Monte Carlo event generators&the top quark forward--backward asymmetry

The leading-order accurate description of top quark pair production, as usually employed in standard Monte Carlo event generators, gives no rise to the generation of a forward--backward asymmetry. Yet, non-negligible -- differential as well as inclusive -- asymmetries may be produced if coherent parton showering is used in the hadroproduction of top quark pairs. In this contribution we summarize the outcome of our study of this effect. We present a short comparison of different parton shower implementations and briefly comment on the phenomenology of the colour coherence effect at the Tevatron.

Abstract. The leading-order accurate description of tt production, as usually employed in standard Monte Carlo event generators, gives no rise to the generation of a forward-backward asymmetry, A FB . Yet, non-negligible -differential as well as inclusive -asymmetries may be produced if coherent parton showering is used in the hadroproduction of top quark pairs. In this contribution we summarize the outcome of our study [1] of this effect. We present a short comparison of different parton shower implementations and briefly comment on the phenomenology of the colour coherence effect at the Tevatron.  Figure 1. The top quark forward-backward asymmetry A FB in dependence on the transverse momentum of the pair. Results from MC@nlo [2], Pythia [3], Powheg [4], MCfm [5] and tt+jet production at NLO [6] are shown as reported by the DØ and CDF Collaborations in Refs. [7] and [8], respectively.

Introduction
In a number of different measurements the Tevatron Collaborations both CDF and DØ reported results on the top quark forward-backward asymmetry that clearly lie bea e-mail: jan.winter@cern.ch b e-mail: peter.skands@cern.ch c e-mail: webber@hep.phy.cam.ac.uk yond the Standard Model expectations for this quantity, see Refs. [7][8][9][10]. They did not only determine inclusive asymmetries but presented also differential distributions, such as A FB (m tt ), showing the asymmetry in different bins of observables O that reflect the kinematic properties of the reconstructed top quark system. One such observable is the transverse momentum of the pair, p T,tt , for which DØ made an interesting observation, documented in Ref. [7], while studying various Monte Carlo (MC) predictions: the leading-order (LO) event generator Pythia [3], using the option for approximate angular coherence, displays a qualitatively similar p T,tt dependent asymmetry to that of the next-to LO (NLO) matched parton shower, MC@nlo [2]; cf. the top panel in Fig. 1. Disabling the coherent shower option will restore the expected behaviour for Pythia as indicated by the dashed line (magenta) in the figure. CDF later on published similar results [8] confirming DØ's observation; this is also shown in Fig. 1, at the bottom.
The definition of A FB used by the Tevatron experimentalists is based on taking the rapidity difference between the top and the antitop quark, ∆y = y t − yt, and dividing the events according to their ∆y hemisphere: The terms (dσ/dO) ± denote the differential (or, if O ≡ 1, inclusive) cross sections measured for forward/backward (±∆y > 0) top quark pair configurations; note that ∆y is a longitudinally boost invariant quantity. We shall now investigate the effects of colour coherent parton showering on A FB in more detail.  [11,12] predictions for all and separately generated forward/backward (fwd/bwd) tt configurations at LO. The colour coherence effect is illustrated after the first and finally all shower emissions have occurred by means of the p T,tt distribution. tt processes, the by far dominant contributions to tt production at the Tevatron. The (leading-)colour flows associated with these partonic interactions stretch from the initial to the final state connecting the incoming (anti)quark with the outgoing (anti)top quark. They form initial-final colour dipoles as shown in the upper part of Fig. 2. The deflection of the top quark out of the incoming quark's direction is a measure of the acceleration of the colour charge. Colour coherence manifests itself through this deflection angle: a mild scatter characterized by a small opening angle, keeping the forward motion of the top quark, only induces weak radiation off this qt dipole, see Fig. 2 upper left. In contrast, strong scattering and re-direction of the top quark leads to a more violent acceleration of colour and hence a larger number of potentially also harder QCD emissions. Top quark pairs in backward configurations therefore experience an increased recoil which we illustrate in the upper right of Fig. 2. Consequently, we find a larger number of soft p T,tt events correlated with positive values of the asymmetry while hard p T,tt events often contribute to negative asymmetries -as well known from the NLO calculation, cf. Fig. 1.
We can easily test this rationale in a MC simulation which we do by utilizing Sherpa's CSshower implementation [11]. 1 We track the evolution of top quark pairs generated at leading, fixed order in forward (∆ỹ > 0) and backward (∆ỹ < 0) configurations independently. The  Leading-order QCD predictions (calculated with MCfm [5] at NLO in tt production) for the forward-backward asymmetry as a function of p tt T imposing an upper, a lower and no bound at all on the tt pair mass. tilde sign in ∆ỹ is used to flag an evaluation at the matrixelement level. The results in the lower part of Fig. 2 clearly demonstrate that after the coherent showering phase, initial backward configurations yield a harder p T,tt spectrum than initially forwards moving top quark pairs. The latter preferably populate the soft region of p T,tt 10 GeV, and both observations thus agree well with our qualitative explanation from above.

