The anomalous top quark coupling tqg and tW production at the LHC

Many new physics models beyond the standard model ($SM$) can give rise to the large anomalous top couplings $tqg$ ($q=u$ and $c$). We focus our attention on these couplings induced by the topcolor-assisted technicolor ($TC2$) model and the littlest Higgs model with $T$-parity (called $LHT $ model), and consider their contributions to the production cross section and the charge asymmetry for $tW$ production at the $LHC$. We find that the anomalous top coupling $tqg$ induced by these two kinds of new physics models can indeed generate sizable charge asymmetry. The correction effects of the $LHT $ model on the production cross sections of the processes $pp\rightarrow tW^-+X$ and $pp\rightarrow \bar{t}W^++X$ are significant large, which might be detected at the $LHC$.


Introduction
One of the main goals of the current or future high energy experiments, such as the LHC and ILC, is to search for new physics beyond the standard model (SM) [1]. Because of the largest mass of the top quark among all observed particles within the SM, it may be more sensitive to new physics than other fermions and it may serve as a window to probe new physics. Thus, studying the correction effects of new physics on observables about top quark is a good way to test the SM flavor structure and to learn more about the nature of electroweak symmetry breaking (EW SB) [2].
In the SM, top quark can be produced singly via electroweak interaction at hadron colliders. At leading order, there are three kinds of the partonic processes: the s-channel process (q ′q → tb) involving the exchange of a time-like W boson, the t-channel process (bq → tq ′ ) involving the exchange of a space-like W boson, and the tW production process (gb → tW − ) involving an on-shell W boson. These processes have completely different kinematics and can be observed separately [2]. Furthermore, the t-channel process is the main source of single top production, both at the T evatron and the LHC. At the T evatron, the contributions of the tW production process are very small, while the contributions from the s-channel production process are very small at the LHC. Thus, an accurate description of all the three production processes is important.
tW production at hadron colliders has been calculated at next leading order (NLO) in the SM [3] and been extensively studied in Refs. [4,5]. It has been shown that this process is observable at the LHC using the fully simulated data at the CMS and AT LAS detectors [6,7]. In the SM, the tW production channel is charge symmetric, which means that the production cross section for the process pp → tW − + X is equal to that for the process pp →tW + +X. However, the charge asymmetry in the tW production process can be generated by non-SM values of V td and V ts of CKM matrix [8] and by the anomalous top coupling tqg (q = u or c) [9].
In the SM, the anomalous top quark coupling tqg is absent at tree level and is extremely suppressed at one loop due to the GIM mechanism [10], which can not be de-tected in current or future high-energy experiments. However, it may be large in some new physics models beyond the SM, such as the topcolor-assisted technicolor (T C2) model [11,12], the littlest Higgs model with T -parity (called LHT model) [13], etc. In this paper, we will focus our attention on the anomalous top couplings induced by the T C2 model and the LHT model, and calculate their contributions to the production cross section and the charge asymmetry for tW production at the LHC with the center-of-mass (c.m.) energy √ s = 14T eV . Our numerical results show that the contributions of the anomalous top coupling tqg induced by the T C2 model to the tW process are generally smaller than those for the LHT model. With reasonable values of the free parameters of the LHT model, its corrections to the production cross sections of the processes pp → tW − + X and pp →tW + + X are in the ranges of 14% ∼ 32% and 11% ∼ 24%, respectively. The value of the charge asymmetry parameter R = σ(tW − )/σ(tW + ) can reach 1.05.
After discussing the anomalous top couplings tqg induced by the T C2 model and the LHT model, we calculate the additional contributions of these anomalous top couplings to the tW production channel at the LHC in sections 2 and 3. Our conclusions are given in section 4.
2. The T C2 model and tW production at the LHC The T C2 model [11] is one of the phenomenologically viable models, which has almost all essential features of the topcolor scenario [12]. This model has two separate strongly interacting sectors in order to explain EW SB and the large top mass. Technicolor interaction is responsible for most of EW SB via the condensation of technifermions, but contributes very little to the top mass εm t with the parameter ε ≪ 1. The topcolor interaction generates the bulk of m t through condensation of top pairs < tt >, but makes only a small contribution to EW SB.
The T C2 model predicts the existence of a number of new scalar states at the electroweak scale: three top-pions (π ± the topcolor interaction is not flavor-universal and mainly couples to the third generation fermions, the couplings of top-pions or top-Higgs to the three family fermions are non-universal, and they have large Y ukawa couplings to the third generation and can induce flavor changing (F C) couplings. The couplings of the top-pions (π 0 t , π ± t ) to ordinary fermions, which are related to our calculation, can be written as [11,12,14] is the physical top-pion decay constant, which can be estimated from the Pagels-Stokar formula. To yield a realistic form of the CKM matrix V CKM , it has been shown that the values of the matrix elements K ij U L(R) can be taken as [14] K tt In the following numerical estimation, we will assume K tc U R = √ 2ε − ε 2 and take ε as free parameter.
