A general analysis of Wtb anomalous couplings *

: We investigate new physics e(cid:11)ects on the Wtb e(cid:11)ective couplings in a model-independent framework. The new physics e(cid:11)ects can be parametrized by four independent couplings, f L 1 , f R 1 , f L 2 and f R 2 . We further introduce a set of parameters x 0 , x m , x p and x 5 which exhibit a linear relation to the single top production cross sections. Using recent data for the t -channel single top production cross section (cid:27) t , tW associated production cross section (cid:27) tW , s -channel single top production cross section (cid:27) s , and W-helicity fractions F 0 , F L and F R collected at the 8 TeV LHC and Tevatron, we perform a global (cid:12)t to impose constraints on the top quark e(cid:11)ective couplings. Our global (cid:12)tting results show that the top quark e(cid:11)ective couplings are strongly correlated. We show that (i) improving the measurements of (cid:27) t and (cid:27) tW is important in constraining the correlation of ( f R 1 ;f R 2 ) and ( f L 2 ;f R 2 ); (ii) f L 1 and f R 2 are anti-correlated, and are sensitive to all the four experiments; (iii) f R 1 and f L 2 are also anti-correlated, and are sensitive to the F 0 and F L measurements; (iv) the correlation between f L 2 and f R 2 is sensitive to the precision of the (cid:27) t , (cid:27) tW and F 0 measurements. The e(cid:11)ective Wtb couplings are studied in three kinds of new physics models: the G (221) = SU (2) 1 (cid:10) SU (2) 2 (cid:10) U (1) X models, the vector-like quark models and the Littlest Higgs model with and without T -parity. We show that the Wtb couplings in the left-right model and the un-uni(cid:12)ed model are sensitive to the ratio of gauge couplings when the new heavy gauge boson’s mass ( M W 0 ) is less than several hundred GeV, but the constraint is loose if M W 0 > 1 TeV. Furthermore, the Wtb couplings in vector-like quark models and the Littlest Higgs models are sensitive to the mixing angles of new heavy particles and SM particles.


Introduction
The discovery of the Higgs boson at the Large Hadron Collider (LHC) [1,2] completes the Standard Model (SM) of particle physics. So far, no evidence of new physics (NP) has been observed at the LHC. It is possible that the scale of new physics Λ NP is much higher than the electroweak symmetry breaking (EWSB) scale (v = 246 GeV). If so, we expect to look for indirect effects of NP with the SM particles which we know. The top quark, discovered at the Tevatron [3,4], is the heaviest particle in the SM. With its mass around the EWSB scale, the top quark is believed to play an important role in connecting the SM and NP. The Wtb coupling plays a pivotal role in top quark physics. The top quark decay and single top quark production processes are sensitive to the Wtb coupling. In addition, it offers a very promising way to probe for NP at the LHC. For instance, the production rate of single top quarks and the polarization of the top quark can be modified by NP beyond the SM, such as new gauge boson W s, vector-like fermions, etc. Those heavy particles are predicted in many NP models, such as SU (2) 1 ⊗SU (2) 2 ⊗U (1) X , often denoted as G(221) models [5,6], vector-like quark models [7][8][9][10], Little Higgs models [11][12][13], warped/composite simpli-fied models [14] and many others. It is convenient to classify the underlying theories according to different approaches that modify the Wtb coupling. A simple case is that the new heavy particles mix with SM particles at tree-level, like a new gauge boson W or a new fermion T(B). Another way to generate the anomalous Wtb coupling is through the loop-level. Typical examples are supersymmetric models [15,16] and two-Higgs-doublet models [17].
The top quark decays before hadronization as its lifetime is much smaller than the typical hadronization time scale. In the SM, the dominant decay mode of top quark is t → W + b, so the Wtb coupling governs the top quark decay process. One way to probe the structure of the Wtb coupling is to study the helicity fractions of the W boson in top quark decay. The W-boson helicity fractions are defined as the partial rate for a given helicity state divided by the total decay rate: F L,R,0 ≡ Γ L,R,0 /Γ , where F L , F R , F 0 are the left-handed, right-handed and longitudinal helicity fractions, respectively. The W helicity fractions, calculated to the accuracy of the next-to-nextto-leading order (NNLO) QCD corrections in the SM, are F 0 = 0.687±0.005, F L = 0.311±0.005, F R = 0.0017±0.0001 for a top quark mass of m t = 172.8 ± 1.3 GeV [18,19]. Recently, the ATLAS and CMS collaborations have measured the W helicity fractions in top quark decays at √ s = 7 TeV, and CMS 8 TeV results are also available. These results are consistent with the SM predictions [19,20].
At hadron colliders, the top quark can be produced singly in three channels: the t-channel process (ub → td), the tW associated production process (bg → tW − ), and the s-channel process (ud → W + → tb). The t-channel cross section is predicted at next-to-next-toleading order plus the contribution due to the resummation of soft-gluon bremsstrahlung (NNLO+NNLL) to be σ t = 87.8 +3.4 −1.9 pb [21,22] at the LHC with √ s = 8 TeV. The theoretical predictions for the tW-channel and s-channel at √ s = 8 TeV at NLO+NNLL precision in QCD are σ Wt = 22.4 ± 1.5 pb [23] and σ s = 5.6 ± 0.2 pb [24], respectively. The cross sections at √ s = 13 TeV are σ t = 221 +6 −2 ± 3 pb, σ Wt = 72.6 ± 1.3 ± 1.3 pb and σ s = 11.29 ± 0.18 ± 0.26 pb [25]. Recently, both ATLAS and CMS collaborations have measured the single top quark production cross sections at √ s = 7 TeV, √ s = 8 TeV and √ s = 13 TeV. These are summarized in Sec. 3. In this work, we use the effective field theory (EFT) to calculate the single top quark production cross sections and W helicity fractions, and use the recent experimental data (denoted as Wtb measurements) to determine the general Wtb couplings. Model independent analyses of the Wtb couplings have already been performed using the EFT approach, see for example in Refs. . In our work, we compute the deviation from the SM by including the dimension-6 operators for top quark decay and single top quark production processes. We incorporate the quadratic terms of dimension-6 operators to obtain the correlations among different operators in the single top quark cross section and W helicity fraction calculations. The constraints on Wtb couplings based on some of the recent experimental data were studied in Refs. [46,47]. We perform a global fit of the general Wtb couplings by analysing the correlations among different couplings and discuss the implication of the top quark effective theory for several NP models.
The paper is organized as follows. In Section 2 we calculate the single top quark production cross sections via an effective field theory approach. In Section 3 we present the allowed parameter space of the general Wtb couplings after incorporating the most recent ATLAS and CMS results. In Section 4 we discuss the constraints on various new physics models from the Wtb couplings. Finally, we conclude in Section 5.

