Bottom-quark Forward-Backward Asymmetry, Dark Matter and the LHC

The LEP experiment at CERN provided accurate measurements of the $Z$ neutral gauge boson properties. Although all measurements agree well with the SM predictions, the forward backward asymmetry of the bottom-quark remains almost 3$\sigma$ away from the SM value. We proposed that this anomaly may be explained by the existence of a new $U(1)_D$ gauge boson, which couples with opposite charges to the right-handed components of the bottom and charm quarks. Cancellation of gauge anomalies demands the presence of a vector-like singlet charged lepton as well as a neutral Dirac (or Majorana) particle that provides a Dark Matter candidate. Constraints from precision measurements imply that the mass of the new gauge boson should be around $115$~GeV. We discuss the experimental constraints on this scenario, including the existence of a di-jet resonance excess at an invariant mass similar to the mass of this new gauge boson, observed in boosted topologies at the CMS experiment.

The Standard Model (SM) provides an accurate description of all experimental observables. The discovery of a 125 GeV resonance with properties consistent with a 125 GeV Higgs boson [1,2] provides evidence of the realization of the Higgs mechanism as a source of gauge boson and fermion masses. However, the exact properties of the Higgs sector are still unknown. The minimal model postulates the existence of just one Higgs, transforming as a doublet under the gauge interactions. Precision measurements of the charged and neutral gauge boson properties [3] show the preference towards a doublet Higgs state. Similar properties would be obtained, however, if there were more than just one Higgs doublets. Finally, the presence of extra singlet scalar Higgs states is not constrained by these considerations.
Another outstanding question is the origin of the Dark Matter (DM) observed in astrophysical configurations. The Standard Model does not provide any DM candidate and its nature is unknown. Among the many DM candidates, weakly interactive massive particles (WIMPs) are particularly attractive since they can easily be incorporated in beyond the SM scenarios. Moreover, it is well known that WIMPs with mass of the order of the weak scale and interactions of about the weak scale one provide a good candidate of thermal DM candidate [4].
Precision measurements of the gauge sector have shown agreement with expected SM properties at the per-mille level. Such a precision leads to sensitivity to radiative corrections which depend in a relevant way on the top-quark and the Higgs mass. Among the many observables measured, the bottom forward-backward asymmetry measured at LEP presents a 3 σ deviation with respect to the values expected in the SM [3]. Although this deviation could be just due to statistical fluctuations, its nature is intriguing since it could be associated with a large correction to the right-handed bottom quark coupling to the Z boson, which may only be explained by either mixing of the bottom-quark with additional (vector-like) quarks, or by mixing of the Z gauge boson with additional neutral gauge bosons. The first possibility led to the proposal of what are called Beautiful-Mirror scenarios [5], and their properties have been studied in detail [6][7][8]. The second possibility, namely the existence of additional gauge bosons contributing via mixing to a variation of the bottom quark coupling has also been explored, within the context of left-right models and warped extra dimensions [9,10].
In this article, we study the properties of a neutral gauge boson with preferential couplings to the bottom and charm quarks. We shall show that it leads naturally to the existence of a low energy spectrum that includes two Higgs doublets, a singlet, and a charged and a neutral vector-like singlets, the latter being a good DM candidate.
This article is organized as follows. In section II, we describe the properties of the proposed SM gauge extension. We present the tree-level couplings of the new gauge boson to SM particles, as well as the necessary fermion content in order to cancel the gauge anomalies.
The new Higgs bosons are introduced in order to induce the necessary mixing and provide masses to all chiral fermions in the theory. In section III, we study the constraints on this model coming from precision electroweak measurements. In section IV, we study the collider constraints on this model and in section V we study the constraints coming from the requirement of obtaining the proper DM relic density without being in conflict with direct and indirect detection constraints. We reserve section VI to our conclusions.

