4.2.3. Measurements at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 13 TeV

CMS performed already two measurements at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 13 TeV using a partial Run 2 dataset of 35.9 fb−1.

In the dilepton channel [116], the normalized distribution of Δ|*y*|*t*¯*<sup>t</sup>* is measured at parton and particle level while the distribution of Δ|*η*|- is measured at particle level, see Figure 39. Using these distributions, charge asymmetries are obtained: *At*¯*<sup>t</sup>* <sup>C</sup>(*parton*)=(1.0 ± 0.9(stat. + syst.))%, *At*¯*t* <sup>C</sup>(*particle*)=(0.8 <sup>±</sup> 0.9(stat. <sup>+</sup> syst.))%, and *<sup>A</sup>*-- <sup>C</sup> (*particle*)=(−0.5 ± 0.4(stat. + syst.))%, which are compared to various SM predictions in Figure 40.

**Figure 39.** The normalized differential *<sup>t</sup>*¯*<sup>t</sup>* production cross-section as a function of <sup>Δ</sup>|*y*| at the parton level in the full phase space (**left**) and as a function of Δ|*η*| in the fiducial phase space at the particle level (**right**) [116].

**Figure 40.** The results of the *A*<sup>C</sup> extraction from integrating normalized parton level and particle level differential cross-section measurements as a function of Δ|*y*| and Δ|*η*| are shown [116].

CMS also measured the forward–backward asymmetry in the -<sup>+</sup>jets channel at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 13 TeV using 35.9 fb−<sup>1</sup> [115]. This is a bit different measurement compared to all the other LHC measurements. The approximate forward–backward asymmetry *A*(1) *FB* is determined instead of edge–central charge asymmetry as measured in all the other LHC measurements. The template method is used based on *mt*¯*t*, *xF* = 2*pL*/ <sup>√</sup>*s*, and cos *<sup>θ</sup>*<sup>∗</sup> variables, where *pL* is the scaled longitudinal momentum *pL* of the *<sup>t</sup>*¯*<sup>t</sup>* system in the laboratory frame, and *θ*∗ is the production angle of the top quark relative to the direction of the initial-state parton in the *<sup>t</sup>*¯*<sup>t</sup>* center-of-mass frame. The *qq*¯ → *<sup>t</sup>*¯*<sup>t</sup>* differential cross-section in cos *<sup>θ</sup>* can be expressed as a linear combination of symmetric and antisymmetric functions, where the antisymmetric function can be approximated as a linear function of cos *θ* and parameter *A*(1) *FB* . Such approximation describes the LO terms and interference terms expected from an *s*-channel resonance with chiral couplings. In such approximation, *AFB* <sup>=</sup> *<sup>A</sup>*(1) *FB* . The generator level distributions for the above mentioned variables for the *t*¯*t* production initiated by different processes are shown in Figure 41. The application of fitting procedure yields *A*(1) *FB* = (4.8+9.5 <sup>−</sup>8.7(stat.) +2.0 <sup>−</sup>2.9(syst.))%. The result is consistent with the NLO QCD [13,21,118] and NNLO QCD prediction [32], although the statistical uncertainty is quite large.

**Figure 41.** The generator-level cos *θ* (labeled here as *c*∗) in (**a**), *x*<sup>F</sup> in (**b**), and *mt*¯*<sup>t</sup>* normalized distributions in (**c**) for the subprocesses *qq*¯, *qg*, and *gg*. These distributions correspond to the CMS measurement in the -<sup>+</sup>jets channel performed at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 13 TeV using 35.9 fb−<sup>1</sup> [115].