Dimension-6 operators and effective Wtb couplings
Using the EFT approach to explore possible NP effects has been discussed widely, see for example Refs. [27,29,30,32,40,[50][51][52][53][54][55][56][57]. A model-independent way to parametrize the low energy effects of NP theories is the linearly realized effective Lagrangian, which incorporates the particle content and symmetries of the SM. The nonlinearly realized mechanism of the electroweak symmetry is studied in Refs. [27,51,53,54]. In this paper, we assume the new scalar particle observed at the LHC is the SM Higgs boson, and use the linear realization to parametrize the NP effects [50,55]. The effective Lagrangian before electroweak symmetry breaking is where L SM is the SM Lagrangian, Λ is the characteristic (2) where q T L = (t, b) L denotes the SU (2) L weak doublet of the third generation left-handed quark fields, t R and b R are SU (2) L weak singlets of right-handed top and bottom-quark fields respectively, φ is a SU (2) L weak doublet of the Higgs field, defined φ T = 1 √ 2 (0, v+h) with v = 246 GeV in the unitarity gauge withφ = iτ 2 φ * , and D µ = ∂ µ −ig(τ I /2)W I µ −ig B µ Y is the covariant derivative, where g and g are gauge couplings of SU (2) L and U (1) Y , respectively, and Y is the hypercharge of the field to which D µ is applied.
are the strength tensors of the SU (2) L gauge fields, ε IJ K denotes the structure constants, and τ I is the usual Pauli matrix. Three types of dimension-6 operators contribute to the Wtb couplings. The first type is operators involving a scalar field carrying one covariant derivative, the second is operators involving fermion and scalar fields both carrying one covariant derivative, and the third is operators involving field strength tensors.
The operators O (3) φq and O φφ are of the first type. They can be generated at tree-level after integrating out the new heavy particles, such as a heavy charged vector boson (W ± ) that mixes with SM gauge boson W ± [5,6] or a heavy quark that mixes with top quark or bottom quark [9,10]. A pictorial illustration of the relation between the tree-level effective operators and the possible NP models is shown in Fig. 1(a) and Fig. 1(b). We use the bold-red line to denote the NP particles. The anomalous Wtb couplings arise at tree-level after spontaneous symmetry breaking with φ = v/ √ 2. The operators O Dt , OD t , O Db , OD b fall into the second category. Those operators give a contribution of order p 2 /Λ 2 , where p is the typical momentum scale in the process. Such operators, corresponding to the vertices involving three external lines, can be induced only at loop-level after integrating out the heavy particles [58].
The operators O tW , O bW , O qW are of the third type. Usually, effective operators which involve the field strength tensor are generated only at loop level if the complete theory is a gauge theory [58]. Typical examples include supersymmetric models [15,16,59], two-Higgsdoublet models [17], etc.
A pictorial illustration of the relation between loopinduced effective operators and the possible underlying theories is shown in Fig. 1(c-e). The Wtb coupling can be induced in the NP models with extended gauge structure, which yields extra gauge bosons (see Fig. 1(c)), or new scalar particles which consist of a new charge scalar and neutral scalar (see Fig. 1(d)), or new fermions which carry one discrete quantum number to avoid mixing with the top quark and bottom quark at tree-level (see Fig. 1(e)). We will comment on the impact of Wtb measurements on several NP models in Section 4.
As usual, the effective Lagrangians consist of redundant terms, which can be removed by the classical equations of motion [60]. This is based on the equivalence theorem of the S matrix. After we use the equations of motion to remove redundant operators, the relevant operators reduce to Assuming new anomalous couplings arise from the above dimension-6 operators, we can parametrize the general effective Wtb couplings as [36] where P L,R = (1∓γ 5 )/2 are the usual chirality projectors, m W is the W-boson mass and the Cabibbo-Kobayaski-Maskawa matrix element V tb is taken to be 1 in our analysis. In the SM, the values of the coefficients f i vanish at tree-level. Those couplings could be generated if NP exists. Although those couplings can in general be complex quantities, we assume the four coefficients are real in our calculation 1) . The coefficients of the effective Wtb couplings are related to the Wilson coefficients of the dimension-6 operators as follows [29]:

