. We are not aiming for high accuracies. After all, thisis a difficult case so even a 30%–50% measurement is infinitely better than not beingable to determine them at all. Any information obtained or techniques developed herecan also be used in more complicated event topologies which contain this decay chainas a subprocess to provide additional constraints. Furthermore, the variables used formass determinations are often useful to separate signals from backgrounds and hencecan be used in new physics search in the first place.A crucial observation is that the masses are mostly determined by certain specialevents (and their nearby events). Those events often lie at the end point or the peak ofsome kinematic distribution. Mathematically, those special events make the kinematicconstraint equations degenerate. In order for the degenerate equations to have solu-tions (since the correct masses should be compatible with those events), this impliescertain relations among the coefficients of the constraint equations, which depend onthe invisible particle masses. In this way we obtain mass relations among the invisibleparticles. It has been shown that the common variables such as the end points of theinvariant mass and the transverse mass variables M T , M T can all be understood inthis way [54]. There some constraint equations used are quadratic so the special eventslie at the end point of a kinematic distribution. The regions near these points are oftenwhere signal events accumulate due to the phase space restriction, so sometimes theycan also be used to select signals over backgrounds.For the decay chain shown in Fig. 1, the constraint equations are p Y = m Y , (1.1)( p Y − p ) = m X , (1.2)( p Y − p − p ) = m N , (1.3)where p and p are the 4-momenta of the visible particles from the decays of Y and X respectively and for simplicity we take them to be massless p = p = 0 which is a goodapproximation for most SM visible particles. Since we do not use the information fromthe other decay chain, there is no constraint from the missing transverse momentum.Taking the differences of these three equations, we can obtain two linear equations inthe unknown momentum p Y , 2 p p Y = m Y − m X ≡ ∆ , (1.4)2 p p Y − p p = m X − m N ≡ ∆ , (1.5)where ∆ , are defined as the corresponding mass-squared differences. Together withany one of the quadratic equation above, we have all the independent kinematic con-– 4 –traints. From the discussion in the previous paragraph we should look at the eventswhich make these equations degenerate. (For the quadratic equation we just take thetangent on the curve.) The events with visible momenta p , p which make the threeequations (including the tangent of the quadratic equation) degenerate can be shownto satisfy ( p + p ) = ( m Y − m X )( m X − m N ) m X = ∆ ∆ m X , (1.6)which is nothing but the well-known end point of the invariant mass distribution of thevisible particle system, m , max = ( p + p ) . By locating the end point of the invari-ant mass distribution, we obtain one relation among the three masses m Y , m X , m N ,eq. (1.6).Because this decay chain can have any boost and orientation in the laboratory(lab) frame, one might think that the Lorentz-invariant mass is the only meaningfulquantity in this process and eq. (1.6) is the only mass-dependent kinematic variablethat we can get. However, by examining the two linear equations, eqs. (1.4),(1.5), onecan see that there is another case where the two equations become degenerate, thatis, when the two visible particles are parallel, p ∝ p . In this case the invariant massvanishes, ( p + p ) = 2 p p = 0. If we take the ratio of the two linear equations inthis case, we have E E = ∆ ∆ , (1.7)where E and E are the energies of the particle 1 and particle 2 respectively. Becausethe two 4-momenta are parallel, the ratio is invariant under any Lorentz transformation,and it gives the second relation among the three unknown masses. One may questionwhether such a co-linear degeneracy is useful in practice, because it rarely happens anddepending on the nature of the visible particles, the two visible particles may not beresolvable in a real experiment. So, instead of using only the extremely rare specialevents when the two visible particles are parallel, we will look at the event distributionin the E /E vs. m space (more precisely E T /E T where the subscript denotes thetransverse energy as it is invariant under longitudinal boosts). We find that the two-dimensional distribution contains useful information and under certain circumstances,it can even provide one additional mass relation which allows us to solve for all threeinvisible particle masses. Even in the cases where three masses can not be independentlyand accurately determined, the two-dimensional event distribution provides informationabout the decay chain beyond what is contained in the one-dimensional invariant massdistribution. This relation was also used in Ref. [55]. We thank M. Nojiri for bringing it to our attention. – 5 –
X N
Table 1 . A summary of the decay chain studied in Section 2. The visible Standard Modelparticles are treated as being massless.
This paper is organized as follows. In section 2 we analyze at the parton level theevent distribution of the decay chain in the log( E T /E T ) vs. invariant mass-squaredplane. We show that the masses of the three invisible particles can be obtained byfitting the distribution with a simple curve. Some more examples are presented inappendix A to show that the method applies to a wide range of models and exceptionsare included in appendix B. In section 3, we consider the method in more realisticsituations by including experimental smearing effects and combinatorial backgrounds.The case when the order of the two visible particles cannot be identified on an event-by-event basis is discussed in section 3.1. The combinatorial problems in the case whenthere are other visible particles identical to one of the visible particles from the decaychain in the same event are discussed in section 3.2. Conclusions are drawn in section 4. In the Introduction, we have argued that for the decay chain in Fig. 1, in addition tothe invariant mass end point, a second mass relation can be obtained by looking at theenergy ratio of the events where the two visible particles are parallel. However, suchevents are rare and may not be usable in real experiments. To avoid these problems, anatural thought is to look at the “nearby” events to see if one can extrapolate to thepoint that we are interested in. To do that we should examine the event distributionin the two-dimensional space of the energy ratio of the two visible particles and theirinvariant mass.To study this with a concrete example, a sample of events from such a decay chainare generated with the SUSY LM2 point [56, 57] chosen as the underlying model. Themasses of the particles in the decay chain are given in Table 1. events are generatedusing MadGraph 4.4.49 [61] at the parton level to reduce the statistical fluctuations. Inthis section we do not include experimental smearing and backgrounds, and assume nocombinatorial problem. These issues will be discussed in the next section when we deal This point has been ruled out at the LHC [58–60]. However, we just use this point for the purposeof illustration. The method that we develop is independent of the overall mass scale of the spectrum. – 6 – p p ð of events Figure 2 . The histogram of the invariant mass-squared of the two visible particles. In themassless limit ( p + p ) = 2 p p . The vertical line indicates the value of ∆ ∆ m X . p p @ GeV D - - - - log H E (cid:144) E L Figure 3 . The scatter plot of log ( E /E ) vs. 2 p p for all the events. with realistic experimental situations. The invariant mass-squared distribution of thetwo visible particles is shown in Fig. 2. The distribution has an end point which canbe clearly identified. The triangular shape of the distribution is a characteristic of thespin-1/2 intermediate state (chargino) in the decay chain. Now we would like to lookat the distribution of the energy ratios. We take the logarithm of the energy ratio tomake it more symmetric between the two particles. The two-dimensional distributionin the space of log( E /E ) vs. invariant mass-squared 2 p p is shown in Fig. 3. One canclearly see some interesting pattern of the distribution. As expected, the distributionof log ( E /E ) converges to a point at log(∆ / ∆ ) when the invariant mass goes tozero. Away from that point, the log ( E /E ) distribution spread out and for a fixedinvariant mass, it is more or less symmetric about some center point which moves up asthe invariant mass increases. This qualitative feature already tells us some importantinformation about the decay chain. It allows us to figure out which visible particlecomes from the first step decay and which comes from the second in the case where the– 7 –wo visible particles are distinct, because if we switch E and E the distribution willmove down instead. In our example it means that the quark jet being emitted beforethe lepton can be determined rather than assumed.To understand this distribution, let us rewrite eqs. (1.4), (1.5) as2 p p Y = ∆ + 2 p p , (2.1)2 p p Y = ∆ , (2.2)and take the ratio between these two equations. We have∆ + 2 p p ∆ = p p Y p p Y = E E Y − ~p · ~p Y E E Y − ~p · ~p Y = E E Y − | ~p || ~p Y | cos θ Y E E Y − | ~p || ~p Y | cos θ Y (cid:12)(cid:12)(cid:12)(cid:12) lab = E E Y (1 − β Y cos θ Y ) E E Y (1 − β Y cos θ Y ) (cid:12)(cid:12)(cid:12)(cid:12) lab = E (1 − β Y cos θ Y ) E (1 − β Y cos θ Y ) (cid:12)(cid:12)(cid:12)(cid:12) lab , (2.3)where the cos θ Y is the angle between particle 1(2) and particle Y , the subscript“lab” indicates that the angles are measured in the lab frame and β Y is the magnitudeof the velocity (boost) of particle Y , defined as β Y = | ~p Y | /E Y . It is easy to see thateq. (2.3) reduces to the simple relation, ∆ / ∆ = E /E , when particle 1 & 2 areparallel to each other. Now taking the logarithm of eq. (2.3) we obtainlog E E = log ∆ + 2 p p ∆ + log 1 − β Y cos θ Y − β Y cos θ Y (cid:12)(cid:12)(cid:12)(cid:12) lab = log ∆ + 2 p p ∆ + log 1 + β Y cos θ Y β Y cos θ Y (cid:12)(cid:12)(cid:12)(cid:12) Y , (2.4)where the