Signal-Event Generation and Normalization

g q

¯b t q⁰

b q

t q⁰

Figure 3.17: Leading-order Feynman diagrams for t-channel single-top-quark production in the 4-flavor scheme (2→3, qg→q⁰t¯b, left) and 5-flavor scheme (2→2, qb→q⁰t, right).

The LO-Feynman diagrams for t-channel single-top-quark production in both 4-flavor scheme (left) and 5-flavor scheme (right) are shown in fig. 3.17. The 4-flavor scheme does not contain b partons in the PDF, and the b quark has to be massive within the matrix-element calculation. In the 5-flavor scheme, logarithms (log(µ²_f/m²_b)) that arise from (collinear) initial-state-gluon splitting are resummed into the b-parton PDF [115, 187]. Here, the b-quark is as-sumed to be massless in the matrix-element calculation.

The 2→2 (qb→q⁰t) process (fig.3.17, right) implies that an additional final-state-b parton is generated within the parton shower, since the initial-state-b parton can be produced only via off-shell-gluon splitting (g^∗→b¯b) in the proton. Initial-State Radiation (ISR) modeling, e.g. with

PYTHIA, provides a way to generate the second b parton. However, the PYTHIA generator

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accurately models the additional-b-quark kinematics only for low transverse momenta. The 2→3 (qg→q⁰t¯b) process (fig.3.17, left) instead is suited most for modeling the hard-pT(2^nd b) region. The additional final-state-b quark, which does not stem from the top-quark decay, is also referred to as “2^nd-b quark” or “spectator-b quark”.

The 2→3 process can also be interpreted in terms of a NLO correction to the 2→2 LO-Feynman diagram. Event generation according to both diagrams gives an “effective” NLO description of the t-channel events. However, part of the phase space is double-counted when simply adding events from both 2→2 and 2→3 contributions. Moreover, different kinemat-ics for the spectator-b quark, as well as different production cross sections, are obtained for events from both diagrams. Therefore, event generators implement special procedures to match events from 2→2 and 2→3 diagrams.

Event generators that simulate t-channel events in NLO+Parton-Shower (PS) accuracy use either the 4-flavor scheme (with massive initial-state-b quarks) or the 5-flavor scheme (with a massless-initial-state-b-quark approximation) for event simulation. A full NLO description for the second-b quark is obtained only in the 4-flavor scheme, which is, however, not available yet (cf. the discussion in the following paragraphs).

In this analysis, the default signal modeling is done with thePOWHEGBOXevent generator [60–62, 188] in the 5-flavor scheme with NLO+PS accuracy. A generator based on COMPHEP [189, 190] (SINGLETOP) is further used to study the influence of the choice of the generator on the t-channel-signal modeling. This generator implements a matching of the 2→2 and 2→3 di-agrams according to the transverse-momentum distribution of the spectator-b quark, and uses massive b quarks. This generator is simply referred to as “COMPHEP” for unambiguousness in the following. COMPHEP was also used as the central signal generator for the observation of single-top-quark production by the D0 collaboration in 2009 [6]. COMPHEP generates events in an “effective NLO approximation“ [190]. The CMS experiment provides events for these two generators only.

t-channel events are generated using the CTEQ6M NLO PDF set [52]. Moreover, both

POWHEGBOXand COMPHEP preserve correlations of the top-quark spin between its produc-tion and decay.

Signal modeling with these generators is explained briefly in the following paragraphs. Fur-thermore, an overview of event generators that produce t-channel events in LO accuracy, 4-flavor scheme in NLO+PS accuracy, 5-4-flavor scheme in NLO+PS accuracy, and effective-NLO accuracy with matched 2→2 and 2→3 processes is given.

Leading-Order-Event Generation Event generators that are able to model t-channel events in the 4-flavor scheme (2→3) or 5-flavor scheme (2→2) in LO+PS accuracy include M^ADGRAPH

[59], WHIZARD[191, 192], andPYTHIA[63]. MADGRAPHand WHIZARDtake spin correlations between the top-quark production and decay into account. Furthermore, the WHIZARDevent generator is able to generate events with anomalous W tb couplings.

4-flavor scheme (2→3) at NLO The event generation of the t-channel 4-flavor scheme (2→3) with massive b quarks in the initial state is available at NLO with the POWHEG BOX[187] or aMC@NLO⁵[187] generators. Event generation in the 4-flavor scheme with massive b quarks yields a more precise description (NLO) of the spectator-b quark [187], and is, in principal, the

5aMC@NLO is implemented within the MADGRAPH5 framework and provides an automated matching between events generated in NLO-QCD accuracy and parton-shower simulations.

preferred choice to generate events. However, in the current implementations, the spin corre-lations between top-quark production and decay are not yet preserved. The total production cross section and differential distributions can also be obtained withMCFM[76, 115].

5-flavor scheme (2→2) at NLO ^POWHEG BOX[60–62, 188] and MC@NLO [67, 193] generate events in the 5-flavor scheme with the massless initial-state-b-quark approximation in NLO+PS accuracy. Hard and wide-angle emissions are calculated in NLO. Soft and collinear emissions are subject to PS modeling. POWHEG BOX and MC@NLO use different schemes to avoid a double- or under counting of phase space between matrix-element calculation and parton-shower simulation.

