Top-quark charge asymmetry and polarization in $t\bar{t}W^\pm$ production at the LHC

We study the charge asymmetry between the $t$ and $\bar t$ quark at the LHC, when they are produced in association with a $W$ boson. Though sizably reducing the cross section with respect to the inclusive production, requiring a $W$ boson in the final state has two important implications. First, at leading order in QCD, $t \bar t W^{\pm}$ production can only occur via $q \bar q$ annihilation. As a result, the asymmetry between the $t$ and $\bar t$ generated at NLO in QCD is significantly larger than that of inclusive $t \bar t$ production, which is dominated by gluon fusion. Second, the top quarks tend to inherit the polarization of the initial-state quarks as induced by the $W$-boson emission. Hence, the decay products of the top quarks display a sizable asymmetry already at the leading order in QCD. We study the relevant distributions and their uncertainties in the standard model, compare them to those obtained in a simple axigluon model and discuss prospects for measurements at the LHC and beyond.


Introduction
The charge asymmetry in tt production at pp colliders is defined by the quantity: where t η = |η t | − |η¯t|. Quantum chromodynamics (QCD) predicts that radiative corrections to the leading-order (LO) tt production process induce a non-vanishing A t c , implying that top quarks are produced with a rapidity distribution wider than anti-top quarks. At next-to-leading order (NLO), this effect was first calculated in Refs. [1,2]. The interest in this asymmetry stems from measurements of the corresponding forward-backward asymmetry ( A FB ) performed in pp collisions at the Tevatron by the CDF and D∅ Collaborations [3][4][5][6][7]. These measurements point to a departure from the SM predictions [1,2,[8][9][10][11][12][13], giving rise to a large literature of possible interpretations based on physics beyond the Standard Model (BSM) (for a recent review, see [14]).
Unfortunately, the charge asymmetry predicted by the Standard Model (SM) at the LHC is much smaller than A FB at the Tevatron. * Corresponding author.
The main reason is that both A FB and A c are induced only by the fraction of tt final states generated by qq collisions, which at the LHC represent only ∼ 15% of the total rate, compared to ∼ 85% at the Tevatron. The smallness of the effect makes it difficult at the LHC to reach the sensitivity required to measure A t c , and to probe the possible existence of BSM contributions, unless the BSM departures from the SM prediction were rather large.
The measurements of top charge asymmetry performed so far by the ATLAS and CMS Collaborations [15][16][17][18][19], have reached uncertainties at the level of δ A t c ∼ 0.01, which is of the order of the SM value of A t c . For example, in their latest publications relative to data taken at √ s = 7 TeV, ATLAS [15] and CMS [19] report the following results 1 : ATLAS: A t c, y = 0.006 ± 0.010 stat+syst , CMS: A t c, y = −0.010 ± 0.017 stat ± 0.008 syst ,
(4) 1 The LHC measurements are reported in terms of rapidity differences t y = |y t | − |y¯t |, and we shall refer to these asymmetries as A c, y . Charge asymmetries based on rapidity differ by about 10-20% from those based on pseudorapidity, but have otherwise the same features. Here the leptonic asymmetries A c are defined by replacing in Eq. (1) t η with η = |η + | − |η − |. In the following we shall also consider A b c , defined by b η = |η b | − |ηb|.  A combination of the ATLAS and CMS results has also been performed by the Top LHC Working Group [20] A t c, y = 0.005 ± 0.007 ± 0.006.
The SM result used as a comparison in the experimental papers, including both QCD and electroweak (EW) radiative corrections at the one-loop level, is obtained from Ref. [13]: A c (7 TeV) = 0.0070 ± 0.0003.
