**2. Charge Asymmetry in** *tt***¯ Production**

In this section, the top quark production within the SM is described. Afterwards, the charge asymmetry is discussed within the quantum electrodynamics (QED), the electroweak (EW) theory, the quantum chromodynamics, and specifically in the top quark pair production. Finally, a few different definitions of the asymmetry will be mentioned which have been used in the measurements and theoretical predictions.

#### *2.1. Top Quark Production in the SM at Hadron Colliders*

At hadron colliders, the dominant production of the top quark is via the top–antitop (*t*¯*t*) pair production through strong interactions described by QCD. At the lowest, leading order (LO) of the perturbative QCD, the top-quark pair (when talking about the top quark pair, it is always assumed the top–antitop pair, unless stated otherwise) production is possible through the quark–antiquark annihilation (*qq*¯ → *<sup>t</sup>*¯*t*) or the gluon fusion (*gg* → *<sup>t</sup>*¯*t*). Feynman diagrams of the LO *<sup>t</sup>*¯*<sup>t</sup>* production are shown in Figure 1. At higher orders, e.g., next-to-leading (NLO) or next-to-next-to-leading (NNLO), the *<sup>t</sup>*¯*<sup>t</sup>* production is possible also through the quark–gluon interaction (*gq* → *<sup>t</sup>*¯*<sup>t</sup>* or *gq*¯ → *<sup>t</sup>*¯*t*).

**Figure 1.** The Feynman diagrams of leading order processes contributing to the top quark pair production at hadron colliders.

The top (antitop) quark alone can not be produced in QCD due to the flavour conservation. However, the flavour is not conserved in weak interactions. Therefore, a single top quark can be produced, see Figure 2. Since the strong coupling constant *α<sup>s</sup>* is the largest of all couplings, the *t*¯*t* pair production has a cross-section larger than the single-top production even though there are two top quarks produced.

**Figure 2.** The Feynman diagrams for the single-top quark production: (**a**) *s*-channel, (**b**) *t*-channel, and (**c**) Wt-channel.

The *t*¯*t* pair production mechanism is quite different at the Tevatron and at the LHC. In *pp*¯ collisions at the Tevatron in Run II at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 1.96 TeV, the dominant *<sup>t</sup>*¯*<sup>t</sup>* production is through the *qq*¯ annihilation (85% *qq*¯ and 15% *gg* at LO in QCD [6]). At the LHC in *pp* interactions, it is almost opposite. The gluon fusion production channel is dominant, being about 80–90% at LO when going from <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 7 TeV to <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 14 TeV [6].

There are a few *t*¯*t* inclusive cross-section predictions available at NNLO or higher in QCD. The initial NNLO calculation became available in Ref. [47]. This included later the higher-order soft-gluon corrections through the resummation at the next-to-next-leading-logarithm (NNLL) accuracy, see [48] and references therein. Based on the above NNLO calculation and adding next-to-next-to-next-to-leading-order (N3LO) soft-gluon corrections by applying a different method, the approximate N3LO (aN3LO) prediction became available [51]. Recently, there was performed another independent calculation of the *t*¯*t* production at NNLO QCD using the MATRIX framework [52–54].

The cross-section predictions of the *t*¯*t* pair production are shown in Table 1 for both the Tevatron and the LHC, for different center-of-mass energies, and for different calculations. The values are very similar and consistent between different calculations although different parton distribution function (PDF) sets were used for different predictions. The comparison of measured *t*¯*t* inclusive cross-sections at the Tevatron and at the LHC with the NNLO + NNLL theoretical predictions is shown in Figure 3. The agreement between predictions and measurements is excellent.

**Table 1.** The predicted next-to-next-to-leading order (NNLO) *t*¯*t* production cross-sections in pb for various energies at the Tevatron and at the Large Hadron Collider (LHC) and for different available calculations. The uncertainties include the factorization and renormalization scale and parton distribution function (PDF)+*α<sup>s</sup>* uncertainties. The assumed top quark mass is always *mt* = 173.3 GeV except for NNLO + next-to-next-leading-logarithm (NNLL) prediction at the LHC where it is *mt* = 173.2 GeV.


**Figure 3.** Measured inclusive cross-sections at the Tevatron and at the LHC compared to NNLO + NNLL predictions [55].

The top quark decays almost always into *<sup>t</sup>* → *<sup>W</sup>* + *<sup>b</sup>*. The *<sup>t</sup>*¯*<sup>t</sup>* decay channels are thus characterized by decays of *W* boson which could be leptonic *W* → *<sup>ν</sup>* or hadronic *<sup>W</sup>* <sup>→</sup> *<sup>q</sup>* ¯ *q*- . The *t*¯*t* decay chain is shown in Figure 4. There are three decay channels according to the number of charged leptons (the inclusion of the charge-conjugate mode is implied): the dilepton (*t*¯*<sup>t</sup>* → -<sup>+</sup>*νb*-<sup>−</sup>*ν*¯¯ *b*) (11%), lepton+jets (-+jets, *<sup>t</sup>*¯*<sup>t</sup>* → -<sup>+</sup>*νbq* ¯ *q*-¯ *<sup>b</sup>*) (44%), and all-hadronic channel (*t*¯*<sup>t</sup>* <sup>→</sup> *<sup>q</sup>* ¯ *qbq* ¯ *q*-¯ *b*) (44%). The quark is color particle which hadronize to create a spray of colorless final state particles (mostly hadrons) flying in about the same direction, a jet.

**Figure 4.** The top–antitop quark pair decay chain.

#### *2.2. Charge Asymmetry in QED and EW Theory*

The angular asymmetry in the differential cross-section of the pair production is the difference in production rates for a fermion and an antifermion flying along a given direction. In QED, it was noticed and calculated a long time ago for e.g., the *<sup>e</sup>*+*e*<sup>−</sup> <sup>→</sup> *<sup>μ</sup>*+*μ*<sup>−</sup> production [56]. At LO in QED, the *<sup>μ</sup>*+*μ*<sup>−</sup> pair production is symmetric under the transformation *<sup>μ</sup>*<sup>+</sup> <sup>↔</sup> *<sup>μ</sup>*−, i.e., under the charge conjugation (*C*), with respect to the incoming *e*<sup>+</sup> and *e*<sup>−</sup> beams. The asymmetry is present at NLO due to the interference of processes that differ under *C*-conjugation, i.e., between the lowest order and two-photon box graphs and between *C*-odd and *C*-even breamshtrahlung diagrams, see Figure 5. The overall effect is that the positive muons *μ*<sup>+</sup> fly a little bit more often in the same direction as incoming positive electrons *e*<sup>+</sup> while negative *μ*<sup>−</sup> fly preferentially in the direction of negative *e*−. It should be stressed that no parity-violating interactions are involved. The QED asymmetry prediction has been confirmed in the experiment [57], see Figure 6.

**Figure 5.** Diagrams of processes contributing to the quantum electrodynamic (QED) charge asymmetry for the *<sup>e</sup>*+*e*<sup>−</sup> <sup>→</sup> *<sup>μ</sup>*+*μ*<sup>−</sup> production: the box diagram in (**a**) interfering with the leading order (LO) diagram in (**b**), and breamshtrahlung diagrams with *C*-odd in (**c**) and *C*-even state in (**d**) [56].

**Figure 6.** The measured angular distribution for the *<sup>e</sup>*+*e*<sup>−</sup> <sup>→</sup> *<sup>μ</sup>*+*μ*<sup>−</sup> production together with the QED prediction [57]. The angle *θ* is the angle between the incoming *e*<sup>+</sup> direction and the outgoing *μ*<sup>+</sup> direction.

In the electroweak theory, the angular asymmetry is already predicted at LO due to the *Z*-boson axial-vector coupling to fermions [58]. It was precisely measured at LEP experiments for *<sup>e</sup>*+*e*<sup>−</sup> <sup>→</sup> *qq*¯, *q* = *c*, *b* reactions [59].

#### *2.3. Charge Asymmetry in QCD*

Similarly to QED, there is no charge asymmetry at LO in the QCD production of *qq*¯ <sup>→</sup> *QQ*¯ while it is expected at NLO [60]. The asymmetry is thus of the order of *α<sup>s</sup>* relative to the dominant production process. The corresponding QCD diagrams similar to QED diagrams contribute and again the asymmetry is induced through the interference between the amplitudes which are relatively odd under the *<sup>t</sup>* ↔ ¯*t*, i.e., the interference of box and LO Born diagrams and the interference of final-state and initial-state radiation diagrams, see Figure 7. The interference of virtual (box) diagrams and LO (Born) diagrams (Figure 7) contributes to a positive asymmetry while the interference of the diagrams with real-corrections has a negative asymmetry with the former to be dominant. The overall net effect is thus a positive asymmetry.

