) = 34.3 µ
incl( ε
Whizard γ t t
) [GeV]
(γ ET
50 100 150 200 250 300 350
[%]ε
0 20 40 60 80
100 Efficiencies
Electron channel Muon channel
0.4 % ) = 18.5 ± (e εincl
0.6 %
± ) = 35.3 µ
incl( ε
MadGraph γ t t
Figure 4.11: Efficiency correction factor evaluated as a function ofpT(γ) from theWHIZARD (left) andMadGraph(right) samples respectively. Bin-by-bin migration effects are included in the calcu-lation. The uncertainties are statistical only.
obtained from the two simulation programs are in excellent agreement.
4.4 Next-to-leading order theoretical prediction
Thet¯tγ cross section result is compared to that of the Next-to-Leading-Order (NLO) calculation in the narrow-width approximation [56], see Sec. 1.4.2.
This calculation is based on the method of generalised D-dimensional unitarity extended to massive particles and on the dipole formalism which are used, respectively, to calculate one-loop virtual amplitudes and real emission corrections. Top quarks are treated in the narrow-width approximation with all spin correlations retained. The hadronic decays ofW-bosons are considered into two families of light quarks, always treated as massless. TheW-bosons are considered in their mass-shells and no Quantum Chromodynamics (QCD) radiative corrections to the hadronic decays are considered. The strong coupling constant is evaluated using one- and two-loop running with five massless flavours.
Because of the treatment of top quarks as unstable particles, this calculation is less dependent on the kinematics of the phase-space defining it, see Fig. 4.12. In particular, the k-factor is considered to be stable with respect to the photon transverse momentum. Whilst the original
calculation (pp→t¯tγ →bµ+νµ¯bjjγ) was performed at a centre-of-mass energy (√
s) of 14 TeV, a dedicated prediction [54] at√
s= 7 TeV has been calculated.
The phase-space is defined by a single high-pTmuon, at least four jets (j) and a neutrino with large momentum. Specifically, the particles are subject to the following definition:
• The muon (µ) is required to havepT(µ)>20GeV and rapidity (y) |y(µ)|<2.5.
• Jets are clustered with the anti-kT [103] algorithm with radius parameter R= 0.4, and they are required to havepT(j)>25 GeVand |y|<2.5
• The photon is constructed using the infra-red safe Frixione prescription [122] with radius parameter of R = 0.5. This method guarantees photon isolation from near-by hadronic activity. A good photon is required to havepT(γ)>15 GeVand y <2.37.
• A good neutrino is required to havepT(ν)>25 GeV.
• Ab-jet is defined as a jet that containsb-quarks from top quark decays in the clusterisation of the jet.
Events are selected applying the following selection criteria:
• The event must contain only one good muon and only one good neutrino.
• The event has to contain at least four good jets, at least two of them must beb-jets.
• Any pair ofiand j jets needs to be separated by ∆R(ji, jj)>0.4.
• The muon is required to have an angular separation with any good jet of∆R(j, µ)>0.4.
• TheW transverse mass, defined asmT(W) =p
2pT(ν)·pT(µ)(1−cos ∆φ), must fulfil the condition: pT(ν) +mT(W)>60 GeV.
• The final-state photon is required to be separated with the muon by ∆R(γ, µ) < 0.4 and with any jet by∆R(γ, j)<0.5
For the renormalisation and factorisation scalesµR =µF =µ=mt. The calculated Leading-Order (LO) and NLO cross sections are:
σLOt¯tγ = 14.7±0.1 (stat)+5.8−3.8 (syst) fb (4.11) and
σNLOttγ¯ = 24.5±0.1 (stat)+5.6−4.5 (syst) fb (4.12) The upper- and lower- bounds correspond to scale variations by a factor of two around the central valueµ=mt. The quark-gluon annihilation, appearing only at NLO, is assumed to be at the origin of this scale dependence [56]. The large value of thek-factor is speculated to be caused by additional radiation originated by high-pT jets [56]. The Parton Density Function (PDF) set used for the LO (NLO) calculation is the MSTW2008 [24–26]. αswas evaluated using a two-loop running fromαs(mZ). The top has a mass ofmt= 172 GeVand a decay width ofΓt= 1.3237 GeV. The fine structure constant used in all cases isαQED = 1/137.