These values 2 result from using the LO total cross sections in the denominator of Eq. (1). This is justified by the fact that, at the one-loop level, the asymmetry is an LO effect. Using the NLO total cross section, which is ∼ 50% larger than the LO one, the calculated asymmetries would be reduced to ∼ 2/3 of the above values. We believe that, in absence of a complete NLO calculation of A t c , the difference between the use of LO and NLO cross sections in the denominator of Eq. (1) should be included in the estimate of the overall theoretical uncertainty. Should the true SM value of A t c end up being closer to the smaller values obtained using the NLO cross sections (e.g. A t c ∼ 0.004 at √ s = 14 TeV), a robust and accurate measurement will be a hard experimental challenge. Alternative observables are known to enhance the size of the asymmetry. For example, Ref. [13] estimates that the asymmetry can increase by a factor of 2-3 placing proper cuts on the invariant mass of the tt system. The smaller rates due to the extra cuts will be compensated by the much larger statistics to become available at 13-14 TeV. But the theoretical systematics will, by and large, remain correlated with those of the predictions for the underlying fully inclusive A t c . In this work, we therefore consider an alternative production mechanism for top quark pairs, which can provide a complementary handle for the determination of the SM charge asymmetry, as well as an independent probe of possible BSM sources of a deviation from the SM result. The mechanism we propose is the production of a tt pair in association with a W boson (Fig. 1). This production process is indeed quite peculiar. At the LO in QCD it can only occur via a qq annihilation, and no contribution from gluons in the initial states is possible. This is at variance with respect to tt Z or ttγ , where the vector boson can also couple to the top quark in the subprocess gg → tt. As it can be seen from Fig. 1, ttW ± can be simply thought of as the standard qq → tt LO diagram, with the W ± emitted from the initial state. At the NLO, the qg channels can open up, yet the gluon-gluon fusion production is not accessible until NNLO. As in qq → tt the top and the antitop are produced symmetrically at LO and an asymmetry arises only starting at NLO due to interference effects. As we will show in the following, the absence of the symmetric gluon-gluon channel makes the resulting asymmetry significantly larger than in tt production. 2 The asymmetries for higher beam energies are determined in Ref. [13] to be A t c, y (8 TeV) = 0.0111 ± 0.0004 and A t c, y (14 TeV) = 0.0067 ± 0.0004. The second key feature of ttW ± is that the emission of the W boson from the initial state acts as a polarizer for quark and antiquarks, effectively leading to the production of polarized top and anti-top quarks. In other words, the W -boson emission makes the production of a tt pair similar to that in polarized e + e − collisions [21][22][23][24]. As a result, the decay products of the top and anti-top display very asymmetrical distributions in rapidity already at the leading order. We shall call this the EW component of the asymmetry. In new physics scenarios, the emission of a W boson might also act as a discriminator of the chirality structure of new interactions, such as that of an axigluon with light quarks, as already advocated in different studies [25][26][27].
Results at the NLO and NLO + PS for the processes tt V (V = W ± , Z ) have appeared in the literature [28][29][30][31][32][33][34] yet no special attention has been given to asymmetries, whether EW or QCD. The effect on the asymmetry due to the emission of a photon has been recently studied in Ref. [35]. Measurements of total rates are also becoming available from the LHC experiments [36].
The plan of this article is as follows. In Section 2 we present the predictions, at NLO in QCD (with and without including parton shower and hadronization effects) for A t c in both tt and ttW ± production, and, in the latter case, for the asymmetries of the decay products A b c and A c . In Section 3, we compare the SM predictions to a simple benchmark model featuring an axigluon compatible with the Tevatron A FB measurements, along the lines of what done in Ref. [37], to illustrate the peculiar discriminating power of ttW ± . In the final section we discuss the prospects at present and future colliders and present our conclusions. In Appendix A, we review the main features of the polarized qq annihilation into tt, highlighting the close similarity of angular distributions with those predicted in qq → ttW ± .

tt and tt W ± at NLO and NLO + PS
In order to study the top charge asymmetry at NLO for both tt and ttW ± , we employ MadGraph5_aMC@NLO, a framework [38] which allows to automatically generate the code needed to compute the cross section and any other observable for these (and any other SM) processes at LO, NLO and NLO + PS. We present results computed using the MSTW 2008 (N)LO PDF set [39] with five massless flavors. The pole mass of the top quark is set to 173 GeV and the W -boson mass to 80.41 GeV. The renormalization and factorization scales are kept fixed and set to μ f = μ r = 2m t , and the corresponding uncertainty is obtained by varying the two scales independently in the interval [m t , 4m t ]. PDF uncertainties are calculated following the Hessian recipe given in Ref. [39]. As we have found that they are negligible in the case of A t c (at the level of 0.01 percent), we do not display them in the tables.
