Probing anomalous Wtb couplings at the LHC in single top quark production through t-channel

Probing anomalous couplings at the LHC in single top quark production through -channel

Adil Jueid adil.jueid@sjtu.edu.cn INPAC, Shanghai Key Laboratory for Particle Physics and Cosmology, Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, ChinaDépartement de Mathématiques, Faculté des Sciences et Techniques, Université Abdelmalek Essaadi, B. 416, Tangier, Morocco
Abstract

We study the sensitivity of certain observables to the anomalous right tensorial coupling in single top production at the LHC at . The observables consist of asymmetries constructed from the energy and angles of the decay products of the top quark produced in single top production through -channel. The computation is done at Leading Order (LO) and Next-to-Leading Order (NLO) in the strong coupling in the flavor scheme. We have estimated projected limits on the anomalous coupling, both at the parton level without cuts and at the particle level with cuts. We find that the asymmetries are robust with respect to the higher order QCD corrections and are indeed a very good probe of this anomalous coupling of the top. Hence they can be be used as experimental probes of the same.

1 Introduction

Top quark is the heaviest among all the SM particles. This particle was discovered at the Tevatron-Fermilab by CDF Abe:1995hr () and D0 Abachi:1995iq () collaborations. It has a pole mass Patrignani:2016xqp () which is very close to the electroweak symmetry breaking scale. Due to its large mass, it can only be created at high energy experiments such as the Tevatron or the LHC with a reasonable number. The top quark plays an important role in high energy physics as it is believed that, due its large mass, effects of new physics beyond the SM can be easily shown Beneke:2000hk (); Han:2008xb (); Bernreuther:2008ju (). Top quark is dominantly produced at the LHC, through QCD, in the pair mode with a cross section approaching one nanobarn at . Due to the vector nature of the strong interaction, the produced top quark pairs are unpolarised. In addition to the pair production mode, top quark can be produced in association with a lighter particle. This production mechanism proceeds through electroweak interaction. Hence it has smaller cross section, the maximum being pb. However, the - nature of the charged current interaction implies that the top quark produced in association are polarised. The much smaller cross section of the single top production along with the very large background from the top pair production meant that the first observation of single top production at Tevatron was made 14 years after the discovery of the top quark Abazov:2009ii (); Aaltonen:2009jj ().
There are three separate modes for single top production. They differ according to the associated particle produced with the top and the initial particles producing the top. These processes can be categorised according to the the virtuality of the boson. All these processes involve the coupling; single top production through -channel (which has the largest cross section at the LHC), through -channel and in association with a -boson. The corresponding Feynman diagrams are depicted in Fig. 1.

Figure 1: LO parton level Feynman diagrams of single top production at hadron colliders.

Single top production, although has smaller rate than production, is phenomenologically very interesting. First, it allows a direct measurement of in the Cabbibo-Kobayashi-Maskawa mixing matrix Alwall:2006bx (); Cao:2015qta (). Inference on the -quark density within the proton is possible as well by measuring single top production cross section both through -channel and process. In single top quark production, the produced top is highly polarised allowing for a direct test of the structure of weak interaction Mahlon:1999gz (). Finally, single top production is one of the interesting channels to look for new physics beyond the Standard Model Tait:2000sh (); Cao:2007ea (); Cao:2006wk (); Huang:2017yso (). Extensive studies of single top production at hadron colliders including radiative corrections have been performed at NLO Campbell:2009ss (); Bordes:1994ki (); Stelzer:1997ns (); Sullivan:2004ie (); Campbell:2004ch (); Campbell:2005bb (); Cao:2004ap (); Cao:2005pq (); Bothmann:2017jfv () and NNLO Brucherseifer:2014ama (); Berger:2016oht (); Berger:2017zof () in the strong coupling. Furthermore, NLO calculations of single top production matched with parton showers become available within MC@NLO Frixione:2005vw (); Frixione:2008yi () and POWHEG Alioli:2009je (); Re:2010bp (). Recently, a transverse momentum resummation at NLO+NLL for single top production through -channel has been proposed in Cao:2018ntd (). Single top production cross sections have been measured by the ATLAS and CMS collaborations at Aad:2014fwa (); Aad:2015upn (); Aad:2015eto (); Khachatryan:2014iya (); Chatrchyan:2012ep (); Chatrchyan:2011vp (); Chatrchyan:2014tua (); Chatrchyan:2012zca (); Khachatryan:2016ewo () and Aaboud:2016lpj (); Aaboud:2016ymp (); Sirunyan:2016cdg (). These results were found in agreement with the SM predictions.