ATLAS already performed a preliminary *A*<sup>C</sup> measurement in the -+jets channel using the full Run 2 statistics (139 fb−1) [50]. Altogether, more than four millions of *t*¯*t* candidates were selected in data events with the expected background of about 15%. The asymmetry is measured to be *A*<sup>C</sup> = (0.60 ± 0.15(stat. + syst.))%, consistent with the NNLO QCD + NLO EW prediction of (0.64+0.05 <sup>−</sup>0.06)%. Differential measurements in *mt*¯*<sup>t</sup>* and *<sup>β</sup>z*,*t*¯*<sup>t</sup>* were also performed, see Figure 42. Moreover, the charge asymmetry measurement was interpreted in the framework of an effective field theory (EFT). In EFT formalism the SM Lagrangian is extended with operators that encode the new physics phenomena. The Warsaw basis includes a complete set of dimension-six operators [119]. The charge asymmetry is affected by the difference *<sup>C</sup>*<sup>−</sup> <sup>=</sup> *<sup>C</sup>*<sup>1</sup> <sup>−</sup> *<sup>C</sup>*2, where *<sup>C</sup>*<sup>1</sup> <sup>=</sup> *<sup>C</sup>*<sup>1</sup> *<sup>u</sup>* = *C*<sup>1</sup> *<sup>d</sup>* and *C*<sup>2</sup> = *C*<sup>2</sup> *<sup>u</sup>* = *C*<sup>2</sup> *<sup>d</sup>* are Wilson coefficients which are obtained from seven four-fermion operators in Warsaw basis by using a flavour-specific linear combination [120]. The constrains on *C*− are shown in Figure 43.

**Figure 42.** Differential charge asymmetry measurements as a function of *βz*,*t*¯*<sup>t</sup>* (**left**) and *mt*¯*<sup>t</sup>* (**right**) [50].

**Figure 43.** Constraints on linear combination *C*−/Λ<sup>2</sup> of Wilson coefficients of dimension 6 operators from inclusive and *mt*¯*<sup>t</sup>* differential charge asymmetry measurements [50].

#### 4.2.4. Summary of LHC Measurements

All inclusive charge asymmetry measurements performed at the LHC are summarized in Table 8. The *At*¯*<sup>t</sup>* <sup>C</sup> asymmetries should be compared with NLO QCD including electroweak corrections prediction [24] (1.23 <sup>±</sup> 0.05)% at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 7 TeV, NNLO QCD + NLO EW prediction [32] (0.97+0.02 <sup>−</sup>0.03)% at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 8 TeV and (0.64+0.06 <sup>−</sup>0.05)% at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 13 TeV. The *<sup>A</sup>*-- <sup>C</sup> asymmetries should be compared with NLO QCD + EW prediction (0.70 <sup>±</sup> 0.03)% at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 7 TeV [24], (0.64 <sup>±</sup> 0.03)% at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 8 TeV [24], and NLO QCD + EW prediction (0.55 <sup>±</sup> 0.03)% at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 13 TeV [25].

All LHC measurements at all energies are well within 2 SD consistent with the SM prediction. The measurements at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 7 TeV are limited by the statistics with all of them at least to have the absolute uncertainty of 1%. The exception is the combination of ATLAS and D0 in the -+jets channel which has the total uncertainty of about 0.9%. At <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 8 TeV, there are already many measurements which have comparable statistical and total systematic uncertainty. The most precise is the combination of the ATLAS and CMS -+jets channel measurements which has the overall uncertainty of about 0.34%

with the dominant systematic uncertainties due to calibration of jets and signal modeling. Finally at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 13 TeV, the full statistics measurement are not yet available except for the preliminary ATLAS -+jets measurement. This measurement is already very precise at the absolute level of 0.15%, very well consistent with NNLO QCD + NLO EW prediction, and differs from zero by 4 standard deviations. This is the first evidence for non-zero charge asymmetry at the LHC. The early measurements are not precise enough to be able to observe the expected decrease of the asymmetry with the energy of interactions.

The leptonic asymmetries have for now uncertainties larger than 0.4% (particle level) and are all consistent with SM predictions.

The differential measurements are also consistent with the SM prediction. Most of the time, the statistical uncertainties are dominant, although in the latest ATLAS measurement at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 13 TeV the total systematic uncertainties are comparable to statistical uncertainties except for high *mt*¯*<sup>t</sup>* bins.