We first show in Table 1 the cross section and asymmetry A t c for pp → tt, computed at the LHC with a center-of-mass energy √ s = 8 TeV. At the LO there is no top-quark charge asymmetry, while including the parton shower generates a small asymmetry, as shown in Ref. [40]. At NLO a small asymmetry appears (less than 1%) both in the fixed order as well as in the NLO + PS computation, the latter being slightly larger [40]. Not surprisingly, a rather strong scale dependence affects the asymmetry predictions, these being de facto LO quantities. Here and below, we shall always use the NLO cross sections in the denominators of the asymmetries, leading to a possible underestimate of the real asymmetry. As stated above, we believe that the difference between using LO and NLO cross sections in the denominators should anyway be considered as an additional component of the theoretical systematics (though, this is not accounted for in the scale uncertainties quoted throughout the paper).
We now turn to the corresponding results for ttW ± , which are shown in Table 2. As in the previous case, A t c vanishes at the LO, but at NLO we obtain A t c ≈ 2-3%, a considerably larger value than in the tt inclusive production. The effect of the asymmetry can be visualized by superimposing the pseudorapidity of the t and t quarks, as shown in Fig. 2. At LO the two distributions are not distinguishable, while at NLO the asymmetry is manifest: the antitop quark tends to be more central, whereas the top quark has a broader spectrum, with a dip at η = 0. Again, the scale dependence of the asymmetry is quite large, consistently with the fact that NLO corrections only provide its LO contribution. The scale dependence of the asymmetry is shown in Fig. 3, varying the renormalization and factorization scales together. It is also worth to briefly comment on the fact that the asymmetry is larger for ttW + than for ttW − . This can be understood using an argument based on PDF's: the main subprocesses in these two channels are ud → ttW + and dū → ttW − , respectively.
The longitudinal momenta of the initial partons are on average In both cases the momentum of the t(t) quark is connected to the momentum of the q(q). The large longitudinal momentum transferred to the t quark from the initial u quark (ttW + ) increases the corresponding |η t | value. As a result the asymmetry A t c is enhanced compared to the ttW − final state.
As a next step, we consider the case of an NLO + PS simulation, obtained by matching the NLO calculation to Herwig6 [41] via the MC@NLO method [42]. We show the corresponding results in the third line of Table 2. The asymmetry at LO + PS (not shown in the table) remains zero within uncertainties. At the NLO + PS level a small decrease compared to fixed NLO is found. Finally, we analyze the results obtained including the decays of the top quarks and the W -boson. In order to keep spin correlations intact for the final lepton and b, b distributions, Madspin [43] is employed. In so doing parton-level events are decayed using the full tree-level matrix element 2 → 8 for the Born-like contributions and 2 → 9 for those involving extra radiation, before they are passed to Herwig6.
At this exploratory stage, we use the MC truth in order to correctly identify leptons and b-jets coming from the top and antitop quark decays, without considering issues related with the top quark reconstruction. Furthermore we ask that the leptons coming from top (anti) quark decays are positrons (electrons), while the extra W bosons decay into muons, requiring the following decay chains: We present the asymmetries A b c and A c in Table 3. The former is computed by reconstructing the b-jets in the event which come from the top and anti-top quarks. We cluster hadrons into jets using the k T algorithm as implemented in FastJet [44], with R = 0.7, p T > 20 GeV and |η| < 4.5. Smaller values of the R parameter have been checked not to alter significantly the results. For the computation of A b c , events that do not feature two b-jets coming from the top quarks have been discarded.