The top quark has a very short lifetime, , which implies that it decays before hadronization effects take place. Hence, all its properties can be studied by looking at the kinematical distributions of its decay products. The top quark decays with almost a branching fraction into . Furthermore, due to the weak interaction universality, this process has certain pattern, the so-called structure which manifests itself at the lowest order in perturbation theory. Departures from this universal structure are possible through radiative corrections in the SM and/or new physics effects. This departure might be seen as the presence of the so-called anomalous couplings. One possible parametrization of these couplings is the use of effective field theoretical approach where gauge invariant and -dimensional operators are added to the SM Lagrangian Buchmuller:1985jz (); Grzadkowski:2003tf (); Grzadkowski:2010es (); AguilarSaavedra:2008zc (). A global fit of these operators to the existing data has been recently done in Durieux:2014xla (); Rosello:2015sck (); Buckley:2015lku (); Buckley:2015nca ().

The transition amplitude for top quark decay into a -boson and a -quark can be written as

(1)

with

where is the momentum of the boson (which is assumed to be on-shell) and are the projection operators, are called anomalous couplings. In the SM, at tree level, and whereas EW and QCD radiative corrections induce non-zero values of the anomalous couplings. Computations of the anomalous couplings have been performed in Czarnecki:1990kv (); Li:1990qf (); Bernreuther:2008us (); Drobnak:2010ej (); GonzalezSprinberg:2011kx (); Duarte:2013zfa (); Gonzalez-Sprinberg:2015dea (); Arhrib:2016vts (); Ayala:2016djv (); Martinez:2016fyd () both in the SM and certain extensions of it. It was found that SM corrections to anomalous couplings are extremely small, i.e , and Arhrib:2016vts (). Furthermore, the dominant contribution comes from QCD whereas EW corrections are subleading accounting about of the total contribution. Contributions of the anomalous couplings to various flavour observables have been considered in Drobnak:2011wj (); Drobnak:2011aa ().

As mentioned above, the top quark produced in association with a quark or , is highly polarised. This polarisation is decided by the anomalous couplings involved in the production of the single top. The production cross-section, the polarisation of the top and the energy distributions of the decay products of the single top, all carry information about these anomalous couplings. In fact, recently a study Godbole:2006tq () had shown how one can simultaneously study the top polarisation as well as the anomalous coupling and constructed some asymmetries in the observables which are sensitive to both. However, it did not make any reference to a specific top production mechanism. It was pointed out in Cao:2015doa () that measuring accurately single top production cross sections throught - and channels would constrain effectively the anomalous couplings. In this paper we wish extend the analysis of the observables suggested in Godbole:2006tq () as well as the cross-section information, to the single top production via -channel.

Thus the aim of this paper is to make a study of the sensitivity of the LHC to the anomalous coupling using energy and angular observables in -channel single top production at TeV. The observables used in our analysis consist of asymmetries based on energy and angular distributions of the top quark decay products. As we will see in this paper, their use will give additional information about the existence of possible anomalous couplings in the vertex. Since these observables depend on polarisation, it is expected that they will be robust against QCD radiative corrections. In fact this was demonstrated explicitly in an analysis of the charged Higgs production Godbole:2011vw (). In this work also we perform a comparative study at LO and NLO in the strong coupling constant.

We use observables suggested in Prasath:2014mfa (), but many of them were proposed long time ago in Godbole:2006tq (); Shelton:2008nq () and were used for several studies (see Rindani:2011pk (); Godbole:2011vw (); Belanger:2012tm (); Godbole:2015bda (); Rindani:2015vya ()). In addition, we separate the study into two different categories; parton and particle level. In the former case, no cuts are imposed on the particles’ momenta while in the latter loose cuts are imposed. The effects of such kinematical cuts were found to be quite important because these affect the shape of the distributions and hence also the projected limits from the results at the parton level. We stress out that these asymmetries are of extreme importance for future experimental analyses and can be used in several channels that involve the top quark and in different collider machines. We will show, in this paper, that using NLO matrix elements is mandatory for future experimental analyses. In this study, following the limits from , we set and will be taken to be real.