Cross section definition 84
) [GeV]
(l+
pT
0 50 100 150 200
)+ (l Tdp
LOσd
/ )+
(l Tdp
NLOσd
2.6 2.8 3
,bjet) R(γ
∆
0.5 1 1.5 2 2.5 3 3.5
,bjet)γR(∆
LOσd / ,bjet)γR(∆
NLOσd
2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8
[GeV]
miss
ET
0 50 100 150 200
miss TdE
LOσd / miss TdE
NLOσd
2.6 2.8 3
) [GeV]
(γ pT
0 50 100 150 200
)γ( Tdp
LOσd / )
γ( Tdp
NLOσd
2.2 2.4 2.6 2.8 3
) y(γ -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
)γdy(
LOσd / )
γdy(
NLOσd
2 2.2 2.4 2.6 2.8 3 3.2 3.4
Figure 4.12: k-factor σtNLO¯tγ /σLOt¯tγ, with µR = 2mt and µF =√ ˆ
s, as a function of: the lepton pT on the upper left frame, the ∆R(γ, b-jet) on the upper right frame, the of missing transverse momentum in the middle frame, thepT(γ)in the bottom left frame and they(γ)in the lower right frame [54] forpp collisions at √
s= 7 TeV.
The authors of this result provided also a dedicated calculation withµR = 2mt and µF =√ ˆ s with all other settings kept as above, of which the results can be seen in. The results are:
σLOt¯tγ = 10.9±0.1 (stat)+4.3−2.8 (syst) fb (4.13) and
σNLOttγ¯ = 27.5±0.1 (stat)+6.3−5.1 (syst) fb (4.14) For the calculation performed withµR =µF =mt and for the calculation performed with µR = 2mt and µF =√
ˆ
sthe corresponding k-factor are1.67 and 2.55respectively.
4.4.1 Comparison with the theoretical phase-space
The LO theoretical calculation has been compared with that obtained with the WHIZARD and MadGraphsimulations. The same phase-space cuts as used in the √
s= 7 TeV theory calculation have been applied to the generated event four-vectors. However, the theoretical calculation was originally performed in the µ+ channel, consequently only positive muons are selected for this comparison. All the cuts are made as consistent as possible with the theoretical calculation [56]:
Muons Muons are required to havepT(µ)>20 GeVand |η(µ)|<2.5. The event must contain a single-muon fulfilling these requirements.
Jets At least four jets, each constructed with the anti-kT algorithm, with radius parameter R= 0.4, and each required to havepT(j)>25 GeV and|η(j)|<2.5.
Photons Photons are required to have ET(γ) > 15 GeV and |η(γ)|< 1.37 or 1.52 < |η(γ)|<
2.37.
Missing transverse energy definition. The magnitude of the four-vector sum of all neutrinos in the event (ETmiss) is required to be ETmiss>25 GeV.
W transverse mass. A W transverse mass and ETmiss requirement are imposed: ETmiss + mT(W)>60 GeV. ThemT(W) is built from the lepton and from the above defined ETmiss. Angular separations. Any pair of jets (i, j) is required to have ∆R(i, j) > 0.4 and each jet is required to have ∆R(j, µ) > 0.4. The photon is separated from the lepton with
∆R(γ, µ)>0.4. Any jet is separated from the photons with∆R(γ, j)>0.5.
The LO cross section obtained withMadGraphor WHIZARDafter applying the theory cutsσtLO,cuts¯tγ is:
σLO,t¯tγ cuts=
Ngen, cuts Ngen, all
×σtLO¯tγ (4.15)
where: Ngen, cuts is the total number of events at generator level after applying the phase-space cuts used in the theoretical calculation. Ngen, all is the total number of events generated in the single-positive-muon channel, σtLO¯tγ fb is the LO single-lepton cross section of the generated t¯tγ simulation (MadGraphor WHIZARD) sample.
A reasonable agreement between theory and the two generators is obtained, as shown on Tab. 4.4.
Cross section definition 86
Sample σte,LO¯tγ [fb] σtµ,LO¯tγ [fb] σtLO¯tγ Theory µR µF
WHIZARD 8.2±0.5 (stat) 9.2±0.6 (stat) 10.9±0.1 2mt √ ˆ s MadGraph 14.7±0.6 (stat) 16.3±0.6 (stat) 14.8±0.4 mtop mtop
Table 4.4: Leading order cross sections as obtained forWHIZARD and MadGraphin the theoretical phase-space. Numbers are based on samples of 2.5×104 events. A reasonable agreement is observed.