T-odd anomalous interactions of the top-quark at the Large Hadron Collider

We study the effects of T-odd interactions of top-quark via the pair production of top-quark in the semileptonic detection modes at the Large Hadron Collider by means of the T-odd observables constructed through the momenta of the observed decay products of the top (and anti-top)-quark for a wide range of CP-violating scale, $\Lambda$. Estimates on the sensitivities of the coupling strength of such interactions for 13 and 14 TeV LHC energies are also presented for $\Lambda$ ranging between $M_W$ and 2 TeV.


T-odd observables and top-pair production
CP-violation in the quark sector face an observational difficulty which partially lies in the fact that due to relatively longer life-time than the hadronisation scale, which is of about 140 GeV ( m π 0 , mass of the pion), quarks form bound states and thereby leave no scope for studing pure CP-violation. By being much heavier than the other quarks, also much energier than the hadronisation scale top-quark turns out to be the only expectation. The life-time of a top quark is less than the time required for a quark to hadronise therefore it does not form any bound state. At hadron colliders, processes involving top-quarks have further advantage in having larger cross-sections due to the strong interactions. This therefore enables us to directly investigate the effects of such interactions via the pairproductionn of the top-quarks and their subsequent decays into a pair of leptons and b-quarks.
In the presence of T-odd interactions of top-quark with gluon, the SM Lagrangian coulld be modify tt by the following interaction term with g s being the strong coupling constant, G µν the gluon field-strength tensor, d g and Λ being the inteaction strength and energy scale of the CP-violation respectively. This interaction term besides contributing to the SM ttg vertex also provides a new dimension five vertex ttgg which are obviously CP-odd in nature according to the above equation.
These would clearly have a significant contribution to the top-pair production processess at hadron colliders, particularly for collider alike LHC where the fusion of gluons emerging from the colliding protons make about 90% contribution; the rest being the annihilation of light-partons of opposites charges. A schemetic representation of various parton-level processess describing the production of tt at the LHC where the modification occurs due to the presence of addtional T-odd inteactions given by Eq. 2.1 are shown in Figs. 1.
It is also worthwhile to mention that as the semileptonic decays of the top (anti-top) takes place due to weak-inteactions, the branching ratio of the top-quark will remain intact as of the SM.
Our study of finding CP-violation is based on assymmetry calculation and we vary both d g and Λ and see how this will effect the CP-violation sensitivity of the observables i.e. we have two free parameters, we allow d g and Λ to be free parameter. Also we allow W ± to decay in dilepton channel i.e. both electrons and muons. Our main focus in this paper is the study of CP-violation sensitivity at √ S = 13 TeV and 14 TeV energy at LHC. CP-odd observables can be formed using T-odd correlations. T-odd correlations are not necessarily CP-odd they can be CP-even as well because T-odd is not for time-reversal here, it represents naive-T-odd [24].
At first we start our calculation with the observables defined in Eq. 2.2 Here p bb represents the four-momenta of b andb quark, p l + l − represents the four-momenta of lepton and anti-lepton and p tt represents the four-momenta of top quark and anti-top quark. P is the sum of top quark and anti-top quark four-momenta (p 1 + p 2 ),q is the difference of top quark and anti-top quark four-momenta (p 1 − p 2 ), p t is the sum of b quark four-momenta and lepton four-momenta (p b + p l + ) andt is the sum of anti-baryon and anti-lepton four-momenta (pb + p l − ). Let us consider observable C 1 to check its CP properties In the above equation, left side of arrow describes the correlation in any frame and the right side in a particualar centre-of-mass reference frame. In first line of Eq. 2.3 we go through bb center-of-mass frame which results in the triple product form. The obtained triple product form after this operation has gone through charge congugation and parity operation in the second and third lines to confirm its CP properties which results that it is CP-odd. Similarly, if we consider (l + l − ) centre of mass frame proceeding in the same way as in Eq. 2.3, here too we find the same result i.e. CP-odd. We include three new observables The advantage of considering these additional observables lies in the fact that these require much lesser information than the observables defined in Eq. 2.2. For example, observable C 6 requires information regarding the beam direction, the direction of the centre-of-mass enegy and a lepton having positive charge and the associated b quark and identifying a lepton having negative charge and the associated b quark. Observable C 7 requires information of the beam direction, direction of the centre-of-mass energy and the leptons having positive and negative charge. Similarly observable C 8 requires information of the beam direction, centre-of-mass energy, b quark and anti-b quark. The reason behind considering so many observables is to over confirmed that the CPviolating asymmetry is indeed CP-odd or it is not faked by the CP-even observables. In next section we will discuss numerical simulation in detail.