subscript “ Y ” denotes that the angles are measured in the rest frame ofparticle Y . More specifically, cos θ Y (cid:12)(cid:12) Y is the angle between particle 1(2) (measuredin the rest frame of particle Y ) and particle Y (measured in the lab frame). From thefirst line to the second line of eq. (2.4) we simply perform a Lorentz transformationand use (assuming particle 1 & 2 are massless)cos θ Y (cid:12)(cid:12) lab = cos θ Y + β Y β Y cos θ Y (cid:12)(cid:12)(cid:12)(cid:12) Y . (2.5)The reason for writing the expression in terms of the angles in the rest frame of particle Y is that cos θ Y (cid:12)(cid:12) Y is directly related to the polarization of particle Y .The left-hand side of eq. (2.4) can be measured experimentally. The first termon the right-hand side of the equation involves the unknown masses to be determined– 8 – p p @ GeV D - - - log 1 + Β Y cos Θ Y + Β Y cos Θ Y Y Figure 4 . The distribution of log β Y cos θ Y β Y cos θ Y (cid:12)(cid:12) Y as a function of the invariant mass-squared.The average of log β Y cos θ Y β Y cos θ Y (cid:12)(cid:12) Y is − . and the invariant mass of the visible particles which is also measurable. The secondterm on the right-hand side, on the other hand, is not measurable on an event-by-eventbasis. It involves the unknown momentum of the invisible particle Y . When we plotthe events on the log( E /E ) vs. 2 p p plane, this term causes the spread in the verticaldirection. Nevertheless, if the directions of particles 1 and 2 measured in the rest frameof Y are not correlated with the direction of Y itself, we expect that the second termwill be evenly distributed around zero and peak at zero. Fig. 4 shows the distributionof log β Y cos θ Y β Y cos θ Y (cid:12)(cid:12) Y as a function of the invariant mass-squared and one can see thatthis is roughly true for any invariant mass. In this case, we can fit the distribution witha two-parameter curve log E E = log e ∆ + 2 p p e ∆ , (2.6)which has the least total χ measured in vertical distances. Consequently, the twofitted parameters e ∆ , e ∆ give estimates of the two mass-squared differences ∆ , ∆ .As a result, ∆ and ∆ can be determined individually, not just their ratio. Combinedwith the end point of the invariant mass-squared, ∆ ∆ /m X , they can be inverted tosolve all 3 unknown masses.Actually, eq. (2.4) can be applied in any longitudinal frame as we can give thissystem an arbitrary longitudinal boost. In particular, one can boost each event to aframe where E /E = E T /E T . This in general removes a large longitudinal boost ofthe mother particle Y which happens when the center of mass of the collision is highlyboosted. Therefore, we can use the distribution in the log( E T /E T ) vs. 2 p p space– 9 – p p @ GeV D - - - log H E T (cid:144) E T L fittedtrue Figure 5 . The scatter plot of log ( E T /E T ) vs. 2 p p for all the events. The red curveis log ( E T /E T ) = log ( ∆ +2 p p ∆ ). The blue curve is a least square fit to the data with thesame function, treating ∆ and ∆ as unknown parameters. The standard deviations for allthe points are assumed to be the same. instead. Such a distribution is invariant under the longitudinal boosts and hence it isless sensitive to the center of mass of the production mechanism. It is also obviousfor parallel massless visible particles, E T /E T = E /E . As shown at the end of thissection, we find that the fit in the transverse energy ratio space works somewhat betterthan in the total energy ratio space for most of the cases, so we will use the transverseenergies in our variable log( E T /E T ) throughout the paper.Fig. 5 shows the fitted result of the scatter plot in the log( E T /E T ) vs. 2 p p plane.The red curve is the function log E T E T = log ∆ + 2 p p ∆ (2.7)for true ∆ and ∆ . The blue curve is the least square fit to the data of the samefunction (2.7) but treating ∆ , ∆ as fitting parameters. We can see that indeed thetwo curves are very close to each other. The term log β Y cos θ Y β Y cos θ Y (cid:12)(cid:12) Y is indeed evenlydistributed around zero in this frame for this case. Assuming that we can measure theinvariant mass end point accurately, we can then reconstruct the 3 invisible particlemasses m Y , m X , m N with the fitted values for ∆ and ∆ . The result is shown inTable 2. We see that, compared to the true values, the reconstructed masses are quiteaccurate.We have checked this mass reconstruction method for models with different massspectra and different particle spins in the decay chain, and we find that it works well– 10 – [GeV ] ∆ [GeV ] log (∆ / ∆ ) m Y [GeV] m X [GeV] m N [GeV]true 1 . × . × − .
08 777 465 292reconstructed 1 . × . × − .
03 780 473 295error +4 . − .
96% +5 .
5% +0 .
34% +1 .
8% +1 . Table 2 . True values, reconstructed values and the errors of the six quantities for the fitto all the events. The errors are calculated using reconstructed − truetrue (except for log (∆ / ∆ ),which is reconstructed − true) and do not represent the statistical fluctuation. In solving forthe masses, we have used the true invariant mass end point value assuming that it can beaccurately determined. The uncertainly in determining ∆ ∆ m X will add additional error on thereconstructed values. for a wide range of different models and spectra. A few more examples can be foundin Appendix A. However, there are two cases where our method falters:1. The mother particle Y is polarized and preferentially emits particle 2 in theforward or backward direction when it decays. In this case it is clear that cos θ Y (cid:12)(cid:12) Y willhave a nonzero average value. On the other hand, the direction of the particle 1 is lesscorrelated with the polarization of the mother particle Y . The term log β Y cos θ Y β Y cos θ Y (cid:12)(cid:12) Y will no longer be distributed evenly around zero, but has a bias depending on thecos θ Y (cid:12)(cid:12) Y distribution. By Taylor expanding the expressionlog ∆ + 2 p p ∆ = log ∆ ∆ + 2 p p ∆ + · · · , (2.8)we see that the curve has the intercept log(∆ / ∆ ) at 2 p p = 0 and the slope 1 / ∆ tothe first order. If h cos θ Y (cid:12)(cid:12) Y i >
0, we will obtain a fitted ∆ (also ∆ since their ratiois fixed by the intercept) greater than the true value, then the reconstructed masseswill be too large, and vice versa.2. The mass difference between two invisible particles (in particular between Y and X ) is small. The visible particle coming from the decay will be soft. There aretwo effects which can affect the mass determination. First, if the mass difference is notmuch larger than the width of one of the particles, the on-shell approximation does notwork well. The off-shell effects can be asymmetric, which makes the “effective” valueof ∆ or ∆ significantly different from the true value and introduces a bias to the termlog ( ∆ +2 p p ∆ ) in eq. (2.4). Second, if ∆ is small and the typical energy of the visibleparticle 2 is not much larger than the E T cut (which is necessary in a real experiment),those emitted in the forward direction of particle Y have a larger chance to pass the thetrigger and the E T cut than those emitted in the backward direction. This introduces a“fake” forward polarization to particle Y , which also causes a problem as we discussed– 11 – - - log H E T (cid:144) E T L ð of events - - - - - Figure 6 . Histograms of log ( E T /E T ) for different ranges of invariant masses. The charton the righthand side indicates the range of invariant masses as a multiple of ∆ ∆ m X for eachhistogram. The histograms have approximate Gaussian shapes. in point 1. In practice, the mass determination is more of a problem for a small ∆ than a small ∆ .In these cases, our simple method does not give an accurate determination of thethree invisible particle masses. However, the ratio ∆ / ∆ is still usually well determinedby the interception of the fitted curve with the axis of the zero invariant mass. If athird mass relation can be obtained in some other ways (e.g., double-chain events), thisdistribution can provide some other non-trivial information about this process such asthe polarization of the particle Y . More detailed studies of these two cases will be givenin Appendix B.For the fit in Fig. 5, every event contributes with the same weight. In reality,backgrounds and noises are always present. There is no reason to expect that thebackground events will be distributed evenly around the log ( ∆ +2 p p ∆ ) curve. Thebackgrounds can hence cause a bias in fitting. On the other hand, as long as backgroundevents are subdominant in the event sample, the central peak of the log( E T /E T )distribution for a fixed 2 p p may not be significantly affected. Therefore, a curvefitted to the peak locations of the log( E T /E T ) distributions for different invariantmasses may be less sensitive to backgrounds than a direct χ fit to all events. Wewould like to check whether such a procedure gives the result as good as the result inTable 2 in the absence of backgrounds first.In Fig. 6 we divide the events into five sets with different ranges of invariant masses.For each set of events, the histogram of log ( E T /E T ) has an approximate Gaussianshape. This indicates that we could divide the events according to their invariantmasses and for each set we extract the peak value using a Gaussian fit. After obtainingthe peak point for each set, we then fit these peak points to the curve log ( E T /E T ) =– 12 – p p @ GeV D - - - - - log H E T (cid:144) E T L fittedtrue Figure 7 . The events are divide into 20 sets with equal width (0 . × ∆ ∆ m X ) of invariantmass. For each set we extract the peak value by fitting with a Gaussian distribution. Thehorizontal coordinate (2 p p ) of each point is the middle point of each division. By doing thiswe have 20 points of peak values and we fit it with the curve log ( E T /E T ) = log ( ∆ +2 p p ∆ ),treating ∆ and ∆ as unknown parameters. The peak points and fitted curve are shown inblue. The red curve is log ( E T /E T ) = log ( ∆ +2 p p ∆ ) with true values of ∆ and ∆ . Theerror bar of each points is estimated by the formula σ [¯ x ] = σ [ x ] √ N , where σ [ x ] is obtained fromthe Gaussian fit and N is the number of events in the set. ∆ [GeV ] ∆ [GeV ] log (∆ / ∆ ) m Y [GeV] m X [GeV] m N [GeV]true 1 . × . × − .
