Transfer Functions

For each event, the one permutation which maximizes the log likelihood compared to the other eleven permutations is chosen as the proper jet ordering for this event. The jets in the event are then assigned by the jet ordering in this permutation.

7.3. Transfer Functions

The transfer functions account for the energy difference between the reconstructed objects and their original parton energy in a LO picture. The transfer functions are parametrized by a double-Gaussian. This function accounts for tails in the energy differ-ence between reconstructed and parton level. The transfer functions are defined by:

W

E˜jet |Eq

= 1

2π(p2+ p3p5) ·

"

e⁻

(∆E−p1)2

2p22 + p3·e⁻

(∆E−p4)2 2p25

#

, (7.2)

where ˜E_jet is the energy of the measured jet and E_q is the original particle energy. The

∆E is given as:

∆E = E_truth−Ereco

E_truth . (7.3)

In each object, there are five fit parameters which are determined from MC. They are all functions of the parton energy. The five fit parameters for light jets and electrons are:

p1 = a1+b1E;

p2 = a2/√

E+b2; p₃ = a₃+b₃E;

p₄ = a₄+b₄E;

p5 = a5+b5E.

The parametersa_i andb_i are fit and assumed to be continuous for the jet energy range up to 700 GeV. Beyond this energy, the transfer functions are extrapolated using the same fit.

For muons, all five fit parameters are linearly dependent on the parton energy. Forb jets, parameters p₁ andp₃ are given as: p_i=a_i/√

E+b_iE, whereas the remaining parameters are identical to the light jet parameters. The transfer functions are normalized to one for a given truth energy, such that:

Z

dEreco W(Ereco|Etruth) = 1. (7.4)

Not only is the fit divided into ranges of energy (or p_T for the muons) but also for different detector regions in|η|. The four different regions are given as: 0<|η|<0.8, 0<|η|<1.37, 1.37<|η|<1.52, and 1.52<|η|<2.5. For muons, the detector regions are not based on calorimeter sections; therefore, the division in|η|is given by three sections: 0<|η|<1.11, 1.11<|η|<1.25, 1.25 <|η|<2.5. An example of the transfer function double Gaussian fit for a given energy range in the barrel section of the detector is found in Figure 7.1.

7. Reconstruction of Top Quark Pairs

Figure 7.1.: (Left): b jet transfer function in the central detector region (|η|< 1) and parton energy 145<E <175 GeV. (Right): Light jet transfer function in the central detector region (|η|<1) and parton energy 64<E<87 GeV.

In both cases, the shape of the double Gaussian is presented by the two curves, plus the combination.

The transfer functions are derived using a sample of signal MC only at a top mass equal to (mtop = 172.5 GeV/c²). Reconstructed jets are matched with the truth quarks using the criteria:

∆R(reco,truth)<0.3. (7.5)

If an event contains a jet which is matched to multiple quarks or a quark which is matched to multiple jets, the event is not used in the calculation of the transfer functions.

It is also required that the event contain at least four matched jets. The four matched jets must be the jets from the leading four p_T jets. The additional jets in the events are disregarded. Therefore, transfer functions are constructed in very clean sub-samples of the total number of events. The sub-sample from which the performance of the kinematic fitter is tested accounts for about only 27 % of the original sample size. The same transfer functions derived on signal are used for background as well.

Examples of the evolution of the light andb jet transfer functions for several different parton energies is given in Figure 7.2. The evolution shows the change in width and shape as the jet in question has higher energies. The transfer functions are validated up to jet energies of about 700 GeV. Comparing the transfer functions of the jets and leptons, the transfer functions for jets are much wider than for the leptons.

84

7.3. Transfer Functions

Jet Energy for parton E = 100 GeV Jet Energy for parton E = 150 GeV Jet Energy for parton E = 200 GeV Jet Energy for parton E = 250 GeV Jet Energy for parton E = 300 GeV Jet Energy for parton E = 350 GeV Jet Energy for parton E = 400 GeV Jet Energy for parton E = 450 GeV Jet Energy for parton E = 500 GeV Jet Energy for parton E = 550 GeV Jet Energy for parton E = 600 GeV Jet Energy for parton E = 650 GeV

[GeV]

0.035 Jet Energy for parton E = 50 GeV Jet Energy for parton E = 100 GeV Jet Energy for parton E = 150 GeV Jet Energy for parton E = 200 GeV Jet Energy for parton E = 250 GeV Jet Energy for parton E = 300 GeV Jet Energy for parton E = 350 GeV Jet Energy for parton E = 400 GeV Jet Energy for parton E = 450 GeV Jet Energy for parton E = 500 GeV Jet Energy for parton E = 550 GeV Jet Energy for parton E = 600 GeV Jet Energy for parton E = 650 GeV

| < 1.5 0.8 < | η

Figure 7.2.: Evolution of the transfer functions for the entire fitted parton energy range.

(Top): Light jet transfer functions for the central |η| region. (Bottom):

b jet transfer functions for the middle |η| range with the calorimeter gap removed. All transfer functions are normalized to 1.

7. Reconstruction of Top Quark Pairs

7.4. b-Tagging

To improve the reconstruction efficiency of the fitter, the b-tagging information is also used in the likelihood. Each jet is given a b-tagging weight, and it is required that at least one jet has a weight over the 70 % efficiency working point in each event. Therefore an additional term is added to the end of the likelihood expression in Formula 7.1, (wbtag), given by:

w_btag =

ε , bhad has b-tag (1−ε), b_had has no b-tag

·

ε , blep has b-tag (1−ε), b_lep has no b-tag

· ₁

R , q₁ has b-tag (1−_R¹), q1 has no b-tag

·

₁

R , q₂ has b-tag (1−_R¹), q2 has no b-tag

, (7.6) where εis the efficiency andR the rejection factor of light jets. The working point of the tagger contains a 70 % efficiency of selecting true b jets and a rejection of 99. Therefore, the additional likelihood expression for b-tagging gives 0.7 for a b-tagged jet in thebquark position, 0.3 for a non b-tagged jet in the same position, and 1/99 for a b-tagged jet in the light quark position and 1 - 1/99 for a non b-tagged jet in the light quark position.

The likelihood thus favours the permutation for which a b-tagged jet is placed in the leptonic or hadronic bquark position and where the non b-tagged jets are placed as light quarks from the W decay. This information complements the kinematic likelihood part of the likelihood.