Numerical Analysis
In order to perform our study we first produced tt pair through the process pp → tt and allowed them to decay semileptonically into (bl + ν l )(bl −ν l ) subsequently with the aid of MadGraph5 [25][26][27] using the decay chain feature described in Ref. [27]. The CP-violating interactions discussed in Eqs. 2.2 and 2.5 have been incorporated in the MadGraph via FeynRules [28]. The experimental values of the input parameters considered for our study are presented in Table 1, the renormalisation and factorisation scale has been set to 91.188 GeV and the parton distribution functions had been considered to be nn23lo1 [29,30]. The events are generated with the MadGraph using following selection criteria. After applying all the changes in the MadGraph as described in Eq. 3.1, we generate 10 7 events at 13 TeV and 14 TeV LHC energy with different choices of coupling constant (d g ) and scale parameter (Λ) for all the observables given in Eqs. 2.2 and 2.5. We vary d g between interval 0 to 5×10 −2 and Λ between M W to 2 TeV. d g = 0 corresponds to the SM and other values correspond to CP-violation in the production. The CP-violating asymmetries are calculted using the formula where N is the number of events. However, the experimental sensitivity is calculated using the following formula As asymmetries are induced by CP-violating couplings and vanish for CP-conserving ones, a non zero value of A CP confirms violation of CP. The sensitivity for a particular confidence level is calculated using the formula where n cl is the number of confidence level. It is clear from the obove mentioned formula that in order to achieve 3σ statistical sensitivity n cl should be equal to 3 and asymmetry should be 0.1 % for N = 10 7 events. We find that asymmetries corresponding to observables C 1 , C 3 , C 4 and C 5 are larger than 3σ error so we presents the asymmetries corresponding to these observables for three different values of d g at the CP-violating scale M W , 0.5 TeV, 1 TeV and 2 TeV for √ S = 13 TeV and 14 TeV at LHC given in the Tables 2 and 3 respectively.
Looking at the Table 2 we see that for d g = 0 i.e. for SM, asymmetries corresponding to all the observables are almost zero or near to zero that means no CP-vioaltion for this value, this result is in favour of various previous studies. As we start increasing the value of d g we see a considerable increase in the value of asymmetries which means that CP is violated. We see that for Λ = M W and d g = 5×10 −3 , asymmetry A 1 = 1.2%, it then increases to 2.2% for d g = 1×10 −2 and 6.3% for d g = 5×10 −2 at same Λ. When we increase Λ from M W to 0.5 TeV we see that asymmetry drops to 0.2% for d g = 5×10 −3 and then increase to 0.4% and 1.8% for d g = 1×10 −2 and 5×10 −2 respectively keeping Λ same. Again increasing the Λ from 0.5 TeV to 1 TeV we see a drop in asymmetry to 0.1 for d g = 5×10 −3 and then increases to 0.1% and 0.9% by increasing d g , Similarly other values can be checked. We observed that if we increase energy scale Λ for the same value of d g then asymmetry decreases, for example A 1 = 1.2% for d g = 5×10 −3 and Λ = M W , if we increase Λ to 0.5 TeV keeping d g constant, A 1 decreases to 0.2% and then decreases to 0.1% and 0.03% for Λ = 1 TeV and 2 TeV respectively. Similarly for observable C 3 asymmetry A 3 = 0.4% for d g = 5×10 −3 and Λ = M W , it decreases to 0.02%, 0.1% and -0.01% by increasing Λ to 0.5 TeV, 1 TeV and 2 TeV respectively. Similar behaviour is observed for observables C 4 and C 5 as can