**Figure 7.** The diagrams contributing to the QCD charge asymmetry in the production of heavy quarks at hadron colliders: interference of final-state (**a**) with initial-state (**b**) gluon breamshtrahlung plus interference of the box (**c**) with the Born diagram (**d**) [13].

The asymmetry is present in the production for all quark pairs, not only top quark pairs. For light quarks (*u*, *d*) when the initial and outgoing quark is the same, the *t*-channel *qq*¯ → *qq*¯ must be considered too [60]. The charge asymmetry in the *pp* <sup>→</sup> *<sup>b</sup>*¯ *b* production was measured at the LHC by the LHCb experiment [61], while the forward–backward asymmetry in the *pp*¯ <sup>→</sup> *<sup>b</sup>*¯ *b* production was measured at the Tevatron by the CDF experiment [62,63] and by the D0 experiment in the production of *B*± mesons [64]. At the Tevatron, the *b*¯ *b* production is dominated by the *gg* fusion unlike the top quark pair production due to the much lower *b*-quark mass. Therefore, the asymmetry is expected to be much smaller for *b*¯ *b* compared to *t*¯*t*. The *A*FB measurement was performed at the CDF experiment at both low and high *mb*¯ *<sup>b</sup>*, see Figure 8. Both CDF measurements and also LHCb measurement are consistent with the SM predictions, although typically with quite large uncertainties. The D0 measurement shows the discrepancy of about 3 SD between the measurement and the NLO QCD estimate from MC@NLO with a large theoretical uncertainty for the prediction which suggests that more precise prediction is needed to interpret this result.

**Figure 8.** The *b*¯ *b* forward–backward asymmetry measured by the CDF experiment at low (**left**) and high (**right**) invariant mass of *b*¯ *b* pair [62,63].

#### *2.4. Top Quark Pair Charge Asymmetry*

From the above description in Section 2.3, it follows that the *qq*¯ → *<sup>t</sup>*¯*<sup>t</sup>* production is charge symmetric at LO and becomes asymmetric at NLO, i.e., the production of the top and antitop quark along a given direction is different. For the *gg* → *<sup>t</sup>*¯*<sup>t</sup>* production, the initial state is symmetric and thus no asymmetry is predicted. For the *qg*(*qg*¯ ) → *<sup>t</sup>*¯*<sup>t</sup>* production, the asymmetry is also expected due to interference terms.

Moreover, there is an asymmetry already at LO in the EW production of the top quark pair *qq*¯ → *<sup>γ</sup>*∗/*<sup>Z</sup>* → *<sup>t</sup>*¯*t*. In addition, QCD–EW interference terms contribute to the asymmetry. However, since the EW production of the *t*¯*t* pair is small compared to the QCD production, its contribution to the charge asymmetry is subdominant although important as it will be seen later.

The fraction of *gg* <sup>→</sup> *<sup>t</sup>*¯*<sup>t</sup>* is increasing with increasing of the energy <sup>√</sup>*s*. Therefore, the overall asymmetry in *pp* or *pp*¯ collisions is decreasing as energy of collisions is increasing.

The *t*¯*t* charge asymmetry is quite different at the Tevatron and at the LHC. As mentioned above, *qq*¯ is the dominant production process at the Tevatron, and it is *pp*¯ collider, so the axis of initial quark largely coincides with the axis of initial proton. The asymmetry in *qq*¯ → *<sup>t</sup>*¯*<sup>t</sup>* is thus largely preserved in *pp*¯ → *<sup>t</sup>*¯*<sup>t</sup>* collisions. At the LHC, the dominant production process is *gg* → *<sup>t</sup>*¯*<sup>t</sup>* which has no asymmetry. Therefore, the charge asymmetry is largely suppressed. Moreover, since *pp* is a charge symmetric initial state, there is no overall charge asymmetry in *pp* → *<sup>t</sup>*¯*t*. However, the interacting initial quark is a valence quark, so it has in average a larger longitudinal momentum compared to an antiquark from the sea of quarks in the proton. Since top quarks fly a bit more often in the quarks direction, as a consequence, the top quarks will fly more in forward/backward direction compared to more central antitops.

#### *2.5. Asymmetry Definitions*

In general, it is important to state how the asymmetry is defined, in which frame and which observable is used for the definition.

As was already mentioned, the charge asymmetry present in a process *qq*¯ → *<sup>t</sup>*¯*<sup>t</sup>* means that in the center-of-mass frame, the number of top quarks flying in a certain direction is different compared to the number of antitop quarks. Therefore, the asymmetry can be defined as

$$A\_{\bar{C}}^{t\bar{t}}(\cos\theta) = \frac{N\_l(\cos\theta) - N\_{\bar{l}}(\cos\theta)}{N\_l(\cos\theta) + N\_l(\cos\theta)},\tag{1}$$

where *Nt*(cos *θ*) and *N*¯*t*(cos *θ*) are the number of top quarks and antitop quarks flying along cos *θ* direction, where *θ* is computed with respect to the direction of the initial quark (and not the antiquark) in both cases. Since the asymmetry is defined as the ratio of two numbers, it is often quoted as a percentage. Such convention is adopted also in this article.

Assuming CP-invariance in strong interactions, *N*¯*t*(cos *θ*) = *Nt*(− cos *θ*), and the charge asymmetry definition in Equation (1) can be rewritten using only the top quark without a need for the antitop quark:

$$A\_C^{\rm eff}(\cos \theta) = \frac{N\_t(\cos \theta) - N\_t(-\cos \theta)}{N\_t(\cos \theta) + N\_t(-\cos \theta)}.\tag{2}$$

This can be then used to define the inclusive charge asymmetry by summing events from opposite hemispheres and it becomes forward–backward asymmetry:

$$A\_{FB}^{t\bar{t}} = \frac{N(\cos\theta > 0) - N(\cos\theta < 0)}{N(\cos\theta > 0) + N(\cos\theta < 0)},\tag{3}$$

where *N*(cos *θ* > 0) is the number of events with the top quark fulfilling condition cos *θ* > 0.

The experiments at the Tevatron and at the LHC use a right-handed coordinate system with its origin at the nominal interaction point in the center of the detector and the *z*-axis along the beam pipe. At the Tevatron experiments, the *z*-axis points along the proton direction, while the *y*-axis points vertically upward and the *x*-axis points radially outwards. At LHC experiments, the x-axis points radially inward toward the center of the LHC, while the y-axis points vertically upward. Cylindrical coordinates (*r*; *φ*) are used in the transverse plane, *φ* being the azimuthal angle around the beam pipe. The polar angle is calculated with respect to *z*-axis. The pseudorapidity is defined in terms of the polar angle *θ* as *η* = − ln(tan *θ*/2), while the rapidity *y* is defined as *y* = 1/2 ln[(*E* + *pz*)/(*E* − *pz*)].

Instead of cos *θ*, other observables can be also used to define the asymmetry, for example the rapidity (*yt*) or the pseudorapidity (*ηt*) of the top quark, or Δ*yt*¯*<sup>t</sup>* = *yt* − *y*¯*t*. Clearly, the inclusive forward–backward asymmetry stays the same in *qq*¯ center-of-mass frame independently of which variable is used.

Experimentally, the top and antitop quark are reconstructed using their decay products registered within a detector. There are numerous methods which have been developed for such task in the past [65–74]. For charge asymmetry measurements, it is needed to determine which of them is the top quark and the antitop quark. In the dilepton channel, it is relatively simple. The quark which decay has assigned the positive lepton is labeled as the top quark while the quark which has assigned negative lepton is labeled as the antitop quark. In the -+jets channel, there is just one final state lepton. After the *t*¯*t* reconstruction, the hadronic (*th*) and leptonic (*t*-) top quark is labeled depending on which one has the hadronic and leptonic decay assigned, respectively. If the final state lepton has a positive electromagnetic charge, the leptonic top quark is labeled as the top quark and the hadronic top quark as the antitop quark. If the final state lepton has a negative charge, the leptonic top quark is labeled as the antitop quark.