08 777 465 292reconstructed 1 . × . × − .
05 776 468 291error +2 . − .
2% +3 . − .
17% +0 . − . Table 3 . True values, reconstructed values and the errors of the six quantities for the fitto the peak points. The errors are calculated using reconstructed − truetrue (except for log (∆ / ∆ ),which is reconstructed − true) and do not represent the statistical fluctuation. log ( ∆ +2 p p ∆ ). The result of this fit is shown in Fig. 7, where the events are divided into20 sets with equal width (= 0 . × ∆ ∆ /m X ) in invariant mass-squareds. The error barof each point is estimated by the formula σ [¯ x ] = σ [ x ] / √ N , where σ [ x ] is obtained fromthe Gaussian fit and N is the number of events in that set. The reconstructed massesobtained from this fit is shown in Table 3. We see that the results are comparable tothe ones in Table 2.The errors in Table 2 and Table 3 could be a combination of both systematic and– 13 – p p @ GeV D - - - - log H E (cid:144) E L fittedtrue p p @ GeV D - - - - - log H E (cid:144) E L fittedtrue Figure 8 . The scatter plot (left) and the peak-point plot (right) of log ( E /E ) vs. 2 p p for the same set of events as in Fig. 5 & Fig. 7. In both case the red curve is log ( E /E ) =log ( ∆ +2 p p ∆ ) with true ∆ and ∆ and the blue curve is the fitted one. m Y [GeV] m X [GeV] m N [GeV]true 777 465 292using transverse energiesreconstructed 764 ±
11 456 ±
12 280 ± − . ± . − . ± . − . ± . ±
33 511 ±
34 336 ± . ± .
2% +10 . ± .
4% +15 . ± . Table 4 . The reconstructed masses from 5 sets of events, each with 10 events, in theform of mean ± standard deviation. The reconstruction is obtained by fitting with the peakpoints, using both the transverse energies and the actual energies. The standard deviationsare estimated using the unbiased estimator (with a factor of q NN − ) throughout this pa-per. The results suggest that using transverse energies results in smaller fluctuations in thereconstructed masses. statistical errors. Nevertheless, they are quite small and both reconstructions are good.To further verify the goodness of our results, we repeated the event generation 5 times(with 10 events for each set), and applied fits to the the peak points for the 5 sets ofevents. The mean and the standard deviation values of the reconstructed masses forthe 5 sets are shown in Table 4. We also compare with the results obtained from thesame procedure on the same 5 sets of events, but using the ratio of the actual energies,log( E /E ) in the lab frame, instead of the transverse energies. Fig. 8 shows the scatterplot and the fit to the peak points of log ( E /E ) vs. 2 p p for the same set of eventsas in Fig. 5 & Fig. 7. The results from the 5 sets of events using the actual energies– 14 –re also listed in Table 4. We can see that using transverse energies gives better resultsand smaller fluctuations in the reconstructed masses. In Sec. 2 we saw that the event distribution in the transverse energy ratio vs. invariantmass-squared space can be used to determine all the invisible particle masses in thedecay chain. It works very well under quite general conditions at the parton level.However, there are many complications in performing such a measurement in a realexperiment. First, we must have a relatively clean sample of signal events to startwith. Selecting signal events from backgrounds is non-trivial and depends on the typeof signal events. This is beyond the scope of this paper and we assume that it canbe achieved for the cases that we are interested in. Even if we have a pure sampleof signal events, in general we have to face some combinatorial problems. If the twovisible particles are not distinct or can appear in either order in the decay chain, e.g.,two leptons from a heavier neutralino decaying down to a lighter neutralino througha slepton in a SUSY theory, we do not know which order to take in the energy ratiolog( E T /E T ). Another combinatorial issue is that there can be other experimentallyidentical particles appearing in the signal events and there is no absolute way to selectthe correct one. For example, if one of the visible particle is a jet, then we do expectother jets to be present in the same event, which can come from the other decay chain orinitial state radiation (ISR). We will discuss these two types of combinatorial problemsin the following two subsections. In subsection 3.1, we consider an example of signalevents with 2 leptons of the same flavor and opposite charges, which can appear ineither order. We will see that mass determination still works for some cases but notfor the others, depending on the mass parameters. In subsection 3.2, we consider adecay chain which produces 1 jet and 1 lepton with a definite order, but there areother jets coming from the other decay chain and/or from the initial state radiation inthe same event. It is possible that both types of combinatorial problems are presentsimultaneously, as in the case when both visible particles in the decay chain are jets.Such cases will be difficult and we do not expect to achieve good mass measurementsthere.Finally, experimental smearing of the visible particles from detector resolutions,fragmentation and hadronization in the case of a jet, will also deteriorate any massmeasurement. The smearing effect is more important for jets than leptons. To takeinto account the experimental smearing effect, the parton level events are smearedaccording to the Gaussian errors listed in Table 5 in our studies in this section. Theyroughly correspond to the performance of the CMS detector [62–64].– 15 –eptons: | η | < . p T >
10 GeV, δp T p T = 0 . ⊕ . p T , δθ = 0 . δφ = 0 . | η | < . p T >
20 GeV, δE T E T = (cid:26) . E T ⊕ . √ E T ⊕ . , for | η | < . , . E T ⊕ . √ E T ⊕ . , for | η | > . ,δη = 0 . δφ = 0 .
02 for | η | < . δη = 0 . δφ = 0 .