II. A MODEL WITH TWO HIGGS DOUBLETS AND A SINGLET
In this section, we shall describe the precise gauge extension of the SM we propose to explain the anomalous value of the bottom-quark forward-backward asymmetry. We consider a new gauge group U (1) D with gauge boson field K µ [11], under which, the right-handed bottom and right-handed charm quark have opposite charge ±X. This ensures the automatic cancellation of the SU (3) 2 c ×U (1) D , U (1) 3 D gauge anomaly. In order to cancel the gauge anomalies involving the hyper-charge gauge field, we introduce two SU (2) singlet SM-vectorlike leptons χ 1,2 with hyper charge -1 and 0, where only the right-handed components are charged under U (1) D , carrying charges ±X, respectively. The neutral state χ 2 will be naturally a dark matter candidate, provided we impose a Z 2 parity, under which χ 2 transforms non-trivially while SM-particles are neutral under this symmetry transformations.
A modification of the forward-backward asymmetry, consistent with the one observed experimentally, may be obtained by a sizable variation of the coupling of the Z to righthanded bottom quarks [5]. Such a variation of the Z gauge boson couplings may be the result of mixing between the Z and the K gauge bosons. Such mixing may be induced by a new SU (2) Higgs doublet Φ 1 with hyper-charge Y = 1/2 and U (1) D charge equal to the b R one, which is needed to make sure that we obtain the enhanced Zb RbR coupling for The SM Higgs-like doublet which gives the other SM fermions and the gauge bosons masses will be denoted as Φ 2 . Another SM gauge singlet scalar Φ 3 charged under U (1) D is needed to give mass to the K gauge boson. It is clear that within this setup, we can not write down the normal Yukawa interaction for the bottom and charm quark directly. To solve the problem, we add two vector-like quarks ψ b , ψ c , which have the same SM charges as b R and c R , but without U (1) D charge. The masses of the bottom and charm quarks are obtained by their mixing with the heavy vector-like quarks, which is in the same spirit of partial compositeness [12]. The particle contents of our model and their gauge group charges are listed in Table I.
anomaly-free condition is applied for this model. The U (1) D charge of b R , c R , χ 1,R and χ 2,R is determined by the anomaly-free condition. We choose X = 1 for the model without loss of generality.
The whole Lagrangian in our models can be written into three parts: where L Φ,q, denotes that Lagrangian in the Higgs sector, the quark sector and the lepton sector respectively. For the Higgs part, the Lagrangian is simply as follows: where the covariant derivative is defined as: where K µ is the U (1) X gauge boson, W a µ are the SM SU (2) L gauge bosons and B µ is the U (1) Y hypercharge gauge boson. The gauge bosons denoted without tildes are gauge eigenstates. After considering mixing effects, we shall later use tildes to denote mass eigenstates.
For W ± µ and the photon A µ , since they do not mix with K µ , the notation is the same as in the SM and there is no need to add tildes. The Higgs potential will be fully discussed in next subsection, and here we just assume that the fields associated with the three CP-even neutral Higgs bosons obtain vacuum expectation values (vev), i.e.
The vev's do not break the electromagnetism symmetry, and Φ 1 induce the mixing between the neutral massive gauge bosons K µ and Z µ , which are proportional to v 2 1 . Since the W boson mass is not modified, the custodial symmetry is explicitly broken by the mixing and this will be reflected in T parameter. The high-precision constraints on the T parameter tell us that the mixing should be very small, which favors a small vev, v 1 v 2,D . For later convenience, it is useful to define the ratio angle β: which controls the charged Higgs mixing by Goldstone equivalence theorem and has to be large. In this limit, the neutral CP-even Higgs h 0 2 will roughly be the SM-like 125 GeV Higgs boson observed at the LHC [1,13], and mixes with the CP-even Higgs boson h 0 1 . The remaining physical charged Higgs and CP-odd Higgs bosons will be Φ 1 -like, while h + 2 , a 0 2 , a 0 3 becomes the dominant longitudinal part of the massive W, Z, K gauge bosons. Φ 1 -like physical Higgs will couple to SM gauge bosons and fermions suppressed by mixing angle to SM-like Higgs cot β . The last CP-even Higgs boson will be Φ 3 -like and only couple largely to U (1) D charged particles and the U (1) D gauge boson K µ . As its vev v D is the source of bottom and charm masses, it couples with them proportional to their masses, i.e. m b,c /v D . In the absence of mixing with the other CP-even states it will be produced in bottom-fusion and gluon fusion processes and it will decay mostly to bottom quarks. Hence, provided the mixing with the SM-like Higgs boson is small, the LHC constraints on it are expected to be very weak.
The most general interactions in the quark sector are given by: The vev of Φ 3 will induce the mixing between the right-handed bottom and charm quarks, b R , c R , and their corresponding vector-like quark partner. As a result, the bottom and charm quarks obtain masses after Electroweak spontaneous Symmetry Breaking (EWSB). In this sense, it is very similar to the partial compositeness scenario of the composite Higgs models except that our vector-like quark partners can be fundamental. It is not difficult to embed our model to a composite Higgs model, where all the Higgs bosons are Goldstone bosons associated with spontaneously broken global symmetry of a new strong sector.
As described above, the masses of the bottom and charm quark arise from the spontaneously broken U (1) D gauge symmetry and electroweak gauge symmetry, which can also been seen by integrating out the heavy vector-like quark ψ b,c at the tree level using equation of motion: then we have the effective Yukawa interaction Lagrangian: It is clear that the flavor interaction structure of h 0 2 is of SM-like and the effective Yukawa couplings may be diagonalized at the same time as the mass matrices. Although the last two terms in Eq. (8) can in principle induce flavor changing neutral current (FCNC) in the quark sector, it is very model dependent. In the following, we will assume the flavor-off-diagonal interactions are very small, which is equivalent to start with the Lagrangian with following parameters: For the leptons, we will focus on the third generation and similarly neglect off-diagonal terms between different generations. The Yukawa interaction Lagrangian reads: where we have imposed the Z 2 parity for the neutral lepton χ 2 → −χ 2 and assumed that χ 1 only mix with the third generation charged lepton τ R by the direct Dirac mass m τ 1 , which is the only source of χ 1 decay.

A. The gauge sector
In this subsection, we will discuss the mixing in the gauge sector and the couplings of the dark gauge boson. After the gauge symmetry breaking, the charged gauge boson sector is the same as SM with v 2 = v 2 1 + v 2 2 : For the neutral sector, we first apply the rotation to transform W 3 , B gauge bosons into Z, A gauge bosons as in the SM. The Φ 1 is charged under both SM SU (2) L × U (1) Y and U (1) D , thus induces off-diagonal mass terms for Z µ and K µ , but the photon state A µ is not affected and stays massless, as it should be. Factoring out the photon state A µ , the Z µ and K µ will mix with each other and the mixing mass-square matrix is given by where we have defined: The matrix can be easily diagonalized by an 2 × 2 orthogonal matrix with mixing angle α: whereZ µ ,K µ are the final mass eigenstates. As will be discussed in detail in Sec. III, the Electroweak precision test (EWPT), including the T parameter and Z-pole measurements, put a strong constraint on the mixing angle thus the mixing should be very small, which further indicates c 2 β 1. Then the value of sin α can be approximately given by: where we have kept the leading terms in a c 2 β expansion. The mass eigenvalues of the gauge bosons are simply: Due to the mixing betweenK andZ, the coupling ofZ to SM particles and also theZ mass will be modified with respect to their SM values. We will carefully discuss it afterwards.
At 1-loop level, the kinetic mixing term K µν B µν can be induced from the fermions which charged under both U (1) Y and U (1) D , with ∼ g D g /(16π 2 ). Given it is much smaller than the direct mixing sin α from vev of Φ 1 , we can neglect this term.

