Constraining parameter space of the Little Higgs model by data of tera-Z factory and ILC

The Standard Model prediction on the forward-backward asymmetry for $b\bar b$ production ($A^b_{FB}$) is well consistent with the data of LEP I at the Z-pole, but deviates from the data at $\sqrt s=89\; 93$ GeV which are slightly away from the pole. This deviation implies that there still is room for new physics. We calculate the $A^b_{FB}$ at vicinity of Z-pole in the Little Higgs Model as well as other measurable parameters such as $R_b$ and $R_c$, by which we may constrain the parameter space of the Little Higgs Model. This can be tested in the newly proposed tera-Z factory. With the fitted parameters we further make predictions on $A^b_{FB}$ and $A^t_{FB}$ for $t\bar t$ production at the ILC energy.


I. INTRODUCTION
As well recognized, the hadron colliders are machines for discovery. On other aspects, the electron-positron collider, muon-collider and even the proposed photon collider will provide detailed information about the discovered new physics candidates. Once some peculiar phenomena are observed at the hadron colliders such as TEVATRON or LHC beyond the expectation of the Standard Model (SM), one is tempted to associate them to new physics.
Generally, making conformation is difficult, especially there are too many models about the new physics available and most of them can offer a plausible interpretation towards the new observation. One of the reasons is that by the data obtained at hadron colliders, it is difficult to study the details which are crucial for identifying the new interaction and/or new particles observed in the physical process accompanied by an enormous background. That is why people will turn to invoke high-energy lepton colliders after successful operation of hadron colliders, especially electron-positron colliders which are more favorable because the technique for building such machines are more mature.
To discover new physics, one is looking for phenomena beyond the SM expectation through experimental measurements carried at hadron colliders. Confirming or at least claiming existence of new physics needs to measure several characteristic quantities which do not meet the SM predictions.
The forward-backward asymmetry (A F B ) in top-antitop production at TEVATRON is one of such measurements. It is defined as where θ is the angle between the outgoing top quark and the injecting proton beam and N Q is the number of top quarks.
The data of TEVATRON at the Fermilab observed a large A F B [1]. The measurements of the CDF and D0 Collaborations yield A F B = 0.158 ± 0.075 [2], A F B = 0.162 ± 0.047 [3] and A F B = 0.196±0.065 [4], which are significantly larger than the SM prediction A SM F B = 0.089 [5]. This discrepancy would compose a hint of existence of new physics beyond SM. Numerous models beyond SM have been proposed to explain the deviation from the SM prediction, and we list a few of them in our references as examples [6][7][8][9][10][11][12][13][14][15][16].
We also showed [17] that the deviation of the theoretical prediction and the data can be mended in the little Higgs model (LHM). Definitely, as commonly conjectured, since top quark is much heavier than rest members of quark families, its mass could be close to the scale of new physics, so that observation on processes associated with top quark should be more favorable for discovering new physics.
However, on another aspect, it is also proposed to observe the A F B in bb production [18].
Indeed, comparing the asymmetries for top (A t F B ) and bottom (A b F B ) productions would be interesting. If the new physics beyond SM indeed contributes to the A F B , and the scale of new physics is high and close to the top quark, then the new physics should make a larger contribution to A F B in tt production than in bb production. Namely, if we calculate the A F B within the framework of only SM, the theoretical prediction on the A F B for bottom should be closer to the data than for top. This general analysis might be violated due to so far unknown behaviors of new physics BSM. Indeed all depends on the scales of new physics (see below in the context).
In this work, we study the A F B at e + e − colliders within the framework of SM and LHM which is one of the models beyond SM. For a proposed Z factory the process under consideration is only e + e − → bb because of the constraint of the phase space, while at ILC both the processes e + e − → tt and e + e − → bb would be accounted.
By a direct observation, the A F B is induced by the odd power of cos θ in the amplitude square. Obviously, such terms imply parity in the process is violated (PV). In the SM, the parity violation in the process e + e − → bb is due to Z 0 boson exchange, whose interaction with fermions has both vector and axial vector components. For next-to-leading order (NLO), the box diagrams also generate the asymmetry, because it is equivalent to a t-channel tree diagram, thus lead to odd powers of cos θ.
Similar to the case of pp collision at TEVATRON, the production angle θ at the e + e − collider is defined as the angle between the outgoing bottom or top quark and the incoming electron beam. The difference of the rapidities of the Q andQ which is Lorentz invariant is written in the e + e − center-of-mass frame as where s = (p 1 + p 2 ) 2 with p 1 and p 2 being the momenta of e − and e + . Obviously, the sign of y Q − yQ is the same as cos θ, the asymmetry in Eq.(3) which is experimentally measurable, can be recast as The bb asymmetry A b F B was theoretically predicted with the SM as 10.34 ± 0.07% [19] and experimentally measured value at LEPI is 9.92 ± 0.16% [19]. There indeed exists an observable distinction between the prediction and data. Moreover, the theoretical estimate did not involve the higher order corrections and interference with the photon contribution, when such corrections are taken into account, the situation becomes even worse, namely the theoretical prediction gets larger above the data (see the details in next sections). It implies that new physics beyond the standard model whose contribution to the asymmetry destructively interferes with that of SM is needed to reduce the value. Combining with the observation about A F B at TEVATRON where the SM prediction is also apart from the data, we would take this deviation as a hint of existence of new physics BSM.
Of course the distinction might be due to the measurement errors, but one cannot exclude a possible contribution from new physics beyond SM. The strategy of this work is to investigate the contributions of both SM and BSM to the asymmetries in e + e − → bb and e + e − → tt with a special BSM, i.e. the LHM which we used to explain the A t F B observed at TEVATRON. The energies we set are that of the Z-factory and ILC (or CLIC) respectively.
Then we compare the asymmetries obtained for tt and bb to investigate their differences.
Even though we employ a special model BSM, the obtained results can make sense about the role of BSM for the asymmetries. It is noted that there is an obvious difference in the two cases, even though we employ the same model: LHM. For the TEVATRON case, the main contribution is from an exchange of the heavy Z boson Z H whose mass is generally believed to be around 400 to 500 GeV, whereas for the LEP cases, the main contribution comes from the heavy photon A H whose mass is within a range of a few tens of GeV to 100 GeV.
The future experiments will provide us more definite information. Especially, a comparison of theoretical predictions and the data for both TEVATRON and e + e − collider may tell us consistency degree of the model and enrich our understanding of the nature, namely help us to search for new physics.
This paper is organized as follows. After this introduction, in Section II, we formulate the total scattering cross section as well as A F B to NLO within the frameworks of SM and LHM. The numerical results along with all the input parameters are shown in Section III. The obtained results are shown explicitly in several figures and tables, as possible interpretations on those curves are made. The last section is devoted to a simple discussion and conclusion.

