Bottom partner B' and Zb production at the LHC

Some new physics models, such as"beautiful mirrors"scenario, predict the existence of the bottom partner $B'$. Considering the constraints from the data for the $Z\rightarrow b\bar{b}$ branching ratio $R_{b}$ and the $FB$ asymmetry $A_{FB}^{b}$ on the relevant free parameters, we calculate the contributions of $B'$ to the cross section $\sigma(Zb)$ and the $Z$ polarization asymmetry $A_{Z}$ for $Zb$ production at the $LHC$. We find that the bottom partner $B'$ can generate significant corrections to $\sigma(Zb)$ and $A_{Z}$, which might be detected in near future.


Introduction
Over the past several decades, the standard model (SM) has provided a consistent description of particles physics and is tested to per-mille precision by experimented data.
Recently, the AT LAS and CMS collaborations have independently reported the discovery [1] of a neutral scalar particle that seems consistent with the SM Higgs boson with a mass of about 125 ∼ 126GeV . However, some observables related to the sector of third generation quarks have been observed large deviations from their SM predictions. The first is the forward-backward (F B) asymmetry of the bottom-quarks, A b F B , which differs by about 2.5σ deviation from the SM value at the Z boson pole according the recent global fit result [2]. The second is the F B asymmetry A t F B in top quark pairs produced at the Tevatron, which has larger value than the SM prediction [3]. Furthermore, a recent calculation of the Z → bb branching ratio R b , which includes new two-loop electroweak corrections, now puts the prediction in tension with the measured value [4].
It is well known that the top loop in the SM is the largest contribution to the Higgs mass quadratic divergence. Thus, for the new physics models to solve the fine tuning problem, there must be some new particles constrained by symmetry, which cancel this loop. Most of these new physics models should contain a heavy particle which shares the gauge quantum numbers of the top quark, generally called "top partner" [5]. This new particle should be in an electroweak doublet in order to properly cancel the divergences to the Higgs mass produced by the top loop. So, this kind of new physics models beyond the SM predicts the existence of the heavy partner B ′ of the bottom quark. Furthermore, if the top and bottom partners have the same mass hierarchy as the SM top and bottom, the new quark B ′ may be the first to be discovered, which has began to be searched at the Tevatron and LHC [6].
Production of the electroweak gauge boson Z associated with a bottom quark at the LHC is an important background process not only to Higgs boson production and single top production, but also to the search for signals of new physics beyond the SM, which has been calculated at next-to-leading order (NLO) [7]. Recently, Ref. [8] has defined the Z polarization asymmetry A Z in the subprocess gb → Zb at the LHC and has shown that A Z is strictly connected to the F B asymmetry A b F B and is almost free from the theoretical uncertainties related to QCD scale and parton distribution function (P DF ) set variations.
Considering the constraints of the data from LEP for the Z → bb branching ratio R b and the F B asymmetry A b F B [9] on the Zbb couplings g b L and g b R , we are model-independent of calculating the contributions of the new physics beyond the SM to Zb production at the LHC in section 2. We find that the correction terms δg b L and δg b R generated by new physics cannot give significant contributions to the production cross section σ(Zb).
While it is not this case for the Z polarization asymmetry A Z . In section 3, we study the correction effects of the bottom partner B ′ on the production cross section σ(Zb) and the Z polarization asymmetry A Z . Our numerical results show that, with reasonable values of the relevant free parameters, B ′ can generate large corrections to σ(Zb) and A Z . Our conclusion is given in section 4.

The new physics and
Zb production at the LHC For the 5-flavor scheme [10], production of the electroweak gauge boson Z associated with a bottom quark at the LHC proceed via two Feynman diagrams with b-quark exchange in the s-channel and the t-channel at leading order. Its production cross section σ(Zb) is proportional to the factor [(g b L ) 2 + (g b R ) 2 ]. Thus, new physics can produce contributions to σ(Zb) via correcting the Zbb couplings g b L and g b R . The effective Zbb couplings can be parameterized by the Lagrangian with S W = sinθ W and C W = cosθ W , in which θ W is the electroweak mixing angle.  Figure 1: The relative correction parameter R 1 is presented as a function of δR b . The regions between dashed lines and between dotted lines correspond 1σ and 2σ allowed regions from R b constraints, respectively.