Analytic approximation of the effect
The qualitative picture which we argued for in Sec. 2 can be made more explicit. Analyzing the real-emission contribution to the asymmetry, mainly arising from qq → ttg processes, one can show, see Ref. [1], that the differential dependence of the asymmetry on the pair transverse momentum can be written as where p T ≡ p T,tt and β = 1 − 4m 2 t /ŝ denotes the top quark center-of-mass velocity. The Born and asymmetry cross sections are given byσ B andσ A , respectively.
We can now discuss to what extent a coherent branching algorithm reproduces this functional form given in Eq. (2). Such shower generators treat the gluon radiation as coherent emission from the top quarks lines of the Born process, therefore overestimate the colour factor in Eq. (2) by 2 C F = (N 2 C − 1)/N C (a 60% overestimate) and approximate the kinematic part of the asymmetry amplitude by the corresponding Born term times  dipole-like/eikonal factors. That means we effectively obtain a description of the asymmetry which is correct in the soft gluon limit (apart from the colour factor), but only approximate for p T > 0, i.e. in the coherent shower generators the coupling-and colour-stripped asymmetry function F(β, p T ) given in Eq. (2) can only be "underestimated" by F(β, 0) which reads F(β, 0) = −4 β − β 3 + O(β 5 ) when Taylor expanded in β. Note that F(β, p T ) becomes less negative the further away from the p T = 0 limit. This is depicted in Fig. 3 for different fixed values of β where we observe larger deviations for increasing top quark pair invariant masses. Both the colour factor and kinematic approximations lead to a more pronounced A FB (p T,tt ) dependence, as seen in Fig. 1 when comparing the Pythia coherent shower prediction with that given by MC@nlo.
Finally, Fig. 4 shows the p T > 0 dependence of the asymmetry as defined in Eq. (1) which one obtains from a full NLO calculation for tt production at the Tevatron. 2 It is interesting to note that the asymmetry approaches zero 2 The finite part of the virtual correction leads to a positive delta peak contribution at p T ≡ 0 which however is not shown here. but does not become positive as p T → 0. This is because the singularity structures present in the numerator and denominator are different. The denominator diverges faster owing to initial-state collinear singularities that cancel in the numerator. In contrast to the behaviour at fixed order, NLO matched or coherent parton showers predict an A FB (p T,tt ) cross-over, as shown in Fig. 1, at low but nonnegligible values of p T . This is a consequence of the Sudakov suppression of small p T occurring due to multiple soft gluon emission, an effect absent in the fixed-order description. This Sudakov suppression yields a spreading of the positive asymmetry over a finite region of p T 0.

Differential asymmetry predictions
Considering the impact colour coherence has on generating differential asymmetries, it is important to study the response of standard MC event generators using tt production (without decays) at the Tevatron. For a representative collection of parton showers, we present their respective asymmetry predictions in Fig. 5 together with the associated differential cross sections as functions of p T,tt (upper row) and m tt (lower row). Far more details regarding this MC tools comparison and the choice of parameters can be found in Ref. [1]. Here we only highlight a small fraction of the results of this comparison. As one can see from Fig. 5, default Herwig++ [13] and Sherpa [12] produce differential A FB sufficiently similar to what one expects from the approximate NLO treatment mentioned above, especially the rise of A FB with increasing m tt is remarkable. Using the m tt observable the Sudakov region is stretched/applied over the entire mass range which leads to a (an almost entirely) positive A FB dependence. Both Herwig++ (through angular ordering) and Sherpa (through dipole showering) have implemented QCD coherence in a proper way, however rely on different strategies regarding initial shower conditions and treatment of recoils. This induces differences between their predictions as seen in Fig. 5. While the version of Pythia 8 [14] used here does not yet account for QCD coherence effects, hence produces rather small, p T,tt as well as m tt insensitive asymmetries, the Pythia 6 [3] predictions in Fig. 5 are examples for the fact that the older generation provides options (tunes) with varying amount of coherence. The D6T and P(erugia)0 tunes [15] emerged from efforts to improve the description of Tevatron and LHC data, respectively. In particular the D6T tune amplifies the effect of colour coherence as explained in Ref. [1].

