Light axigluon and single top production at the LHC

The light axigluon model can explain the Tevatron $t\overline{t}$ forward-backward asymmetry and at the same time satisfy the constraints from the electroweak precision measurement and the $ATLAS$ and $CMS$ data, which induces the flavor changing ($FC$) couplings of axigluon with the $SM$ and new quarks. We investigate the effects of these $FC$ couplings on the s- and t-channel single top productions at the $LHC$ and the $FC$ decays $Z\rightarrow \overline{b}s+b\overline{s}$, $t\rightarrow c\gamma$ and $cg$. Our numerical results show that the light axigluon can give significantly contributions to single top production and the rare top decays $t\rightarrow c\gamma$ and $cg$.


Introduction
The standard model (SM) of particle physics has been proven to be extremely successful describing collider experimented data so far. Even the discovery of a Higgs-like particle [1,2] has confirmed the validity of the SM at the Fermi scale. However, the SM suffers from a key theoretical drawback, the so-called "hierarchy" problem, which means that it could be a low-energy effective theory valid only up to some cut-off energy scale Λ, about T eV scale. So new physics beyond the SM would be in an energy range accessible at the LHC and might be discovered in coming years, although, at the moment, there is not any collider hint of new physics at the LHC.
There are various new physics models extending the gauge group of the strong interaction sector give rise to massive color-octet vector boson, for example, the topcolor models [3] and chiral color models [4]. Other examples include the extra dimensional models [5] and technicolor [6], which predict the existence of the Kaluza-Klein (KK) gluons and technirhos, respectively. Among these color-octet vector bosons, the new paricles with axial-vector couplings to the SM quarks are called "axigluons", which might explain the anomalous forward-backward asymmetry (F BA) in the tt production observed at the Tevatron [7]. So far, there has been a significant amount of works to explain the tt F BA via axigluons, for example see [8,9,10,11,12,13]. Furthermore, the light axigluon A with a mass M A in the range from 100GeV to 400GeV can explain the tt F BA and satisfy the constraints from the AT LAS and CMS data [14,15], as long as its decay width is large and its couplings to the SM quarks are relatively small [9,10,11,12].
Top quark physics is expected to be a window to any new physics beyond the electroweak scale. At LHC energies, top quark is copiously produced both in pair and single productions, which allows for an unprecedented precision in the study of top observables, such as its couplings and rare decays [16]. At hadron colliders, single top quark production is an important process in probing the mechanism of electroweak symmetry breaking (EW SB), providing informations complementary to those that can be obtained from top pair production [17]. Single top production is also very sensitive to new physics effects, whose strength can be assessed by precise measurement of the production cross section.
Single top production at hadron colliders has been observed in three channels: schannel, t-channel [18,19] and tW associated production channel [20], which accord with the SM predictions within experimental uncertainties. AT LAS and CMS collaborations have started searching for the new physics effects on single top production.
Inspired by the solution of the light axigluon to the tt F BA, some axigluon-mediated phenomena are studied in this paper. We consider the contributions of the light axigluon with flavor changing (F C) couplings to the SM and new quarks to the F C decays Z → bs(bs), the s-and t-channel single top productions, and rare top decays t → cγ and cg in the context of the light axigluon model proposed by Tavares and Schmaltz [10].
The constraints on this new physics model from the electroweak precision observables and the relevant data given by hadron colliders are taken into account in our numerical calculations.
The rest of this paper is organized as follows: After reviewing the basic ingredients of the light axigluon model, in section 2, we calculate the contributions of the light axigluon to the F C decays Z → bs and bs. Corrections of the light axigluon to the cross sections of the s-and t-channel single top productions at the LHC are studied in section 3. The branching ratios of the rare top decays t → cγ and cg induced by light axigluon exchange are given in section 4. Section 5 is devoted to simple summary.

Light axigluon and the F C decays Z → bs and bs
The light axigluon model [10] is based on the gauge group G = SU (3) is the conventional electroweak group and the extended gauge group SU(3) 1 × SU(3) 2 is spontaneously broken to the QCD gauge group SU(3) C by the vacuum expectation value (V EV ) of a bifundamental scalar φ. This breaking pattern yields two mass eigenstates of color-octet gauge bosons. One is massless particle, which can be identified with the SM gluon, and the other is massive particle, which is called the light axigluon A. For its couplings to the SM quarks, there are the vector coupling g V ≈ 0 and the axial-vector coupling g A = 0 in the case of assuming approximately parity symmetry. In order to cancel the gauge anomaly, the extra up-and down-type quarks are introduced into this model, and the lepton sector is exactly same as that of the SM. To explain the tt F BA, the axigluon A should have mass below 450GeV , while should be broad with Γ A /M A ∼ 10 ∼ 20%, where Γ A and M A represent its total decay width and mass, respectively.