Asymmetries can be measured also for decay products of top quarks. They will carry the information about a direction of flight of the top quarks due to the boost given by top quarks, but the direction will not be 100% correlated. Therefore, the asymmetry in decay products will be a little bit diluted. The potential decay products are *W* bosons, *b* quarks, light quarks, or charged leptons and neutrinos from *W* boson decay. The good candidates must have a well reconstructed direction of flight and an electromagnetic charge must be well measured for them. The light quarks are almost hopeless since they are reconstructed as jets which direction is not precisely measured and more importantly the charge determination is very hard. Better candidates would be *b*-jets since there are methods to measure the *b*-jet charge [75–77] but these are not very precise. The *W* boson is also not particularly good object because it can be reconstructed from either jets (hard to get a charge) or from the charged lepton and the neutrino (hard to precisely measure a direction). Therefore, the best candidates for measuring the asymmetry are charged leptons, since the reconstruction of both their charge and direction is excellent. This is also reason why the all-hadronic channel is not used for charge-asymmetry measurements.

Similarly as for the top quark, more variables can be used to define the leptonic asymmetry. Typically, only electrons and muons are used in measurements since the *τ* lepton reconstruction is more complicated. For these leptons, the pseudorapidity is used rather then the rapidity since it is easier to be measured. For practical purposes, the values are the same, given their large energies compared to their mass. Therefore, the *η* of leptons or Δ*η* = *η*-<sup>+</sup> − *η*-<sup>−</sup> are used in the definition of the asymmetry.

#### 2.5.1. Asymmetry Definitions for Tevatron

At the Tevatron *pp*¯ collider, where *qq*¯ → *<sup>t</sup>*¯*<sup>t</sup>* is the dominant production process, the valence quark from the proton interacts with the valence antiquark from the antiproton most of the time. Therefore, the charge and forward–backward asymmetry in *qq*¯ → *<sup>t</sup>*¯*<sup>t</sup>* is mostly preserved in the *pp*¯ → *<sup>t</sup>*¯*<sup>t</sup>* production although it is a little bit smaller due to a symmetric contribution of the *gg* fusion.

The Tevatron measurements are performed in the laboratory frame (*pp*¯ center-of-mass frame) or in the *t*¯*t* rest frame. There are advantages and disadvantages for both frames. The advantage of the *t*¯*t* rest frame is that the asymmetry is larger than the asymmetry in the laboratory frame. At the Tevatron in Run II, the laboratory frame asymmetry is diluted by 30% [42]. That is because in a given *qq*¯ → *<sup>t</sup>*¯*<sup>t</sup>* interaction, the interacting quark and antiquark has in general a different longitudinal momentum in the laboratory frame which will give a boost to the *t*¯*t* system. It can then happen that even that the top quark flies in the forward direction in the *t*¯*t* rest frame, it will fly in the backward direction in the laboratory frame. For that reason, there is an advantage to use the Δ*yt*¯*<sup>t</sup>* variable. It is a Lorentz invariant, so it is independent of the *t*¯*t* longitudinal motion and it is simply related to the top quark rapidity in the *t*¯*t* rest frame: *yt*¯*<sup>t</sup> <sup>t</sup>* = 1/2Δ*y*. The advantage of the laboratory frame is that it can be defined to rely only on the measured hadronically decaying top quark rapidity *yth* which has a much better resolution compared to the leptonically decaying top quark which includes only indirectly and partially (no z component) measured neutrino. Consequently, the statistical precision of *A*FB is better in the *pp*¯ frame.

The forward–backward asymmetry in the laboratory frame is usually defined using the rapidity of the top quark in the laboratory (*pp*¯) frame, *yt*:

$$A\_{FB}^{pp} = \frac{N(y\_t > 0) - N(y\_t < 0)}{N(y\_t > 0) + N(y\_t < 0)} = \frac{N(-q\_\ell y(t\_h) > 0) - N(-q\_\ell y(t\_h) < 0)}{N(-q\_\ell y(t\_h) > 0) + N(-q\_\ell y(t\_h) < 0)}\tag{4}$$

where the rapidity of the hadronic top *y*(*th*) is used. This has the advantage that it has a better resolution than Δ*y*, but the disadvantage is that it measures the diluted laboratory frame asymmetry. Similarly, another variable can be used in the forward–backward asymmetry definition: cos *θ* = *q*- · cos *α<sup>p</sup>* is used, where *θ* is the polar angle between the top quark and the proton beam while *α<sup>p</sup>* is the polar angle between the hadronic top quark and the proton beam.

The *<sup>t</sup>*¯*<sup>t</sup>* rest frame asymmetry is defined using <sup>Δ</sup>*<sup>y</sup>* = *yt* − *<sup>y</sup>*¯*<sup>t</sup>*

$$A\_{FB}^{\text{if}} = \frac{N(\Delta y > 0) - N(\Delta y < 0)}{N(\Delta y > 0) + N(\Delta y < 0)} = \frac{N(q\_\ell \cdot (y\_{t\_\ell} - y\_{t\_h}) > 0) - N(q\_\ell \cdot (y\_{t\_\ell} - y\_{t\_h}) < 0)}{N(q\_\ell \cdot (y\_{t\_\ell} - y\_{t\_h}) > 0) + N(q\_\ell \cdot (y\_{t\_\ell} - y\_{t\_h}) < 0)},\tag{5}$$

where *q* is the electric charge of the lepton, and *yt*- , *yth* is the rapidity of the hadronically, leptonically decaying top, or antitop quark. The disadvantage is that Δ*y* has a worse resolution compared to *th* since it combines the uncertainties of both quark reconstructions, including neutrino-related complications of the *t*quark system.

For the leptonic defined asymmetry in the laboratory frame, the asymmetry is usually defined as:

$$A\_{\rm FB}^{\ell} = \frac{\mathcal{N}(q\_{\ell}\eta\_{\ell} > 0) - \mathcal{N}(q\_{\ell}\eta\_{\ell} < 0)}{\mathcal{N}(q\_{\ell}\eta\_{\ell} > 0) + \mathcal{N}(q\_{\ell}\eta\_{\ell} < 0)}.\tag{6}$$

Similarly to the top quark asymmetry in the *t*¯*t* rest frame defined using Δ*y*, the asymmetry in the dilepton channel can be defined this way:

$$A\_{\rm FB}^{\ell\ell} = \frac{N(\Delta\eta\_{\ell\ell} > 0) - N(\Delta\eta\_{\ell\ell} < 0)}{N(\Delta\eta\_{\ell\ell} > 0) + N(\Delta\eta\_{\ell\ell} < 0)} = \frac{N(\eta\_{\ell+} - \eta\_{\ell-} > 0) - N(\eta\_{\ell+} - \eta\_{\ell-} < 0)}{N(\eta\_{\ell+} - \eta\_{\ell-} > 0) + N(\eta\_{\ell+} - \eta\_{\ell-} < 0)}.\tag{7}$$

The *A*- FB is related to *<sup>A</sup>pp* FB but the effect is smaller thanks to the dilution due to leptons not following the top quark direction precisely, which in turn is smaller than *At*¯*<sup>t</sup>* FB. The *<sup>A</sup>*-- FB is related to *At*¯*t* FB but a bit smaller due to the same reason. Therefore, there is an advantage in using *<sup>A</sup>*-- FB since its value is not diluted by laboratory frame. However, the disadvantage is that *A*-- FB can be measured only in the dilepton channel, which has the smallest statistics.

#### 2.5.2. Asymmetry Definitions for LHC

As it was already pointed out above, top quarks are more often produced in the forward/ backward direction while antitops are produced more often in the central region. This can be explored to define the edge-central charge asymmetry, which will be called just simply the charge asymmetry in the following. Most often, the variable Δ|*y*| = |*yt*|−|*y*¯*t*| is used to define the charge asymmetry:

$$A\_{\mathbb{C}} = \frac{N(\Lambda|y| > 0) - N(\Lambda|y| < 0)}{N(\Lambda|y| > 0) + N(\Lambda|y| < 0)}. \tag{8}$$

For leptons, Δ|*η*| = |*η*-+|−|*η*-−| is used instead to define the dileptonic asymmetry:

$$A\_{\mathbb{C}}^{\ell\ell} = \frac{N(\Delta|\eta|>0) - N(\Delta|\eta|<0)}{N(\Delta|\eta|>0) + N(\Delta|\eta|<0)}.\tag{9}$$

#### **3. Theory Overview**

The overview of theoretical predictions of the charge asymmetry in the top quark pair production at hadron colliders is presented. In the first part, the evolution of SM predictions is described while in the second part are mentioned various BSM models which could affect the *t*¯*t* charge asymmetry.