B. Higgs sector
In this subsection, we will discuss the Higgs sector and get the mass eigenstates of Higgs.
First, we write down the general scalar potential which is gauge invariant under SU (3) C × SU (2) L × U (1) Y × U (1) D as follows: The minimum condition of V can be always satisfied by requiring the mass terms have the following relationship where the vevs of the Higgs are defined in Eq. (4). Let's start from the charged Higgs mass matrix, which is straightforward to obtain by the second derivative of the potential V : The mass of the physical charged Higgs is: The two charged Higgs fields h ± 1 and h ± 2 mix to form the mass eigenstates H ± and G ± according to Similarly, we can obtain the mass eigenvalue of physical CP-odd Higgs as the trace of the mass matrix: whose value is given by: where we have abbreviated s β ≡ sin β, c β ≡ cos β. The mass mixing matrix is listed in Appendix B. From the masses, we can easily see that in the large t β limit, which is required by the smallK,Z mixing, the mass scales of the heavy charged Higgs and CP-odd Higgs can be as large as TeV if µ 8 is around the electroweak scale. In this limit, both heavy charged Higgs and CP-odd Higgs dominantly come from Φ 1 .
Finally we consider the CP-even sector, which involves three physical states. The mass matrix is obtained as follows: As discussed before, in order to decouple the heavy charged Higgs and not induce the large mixing between SM Higgs and the other CP-even Higgs , we require that µ 8 is roughly of O(v 2 ) and c β 1. In order not to induce large mixing between the SM Higgs h 2 and the singlet h 3 , we further require that λ 6 is small and of the same order as c β . Under the above assumption, we can simplify the mass matrix by eliminating the quadratic and linear term of v 1 , except v 1 µ 8 terms, which since v D is of the same order as v, are of the same order as This is equivalent to set λ 1 , λ 4 , λ 5 , λ 7 to 0 and the CP even mass matrix is now: The mass eigenvalues at leading order in cot β and λ 6 are simply as following: The unitary mixing matrix is define as:  where h (H) denote flavor (mass) eigenstates respectively. The entries can be obtained at the leading order in cot β: where the expression of U 23 proceeds from a combination of terms proportional to cot β and λ 6 , and we set it as a free parameter. The more detailed expressions for the mass of CP-even Higgs and mixing matrix U are given in the Appendix A. We can easily see that, in the decoupling limit, the modifications to SM Higgs couplings with massive gauge bosons and the fermions arise at second order in cot β, which are therefore at the percent level in our scenario since cot β ∼ 0.1.

C. Fermion sector
Let's now turn to mixing in the fermion sector, where we especially focus on the b and c quarks. As explained above, the masses of the b and c quarks come from the mixing with heavy vector like fermions ψ b,c . We first consider the mixing between ψ b and b. The 2 × 2 mass matrix in (ψ b , b) basis simply reads: where we simply treat the off-diagonal terms as small variables m b 12 m b,ψ . It is straightforward to diagonalize the mass matrix by the orthogonal rotation of the left-handed and right-handed quark fields: where the mixing angles are approximately given by: and the mass eigenvalues are: where the mass formula for the bottom quark is similar to the partial compositeness scenario [12]. The same analysis applies to the charm quark except the parameters are in the charm sector. The mass formula and the mixing angle are given by: We now consider the mass eigenstates of χ 1,2 . The Dirac mass term for χ 2 is simply: without any mixing with SM particles and this will be our dark matter candidate. At current stage, we assume the elastic DM scenario that Majorana mass M m = 0, which can be originated from a global continuous symmetry for χ 2 . We will come back to Majorana DM later. There is a mixing between χ 1 and τ induced by the Dirac mass m τ 1 , which we assume to be tiny. So the mass eigenvalues at leading order are simply: At the linear order in m τ 1 /m χ 1 , only the right-handed part mix with each other: where the mixing angle are: and we see clearly s τ,L s τ,R and can be neglected.
The relevance of s τ,R mixing is to let the χ 1 decay, so in principle we can make it as small as we want unless the lifetime of χ 1 is long enough to have cosmological problems.
For example, if we make it as small as 10 −4 , it will not affect the SM τ interactions in any significant way and χ 1 will have a decay width ∼ α em m χ 1 s 2 τ,R ∼ 10 eV, implying that it will still decay promptly at the LHC.

D. Gauge bosons interactions with fermions
In this section, we will review the interactions between the fermions and the gauge bosons.
Let us emphasize again that the gauge eigenstates of gauge bosons (e.g. Z and K) are denoted without tildes, while the mass eigenstates (e.g.Z andK) are denoted with tildes.
For the gauge bosons W ± and photon A, no further mixing are induced by U (1) D and thus they are the same as in SM. First, we notice that in the gauge basis, the interaction Lagrangian in the quark sector reads: where we neglect the photon couplings as it is only determined by the electric charge of the fermions, not changing the couplings of K and Z. To determine the couplings of Z, we separate the electric-charge (Q) part and the weak isospin part T 3 . Because the electromagnetic gauge symmetry is unbroken, only particles with the same electric-charge can mix with each other after EWSB, making the Q part of the Z couplings flavor diagonal. Then the only flavor off-diagonal Z coupling comes from the T 3 contribution, namely which are purely left-handed. In contrast, the K couplings are purely right-handed. Now It is easy to obtain the gauge boson couplings in the mass mass eigenstate by performing the orthogonal rotation to the gauge bosons and the fermions. The results for the SM charge gauge bosons read: and for the neutralZ µ state the interactions read where the mixing angles are defined in the previous two sections. We can clearly see that the modifications to the Zb R b R and the Zc R c R couplings come at linear order in sin α and are of opposite sign, while for the left-handed couplings, they arise from the normalization of the quark fields starting at the square order of the mixing parameters sin 2 α, s 2 c,L , s 2 b,L . As we will see later, a small modification to the left-handed bottom and charm Z boson couplings is necessary in order to satisfy the total b, c hadronic cross section measurements on the Z-pole.
For the U (1) D gauge boson interactions at lowest order, we have: where J µ Z,q is the SM quark neutral currents except the bottom and charm quarks: We can see thatK µ mainly couples to the SM right-handed bottom and charm quarks with gauge coupling g D and couples universally to other quarks and leptons through its small mixing with Z boson. We finally comment that due to the existence of a Dirac mass for the vector-like quark ψ b and ψ c , one can lift these vector-like fermion masses ( 1 TeV) to decouple ψ b and ψ c from LHC physics, while choose appropriate mixing angles to give the right mass to the b and c quarks.
Next we consider the gauge boson interactions in the lepton sector including τ and χ 1,2 .
The interaction Lagrangian in gauge basis reads: In the mass eigenstate basis, the Lagrangian at leading order mixing is, where J µ Z, is the SM lepton neutral currents except the τ : As explained in previous subsection, s τ,R can be chosen to be very small to make χ 1 decay promptly at LHC while not affecting the early cosmology. We note that χ 1 has mass around ∼ v D , thus is relevant for LHC physics. Later we will show that due to its coupling only to hypercharge, it is not constrained by current LHC limits.