Single top production
In this section we discuss the contribution of general effective Wtb couplings to the total cross sections of single top production and the W helicity fractions in top quark decay. In this work, we focus on the four independent operators, see Eq. (3). The top quark can be produced singly through the electroweak interaction. Depending on the kinematics of the W boson involved, single top production is usually sub-categorized into three channels: s-channel production (q 2 W > 0), tchannel production (q 2 W < 0), and tW associated production (q 2 W = m 2 W ), where q W denotes the four momentum of the W boson.
We separate the total cross section of the single top production into the SM contribution plus the contributions from anomalous Wtb couplings where σ SM i denotes the cross section of the i-channel (i = s, t, tW) single top production in the SM with the approximate NNLO QCD correction, and ∆σ i denotes the variation from the SM prediction induced by the anomalous couplings at tree level. The K-factor, defined as K ≡ σ SM NNLO /σ SM LO , describes the approximate NNLO QCD corrections in the SM. We assume the anomalous couplings receive exactly the same corrections as the SM processes. Following Ref. [36] we write the contributions of anomalous couplings to the single top productions as where we reparametrize the four coefficients f L,R with a t ≡ m t /m W . Note that the terms proportional to the bottom quark mass have been ignored in our calculation due to the suppression factor (m b /m W ) 2 2) . The coefficients (a i , b i and c i ) depend on the collider type and energy and have to be calculated numerically. Ref. [36] calculated the a i 's and b i 's at the Tevatron Run II and 14 TeV LHC. In this work we update both the a i 's and b i 's using the CTEQ6L parton distribution functions (PDFs) [62] at the 7 TeV, 8 TeV and 13 TeV LHC. The numerical values of the coefficients are shown in Table 1.
1) The complex anomalous couplings have been studied in Ref. [47], and the ATLAS collaboration also give a constraint on the imaginary part of f R 2 in Ref. [61]. 2) We deem that neglecting the bottom quark mass in the present work is reasonable because inclusion of the bottom mass only induces a tiny asymmetry of about 0.01 ∼ 0.02 [46,47] in the allowed region of f L,R 1,2 , whereas the length of the marginal allowed region of one variable in the global fit is typically O(1), see Fig. 2. Even if only one parameter such as f R 1 is allowed to vary and all the other parameters are turned off, the length of the allowed region reaches about 0.2 (see Fig. 5) which is still larger than the amount of asymmetry induced by the bottom mass. To obtain the coefficients (c 0 , c m , c p , c 5 ) of the tWchannel single top production, we rewrite the cross section in terms of different combinations of the effective couplings where σ 0 denotes the SM LO prediction while σ N 1 and σ N 2 represent the partial cross section that is proportional to f R 2 and (f R 2 ) 2 , respectively. The cross sections σ 0 , σ N 1 and σ N 2 are obtained by integrating out the final state phase space, and then convoluting with the initial state PDFs. The coefficients (c 0 , c m , c p , c 5 ) are then determined from σ 0 , σ N 1 and σ N 2 as follows: The numerical values of the coefficients are given in Table 1 at the 7 TeV, 8 TeV and 13 TeV LHC. With the help of the coefficients a i , b i and c i , the contributions of the Wtb anomalous couplings can be written as: where β i = a i , b i , c i denote the single top quark cross section coefficients of different channels.

W helicity fractions in top decay
For completeness, we also list the fractions of the W helicity in the top quark decay in terms of x i [36], where F 0 , F L and F R represent the fractions of W bosons with longitudinal polarization (W 0 ), left-handed polarization (W L ) and right-handed polarization (W R ). x 0 represents the contribution from the W boson longitudinal polarization, and x m and x p denote the contributions from W boson left and right handed polarization, respectively.
Neglecting terms which are proportional to the bottom quark mass, the tree level results of the W helicity  fractions in the SM are In the SM the top quark decays predominantly into the longitudinal W boson because the coupling of the top quark to the longitudinal W boson is similar to the Yukawa coupling, which is proportional to the top quark mass. The top quark cannot decay into a right-handed W boson owing to the purely left-handed Wtb coupling in the SM. When the bottom quark mass is ignored, the right-handed W boson is forbidden by angular momentum conservation. However, the anomalous coupling f R 1 or f L 2 can yield a right-handed W boson in top quark decay. Different from the f R 1 coupling, the f L 2 contribution to a W R involves flipping the chirality of the top quark, which gives rise to a factor of a t . Therefore, F R is proportional to Similarly, the f R 2 coupling can also produce a W L in the top quark decay by flipping the top quark's chirality. As a result, F L is proportional to All of the four effective Wtb couplings can generate a W 0 in the top quark decay.
3 Global fit of the effective Wtb couplings

Experimental data and statistical analysis
The single top production cross sections and the W helicity fractions have been measured at the Tevatron and the LHC. The best measurement of the cross section of the s-channel single-top production is given at the Tevatron at √ s = 1.96 TeV with luminosity 9.7 fb −1 [75].
We also consider the updated experimental results of the t-channel and tW-channel cross sections and W -helicity measurements at both the CMS and ATLAS collaborations. All the experimental data are summarized in Table 2.
We perform a global χ 2 test to obtain the present constraints on the effective Wtb couplings. In the statistical analysis, the χ 2 is defined as where O exp i and O th i are the experimental values and the theoretical predictions for the experimental observable i, respectively. δσ i represents the total error of the experimental measurement, which is defined as The CERN library MINUIT [83] is used in our analysis to obtain the best-fit values of the effective Wtb couplings and the contours at different confidence levels (C.L.).
In this work we consider both direct and indirect constraints on the effective Wtb couplings. The direct constraints arise from the experimental measurements of top quark productions and decays, while the indirect constraints arise from precision measurements of flavor physics. For example, the anomalous Wtb cou-plings can contribute to flavor changing neutral current (FCNC) processes through quantum effects involving top quarks inside the loop. In particular, the inclusive decaȳ B → X s γ provides very stringent bounds on the anomalous Wtb couplings [84].