#### *3.1. SM Predictions*

First, it should be noted that two charge asymmetry definitions are used in theoretical calculations. In the first calculation (unexpanded), the most precise calculation available for the numerator and the denominator in the asymmetry definition is used. In such case, the denominator is effectively calculated at the higher order than the numerator since the asymmetry is non-zero starting only at NLO. For example, using the NLO calculation, the total cross-section (denominator) is at NLO while the numerator is effectively at LO. Therefore, it has been argued that it is better to define the charge

asymmetry in a way that both the numerator and the denominator are at the same order, e.g., using the expansion in *α<sup>s</sup>* (expanded definition). For example, use the LO cross-section in the denominator for the asymmetry calculation at NLO. In such way, uncalculated higher-order corrections should be at about the same level in both the numerator and the denominator. Using these two definitions provide quite different predictions for the asymmetry at NLO which is due to the large change between the LO and NLO cross-section. However, at NNLO, these definitions already provide very similar values, therefore it does not matter much which definition is used.

The initial attempt at predicting the forward–backward asymmetry in the pair production of top quarks was made more than 30 years ago even before the top quark was discovered [11]. The top quark mass used in the calculation was 45 GeV. In this calculation, only contributions from the initial and final state gluon radiation interference were considered, i.e., only the asymmetry in the *t*¯*t*+jet process at LO was calculated. This was not the full NLO QCD correction to the inclusive *t*¯*t* asymmetry. This calculation required the introduction of cuts on the energy and the rapidity of gluons to avoid singularities. The predicted asymmetry was negative: up to about −2% for a given kinematic criteria on heavy quarks and gluons.

After the discovery of the top quark, but well before first measurements became available, the charge asymmetry was studied in a more detail using the full NLO QCD prediction [12,13]. This means including the interference terms of Born and virtual box corrections. It turned out that these contributions to the asymmetry were larger than the initial and final state radiation interference and they were in the opposite direction. The overall prediction for the forward–backward asymmetry thus changed the sign compared to the first partial NLO prediction and was positive. At the Tevatron and <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 1.8 TeV, the asymmetry was predicted to be up to 15% in *qq*¯ <sup>→</sup> *<sup>t</sup>*¯*<sup>t</sup>* process in certain kinematic regions while the integrated forward–backward asymmetry was about 7–8% in the *t*¯*t* rest frame, see Figure 9, and about 4–5% in the *pp*¯ laboratory frame.

**Figure 9.** The differential charge asymmetry (**left**) in the *<sup>t</sup>*¯*<sup>t</sup>* pair production *qq*¯ → *<sup>t</sup>*¯*<sup>t</sup>* for the fixed partonic center-of-mass energy <sup>√</sup>*s*<sup>ˆ</sup> <sup>=</sup> 400 GeV. The integrated charge asymmetry (**right**) for *qq*¯ <sup>→</sup> *<sup>t</sup>*¯*<sup>t</sup>* as a function of <sup>√</sup>*s*<sup>ˆ</sup> [12].

As it was mentioned already, the asymmetry appears first at NLO in the inclusive *t*¯*t* production. This means that using the NLO QCD prediction, the asymmetry is known only at leading order accuracy since numerator has only the leading order contribution. In the *t*¯*t*+jet production, the asymmetry is already present at LO QCD. Therefore, the NLO calculation of such process provides a true NLO prediction of the asymmetry. It was shown that NLO QCD contributions to the *t*¯*t*+jet production provide very large corrections to the *t*¯*t*+jet asymmetry which is then drastically reduced from about –8% at LO to about –2% at NLO for jet *p*<sup>T</sup> > 20 GeV for the Tevatron Run II [14,15]. It was not clear at that time whether a similar shift would not happen for the inclusive *t*¯*t* prediction. Therefore, the NNLO prediction for the inclusive *t*¯*t* process was highly desirable.

It was believed for quite some time that EW corrections are small similarly as EW corrections to the inclusive cross-section are small due to *α<sup>s</sup> α*. It turned out this was not the case. It was found that EW corrections of the order <sup>O</sup>(*α*2) and <sup>O</sup>(*αα*<sup>2</sup> *<sup>s</sup>*) have a surprisingly large effect. In general, about 20% of the enhancement with respect to the NLO QCD prediction is observed when including EW corrections [20,21,24,25]. Similarly, leptonic and dileptonic asymmetries were computed for the Tevatron and for the LHC too [23,24].

Furthermore, the higher-order corrections from the soft gluon QCD resummation have been studied at various accuracies. Initially, the soft gluon corrections at next-to-leading-logarithm (NLL) level were computed [16]. It was found that the asymmetry is stable with respect to these corrections: the inclusive asymmetry changed from 6.7% to 6.6% at the Tevatron Run II. Later on, the next-to-next-to-leading logarithm (NNLL) corrections were computed in [18,19]. Here, the change in the inclusive asymmetry is from (7.4+0.7 <sup>−</sup>0.6)% at NLO to (7.3+1.1 <sup>−</sup>0.7)% in Ref. [19], so again negligible change, while a modest change from 4.0% to 5.2% was found in Ref. [18].

Moreover, further understanding of soft-gluon emissions came from parton shower studies. It was shown that a coherent QCD radiation in the *t*¯*t* production leads to a forward–backward asymmetry that grows more negative with the increasing transverse momentum of the pair [78].

Finally, full NNLO QCD corrections were calculated [30]. These provided large, 27%, increase relative to the NLO QCD prediction for the inclusive asymmetry. The evolution of various calculations is shown in Figure 10. The EW contributions considered here are <sup>O</sup>(*α*<sup>2</sup> *<sup>s</sup> <sup>α</sup>*) and <sup>O</sup>(*α*2). The detailed studies of NNLO QCD predictions for various kinematic distributions have been presented in Ref. [31].

**Figure 10.** Various levels predictions of the inclusive forward–backward asymmetry at the Tevatron compared to the CDF and D0 measurements. Capital letters (NLO, NNLO) correspond to the unexpanded definition, while small letters (nlo, nnlo) to the expanded definition [30].

Later on, the 'complete NLO' corrections were added to the NNLO QCD prediction [32]. The complete NLO contributions include NLO QCD corrections at <sup>O</sup>(*α*<sup>3</sup> *<sup>s</sup>*), the NLO EW at <sup>O</sup>(*α*<sup>2</sup> *<sup>s</sup> α*) as well as contributions at <sup>O</sup>(*αsα*2) and <sup>O</sup>(*α*3) together with LO corrections at <sup>O</sup>(*αsα*) and <sup>O</sup>(*α*2). The comparison of various predictions of the charge asymmetry *A*<sup>C</sup> with the experiments for the LHC at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 8 TeV is shown in Figure 11.

**Figure 11.** Various predictions of the inclusive charge asymmetry for the LHC at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 8 TeV compared to the A Toroidal LHC ApparatuS (ATLAS) and the Compact Muon Solenoid (CMS) experiment measurements. *A*<sup>C</sup> corresponds to the unexpanded definition while *Aex* <sup>C</sup> corresponds to the expanded definition [32].

Similarly as for the inclusive cross-section, the calculation of N3LO soft-gluon contributions together with the inclusion of above NNLO QCD calculation allowed to obtain the aN3LO prediction of *A*FB [29]. The increase in the *A*FB at the Tevatron due to N3LO soft-gluon contributions is about 5% compared to the pure NNLO QCD calculation.

As it was already mentioned above, the independent NNLO QCD prediction for the *t*¯*t* process has become available recently with the MATRIX program which provides fully differential *t*¯*t* predictions. The Δ*y* distribution was obtained and the forward–backward asymmetry for the Tevatron calculated for the following set of parameters: *mt* = 172.5 GeV, *μ<sup>R</sup>* = *μ<sup>F</sup>* = *mt*, NNPDF3.1 NNLO with *<sup>α</sup>s*(*MZ*) = 0.118 parton distribution function [79]. The result is: *<sup>A</sup>t*¯*<sup>t</sup>* <sup>C</sup> = (7.4+0.3 <sup>−</sup>0.8)%. It should be noted that the uncertainty estimated here is only approximate since the MATRIX program provides only minimal and maximal deviations in the cross-section for scale variations for a given bin and the maximal potential difference to estimate the scale uncertainty was used. Nevertheless, the result is in excellent agreement with the above mentioned pure NNLO QCD prediction (7.49+0.49 <sup>−</sup>0.86)% from Ref. [30]. Similarly, the charge asymmetry at the LHC <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 7 TeV was calculated *<sup>A</sup>*<sup>C</sup> = (0.95 <sup>±</sup> 0.08)%.

