The t¯tγ cross section measurement is based on the template fit method, using as discriminating variable the track isolationpisoT distributions for two different kind of photon candidates: prompt-photons and prompt-photons from the decay of high-pT hadrons from jet fragmentation (e.g. π0, η neutral mesons decaying to diphotons). The track isolation provides a good discrimination between prompt-photons and hadrons faking photons, and is favoured over other isolation criteria (e.g.
calorimeter isolation with fixed cone) because of its smaller dependence with η (the calorimeter transverse isolation energy depends on the photon η due to the varying amount of material in front of the presampler) and its robustness against pile-up.
The pisoT absolute track isolation is generally defined as the scalar sum of the transverse
mo-Physics objects definition 68
menta of all selected tracks(ξ)in a cone ∆R <0.2around the photon candidate minus theET of the photon candidate:
pisoT :=p20T(γ) =
"
Z 0.2
0
dR Z ∞
pT(ξ)>1 GeV
dξ pT(ξ, R)
#
−ET(γ) (3.6)
For electron candidates, the tracks are required to have a transverse momentumpT >1GeV, a transverse impact parameterd0 ≤1 mm, a longitudinal impact parameterz0 ≤1 mm, at least six hits in the SCT and Pixel detectors and at least one hit in the B-layer (to avoid including tracks from conversions). Bothd0 and z0 impact parameters are computed with respect to the primary vertex. The minimumpT-cut minimises the effect of pileup and underlying events.
Contrary to electron candidates, the defaultpisoT track isolation for photons is computed without a vertex constrain on the tracks, as in general the vertex associated with the photon is not known (or subject to large uncertainties). Since the signal isolation template for prompt-photons is extracted by extrapolating the electron template fromZ →eedecays using thet¯tγ MC sample, a consistent definition ofpisoT for both electrons and photons is required in this case for consistency in the isolation definitions.
The photon track isolation is thus recomputed by excluding all tracks that fail a minimum z0 = 1 mm cut, in the same way as it is done for electrons. Since only the total number of tracks that entered into the calculation of the photonpisoT is known, but not the tracks themselves, permutations among all reconstructed tracks are performed in order to extract the subset of them that give rise to the original photonpisoT . Within the selected subset, the pT of each track within the cone ∆R < 0.2 not passing the z0 requirement is subtracted from original track isolation of the photon candidate, see Fig. 3.21.
0 1 2 3 4 5 6 7 8 9 10
102
103
104
105 ttγWhizard
Uncorrected Corrected
pTiso [GeV]
Events / GeV
0[mm]
z
-5 -4 -3 -2 -1 0 1 2 3 4 5
1 10 102
103
104
Whizard γ t t
Events / mm
Figure 3.21: Comparison of the track isolationpisoT for photons candidates (without any additional event selection) before and after correction by subtracting the transverse momentum of all tracks with |z0| > 1 mm (left) and distribution of the longitudinal impact parameter for all tracks found within a cone∆R = 0.2 around the photon direction (right). Distributions are evaluated usingWHIZARD. The vertical dashed lines in the right plot correspond to the cut applied to the z0 distribution [86].
Cross section definition
The phase-space in which the cross section is reported needs a definition. In order to compare the analysis results with any theoretical prediction, the cross section measurement is made within a fiducial phase-space defined from simulations oft¯tγ decays in the single-lepton (electron or muon) final state. This chapter gives the formal definition of, and the motivation for, the phase-space used. The prediction from theory is also reviewed and its projection, into the volume in which the measurement is performed, is calculated.
The chapter starts with Sec. 4.1, which defines, generally, the relation between the cross section and the number of observedt¯tγevents. A distinction is made between the cross section measurable within the detector phase-space and its extrapolation to larger regions. The construction of the phase-space follows. At first, the definition of the particles constituting the phase-space (based upon observable quantities) is given (Sec. 4.2), then the event selection criteria (closely following those applied on data) are applied (Sec. 4.2.5). At each step the correlation between the simulation-based definitions and the reconstructed quantities is reviewed.
Simulated events are categorised in exclusive ensembles which are based on the fulfilment (or not) of the definitions for both the simulation-level particles and the reconstructed quantities (Sec. 4.3). Consequently, the detection and reconstruction efficiency with respect to the phase-space is extracted.