The paper is outlined as follows: In section 2, we review the limits on the anomalous couplings from direct experimental searches and from statistical fits to several measurements. The setup of the calculation and details about event selection are summarized in section 3. In section 4, we show the computations of single top production through -channel at LO and NLO both in the SM and the SM augmented by anomalous coupling. The studied observables are briefly discussed in section 5. Numerical results and sensitivity projections are shown in section 6. In section 7, we draw our conclusions. Details about the interpolation procedure are shown in appendix A.

2 Limits on anomalous couplings

Anomalous couplings are constrained both indirectly from flavour changing decays such as as well as from measurements at both the Tevatron and the LHC: those of helicity fractions of the produced in top decay, both in pair production of the top as well as the single top production, as well as the spin-spin correlations in top pair production etc. A summary of early constraints can be found in Bernreuther:2008ju (). Measurement of branching ratio constrains strongly, however is rather weekly constrained Grzadkowski:2008mf (). Measurements of the -boson helicity fraction in top pair production at the Tevatron Abazov:2012uga () as well at the LHC Aad:2012ky (); Chatrchyan:2013jna (); Aaboud:2016hsq (). In fact an analysis of a combination of the measurements of the BR() Grzadkowski:2008mf () and the helicity fractions at the Tevatron Abazov:2012uga () together, had shown that is the only coupling that could have nontrivial values. The large increase of single top production cross-sections with increasing energy, from Tevatron to the LHC, meant that one could also use the single top processes to this end as well Chatrchyan:2013jna (); Khachatryan:2014vma (); Aad:2015yem (); Khachatryan:2016sib (); Aaboud:2017aqp (). LHC experiments have used helicity fraction of the produced in the decay Aad:2012ky (), the double differential decay rate of the singly produced top quark Aad:2015yem (), asymmetries constructed out of -boson angular distributions Aaboud:2017aqp () as well as the triple differential decay distributions for the quark Aaboud:2017yqf (). In these analyses limits on various anomalous couplings are obtained under different assumptions; sometimes letting all the couplings vary around their SM values or sometimes keeping some of the couplings fixed at their SM values and so on. Furthermore, depending on the variables used, limits can be obtained on the real or imaginary parts of these anomalous couplings. Analyses which tried to constrain all the anomalous couplings simultaneously, using only the data on single top at the LHC Khachatryan:2016sib (); Aaboud:2017yqf () with no assumptions on value of still allow values of . Fits assuming from production Aaboud:2016hsq () by ATLAS also allow large values of but these are in conflict with the cross-section measurements of single top processes and hence cannot be taken seriously. On the other hand, a phenomenological analysis of only the collider data, viz. the single top cross-sections and helicity fractions from both the Tevatron and the LHC Fabbrichesi:2014wva (), obtains mildest constraints on and .

In addition, a combination of different measurements corresponding to electron and neutron electric dipole moments (EDM), top quark observables and oblique parameters, constrains both the imaginary and real parts of the tensorial couplings and Cirigliano:2016njn (); Cirigliano:2016nyn (). In the end we note that while the imaginary parts of anomalous tensorial couplings are most severely constrained by EDM’s of neutron and electron, BR() constrains and -helicity fraction from the collider data constrain the most effectively. Recentl global fits to the LHC and Tevatron data Birman:2016jhg (); Castro:2016jjv (); Hioki:2016xtc (); Deliot:2017byp (); Xiong:2018mkc () and the limits on are not very strong.

A summary of current limits and analyses can be found in Patrignani:2016xqp (). For reference, we depict some limits from experimental searches and from BR() in Table 1. All this discussion thus tells us that it is of great interest to see how the collider data on cross-section and top polarisation in single top production can constrain . We proceed now to discuss the procedure to do so using new observables.

