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On Measuring the top quark mass using the dilepton decay modes

We demonstrate a new likelihood method for extracting the top quark mass from events of the type ttbar-->bW(l+nu)bW(l+nu) This method estimates the top quark mass correctly from an ensemble of dilepton events. The method proposed by Dalitz and Goldstein [1] is shown to result in a systematic underestimation of the top quark mass. Effects due to the spin correlations between the top and anti-top quarks are shown to be unimportant in estimating the mass of the top quark.

and independently Kondo [2] et al have shown, it is possible to extract mass information from these events using a likelihood method. For each event, solutions are obtained for the kinematic quantities for a range of top quark masses. Each solution is weighted by a product of structure functions which estimates the probability of producing a tt pair consistent with the event at that top quark mass and a decay probability factor which neglects the polarization of the top quark. In this paper we show that the Dalitz-Goldstein weighting scheme leads to a systematic underestimation of the top quark mass. We propose a likelihood scheme which involves no kinematic weighting that is shown to estimate the top quark mass correctly. Finally we show that not allowing for the spin correlations in the decay of top quarks in the Dalitz-Goldstein scheme does not further bias the mass estimate significantly.
With the proposed luminosity upgrades of the Tevatron [3], it is possible to acquire thousands of events of the type tt → bW + (lepton + ν)bW − (lepton + ν), where both the b quark jets are identified. The number of jet permutations in these channels is smaller than the lepton + jets decay modes of the top quark. It may then become possible to measure the top quark mass using the dilepton channels with the least amount of systematic error.

II. METHOD
Each dilepton event is characterized by 14 measurements, namely the three vectors of the two b jets, leptons and the missing E T vector of the event. We denote these measurements collectively by the configuration vector c. Kinematically, each event is characterized by 18 variables namely, the three vectors of the b jets, leptons and the two missing neutrinos.
For any given top quark mass, there are four constraints, that constrain the lepton and neutrino pairs to the W mass and the W and b pairs to the top quark mass. Given a top quark mass, this enables one to solve for the neutrinos. This results in a pair of quadratic equations for the transverse components of each neutrino [2]. The solution involves finding the intersection of two ellipses. This can yield zero, two or four solutions for a given top quark mass. The likelihood P(m|c) of a solution for a top quark mass m, given the observed configuration vector c, is obtained by using Bayes' theorem.
Where P(m) is the a priori probability distribution of the top quark mass. P(c|m) is the probability of observing the configuration vector c, for a given top quark mass m. If after each event is analyzed, P(m) is updated by P(m|c) iteratively, one gets the familiar multiplicative rule for combining likelihoods. Dalitz and Goldstein [1,4] use the prescription The Monte Carlo integration then yields

B. Choice of the configuration vector
In what follows, we will assume that both the leptons are electrons. We choose the three quantities, energy, pseudo-rapidity and azimuth (E,η, φ) to define the three vectors of the leptons and jets. The electrons are smeared with a typical collider detector fractional resolution of 15%/ (E) in energy and the jets with a fractional energy resolution of 80%/ (E) ⊕ .05. We ignore the fluctuations in direction, as these are dwarfed by the energy fluctuations. The p T of the rest of the event after removing the leptons and jets is also a measured quantity and is smeared as though it were a small jet. The E / T is a deduced quantity from the measured quantities listed. The case when one or both of the leptons is a muon is handled by smearing the inverse momentum of the muon as a Gaussian, but will not be further discussed here.
We do not a priori know which lepton is associated with which b quark. We consider both combinations and add the likelihoods from either combination to form the total likelihood for each event, which is normalized to unity when integrated over the top quark mass m. In general Monte Carlo events have weights associated with them. These were normalized so that the average weight in the event sample was unity. Events with weights outside the window 0.3-3.0 were rejected. The likelihood distribution for each event was raised to the power given by its weight before being used to form the combined likelihood.

D. Event selection criteria
We select only those events with E T > 15 GeV for both the leptons and jets and E / T > 25GeV . We demand that both the b jets are explicitly identified by a tagging algorithm.
While smearing, we only admit smeared configurations that satisfy the same criteria as the event selection.
In what follows we smear each Monte Carlo generated event once to simulate the mea-surement process and subsequently 1000 times to do the Monte Carlo integration.

III. RESULTS
We generate Monte Carlo events with a top quark mass of 175 GeV/c 2 . We neglect top quark polarization in generating these events, but treat the subsequent W decays according to the standard model [5].   Figure (1(b)) shows the probability distribution for this event using the Dalitz-Goldstein prescription of equation (2.2). The structure function weighting in equation (2.2) makes the high mass solutions less likely yielding a likelihood distribution that has a distinct peak. We now proceed to analyze a sample of ≈ 1000 such Monte Carlo events that decay into dileptons. Because of measurement errors, not all of these events will give solutions consistent with a top quark in the mass range 100-250 GeV/c 2 . Figure (2) is a histogram of the quantity R defined by where N i is the number of solutions for top quark mass i , tot M is the total number of top quark masses considered and N smear is the total number of smears per event. The sum extends for top quark masses in a window ± 35 GeV/c 2 of the generated top quark mass.
There is a peak in the histogram for values of R below 0.1. This is due to events that are so mismeasured that they have difficulty solving for a top quark mass in the window considered even when smeared a thousand times. We reject events with R <0.2 since these will have very spiky likelihood distributions. Figure (3(a)) is the combined likelihood of 511 events which survive after event selection criteria and the R cut from an initial sample of 925 events, using the Dalitz-Goldstein weighting scheme [7].
The most likely top quark mass from the event sample is 164.5 ± 0.54 GeV/c 2 . The Dalitz-Goldstein weighting scheme thus introduces a bias of 10.5 GeV/c 2 towards lower masses at this value of the top quark mass.