The next-to-leading-order theoretical prediction is explained in Sec. 4.3, and its leading-order computation is compared to that of thettγ¯ simulation programs used by this analysis. In Sec. 4.5, the prediction of several models is projected into the fiducial phase-space.
4.1 General considerations
Considering two opposite oriented bunched beams of Nb colliding protons then the number of scattered events (dNs) per unit time (dt) and unit volume (dV) is:
dNs=σLLumidV dt (4.1)
where LLumi is the luminosity (see Eq. 2.1) of the two colliding beams. The proportionality constant σ is by definition the cross section. From Eq. 4.1 it is easy to see that σ has the
69
Cross section definition 70
dimension of an area. The proportionality constant must be related to the invariant amplitude Mif(pp→ `νlqqb¯ ¯bγ) and the phase-space (Φ =V t) must be written in a Lorentz-invariant from.
A n-body Lorentz-invariant phase-space Φn, where incoming particles (i) have four-momenta pi
and out-coming particles (f) with four-momenta pf, can be written as dΦ(n)= (2π)4δ(4)
ni
X
i
pi−
nf
X
f
pi
n
Y
j
d3pj (2π)32Ej
(4.2) which defines the integrated cross section over an arbitrary period of time:
dσt¯tγ = 1 RLLumidt
Mif pp→`νlqqb¯¯bγ
2dΦ(n) (4.3)
Reformulating Eq. 4.3 as a function of Nb background events and incorporating the phase-space elementΦ(n)into a geometrical and kinematic acceptance factor(A)one obtains the reduced cross section times the Branching Ratio (BR) :
σt¯tγ×BR= N −Nb
A·C·R
LLumidt (4.4)
where: Ns=N−Nb is the number ofttγ¯ observed data events with`νlqqb¯¯bγ final state (`≡e, µ) and C is a detection efficiency correction, i.e. the fraction of recorded detector events over the total. It follows that a cross section measurement will depend on both C and A, therefore, the reproducibility of the result depends upon the correct definition of those constants. The acceptance defines the phase-space in which the result is reported and it is a measure of the extrapolation from the detector phase-space, to a theoretical phase-space defined by kinematic cuts imposed at simulation level. Cross sections with A = 1 are called fiducial (σfid) since the value is reported within the the geometrical (and kinematic) fiducial marks of the detector. A cross section with A >1will be referred in this document as a total cross section(σtot). One can easily expressσfid as a function ofσtot:
σfidt¯tγ×BR=A×(σttot¯tγ×BR) = N −Nb C·R
LLumidtL. (4.5)
The extrapolation from the phase-space in which the measurement is performed to the total phase-space can be subject to large theoretical uncertainties, ill-defined kinematic regions and simulation-induced model-dependencies. Figure 4.1 illustrates the level of the extrapolation down to the WHIZARD simulation defined phase-space for the photon and lepton transverse momenta respectively. It can be seen that the extrapolation reaches values as large as six times the size of the detector defined phase-space.
As a further example, the WHIZARD phase-space imposes cuts on invariant masses between quarks which are not a detector observable quantity. The exact extrapolation from the detector observable, a jet, to a quark, is not known and can only be defined using simulation programs.
As no experimental data can control the phase-space for A >1 the measurement reported in this document is chosen to be evaluated atA= 1.
The advantage of a total cross section is that, from an experimentalist point of view, no the-oretical prediction needs to be determined. Since the extrapolation is done to the simulation defined phase-space, the result can be directly compared with those values. This is also an advan-tage if two similar experiments want to compare their results to the theory predictions. If both
[GeV]
Figure 4.1: Extrapolation acceptance factors with respect to theWHIZARDphase-space for the pho-ton (top) and the leppho-ton (bottom) transverse momenta. The dotted line represents the minimum transverse energy requirement imposed on reconstructed objects. The results are separated into the electron channel (left) and muon channel (right) fromttγ¯ →`νlqqb¯¯bγ decays.
experiments choose the same simulation program (with same settings), then a direct comparison between the two results is obvious. On the other hand, the disadvantage stands in the fact that the universality of the result is not easy to achieve. Extrapolating to other definitions of phase-spaces will be subject to corrections which are, from one side, difficult to determine and, from the other, not possible to confirm experimentally. The clear disadvantage of a fiducial measurement stands, from the point of view of the experimentalist, that the theoretical prediction has to be re evaluated within the phase-space defined by the experiment.