In the original light axigluon model [10], the authors assume the existence of an exact global symmetry of the axigluon couplings, and thus the light axigluon only has flavor universal couplings to the SM quarks. In fact, this global symmetry is only approximate and there is mixing between new and ordinary quarks, which can induce flavor changing neutral currents (F CNCs) at tree level [21]. The new and ordinary quarks have same SU(2) × U(1) charge, their mixing does not give rise to the F C Z couplings at tree level.
The new scalars can not induce F CNCs, thus the non-universal axigluon couplings are the main source of F CNC for this model.
In this paper we will not assume the existence of an exact global symmetry of the axigluon couplings, which allows F C couplings of the axigluons to the SM quarks. If one assumes that these F C couplings are only axial-vector couplings, which are similar with their flavor conserving couplings to the SM quarks, then the axial-vector couplings of the light axigluon to the SM quarks can be general given by the Lagrangian where A µ is the light axigluon, g s is the QCD coupling constant, u i and d i are the SM up-and down-type quarks, respectively. In above equation, we have neglected the color and spinor indices. g u i A and g d i A are the flavor independent coupling constants and there are g u i A = g d i A = g q A [10]. The F C coupling constants ε ij u and ε ij d , which arise from flavor symmetry breaking of new and light quarks, are given by the matrices The couplings of the axigluon to a pair of ordinary quarks and to the corresponding partners have opposite sign. So, in order to get suppressed couplings of the ordinary quarks to the axigluon, the extra quarks and the SM quarks should have mixing [10,12,22]. The mixing can be obtained by adding a Yukawa coupling involving a scalar field φ in addition to the quark field of Q ′ with Q. After the spontaneous breakdown of (3) C induced by the V EV for φ, the new quarks from the line combinations of Q ′ and Q get masses, while their orthogonal combinations correspond to the SM quarks remain massless, which get masses from the SM Higgs V EV via Yukawa couplings. In the mass eigenstates, the mixing couplings of the axigluon to ordinary and new quarks, which are assumed to be axial-vector couplings, can be general written as U Hi and D Hi represent the up-type and down-type new quarks, respectively. For the mixing coupling constant g mix A , there is the relation (g mix A ) 2 + (g q A ) 2 = 1. For the two matrices ε Hu and ε Hd , they are related through the SM CKM matrix: ε + Hu ε Hd = V CKM , which is similar with the case for the mixing between the T-odd and T-even quarks in the LHT model [23]. In this paper, we assume that both ε Hu and ε Hd are nearly equal to the identity matrix, which provides us with a set of minimal flavor mixing scenarios.
We take as examples two simple cases: In case I, the mixing coupling g Qq A has no contributions to D 0 − D 0 mixing, while contributes to B 0 q − B 0 q and K 0 − K 0 mixings. For case II, it is obvious that the mixing coupling g Qq A can only contribute to D 0 − D 0 mixing. Reference [21] has obtained the constraints on the mixing matrix ε d by using the available data from neutral meson Taking into account of these constants, in this section, we calculate the branching ratios of the F C decays Z → bs and bs given by axigluon exchange as shown in Fig.1. The self-energy diagrams Fig.1(b) and (c) contribute a finite field renormalization and the individual diagrams are finite [24]. To fulfill the broad width of the axigluon, the first and second generation new quarks should be degenerate and lighter than the axigluon, while the third generation new quarks must be heavier [10]. So we think that the contributions of the third generation new quarks to the F C decays Z → bs(bs) decouple and only consider the contributions of the first and second generation new quarks. In our numerical estimation, we will take In this case, one can safely neglect the phase space suppression effect for the axigluon decaying to one new quark and one ordinary quark The light axigluon model predicts the existence of new scalar, which also has the mixing couplings to new and ordinary quarks. However, it can not induce F C couplings at tree level and thus in this paper we neglect the effects of the new scalar on the F C processes Z → bs and bs.
The corrections of color-octet gauge boson to the Zbb coupling are firstly studied by Ref. [25] in the context of topcolor models, which contain only the leading-logarithmic contributions. The full one-loop results for the corrections of the axigluon to the Zbb coupling are given in Refs. [11,12] in the case of neglecting the bottom quark mass.