be seen in the given Table 2. Also we note that that by increasing d g keeping Λ constant, the asymmetry increases approxiamately linearly, e.g., for Λ = M W and d g = 5×10 −3 , A 1 = 1.2% and then increases to 2.2% and 6.3% as we increases d g to 1×10 −2 and 5×10 −2 respectively for the Λ. Similarly for other values of Λ we observe same behaviour. Observables C 3 , C 4 and C 5 also show the similar behaviour. Table 3 which represents the results at √ S = 14 TeV energy at LHC we see that the observations are same as in case of 13 TeV results, as an example let us consider asymmetries corresponding to observable C 1 we see that if d g = 5×10 −3 then A 1 = 1.1% for Λ = M W and then decreases to 0.2%, 0.1% and 0.1% for Λ = 0.5 TeV, 1 TeV and 2 TeV respectively. Similarly for C 3 , C 4 and C 5 we observe the similar behaviour. If for Λ = M W , we increase d g then A 1 increases for example for d g = 5×10 −3 , A 1 = 1.1% and then increases to 2.2% and 6.4% for d g = 1×10 −2 and 5×10 −2 respectively and similar behaviour is observed at other values of Λ. Observables C 3 , C 4 and C 5 show the same behaviour. After gone through these observations we conclude that asymmetry increases on increasing d g keeping Λ constant and decreases on increasing Λ for same d g . Also we observe that asymmetries increase linearly with d g , for example look at Table 2., A 3 = 0.4% for d g = 5×10 −3 and then increasing d g to 2 times i.e. 1×10 −2 , A 3 also increases double to its previous value and for d g = 5×10 −2 which is 5 times larger than previous value, A 3 increases almost 5 times to its previous value and similarly for other observables. Table 3 shows the similar behaviour.

Now looking at
With the results presented in Tables 2 and 3, we find expressions of cross-section and asymmetries A 1 , A 3 , A 4 and A 5 as a function of d g and Λ given in Eqs. 3.5 and 3.6.  Table 2: Integrated asymmetries (in %) for √ S = 13 TeV at LHC for the process pp → tt → (bl + ν l )(bl −ν l ) corresponding to various observables for three different choices of d g at Λ = M W , 0.5 TeV, 1 TeV and 2 TeV with 0.03 % theoretical uncertainties at 1σ confidance level.  Looking at Eqs. 3.5 and 3.6 we notice that first term is a constant with no coupling and second term is dependent on coupling (d g ) and scale parameter (Λ) which shows that first term corresponds to SM and second term is CP-violating term at production. Hence it is the second term which is responsible for the violation of CP-symmetry at the production.
For estimating the experimental uncertaintlies in event rates we first combined the ATLAS [31] and CMS [32] experimental uncertainties observed with 2015 and 2016 data during LHC Run II for the top pair at √ S = 13 TeV for 36.1 fb −1 presented in Ref. [33]. In order to calculte experimental sensitivity we first combined the ATLAS and CMS cross-sections including which are as follows.
event rate then estimated by multiply it by the luminosity, branching ratios for the t → blν l and the b tagging efficiency which is assumed to be 56 %. A similar calculation has been performed for √ S = 14 TeV with a theoretical cross-section 953.6 +22.7+16.2 −33.9−17.8 pb at NNLO + NNLL level [34] for the projected integrated luminosities of the LHC of Ldt = 300 and 1000 fb −1 . We show the reults for 13 and 14 TeV c.m. energies at LHC for d g vs Λ at various confidence levels in Figs. 2 and 3. We present the results for Λ between the range 0 to 5 TeV but we had actually performed the study between the range M W to 2 TeV.