The MC@NLO generator subtracts double-counted phase space by generating negatively-weighted events. MC@NLO can be interfaced to HERWIG [64] for PS modeling. However, the current implementation faces a technical feature that leads to an unphysical description of the additional b-quark (cf. [187]). POWHEGBOX is interfaced toPYTHIA, which uses a pT -ordered shower to resum all remaining soft and collinear corrections. Renormalization- and factorization scale are set to the transverse momentum (relative to the beam axis) of the hardest emitted parton. Then, the first emission is always calculated byPOWHEGBOX, and subsequent radiation is performed byPYTHIA. Only positively-weighted events are obtained in this way.

In the matrix-element calculation, u, d, s, c, b quarks are assumed to be massless. However, the quark masses are considered as lower thresholds (p^min_T ) for parton radiation of a certain flavor (cf. [188] for more details). The default t-channel signal in this analysis is modeled with the

POWHEG BOXgenerator.

Matching of the 2→2 and 2→3 contributions at “effective-NLO” An alternative procedure involves the matching of 2→2 and 2→3 contributions in order to avoid a double counting of phase space. Matching procedures enable an “effective-NLO” description with massive b quarks. Automated matching procedures are implemented in COMPHEP (SINGLETOP) [189, 190], ACERMC [194, 195], and PROTOS [13]. Furthermore, PROTOS and COMPHEP are able to generate events with anomalous W tb couplings.

The COMPHEP (SINGLETOP)event generator matches events from both diagrams such that a smooth transverse-momentum distribution of the spectator-b quark is obtained. The match-ing procedure is described in detail in ref. [190], and briefly summarized in the followmatch-ing paragraphs.

For the 2→2 (qb→q⁰t) process (fig.3.17, right), the final-state-b parton is generated within thePYTHIAparton shower, which provides a good approximation at low transverse momenta.

This process is referred to as pp → tq + bISR,PYTHIA in the following. For the 2→3 (qg→q⁰t¯b) process (fig.3.17, left), the final-state-b parton is modeled within the matrix-element calculation.

The matrix-element calculation provides a good estimate for large transverse momenta of the additional b parton. This process is referred to as pp→ tq + bLO, COMPHEPin the following.

However, the kinematics of the spectator-b quark, i.e. the pT and η distributions, differ sig-nificantly between both processes pp→ tq +bISR,PYTHIAand pp→ tq +bLO, COMPHEP. A matching of both processes is done w.r.t. the kinematics of the spectator-b quark in the final-state,

σ_NLO= K × σpp→tq+b_ISR,PYTHIA

_pT(2ndb)<Q + σpp→tq+b_{LO, COMPHEP}

_pT(2ndb)>Q .

(3.16)

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Here, the matching threshold Q is optimized such that a smooth distribution of the transverse momentum of the spectator-b quark (pT(2^ndb)) is obtained. Then, also a smooth η-distribution is achieved. The overall normalization is kept constant at the total NLO cross section, while events with soft second-b quarks are enhanced by a factor K. The K-factor effectively resums higher-order-loop corrections [190]. A threshold of Q = 28 GeV/c is used for the simulated event samples at√

s = 7 TeV. In particular, this threshold is close to pT threshold for recon-structed jets as used in this analysis (30 GeV/c).

Another matching technique [195] is implemented in the ACERMC event generator [194].

Here, the full phase space is described by 2→2 and 2→3 diagrams, and the double-counted phase space is subtracted (eq. 3.17). Events of the 2→3-subtraction term (order α^(1)s ) obtain negative event weights.

σ = 2→ 2 ⊕ 2 → 3 (2 → 3)(subtraction term) (3.17)

The PROTOS event generator [13] implements two matching procedures. The first match-ing algorithm exploits a matchmatch-ing based on the pT of the 2^nd b quark, similar to the one im-plemented in the COMPHEP (SINGLETOP) event generator. The second matching procedure performs a subtraction to the b-quark PDF, which only has an effect on the 2→2 contribution.

Normalization The inclusive cross sections for the single-top-quark processes [8–10] are cal-culated with a top-quark mass of mt = 172.5 GeV/c², which corresponds to the value used in the event simulation. Calculations are available in approximate NNLO accuracy. Renormaliza-tion scale µrand factorization scale µf are set to a common scale µ≡ mt, i.e. µ = µf = µr = mt. The MSTW2008 NNLO PDF set [75] is used. Theoretical uncertainties of the calculated cross section arise from scale variations and the parametrization of the PDF set. The scale µ is varied between mt/2 and 2mt, and µf and µrare taken as fully correlated. The PDF uncertainty covers (eigenvector) variations within the MSTW2008 NNLO PDF set [75] at 90% CL. The t-channel-production cross section is predicted to be (41.92^+1.59_−0.21^+0.83_−0.83) pb for events with top quarks and (22.65^+0.50_−0.50^+0.68_−0.91) pb for events with top-anti quarks.