**Table 8.** Summary of inclusive *t*¯*t* and leptonic charge asymmetry measurements performed at the LHC. For a given measurement, if there is just one uncertainty, it is combined statistical and systematical uncertainty. If there are two uncertainties, the first one is statistical and the second one is systematic uncertainty. All measurements used Δ|*y*| variable except for the measurement with <sup>∗</sup> which used Δ|*η*|. All measurements were performed at the parton level except for the measurement with ∗∗ which was performed at particle level.


#### **5. Discussion and Outlook**

It is clear from the description in Sections 3 and 4 that the long path and large effort in improving the theory and experiments has paid off. Although, some may be unhappy that tensions between theoretical calculations and experimental measurements mostly disappeared, the understanding of the *t*¯*t* charge asymmetry is much better now.

On the theoretical side, the progress has been enormous from only a partial NLO prediction for *A*FB at the Tevatron which predicted negative asymmetry, through the full NLO prediction in the laboratory frame of about 5%, to the latest full NNLO QCD + NLO EW prediction for both *A*FB at the Tevatron and *A*<sup>C</sup> at the LHC and the aN3LO QCD + NLO EW prediction at the Tevatron. At the Tevatron, the predicted asymmetry is about 10% while it is around 1% at the LHC. Moreover, differential asymmetries have been also calculated at NNLO QCD + NLO EW too as a function of many variables such as *mt*¯*t*, Δ*y*, *p*T,t¯t, *βz*,*t*¯*t*, and cos *θ*. The leptonic asymmetry has been calculated at NLO+EW order.

On the experimental side, there has been performed a full set of measurements for various observables. The very early measurements were performed just at the reco level. Later, this has been improved to perform measurements at the parton level and lately also at the particle level. There are now available not only inclusive measurements of both forward–backward and charge asymmetries, but also detailed differential measurements as a function of a few variables such as *mt*¯*t*, *p*T,t¯t, Δ*y*, *βz*,*t*¯*t*. All inclusive *t*¯*t* asymmetry measurements of CDF, D0, ATLAS, CMS show a very good agreement with the NNLO QCD + NLO EW prediction with the largest disagreement of about 1.6 SD. The leptonic asymmetry measurements with the full Tevatron dataset and at the LHC also agree with the NLO QCD + EW prediction with the largest disagreement of about 2.3 SD for the CDF leptonic asymmetry measurement. However, it should be mentioned that all inclusive Tevatron measurements are higher than the NNLO QCD + NLO EW prediction, so it is possible that some non-negligible correction is still not calculated. At the LHC, the asymmetries both higher and lower compared to the best prediction have been measured. At the Tevatron, the non-zero forward–backward asymmetry (*δA*FB/*A*FB = 20%) has been observed now (with a significance of about 5 SD) and the leptonic asymmetry is measured with the relative precision of about 26%. For *<sup>A</sup>*<sup>C</sup> at the LHC at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 13 TeV, the evidence (significance of at least 3 SD) of non-zero charge asymmetry has been obtained and the relative precision is about 25%. Given that the dileptonic asymmetry has not been measured yet with the full LHC Run 2 statistics, the fact that dileptonic asymmetry is supposed to be smaller than *A*C, and the fact it can be measured only in the dilepton channel, its relative precision is for now only around 80%. Most of the inclusive measurements at both the Tevatron and the LHC have been statistically limited although the statistical and total systematic uncertainties are about the same in the LHC combination at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 7 TeV and <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 8 TeV and in the latest measurement at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 13 TeV. The *A*FB and *A*<sup>C</sup> asymmetries and their leptonic versions have been measured also differentially as a function of a few variables. Most of the measurements have been statistically limited, but this starts to change with the full LHC Run 2 statistics. The Tevatron results are very probably final, since the data taking finished already in 2011.