The relevant couplings for the top-Higgs h 0 t are similar with those of the neutral toppion π 0 t [14]. However, the coupling h 0 tc tt is very small, which is proportionate to a factor of ε/ √ 2 [15]. Furthermore, the mass of the techni-Higgs h tc is at the order of 1T eV .
Thus, the contributions of h tc to the tW production process can be safely neglected.
From the above discussions we can see that the neutral top-pion π 0 t and the top-Higgs h 0 t can generate the anomalous top coupling vertex tcg, which are shown in F ig.1. It is obvious that the effective vertex tcg can generate additional contributions to the tW production channel at the LHC. The relevant Feynman diagrams are shown in F ig.2.
Certainly, the neutral scalars π 0 t and h 0 t can also generate the anomalous top coupling vertex tūg via the F C couplings π 0 t (h 0 t )tū . However, it has been argued that the maximum F C mixing occurs between the third and second generation fermions, and the F C couplings π 0 t (h 0 t )tū is very small which can be neglected [14]. Similar to π 0 t , the charged Figure 1: Feynman diagrams for the effective vertex tcg in the TC2 model.
top-pions π ± t can also give rise to the anomalous top coupling tcg via the F C couplings π ± t bc. However, compared with those of π 0 t , the contributions of π ± t to the tcg coupling are approximately suppressed by the factor m 2 b /m 2 t , which can be safely neglected. Hence, in the following numerical estimation, we will ignore the contributions of π ± t to the tW production process. One of the authors for this paper has discussed the anomalous top coupling tcg induced by the T C2 model in Ref. [16]. The explicit expressions for the effective vertex tcg has been given in Ref. [16]. In this paper, we will use LoopT ools [17] and the CT EQ6L parton distribution functions (P DF s) [18] to calculate the contributions of the T C2 model to the tW production process. The renormalization and factorization scales (µ R and µ F ) have been taken equal to µ F = µ R = m t + m W . The masses of the top quark and the gauge boson W are taken as m t = 170.9GeV and m W = 80.42GeV [19]. It is obvious that the cross sections for the processes pp → tW − + X and pp →tW + + X are dependent on the free parameter ε and the masses of the top-pion and top-Higgs boson. From the theoretical point of view, ε with value from 0.01 to 0.1 is favored [11]. In this paper we will assume that its value is in the range of 0.03 ∼ 0.08. The masses of the neutral top-pion and top-Higgs boson are model-dependent and are usually of a few hundred GeV [12]. In our numerical estimation, we will take m π 0 t = m h 0 t = M and assume that the value of M is in the range of 200GeV ∼ 500GeV . To see whether the contributions of the anomalous top coupling tcg induced by the T C2 model to the tW production channel can be detected at the LHC, we define the relative correction parameters as where σ(tW + ) and σ(tW − ) denote the total production cross sections including the contributions from the SM and the T C2 model for the processes pp →tW + + X and pp → tW − + X, respectively. The charge asymmetry parameter R is defined as R = σ(tW − )/σ(tW + ). Since the P DF for the bottom quark in proton is same as that for the anti-bottom quark, there is R = 1 in the SM.
Our numerical results are summarized in F ig.3 and F ig.4, in which we plot the parameter R i as function of the mass parameter M for the c.m. energy √ s = 14T eV and three values of the free parameter ε. One can see from F ig.3 that there is a peak at M ∼ 330GeV , which is due to the effect of the tt in the loop going on-shell and the anomalous top coupling tcg increasing. In all of the parameter space of the T C2 model, the value of R + is smaller than that of R − and the value of the parameter R is larger than 1, which leads to an charge asymmetry for the tW production process. For 0.03 ≤ ε ≤ 0.08 and 200GeV ≤ M ≤ 500GeV , the corrections to the production cross sections of the processes pp →tW + + X and pp → tW − + X are in the ranges of 2.5% ∼ 5.2% and 3.7% ∼ 7.2%, respectively. The value of the charge asymmetry parameter R is in the range of 1.011 ∼ 1.018. It has been shown [6,7] that the production cross section of tW production at the LHC can be measured with precision of about 9.9% and 2.8% for 10f b −1 and 30f b −1 of integrated luminosity of data, respectively. Thus, it is impossible to detect the charge asymmetry induced by the T C2 model for the tW production process at the LHC even for the c.m. energy √ s = 14T eV .