01 for | η | > . Table 5 . Parton level events are smeared according to the above Gaussian errors. The cutson p T and η are consistent with the default cuts in MadGraph. The observables of energydimension are in GeV units and the angular and the rapidity variables are in radians. In this subsection we consider the combinatorial problem between the two visible par-ticles. As jets will have other problems, we consider that the two visible particles in thedecay chain are leptons of the opposite charges and the same flavor. This happens inSUSY if a heavier neutralino decays through a slepton to the lightest neutralino. Thetwo leptons emitted can be in either charge order because the neutralinos are Majoranaparticles and the intermediate state can be either a slepton or an anti-slepton. To focuson this combinatorial problem, we assume that there is no other lepton of the sameflavor in the event. Because we do not know which lepton is particle 1 and which isparticle 2 on an event-by-event basis, we can not measure log( E T /E T ) but only itsabsolute value | log( E T /E T ) | (i.e., the ratio between the larger E T and the smaller E T ). In other words, the scatter plot in the log ( E T /E T ) vs. 2 p p plane is foldedalong the log ( E T /E T ) = 0 axis. This would certainly make the reconstruction morechallenging.Because the scatter plot is folded, the position of the distribution become crucial forreconstruction. If the center of distribution is far away from the log ( E T /E T ) = 0 line,the folding will not cause too much of a problem. One can still easily identify the peakof the | log( E T /E T ) | distribution of each invariant-mass interval, and the slope of thefitted curve connecting the peak points tells us the sign of log(∆ / ∆ ). On the otherhand, if the center of distribution is close to the log ( E T /E T ) = 0 line, the foldingmakes it difficult to identify the pattern and the peak of the unfolded distribution, thenreconstructing ∆ and ∆ individually by fitting a curve through the peak points may– 16 – Y [GeV] m X [GeV] m N [GeV] log ∆ ∆ case 1 468 187 140.5 -2.5case 2 468 237 140.5 -1.5case 3 (LM2) 468 304 140.5 -0.56case 4 468 373 140.5 0.40case 5 468 407 140.5 1.0 Table 6 . A summary of the mass spectra in subsection 3.1. The leptons (particles 1 & 2)are treated as massless. become impossible.As the event distribution is folded, it is not good to fit a curve with minimal χ onthe distribution directly. We will use the second technique discussed in the previoussection by dividing the data into small intervals of invariant mass-squareds and thenfinding the peak point of the distribution for each set. To extract the peak point of theunfolded distribution, we fit each set with a folded Gassian disribution in log ( E T /E T ),i.e., the Gaussian distribution also folded along the log ( E T /E T ) = 0 point. However,it is important to notice that the error for a Gaussian fit, σ [¯ x ] = σ [ x ] √ N , is no longera good estimate of the uncertainty if the distribution is folded. If the center of thedistribution is close to log ( E T /E T ) = 0, the folded distribution may not be sensitiveto the original peak position at all, and the uncertainty is much larger than that ofthe corresponding unfolded Gaussian distribution. To estimate this uncertainty, wefirst use the maximum likelihood method to fit the distribution to a folded Gaussiandistribution with two parameters, the mean µ and the standard deviation σ of theunfolded Gaussian distribution. We then find the contour log L = log L max − / µ – σ plane, and the tangent of this contour parallel to the σ -axis corresponds(approximately) to the boundary of the 68.3% central confidence interval of µ . Thismethod has some limitations, however, as the likelihood method requires knowledge ofthe full model up to a few free parameters. We do not know the exact shape of theoriginal distribution, but simply treat it as approximately Gaussian. The estimations ofthe peak point and its uncertainty may be off if the original distribution is not close to aGaussian. Once we obtain the estimations of the peak point and the uncertainty for eachbin, we can fit the peak points with the folded curve | log ( E T /E T ) | = | log ( ∆ +2 p p ∆ ) | to extract the values of ∆ and ∆ . To examine how the mass determination depends on the value of log ∆ / ∆ , we To account for the asymmetric uncertainties but without too many complications, we assume aasymmetric Gaussian distribution for the peak point values and use a modified least square fit, i.e.,the uncertainty used for the fit depends on whether the curve is above or below the point. – 17 –onsider a SUSY process with a heavy neutralino (particle Y ) decaying through anon-shell slepton (particle X ) to the lightest neutralino (particle N ), emitting a pair ofleptons in the decays. The five examples of spectra that we choose to study are listed inTable 6. We fix M Y , M N and vary M X to obtain different values for log ∆ / ∆ . The case3 corresponds to the LM2 point with Y being the 4th neutralino, but the actual modelis not important. The events are generated through neutralino pair production andthe polarization of the neutralino is close to zero. Since the neutralino is a Majoranaparticle, by the CP symmetry its decay is symmetric if all final states are includedanyway.Before studying the combinatorial problem we first look at the effect of smearing.For case 3, Figs. 9, 10 and 11 show the invariant mass distributions from the twovisible particles, the scatter plots and the peak points plots in the log ( E T /E T ) vs.2 p p plane before and after smearing. We see that the invariant mass end point is lesssharp after smearing but its existence is still eminent. The value of the end point maybe obtained with a template fit and we assume that it can be accurately determined.Consequently we will use the true end point value in the mass reconstruction as weexpect much larger uncertainties coming from other quantities. As the smearing effectsare small for leptons, the distribution in the log ( E T /E T ) vs. 2 p p plane does notchange much. The fitted curve also agrees with the true curve quite well after smearing,if the ordering of the 2 visible particles were known.We now study the folded distributions for the five cases listed in Table 6. We firstlook at the distribution of 10 events for each case to reduce the statistical fluctuations.The experimental smearings are included but they do not have a significant effect aswe see in the above discussion.For case 1 (log ∆ / ∆ = − .
5) the scatter plot and the folded peak-point plot areshown in Fig. 12. In the scatter plot one can see that a small portion of the events isabove the log ( E T /E T ) = 0 axis, which will be folded. Nevertheless, as the center ofdistribution is far away from the folding line, the folded Gaussian fit works well for allbins. The error bars are larger and some touch the folding axis for the last few pointsbecause the peak points are closer to the folding axis. In this case a fit to all the peakpoints is quite close to the true curve and we can get a good determination of ∆ and∆ . Fig. 13 shows the plots for case 2 (log ∆ / ∆ = − . p p ð of events p p ð of events Figure 9 . The histogram of the invariant mass-squared of the two visible particles before(Left) and after smearing (Right) for case 3. The vertical line indicates the value of ∆ ∆ m X .Both visible particles are leptons. p p @ GeV D - - log H E T (cid:144) E T L fittedtrue p p @ GeV D - - log H E T (cid:144) E T L fittedtrue Figure 10 . The scatter plots in the log ( E T /E T ) vs. 2 p p plane before (Left) and aftersmearing (Right) for case 3. In both case the red curve is log ( E T /E T ) = log ( ∆ +2 p p ∆ )with true ∆ and ∆ and the blue curve is the fitted one. p p @ GeV D - - log H E T (cid:144) E T L fittedtrue p p @ GeV D - - log H E T (cid:144) E T L fittedtrue Figure 11 . The peak-point plots in the log ( E T /E T ) vs. 2 p p plane before (Left) andafter smearing (Right) for case 3 (no folding). In both case the red curve is log ( E T /E T ) =log ( ∆ +2 p p ∆ ) with true ∆ and ∆ and the blue curve is the fitted one. turns out that their unfolded distributions are somewhat asymmetric and hence are notGaussian-like. Therefore, the likelihood method assuming a Gaussian distribution didnot work very well for these points. The fit will deteriorate if these points are included.– 19 – p p @ GeV D - - log H E T (cid:144) E T L fittedtrue p p @ GeV D - - - log H E T (cid:144) E T L fittedfolded truetrue Figure 12 . The scatter plot (Left) and the folded peak-point plot (Right) in thelog ( E T /E T ) vs. 2 p p plane for case 1 (log ∆ ∆ = − . E T /E T ) = log ( ∆ +2 p p ∆ ) with true ∆ and ∆ and the blue curve is the fitted one. Inthe folded plot the purple curve is the folded true curve. p p @ GeV D - - log H E T (cid:144) E T L fittedtrue p p @ GeV D - - log H E T (cid:144) E T L fittedfolded truetrue Figure 13 . The scatter plot (Left) and the folded peak-point plot (Right) in thelog ( E T /E T ) vs. 2 p p plane for case 2 (log ∆ ∆ = − . E T /E T ) = log ( ∆ +2 p p ∆ ) with true ∆ and ∆ and the blue curve is the fitted one. Inthe folded plot the purple curve is the folded true curve. However, as the nearby points have large errors and touch the folding axis, the centerof the distribution for this range of invariant mass should be quite close to the foldingaxis. As the points with large error bars are not very useful in the fitting, we chooseto fit the curve only in the “good region,” which contains the first ten points from theleft, and a reasonable fit can be obtained.Fig. 14 shows the plots for case 3 (log ∆ / ∆ = − . / ∆ , whichmay be figured out by a more careful examination of the folded distribution. Anyway,this is a bad case for our mass determination method.– 20 – p p @ GeV D - - log H E T (cid:144) E T L fittedtrue p p @ GeV D - - - log H E T (cid:144) E T L fittedfolded truetrue Figure 14 . The scatter plot (Left) and the folded peak-point plot (Right) in thelog ( E T /E T ) vs. 2 p p plane for case 3 (log ∆ ∆ = − . E T /E T ) = log ( ∆ +2 p p ∆ ) with true ∆ and ∆ and the blue curve is the fitted one.In the folded plot the purple curve is the folded true curve. p p @ GeV D - - - log H E T (cid:144) E T L fittedtrue p p @ GeV D - - log H E T (cid:144) E T L fittedtrue Figure 15 . The scatter plot (Left) and the folded peak-point plot (Right) in thelog ( E T /E T ) vs. 2 p p plane for case 4 (log ∆ ∆ = 0 . E T /E T ) = log ( ∆ +2 p p ∆ ) with true ∆ and ∆ and the blue curve is the fitted one. Inthe folded plot the purple curve is the folded true curve. Fig. 15 shows the plots for case 4 (log ∆ / ∆ = 0 . / ∆ = 1 . is small and hence particle 2 is quite soft. As discussed inAppendix B.2 our reconstruction method starts to produce some bias even without thefolding.The results of mass determination for the five cases are shown in Table 7. For cases– 21 – p p @ GeV D - - log H E T (cid:144) E T L fittedtrue p p @ GeV D log H E T (cid:144) E T L fittedtrue Figure 16 . The scatter plot (Left) and the folded peak-point plot (Right) in thelog ( E T /E T ) vs. 2 p p plane for case 5 (log ∆ ∆ = 1 . E T /E T ) = log ( ∆ +2 p p ∆ ) with true ∆ and ∆ and the blue curve is the fitted one. Inthe folded plot the purple curve is the folded true curve. (Note: The origin is not at 0 in thepeak plot.)