E. Higgs interaction with Fermions and Gauge Bosons
After we consider the mass eigenstates of Higgs and fermions, we can have the following interactions: where we have abbreviated c β ≡ cos β, ct β ≡ cot β, · · · etc and substituted the leading values for U 12 and U 13 in Eq. (29). Note that we have only kept the leading term in the H 0 2 (H 0 3 )bb(cc) couplings in the limit c β 1. Since s β 1, the SM-like Higgs boson H 0 2 will couple to SM fermions the same as Standard Model except from O(c 2 β ) corrections, which are at the percent level in our model. This implies that this model cannot be tested through Higgs fermion coupling measurements at the LHC and hence we shall not discuss these constraints anymore. We also see that the H 0 3 is Φ 3 -like and coupled to bottom and charm quark proportional to their mass as discussed before. Note that it also couples to top quark through its mixing with h 0 2 , which maybe relevant due to the large top Yukawa coupling and the mixing size of order ct β .
In the following, we will consider the mass hierarchy Hence, the heavy charged lepton χ 1 can decay to scalars plus τ leptons, where the leading channel is τ H 0 3 which is only suppressed by s 2 τ,R , while the channel τ H 0 2 is further suppressed by tau mass. Given Eq. (46), the other dominant decay channel for χ 1 is τK which is also of order s 2 τ,R . Therefore,χ 1 decays into τ (bb) and τ (cc), which could be a new signature to look for at LHC depending on the production cross section of χ 1 .
For completeness, we list the leading interaction between ψ and c, b, and neglect the Note that the couplings to diagonal heavy quarkψψ are neglected at O(s 2 b,c ). The vectorlike quarkψ b,c can decay intob,c quarks plusZ,K and scalars. The decay width toZ,K, Given that the Dirac mass of ψ is much larger than v 2 ∼ v D , the dominant decay channels for ψ b,c are b, c plus scalars.
Since one can give a large enough Dirac mass forψ b,c to evade the collider constraints, we will not further discuss their search at LHC.
Next, we consider the Yukawa interaction with charged Higgs H ± and CP odd Higgs A 0 .
The Lagrangian for the charged Higgs in the mass eigenstates reads: The fermion interaction with A 0 is given in the Appendix B. As discussed before, H ± and A 0 can be made as heavy as TeV, therefore we are not going to discuss them further.
We finally list the interactions between one CP-even scalar and two gauge bosons, which maybe relevant for the LHC phenomenology. The Lagrangian in the gauge basis at leading c β order is : where the couplings of gauge bosons with the scalars are determined by the scalars' contributions to the mass of the gauge bosons. The Lagrangian for the mass eigenstates are: where we have kept leading terms in c β and s α for H 0 1,2,3 term respectively. We can see that H 0 2 couplings to gauge bosons are modified at the percent level ∼ c 2 β , which is consistent with the present precision at the LHC. The H 0 1 couplings are further suppressed at quadratic or cubed order, O(c 3 β , c β s α , s 2 α ), though linearly suppressed by c β forZK coupling, while H 0 2 , H 0 3 are at most suppressed by linear c β or s α . This fact reveals that it is much more difficult to search for H 0 1 at the LHC. For the H 0 3 , it couples largely to theK gauge boson as it is the main source ofK gauge boson mass. As a result, if m χ 1 ,χ 2 > m H 0 3 /2, it will dominantly decay intoK pair if this decay channel is kinematically open. It can also decay intobb,cc pairs which may be dominant if theK decay channel is closed. It could decay into other SM fermions pair but will be suppressed by the mixing between H 0 2 and H 0 3 . Concerning its production at the LHC, we expect that it is mainly produced through gg fusion due to top and bottom loops. If U 23 is of order ct β , top loop will dominate. In this case, its production cross section at the LHC will be suppressed by ct These cross sections are too small to discriminate the H 0 3 production from the multi-jet QCD background. If m χ 2 < m H 0 3 /2, the most promising scenario for searching H 0 3 is H 0 3 jj production, following by the nearly 100% invisible decay to χ 2χ2 , if m χ 1 , mK > m H 0 3 /2. Comparing to the cross section of σ(jj(Z → νν)) ∼ 10 3 pb, H 0 3 production is still hard to probe at the LHC.