Two observations on the effects of NLO corrections can be made. The first is that for both A c and A b c NLO corrections tend to shift the EW asymmetries towards positive values, an effect which  The second observation is that the scale dependence of these asymmetries is very small at the LO, while it becomes larger at the NLO, as it can be seen in Fig. 4. This is due to the fact that the asymmetry at LO is purely EW in origin, and it therefore rather stable against scale variations, while the asymmetry at NLO includes the QCD effects, and is directly affected by the scale dependence.

BSM: the axigluon model
The Tevatron experiments (CDF, D∅) have measured the forward-backward asymmetry, which is defined in a similar way to the peripheral-central asymmetry used for the LHC, i.e., The central values of the measurements from the two collaborations [3,6] are larger than the SM [13,11], CDF: A tt = 16.4 ± 4.7%, D∅: A tt = 19.6 ± 6.5%, A simple toy model that is often used to describe the enhancement of the forward-backward asymmetry at the Tevatron features a massive color octet vector boson (axigluon, G ) [45,46] that in general can couple in a different way to light and heavy quarks. The enhancement is a result of interference between the LO SM ( Fig. 1) and the BSM amplitude (Fig. 5). Studies have already been performed in order to calculate the forward-backward (Tevatron) as well as the central-peripheral (LHC) asymmetry predicted by this model for the tt channel [37,47]. We now study how an axigluon would manifest itself in a ttW ± final state. To this aim, and to keep this part as simple as possible, we restrict our BSM predictions to the LO. In general, the axigluon couplings to the quarks are related to the strong coupling g s and can either be universal (same couplings between light and top quarks) or non-universal. The parameter space of the model provides the freedom to choose a light or a heavy axigluon. The term in the Lagrangian that describes the coupling of the axigluon to quarks is L = iG μ,a g i Lq i T a γ μ 1 − γ 5 q i By choosing g i L = 0, g i R = 0, a pure right-handed coupling is possible. However, as argued in Appendix A, in ttW ± the initial quark line can only be left-handed, leading to vanishing amplitudes for right-handed axigluons. This brings the first very important difference with respect to tt production whose total rates are not sensitive to the relative amount L and R chiralities of the couplings of the axigluon. This is just an example of a more general point: comparing asymmetries in tt and ttW ± (and also other associated productions such as with Z and γ ) could provide further key information on the new physics interactions. To illustrate this in the case of the axigluon in a quantitative way, we consider four scenarios, two (left, axial) for a light axigluon and two for a heavy one. The light axigluon is chosen to have universal couplings and a fixed width ΓG = 50 GeV, while the heavy axigluon is chosen to have non-universal couplings and of opposite sign between the light and top quark couplings. The model used for the light axigluon is available from the FeynRules model database [48,49] though it has been slightly modified in order to include nonuniversal couplings for the heavy axigluon. For the light axigluon (mG = 200 GeV, ΓG = 50 GeV) the scenarios considered are: Left-handed (I):   For the heavy axigluon (mG = 2 TeV) the decay width is calculated internally [50] considering the decays of the axigluon to quarks (and using α S (mG )) the scenarios are: Left-handed (III): In order to calculate the asymmetries at the best of our knowledge we combine additively the NLO prediction for the SM to the BSM one at LO, i.e., i.e., the contribution of the diagram featuring the axigluon exchange squared as well as the interference with the SM amplitude. For consistency, we employ NLO PDF's in both SM and BSM terms.
The total asymmetry reads All results include showering and hadronization. The effect of the axigluon BSM is calculated using Eq. (10) and the results are presented in Tables 4 and 5 (at 8 and 13 TeV respectively), with the scales μ f = μ r = 2m t . We include the uncertainties due to scale variation (μ f = μ r = m t , 2m t , 4m t ) separately for the three A i c asymmetries in Fig. 6 (Fig. 7) for To compare with the sensitivity of the standard charge asymmetry in tt final states, we show the results for 13 TeV in Fig. 8. These plots, compared to those in Fig. 7, show that the relative impact of BSM modifications is larger for the tt W ± asymmetries than for the tt ones. The reason is that any asymmetry in the tt final state, whether induced by QCD effects or by BSM physics, is largely washed out by the symmetric contribution due to the gg initial state. Of course the ultimate reach of measurements in the ttW ± is challenged by the reduced statistics; we shall show in the next section that the high luminosities expected in future runs of the LHC are sufficient to precisely measure the SM asymmetries, and to expose possible BSM contributions.