Constraints on the effective couplings and operators
Below we present the allowed regions of the effective Wtb couplings based on a global-fit analysis of all the four effective Wtb couplings f L,R 1,2 . We plot in Fig. 2 the allowed parameter space for the effective Wtb couplings at 68%, 90% and 95% C.L., and investigate the correlations among the effective Wtb couplings. We note that the allowed parameter space on the plane of the (f L 1 , f R 1 ) and (f L 1 , f L 2 ) contours is mainly driven by the t-channel and tW-channel cross sections. Figure 2(a) shows the allowed parameter space of (f L 1 , f R 1 ). The cross sections of the single top quark production processes are proportional to (1 + f L 1 ) 2 and (f R 1 ) 2 ; see Eq. (11). It yields a circular contour region on the plane of f L 1 and f R 1 and favors a negative f L 1 . Figure 2(b) displays the contour on the plane of (f L 1 , f L 2 ), which is elliptical. Even though the cross sections are proportional to (1+f L 1 ) 2 and (f L 2 ) 2 , the (f L 2 ) 2 term contributes less to the cross section than the (1 + f L 1 ) 2 term. This generates the ellipse shape. We also note that the differences among the parameter spaces at the three confidence levels are very small. This is because the contours of the effective couplings on the plane of two effective Wtb couplings are a projection from the four dimension parameter space down to a two dimension subspace. That projection leads to the small differences.  Fig. 2(c) clearly indicates a strong anti-correlation between f L 1 and f R 2 . Such a behavior can be understood from the single top production cross sections, which are approximately proportional to (1+f L 1 +a t f R 2 ) 2 . As a result, the two islands of the allowed parameter space are symmetric around the in the cross sections, yielding the anti-correlation band in Fig. 2 contours come from the tW-channel cross section and the W helicity fractions. Figure 2(c) shows a positive f R 2 is preferred when contours only allow positive f R 2 . Due to the interference effect between the anomalous Wtb couplings and the SM, the linear term which is proportional to f R 2 is valid in the cross sections. Therefore the relation of f R The current experimental data of σ s and σ tW exhibit ∼ 20% uncertainties. The accuracy of σ tW is expected to be improved at the forthcoming LHC Run 2 , but the precise measurement of σ s is still challenging. We thus consider the σ t , σ tW and the W-helicity fractions in our global analysis to constrain the effective Wtb couplings.
Using the x i 's while neglecting O(x 2 i ) and higher order contributions, we can extract the correlation between different channels:  where ∆σ NP where the coefficients a i and c i are given in Table 1. K(t) denotes the K-factor of the t-channel single top quark production, K(t) denotes the K-factor of the t-channel single anti-top quark production and the K(tW) is the K-factor of the tW associated production. Solving the systems of linear equations shown in Eq. 15 gives rise to the following relations where σ exp ( in which the σ t and σ tW measurements dominate over the W-helicity measurements. Improving the measurements of σ t and σ tW is important to test the correlations of (f All coefficients are comparable, such that one has to consider all the four experiments to determine δx m . As f L 1 and f R 2 are anti-correlated in improving δx m would further constrain the correlation between f L 1 and f R 2 , i.e. the band in Fig. 2(c) tends to be narrower. The is directly linked to the righthanded W-helicity fraction F R , which is inferred from F L and F 0 measurements. As a result, δx p depends only on the W-helicity measurement as following: As a result, a strong anti-correlation between f R 1 and f L 2 can be obtained from the F 0 and F L measurements.

063101-8
The variance of x 5 is given by . (22) It is sensitive to the precision of σ t , σ tW and F 0 measurements.
From the precision measurement of the Wtb couplings, one can also derive conservative bounds on the NP scales when no deviation is seen compared to SM predictions. Though we expect C i = O(1), their precise values are unknown. Measurements such as the ones described above can be used to obtain the ratios of these coefficients, but the values of Λ i cannot be obtained separately. Therefore, we define dimensionless parameters C i according to the Wilson coefficient C i and normalize to 1 TeV, The allowed parameter contours for the parameters C (3) φq , C φφ , C bW , C tW are shown in Fig. 3. Although the values of effective couplings f L 1 , f R 1 , f L 2 and f R 2 are of the same order of magnitude, each individual parameter C i is different. For example, the C φφ equal to 27 is also allowed at 95%C.L., while the maximal value for C bW and C tW is 5 at the same C.L. The difference comes from the relation between the effective couplings (f L,R 1,2 ) and the Wilson coefficients C i 's in Eq. (5). In principle, after we know the range of the Wilson coefficients at low energy, we can obtain the values at NP scale Λ by renormalization group equations, and further determine the NP parameter space. However, in this paper, we focus on the model independent approach to search for NP effects, and will not calculate the mixing of the different operators. Figure 4 displays the lower bound on the NP scale Λ i obtained from the Wtb measurements at 95% C.L. with |C i | = 1. The red bars represent those limits when all the four operators contribute simultaneously, while the blue bars display those limits obtained when we consider one parameter at a time. The current bound implies Λ φq > 246 GeV, Λ φφ > 188. 6 GeV, Λ bW > 443.2 GeV, Λ tW > 561.7 GeV when all the four operators contribute simultaneously, while Λ φq > 1000 GeV, Λ φφ > 440. 6 GeV when we consider one parameter at a time.