III. ELECTROWEAK PRECISION MEASUREMENTS
The main motivation behind this model is the observed 3 σ deviation of the bottom-quark forward-backward asymmetry A b FB measured at the LEP experiment at CERN. It is well known that this asymmetry may be modified by varying the right-handed bottom coupling to the Z-boson [5,10,[14][15][16][17][18]. In general, the modification of the couplings produces other effects that have relevant implications on the precision electroweak observables, which should be considered simultaneously. In fact, the strongest constraints on this model come precisely from the Electroweak precision measurements [3,[19][20][21][22] including the T parameter and the Z-pole observables. In our setup, the mixing betweenK andZ will induce the custodial symmetry breaking, which modifies theZ mass without changing the mass of the W boson.
The corresponding contribution to the T-parameter is given by: whereα(m Z ) is the value of the fine-structure constant evaluated on the Z-pole, whose value is [23,24]:α The modification of the T-parameter has the same sign as m 2 K − m 2 Z . From the T-parameter measurement T = 0.08 ± 0.12 [25], we can obtain the 95% bound on the modification of the Z mass: which can translated into the bound on the mixing angle sin α for given mass of theK gauge boson.
Next, we consider the Z-pole measurements, including not only A b FB but also the total width of the Z boson Γ tot , the heavy flavor quarks (bottom and charm quark) production ratio R b,c , lepton production ratio R l , and the forward-backward asymmetry of the charm quarks A c F B . They can be roughly written in terms of the left-handed and right-handed Z-couplings as: where we have neglected the masses of SM quarks and leptons. We defined the coupling ratio factor: for any of the SM quarks and leptons. TheZ coupling expressions in Eq. (42) has been used. In particular, the coupling betweenZ andb is changed due to the mixing betweenK Note that the values of the mixing angles for the bottom and charm quarks with the heavy vector-like quark are constrained by the requirement of correctly reproducing the bottom and charm mass: where we have required the masses of heavy vector-like quarks to be larger than 1 TeV to satisfy the LHC direct search bounds, and the running mass of the bottom and charm quark at the 1 TeV scale has been used. This makes all mixing angles naturally small and hence the c b,(c),R 1.
In Fig. 1, we present the 1σ bounds on the different precision measurements, considering the measurement of A b,c FB , R b,c,l and Γ tot . The constraints coming from different measurements are represented by different colors, and the shaded areas are excluded at the 1σ level, with colors corresponding to a superposition of the colors associated to the observables that lead to a constraint in that region of parameters. Most importantly, the white bands are allowed by all precision measurements at the 1σ level and can fit the deviation of the forwardbackward asymmetry A b FB within 1σ. Combing all the electroweak precision measurements and T parameter constraint, we find out the preferred parameter space of g D and sin α is And we also fix the other mixing angels The T parameter constraint can also been rewritten as: This clearly put a bound in the mK − g D plane for fixed value of δgZb RbR , which is shown as orange region in Fig. 3 for δgZb RbR = 0.011. Note such value can solve the A b F B discrepancy. We can see clearly that the constraints on the T parameter almost exclude the lower half of the parameter space. Since we will take g D , mK, sin α as input parameters, the c 2 β can be written as: where we can easily see that in order to modify A b FB at the desired value and be consistent with T parameter constraint, we need c 2 β 0.08. It indicates the vev of Φ 1 should be small, i.e. v 1 75 GeV.

