Colored bosons on top FBA and angular cross section for $t \bar t$ production

With full data set that corresponds to an integrated luminosity of 9.4 fb$^{-1}$, CDF has updated the top quark forward-backward asymmetry (FBA) as functions of rapidity difference $|\Delta y|$ and $t\bar t$ invariant mass $M_{t\bar t}$. Beside the sustained inconsistency between experiments and standard model (SM) predictions at large $|\Delta y|$ and $M_{t\bar t}$, an unexpected large first Legendre moment with $a_1= 0.39\pm 0.108 $ is found. In order to solve the large top FBA, we study the contributions of color triplet scalar and color octet vector boson. We find that the top FBA at $|\Delta y|>1$ and $M_{t\bar t}>450$ GeV in triplet and octet model could be enhanced to be around 30% and 20%, whereas the first Legendre moment is $a^{\bf Di}_1= 0.38$ and $a^{\bf Axi}_1= 0.23$, respectively.

It is believed that the standard model (SM) is just an effective theory of a fundamental one yet to be discovered, even it is consistent with most experimental data. For pursuing more basic elements which are made of our universe, the search of new physics beyond SM is inevitable. If such new physics exists, it can be probed either directly at collider or indirectly through precise measurements. The recent measurements at Tevatron on the forward-backward asymmetry (FBA) in the top-quark pair production at √ s = 1.96 TeV may provide the clue for the existence of new physics. The FBA of top pair system is defined by where ∆y = y t − yt, y t(t) is the rapidity of top (anti-top) quark and N is the number of events for ∆y ≷ 0.
With full Tevatron data set which corresponds to an integrated luminosity of 9.4 fb −1 , CDF Collaboration measures the top-quark FBA with lepton+jets topology at parton level to be [1] A F B = 0.164 ± 0.047 .
The result is roughly in agreement with early CDF and D0 data [2,3]. CDF also reports the linear mass (A F B (M tt )) and rapidity (A F B (|∆y|) ) dependences and the associated slope as (15.2 ± 5.0) × 10 −4 and (28.6 ± 8.5) × 10 −2 , respectively. The former is 1.3σ deviation from the SM prediction and the latter is 2.1σ. Additionally, CDF also analyzes the angular differential cross section for tt production and observes an unexpected large first Legendre moment, where it is associated with top FBA [4].
In the SM, since the top-quark pair production is dominated by the strong interaction QCD contribution, due to C-parity invariance, a vanishing FBA at the leading order (LO) is expected. However, a nonvanishing FBA can be induced at the next-to-leading order (NLO) [5] and beyond [6][7][8][9][10][11][12]. The SM predictions have been improved and the resultant range could be from around 6% to around 10% [12,13], however, by comparing with the central value of Eq. (2), the inconsistency between SM and data does not disappear. Although the anomaly of FBA has derived many studies of new physics in the literature [14][15][16][17][18][19][20][21][22][23][24][25], based on the new measurements and updated data, we investigate the issue by introducing new u-channel and s-channel effects. For illustration, we study the color triplet [14] and color octet models [15,16,26], where the former is a representative of u-channel and the latter is a s-channel.
In order to study the angular cross section for tt production and the unexpected large top FBA as functions of |∆y| and M tt , here we consider two extensions of the SM. One is to introduce a color triplet scalar, called diquark, to the SM. Although there are many possible representations of diquark in SM gauge symmetry, for simplicity, we only focus on the representation (3, 1, −3/4). The other is color octet gauge boson which could be arisen from an extended SU(3) R ×SU(3) L gauge symmetry, called chiral color gauge model [27,28].
In such model, the SM QCD could be taken as an unbroken symmetry of the extended one.
Since the couplings of the new color gauge boson to quarks have the axial vector currents, hereafter, we call the massive color octet gauge boson as axigluon.
Now we briefly introduce the relevant pieces for the new models. Firstly, we discuss the color triplet model. The SM gauge invariant interactions of color triplet diquark with quarks are written by where the indices i and j are the quark flavors, f 3 ij = −f 3 ji , (α, β, γ) stand for the color indices and P L(R) = (1 ∓ γ 5 )/2 is the helicity projection. The antisymmetric tensors in color space are defined asK αβ γ = (K γ αβ ) † and the Ks are given by [29] where the antisymmetric tensors satisfy T r(K aK b ) = δ a b and K a αβK ρσ a = 1 2 (δ σ α δ ρ β − δ ρ α δ σ β ). As a result, the process uū → tt could proceed through the following interactions For color octet gauge boson of SU(3) R ×SU(3) L , we write the interactions of the axigluon with quarks as where the flavor and color indices are suppressed, q T = (u, c, t), g V,A are the gauge couplings of axigluon to the first two generation quarks, T b are the Gell-Mann matrices which are denotes the gauge coupling of the third-generation quark and its value depends on a specific model, e.g. ζ 1(2) = 1(−1) in Ref. [15]. Although flavor changing neutral currents (FCNCs) at tree level could be induced by g t V (A) = g V (A) and have an interesting influence on low energy physics [30], since the small effects do not have a significant contributions to the studying phenomena, we will not further discuss the FCNC effects. In our numerical analysis, the parametrisation of free parameters is the same as that in Ref. [15]. Therefore, the gauge coupling of SM QCD is given by g s = g sin φ cos φ, If we take the value of g s as input, the mixing angle φ and mass of axigluon are the main free parameters.