Constraints on the effective Wtb couplings and operators in 2-dimensional subspaces
So far we have considered the full correlations among the four effective couplings and explored the allowed parameter space of the effective Wtb couplings. However, not all but only some of the effective couplings are nonzero in many NP models.
In those cases, we limit the parameter space to subspaces and redo the global analysis in those subspaces. Based on the NP models to be discussed, we perform the following parameter scans: 1) We scan the 1-dim parameter space by considering either f L 1 or f R 1 only. For example, in the G(221) models and vector-like quark models, the dominant correction is in the coupling f L 1 or f R 1 , depending on the detail of the models (see Tables 4 and 6).
2) We also consider the case where both f L 1 and f R 1 are modified, e.g., the vector-like quark doublet (T, B) model and triplet cases, shown in Table 6. 3) Another case is where both f L 1 and f R 2 are modified. A typical example is MSSM. Ref. [85] has shown that the anomalous Wtb couplings have the following The results of the parameter scan are plotted in Fig. 5. In our analysis, we include the constraints from b → sγ [84] and update the limits of those anomalous couplings using the updated experimental data [86], see Table 3. From Table 3, we note that b → sγ imposes strong constraints on f R 1 and f L 2 , which can be viewed as the results of m b suppression for the right-handed bottom quark in the pure left-handed Wtb vertex, while the amplitude is enhanced by m t if the right-handed Wtb vertex exists [33,84,[87][88][89][90]. We also note that the central value of f L 1 is negative, while f R 1 is zero in that case (see Fig. 5(a) and Fig. 5(b)). This is because the cross section is proportional to (1 + f L 1 ) 2 , and the minimal χ 2 is dominant, determined by the most precise experiment. In this case, the most precise experiment is the measurement of the t channel cross section (see Table 2). However, the central value of the experiment is smaller than the approximate NNLO SM prediction (see Table 1). Therefore the best-fitted value of f L 1 is negative. For the f R 1 case, the cross section is proportional to (1 + (f R 1 ) 2 ), thus the minimal χ 2 corresponds to the SM case, and the central value of f R 1 is zero. As shown in Fig. 5(a) and Fig. 5(b), f L 1 and f R 1 are constrained to be −0.05 < f L 1 < 0.025 and |f R 1 | < 0.13 at 90% C.L. by direct experimental measurements of the top quark. Similarly, we translate the allowed region of the effective couplings to the coefficients C i of dimension-6 operators, with the results shown in Fig. 6.

Top couplings in new physics models
As mentioned in Section 1, the Wtb coupling can be modified by many kinds of NP models. In this section, we discuss the constraints from the effective Wtb couplings in several NP models.
At the electroweak scale, the symmetry is further broken by a bi-doublet scalar field H ∼ (2,2) 0 with two VEVs v 1 and v 2 . We introduce v = v 2 1 + v 2 2 and a mixing angle tan β = v 1 /v 2 for convenience. In BP-II, the breaking of SU (2) 1 ⊗SU (2) 2 → SU (2) L is induced by a scalar bi-doublet Φ ∼ (2,2) 0 with one VEV u at TeV scale, and the electroweak symmetry breaking is induced by a Higgs doublet H ∼ (2, 1) 1/2 with VEV v. After symmetry breaking, the new gauge boson W obtains mass and mixes with the SM gauge boson W. For simplicity, we define a new mixing angle φ, where g 1 , g 2 and g x are the gauge couplings of SU (2) 1 , SU (2) 2 and U (1) X , respectively. The gauge bosons' masses are where e denotes the electron charge and x = u 2 /v 2 . We also abbreviate the trigonometric functions as c φ ≡ cosφ, s φ ≡ sinφ, s 2β ≡ sin2β, c W ≡ cosθ w and s W ≡ sin θ W where θ W is the SM weak mixing angle.
The third generation quarks play a special role in several G(221) models, and the Wtb couplings are modified through the mixing effects between the new gauge boson W and SM gauge boson W at tree-level. In this work, we will use the left-right model [91][92][93], un-unified model [94,95] and top-flavor model [96][97][98]103] as examples to discuss the impact of the Wtb measurements on those NP models. The charge assignments of the third generation quark fields under the G(221) gauge groups and the detailed expressions of f L 1 and f R 1 of those NP models are listed in Table 4.
We discuss the impact of the effective Wtb coupling measurements on the parameter space of the G(221) models. Figure 7 presents the allowed parameter space at 68%, 90% and 95% C.L., respectively in the left-right model in BP-I and the un-unified and top-flavor models in BP-II. In the left-right model, we show the allowed parameter space in the (M W , s β ) plane for s φ = 0.1 and s φ = 0.9 in Fig. 7(a) and Fig. 7(b), respectively. Figure 7(a) shows that, for s φ = 0.1, a vast parameter space of moderate s β and smaller M W is not allowed by the highly constrained right-handed Wtb coupling, see Table 4. For a larger s φ = 0.9, the constraint for M W tends to be looser, see Fig. 7(b). The reason is that the gauge coupling of the SM U (1) Y is related to the gauge couplings g 2 and g x as 1/g 2 Y = 1/g 2 2 + 1/g 2 X . Thus, g Y is approximately equal to g X in the limit s φ → 0, which corresponds to the decoupling region. The shape of the allowed parameter space for s φ = 0.9 is the same as that of s φ = 0.1 in the small M W region. Figures 7(c-e) display the allowed parameter space in the plane of (M W , s φ ) for s β = 0.1, 0.5, 0.9, which shows that M W is not sensitive to s β . This is because s β only induces mixing between the gauge boson W and W (see Eq. (26)), and the primary source of the mass of the W is from the first step breaking, which is proportional to the VEV u [5]. In the un-unified and top-flavor models, only the lefthanded Wtb coupling is modified. The allowed parameter space in the (M W , s φ ) plane is shown in Fig. 7(f) (ununified model) and Fig. 7(g) (top-flavor model). We note that the shape in the large s φ region is different between the un-unified model and top-flavor model. The difference can be understood from the effective Wtb couplings shown in Table 4. The effect consists of two factors: one is W tb coupling, the other is W-W mixing. The former is different in the two models, but the latter is the same. The gauge coupling of the heavy gauge boson W with the top and bottom quarks is proportional to tan φ in the un-unified model, while it is proportional to cotφ in the top-flavor model. Thus, larger s φ is allowed in the top flavor model compared with the un-unified model. Another important feature in the top-flavor model is that the constraint on M W is not sensitive to the parameter s φ , as shown in Fig. 7(g). In the model, f L 1 is modified by the W mass, which is proportional to x/(s 2 φ c 2 φ ). In order to validate our EFT prescription of the NP effects, W should be heavy. Figure 7 shows the constraints of G(221) models from Wtb measurements are weak when new gauge bosons are heavier than 800 GeV. The constraints from low energy precision measurements and direct searches at the Tevatron and LHC have pushed the new heavy gauge bosons to several TeV [6]. Therefore, it is difficult to further constrain the parameter space by the Wtb measurements.