II. THE CONTRIBUTIONS OF SM AND LHM TO THE ASYMMETRY UP TO NLO
In this section we formulate the contributions up to NLO to the A F B and total cross section of the e + e − → QQ system in the frameworks of SM and LHM. The derivation in SM is standard and straightforward, here we just repeat the calculation which was done by number of authors to check our programs. Then we turn to the contribution of new physics, concretely the LHM.
In LHM [20] only two vector bosons A H and Z H can contribute asymmetry through schannel. The relevant Lagrangian is and In the LHM, the masses are separately f is the vacuum exception value (vev) in LHM and a is a model parameter equal to s ′ c ′ where s ′ ≡ sin θ ′ and c ′ ≡ cos θ ′ defined in Ref. [20]. The relations of the coupling constants are listed in the appendix of Ref. [20] and those numerical values are presented in next section.
The tree and NLO Feynman diagrams are shown in Figs.1, 2 and 3 respectively.
The amplitude of the first two diagrams of Fig.1 is: where θ W is the Weinberg angle, p 1 and p 2 respectively stand for the four-momenta of the initial electron and positron, e Q is the electric charge of the heavy quark (b or t), and p 3 , p 4 denote the four-momenta of the final Q andQ. Here, p 1 + p 2 = p 3 + p 4 is four-momentum of Z 0 or photon at s-channel. For the rest three diagrams, the amplitudes are similar, so we omit them for saving space of the text.
In Fig.2 all SM and LHM box diagrams are presented. The diagrams (c) and (d) where charged W-bosons are exchanged stand for bb and tt productions respectively. To explicitly   demonstrate the procedure of the derivation, let us present the amplitude of the first two diagrams in Fig.2 where only photons are exchanged as an example, that is: For the rest diagrams, the amplitudes are similar but the coupling constants are different.
We carry out the complete calculation of four diagrams in Fig.2 with the software LoopTools.
where m stands for the mass of the fermion in fermion loop, and M 3b corresponds to the boson loop where m W is the mass of W-boson.
After averaging the initial spin-states and summing over the finial spin-and color-states, the differential cross section with respect to the production angle θ is: The asymmetry is obtained by integrating over the positive and negative ranges of cos θ separately. We use the Lorentz invariant rapidity difference y Q − yQ to calculate the asymmetry as shown in Eq.(2) and Eq.(3). The numerical results will be presented in next section.