1, 2, 4 & 5 we also wish to know how well the method works with a smaller samplesize. It turns out that in terms of statistical fluctuations, for cases 1 and 2 the methodworks reasonably well with 10 events. The results are presented in Table 8.For cases 4 & 5, some care needs to be taken. As seen in eq. (2.8), the slopecorresponds to the value of 1 / ∆ at the first order. When ∆ ≫ ∆ , the slope issmall and a small fluctuation in the slope could cause a large fluctuation in the valueof ∆ (also ∆ since their ratio can be well determined) and the mapping is nonlinear.As a result, a larger statistics for cases 4 & 5 is needed to get a good measurement.Furthermore, a Gaussian-like uncertainty of the reconstructed slope will not result ina Gaussian-like uncertainty in the reconstructed ∆ when the slope is close to zerodue to the nonlinear mapping. In particular, the value of ∆ goes to infinity whenthe slope goes to zero. This behavior shows up when we look at case 4 with 50 setsof 10 events. Among the 50 sets of reconstructed masses, 44 of them are relativelyclose to each other but 6 sets has very large reconstructed masses ( m Y > × events)is needed for a reasonable reconstruction. Among 25 sets of 2 × events, 5 setshas m Y > ± standard deviation. However, it is hard to obtain the formof the probability distribution function without generating a large number of sets ofevents. Here we simply present the means and standard deviations of the reconstructedmasses excluding the bad sets in Table 8.– 22 – [GeV ] ∆ [GeV ] log (∆ / ∆ ) m Y [GeV] m X [GeV] m N [GeV]case 1true 1 . × . × − .
49 468 187 140 . . × . × − .
34 494 217 167error +25% +7 .
3% +15% +5 .
6% +16% +19%case 2true 3 . × . × − .
50 468 237 140 . . × . × − .
41 582 340 247error +50% +37% +8 .
8% +24% +43% +75%case 3true 7 . × . × − .
555 468 304 140 . . × . × .
37 734 614 379error +221% +27% +93% +57% +102% +170%case 4true 1 . × . × .
402 468 373 140 . . × . × .
35 407 312 0 . − − − . − − − . × . × .
01 468 407 140 . . × . × .
02 511 450 200error +11% +9 .
7% +1 .
6% +9 .
1% +11% +42%
Table 7 . The results of mass reconstruction from fitting the folded peak points for the 5 casesin Section 3.1 with 10 events. The errors are calculated using reconstructed − truetrue (except forlog (∆ / ∆ ), which is reconstructed − true) and do not represent the statistical fluctuation. In this subsection we study another type of the combinatorial problem when there areother particles which can not be distinguished experimentally from one of the particlesfrom the decay chain in the event. A typical example is that the two visible particlesfrom the decay chain are one jet and one lepton. In general we expect there will beother jets present in the same event. They can come from the other decay chain(s)in the event or simply the initial and final state radiations. In this case we face theproblem of not knowing which jet is the correct one to be paired with the lepton inthe same decay chain. It is possible that there are also other leptons in the sameevent to cause further confusions. To understand the basic issues of this type of the– 23 – Y [GeV] m X [GeV] m N [GeV]case 1 (10 events)true 468 187 140 . ±
38 245 ±
34 195 ± ±
8% +31% ±
18% +39% ± events)true 468 237 140 . ±
169 373 ±
160 280 ± ±
36% +57% ±
67% +99% ± events, 44 out of 50 sets have m Y < . ±
62 347 ±
62 26 ± − . ± − . ± − ± × events, 20 out of 25 sets have m Y < . ±
119 437 ±
119 151 ± . ±
25% +7 . ±
29% +7 . ± Table 8 . The results of reconstruction of case 1, 2, 4 & 5. The errors are in the form“bias ± uncertainty”. The means and uncertainties are estimated from a total number of5 × events. (i.e. For case 1 & 2, we divided the sample into 50 sets, each with 10 events,reconstructed the masses for each set and obtained the means and standard deviations of thereconstructed masses. For case 4 we did the same but we only used the 44 “good” sets outof 50 sets. For case 5 we divided the sample into 25 sets, each with 2 × events, and onlyused the 20 “good” sets among the 25 sets.) For case 2, only the first ten points (countingfrom the left, see Fig. 13) are used for fitting. combinatorial problem we will keep it simple by restricting ourselves to the case whereonly additional jets are present. The analysis should be readily generalizable to morecomplicated situations.The jets coming from decays of some heavy particles and the jets coming frominitial state radiation have very different characteristics. To consider both possibilities,we generated the following process using MadGraph. A left-handed down squark andits anti-particle are pair produced with the left-handed down squark decaying throughthe same chain studied in Sec. 2 (with the spectrum shown in Table 1), while the antileft-handed down squark decays to an anti-down quark and a lightest neutralino. Inaddition, an extra gluon is produced, either through ISR or FSR or elsewhere. The– 24 – igure 17 . The histogram of the invariant mass-squared of the three possible combinationsbefore smearing (left) and after smearing (right). The vertical line indicates the value of ∆ ∆ m X . events are smeared according to the Gaussian errors listed in Table 5. The signal ofthis process is 3 jets + 1 lepton + E T and we do not know which jet is particle 2. Thiscase is, of course, an over simplified version compared with the real processes at theLHC, but it should help us understand how to apply our method to the realistic cases.Fig. 17 shows the invariant mass-squared distributions for different combinationsbefore and after smearing. For convenience we label the anti-down quark from the otherchain particle 3 and the extra gluon particle 4. The invariant mass-squared distributionof the three different combinations are shown in the same plot. In reality we will not beable to distinguish them, but only obtain a distribution in which all three are stackedtogether. Fig. 18 shows the scatter plot in the log ( E T /E T j ) vs. 2 p p j plane where j = 2 , , j = 4, the events are mostlyin the upper left region due to the fact that most gluons are soft. For j = 3 the eventsare more spread on the 2 p p j axis and have less correlation between log ( E T /E jT ) and2 p p j .Before smearing, the invariant mass-squared distributions have two distinct fea-tures. First, for the correct combinations ( p , p ), the distribution has a sharp edge at ∆ ∆ m X , while the wrong combinations do not have such an edge. Second, for invariantmass below the edge, the correct combination is more likely to have large invariantmass than that of the wrong combinations. The shape of the invariant mass-squaredis model-dependent. In particular, it depends on the spin of the particle X and thechiralness of its couplings. If particle X is a scalar, the invariant mass-squared will have The ISR jet may be identified with some probabilities through some sophisticated methods [65, 66],but will not be attempted here. – 25 – p p j @ GeV D - - - log H E T (cid:144) E jT L j = = = p p j @ GeV D - - - log H E T (cid:144) E jT L j = = = Figure 18 . The scatter plots in the log ( E T /E jT ) vs. 2 p p plane before smearing (left)and after smearing (right) where j = 2 , , a flat distribution below the edge, but compared with the distributions of the wrongcombinations, the two features still hold. On the other hand, if the left-handed downsquark in the above process is replaced by an up anti-squark, then the quark and thelepton in the decay chains will have opposite helicities (in contrast to the same helicityin our example). Consequently they tend to go in the same direction and have a smallinvariant mass. In this case there will be no sharp edge at the end point. However,given that we need a good measurement of ∆ ∆ m X from the end point, we will only con-sider cases where a sharp edge is present, then generally the second feature also holds.Based on these two features we will use a simple and na¨ıve way to select the jet-leptonpair by choosing the combination with the largest invariant mass below the edge.Because jets suffer more experimental smearing than leptons, the edge is morewashed out compared to the pure leptonic case in the previous subsection. Neverthe-less, the existence of an edge in the invariant mass-squared distribution should still beidentifiable from the right panel of Fig. 17. We again assume that the location of ∆ ∆ m X can be obtained from a template fit. Any uncertainty for this quantity will add to thetotal uncertainties of the reconstructed masses but we do not expect it to be the mainsource of the final uncertainties. We will not include it in our following discussion.To select the correct jet, we first make the following cuts: • A p T cut on the jets is imposed. Such a p T cut is inevitable experimentally toreduce the QCD backgrounds. It mostly help to remove the ISR jets. However, ifthe p T cut is too high, it may create a fake polarization for the mother particle Y as it will favor the events with Y decaying to particle 2 in the forward direction,and hence causes a bias in the mass determination. In our example we require– 26 –ets to have p T >