Vector-like quark models: T and B
The vector-like quark (VLQ) is a common ingredient of many NP models. In order to keep the discussion general, we employ an effective Lagrangian approach to parametrize the effects of vector-like quarks. The quantum numbers of the new VLQ with respect to the SU (2) L ⊗U (1) Y gauge group are summarized in Table 5 [7][8][9][10]. Table 5. The quantum numbers of the vector-like quarks under the SM SU (2)L⊗U (1)Y gauge symmetry. The electric charge of the quark is obtained by Q = T3 + Y , where T3 is the third component of the isospin, and Y is the hypercharge The Wtb coupling is modified by the mixing between the top quark or bottom quark with their corresponding vector-quark partner, e.g. the top partner T or bottom partner B. After spontaneous symmetry breaking, the mass terms of the top quark and top partner T in the 063101-13 singlet or triplet models are [9] L t mass = − where y t is the Yukawa coupling of top quark in the SM, x t represents the mixing parameter between t and T , and M denotes the VLQ's mass.
In the case of doublet VLQ models, the mass terms are [9] L t mass = − The weak and mass eigenstates can be related by 2 × 2 unitary matrices, where c t L,R and s t L,R denote the cosine and sine of the mixing angles between left-handed and right-handed top quark t L,R with the top partner T L,R . In this section, we use the prime in the superscript on mass eigenstates to distinguish from the weak eigenstates. After diagonalizing the mass matrices, we can rewrite the mixing angles according to the parameters in Eqs. (28) and (29).
In the case of singlet and triplet VLQ models For the doublet VLQ models where m t is the top quark mass. The mass of the heavy top partner is We can obtain similar formulae in the bottom quark sector, and use m b and x b to denote the bottom quark mass and mixing parameter between bottom quark and bottom partner B hereafter. In the triplet VLQ models, s t L/R is correlated with s b L/R as the mixing parameters x t and x b are linearly related to each other. For example, in the (X, T, B) model [9], with x b = √ 2x t , and in the (T, B, Y ) model [9], After diagonalizing the mass matrices, we obtain the couplings of the gauge boson fields to the third generation quarks, The couplings f L,R 1 and f L,R b for all 7 models are listed in Table 6, see also Ref. [10]. Table 6. The Wtb and Zbb couplings in 7 models, where c t L (s t L ) and c b L (s b L ) denote the cosine (sine) of the mixing angles of left-handed top quark and bottom quark with their heavy partners, respectively. Similarly, s t R and s b R are the sine of the mixing angles of right-handed top quark and bottom quark with their heavy partners, respectively.

Mixing angles
Using the results of Section 3, we translate the allowed region of the effective Wtb couplings to the parameter space of VLQ models at 68%, 90% and 95% C.L., respectively; see L in the doublet model. It is, therefore, convenient to neglect the smaller mixing angles in our parameter scan. Figure 8(a) shows the allowed parameter space of (s t R , s b R ) in the doublet (T, B) model. It yields the hyperbola contour region on the plane, which is determined by the right-handed Wtb coupling f R 1 = s t R s b R . Figure 8(b) displays the contour on the plane of (s t L , s b L ) in the triplet models, and the allowed parameter space is symmetric about s t L ↔ s b L and s t L ↔ −s b L . Such behavior can be understood from 063101-14 the deviation of the left-handed Wtb coupling between the triplet models and the SM Furthermore, the upper limit of f L 1 determines the shape of the parameter contour in the top-right and bottom-left regions, and the lower limit of f L 1 determines the boundary of the parameter space in the top-left and bottomright regions. Note that only one mixing angle exists in the singlet models T and B, and in the doublet models (X, T ) and (B, Y ). Figure 8(c) shows the allowed parameter space of the sole mixing angle, s t L or s b L , which indicates |s t,b L | < 0.34 at 95% C.L..