Ref. [12] have further computed the contributions from new quarks and new scalar to the Zbb coupling and find that the two kinds of contributions have opposite sign and the effect of new scalar is much smaller than that of new quarks. Following Refs. [11,12], we can straightforwardly calculate the contributions of the light axigluon model to the F C couplings Zbs and Zbs. Then, the effective Zbs coupling can be written as where P = L and R. g Zbb P and g Abb P represent the couplings of the gauge boson Z and axigluon A to the bottom quark pairs, respectively. The explicit expressions of the factors κ(x Z ) and κ(x z , x h ) have been given in Ref. [12]. Since the couplings of the axigluon to pair of ordinary quarks and pair of new quarks are flavor universal and the new and ordinary quarks have same SU(2) × U(1) charge, in above equation we have added the contributions of the ordinary quarks b and s, and taken where i = 1 and 2, S W = sin θ W and C W = cos θ W , θ W is the Weinberg angle. The F C coupling g Abs P can contribute to B 0 s − B 0 s mixing at tree level and its upper bound has been obtained by Ref. [21] as |g bs L | = |g bs R | = |g bs A | ≤ 1.83 × 10 −3 . In fact, for the case I, the new quarks can also generate contributions to B 0 s − B 0 s mixing via box diagrams that contain the light axigluon and new quark. However, the contributions from box diagrams are suppressed with respect to axigluon tree-level contributions by a loop factor 1/(16π 2 ) and two additional mixing matrix elements ε i3 Hd and ε i2 Hd . Therefore they cannot compete with the latter and are negligible. As numerical estimation, we will take g bs In the SM, the F C decay Z → bs + bs originates from one loop diagrams with branching ratio ∼ 3 × 10 −8 [26]. For future linear collider (ILC), the expected sensitivity to the branching ratios of rare Z decays can be improved from 10 −5 at the LEP to 10 −8 at the Giga Z [27]. The new physics effects might be detectable via Z → bs if it indeed affects this decay. A lot of theoretical studies involving the F C decay Z → bs have been given within some popular models beyond the SM, where its branching ratio can be significantly enhanced [28].
Using the effective couplings g Zbs L and g Zbs R given by Eq.(4), we can easily calculate the partial width Γ(Z → bs + bs). The numerical results for the branching ratio Br(Z → bs + bs) = Γ(Z → bs + bs)/Γ total are shown in Fig.2, in which we have taken the SM input parameters as: α s (m Z ) = 0.118, S 2 W = 0.231, Γ total = 2.4945GeV , and M Z = 91.1875GeV [29]. If the light axigluon can explain the tt F BA and at the same time satisfy the constraints from the electroweak precision observables and the relevant data given by hadron colliders, its mass should be in the range of 100GeV ∼ 400GeV , its total decay width Γ A t = (0.1 ∼ 0.2)M A and the flavor conserving coupling g q A might be in the range of 0.3 ∼ 0.5 [9,10,11,12]. In our numerical estimation we have considered the effects of the axigluon width and taken Γ A t = 0.1M A . For the mixing between the SM and new quarks, we have taken case I and assumed M H = 0.2M A . One can see from Fig.2 that, in most of the parameter space, the value of the branching ratio Br(Z → bs + bs) is smaller than 1 × 10 −8 , which is still below the SM prediction. So considering the constraints of B 0 s − B 0 s mixing on the F C coupling g bs A , the contribution of the light axigluon to the rare decays Z → bs and bs is very difficult to be detected in near future. Certainly, if we assume ε Hd = V CKM , the numerical results should has some changes.
3. The F C couplings of the light axigluon A and single top production at the Figure 3: Leading order Feynman diagrams for tb and tj production contributed by the F C couplings g tq A , in which q = u, c, q ′ = d, s, b, and q ′′ = d, s.
In the SM, single top production dominantly occurs through electroweak processes, which are customary divided into three production channels: t-channel exchange of a space-like W boson, s-channel production and decay of a time-like W boson, and associated production of a top quark and an on-shell W boson. These partonic processes have their own distinct kinematics and do not interfcere with each other. Both at Tevatron and the LHC, the t-channel process is dominant one, which in five flavor (5F ) scheme proceeds via the partonic processes qb → q ′ t and qb → q ′ t for single top production, and qb → q ′ t and qb → q ′ t for single antitop production. The s-channel partonic processes are qq ′ → tb and qq ′ → tb for single top and antitop productions, respectively. The contributions of charged and neutral color-octet vector bosons to top pairs and single top production has been studied in Refs. [13,30]. In this section we will consider the corrections of the light axigluon to the s-and t-channel single top productions via the F C couplings g tq A with q = u or c. The relevant Feynman diagrams are shown in Fig.3.