50 GeV. • We require the jet when combined with the lepton to have an invariant mass-squared smaller than the position of the edge, ∆ ∆ m X . This dominantly reducesthe wrong combinations coming from the jet of the other decay chain. Becauseof smearing a small fraction of the correct combinations will be removed too.If no jet survives the above cuts, the event is dropped. If there are more than one jetspassing both cuts, we select the jet which forms the largest invariant mass togetherwith the lepton.After selecting the jet in each event, we can proceed as usual by dividing the datapoints according to their invariant mass-squareds and finding the peak location in thelog( E T /E T ) distribution for each invariant mass-squared bin. The peak-point plotafter smearing is shown in Fig. 19. The blue points are for all correct combinations andthe green points are for selected combinations using the method we described above.From the blue points we can see the effects of smearing: most points at lower invariantmassed are shifted slightly upwards while the points with invariant mass close to ∆ ∆ m X are shifted downwards. The wrong combinations mostly come from the extra gluon atlow invariant masses and the jet from the other decay chain at high invariant masses.We see that the green points at small invariant masses are shifted up significantly dueto the softness of the extra gluon. At large invariant masses close to the edge, the greenpoints are further shifted down compared with the blue points because in this examplethe jet from the other chain on average has a larger energy than the correct jet does inthe large invariant mass region.The wrong combinations make the mass determination more challenging. We seethat the peak points at both ends of the invariant mass range deviate from the truevalues significantly. Because the distribution of the ISR jets mostly lies at the upperleft corner and the distribution of the jets from the other decay chain is flat in thelog( E T /E T ) vs. 2 p p plane, the effects of the wrong combinations from both sourceswill make the extracted slope smaller than the true value, which will result in too bigreconstructed masses. To reduce the bias caused by wrong combinations, one may wantto use only points in the middle part of the invariant mass-squared interval to performthe fit. However, the good range for the fit seems to be model-dependent when we checkmodels with different spins and spectra. A fixed range in the invariant mass-squaredinterval (e.g., middle 1/3 of points) which works for some models does not work for theothers. The best strategy that we come up with so far is to have a fixed size of theinvariant mass-squared range, but keep the upper and lower ends floating to maximizethe fitted slope. This is motivated by the fact that wrong combinations tend to reducethe slope. The range used for fitting should not be too small to avoid a too large slope– 27 – p p @ GeV D - - - - - log H E T (cid:144) E T L fitted to selected combinationsfitted to true combinationstrue Figure 19 . The peak-point plot of log ( E T /E T ) vs. 2 p p after smearing. The events aredivide into 20 sets with equal width (0 . × ∆ ∆ m X ) of invariant mass and there are 20 pointsin the plot. The red curve is log ( E T /E T ) = log ( ∆ +2 p p ∆ ) with true ∆ and ∆ . Theblue points are for all correct combinations and the blue curve is the fitted curve with last5 points dropped. The green points and curve are for the selected combinations with themethod described in Section 3.2. For the green curve, point 7 to 13 are used in the fit. coming from the statistical fluctuations, especially when the sample size is small. InFig. 19 we fix the length of the fitting range to be 7 points and find that point 7 to13 (counting from the left) gives the largest slope after scanning through all possiblechoices. The result obtained from fitting these points is shown in Table 9.The fitting method is somewhat heuristic so one would like to know whether theerrors obtained in Table 9 is typical and how they depend on the size of the eventsample. To study that we generate 10 events which are smeared according to theGaussian errors listed in Table 5. We first divide the events into 10 sets with 10 events in each set, then reconstruct the masses for each set using the above method.The results are shown in Table 10. In practice, 10 is a very large number in terms ofnumber of events which can obtained in real experiments, so we also check how wellthe method works with less events. To do this, we divide the events into 50 sets, eachcontaining 2000 events. The results of the mass reconstruction in this case are shown inTable 11 using the same method, except that the length of the fitting range is increasedto 8 points to compensate for the effect of the larger fluctuations due to the smallersample size. – 28 – [GeV ] ∆ [GeV ] log (∆ / ∆ ) m Y [GeV] m X [GeV] m N [GeV]true 1 . × . × − .
08 777 465 292true combinationsreconstructed 1 . × . × − .
97 772 473 287error +7 . − .
0% +12% − .
68% +1 . − . . × . × − .
939 876 570 390error +32% +14% +15% +13% +23% +34%
Table 9 . Results of reconstruction from the fit to the peak points after smearing withtrue and selected combinations. The errors are calculated using reconstructed − truetrue (except forlog (∆ / ∆ ), which is reconstructed − true) and do not represent the statistical fluctuation. m Y [GeV] m X [GeV] m N [GeV]true 777 465 292reconstructed 931 ±
100 624 ±
103 445 ± ±
13% +34% ±
22% +52% ± Table 10 . The reconstructed masses from 10 sets of events, each with 10 events, in the formof mean ± standard deviation. The reconstruction is obtained by fitting with 7 consecutivepeak points which maximize the fitted slope. m Y [GeV] m X [GeV] m N [GeV]true 777 465 292reconstructed 754 ±
157 442 ±
162 265 ± − . ± − . ± − . ± Table 11 . The reconstructed masses from 50 sets of events, each with 2000 events, in the formof mean ± standard deviation. The reconstruction is obtained by fitting with 8 consecutivepeak points which maximize the fitted slope. The uncertainties of the results in Table 11 are quite substantial, though theyalready represent significant progress as there was no existing method which can deter-mine the invisible particle masses for this topology. We have used a rather simple andsomewhat heuristic method in treating the combinatorial problem and the results maybe improved if more sophisticated techniques are employed. Some possible improve-ments include:1. We use a na¨ıve method to select the jet to pair with the lepton. We expect that– 29 –he correct rate can be improved with a better algorithm. The ISR jet may bebetter identified with more sophisticated methods [65, 66]. One can also employa full likelihood method to distinguish the correct jet from the wrong ones usingmore variables. With better selections we may extend the fitting range to reducestatistical uncertainties and model dependence.2. Our heuristic method in choosing the fitting range is concerted in such a waythat it can be applied to a wide range of models. If more detailed informationis available about the process and the impostor particles from other part of theevent which contains the decay chain, one can design a fitting procedure bestsuit for the specific set of signal events using all available information and henceobtain better results.