Mixing angles versus M T,B
We note that the constraints from the Wtb measurements on the parameter space of the VLQ models are weak. However, the left-handed top quark and bottom quark form a SU (2) L weak doublet, and the lefthanded Wt L b L coupling is always related to the lefthanded Zt L t L and Zb L b L couplings. The R b and A b F B measurements at LEP-II [104] impose severe constraints on the Zb L b L couplings, which yields a correlation between Zt L t L and Wt L b L [40,105]. To fully constrain the allowed parameter space of the VLQ models, it is necessary to include the Zbb and Ztt couplings in the analysis. The complete study will be presented elsewhere. In this work, we will allow a variation of −0.2% and +1% for the left-handed Zb L b L coupling, while in the right-handed case, −5% and +20% is used [9].
Another important constraint comes from the Peskin-Takeuchi parametersŜ,T andÛ [106]. The contribution of an arbitrary number of vector-like singlet and doublet quarks to theŜ,T andÛ parameters have been calculated in Ref. [107], and is generalised to arbitrary couplings in Ref. [108]. In our work, we calculate theT parameter of all possible VLQ models. Our analytical results are consistent with Ref. [109]. The definition of theT parameter is where the notation Π WW (0) and Π ZZ (0) denotes the vac-uum polarization amplitudes of the W loop and Z loop at zero momentum, respectively, α is the fine-structure constant and m W/Z is the mass of the W or Z boson. In the NP model, the contribution of theÛ -parameter is usually very small and can be neglected. FixingÛ = 0, theT parameter is obtained [110] ∆T =T −T SM = 0.10 ± 0.07, where the reference top quark and Higgs boson masses are m t = 173 GeV and m H = 125 GeV. We present the allowed regions of the VLQ models in Figs. 9-12. In order to better understand the impact of various bounds, we separate the constraints into different categories: the Wtb coupling constraint at 95% C.L. (green region),T parameter constraint at 95% C.L. (red region) and Zbb coupling constraint (black line). The top and bottom quark partners must be heavy in order to validate our EFT prescription of the NP effects. Currently, both the ATLAS and CMS collaborations have searched for various VLQs and imposed bounds on the heavy quark's mass [111,112] We plot in Figs. 9-12 the effects of the Wtb mea-surements on the parameter space of the VLQ models. We note the following common features when comparing different experimental constraints. There is no direct constraint on the heavy VLQ's mass from the Wtb measurements, as f L 1 and f R 1 depend only on the quark mixing angles (see Table 6). For most of the parameter space of the VLQ models, the bounds from the Wtb cou-pling measurements are weak. We also consider the measurements of the Zbb coupling if a bottom-quark partner is present. The Zbb coupling was measured very precisely at LEP II [104], such that it leads to a much tighter bound than the Wtb coupling measurements at the Tevatron and LHC.  In the singlet T model, the light T quark contributes largely to theT parameter, thus a large quark mixing angle is needed to respect theT parameter constraints [113], and the bound from the Wtb measurements plays an important role for a light T quark, say M T ∼ 300 GeV, whereT F SM ≡ 3/(16πs 2 W c 2 W )(m 2 t /m 2 Z ) denotes the contributions from the third generation SM quarks. For a heavy T quark, theT parameter constraint dominates over the Wtb coupling, Fig. 9 also shows the allowed parameter space of the singlet B model. We note that, different from the singlet T model, a two-fold contour of theT parameter occurs. The reason is that ∆T T > 0 in the singlet T model, and there is no constraint from the lower limit of theT parameter. That result is obvious in the heavy mass region M T m t . In the small mass region (M T ∼ m t ), Eq. (40) can be written as which shows the ∆T T is positive if M T > m t . But the ∆T is not always positive in the singlet B, thus both the upper and lower limits of theT parameter give a constraint on the parameter space of the singlet B [107], We also note that the two-fold contours of theT parameter in the singlet B model tend to overlap each other when M B > 5.5 TeV due to the decoupling of the heavy VLQ. Figure 10 shows the allowed parameter space of the doublet (X, T ) model (left) and the doublet (B, Y ) model (right), respectively. The bound on s L is dominated by theT parameter in the heavy VLQ's mass region. It mainly stems from the fact that the top partner or bottom partner of doublet models mainly have right-handed couplings to SM particles, and the left-handed mixing angle is highly suppressed by factor m (t,b) /M in the heavy mass region, see Eq. (32).
The (X, T ) quark doublet contributes to theT parameter as follows: The quark mixing yields a mass splitting between the two vector-quarks in the same doublet, which breaks the SU (2) symmetry. Such a breaking effect lead to a nonzero contribution to theT parameter.  Figure 11 shows the allowed region of a doublet (T,B) model. We note that the Zbb constraint is different between the (s b R , M B ) and (s t R , M T ) plane. The reason is that the dangerous bound of the Zbb coupling is sensitive to the mixing angle of the bottom quark sector, whereas for the top quark sector, the constraint is indirect and comes from the allowed range of the mass parameter M . However, the mixing parameter x t is not fixed in the (s t R , M T ) plane, therefore the constraint is very weak in the (s t R , M T ) plane, see Eq. (32). The constraint from theT parameter is sensitive to the parameter x t , and the constraint is weaker for the smaller x t . This is because x t represents the mixing between the top quark and top partner, and the smaller x t corresponds to the decoupling limit. Note that, even in the case of s b R = 0, theT parameter constraint demands M B to be larger than several hundred GeV for x t = 100 GeV; see the left-hand figure in Fig. 11. This arises from the non-decoupling effect of the top-quark partner.
The (X,T,B) quark triplet contributes to theT parameter as follows: The allowed regions of the triplet (X, T, B) and (T, B, Y ) model are shown in Fig. 12. It shows that the allowed parameter space of s t L is smaller than that of s b L in the heavy mass region. The reason is that the mixing parameter of the bottom quark sector ( √ 2x t ) is larger than the top quark sector (x t ), see Eq. (34). A similar result holds for the triplet (T,B,Y) model except that s b L < s t L in the heavy mass region and the sign of the mixing parameter is opposite (