The usual theoretical predictions set a renormalization and factorization scale to some value typical for the process, e.g., for the *t*¯*t* process it is typically a top quark mass. The uncertainty is then evaluated by changing the renormalization and factorization scale by a factor of two which is essentially just a consensus within the theoretical community, but has no deep foundation within the theory. The alternative calculation of the charge asymmetry is based on the Principle of Maximum Conformality (PMC) scale-setting approach where the renormalization scale is automatically determined and the corresponding uncertainty is essentially eliminated [80,81]. The PMC predictions were computed at NLO QCD with partial NNLO terms and also including NLO EW corrections (aNNLO + NLO EW). They were computed for the Tevatron and for the LHC [26–28]. The large difference between the PMC prediction and the conventional scale-setting NNLO prediction is seen for the Tevatron *A*FB(*mt*¯*<sup>t</sup>* > 450 GeV) where the PMC predicts *A*FB(*mt*¯*<sup>t</sup>* > 450 GeV) = 29.9% which is much larger compared to the NNLO prediction *A*FB(*mt*¯*<sup>t</sup>* > 450 GeV) ≈ 11% [31]. It should be noted that the PMC method has a residual scale dependence due to the unknown perturbative terms which could be relatively large in the *t*¯*t* pair production [82] while the updates of the PMC method try to overcome this limitation [83,84].

When the direct theoretical prediction is not available, e.g., in a specific fiducial phase space, the charge asymmetry predictions are calculated using Monte Carlo (MC) programs or generators of particle collisions. Such programs have typically only NLO QCD corrections implemented, e.g., MC@NLO [85], POWHEG [86], or MCFM [87].

#### Summary of SM Predictions

The SM inclusive predictions at various orders, in various frames, and using different definitions are shown in Tables 2 and 3. At the Tevatron, the predictions have increased significantly by almost factor of two when going from NLO in the laboratory frame to NNLO QCD + NLO EW in the *t*¯*t* frame. At the LHC, the charge asymmetry decreases with the increase of the energy of interactions due to the increase of the symmetric *gg* production process fraction.

**Table 2.** The summary of Standard Model (SM) predictions for *t*¯*t* and leptonic forward–backward asymmetries at the Tevatron at various levels of the perturbation theory. Some predictions are in the laboratory frame (lab) while some are in the *t*¯*t* rest frame (*t*¯*t*). Some of the predictions are using the unexpanded definition while the others use the expanded (ex) definition.


**Table 3.** The summary of SM predictions for charge asymmetry at various levels of perturbation theory at the LHC for different center-of-mass energies. All of these predictions are in the laboratory frame. Some of the predictions are using the unexpanded definition while the others use the expanded (ex) definition.


It was realized already very early in the initial predictions [11–13,60] that the charge asymmetry depends on the initial/final state gluon radiation due to the interference of these diagrams contributing to the asymmetry. It is therefore expected the asymmetry depends on *p*<sup>T</sup> of the final-state jet which is related to *p*T,t¯t (the size of *p*<sup>T</sup> of the jet will be the same as *p*<sup>T</sup> of *t*¯*t*, the direction will be opposite in the transverse plane). The contribution from such interference to the asymmetry is negative, so the larger *p*<sup>T</sup> the more negative asymmetry is expected. The variable |*yt*¯*t*| is sensitive to the ratio of the contributions from the *qq*¯ and *gg* initial states. The charge-symmetric *gg* initial state produces more central *t*¯*t* events while *qq*¯ contributes more in the forward direction. Therefore, it is expected the asymmetry will rise with the increasing value of Δ*y*. The charge asymmetry is expected to also rise for the *mt*¯*<sup>t</sup>* variable since the *qq*¯ initial state is enhanced for larger values of this variable. Finally, the charge asymmetry is expected to rise steeply for high boost of the *t*¯*t* system along the longitudinal axis [33]. It is due to the much higher average momentum fractions for quarks than for antiquarks in *pp* collisions. Requiring the high boost of *t*¯*t* system thus increases the *qq*¯ fraction and consequently also the charge asymmetry. As a consequence, the predictions of differential asymmetries as a function of the above mentioned variables were calculated (and also measured), see Figure 12.

**Figure 12.** The NNLO predictions for the differential charge asymmetry as a function of *mt*¯*<sup>t</sup>* in (**a**), *yt*¯*<sup>t</sup>* in (**b**), *<sup>p</sup>*T,t¯t in (**c**), and *<sup>β</sup>z*,*t*¯*<sup>t</sup>* in (**d**) at the LHC at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 8 TeV [32].

#### *3.2. BSM Models*

After CDF claimed an evidence for 3 SD deviation from the SM prediction at that time in one of their *A*FB measurements (for large *mt*¯*<sup>t</sup>* > 450 GeV), there have been lots of new physics BSM models proposed to explain such measurement, see Refs. [33–39,45,46].

The BSM models also give us a strong reason to measure all different combinations of asymmetries, e.g., *A*C, *A*FB, *A*- FB. While in the SM, the *A*FB and *A*<sup>C</sup> have the same underlying cause and there is a specific relation between *A*FB and *A*C, this relation can be largely changed in BSM models. It was shown that using for example the axigluon model, it is possible to obtain a negative *A*<sup>C</sup> at the LHC for positive *A*FB asymmetry at the Tevatron [36,37]. The correlation between *A*FB and *A*- FB is given in the SM. It is due to the fact that there is about zero top quark polarization in *t*¯*t* events, i.e., there is

an equal number of positive and negative helicity top quarks produced. Models with different top quark polarization could change the *A*FB and *A*- FB relations in both directions. This was studied in Refs. [46,89] for axigluon and *W* models, where it was shown that for the same *A*FB there was different *A*- FB predicted.

There are a few models which can change the charge asymmetry [34,35]:


The diagrams showing potential contributions from BSM models are shown in Figure 13. The potential values of the charge asymmetry at the LHC and the forward–backward asymmetry at the Tevatron for the above models with various parameters are shown in Figure 14.

**Figure 13.** The interference of various beyond the Standard Model (BSM) particles which contribute to the charge asymmetry with the gluons [90].

**Figure 14.** The measured inclusive charge asymmetry *<sup>A</sup>*<sup>C</sup> at the LHC at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 8 TeV (horizontal line) plotted against the forward–backward asymmetry *A*FB (vertical lines) at the Tevatron. The data are compared with the SM prediction at NNLO QCD + NLO electroweak (EW) and predictions incorporating various potential BSM contributions: a *W* boson, a heavy axigluon (*Gμ*), a scalar isodoublet (f), a color triplet scalar (*ω*4), and a color sextet scalar (Ω4) [91].

#### **4. Experimental Measurements**

The measurements performed at the Tevatron and at the LHC are reviewed in this section. At the Tevatron, being the *pp*¯ collider, it was possible to measure forward–backward asymmetry. At the LHC, being *pp* collider, the edge–central charge asymmetry *A*<sup>C</sup> has been measured.

All *A*FB measurements at the Tevatron were performed in Run II data taking period. Two general-purpose experiments were collecting the data: CDF [92] and D0 [93]. The charge asymmetry measurements at the LHC have been performed in both the Run 1 and Run 2 data taking periods by both general-purpose experiments, ATLAS [94] and the Compact Muon Solenoid (CMS) [95].

At the LHC, there are more experiments running. The LHCb experiment, designed to study *b*-quark interactions, is one of them and it also observed the top quark [96]. However, up to now, only cross-section measurements of the top quark have been performed at the LHCb and no *t*¯*t* charge asymmetry related studies.

As it was mentioned above, it is important to measure all possible combinations of the asymmetries, i.e., measure both *A*FB at the Tevatron and *A*<sup>C</sup> at the LHC and also both *t*¯*t* and leptonic asymmetries, since the relation between them is model dependent. Experiments at the Tevatron and at the LHC therefore performed full set of these measurements where possible, in both -+jets and dilepton channels.

Most of the measurements follow this typical procedure. First, selection criteria are applied to select the sample which is enhanced in *t*¯*t* events. The backgrounds are estimated using MC or data driven methods. For the *t*¯*t* asymmetry, the kinematic reconstruction of top and antitop quark 4-momenta is performed using the measured top quark decay products. For lepton-based asymmetries, this is not needed. The kinematic variable which is used to define the asymmetry is calculated. At the Tevatron, this is mostly Δ*y* (for the top quark asymmetry), *η*- (for the lepton-based asymmetry), and Δ*η*-- (for the dileptonic asymmetry). At the LHC, this is mostly Δ|*y*| (for the top quark asymmetry) and Δ|*η*| (for the dileptonic asymmetry). The kinematic distribution of the observable of interest is plotted for data, see e.g., Figure 15a. This corresponds to 'reco level' before the background subtraction. Subsequently, the expected distribution for the background process is subtracted from the distribution in data, corresponding to 'reco level' after background subtraction, see e.g., Figure 15b. The resulting distribution is assumed to correspond to the *t*¯*t* distribution after the reconstruction and event selection.