The LHC running will continue, mostly at the energy of <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 14 TeV and about 20 times more data (3000 fb−1) are expected to be delivered by the end of the LHC lifetime. This will allow to improve the statistical uncertainty by at least a factor of 4–5 and the systematic uncertainties will become dominant. Based on the ATLAS measurement at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 13 TeV, it can be expected the dominant systematic uncertainties will be the *t*¯*t* modeling, the jet energy calibration related uncertainties and the *W*+jets background modeling. These systematic uncertainties will become dominant also for differential measurements and this will allow to measure them in a more detail using more bins and the larger range. Eventually, the dileptonic asymmetry should be more precisely measured because the leptons are more precisely measured than top quarks and typically have smaller systematic related uncertainties. Moreover, it is expected that another LHC experiment, the LHCb, will be able to observe a non-zero *t*¯*t* charge asymmetry at the high-luminosity LHC [121]. Additionally, there is a possibility to measure different types of asymmetries, such as energy asymmetry between the top and antitop quarks [122].

At the potential Future Circular Collider (FCC) in *pp* collisions at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 100 TeV, the charge asymmetry is greatly diluted by the dominance of the *gg* initial state. The SM expected value is *A*<sup>C</sup> = 0.12% [123] which will make it very hard to measure. However, the asymmetry is enhanced in associated processes *t*¯*t* + *Z*, *t*¯*t* + *γ* and mainly in *t*¯*t* + *W*, where the asymmetry is enhanced by about a factor of ten due to the *t*¯*t* + *W* process being dominated by a *qq*¯ initial state [123,124]. A relative statistical precision of about 3% is expected in the determination of *A*<sup>C</sup> in the *t*¯*t* + *W* process [124].

At the linear *<sup>e</sup>*+*e*<sup>−</sup> collider, the EW based forward–backward asymmetry in *<sup>e</sup>*+*e*<sup>−</sup> <sup>→</sup> *<sup>t</sup>*¯*<sup>t</sup>* is expected [125,126]. The preliminary studies for the potential International Linear Collider at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 500 GeV show that for the large asymmetry of about 40% (depending on the polarization of the beams), the expected relative precision of about 2% can be achieved [125].

The asymmetry measurements should also help in the model independent search for a new BSM physics within the effective field theory approach by constraining the EFT coefficients related to the top quark production.

#### **6. Conclusions**

As the heaviest known elementary particle, the top quark and studies of its properties is a promising portal to the new physics beyond the Standard Model. The charge asymmetry in the *t*¯*t* production is the effect which is predicted to be present at higher orders in perturbative quantum chromodynamics and by necessity to be small, but it is highly enhanced in various theories beyond the Standard Model.

After unexpectedly large values of the forward–backward asymmetry in the top quark pair production were observed in initial measurements at the Tevatron, a lot of attention has been paid to it by the experimental and theoretical community. This allowed to perform precise and detailed tests of the SM at high energies. At present, the prediction is known at full next-to-next-to-leading order in perturbative QCD with complete next-to-leading order electroweak corrections. The full statistics Tevatron forward–backward and the LHC charge asymmetry results for the inclusive and differential measurements agree with the predictions very well, mostly within two standard deviations, with the largest deviation of about 2.3 standard deviation. The predicted forward–backward asymmetry at the Tevatron of about 10% is now measured with a relative precision of 20%. At the LHC, although the effect is much smaller (≈1%), the relative precision of the latest measurement is already at the level of about 25%.

In the coming years at the LHC and potential future colliders, it can be expected that more measurements will be performed at higher energies and in the processes like *t*¯*t* + *W* boson where the relative precision at the level of a few percent can be potentially achieved. Moreover, there is a possibility to measure a very large *t*¯*t* asymmetry in electroweak interactions at the lepton collider in polarized beams with a relative precision of a couple of percent. This will allow to precisely test the present theory at high energies and to potentially observe the presence of BSM effects or to constrain the BSM physics either by excluding particular models or by constraining parameters of effective theories.

**Funding:** This research was supported by the project LTT17018 of Ministry of Education, Youth and Sports of Czech Republic.

**Acknowledgments:** The author would like to thank Alexander Kupˇco and Jaroslav Antoš for reading the manuscript and providing useful comments.

**Conflicts of Interest:** The author is a member of the CDF and ATLAS collaboration. The funders had no role in the writing of the study.

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