Little Higgs models
Little Higgs models have been proposed to solve the hierarchy problem [114]. The Higgs boson in the Little Higgs models is a pseudo-Goldstone boson arising from the global symmetry breaking at TeV scale and is light due to the collective symmetry breaking mechanism. The one-loop divergence to the Higgs boson mass is cancelled between top-quark and SM gauge bosons and their partners. The Littlest Higgs model was proposed in Ref. [114], based on the SU (5)/SO(5) nonlinear sigma model, with a locally gauged subgroup The global symmetry SU (5) is spontaneously broken down to the subgroup SO(5) at the scale of f . At the same time, the gauge symmetry G 1 ⊗G 2 is broken down to the diagonal subgroup SU (2) L ⊗U (1) Y , which is identified as the SM electroweak symmetry.
In the Littlest Higgs model, a vector-like T quark and new heavy gauge boson W mix with SM particles and modify the Wtb vertex [12], where c φ = g 1 / g 2 1 + g 2 2 with g 1 and g 2 are the gauge couplings of the SU (2) 1 and SU (2) 2 , respectively, and c L is the cosine of the mixing angle of the top quark and T quark.
After considering the Wtb measurements, we plot the allowed parameter space of the Littlest Higgs model in Fig. 13, which shows the allowed parameter space at 68% (blue region), 90% (purple region) and 95% (red region) 063101-18 C.L., respectively for f = 1 TeV. It shows that the constraint on the mixing angle s φ is weak but the constraint on s L is much tighter. This can be understood from Eq. (45), as the contribution of s φ to the Wtb coupling is suppressed by the scale f , while s L directly modifies the Wtb coupling. The W mixing with the W boson at tree-level modifies the obliqueT parameter significantly. In order to respect the electroweak precision tests (EWPT), the scale f has to be above several TeV. The current lower bound of f in the Littlest Higgs model at 95% C.L. is 5.1 TeV [115]. Refs. [116][117][118] introduce a new discrete symmetry to forbid tree-level mixing between W and W and relax f down to hundreds of GeV [119]. A simple case is the Littlest Higgs with T-parity (LHT) model, in which the SM particles are even and the new heavy gauge bosons and scalars are odd under T-parity. The only T-parity even non-SM particle is the top partner T and it will contribute to the Wtb coupling through its mixing with the top quark. In the LHT model, the Wtb coupling is [11], Figure 13(b) shows that the effective Wtb coupling measurements require |s L | < 0.34 at 95% C.L. In order to further constrain the mixing angle, we include the Zbb coupling and oblique parameter constraints. To leading order and in the limit M T m t m W , the variation of Zb L b L is given by [120], where M T denotes the T quark mass, and R = λ 1 /λ 2 = s L M T /m t . The correction to the right-handed Zb R b R vertex is negligible in this case. The electroweak precision constraints on the LHT model have been calculated in Ref. [120], which shows that theT parameter induced by the T quark loop is much large than theŜ andÛ parameters for the same model parameters. Therefore we will only include theT parameter bound in our analysis. The contribution from the heavy T quark is, where x t = m 2 t /M 2 T . The convention is the same as in Fig. 9.

063101-19
in all the vector quark models except the (T, B) doublet model. To fully constrain the allowed parameter space of vector quark models, we included the constraints of obliqueT parameter and Zbb couplings. We note that theT parameter and Zbb couplings impose much tighter constraints on the mixing angles than the effective Wtb couplings obtained from σ s/t/tW and F 0/L/R . The Wtb coupling in the Littlest Higgs model is modified by both quark and gauge boson mixing. The Littlest Higgs model is severely constrained by W-W mixing at tree-level. We then considered the Littlest Higgs model with T-parity, which forbids the tree-level mixing of W and W bosons. The Wtb coupling is modified by the mixing of the top quark and top quark partner, which gives rise to constraints similar to the vector-like quark models.
The forthcoming LHC Run-II will collect more top quark pairs and single top quark events, which will improve the measurements of single top production and the W-helicity fraction. We expect tighter limits will be made on the effective top quark couplings and the dimension-six operators when new data is available. That will help us to probe new physics beyond the standard model.