In searching for new physics at colliders there are different approaches. Given a spe-cific model, one can design cuts and strategies to obtain the maximal reach for thatparticular model, and perform the most likelihood fit to extract model parameters.However, they may not be suitable for a different model. As there is a vast number ofpossibilities for new physics at the TeV scale, one can only perform specific searchesfor a limited number of models and there is no reason to expect that they are favoredover the other models. On the other hand, model-independent global searches comparedata and standard model predictions in a global sample of high- p T collision events andlook for excess in the high- p T tails as signals for new physics [67]. They do not requirespecification of any particular model. The drawback is that new physics signals maybe subtle and hide in the correlations of observables, then the global searches are noteffective to uncover them.An intermediate approach is to focus on the possible event topologies of new physicswithout completely specifying the underlying model. An example in this direction is thesimplified model approach for new physics searches [68]. For any given event topology,one can look for kinematic variables and relations which exhibit particular features forthat topology, so that one can use them to distinguish new physics from the standardmodel backgrounds and also extract relevant parameters of new physics. Because manydifferent models can produce signal events of the same topologies, the study of anyparticular event topology can apply to a wide range of possible new physics models.In this paper we consider the event topology of a 2-step decay chain ending with amissing particle. This topology appears in many models which contain a stable or long-lived neutral particle that escapes detection. Out of the two visible particles coming– 30 –rom the decay chain, the obvious kinematic variable to look at is the invariant masscombination of the 4-momenta of the two visible particles, and one might thought thatit is the only relevant kinematic variable for this topology. We show that the logarithmof the transverse energy ratio of the two visible particles is also a useful variable. Theevent distribution in the log( E T /E T ) vs. invariant mass-squared space carry usefulinformation beyond what is contained in the one-dimensional invariant mass distribu-tion. In many cases it allows the extraction of masses of all three invisible particles inthe decay chain. The shape of the distribution may also be used to distinguish signalsfrom backgrounds.The method of searching for new useful kinematic variables may be generalized toother event topologies. Digging new physics signals out of the tremendous backgroundsat hadron colliders is always a challenging task. Finding the most effective kinematicvariables for new physics signals will improve the search reaches and help us to figureout the properties of the underlying theory once it is discovered. Acknowledgments
We would like to thank Spencer Chang, Max Chertok, Zhenyu Han, Ian-Woo Kim,Markus Luty and Jesse Thaler for discussion. H.-C. C. would like to thank the KavliInstitute for Theoretical Physics and CERN TH-LPCC Summer Institute where partof this work was done. This work was supported by the US Department of Energyunder contract DE-FG02-91ER406746.
A More Parton-level Examples
In this Appendix we present a few more parton-level examples for our mass determina-tion method. It works well for a variety of models with different mass spectra and spins.In all examples in this Appendix, the mother particle Y decays isotropically and theorder of the visible particles are assumed to be known. Anisotropic decays of particle Y will cause a bias in mass determination which will be discussed in Appendix B. Thecombinatorial problems of indistinguishable particles or orders are discussed in sec. 3. Example 1
In this example we consider a SUSY process with a heavy neutralino (particle Y )decaying through an on-shell slepton (particle X ) to the lightest neutralino (particle N ). The mass spectrum and spin configuration are summarized in Table 12. Theinvariant mass distribution is shown in Fig. 20. The scatter plot and the peak-point– 31 –lot are shown in Fig. 21. The result of mass reconstruction from fitting the peakpoints is shown in Table. 13. Y X N spin 1/2 0 1/2mass[GeV] 468 304 140.5
Table 12 . The summary of the mass spectrums and spin configurations of Example 1 inAppendix A. p p ð of events Figure 20 . The histogram of the invariant mass-squared of the two visible particles forExample 1 in Appendix A. In the massless limit ( p + p ) = 2 p p . The vertical lineindicates the value of ∆ ∆ m X . p p @ GeV D - - log H E T (cid:144) E T L fittedtrue p p @ GeV D - - log H E T (cid:144) E T L fittedtrue Figure 21 . The scatter plot (left) and the peak-point plot (right) of log ( E T /E T ) vs. 2 p p for Example 1 in Appendix A. In both case the red curve is log ( E T /E T ) = log ( ∆ +2 p p ∆ )with true ∆ and ∆ and the blue curve is the fitted one. – 32 – [GeV ] ∆ [GeV ] log (∆ / ∆ ) m Y [GeV] m X [GeV] m N [GeV]true 7 . × . × − .
555 468 304 140 . . × . × − .
544 477 313 150 . .
4% +2 .
3% +1 .
1% +1 .
9% +2 .
9% +7 . Table 13 . The result of mass reconstruction from fitting the peak points for Example 1in Appendix A. The errors are calculated using reconstructed − truetrue (except for log (∆ / ∆ ),which is reconstructed − true) and do not represent the statistical fluctuation. The statisticalfluctuation are expected to be at the same order as the example in Section 2. Example 2
This is a simplified model. The interactions are vector-like so that there is no preferreddirection for the particle Y decay. The mass spectrum and spin configuration of theparticles in the decay chain are summarized in Table 14. The invariant mass distributionis shown in Fig. 22. The scatter plot and the peak-point plot are shown in Fig. 23. Theresult of mass reconstruction from fitting the peak points is shown in Table. 15. Y X N spin 1 1/2 1mass[GeV] 1000 800 500
Table 14 . The summary of the mass spectrums and spin configurations of Example 2 inAppendix A. p p ð of events Figure 22 . The histogram of the invariant mass-squared of the two visible particles forExample 2 in Appendix A. In the massless limit ( p + p ) = 2 p p . The vertical lineindicates the value of ∆ ∆ m X . – 33 – p p @ GeV D - - - log H E T (cid:144) E T L fittedtrue p p @ GeV D log H E T (cid:144) E T L fittedtrue Figure 23 . The scatter plot (left) and the peak-point plot (right) of log ( E T /E T ) vs. 2 p p for Example 2 in Appendix A. In both case the red curve is log ( E T /E T ) = log ( ∆ +2 p p ∆ )with true ∆ and ∆ and the blue curve is the fitted one. ∆ [GeV ] ∆ [GeV ] log (∆ / ∆ ) m Y [GeV] m X [GeV] m N [GeV]true 3 . × . × .
080 1000 800 500reconstructed 4 . × . × .
078 1023 822 525error +2 .
6% +2 . − .
25% +2 .
3% +2 .
8% +5 . Table 15 . The result of mass reconstruction from fitting the peak points for Example 2in Appendix A. The errors are calculated using reconstructed − truetrue (except for log (∆ / ∆ ),which is reconstructed − true) and do not represent the statistical fluctuation. The statisticalfluctuation are expected to be at the same order as the example in Section 2. Example 3
This is also a simplified model with vector-like couplings. The mass spectrum and spinconfiguration are summarized in Table 16. The invariant mass distribution is shown inFig. 24. The scatter plot and the peak-point plot are shown in Fig. 25. The result ofmass reconstruction from fitting the peak points is shown in Table. 17.
Y X N spin 1/2 1 1/2mass[GeV] 800 300 100
Table 16 . The summary of the mass spectrums and spin configurations of Example 3 inAppendix A. – 34 – p p ð of events Figure 24 . The histogram of the invariant mass-squared of the two visible particles forExample 3 in Appendix A. In the massless limit ( p + p ) = 2 p p . The vertical lineindicates the value of ∆ ∆ m X . p p @ GeV D - - - - log H E T (cid:144) E T L fittedtrue p p @ GeV D - - - log H E T (cid:144) E T L fittedtrue Figure 25 . The scatter plot (left) and the peak-point plot (right) of log ( E T /E T ) vs. 2 p p for Example 3 in Appendix A. In both case the red curve is log ( E T /E T ) = log ( ∆ +2 p p ∆ )with true ∆ and ∆ and the blue curve is the fitted one. ∆ [GeV ] ∆ [GeV ] log (∆ / ∆ ) m Y [GeV] m X [GeV] m N [GeV]true 8 . × . × − .
93 800 300 100reconstructed 8 . × . × − .
90 815 313 115error +6 .
0% +2 .
8% +3 .
0% +1 .
8% +4 .