Constraint Limits Reference
BR() Grzadkowski:2008mf ()
helicity fractions Aad:2012ky ()
Chatrchyan:2013jna ()
Khachatryan:2014vma ()
double differential cross section Aad:2015yem ()
-channel cross section Khachatryan:2016sib ()
-boson polarisation Aaboud:2016hsq ()
Aaboud:2017aqp ()
triple differential cross section Aaboud:2017yqf ()
Table 1: Summary of the limits on anomalous couplings from BR() and from direct experimental searches. A complete listing can be found in the PDG Patrignani:2016xqp ().

3 Setup and event selection

For this study, we consider single top production through -channel in collisions at . For electroweak couplings, we use the -scheme, in which the input parameters are . We choose , and . From these input parameters, and are obtained. Furthermore, the top quark pole mass is chosen to be . The computation of single top production cross section was done in the FS with massive (massless) -quark at LO (NLO). We use the NNPDF30 PDF sets Ball:2014uwa () with the LHAPDF6 interpolator tool Buckley:2014ana () with . Throughout this study, we will use fixed factorization and renormalization scales, i.e .

Events are generated with Madgraph5aMC@NLO Alwall:2011uj (); Alwall:2014hca () in the SM at Leading Order (LO) and Next-to-Leading Order (NLO) and the SM with anomalous couplings at LO. The right tensorial anomalous coupling is implemented by hand in a UFO model file Degrande:2011ua (). The model file were validated by comparing some calculations to several results existing in the literature Rindani:2011pk (); Cao:2015doa () concerning cross section calculations and several distributions in - and channels and we found excellent agreement. The produced events were decayed with MadSpin Artoisenet:2012st () which uses the method developed in Frixione:2007zp () to keep full spin correlations. The decayed events are passed to PYTHIA8 Sjostrand:2014zea () to include parton showers (ISR and FSR) and hadronization. Parton showering algorithm is based on dipole type evolution Sjostrand:2004ef (). The other parameters are set according to the Monash tune Skands:2014pea (). We have adopted the MC@NLO scheme Frixione:2002ik () for consistent matching of hard-scattering matrix elements and parton shower MC. RIVET Buckley:2010ar () was used for analysis of the events. For jet clustering, we use FastJet Cacciari:2011ma () with an anti- algorithm and jet radius Cacciari:2008gp ().

We, first, perform a partonic level analysis (no showers, no soft QCD effects and no cuts on the kinematical quantities). We then, perform a particle level analysis (without detector effects) of the showered events. Throughout this paper, we will show results at both the partonic and the particle levels. For the particle level analysis, we require a topology consisting of exactly one isolated charged lepton (electron or muon), missing energy and at least two jets with at least one of them is -tagged. First, we require exactly one isolated charged lepton with transverse momentum and pseudorapidity . We require at least two jets where one of them is tagged with and . Further isolation requirements are applied to jets, i.e the angular separation should be always for any two jets in the event and .

4 Single top production through -channel

In this section, we discuss single top production cross section in the SM at LO and NLO. We illustrate at the end of the section the method that we used to include anomalous coupling in the production at NLO.

4.1 LO calculation in the SM

At LO, there are two generic contributions to the -channel process. The first contribution corresponds to the subprocess:

(2)

and the second contribution represents the subprocess:

(3)

where and . Furthermore, contributions to the -channel process involving the negligble elements of the Cabbibo-Kobayashi-Maskawa (CKM) mixing matrix such as were not taken into account. We have computed the inclusive LO cross sections at TeV. Due to the dominance of the valence u-quark PDF over the sea anti-quarks, the subprocess 2 gives the dominant contribution which accounts of about of the total cross section.

4.2 The -channel at NLO in the SM

Figure 2: Feynman diagrams contributing to subprocess at NLO in the SM where and .
Figure 3: Feynman diagrams contributing to subprocess at NLO in the SM where and .

Further contributions to the -channel process, at NLO exist. Such contributions include virtual one-loop corrections to the tree level process as well as tree level real emission processes where the additional emitted parton is soft or collinear. Parton level Feynman diagrams are depicted in figures 2 and 3. All the possible flavours that might contribute to this process were included. Due to color conservation, box diagrams do not contribute to the cross section at NLO.