Before presenting the scattering amplitude squares for qq → tt which are mediated by gluon and the colored bosons, let us first define a convenient notation for the following calculations. The momenta of the incoming quark and anti-quark, outgoing top and outgoing anti-top quarks are denoted by p q , pq (q=u,d), p t and pt respectively such that p q + pq = p t + pt. The momentum can be written as: where β 2 = 1 − 4m 2 t /ŝ and θ is the scattering angle in the center-of-mass frame of the tt. Neglecting the light quark masses of the incoming partons, the Mandelstam variables are defined as follows:ŝ Accordingly, the averaged amplitude square for QCD gluon and color triplet can be derived by [14] where we have already summed over final state color and averaged over the initial spin and color, m H 3 is the mass of diquark, f ′ ut = 2f ut and N c = 3. We note that since the propagator in the u-channel diagram depends on the scattering angle θ, the FBA may arise not only from cos θ terms in Eq. (9), but also from the constant terms and the cos 2 θ terms. Although the interference between SM and diquark is negative whenû − m 2 H 3 < 0, however the pure diquark contribution is positive and dominates in the considered M tt range.
The averaged amplitude square for QCD gluon and axigluon contributions can be obtained as Clearly, the FBA is only from the linear cos θ terms and associated with axial-vector coupling.
Unlike diquark model, with g V g t A , g V g t V g A g t A < 0, the interference between SM and axigluon is positive whenŝ − m 2 A < 0, whereas axigluon contribution is negative. Therefore, both contributions to the FBA are in competition.
Beside the experimental limit on the mass of new particles, the direct strict constraint on the free parameters is from the tt production cross section. Based on 8.8 fb −1 of Tevatron data, the recent combination of CDF and D0 results for tt cross section is [31] σ(pp → tt) exp = 7.65 ± 0.20 ± 0.36 pb .
In order to study new physics on tt production cross section and top FBA, we implement the diquark and axigluon matrix elements, shown in Eqs. (9) and (10), in Pythia8175 [32] as semi-internal 2 → 2 processes. Thus, tt pair production in the two models could be dealt with as a normal internal Pythia process. For estimating the SM NLO contributions, we use POWHEG-BOX-1.0 [33].
All estimations in our analysis are performed at parton level and we do not take into account the effects from parton showering, hadronizations and detector conditions. The taken inputs are the experimental value given by Eq. (11) within 2σ errors, m t = 172.5 GeV and α s = 0.1095 at m Z . The renormalization and factorisation scales are fixed at µ R = µ F = m t . The event samples are generated using CTEQ6L1 [34] set parton distribution functions for diquark and axigluon models, whereas CTEQ6M [34] is used for SM NLO.
Before we discuss the contributions of new physics to top FBA, we first study the allowed parameter space, where the main constraint is from the tt production cross section given in Eq. (11). For color triplet, since the free parameters are f ut and m H 3 , we display the scatters of σ(pp → tt) with respect to f ′ ut = 2f ut and m H 3 in left panel of Fig. 1. Similarly, for axigluon model, the scatters of σ(pp → tt) with respect to mixing angle φ and m A are shown in right panel of Fig. 1.  Tables   I and II. Beside M tt ≤ 750 GeV which is presented in CDF paper [1], we display the FBA up to 1400 GeV in Fig. 4, where we have integrated the FBA over the width of 7 bins in [350, 1400] GeV. It is found that when M tt < 1100 GeV, the two models induce a positive asymmetry. When M tt > 1100 GeV, the asymmetry induced by the diquark is positive and  grows with M tt , whereas that induced by the axigluon is negative and falls with M tt [15].
Thus we can use the different behavior at large M tt to rule out one of the two models.
CDF recently reports the measurements of the Legendre moments for differential cross section of tt production with respect to the scattering angle θ of top-quark in the tt center of mass. The moment is defined by with ℓ being the degree of Legendre polynomial. In order to compute the Legendre moments, we normalize a ′ 0 to be unity. Thus, the Legendre moments a ℓ can be estimated from a sample of N ev events as Clearly, diquark fits well CDF result. The first eight Legendre moments a ℓ , ℓ=1-8, are shown in the left panel of Fig 5. For understanding the new physics contributions, we calculate the normalised differential cross section as a function of cos θ and present the results in right panel of Fig. 5, where the solid, dashed and dotted lines represent the SM NLO, diquark with m H 3 = 665 GeV and f ′ ut = 2.7 and axigluon with m A = 1525 GeV and φ = 30 0 , respectively. In order to compare with CDF data, in the calculations we divide [−1, 1] into 10 bins and integrate the angular differential cross section over the width of 10-bin in cos θ.
The right plot in Fig. 5 shows that the diquark mode is in better agreement with the data than axilguon model. Note that, the difference in the calculations of FBA between tt frame and partonic center-of-mass frame is numerically negligible.
In summary, we have studied the top FBA in diquark and axigluon models, where the former is a representative of a u-channel and the latter is a s-channel. According to our analysis, both models could enhance the FBA and fit well in |∆y| and M tt distributions.
We also show that the top FBA induced by s-channel will decrease from positive to negative at M tt > 1100 GeV, while the u-channel is still growing slightly. We also give the first eight Legendre moments in diquark and axigluon models and find that diquark could explain the large a 1 obtained by CDF.