IV.K SEARCHES AT COLLIDERS
In this section, we will consider the phenomenology ofK at the LHC. Since ourK only coupled with bottom and charm quarks before the small mixing between theZ boson, its main production channel will bebb andcc initiated processes. It will also mainly decay into bottom and charm quarks with roughly the same branching ratio ∼ 50%. The decay into leptons will be highly suppressed by the small mixing. We present the decay branching ratios ofK in Fig. 2. There could be another decay channel ofK → χ 2χ2 if m χ 2 < mK/2, which would be around 1/7 due to the color factor counting in low mass limit.
The presence of the light gauge bosonK is subject to several constraints. The first constraint comes from the exotic Z decaying to dijet which associated produced with a jet from CMS [26] at 13 TeV, which is shown as red region in Fig. 3 . We see that there is a deep valley around 115 GeV, which is associated with an interesting 2.9σ local excess in that region of invariant masses. CMS and ATLAS also search for exotic Z decay to b-jet pair [27, 28] at 13 TeV, but focus on the mass region around 550 − 1500 GeV. We only show   searches by D0 and CDF [34,35]. Comparing all the searches, the most stringent constraint comes from the 13TeV ATLAS search [30] (green shaded) which goes down to 170 GeV.
The constraints from D0 and CDF are shown as brown and cyan area. We also show T parameter constraint in Fig. 3 as orange area.
LHC also searched for the low mass scalar in its leptonic decay. For our benchmark point, the branching ratio ofK to e + e − , µ + µ − and τ + τ − are the same, which is 8.7 × 10 −5 . The most recently research is done by ALTAS [29] at 7TeV and the constraint is ∼ 0.1pb around mass m φ = 120GeV, which is the lowest mass they considered in the µ + µ − channel. For our benchmark point, the cross section for pp → (K → µ + µ − ) = 0.08pb at mK = 115GeV, which is again marginal within the constraint from ATLAS. The constraint is shown as the magenta shaded area of Fig. 3.
Before closing this section, we comment on the intriguing hints of lepton flavor non- K [36,37] processes at the LHCb experiment and also in R ( * ) D processes at the BaBar experiment [38,39] and at LHCb [40] in charged lepton decay channel with tau leptons, though only weakly supported by Belle [41,42] and the recent LHCb result [43] from three-prong tau lepton decays.
In our model, the gauge boson K couples flavor diagonally to b and c quark and hence not in a flavor universal way, which is similar to Ref. [44,45]. In this case, the W ± loop effects can introduce flavor changing coupling between the K boson and b, s quarks. However, the leptons couple with K only via Z boson mixing, and hence the gauge boson couplings are lepton flavor universal. Therefore, our model is unlikely to address R ( * ) K , unless we introduce, for example, muon leptons charged under U (1) D . Thus, it needs further considerations to reconcile R K or R * K problems with bottom quark forward-backward asymmetry problem, what is beyond the scope of this paper. For R ( * ) D lepton flavor non-universality, the charged Higgs extension in type-II 2HDM has been excluded by the combination of R D and R * D [38]. In our model, the U (1) D assignment of Φ 1 determines that it is similar to type-I 2HDM. In this case, the charged Higgs coupling to quarks are suppressed by cot β, which we take to be small, and its contributions would be further suppressed by the fact that the masses of our Higgs bosons H 0 , A 0 and H ± are large, of order of a few to several TeV, which further reduces their relevance to R(D * ). Actually one might try changing U (1) D charge of Φ 1 from X to −X in order to write down SM Yukawa coupling for Φ 1 . However, this induces the wrong sign for sin α, which forces us to stay with the current charge assignment in Tab. I. Therefore, we conclude that an extension of this model would be necessary to solve the flavor problems in R  Fig. 1, which especially can also lead to 1σ agreement for the bottom-quark forwardbackward asymmetry. In Fig. 3, the collider limits ofK and T parameter still allow its mass to be around [100, 140]. We do not consider degenerate masses between Z and K which may leads to large mixing. mK = 115 is chosen because of the interesting 2.9σ local excess in Ref. [26], but other mK around this region is also plausible.
Note that the mass of the new gauge bosonK is very close to SM Higgs mass. For this benchmark point, the Drell-Yan cross section forK production at the 13 TeV LHC will be sizable, around 3.1 × 10 3 pb. The associated production cross section at LHC with another one or two jets are also listed in Table II.   [46] to generate the model files and implement it in MadGraph5 aMC@NLO [47]. The crosssections are calculated by MadGraph5 at tree-level to estimate the constraints.
We first consider the SM Higgs searches from VBF production by ATLAS at the 8 TeV [48] and 13 TeV [49] and also by CMS at the 8 TeV [50] . The observed 95% upper limit on SM Higgs cross section times the branching ratio is 4.1 pb from ATLAS and 4.6 pb from CMS at 8 TeV. For our benchmark point, the cross section for the process pp → jjK = 162 pb with p T,j > 20 GeV, |η j | < 5. In order to obtain the rough idea about the constraint by comparing the LCH VBF search, we simplify require m jj > 650 GeV from the Madgraph parton-level simulation for the SM VBF Higgs process and for ourKjj. This cut efficiency forKjj is only 0.006 comparing to the cut efficiency on SM VBF process 0.23. Then the effective cross section after this cut for our benchmark point is only σ(pp → jj(K → bb)) ∼ 0.5 pb by including the branching ratio ofK → bb, which is smaller than the constraint from ATLAS [48] 0.94pb and 1.06pb from CMS [50].
At 13TeV, ATLAS collaboration has explored SM Higgs in VBF production with an associated high energy photon in [49]. The observed 95% confidence level upper limit on the production cross section times branching ratio for a Higgs mass of 125 GeV is 4.0 times the Standard Model expectation. We use Madgraph to produce SM Hjjγ and our model Kjjγ, with both H andK decaying tobb. At the parton level, we estimate the cross-section based on the basic cuts p j T > 40GeV, p γ T > 30GeV, and m jj > 800GeV. After cuts and multiplying the correspondingbb BR, we found SM Hjjγ →bbjjγ andKjjγ →bbjjγ have cross-sections of about 4.5 fb and 4.0 fb respectively. Therefore, we conclude that our benchmark is not excluded by the constraints coming from the SM Higgs search in the VBF channel with an associated high energy photon.
Besides the VBF search, LHC also searched for SM Higgs via ZH and W ± H associated production. The constrain on such scenario is σ(ZH) × BR(H → bb) = 0.57 +0. 26 −0.23 pb from ATLAS [51]. For our model, the corresponding process is pp →ZK, the cross section is suppressed by sin 2 α ∼ 10 −3 , which is much smaller than the SM cross section.
Before closing the section, we make some more comments on the 2.9σ excess in the dijet resonance searches at 13TeV CMS [26], which motivated us to set mK = 115 GeV as the benchmark point. This search is dedicated to look for new vector resonance Z , which only coupled to the SM quarks with universal vector-like coupling, and the largest deviation from the SM background only hypothesis is around m Z = 115 GeV with local significance ∼ 2.9 σ. Comparing the observed 95% CL upper limit cross section ∼ 1.05 × 10 4 pb for the Z with the expected one ∼ 4.5 × 10 3 pb, we can see that roughly one needs 5 × 10 3 pb to fit the excess. The cross-section in our benchmark point at tree level is 3.1 × 10 3 pb, which is capable to explain this excess. The search requires high p T Z that the dijet merged into a single jet. Given that in our model,K decays tobb andcc at equal rate, it is interesting to analyze what could be the significance had CMS performed heavy flavor tagging, something not done in Ref. [26]. At 13TeV LHC [52], CMS collaboration has looked for the high p T fat jet with b-tagging in the inclusive H + j measurement. The tagging efficiency is 33% for H → (bb) as a fat jet and 1% for mis-tagging efficiency from light flavor quarks. If applying b-tagging in Z resonance search in Ref. [26], the increase in S/ which is a moderate increase if the background error is statistic dominant.
The CMS collaboration further used this high p T fat jet with b-tagging technique in related searches for the inclusive H + j process with H →bb, by requiring p H T > 450 GeV [52]. The theoretical cross-section for H(bb) with p j T > 450GeV is 31.7 ± 9.5 fb with 30% uncertainty, while the measured value is 74 ± 50 fb. The mean value is therefore about 2.5 times the Higgs one, with an observed significance of 1.5σ. No other significant resonances have been found. In our benchmark model, the cross-sections after cut forKj → (bb)j and Kb → (bb)b are about 41 fb and 25 fb respectively. Note that Kj has a similar cross-section as Hj, and mK = 115 GeV in our benchmark. Moreover, with an extra b quark inKb, the mis-reconstruct, mis-combination and mis-tagging might result in a smaller contribution, thus we estimate its contribution should be less significant. The m bb distribution in Fig. 4 of [52] presents a broad excess that range from 105 GeV to 140 GeV and therefore, although a dedicated experimental analysis must be performed, we conclude that theK signal is compatible with the current experimental observations in this channel. Higher luminosity LHC measurements in this channel are likely to provide the most effective way of probing this scenario.