Inclusive asymmetry and longitudinal recoil effects
Inspecting the numbers in Tab. 1, it is obvious that colour coherent showering not only produces non-trivial differential asymmetries but also an inclusive A FB . 3 This clearly comes as another surprise, but can be explained via event migrations arising from longitudinal recoil effects. This is specified in Ref. [1] where we also show that the effect, already at LO, may give a relevant contribution to the inclusive asymmetry. Here, we only want to provide evidence that migration indeed occurs in coherent showers, using the CSshower results displayed in Fig. 6. Following the same strategy used to produce the p T,tt spectra presented in Fig. 2, it is straightforward to obtain the corresponding ∆y distributions to examine the migration effect directly. A simple visualization of the longitudinal recoil effect comes in terms of a stretching/widening of the gluon emitting qt initial-final dipole. Independently of the magnitude of the initial scattering angle, the top quark will be slightly pushed forward while the antitop quark retains its direction such that ∆y = ∆ỹ + is a plausible 3 This can also be anticipated by viewing Herwig++'s and Sherpa's positive-valued A FB (m tt ) results given in Fig. 5.  Figure 6. The ∆y distributions for various modes of CSshowering applied after disjoined as well as joined ∆ỹ hemisphere generation at leading, fixed order in top quark pair production. parametrization ( > 0). The results shown in Fig. 6 confirm this conjecture. Several observations are made: (1) the imbalance between ∆y < 0 and ∆y > 0 events, i.e. the generated asymmetry, is clearly visible by comparing the after-with the before-shower result (cf. red vs. black line), (2) the migration of initial backward and forward configurations to higher values of ∆y produces an overflow of events of the former category closing the dip at ∆y 0 generated by shifted events of the latter category (cf. green vs. blue lines), (3) + → − migrations in ∆y (i.e. < 0) are suppressed and (4) the largest contribution to the migration effect already appears with the first emission (cf. dashed vs. solid lines). Generally, migrations are small, occur locally and clearly favour > 0. They therefore fuel the generation of a positive-valued inclusive asymmetry.

Phenomenological tests
With our understanding of the origin of the colour coherence effect, we can study some of its phenomenological implications which we briefly discuss below.

Asymmetry in different phase-space domains
By focusing on certain phase-space regions, one can amplify the inclusive asymmetry, as shown in Tab. 2. The application of cuts isolating large m tt or low p T,tt leads to an increase of the positive asymmetry. The latter cut is particularly useful to efficiently separate off the Sudakov region, i.e. the domain of low p T,tt that generates the positive contribution to the overall asymmetry. Considering the anti-cut (real-emission) region, one can test the most striking feature of the transition in p T,tt domains: that is the sign change in the asymmetry. Because of the approximations discussed earlier, coherent showers predict a stronger asymmetry flip at large p T,tt . For all other cases, they however give enhancements similar to the NLO result.
These single-cut, A FB enhancing effects can also be found at the differential level. We exemplify this in Figs. 7 and 8 using Sherpa CSshower results. Figure 7 is shown to compare the different A FB (m tt ) distributions associated with the two p T,tt domains. It exhibits the sign flip in the asymmetry very clearly. The opposite scenario is presented in Fig. 8 Figure 9. A FB (β) distributions obtained from coherent parton showering (as implemented in the CSshower [11]) for varying "top quark" masses, m, and pp collider energies, √ s.
noticeable in the first few p T,tt bins, those that are close to zero. In both figures we also illustrate the impact of a recoil scheme variation (as provided by Sherpa v.1.4.0): differences only occur in the high p T,tt region where the amount of Tevatron data is too sparse to discriminate between the two options.

Top quark velocity dependence of the asymmetry
We want to briefly discuss how the coherence effect (generating non-zero A FB ) changes under variation of the top quark mass and collider energy √ s. It is thus convenient to analyze the β dependence of the asymmetry, which behaves roughly like A FB (m tt ). Figure 9 contains a number of A FB (β) curves obtained from coherent showering using different values for the m and s parameters. Apart from PDF effects, one finds, as expected, an approximately stable asymmetry dependence, if mass and √ s are scaled equally. Investigating the limiting cases, it is clear that the dramatic increase of A FB occurs as a consequence of reaching the collider phase-space boundaries. The contri- Top quark pair production and semileptonic decays @ Tevatron Run2. Figure 10. Differential forward-backward (A tt ≡ A FB ) and lepton-based (A l ) asymmetries given in terms of various (transverse) mass observables [16]. Here, CSshower results are shown.