The 'reco level' has a disadvantage that it includes detector resolution and acceptance effects, so such results can be only compared to MC generator predictions which pass detector simulation and reconstruction and can not be directly compared to results from other experiments nor to the direct theoretical predictions. In order to be able to compare to the latest, most precise predictions, such distribution needs to be corrected for detector resolution effects and for the event selection acceptance and inefficiency. This correction is typically performed by the procedure of unfolding. Removing the detector effects brings the distributions to the 'parton' or 'particle' level, see e.g., Figure 15c. When extrapolating to the full phase space and to the level of top quarks, it is called a parton level. This allows to compare experimental results directly to theoretical calculations or to other experiments. Most of the results from the experiments are at parton level and it will be not mentioned unless it is otherwise. When unfolding to the level of final state stable particles and typically requiring some fiducial cuts on final state particles such as *p*Tor |*η*| which are similar to event selection criteria, it is called a particle level. The advantage of the 'particle fiducial level' is that there is a much smaller degree of the extrapolation to an unmeasured phase-space compared to the parton level and consequently modeling uncertainties are typically smaller. Moreover, it is less ambiguous than the parton level, since the 'top quark' definition can differ between different MC generators which are necessary when performing the unfolding. However, not many predictions are available at such level.

**Figure 15.** The Δ*y* distribution at the reco level in (**a**), at the reco level after the background subtraction in (**b**), and at the parton level in (**c**), compared to the prediction [97].

The asymmetry can be calculated at each level by counting the events at positive and negative side of the *x*-axis with the statistical uncertainty properly calculated taking into account possible correlations between the bins which arise due to e.g., systematic uncertainties of the unfolding procedure. Since the asymmetry is expected to depend on a few different observables within the SM and even differently in BSM models, the experiments also performed lots of differential measurements of the asymmetry mostly as a function of *mt*¯*t*,*p*T,t¯t, Δ*yt*¯*t*, and *βz*,*t*¯*t*.

There are also some measurements which use different methods to obtain the asymmetry and these will be mentioned later when describing a specific measurement.

#### *4.1. Forward–Backward Asymmetry Measurements at the Tevatron*

The CDF and D0 experiments started to perform measurements of *A*FB from the beginning of Run II. The initial measurements at the Tevatron were performed in the -+jets channel with about 10–20% of eventual Run II statistics [40,41]. The next set of measurements were performed with about half of full Run II statistics by both CDF [42] and D0 [43,44]. These already included also the measurement of leptonic asymmetries.

Both CDF and D0 collaborations performed the full set of measurements in both -+jets and dilepton channels with full Run II statistics [72,97–103]. Moreover, there was performed the combination of the CDF and D0 measurements [49].

All these measurements will be described in the following.

#### 4.1.1. Initial Measurements

The first measurement related to the charge asymmetry in the top quark pair production was performed by the D0 experiment using 0.9 fb−<sup>1</sup> of the integrated luminosity [40]. It was performed only at reco level in the *t*¯*t* rest frame using Δ*y*. The inclusive asymmetry was not measured, only the *A*FB as a function of the number of reconstructed jets. The measured values are listed in Table 4 together with predictions from MC@NLO generator where events passed the full simulation and reconstruction as data.

**Table 4.** The MC@NLO predictions and measured forward–backward asymmetries at reco level as a function of the number of jets in the D0 measurement using 0.9 fb−<sup>1</sup> [40].


In the CDF measurement using 1.9 fb−<sup>1</sup> [41], the acceptance and reconstruction effects were already corrected for and parton-level asymmetries were measured. The asymmetry was measured using two observables, −*q*- · cos *<sup>θ</sup><sup>p</sup>* and <sup>Δ</sup>*y*, which measured the asymmetry in *pp*¯ and *<sup>t</sup>*¯*<sup>t</sup>* rest frame, respectively. Measured distributions for these variables are shown in Figure 16. Inclusive asymmetries

**Figure 16.** The production angle cos *θ* (**left**) and Δ*y* (**right**) distribution at the reco level for the *A*FB measurements in the *pp*¯ and *t*¯*t* frame, respectively. The solid line is the prediction for *t*¯*t* with MC@NLO generator and *σt*¯*<sup>t</sup>* = 8.2 pb, plus the expected non-*t*¯*t* backgrounds. The dashed curve shows the prediction when *t*¯*t* is reweighted according to the form 1 + *A*FB cos *α* using measured values of *A*FB [41].

To summarize these initial Tevatron measurements: only the *t*¯*t* forward–backward asymmetry was measured in the -+jets channel only. Both experiments showed larger than at that time and even presently predicted asymmetries although they were really limited by a small data sample and so still consistent with the predictions. The CDF results pointed to the expected frame dependence. The measurement as a function of the number of jets in both CDF and D0 pointed to the expected trend of decreasing asymmetry with the increase in the number of jets.

#### 4.1.2. Measurements with Half of Run II Statistics

Both CDF and D0 performed the measurement of the *t*¯*t* asymmetry with about a half of the full Run II statistics. CDF performed only the *t*¯*t* asymmetry measurement in the -+jets channel while D0 performed the measurement of *At*¯*<sup>t</sup>* FB, *<sup>A</sup>*- FB, and *<sup>A</sup>*-- FB in the -+jets and dilepton channel.

The CDF measurement [42] was performed with 5.3 fb−<sup>1</sup> in both laboratory and *t*¯*t* rest frames using *yt* and Δ*y*, respectively. The distributions at the reco level are shown in Figure 17 where data are compared to MC@NLO generator predictions. The inclusive measurements at different levels are summarized in Table 5. The measured *App*¯ FB asymmetry exceeds the MC@NLO prediction by more than two standard deviations at all correction levels. The *At*¯*<sup>t</sup>* FB asymmetries are similar in a magnitude to the

*App*¯ FB but they are less significant because of the larger relative uncertainties. The *<sup>A</sup>t*¯*<sup>t</sup>* FB is also measured as a function of Δ*y* and *mt*¯*<sup>t</sup>* in two bins, see Figure 18. At high values of Δ*y* and *mt*¯*t*, the asymmetries are higher than predictions available at that time. While the difference was less than 2 SD for high Δ*y*, it was about 3.4 SD for high *mt*¯*<sup>t</sup>* bin ((47.5 ± 10.1(stat.) ± 4.9(syst.))% = (47.5 ± 11.4)% in data, while (8.8 ± 1.3)% for NLO QCD in MCFM). It should be noted that in this measurement, CDF also tested the CP-invariance assumption by calculating charge separated asymmetries. The asymmetries in both laboratory and *t*¯*t* rest frame are equal and opposite within uncertainties, as expected.

**Figure 17.** The Δ*y* (**left**) and *yt* (**right**) distribution at the reco level corresponding to the CDF measurement performed in the -+jets channel using 5.3 fb−<sup>1</sup> [42].

**Table 5.** The summary of inclusive asymmetries in *t*¯*t* and *pp*¯ rest frames at the reco level with and without including the background, and at the parton level corresponding to the CDF measurement using 5.3 fb−1. Uncertainties include statistical, systematic, and theoretical uncertainties [42].


**Figure 18.** Parton level asymmetries as a function of Δ*y* (**left**) and *mt*¯*<sup>t</sup>* (**right**) compared to the SM prediction of MCFM. The negative going uncertainty for Δ*y* < 1.0 is suppressed [42].

D0 performed a similar measurement in the -+jets channel using 5.4 fb−<sup>1</sup> [43]. Both *t*¯*t* and leptonic asymmetries are measured using Δ*y* and *q* · *y* distributions, respectively, see Figure 19. The measured inclusive asymmetry is *At*¯*<sup>t</sup>* FB = (19.6 <sup>±</sup> 6.5)% and *<sup>A</sup>*- FB = (15.2 ± 4.0)% which disagree with the NLO QCD prediction from MC@NLO (*A*FB = (5.0 <sup>±</sup> 0.1)% and *<sup>A</sup>*- FB = (2.1 ± 0.1)%) by about 2.4 SD and 3.2 SD, respectively. The differential *t*¯*t* asymmetry measured as a function of *mt*¯*<sup>t</sup>* and Δ*y* only at the reco level is summarized in Table 6.

**Figure 19.** The reconstructed Δ*y* (**left**) and the charge-signed lepton rapidity (**right**) corresponding to the D0 measurement in the -+jets channel [43].