3. The LHT model and tW production at the LHC Little Higgs theory [20] was proposed as an alternative solution to the hierarchy problem of the SM, which provides a possible kind of EW SB mechanism accomplished by a naturally light Higgs boson. In order to make the littlest Higgs model consistent with electroweak precision tests and simultaneously having the new particles of this model at the reach of the LHC, a discrete symmetry, T -parity, has been introduced, which forms the LHT model. The detailed description of the LHT model can be found for instance in Refs. [13,21,22], and here we just want to briefly review its essential features, which are related to our calculation.
The LHT model is based on an SU (5) After taking into account EW SB, at the order of v 2 /f 2 , the masses of the T -odd set of the SU(2) × U(1) gauge bosons are given as where v = 246GeV is the electroweak scale and f is the scale parameter of the gauge symmetry breaking of the LHT model.
Where the CKM matrix V CKM is defined through flavor mixing in the down-type quark sector, while the P MNS matrix V P M N S is defined through neutrino mixing. The Feynman rules of the LHT model have been studied in Ref. [22] and the corrected Feynman rules of Ref. [22] are given in Refs. [23,24]. To simplify our paper, we do not list them here.
From the above discussions, we can see that the flavor structure of the LHT model is much richer than the one of the SM, mainly due to the presence of three doublets of mirror quarks and leptons and their interactions with the ordinary quarks and leptons, which are mediated by the T -odd gauge bosons (A H , W ± H , and Z H ) and Goldstone bosons (η 0 , ω 0 , and ω ± ). Such new F C interactions can induce the anomalous top coupling tqg (q = c and u) in quark sector. The relevant Feynman diagrams for the effective vertex tqg are shown in F ig.5. To simplify our paper, we do not give the analytical expressions of the effective vertexes tcg and tūg here. The new coupling tqg can generate significant contributions to the F C top decays t → cg, t → cqg and the F C single top production processes pp →tc + X, pp → t + X, and pp → tg + X [25]. In this section, we will consider its contributions to tW production at the LHC. Similar with section 2, we use the LoopT ools [17] to give our numerical results in the ′ t Hoof t-Feynman gauge. In our calculation, we use the corrected Feynman rules including the high order ν 2 /f 2 terms and neglect the terms proportioning to m c /m t or m u /m t .   Refs. [21,22,26,27] have studied the impact of the LHT dynamics on the K, B, and D systems in considerable detail. They have shown that the LHT model can produce potentially sizable effects on the relative observables and its free parameters should be constrained. To simplify our calculation, in this paper, we only consider two scenarios for the structure of V Hd , which can easily escape these constraints, Case I: V Hd = I, V Hu = V + CKM , Case II: Thus, the correction effects of the anomalous top coupling tqg induced by the LHT model on the tW production cross section might be detected at the LHC. Although the value of the charge asymmetry parameter R induced by the LHT model is larger than that for the T C2 model, its value is smaller than 1.06. So, observing the charge asymmetry of tW production at the LHC induced by the LHT model is much challenge.

Conclusions
The tW production process is one of important single top production channels at the LHC. In the SM, the production cross sections of single top quark and single antitop quark in the tW channel are equal, i.e. R = σ(tW − )/σ(tW + ) = 1. However, the anomalous top coupling tqg can generate contributions to the cross sections σ(tW − ) and σ(tW + ), and further give rise to the charge asymmetry. If the correction effects of the new coupling tqg on the tW production channel are observed at the LHC, it will be helpful to test the flavor structure of the SM and further to probe new physics beyond the SM.
The T C2 model and the LHT model are two kinds of popular new physics models, which can generate the anomalous top coupling tqg. In the context of the T C2 and LHT models, we consider the correction effects of the new coupling tqg on the tW production channel at the LHC with the c.m. energy √ s = 14T eV . Our numerical results show that they can indeed generate significant contributions to the tW production process. The contributions of the anomalous top coupling tqg induced by the T C2 model to the tW production process are generally smaller than those for the LHT model. With reasonable values of the free parameters for the LHT model, its corrections to the production cross sections of the processes pp → tW − + X and pp →tW + + X can reach 32% and 24%, respectively. The value of the charge asymmetry parameter R = σ(tW − )/σ(tW + ) can reach 1.06.
The T C2 model and the LHT model can modify the W tb coupling and further produce correction effects on the tW production cross section [28,29]. However, their contributions to the production cross section of the process pp → tW − + X are equal to those for the production cross section of the process pp →tW + + X. Thus, such modification about the W tb coupling can not cause the charge asymmetry in the tW production process at the LHC.