A. A Critique of the Dalitz-Goldstein weighting scheme
For a given event, the parton momenta (x 1 , x 2 ) needed to produce it will decrease as the top quark mass m is decreased since x 1 x 2 = m 2 /s, where s is the overall center of mass energy squared. This means that the Dalitz-Goldstein weighting scheme will tend to skew the likelihood distribution for each event toward lower top quark masses, since it is proportional to the product of the structure functions. We note that the top quark production cross section is also a product of such structure functions and decreases rapidly as the top quark mass increases, for the same reason. The likelihood scheme proposed by Kondo et al [2] is proportional to the top quark production cross section and also suffers from this defect.
It is this skewing of the likelihood distributions towards lower masses that produces a 10.5 GeV/c 2 bias in the Dalitz-Goldstein scheme. One can indeed ask why the top quark mass measurement has to be coupled to its production mechanism at all. Figure (1(a)) shows the number of solutions for a typical event as a function of the top quark mass. We now make the radical proposal of not using any weights at all, but simply use a likelihood distribution that is shaped like the number of solutions as a function of the top quark mass. If one examines this distribution visually for an ensemble of top quark events, there exist a significant number of events where the likelihood distribution thus formed does show a peak and falls for large top quark masses. Using this scheme, one gets the combined likelihood of Figure (3(b)) which peaks at the input mass, but has a larger standard deviation. The larger standard deviation is due to the fact that we are not suppressing the high mass tail of the individual event likelihood distributions using a weighting scheme.

B. A new likelihood method
This method does not use any extrinsic information of the top quark production mechanism to obtain the mass but relies solely on the measured kinematic quantities of the events in question. We christen this scheme the "no-weights" method. Figure (4(a)) shows the evolution of the mean value of the combined likelihoods for the Dalitz-Goldstein method and the no-weights method as a function of the number of events. Figure (4(b)) shows the evolution of the standard deviation [8] of the combined likelihoods using the two methods as a function of the number of events. An approximate 1/ (N) dependence on the number of events is evident. The "no-weights" mass is slightly sensitive to the value of the R cut, since the events rejected by the R cut tend to favor lower top masses. It is possible to adjust the R cut so that the input top mass is returned by the "no-weights" algorithm. Once tuned at one generated top quark mass, the algorithm works well at all other masses with the cut unchanged. The Dalitz-Goldstein scheme cannot reproduce the generated mass for any value of the R cut. It should be noted that the window chosen around the generated mass in defining the R cut has to be symmetric about the generated mass to avoid bias. This can be done iteratively when dealing with data.

C. Spin correlations and final state radiation effects
We now generate events where both the top and anti-top quark polarizations are taken into account and all spin correlations are kept at the tree level [6]. We use the two weighting methods outlined above to determine the top quark mass. The results are presented in  [9]. We demand that both the b quark jets are identified. Table (I) shows the results using either method. Once again, the Dalitz-Goldstein method underestimates the generated mass. The "no-weights" method can now be used to estimate the effects due to final state radiation as implemented in Isajet. It can be seen that the net effect of the final state radiation is to systematically lower the measured value of the top quark mass.
The amount of lowering increases with the top quark mass, due to the increased amount of final state radiation. At a top quark mass of 180 GeV/c 2 , the effect of final state radiation is to lower the top quark mass by ≈ 5 GeV/c 2 . Finally, we have also studied the effect of the event selection E T cuts for their effect on the result. We get results that are the same within errors, even when no E T cuts are used.

IV. A PROPOSAL FOR A CORRECT WEIGHTING SCHEME
If one insists on weighting events using production and decay information from the standard model, the expression for P(c|m) has to have the following properties.
P (c|m)dc = 1 (4.1) An expression that satisfies this is given by where σ vis (m) is the top quark production visible in the detector. The biasing effect in the top quark mass due to the structure function product is removed by division by the function σ vis (m). The configuration vector can be chosen as any set of measured variables, since the resulting expression for P(m|c) is invariant under a change of variables [10]. However, equation (2.3) , implies a unique set of variables for the configuration vector c, since these are the quantities that are fluctuated in a Gaussian fashion. We will report on results using this weighting scheme in a forthcoming paper.

V. CONCLUSIONS
We have demonstrated a new likelihood method that determines the top quark mass in dilepton decays of the top quark that gives an unbiased estimate of the top quark mass. We demonstrate that weighting schemes that involve products of structure functions such as the Dalitz-Goldstein scheme, give a downward bias to the measured value of the top quark mass.
We demonstrate that spin correlation effects between the top and anti-top decay products do not influence the outcome of the mass measurement. We estimate the effects due to final state radiation as implemented in Isajet.
The statistical precision obtainable using a thousand top to dilepton fully tagged events using this method is of the order of a GeV/c 2 using this technique. Assuming that jet energy scale systematics in the upgraded Tevatron detectors can be controlled to this level, the dilepton channels provide an excellent means of measuring the top quark mass.

VI. ACKNOWLEDGEMENTS
The author wishes to thank Stephen Parke for helpful discussions and for providing the non-Isajet top Monte Carlo samples.