For the partonic process qb → tb as shown in Fig.3 (a), the differential cross section with respect to emerging angle of the single top quark cos θ t can be written as The partonic process qq → tq is composed of the s-and t-channel diagrams corresponding to Fig.3 (b) and 3 (c). Its differential cross section is given by The differential cross section of the t+u channel partonic process qq → t+q can be written The differential cross section for the s-channel partonic q ′ q ′ → tq as shown in Fig.3 (e) is given by The explicit expression of the differential cross section for the t-channel qq ′′ → tq ′′ is same as that for the process qb → tb, as long as replace the initial state b quark by the quark q ′′ (d or s). In above equations, β = 1 − m 2 t s ,ŝ,t, andû are the usual Mandelstam variables, Using above equations we can calculate the cross sections of tb and tj production at the LHC induced by the light axigluon with the F C coupling g tq A . In our numerical calculations, we use the leading order parton distribution function of CTEQ6L1 [31] and choose the factorization and renormalization scales to be µ f = µ r = m t /2 with m t = 173GeV . Our numerical results are added tb and tb for the process pp → tb, and similar for tj production with j = u, c, d, and s. It is obvious that the production cross sections depend on the mass parameter M A , the coupling parameters g tq A and g q A , where we have taken g tu A = g tc A and the flavor conserving coupling g q A being flavor universal. In the SM, single top production at hadron colliders was first considered in Ref. [32]. Now the production cross sections for the s-and t-channels have been calculated up to next-to-next-to leading logarithm (NNLL) accuracy [33]: σ s = 1.04 ± 4% pb and σ t = 2.26 ± 5% pb at Tevatron with the centre-of-mass (c.m.) energy √ s = 1.96T eV and σ s = 12 ± 6% pb and σ t = 243 ± 4% pb at the LHC with √ s = 14T eV . The s-and t-channel cross sections have been measured at Tevatron by CDF and DO collaborations and the measurement precision can reach 18% [18]. The measurement precision for the t-channel cross section at the 8T eV LHC reported by AT LAS and CMS is about 15% [19]. It will be enhanced in coming years. For example, Ref. [34] has shown that the cross section of the t-channel single top production at the 14T eV LHC can be measured with a precision of 5%. From above discussions we can see that the theoretical error of the SM NNLO cross section at the 14T eV LHC for the s-and t-channel productions could be as large as 5%, the same amount of the expected precision at the 14T eV LHC. So if the relative correction of the light axigluon to the single top production cross section is larger than 10%, the 14T eV LHC should detect this correction effect. In Fig.4 and Fig.5 we demand that δσ s /σ s SM = 10% and δσ t /σ t SM = 10%, where σ s SM and σ t SM are the SM NNLO predictions for the s-and t-channel single top production cross sections at the LHC with √ s = 14T eV , δσ s and δσ t are induced by the light axigluon A, and plot the F C coupling g tq A as a function of the mass parameter M A for different values of the flavor conserving g q A . In our numerical calculation, we have taken the central values for σ s SM and σ t SM . From these figures one can see that the contributions of the light axigluon to the production cross sections of the processes pp → tb + X and pp → tj + X increase as the coupling parameters g tq A and g q A increasing, while decrease as M A increasing. For 100GeV ≤ M A ≤ 400GeV and 0.3 ≤ g q A ≤ 0.5, the values of F C coupling g tq A are in the ranges of 0.017 ∼ 0.163 and 0.024 ∼ 0.139 for δσ s /σ s SM = 10% and δσ t /σ t SM = 10%, respectively. We expect that, in near future, the LHC can authenticate this correction effect on single top production or at least give constraint on the F C coupling g tq A .
4. The light axigluon and the rare top decays t → cγ and cg It is well known that in the SM the rare top decays t → qV (q = u, c and V = γ, g, Z) mediated by F CNCs are highly GIM suppressed with branching ratios of Br(t → cV ) ∼ 10 −14 ∼ 10 −12 [35], which are far below the detectable level of current or near future experiments. However, some new physics models can enhance these branching ratios significantly [36]. On the experimental side, rare top decays are being searched for at Tevatron [37] and LHC [38,39]. AT LAS collaboration has set upper limit on the branching ratio Br(t → cg) < 2.7 × 10 −4 at 95% C.L. [39]. The sensitivity of AT LAS to the branching ratio Br(t → cγ) is expected to be of the order of 10 −4 [40].
From discussions given in above sections we can see that the light axigluon with F C couplings can contribute rare top decays. In this section we will calculate the branching ratios Br(t → cγ) and Br(t → cg) induced by the light axigluon. The relevant Feynman diagrams are shown in Fig.6. In this section, we also assume that the contributions of the third generation new quarks to the rare top decays t → cγ and t → cg decouple.