In Figs. 2 and 3 yellow region is ruled out by 2.5σ sensitivity, red region is ruled out by 5σ sensitivity and white region is disallowed by cross-section. Figs. 2 and 3 describes the allowed range of d g and Λ at which we can observe 5σ sensitivity. We have wide range of d g ( dg Λ ) values at which we can observe 5σ sensitivity at 13 and 14 TeV LHC energies. From the figures we can get a rough esimate of minimum bound on d g and Λ and is able to find the lower limit ond g ( dg Λ ) at which 5σ sensitivity could be observed. A precise calculation to calculate the exact limit ond g will be done with the experimental sensitivity for LHC.
1σ experimental sensitivity for the LHC c.m. energy of 13 TeV is 0.2% and the similar value at 5σ would be 1.0%. This translats into a value of -dg Λ -of order ≥ 0.7 × 10 −4 GeV −1 at 13 TeV c.m. energy for observable C 1 . Similarly at 14 TeV c.m. energy at LHC the value of -dg Λ -would be ≥ 0.6×10 −5 GeV −1 and ≥ 0.2×10 −5 GeV −1 for the projected luminosities of 300 and 1000 fb −1 respectively. From the tables we note that the asymmetries (A i ) corresponding to observables (C i ) could be written as Figure 2: d g vs Λ for various observables at √ S = 13 TeV energy at LHC. The yellow and red regions corresponds to 2.5σ and 5σ sensitivities respectively. The white region is disallowed by the cross-section for pp → tt at 5σ CL.
.   Table 4 provides values of b i s defined in Eq. 3.9 for various observables C i for √ S = 13 and 14 TeV energy at LHC. Fig. 4 shows the behaviour of asymmetriy with respect to dg Λ . At d g = 0 i.e. for the SM which is actually low CP-violation, asymmetry is almost zero but even for low values of dg Λ CP-violation is consistent with errors so it is possible to observed it practically. Fig. 4 clearly shows that asymmetry is almost zero up to 10 −3 and then  5 shows the behaviour of differential asymmetry defined in Eq. 3.9. The plot shows that the coefficient of the CP-violating coupling remains constant for the particular observable, this shows that CP-violation effect is observed due to presence of coupling constant (d g ) and scale parameter (Λ).
We have successfully observed 5σ sensitivity for 13 TeV c.m. energy at LHC with an integrated luminosity of 36.1 fb −1 for ∼ 19 k events per month corresponding to 6.3 pb cross-section and predicted that we can achieve 5σ sensitivity at 14 TeV LHC energy with projected luminosity of 300 fb −1 with ∼ 183K events.
In study of Ref. [35] they have calculated asymmetries for SM, production vertex, decay vertex and strong phases in the decay vertex but our study is only in the production vertex. However, our study is more complete as we calculated the counting asymmetry in dilepton channel also we used d g and Λ as a free parameter whereas they used only single muon channel and constructed asymmetries at a fixedd g . Our results at 13 TeV c.m. energy are 4 times improved than their results and good in a manner that we have many choices ofd g to observed 5σ sesitivity. They used 10 fb −1 luminosity with 23 k events at 13 TeV c.m. energy at LHC while our results are based on 36.1 fb −1 luminosity with 19 k events per month.

Results and Discussion
In this article, effects of the T-odd interactions of the top-quark have been studied for the process gg → tt → (bl + ν l )(bl −ν l ) at the LHC for 13 and 14 TeV center-of-mass energies for a wide range of T-odd coupling (d g ) and the CP-violaitng scale (Λ). These results have been presented in Figs. 2 and 3 for 13 and 14 TeV LHC energies respectively. Our estimates for 13 TeV LHC energy with an integrated luminosity of 36.1 fb −1 reveals that dg Λ should be ≥ 0.7×10 −4 GeV −1 at 5σ confidence level. The corresponding 5σ sensitivities for the 14 TeV LHC with projected luminosities of 300 and 1000 fb −1 have been found to be ≥ 0.6 × 10 −5 and 0.2×10 −5 GeV −1 respectively. Similar results for the other T-odd observables could be obtained through the corresponding asymmetries listed in Tables 2  and 3.
Further improvements to include the effects of QCD showering and hadronisation and other detector specific details are highly desirable by the experimentalist. Possible sources which may fake such asymmetries e.g. in the Ref. [36] are also expected to be analysed.