V. DARK MATTER SEARCH
In this section, we will explore in detail the possibility of the neutral vector-like fermion χ 2 being a dark matter candidate. The interaction Lagrangian for χ 2 in the mass basis at leading order of sin α and cos β is where m χ 2 = y χ 2 v D / √ 2. The Majorana mass term 1 2 M mχ2,L χ c 2,L in eq. (10) will split the Dirac fermion into two Majorana fermions, which is similar to the inelastic DM setup considered in Ref. [53]. In the Weyl fermion basis χ 2,L , χ c 2,R T , the mass matrix is given by: where we assume M m , m χ 2 > 0 without loss of generality. This symmetric mass matrix can be diagonalized by an orthogonal rotation: where η 1,2 are the mass eigenstates of two Majorana fermions and the factor −i is to ensure the Majorana masses of η 1,2 have the same value m χ 2 in the limit of M m = 0.
In the small Majorana mass limit M m m χ 2 , the eigenstate masses are where the mixing angle is given by: For large Majorana mass M m m χ 2 , the eigenstate masses are which is a typical see-saw mass, with the mixing angle s χ 2 = −m χ 2 /(M m ) 1. With the mixing angle we can rewrite the light Majorana DM η 2 back into its Dirac form, and also the interaction Lagrangian as follows: We can simplify it by where we keep only the leading order interactions in O(M m /m χ 2 ) or O(m χ 2 /M m ). In this following subsections, we will discuss the phenomenology of Dirac and Majorana DM separately.

A. Dirac Dark Matter
We first consider the case of pure Dirac dark matter, whose Lagrangian is listed in Eq. (65). We will study the condition to obtain the correct relic abundance and explore the dark matter limits from indirect detection, direct detection and collider searches.

DM annihilation
We first calculate the χ 2 χ 2 annihilation cross sections. The DM annihilationχ 2 χ 2 →f f is an s-channel process, mediated byK,Z, H 0 1,2,3 and A 0 . From Eq. (65), only processes with bb(cc) final states and mediated byK and H 0 3 are not suppressed by small mixing angle sin α and cos β. Given that the Yukawa couplings between H 0 3 and b, c quarks are much smaller than 1, we conclude that the dominant DM annihilation process isχ 2 χ 2 →K * →bb,cc with annihilation cross-section where we have neglected the quark mass in the second line. For the annihilation at freezeout, it needs to be averaged over thermal distribution of DM, while for annihilation today, it only needs the substitution s = 4m 2 χ 2 . To reproduce the right relic abundance Ωh 2 = 0.12 [54], the thermal averaged crosssection for Dirac fermion DM is about 6 × 10 −26 cm 3 /s. In Fig. 4, we plot the contours (the orange line) in the m χ 2 − g D plane, which gives the right relic abundance for our benchmark scenario mK = 115 GeV. If we further choose g D = 0.36 as our benchmark point, we obtain two solutions for the DM mass, m DM =14 GeV or 236 GeV, which can satisfy the relic abundance requirement.

DM indirect detection
The Dirac fermion DM χ 2 annihilation tobb andcc have equal rate, with total annihilation cross-section leading to right relic abundance for DM mass 14 (236) GeV. Since the annihilation is s-wave, the final state particles from DM annihilation will inject energy into primordial plasma which would delay recombination and thus leave observable imprints in the Cosmic Microwave Background (CMB) [59][60][61][62]. Given that energy injection efficiency ofbb andcc are similar [63], the constraint from CMB [54] is m χ 2 -m H 0 3 plane (Right). The orange line correspond to parameters that lead to the right relic abundance Ωh 2 = 0.12. The red shaded region gives the CMB limits [54], while the blue shaded region gives the most stringent gamma-ray limits from Fermi observation in dwarf galaxies [55,56] (labeled as "Fermi γ at Dwarf galaxies"). The green area is excluded by Xenon1T [57] for benchmark point parameters. The gray line is limits from jets+MET with 1 b-jet tagging at 13TeV CMS [58].
Making use of the f (z) function from [62], we plot the excluded region (in red) in Fig. 4, where we can see that the low mass benchmark m χ 2 = 14 GeV is excluded, while the high mass m χ 2 = 236 GeV is still allowed .
In addition, the Fermi-LAT gamma-ray observations of dwarf galaxies provide a constraint on the DM annihilation cross-sections based on final states [55,56]. Forbb final states, this tells us that the DM mass should be larger than 100 GeV, i.e. m χ 2 100 GeV, in order to have the right thermal relic density. Since the photon spectrum from final statebb and cc are quite similar [64], it again excludes the light DM benchmark but not for the heavy one. The gamma-ray observation from Galactic Center (GC) by Fermi-LAT gives constraint m DM 50 GeV forbb final states [65], which is less stringent than dwarf galaxies. There is also a gamma-ray constraint from the Virgo cluster [66], but is much weaker than the above two constraints. Therefore, in Fig. 4, we only show the most stringent limits from Fermi dwarf galaxies observation in blue shaded area.