4% +15%
Table 17 . The result of mass reconstruction from fitting the peak points for Example 3in Appendix A. The errors are calculated using reconstructed − truetrue (except for log (∆ / ∆ ),which is reconstructed − true) and do not represent the statistical fluctuation. The statisticalfluctuation are expected to be at the same order as the example in Section 2. – 35 – X N particle 2nd chargino left handed anti up squark 1st neutralinomass[GeV] 1000 770 140.5
Table 18 . A summary of the decay chain in Appendix B.1.
B Problematic Cases for Mass Determination
In this Appendix we discuss cases where the mass determination method does not workeven at the parton level. The two cases mentioned in sec. 2 are discussed in more detailsbelow.
B.1 The mother particle Y is polarized and decays anisotropically Our mass determination method relies on that the distribution of log β Y cos θ Y β Y cos θ Y (cid:12)(cid:12) Y issymmetric around zero. This is obviously not the case if the decay of particle Y is notisotropic and preferentially emits particle 2 in the forward (or backward) direction. Itcan happen if the particle Y has nonzero spin and is polarized. Even if Y is polarized,sometimes a P or CP symmetry may still ensure the decay to be symmetric whenall final states are included, as in the case of a Majorana neutralino, so some chiralstructure of the process needs to be present. In the minimal supersymmetric standardmodel, the only particles that can have asymmetric decays are the charginos, so thestudy case that we consider is a decay chain initiated by a chargino. The particles inthe decay chain are summarized in Table 18. The underlying model is SUSY LM2, withthe mass of the heavy chargino adjusted by hand so that the decay chain can occur.To illustrate the polarization effect we restrict the heavy chargino to be produced onlyfrom the t -channel processes to obtain a larger polarization.The average value of the cosine of the transverse angle between particle 2 in therest frame of Y and particle Y in the lab frame h cos θ Y (cid:12)(cid:12) Y i is 0 .
124 (while h cos θ Y (cid:12)(cid:12) Y i = − . β Y cos θ Y β Y cos θ Y (cid:12)(cid:12) Y after we boost each event to the framewhere E /E = E T /E T is shown in Fig. 26. The distribution has a bias and theaverage value is − . B.2 Soft visible particles and off-shell effects
As mentioned in sec. 2, if the mass difference of two neighboring invisible particles inthe decay chain is small, the off-shell effects and the “fake polarization” created by– 36 – - - log1 + Β Y cos Θ Y + Β Y cos Θ Y Y ð of events p p @ GeV D - - - - log1 + Β Y cos Θ Y + Β Y cos Θ Y Y Figure 26 . The distribution of log β Y cos θ Y β Y cos θ Y (cid:12)(cid:12) Y (Left) and log β Y cos θ Y β Y cos θ Y (cid:12)(cid:12) Y vs. invariantmass-squared (Right) of Appendix B.1. Each event is boosted longitudinally to the frame inwhich E E = E T E T . The mean value of log β Y cos θ Y β Y cos θ Y (cid:12)(cid:12) Y is − . p p @ GeV D - - - log H E T (cid:144) E T L fittedtrue p p @ GeV D log H E T (cid:144) E T L fittedtrue Figure 27 . The scatter plot (left) and the peak-point plot (right) of log ( E T /E T ) vs. 2 p p for Appendix B.1. In both case the red curve is log ( E T /E T ) = log ( ∆ +2 p p ∆ ) with true ∆ and ∆ and the blue curve is the fitted one. ∆ [GeV ] ∆ [GeV ] log (∆ / ∆ ) m Y [GeV] m X [GeV] m N [GeV]true 5 . × . × .
342 1000 770 140 . . × . × .
349 1780 1536 1101error +100% +99% +0 .
67% +78% +99% +683%
Table 19 . The result of mass reconstruction from fitting the peak points for Ap-pendix B.1. The errors are calculated using reconstructed − truetrue (except for log (∆ / ∆ ), whichis reconstructed − true) and do not represent the statistical fluctuation. The statistical fluc-tuation are expected to be at the same order as the example in Section 2. the E T cut may cause our mass determination method to fail. Here we examine theseeffects in more details. We first consider a decay chain with a small ∆ . The spectrumis summarized in Table 20. The on-shell approximation does not seem to work well forparticle Y , which can be seen in Fig. 28. The distribution of δ log ( ∆ +2 p p ∆ ) (defined– 37 – X N particle heavy neutralino slepton lightest neutralinomass[GeV] 468 441.9 140.5
Table 20 . A summary of the decay chain in Appendix B.2.
460 470 480 490 500 510 520 p Y ð of events
438 440 442 444 446 448 p X ð of events Figure 28 . The histograms of q p Y and q p X of Appendix B.2. The mass and width of Y are 468 GeV and 3.115 GeV. The mass and width of X are 441.9 GeV and 0.4076 GeV. p p (cid:144) D D m X - - - ∆ log H D + p p D L Figure 29 . The scatter plot of δ log ( ∆ +2 p p ∆ ) vs. 2 p p / ∆ ∆ m X . δ log ( ∆ +2 p p ∆ ) is defined asthe effective value minus the true value, where the effective value for each event is calculatedusing p Y and p X while the true value is calculated using m Y and m X . The distribution isasymmetric, especially near 2 p p / ∆ ∆ m X = 1. This is due to a strong correlation between themaximum invariant mass and the effective values of ∆ and ∆ . as the effective value minus the true value, where the effective value for each eventis calculated using p Y and p X while the true value is calculated using m Y and m X )vs. invariant mass-squared is shown in Fig. 29. One could see that it is asymmetricespecially near 2 p p = ∆ ∆ m X . This is due to a strong correlation between the maximuminvariant mass and the effective values of ∆ and ∆ .In addition, particle 2 tends to be very soft in this example and the cut on E T – 38 – T cut (GeV) h cos θ Y | Y i (transverse) h cos θ Y | Y i (3-D)0 -0.025 -0.03910 0.041 0.01120 0.269 0.17630 0.439 0.315 Table 21 . h cos θ Y | Y i (transverse and 3-D) for the spectrum in Appendix B.2 with different E T cuts. p p @ GeV D - - log H E T (cid:144) E T L fittedtrue p p @ GeV D log H E T (cid:144) E T L fittedtrue Figure 30 . The scatter plot (left) and the peak-point plot (right) of log ( E T /E T ) vs. 2 p p for Appendix B.2. In both case the red curve is log ( E T /E T ) = log ( ∆ +2 p p ∆ ) with true ∆ and ∆ and the blue curve is the fitted one. introduces a fake polarization for the particle Y decay. In this case, a 10 GeV E T cut is applied to the visible particles and the resulting mean transverse angle betweenparticle 2 in the rest frame of Y and particle Y in the lab frame (cos θ Y (cid:12)(cid:12) Y ) is 0 . E T cut, we generated a fewmore examples with the same spectrum but different E T cuts. The results are shownin Table 21. It is clear that when the visible particles are soft, a large E T cut willintroduce a large fake polarization which could result in poor mass determination.The scatter plot and the peak-point plot are shown in Fig. 30. It is clear that thedistribution is shifted downwards. Furthermore, because the slope is very small, the re-constructed values are very sensitive to fluctuations which brings additional difficultiesto the reconstruction. The result of mass reconstruction is shown in Table 22.As a comparison, we look at the same case, but with the decay widths changedto very small values to force on-shell decays, and also with no E T cut. The result isshown in Fig. 31. It is clear that the bias presented in the previous case is removed.This shows that indeed the E T cut and the off-shell effects are the sources of the bias.The effects discussed above can also occur when ∆ is very small. To explore thiscase we generated events according to the spectrum in Table 23. In addition, the width– 39 –
000 10000 15000 20000 p p @ GeV D - log H E T (cid:144) E T L fittedtrue p p @ GeV D log H E T (cid:144) E T L fittedtrue Figure 31 . The scatter plot (left) and the peak-point plot (right) of log ( E T /E T ) vs. 2 p p of the case with very narrow width and no E T cut in Appendix B.2. In both case the redcurve is log ( E T /E T ) = log ( ∆ +2 p p ∆ ) with true ∆ and ∆ and the blue curve is the fittedone. The fit to the scatter plot works well. For the peak-point plot, the fluctuation is largedue to a small slope. However, it is clear that the bias in Fig. 30 is not present in these twoexamples. ∆ [GeV ] ∆ [GeV ] log (∆ / ∆ ) m Y [GeV] m X [GeV] m N [GeV]true 1 . × . × .
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