Order       [pb]       [%]       [%]
LO                  
                 
NLO                  
                 
Table 2: -channel production cross section at LO and NLO at TeV. Uncertainties due to scale variations and PDF are shown. First rows show the values of the cross section without cuts and in the second rows the cross sections are computed using the cuts highlighted in section 3.

We have computed the total cross section at with . The dominance of the valence -quark PDF implies that the cross section from the contribution of Feynman diagrams in Fig. 2 is dominant. We have also estimated theoretical uncertainties on the inclusive cross section both at LO and NLO from scale variations and from PDF. Theoretical uncertainties which are due to scale variations are estimated by varying simultaneously the hard factorization and renormalization scales around their nominal values, i.e :

(4)

where () is the central renormalization (factorization) scale. Thus, obtaining an envelope of the nine possible variations. PDF uncertainties are obtained using the replicas method where each PDF set has one central and 50 members corresponding to the minimal fit and the eigenvectors respectively. The results for the inclusive cross section at LO and NLO along with their theoretical uncertainties are depicted in Table 2. After cuts were imposed, the total rate decreases by a factor of and at LO and NLO respectively. Furthermore, theoretical uncertainties due to scale variations increase in both LO as well as NLO with the former has even larger relative increase than the latter. The most important consequence of imposing cuts on the decay products is that the -factor of the fiducial cross section increase to about while the theoretical uncertainties at LO do not change much. An immediate consequence of such observation is that the fiducial cross section at NLO is outside the allowed range of scale variations’ uncertainty at LO since pb . Hence, for any future analysis or fit involving the cross section of single top through -channel, at least the calculation at NLO should be used.

4.3 Including anomalous couplings

The presence of anomalous couplings modifies single top quark production cross sections. The production cross section of a top quark through -channel in collisions can be expressed as function of anomalous right tensorial coupling as

(5)

where . They are determined from a fit and are given by

at TeV. In eqn. 5, is the SM cross section at LO (see e.g. Table 2). We can see that imposing cuts strengthen the dependence of the cross section on the anomalous coupling by a factor of . Although the quadratic term is about 5 times larger than the linear one its contribution to the cross section is mild even for the extreme values of , i.e . The results we obtained were compared with those presented in Cao:2015doa () and we found excellent agreement.

Taking into account the anomalous coupling in the production at NLO is not straightforward. The reason is that one cannot renormalize high dimensional operators using the traditional on-shell renormalization schemes. Using alternative schemes for the renormalization of the SM wave functions and parameters will result in large theoretical uncertainties. However, including anomalous couplings in the production is very interesting since they completely change the chiral structure of the vertex and hence the top quark polarisation which in fact improve the sensitivity of most the observables on the anomalous couplings. We include their effects as a shift on the production while neglecting interference between the virtual corrections and tree level amplitudes with anomalous couplings. Hence, three samples will be generated with the first one corresponds to SM amplitude at NLO, the second to the LO amplitude in the SM and the third one to the amplitude with anomalous couplings at tree level. The transition amplitude for the process

can be written as

(6)

where () is the production (decay) matrix elements for the top quark.

For convenience () and () stand for the helicity labelling of all particles. The pure SM tree level amplitude (equivalent to ) is . The contribution of the anomalous coupling will only be taken into account at tree- level and will be denoted . The full tree-level amplitude, anomalous amplitude, will be denoted , such that

(7)

Since the radiative corrections at the level of the decay are very small in the SM (see e.g. Gao:2012ja ()), the decay part will be considered at tree-level only. For the production part, at tree-level, with the inclusion of the anomalous part we have

(8)

The one-loop radiative corrections will only apply to the SM part. The first higher order contribution consists of the one-loop virtual correction that includes also counterterms, . To this one needs to add the pure SM radiative, emission. First of all, for the one-loop virtual correction and the contribution we may write

(9)

where

(10)

Including and integrating over real emission, , and adding it to the virtual correction of Eq. 10 will give the full NLO SM result. What we will refer to as the full NLO (including the LO anomalous part) is

(11)

In order to reproduce the data including both the NLO SM and the anomalous contribution (with its quadratic part) we therefore, for the same phase space point, generate three samples. One for the full anomalous part at tree-level, one for the SM tree-level part (which has to be subtracted to avoid double counting) and one for the SM NLO (which includes the SM tree, virtual and real emission).

