Modeling of associated tW single top production including the interference with tt ̄ diagrams

The associated tW single top production channel is one of the main mechanism of the electroweak production of top quarks at the LHC. Correct simulation of the SM prediction for this process is one of the important tasks in the SM measurements and BSM searches. One of the difficulties to simulate it correctly is a significant contribution of the interference terms with the pair top quark production. In this article authors compare most popular simulation schemes for tW production with simulation of the complete set of diagrams and propose new scheme to simulate interference terms without negative weights of the events.


Introduction
Electroweak top quark production is one of the most promising topic to search for New Physics. These are unique processes to measure some of the SM parameters. There are three different production channels. They are classified using the topology of diagrams which include W-boson [1] as t-channel, s-channel and associated tW production. The associative tW channel becomes important at the LHC and requires proper simulation. The correct simulation of the tW process is difficult due to the well known problem [2]- [8] that at NLO level tW diagrams interfere with pair top quark production diagrams. The top quark pair production has more than ten times larger cross section than the associated tW production and interference is an important part in case of the precise measurements of tW process. There are different approaches to simulate associated tW production with different advantages and disadvantages. It is also possible to simulate complete set of tree level diagrams including pair top quark production and interference terms. In the next sections we discuss different popular approaches, simulation of the complete set of Feynman diagrams and new scheme proposed diagr.8 Figure 1: Diagrams for the process gg → tbW − .
in this article which we call "contact interactions" (CI) scheme. The CI scheme solves number of simulation problems. These schemes were implemented in CompHEP [9,10] package and the necessary simulations have been performed. The results are presented in the next sections.

Modeling approaches
The main LO contribution to the tW process comes from subprocesses with b-quark in the initial state. At NLO level this contribution yields from gluon collisions and leads to the process gg → tWb which includes 8 diagrams. The corresponding diagrams are presented in Fig. 1. Simulation of all these diagrams is the most precise approach to model this process. However, part of these diagrams correspond to the pair top quark production with significantly lager cross section than the other diagrams with single top quark production. It is possible to distinguish double resonant diagrams with pair top quark production and single resonant diagrams with single top quark production. In additional, there is significant interference between double and single resonant diagrams. Accurate simulation of the associated tW electroweak single top quark production have to take into account not only single resonant diagrams, but also the negative interference terms with pair top quark production. Different approximate schemes have been developed to properly model tW channel of single top quark production. Diagram Removal. The most simple recipe is to remove all Feynman diagrams with pair top quark production, diagrams 1-3 in Fig 1. Such approach is called "Diagram Removal" (DR) scheme [2,3,7]. In this approach the interference of single resonant diagrams (diagrams 4-8 in Fig. 1) and double resonant diagrams (diagrams 1-3 in Fig. 1  during the modeling. The remaining diagrams are shown in Fig. 2 for the single resonant contribution and in Fig. 3 for the interference between single and double resonant contributions. This approach allows to retain all of anticipated single top quark production properties in the Wtb vertex. On the other hand this subset of diagrams also introduces extra singularities from an uncoupled top quark decaying into W boson and b quark in either part of squared diagrams. This effect can be clearly observed in a distribution of invariant mass of system of W boson and b quark, and leads to negative weights of the significant part of the simulated events.
Contact interactions scheme. An improvement can be done to previous scheme in order to get rid of described singularities. These singularities can be excluded entirely by replacing top quark decaying into W boson and b quark by a single contact interaction in the double resonant part of the squared interference diagrams in Fig. 3. In this case the distribution of invariant mass of W and b turns up to be single-top-like while keeping all interference in place without negative contribution.
Full scheme. Modeling the complete set of diagrams is the most precise approach. For the SM measurements this is not very interesting approach since for the most of the SM measurements one needs to separate top quark single electroweak production mode and pair production mode in strong interactions. In the full modeling scheme, which will be referenced as a "Full scheme", different terms contribute in total cross section. These are (as shown in Fig. 1) resonant paired production diagrams, resonant single top production diagrams, and their interfere terms. To retain all single top-sensitive interactions, interference terms need to be studied as well as resonant single top terms.

Comparison
In the previous section we discuss different schemes to simulate tW with different contributions. In this section we compare the common kinematic distributions achieved in different schemes.  Fig. 7 and Fig. 11. The significant difference between simulation approaches is visible in the shapes of b-quark and Wb system distributions. The angular distributions are show in Figs. 12-15. The cosine of angle between W and t momenta in the laboratory rest frame and in Wb rest frame are shown in Figs. 12 and Figs. 13 correspondingly. The cosine of angle between t and b quarks momenta are shown in Figs. 14 and Figs. 15 correspondingly. Angular distributions demonstrate significant difference of DS and CI schemes from other simulation approaches. This is reasonable, since the DS and DI schemes takes into account the interference terms. The differential cross section of invariant mass of W boson and b-quark is shown in Fig. 16. The distribution of invariant mass of W boson and b-quark for the interference term is presented in Fig. 17.

Conclusion
Presented techniques allow different approaches to simulate tWb production and may be useful for different particular analyses. However, only modeling of the complete set of Feynman diagrams retains correct distributions of kinematic variables. The "Diagram Removal" scheme discards all interference terms which are significant part of single top quark production processes. The "Diagram Subtraction" scheme provides best differentiation in angular distributions and could be useful for separation of single top quark production, but introduces negative weights for the significant part of the events and barely can be used in the analysis. The improved "Contact interaction" scheme does not have events with negative weights and shows little difference from "Diagram subtraction" scheme for the angular distributions, thus could be used for effective simulation and separation of single top quark production process.  GeV. GeV. Kinematic cut based approach, as shown in Section 4 is generally less sensitive to electroweak properties since important part of phase space is discarded and the impact from paired top quark production is not reduced entirely. Nonetheless, such approach takes into account all necessary contributions and gives a more stable result because it uses purely gauge invariant kinematic reduction procedure. Kinematic separation does not have negative weights of the events and can be used in the searches for the deviations from the SM.