Outlook and conclusions
In the previous sections we have argued that the polarization and asymmetry effects in ttW ± production are large enough to offer a useful handle to constrain new physics effects. The question, however, is whether such effects will be measurable given the expected cross sections and luminosities at present and future colliders. To this aim, we have calculated the cross section for the ttW ± process (Table 6) at various pp collider energies, as well as the corresponding asymmetries A i c , i = t, b, e. For comparison, we also show the results for inclusive tt production.
To start with, we observe the steady reduction with beam energy of the leptonic and b asymmetries. This is due to the growing role of the qg initial-state channel (shown in Table 7), which dilutes the EW component of the asymmetry of the decay products. The charge asymmetry in tt production, viceversa, is reduced by a factor ∼ 6 when increasing the energy from 8 to 100 TeV. This  Table 7 Contribution of the qg parton subprocess at NLO for the ttW + process for μ f = μ r = 2m t . is due to two effects related to the small-x behavior of the PDF's: first the gg channel, which is symmetric and therefore enters only in the denominator of Eq. (1) becomes more and more dominant; second the q and q asymmetry at large rapidities is less and less pronounced.
While a realistic experimental analysis will certainly degrade this optimal precision, these numbers show the great potential of this observable. We remark that the larger sensitivity of A In conclusion, the main motivation of our work has been the observation that the top quark charge asymmetry in pp → ttW ± at the LHC is larger than that of inclusive tt, being of a few percents. In addition, the lepton and b asymmetries are very large and already present at the leading order due to the polarization of the initial fermionic line by the W ± emission. As a simple application, we have shown how the existence of an axigluon that could describe the Tevatron measurements of the forward-backward asymmetry would impact pp → ttW ± and discussed the prospects in LHC Run II, HL-LHC and at future colliders.
The ttW ± final state will not replace the use of the tt asymmetry, particularly while the total integrated luminosity of the LHC is still below the O(100 fb −1 ). In the long term, however, it will provide a powerful probe, complementary to the tt asymmetry, and uniquely sensitive to the chiral nature of possible new physics that were to manifest itself in these measurements.
In the beam line basis, i.e., when the polarization axis of the top is the light antiquark direction in the top rest frame, the polarized differential cross sections dσ tpol,tpol for an initial state q LqR pair read a result which is also valid in the helicity basis [51]. Eq. (A.3) predicts the total number of events with opposite top anti-top polarization to be the same far from threshold. The polarization information is transferred to the decay products angular distributions, and in particular to the leptons that are 100% correlated with the top-quark spins. One therefore expects the lepton polar distributions with respect to the beam axis to show a linear dependence in cos θ e at threshold that flattens out at high energies.
We have explicitly checked the expressions Eq. (A.1) and the analytic computation of the tree-level q LqR → tt → b + νb − ν amplitude numerically to those obtained via MadGraph5_aMC@NLO.
Apart from more complicated analytic formulas the case of qq → ttW ± is totally analogous, as the only non-trivial effect of the W -boson emission is that of selecting a q LqR in the initial state. This is clearly shown in Figs. 9 and 10. In the first set of plots we show the lepton distributions from the top-quarks decay for both q LqR → tt and qq → ttW ± in the beam-axis frame at three values of √ŝ , one close to threshold (400 GeV for tt and 500 GeV for ttW ± ) and increasingly far from threshold (1 and 8 TeV). The two processes lead to very similar distributions. We have then considered the pseudorapidity distributions in the tt rest frame. We find that the t and the t pseudorapidity distributions are equal and symmetric at LO and we do not show them. The lepton distributions, however, see Fig. 10, display an opposite and equal forward-backward asymmetry whose shapes in the center-of-mass frame of the tt pair are again extremely similar in q LqR → tt and qq → ttW ± . The fact that the asymmetry is larger at threshold is a direct consequence of the fact that there the top quarks are fully polarized.