5 Observables

In this section, we review the observables that we will be using for our analysis of anomalous couplings. They consist of asymmetries constructed from the energy and angular distributions of the top quark decay products in single top production through -channel.

We define an asymmetry with respect to a kinematical variable by:

(12)

where , , is the differential cross section of the top quark with respect to the variable and is a reference point around which the asymmetry will be evaluated. will be chosen such that the evaluated asymmetry is sensitive to the anomalous coupling and allows for a comparison of cases of different values of the parameter. In what follows, kinematical quantities written with a superscript "0" are given in the top quark’s rest frame. Otherwise, they are given in the laboratory frame.

5.1 Lepton polar asymmetry

Figure 4: distribution in the top quark rest frame using full partonic information (left) and with the cuts implemented at the particle level (right) at NLO.

The polar angle of the charged lepton, denoted by , is defined by

(13)

with is the three-momentum of the charged lepton in the top quark rest frame and is the top quark three-momentum in the laboratory frame. This observable is a good probe of the top quark polarisation111The top quark direction of flight in the center-of-mass frame defines the spin quantization axis in the helicity basis.. However, given that the presence of anomalous coupling changes the chiral structure of the top quark, we expect that it is a good probe of the anomalous coupling too. This can clearly be seen in Fig. 4 where the distribution is plotted in both the SM and for . We can see that the effect of the anomalous coupling on the is important and which can even change the slope of the distribution for . Although the presence of cuts (right panel of Fig. 4) modifies the sensitivity of this observable, it can be used for searches of the anomalous couplings. However, the measurement of the polar distribution is quite challenging since it requires a full reconstruction of the top quark momentum. This is hard to be achieved due to the presence of missing energy in the semi-leptonic decay of the top quark. Nevertheless, several measurements of the charged lepton angle in the helicity basis exist, e.g. in the system Aaboud:2016bit (). Hence, it can possibly be measured in the -channel process in the future. From Fig. 4, we define the reference point for the to be .

5.2 Charged lepton Energy

In addition to the angular polar distribution in the top quark’s rest frame, two other observables constructed from the charged lepton energy are considered here. We define a dimensionless variable, , by:

(14)

where is the lepton’s energy in a given frame and is the top quark mass. We consider the energy of the charged lepton in two different frames; the top quark’s rest frame and the center-of-mass frame.

It was shown that, in the top quark’s rest frame, this asymmetry is a pure probe of the anomalous coupling regardless the top quark production mechanism (or in other words top quark polarisation) Prasath:2014mfa (). Hence, at the experimental level, full advantage of this observable should be taken by measuring it in several channels. However, for its measurement, a reconstruction of the top quark momentum is needed. We depict the in the SM and for in Fig. 5. The reference point for the corresponding asymmetry, is chosen to be , i.e the value at the peak position in the SM Godbole:2015bda ().

Figure 5: distribution in the top quark rest frame using full partonic information (left) and with the cuts implemented at the particle level (right) at NLO.

The situation is different for in the laboratory frame which is shown in Fig. 6. This observable is sensitive to both the anomalous coupling as well as the top polarisation and has a high sensitivity to the anomalous coupling for small values of for positive values of and in the full range of for negative values of . However, this observable has a lower sensitivity on the anomalous coupling than due to some cancellations which occur between anomalous couplings and other kinematical factors Prasath:2014mfa (). Finally, no reconstruction of the top quark momentum is needed in order to measure this observable. A reference point of is chosen for the evaluation of the corresponding asymmetry Godbole:2015bda ().

Figure 6: distribution in the laboratory frame using full partonic information (left) and with the cuts implemented at the particle level (right) at NLO.

5.3 - and -variables

Figure 7: distribution in the laboratory frame using full partonic information (left) and with the cuts implemented at the particle level (right) at NLO.