**Table 6.** The reco level *At*¯*<sup>t</sup>* FB by subsample in the D0 -+jets measurement using 5.4 fb−<sup>1</sup> [43].

In the dilepton channel, the leptonic and dileptonic asymmetry is measured by D0 using 5.4 fb−<sup>1</sup> [44]. The leptonic asymmetry is measured to be *A*- FB = (5.8 ± 5.1(stat.) ± 1.3(syst.))% while the dileptonic asymmetry is *A*-- FB = (5.3 ± 7.9(stat.) ± 2.9(syst.))%. The combination with the result in the -+jets channel yields *A*- FB = (11.8 ± 3.2)%.

To summarize mid-term measurements: these measurements were performed in both -+jet and dilepton channels and both *t*¯*t* and leptonic based asymmetries were measured. Moreover, the *t*¯*t* asymmetry is measured as a function of *mt*¯*<sup>t</sup>* and Δ*y*. The inclusive *t*¯*t* asymmetry is observed by both CDF and D0 to be larger than the predictions available at that time by 1.4 − 2.2 SD depending on the experiment, the frame, and the prediction. The *t*¯*t* asymmetry at high *yt*¯*<sup>t</sup>* and *mt*¯*<sup>t</sup>* region was measured by CDF to be larger than predicted by up to 3.4 SD at high *mt*¯*<sup>t</sup>* values while D0 did not see any significant disagreement. The leptonic asymmetry was measured only by D0 and was again higher than the prediction available at that time, mostly for the -+jets channel where the deviation is in between 2.6 and 3.3 SD depending on the prediction ( (2.1 ± 0.1)% in MC@NLO and (4.7 ± 0.1)% in MC@NLO + NLO EW corrections, see [43,44]). Moreover, *A*-- FB was also measured to be consistent with the prediction, but with significant uncertainties. The dominant uncertainty in these measurements was still the statistical uncertainty. Although the amount of data analyzed increased by factor 3–6, the systematic uncertainties were largely improved by about factor of two. It was clear that the progress on both the experimental side, to improve the statistical precision, and on the theory side, to make more reliable uncertainties, was needed.

#### 4.1.3. Measurements with Full Statistics

The measurements with the full Tevatron Run II statistics were performed by CDF and D0 in both -+jets and dilepton channels and for the whole set of *At*¯*<sup>t</sup>* FB, *<sup>A</sup>*- FB, and *<sup>A</sup>*-- FB asymmetries.

CDF measured *t*¯*t* asymmetry in the -+jets channel using 9.4 fb−<sup>1</sup> [97] and in the dilepton channel using 9.1 fb−<sup>1</sup> [72]. The Δ*y* distribution in the -+jets channel is plotted in Figure 15 where also the inclusive asymmetry measurements are summarized at various levels. The measured inclusive asymmetry in the dilepton channel is *At*¯*<sup>t</sup>* FB = (12 ± 11(stat.) ± 7(syst.))% = (12 ± 13)%. The asymmetry is also measured as a function of |Δ*y*|, and *mt*¯*<sup>t</sup>* in the -+jets channel, see Figure 20. The dependencies on these kinematic variables are linear and the slope is higher by 2.8 SD and 2.4 SD than expected from the NLO QCD prediction by POWHEG. In the dilepton channel, only the differential asymmetry as a function of Δ*y* is measured in two bins (*At*¯*<sup>t</sup>* FB(|Δ*y* < 0.5|)=(12 ± 3.9)% and *At*¯*<sup>t</sup>* FB(|Δ*y* > 0.5|)=(13 ± 17)%).

**Figure 20.** The forward–backward asymmetry as a function of |Δ*y*| (**left**) and *mt*¯*<sup>t</sup>* (**right**) with a best-fit line superimposed. The shaded region represents the theoretical uncertainty on the slope of the prediction [97].

D0 measured the *t*¯*t* asymmetry in the -+jets channel using full Run II statistics of 9.7 fb−<sup>1</sup> [102]. The inclusive asymmetry is measured *At*¯*<sup>t</sup>* FB = (10.6 ± 3.0)%. The dependence of the asymmetry on the |Δ*y*| and *mt*¯*<sup>t</sup>* was also measured, see Figure 21. The linear fit is performed to these dependencies and the slope is measured to be 15.4 <sup>±</sup> 4.3 and 3.9 <sup>±</sup> 4.4 <sup>×</sup> <sup>10</sup>−4, respectively.

**Figure 21.** The *A*FB dependence on |Δ*y*| (**left**) and on *mt*¯*<sup>t</sup>* (**right**). The dashed line shows the fit to the data with the dotted lines indicating the fit uncertainty. The *x* coordinate of each datum point is the observed average of |Δ*y*| in the corresponding bin [102].

In the dilepton channel, D0 measured simultaneously *At*¯*<sup>t</sup>* FB and the top quark polarization using 9.7 fb−<sup>1</sup> [103]. If the top quark polarization is fixed to its expected SM value, the measured value of asymmetry is *At*¯*<sup>t</sup>* FB = (17.5 ± 5.6(stat.) ± 3.1(syst.))%.

In order to study the source of unexpectedly large forward–backward asymmetry in more detail, CDF measured the cross-section as a function of the top quark production angle d*σ*/d cos *θ<sup>t</sup>* [98]. The shape of such differential distribution is characterized by Legender polynomials and the Legender moments *<sup>a</sup>*<sup>1</sup> − *<sup>a</sup>*<sup>8</sup> are measured. For the *qq*¯ → *<sup>t</sup>*¯*<sup>t</sup>* process at LO, it is expected that there are non-zero *<sup>a</sup>*<sup>0</sup> and *<sup>a</sup>*<sup>2</sup> moments. *gg* → *<sup>t</sup>*¯*<sup>t</sup>* is expected to add only small contributions to all even-degree Legender moments. The measured Legender moments *a*<sup>1</sup> − *a*<sup>4</sup> are shown in Figure 22, the remaining ones are consistent with zero within large uncertainties. A good agreement within the uncertainties with the NLO SM prediction is observed for the moments *a*<sup>2</sup> − *a*8, but *a*<sup>1</sup> showed an excess with respect to the prediction: *<sup>a</sup>*<sup>1</sup> <sup>=</sup> 0.40 <sup>±</sup> 0.12 vs. NLO SM: 0.15+0.07 <sup>−</sup>0.03. It means the excess was observed in the differential cross-section in the term linear in cos *θ*.

**Figure 22.** Measured Legendre moments *a*<sup>1</sup> − *a*4, with various theory predictions overlaid [98].

The leptonic measurements have been performed in both -+jets and dilepton channels by both CDF and D0 experiments. In the -+jets channel, the leptonic asymmetry is measured by CDF using 9.4 fb−<sup>1</sup> [99] while D0 uses 9.7 fb−<sup>1</sup> [104]. Both CDF and D0 measure the single-leptonic asymmetry using the charge-weighted rapidity *qy*-. The CDF measurement separates the rapidity distribution into symmetric part, which is largely independent on the model, and the antisymmetric part which

encapsulates the possible variation between the modes. The inclusive asymmetry is measured to be *A*- FB = (10.5 ± 2.4(stat.) +2.2 <sup>−</sup>1.7(syst.))% = (10.5+3.2 <sup>−</sup>2.9)% which is 2.3 SD away from the NLO QCD + NLO EW prediction ((3.8 ± 0.3)%). D0 measured the asymmetry in restricted region of |*y*-| < 1.5: *A*- FB = (4.2+2.9 <sup>−</sup>3.0)%. The asymmetry is measured also as a function of *<sup>p</sup>*<sup>T</sup> of lepton and <sup>|</sup>*yt*|.

In the dilepton channel, both CDF using 9.1 fb−<sup>1</sup> [100] and D0 using 9.7 fb−<sup>1</sup> [101] measure both single-lepton and dilepton asymmetries using *qη* and Δ*η*-, respectively. In the CDF measurement, similarly to previous measurement in the -+jets channel, the pseudorapidity distributions are splitted into symmetric and antisymmetric parts. The results are *A*- FB = (7.2 ± 5.2(stat.) ± 3.0(syst.))% = (7.2 ± 6.0)% and *A*-- FB = (7.6 ± 7.2(stat.) ± 3.9(syst.))% = (7.6 ± 8.2)%. D0 measured the inclusive asymmetries using the distributions shown in Figure 23. The measured values are *A*- FB = (4.4 ± 3.7(stat.) ± 1.1(syst.))% and *A*-- FB = (12.3 ± 5.4(stat.) ± 1.5(syst.))%. The dependence of asymmetries on |*qη*-| and |Δ*η*-| is also measured but only in the fiducial parton level phase-space (leptons must have |*η*| < 2 and |Δ*η*| < 2.4).