The relative corrections of new physics to R SM b and A b,SM F B can be approximately written as [11] Where In above equations, we have neglected the new physics corrections to the Zee couplings g e L and g e R . The experimental results for The recent SM prediction for R b , including electroweak two-loop and QCD three-loop corrections is R SM b = 0.21474 ± 0.00003, which deviates by 2.4σ deviations below the experimental measured value [2,4], while the recent global fit result for −0.0006 , which is still above the experimental measured value by 2.5σ deviations [2]. Using above experimental and SM prediction values, one can easily obtain the constraints of the electroweak precision data on the new Zbb couplings δg b L and δg b R . It is obvious that the data favor small corrections to δg b L and more large shifts in δg b R . Considering the discovery of a Higgs-like particle at the LHC, Ref. [12] has updated the constraints of the electroweak precision data on δg b L and δg b R and there is We use the relative correction parameter the corrections of the new Zbb couplings δg b L and δg b R to the cross section of the process pp → Zb, in which σ(Zb) denotes the total production cross section including the contributions from the SM, δg b L , and δg b R . In our calculations, the P DF s of the bottom quark and gluon are taken as the CT EQ6L P DF s [13] with renormalization and factorization scales µ R = µ F = M Z . To make our numerical results more realistic, we have applied the cuts on the b − jet with transverse momentum P T > 15GeV and a rapidity range |η| < 2. It is obvious that the radiative corrections to σ(Zb) and σ SM (Zb) are canceled in the relative correction parameter R 1 . In Fig.1 we plot R 1 as a function of δR b for 1σ and 2σ constraints from the R b experimental value. One can see that the value of R 1 allowed by the R b constraints is very small. For the theory value of R b being consistent with its experimental value with 1σ and 2σ error bars, the values of the parameter R 1 are in the ranges of 0.53% ∼ 1.3% and 0.14% ∼ 1.7%, respectively, which are much smaller than the QCD corrections [7].
Searching for the gauge boson Z produced in association with the bottom quark has been performed at the LHC. Recently, the AT LAS collaboration [14] has reported their measurement of the Zb production cross section and found that it is in good agreement with the SM prediction including the NLO QCD corrections. Considering the statistical and systematic uncertainties, the AT LAS data cannot give severe constraints on the new Zbb couplings δg b L and δg b R . The Z polarization asymmetry A Z in Zb production at the LHC can be defined as where σ(Z R b) and σ(Z L b) are the hadronic cross sections of Z R b and Z L b production at the LHC, respectively. Ref. [8] has shown that A Z is connected to the Zbb F B asymmetry A b F B and given its SM prediction value. If the large deviation between the SM prediction and the LEP measurement of A b F B indeed exists and comes from the new Zbb couplings δg b L and δg b R , then these new couplings should generate significant contributions to A Z . To see whether the correction effects of the new Zbb couplings δg b L and δg b R on the Z polarization asymmetry A Z can be detected at the LHC, we define the relative correction Our numerical results are shown in Fig.2, in which we plot R 2 as a function δA b F B to consistent with the experimental value of A b F B with 1σ and 2σ error bars. One can see that the absolute value of R 2 can reach 6.8%. Considering A Z almost free from the theoretical uncertainties, we hope that the LHC might detect this correction effects and confirm or obviate the A b F B anomaly.
3. The bottom partner B ′ and Zb production at the LHC So far, the Zbb F B asymmetry A b F B measured in Z boson decays at LEP experiments still exist 2.5σ deviations from the SM prediction [2]. Considering modification of the SM Zbb couplings g b,SM L and g b,SM R , some new physics models have been proposed to cure the large discrepancy [15,16,17]. Ref. [17] proposed the beautiful mirrors model, which introduces vector-like quarks which mix with the bottom quark subtly affecting its couplings to the gauge boson Z and addressing the observed anomaly in A b F B . This model predicts the existence of the bottom partner B ′ . Some of their phenomenological consequences have been explored in Refs. [17,18]. Taking into account of the constraints on the relevant free parameters from explaining the current R b and A b F B deviations [2,4,12], we consider the contributions of the bottom partner B ′ to the hadronic cross section σ(Zb) and the Z polarization asymmetry A Z for Zb production at the LHC in this section. The beautiful mirrors model [17] extends the SM by introducing two sets of vectorlike quarks, ψ L,R with quantum numbers (3, 2, -5/6) and ξ L,R with quantum numbers (3, 1, -1/3), in which the SM Higgs is the only source of electroweak symmetry breaking (EW SB). In terms of its SU(2) components, ψ L,R decomposes as where ω is a charge -1/3 quark and χ has charge -4/3. It is assumed that the new quarks only couple to the third generation SM quarks, which are governed by the SU (3) In the beautiful mirrors model, the couplings between the gauge boson Z and the down-type quarks may be written in matrix form [17] where and b 3 is mostly ξ. We call b 2 as bottom partner B ′ and consider its contributions to Zb production at the LHC. The coupling matrices L and R are written as where . The unitary matrices U d and W d transform the left-and right-handed gauge eigenstates into the corresponding mass eigenstates, which can diagonalize the mass matrix, where m 1 = m b , m 2 and m 3 are the SM bottom quark mass, and two new quark masses.