The last two variables from which two different asymmetries will be evaluated have been proposed in Shelton:2008nq () as a probe of top quark polarisation for highly boosted top quarks. First, a variable which measures the energy ratio of the charged lepton to the total visible energy (of the top quark decay). This variable is defined by :

(15)

where and are the lepton and b-quark (-jet) energies in the laboratory frame. It was found that the -variable is sensitive to both the top quark polarisation and the anomalous coupling Prasath:2014mfa (). We found that including the anomalous in the production improves the sensitivity of the -variable on . From experimental point of view, it is possible to measure this variable from a simultaneous measurements of the both the charged lepton and -jet energies in the laboratory frame. This implies that there is no need for reconstructing the top quark momentum. We depict the variable for three different models at NLO in Fig. 7. From this figure, we choose the reference point which is the intersection point of the three different curves corresponding to the SM and to .

Finally, the variable, which measures the fraction of the top quark energy taken by the -jet in the laboratory frame, is defined by:

(16)

where and are the energies of the bottom and top quarks respectively in the laboratory frame. In Fig. 8, we depict the -variable for the SM and . We can see that -variable has a lower sensitivity on the anomalous coupling. Furthermore, its measurement requires a determination of the top quark energy, which by itself depends on the decay mode. The hadronic mode, although has a larger background is better for the measurement of the -variable. The reference point will be chosen to be .

Figure 8: distribution in the laboratory frame using full partonic information (left) and with the cuts implemented at the particle level (right) at NLO.
              LO               NLO           NLO/LO
Table 3: Asymmetries and their theoretical uncertainties in the SM at LO and NLO. The first rows for each asymmetry show the partonic level results while the second rows the particle level ones.

6 Results

From the different kinematical distributions shown in section 5, asymmetries are constructed (for more details, see section A). These asymmetries present, except , a high sensitivity on the anomalous right tensorial coupling as it is depicted in Fig. 9 which is weakened at the particle level. We can see that there are some differences between the central values of the asymmetries at LO and NLO of about depending on the particular asymmetry. However, one notices that these differences are within the theoretical uncertainties which are depicted in Table 3 where only the effect of scale variations is shown. The effect of radiative corrections on the asymmetries depends on the particular variable. At the parton level, two asymmetries are perfectly stable against radiative corrections; i.e and while the others can receive corrections of for , for and for . On the other hand, the theoretical uncertainties due to scale variations are quite large except for both at LO and NLO and for at NLO. At the particle level, the corrections to the asymmetries are lower than to the cross sections with again a strong stability against NLO corrections for and . The theoretical uncertainties due to scale variations are lower than in the parton level case with one notable exception, i.e which has very large theoretical uncertainties (see Table 3).

Figure 9: Dependence of different asymmetries as function of the anomalous coupling at the parton level without cuts (left panel) and at the particle level with cuts (right panel). The solid (dashed) lines show the dependence at LO (NLO).

We perform a exclusion to obtain the limits on the anomalous coupling . A is defined as follows

(17)

where . is the sum, by quadrature, of the statistical and theoretical uncertainties on the asymmetry in the SM. The former are defined as

(18)

where is the number of events

we assume a luminosity of fb. Values of are excluded within if the corresponding is larger than and respectively. We show in Table 4, individual limits obtained from the different asymmetries at both at LO and NLO. We can see that all the asymmetries but give strong constraints on the anomalous coupling. Furthermore, the limits placed on the anomalous coupling from and are strenghten, at the particle level, by about a factor of and respectively. This is due to the large theoretical uncertainties on those observables at the partonic level which are significantly reduced when showers and cuts are implemented. Overall, constraints on the anomalous right tensorial coupling are stronger at NLO due to the reduction of theoretical and statistical uncertainties. Before discussing the combination of different asymmetries and its effect on the anomalous coupling, we comment about the experimental uncertainties. Generally, measurement of the asymmetries and introduces additional systematic uncertainties since they involve the reconstruction of the top quark rest frame which is very hard for single top production. Hence, although the asymmertry is resilient to NLO corrections, it might not be very efficient in the determination of the anomalous coupling. On the other hand, among all the laboratory frame asymmetries, is the most sensitive and involving less systematics (both theoretical and experimental) uncertainties222However, it is expected that experimental systematic errors drop when asymmetries are used.. Combining two asymmetries at one time improves significantly the limits on . We found that combining and gives () at LO (NLO) at the parton level at . While at the particle level, those limits are weakened to give () at LO (NLO).