**Figure 23.** The reco level distribution of *q* · *η* (**left**) and Δ*η* = *η*-<sup>+</sup> − *η*-<sup>−</sup> (**right**). The error bars indicate the statistical uncertainty on the data [101].

#### 4.1.4. Full Dataset Combinations

Both CDF and D0 performed individual combinations of their *t*¯*t* and single-lepton asymmetry measurements from both -+jets and dilepton channels using the full Run II data statistics of the actual experiment [72,100,103,104]. Moreover, they performed together the Tevatron combinations of all their results from both channels [49].

CDF combined inclusive *t*¯*t* asymmetry is *At*¯*<sup>t</sup>* FB = (16.0 ± 4.5(stat. + syst.))% while the D0 combination is *At*¯*<sup>t</sup>* FB = (11.8 ± 2.5(stat.) ± 1.3(syst.))%. The combined single-leptonic asymmetry at CDF is *A*- FB = (9.0+2.8 <sup>−</sup>2.6(stat. <sup>+</sup> syst.))% while the D0 combination is (4.7 <sup>±</sup> 2.3(stat.) <sup>±</sup> 1.5(syst.))%. For the differential *At*¯*<sup>t</sup>* FB as a function of Δ*y*, rather then combining the data, the combined fit of the slope to both CDF -+jets and dilepton data was performed. The result is *α* = 0.227 ± 0.057 which is 2 SD larger than the NNLO QCD prediction of 0.114+0.006 <sup>−</sup>0.012 [31].

The Tevatron combination of the CDF and D0 inclusive *t*¯*t* forward–backward asymmetry is *At*¯*t* FB = (12.8 ± 2.1(stat.) ± 1.14(syst.))% = (12.8 ± 2.5)%. The precision of the combination is such that the *A*FB is measured with a significance of 5 SD from zero asymmetry. The combined inclusive single-lepton asymmetry is *A*- FB = (7.3 ± 1.6(stat.) ± 1.12(syst.))%, while the combined dileptonic asymmetry is *A*-- FB = (10.8 ± 4.3(stat.) ± 1.6(syst.))%. All inclusive combined measurements together with the individual measurements used as the inputs to the combination and the theoretical predictions are summarized in Figure 24.

Differential measurements of *At*¯*<sup>t</sup>* FB as a function of *mt*¯*<sup>t</sup>* were measured only in the -+jets channel and combined together, see Figure 25. For the combination, the data are fitted by a linear function.

The obtained slope of 9.71 <sup>±</sup> 3.28 <sup>×</sup> <sup>10</sup>−<sup>4</sup> GeV−<sup>1</sup> is compatible with NNLO QCD + NLO EW prediction of 5.11+0.42 <sup>−</sup>0.64 <sup>×</sup> <sup>10</sup>−<sup>4</sup> GeV−<sup>1</sup> at the level of 1.3 SD. The differential *<sup>t</sup>*¯*<sup>t</sup>* asymmetry as a function of <sup>Δ</sup>*<sup>y</sup>* is available from CDF for both -+jets and dilepton channels, and from D0 for the -+jets channel, see Figure 25. Since the choice of the binning differs for these measurements, the simultaneous fit to a linear function with zero offset was performed for all available measurements employing the correlations. The slope parameter is measured to be 0.187±0.038 which is compatible with NNLO QCD + NLO EW prediction of 0.129+0.006 <sup>−</sup>0.012 at the level of 1.5 SD. The individual CDF and D0 measurements of *A*- FB as a function of |*qη*-<sup>|</sup> together with the individual measurements of *<sup>A</sup>*-- FB as a function of |Δ*η*| are shown in Figure 26 without any quantitative comparison to the prediction. Looking at the plots, there can not be seen any striking disagreement with NLO QCD + NLO EW prediction.

**Figure 24.** Summary of inclusive forward–backward asymmetries used in the Tevatron combination together with their combination [49].

**Figure 25.** The dependence of *A*FB as a function of *mt*¯*<sup>t</sup>* (**left**). The individual measurements and the Tevatron combination are shown together with the NNLO QCD + NLO EW prediction. The dependence of *A*FB as a function of Δ*y* (**right**). Here, the individual measurements are shown together with the simultaneous fit and the NNLO QCD + NLO EW prediction [49].

**Figure 26.** The individual CDF and D0 measurements of *A*- FB as a function of |*qη*-<sup>|</sup> (**left**) and of *<sup>A</sup>*-- FB as a function of |Δ*η*| (**right**) together with the NLO QCD prediction [49].

#### 4.1.5. Summary and Discussion of Tevatron Measurements

All inclusive Tevatron measurements of forward–backward asymmetry at the parton level in the *t*¯*t* rest frame are summarized in Table 7. These should be compared to the latest NNLO QCD + EW prediction of *<sup>A</sup>*FB = (9.5 <sup>±</sup> 0.7)% [30] and the aN3LO QCD + EW prediction of *<sup>A</sup>*FB = (10.0 <sup>±</sup> 0.6)% [29], while for the leptonic and dileptonic asymmetries only NLO QCD + EW predictions exist: *A*- FB = (3.8 ± 0.3)% and *A*-- FB = (4.8 ± 0.4)% [24].

The inclusive *At*¯*<sup>t</sup>* FB are all consistent between them for different channels and experiments, and are consistent with both NNLO QCD + EW and aN3LO QCD + EW predictions. The maximum deviation is 1.6 SD for D0 -+jets measurement using 5.4 fb−<sup>1</sup> while the final Tevatron combination is 1.3 SD higher when compared to NNLO QCD + EW prediction. Similar conclusions hold also for the inclusive dileptonic asymmetry *A*-- FB with the maximum deviation at 1.3 SD in the D0 dileptonic measurement using 9.7 fb−1. However, it should be noted that all measurements are consistently higher than both NNLO QCD + EW and aN3LO QCD + EW predictions. For the inclusive leptonic asymmetry *A*- FB again all measurements are higher than the prediction. The biggest deviation is for the D0 measurement in the -+jets channel using 5.4 fb−<sup>1</sup> at the level of 2.8 SD, while this has been lowered significantly with the full Run II statistics. The final Tevatron combination is only about 1.7 SD higher than prediction. It should be also noted that the prediction here is only at NLO QCD + EW level, so with the potential NNLO it is expected the deviation will be even lower. The dominant uncertainty in inclusive measurements is the statistical uncertainty.


**Table 7.** Summary of Tevatron measurements of inclusive forward–backward asymmetries. For a given measurement, if there is just one uncertainty, it is combined statistical and systematic uncertainty. If there are two uncertainties, the first one is always statistical and the second one is systematic uncertainty.

The forward–backward asymmetry differential measurements at the Tevatron were performed as a function of *mt*¯*<sup>t</sup>* and rapidity related observable |Δ*yt*¯*t*|, |*y*-|, and Δ*η*--. The *mt*¯*<sup>t</sup>* and |Δ*yt*¯*t*| dependencies are a bit stronger than expected but the agreement is within 2 SD.

#### *4.2. LHC Measurements*

Both ATLAS and CMS experiments started to perform the measurements of the charge asymmetry from the beginning of Run 1 at the energy <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 7 TeV. Initially, only the measurements in the -+jets channel and with a small luminosity of about 1 fb−<sup>1</sup> were performed by ATLAS [65] and CMS [69].

Next measurements at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 7 TeV were performed with the full statistics of 2011 year (≈<sup>5</sup> fb−1) in both -+jets and dilepton channels by both ATLAS [105,106] and CMS collaborations [107,108]. Afterwards, the measurements with about four times larger statistics (full 2012 year, <sup>≈</sup><sup>20</sup> fb−1) at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 8 TeV were again performed in both dilepton and -+jets channels and by both ATLAS [109–111] and CMS [112–114] collaborations.

In Run 2 at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 13 TeV, CMS performed the measurement in both -+jets [115] and dilepton [116] channels using partial dataset (2015+2016 years, <sup>≈</sup><sup>36</sup> fb−1). On the other hand, ATLAS performed the measurement in the -<sup>+</sup>jets channel with the full Run 2 statistics (years 2015–2018, <sup>≈</sup><sup>140</sup> fb−1) [50]. However, it is for now only the preliminary measurement.

Moreover, ATLAS and CMS combined their measurements from Run 1 at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 7 TeV and <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 8 TeV [91]. In the following, all these measurements will be briefly described.