The matrix U d can be parameterized as with C L 12 = cosθ L 12 and so on, in which θ ij are the mixing angles. The matrix W d has an analogous expression but with θ L ij → θ R ij . Using above equations, one can write the explicit expression forms for the Zbb, ZB ′ B ′ , ZbB ′ couplings, etc, and further give the correction terms δg b L and δg b R to the SM Zb L b L and Zb R b R couplings. To predigest our calculation, we set S R 12 = S R = 0, S L 13 = S L = 0, and all other mixing angles equal to zero. In this simply case, the couplings, which are related our calculation, can be written as Comparing the experimental measured values of the Z → bb branching ratio R b and F B asymmetry A b F B with their current theoretical prediction values [2,4], one can obtain the constraints on the mixing parameters S L and S R . To make A b F B and R b consistent with their experimental measured values with 1σ and 2σ error bars, the mixing parameters S L and S R must satisfy the relation 1σ : 0 ≤ S 2 L ≤ 0.004, 0.022 ≤ S 2 R ≤ 0.042; (14) 2σ : 0 ≤ S 2 L ≤ 0.006, 0.012 ≤ S 2 R ≤ 0.052. The couplings of the SM quarks and new down-type quarks to the Higgs boson H and the gauge boson W can be obtained from Ref. [17].
The couplings of the new fermions to the SM gauge bosons and ordinary fermions are uniquely fixed by gauge invariance [19]. The general Lagrangian describing the interactions between the SM bottom quark, its partner B ′ and gluon is fixed by SU (3) gauge invariance to be of magnetic moment type [20,21] where G a µν is the gluon field strength tensor with the color index a = 1, ..., 8, and g s is the QCD coupling constant, λ a are the fundamental SU(3) representation matrices. In this Letter, we set the new physics scale Λ to M B ′ and assume that the coupling constants K b L and K b R are both of order one in the strongly interacting theory. It is should be noted that, using this type couplings, Ref. [22] has considered the contributions of B ′ to tW association production and discussed the possibility of detecting the bottom partner B ′ at the LHC.
From above discussions we can see that the bottom partner B ′ can contribute to Zb production at the LHC via s-channel and t-channel B ′ exchanges, as shown in Fig.3. Our numerical results are given in Fig.4, in which we plot the relative correction parameter R 3 = (σ total − σ SM )/σ SM as a function of the bottom partner B ′ mass M B ′ , σ total includes the contributions from the SM and the bottom partner B ′ . Since the contributions of the new couplings δg b L and δg b R to Zb production are very small, we have not included their correction effects in the relative correction parameter R 3 . In our numerical calculation, we have considered the constraints of the electroweak precision measurement, such as R b and A b F B , on the mixing parameters S L and S R , and assumed the total decay width  correction and might be larger than the NLO QCD correction for taking special values of the free parameters.
In the beautiful mirrors model, the correction effects on the Z polarization asymmetry A Z for Zb production at the LHC come from two sources: the new Zbb couplings δg b L and δg b R , and the bottom partner B ′ . The contributions of B ′ to A Z is not related the free parameter S L and the contributions of δg b L are much smaller than those for δg b R and B ′ , so we fix the value of the free parameter S L to S 2 L = 0.004. The relative corrections of the beautiful mirrors model to A Z is presented by the parameter R 4 , which is plotted as a function of S 2 R for K b = 1 and three values of the B ′ mass M B ′ in Fig.5. The absolute value of the parameter R 4 increases as M B ′ decreases and S R increases.

Conclusions
The electroweak precision measurements can generate severe constraints on the new physics beyond the SM. The large deviation between the SM prediction and the LEP measurement of the F B asymmetry A b F B and the Z → bb branching ratio R b require that the new physics has large corrections to the SM Zb R b R coupling g b,SM R and small corrections to the SM Zb L b L coupling g b,SM L . In this Letter, we first consider the contributions of the new Zbb couplings δg b L and δg b R to the hadronic cross section σ(Zb) and the Z polarization asymmetry A Z for Zb production at the LHC. We find that the relative correction of δg b L and δg b R to σ(Zb) is very small, while can reach 6.8% for A Z . Some new physics models beyond the SM predict the existence of the bottom partner B ′ . Considering the constraints from the electroweak precision measurements on this new physics model, we further calculate the contributions of B ′ to the production cross section σ(Zb) and the Z polarization asymmetry A Z . Our numerical results show that the "beautiful mirrors" scenario can give significant corrections to the physical observables σ(Zb) and A Z , which might be detected at the LHC in near future.