These limits can be improved by combining more than one asymmetry at a time. As an example, we estimate projected limits using ten different combinations of the three different asymmetrie. We can see in Tables 5-6 that the limits are improved by about one order of magnitude from those obtained using one asymmetry at a time. In Table 5, we show these limits at the partonic level. We can see that, at LO that six different combinations of the three asymmetries give at . The situation is different for the NLO case, where the limits are improved by a factor of from the combinations and yielding in the interval . On the other hand, including showers and cuts do not change much the limits that we obtain but only the combination of the asymmetries change. In Table 6, we can see that six asymmetries give the strongest limit at LO; i.e . While at NLO, we obtain using the combination. However, as this combination involves the asymmetry which requires full reconstruction of the top quark direction of motion. Hence, that either the or will do a better job in pinning down the anomalous coupling even the obtained limits are milder than .

Asymmetry       LO       NLO

-

Table 4: Individual expected limits at on the anomalous coupling at LO and NLO using full partonic information (first rows) and at the particle level (second rows).
Combination                  
                 
                 
Table 5: Limits on the anomalous coupling at , and at the partonic level without cuts. The first row for each combination represents the interval at LO while the second row represents the NLO exclusion.
Combination                  
                 
                 
Table 6: Same as Table 5 but at the particle level.

Now, let us compare our findings with the other results in the literature which used different observables in different channels; -boson angular observables, single top production cross sections at hadron colliders…etc. In ref. Boos:1999dd (), limits were obtained from single top production at both the Tevatron and the LHC, they obtained for the LHC and for the Tevatron. In their simulation, they considered and processes at tree level and included anomalous couplings using both background contribution as well signal contribution from top quark decays (i.e -channel process). In Hubaut:2005er (), a detailed study of ATLAS sensitivity to the top quark and -boson polarisation in production using both semi-leptonic and dileptonic channels was carried out. This analysis was translated to limits on the anomalous couplings, top quark decay into charged Higgs boson and to constraints on resonances. Using boson polarisation, and assuming the presence of all the anomalous couplings with , they got limits on , i.e at . Limits on were obtained by the authors of AguilarSaavedra:2006fy () using helicity fractions, some angular and energetic asymmetries and spin-spin correlations observables in production at the LHC. They obtained at the level by using the asymmetry defined as

where are -boson helicity fractions and . In their work, they assumed the presence of all anomalous couplings. In AguilarSaavedra:2007rs (), sensitivity of the ATLAS experiment on the anomalous couplings was studied. The study concerned production with semi-leptonic final state. In AguilarSaavedra:2007rs (), -boson helicity fractions, ratios of helicity fractions and new angular asymmetries were studied. A systematic study of the different background contributions was carried including detector effects and particle reconstruction effeciencies. All the anomalous couplings were included and individual limits on were obtained by setting and . The stringent bound on was obtained from the asymmetry, i.e . By setting two couplings to be non-zero at a time and combining four measurements, they got the strongest constraint on , i.e which obtained in combination with and considerably improves the limit from alone. In Najafabadi:2008pb (), limits on were obtained using cross section of single top production through channel at the partonic level assuming and . They obtained assuming a systematic uncertainties. In reference AguilarSaavedra:2008gt (), limits on anomalous couplings were obtained using single top production at the LHC. On the one hand, they obtained limits by combining single top production cross section through -, - and channels with the ratio defined by

They found assuming and . Then, they included top quark decay observables such as -boson helicity fractions and their ratios. The obtained limit is which is about one order of magnitude better than those obtained from cross section measurements alone. In AguilarSaavedra:2010nx (), limits on anomalous couplings were obtained using new proposed observables which consist of angular distributions which probe the boson polarisation and assuming that only one coupling is non-zero at a time or they are either purely real or purely imaginary. They got the following limits at