III. NUMERICAL RESULTS
In our numerical computation, the mass of charm, bottom and top quark are taken as 1.25, 5 and 175 GeV and the masses of light quarks (u, d, s) are neglected. In the center of mass frame, one has p Q = pQ and p 2 Q = m 2 Q , the kinematics is determined  [19,26,27]. At the proposed Z factory the center-ofmass energy is located in vicinity of Z 0 mass, so the on-mass-shell resonance effect would be dominant and the Breit-Winger formulation should be adopted.
In the LHM [20], the relevant parameters depend on parameters a and b via the relations: GeV [20].
It is noted that for the heavy photon, all its couplings to fermions uniquely depends on parameters a and b, which are not determined in the model, so that here we treat them as free parameters. The only way, so far before a more fundamental principle appears, to fix them is by fitting available experimental data. To fit the data of the measured asymmetry, we find that a can only reside in a rather narrow range from 1.1 to 1.3. Thus we let a vary from 1.0 to 1.4 and b vary from 0 to 1. Instead, the mass of Z H is written as f GeV, and for simplicity we use the relation ae and they all vary from −0.0165 to −0.33 [20], thus the coupling constant α l = g ′2 au 4π varies from 0.00002 to 0.00867. Fig.4 shows the dependence of A b F B on √ s (the superscript b refers to bb production) which is theoretically estimated by SM up to NLO, and the experimental data. The narrow band corresponds to the measurement error range. One notices that A b F B predicted by only SM up to NLO is 12.78% above the data at √ s = 92.5GeV which is about 20% larger than the previous LO estimate 10.34 ± 0.07% [19]. This change is mainly caused by the interference between photon and Z 0 and as well NLO corrections. When the contribution of LHM is introduced, the theoretical prediction can be in agreement with experimental data as shown in Fig.5, Fig.6 and Fig.7.  We do see that the predicted A b F B overlaps with the data band. Fig.9 and Fig.10 respectively illustrate the A b F B and A t F B vs the center-of-mass energy for ILC .
It is noted that In Fig.9, the curve of A b F B evaluated with LHM+SM has a minimum near the √ s = 410 GeV, this is understood as a destructive interference between contribution of Z H and that of SM bosons.      GeV, this is also caused by a destructive interference between Z H and SM particles. While, √ s is above 500 GeV and below 400 GeV, theoretical prediction on A t F B tends to be that determined by SM-only. In

IV. DISCUSSION AND CONCLUSION
The observation of the asymmetry of top pair production A t F B at TEVATRON, which is obviously larger than the SM prediction, implies existence of new physics beyond standard model. Many authors tried to explain the discrepancy between theoretical predictions and data in terms of various models BSM. Moreover, some authors also studied asymmetries for b-quark pair production with those models.
Recollecting the theoretical predictions on the asymmetry A b F B at LEP I and II energies, one notices that there is also a gap between SM prediction and the measured value. It is natural to conjecture that such new physics BSM should contribute to A b F B at LEP energies. In our previous work, we investigated possible contributions of the heavy Z-boson which exists in LHM, to A t F B and found that as its mass and coupling to SM particles are within a suitable range, the asymmetry observed at TEVATRON can be reasonably explained.
Extending this scenario to study the asymmetry which can be observed at the e + e − colliders, we notice that the mass of heavy Z-boson is as high as more than 400 GeV, so does not make substantial contributions to A b F B , instead in the LHM the heavy photon whose mass is around 100 GeV does play an important role at the LEP I and II energies. Thus, we employ the LHM to calculate the production rate and asymmetry for e + e − collisions at the  LEP I and II energies. We find that as the model parameter a takes a value of 1.1∼ 1.3, the theoretical predictions on the total cross sections of bb production and the asymmetry A b F B can be well consistent with the data. This is somehow slightly fine-tuning.
By the same model, we further calculate the asymmetries A t F B and A b F B at the ILC energy of about √ s=500 GeV. Moreover, we also investigate A b F B at the proposed Z-factory. All those values should be tested by the more precise experiments which will be carried out at the Z-factory and ILC. But our calculations also indicate that for the B-factory and charm-tau factory, which are running at much lower energy scales than that of LEP, such new physics does not apply. We are expecting the future high energy experiments to confirm or negate the LHM or make more rigorous constraints on its model parameters.