Compared to the F C couplings of the light axigluon A to the new quarks and the SM quarks, the F C couplings of the scalar φ to the new quarks and the SM quarks arise at higher order, their F C effects are much smaller than those induced by the axigluon A.
Thus, in this section, we neglect the contributions of the scalar φ to the rare top decays t → cγ and t → cg as done for Z → bs in section 2.
Considering electromagnetic gauge invariance, the amplitude of the rare decay t → cγ can be general written as where q = P t − P c is the photon momentum and ε is its polarization vector, in which P t and P c represent the momenta of top and charm quarks, respectively. A similar structure is valid for t → cg with form factors A g and B g . For the light axigluon A with zero vector couplings to the SM and new quarks i.e. g tq V ≈ 0, g Q H q V ≈ 0 and g q V ≈ 0 [10,12], there are A γ = 0, A g = 0 and B γ = 0, B g = 0. Recently, Ref. [41] has calculated the contributions of color-singlet gauge bosons predicted by the 331 models to the rare top decay t → cγ and give the explicit expressions for the relevant form factors. In this paper we will use LoopTools [42] to obtain our numerical results.
Using Eq. (11), the partial widths of t → cγ and t → cg contributed by the light axigluon can be written as where C F = 4/3 is a color factor.
To obtain numerical results, we have assumed that the top total decay width is dominated by the decay t → W b. The F C coupling g tc A is determined by the parameters g q A and M A via the relation δσ t /σ t SM = 10%. For calculation the contributions of the first and second generation new quarks, we take the case II: ε Hd = I, ε Hu = V CKM and assume M H = 0.2M A . In Fig.7 and Fig.8 we plot the branching ratios Br(t → cγ) and Br(t → cg) as functions of the axigluon mass M A for three values of the flavor conserving coupling g q A . One can see from these figures that the light axigluon A can indeed enhance the branching ratios Br(t → cγ) and Br(t → cg). For 0.3 ≤ g q A ≤ 0.5 and 100GeV ≤ M A ≤ 400GeV , the values of Br(t → cγ) and Br(t → cg) are in the ranges of 4.8 × 10 −9 ∼ 5.9 × 10 −8 and 1.1 × 10 −8 ∼ 1.3 × 10 −6 , respectively. Replacing the F C couplings g tc A and g U i c A by g tu A and g U i u A , we can easily calculate the contributions of the light axigluon A to the rare top decays t → uγ and ug.

Conclusions
The light axigluon A with a mass M A in the range from 100GeV to 400GeV predicted by the light axigluon model [10] can explain the tt F BA and satisfy the constraints from the AT LAS and CMS data, as long as its decay width is large and its couplings to the SM quarks are relatively small. In order to get suppressed couplings of the light axigluon A to the SM quarks, the new quarks and the SM quarks should have mixing, which can induce the F C couplings to the new quarks and the SM quarks. Furthermore, to fulfill the broad width of the axigluon, the new quarks, at least the first and second generation new quarks, are lighter than the light axigluon. In this paper, we assume the flavor conserving axigluon couplings are universal and pure axial vector-like, and investigate some F C phenomena mediated by the light axigluon.
The contributions of the light axigluon model to the F C decays Z → bs, bs and t → cγ, cg mainly come from the F C quark-quark-axigluon coupling g qq ′ A and the F C quark-new quark-axigluon coupling g qQ H

A
. Considering the constraints of meson mixing on the F C coupling g qq ′ A and assuming that both ε Hu and ε Hd are nearly equal to the identity matrices and satisfy the relation ε + Hu ε Hd = V CKM to give the value of g qQ H A , we calculate the branching ratios Br(Z → bs+bs), Br(t → cγ) and Br(t → cg) in the context of the light axigluon model. Our numerical results show that, in most of parameter space, the value of the branching ratio Br(Z → bs + bs) is smaller than 1 × 10 −8 , which is still below the SM prediction. Compared to the SM predictions, the branching ratios Br(t → cγ) and Br(t → cg) can be significantly enhanced in the light axigluon model, while are still lower than the corresponding current experimental upper limits.
It is well known that single top production is very sensitive to new physics beyond the SM, whose effects can be assessed by precise measurement of the production cross section. In this paper, we study the correction effects of the light axigluon A to the s-and t-channel single top productions at the LHC. We find that, in near future, the LHC should observe this correction effect with reasonable values for the F C coupling g tq A or at least give constraint on the F C coupling g tq A . If one demands δσ s /σ s SM = 10% and δσ t /σ t SM = 10%, the values of the F C coupling g tq A should be in the ranges of 0.017 ∼ 0.163 and 0.024 ∼ 0.139, respectively.