DM direct detection
In this section, we will consider the direct detection (DD) of χ 2 , which are related to the scattering between χ 2 and nucleon. The sum of different flavor quark contribution inside nucleon from scalar mediator should be performed at the amplitude level and the results read: where f TG ≈ 0.917 [69] (see also results from [70,71]). In our model, the scattering between nucleon and χ 2 are mediated by CP even scalars H 0 1,2,3 , CP odd scalar A 0 and neutral gauge bosonK andZ. We will consider the scalar and vector contribution separately in the next two paragraphs. In the non-relativistic and heavy DM limit, both scalar mediation and V-V mediation have the fermion bilinearχ 2 (1 + γ 0 )χ 2N N/2, and we can calculate the SI cross-section for χ 2 scattering with nucleon N [72]. Note that the J Z current involves an isospin violating coupling that f p = g 4cw (1 − 4s 2 w ) and f n = g 4cw (−1), therefore we should average over proton and nucleon in the nuclei. The averaged SI cross-section for χ 2 and nucleon is where µ 2 N = m χ 2 m N /(m χ 2 + m N ) is the DM-nucleon reduced mass. From Eq. (80), we see thatK andZ contribution cancels each other due to mass mixing effect. In our benchmark point g D = 0.36, sin α = −0.03, and mK = 115GeV, with |f p | |f n | and f n < 0, we found that the scalar mediated amplitude and vector mediated amplitude interfere destructively.
If m H 0 3 ∼ v D , then the vector contribution dominates, and σ SI N does not depend on m H 0 the vector contribution gets a reduction of about 1/25 already. The current limits on σ SI N are from PANDAX-II, LUX and Xenon1T [57,74,75], and for DM mass around 10 ∼ 100 GeV is of the order of a few 10 −46 cm 2 . Therefore, in order to satisfy the DD bounds, a cancellation between the vector and scalar contributions is required. We show the constraint from Xenon1T in the right panel of Fig. 4. The green area is excluded by Xenon1T with our benchmark point. For the values allowed by indirect detection, m χ 2 = 236 GeV, the allowed region for m H 0 3 is 103 − 116 GeV. Therefore, if χ 2 is a Dirac fermion, we need to tune the mass of H 0 3 to avoid direct detection limit with a level of ∼ 10% tuning in mass.

DM searches at the LHC
We start by analyzing DM searches at the LHC with mono-jet process pp → jK → j(χ 2χ2 ). The cross section is σ(jK) × BR(K → χ 2χ2 ) for on-shellK production if m χ 2 < mK/2, or σ(pp → jχ 2χ2 ) which is suppressed by 3 body phase space. We first consider the constraint when m χ 2 < mK/2, which is, however, in tension with indirect detection limits.
Then the branching ratio ofK → χ 2χ2 varies from 0.14 to 0 when the mass m χ 2 is varied from m χ 2 = 0 to mK/2. Taking a benchmark point mK = 115 GeV, g D = 0.36, then the cross section of pp → jK at 13TeV LHC is 1.3 × 10 3 pb, and at 8TeV LHC is 960 pb. Then we consider the jet plus MET constraints from LHC 13TeV data with integrated luminosity 36fb −1 [58,76,77]. The ATLAS collaboration [76] studied the mono-jet limits for vector and axial vector mediator between SM quarks and DM. Their inclusive region (IM1) requires / E T > 250 GeV which gives 95% C.L. constraints on cross-section smaller than 0.53 pb.
We calculated parton level process j +K with a requirement P j t > 250 GeV, leading to cross-section of about 2.4 pb at 13TeV. Then we obtain a constraint on the branching ratio BR(K → χ 2χ2 ) < 0.22, which is always satisfied in the low mass region of m χ 2 . The limit is given in Fig. 4 as gray dashed line, showing that m χ 2 10 GeV is excluded.
Aside from mono-jet process, the mono-X (X = A/W/Z) processes are also interesting to look for. However, in s-channel vector mediator type models, usually the mono-jet channel provides the strongest limits [78]. Multi-jets plus missing energy processes have been considered in addition to mono-jet channel to constrain DM simplified models. The usual expectation is that the two type of constraints have comparable limits, which is the case for s-channel vector mediator type models, [79]. This is different from scalar and pseudo-scalar mediators with couplings to quarks which are proportional to quark masses, for which multi-jets process provides stronger limits [80]. The reason is that the production of scalar mediators is typically dominated by gluon fusion, which leads to more events with higher jet multiplicity [81][82][83].
For the other two CMS multi-jet plus MET searches [58,77], the constraints should provide similar limits as mono-jet searches [79]. For the case with no b-tagging, we have checked the signal bin 1 and 2 in Table B.1 of Ref. [77]. However, we found the constraint is weaker than mono-jet search [76], probably because this is a parton level estimation. Adding parton shower and detector simulation should bring a conclusion close to Ref. [79]. Given the K µ are not universally coupled to all quarks but couple specifically with b quark and c quark, it is natural to pay special attention to signal regions with b-jet tagging. The CMS sbottom search [58] looks for di-jet plus MET with b-tagging. The most prominent production mode in our model is a single bottom quark in association withK that correspond to what is called the "Compressed" search region. We checked the two Bins with / p T within [250,300] and [300, 500] with 1 b-jet and H b T < 100 GeV requirements. The 95% C.L. limits on the cross-section are about 14 fb and 18 fb. We calculated the cross-section from parton level analysis respectively, and the corresponding cross sections after cut are 7 fb and 8 fb for our benchmark mK = 115 GeV, respectively. Therefore it does not provide an efficient constraint on BR(K → χ 2χ2 ). For the CMS multi-jet plus MET search [77], we have checked the btagging signal bins 11, 12, 21 and 22 which has N b-jet = 1, 2, and found the sbottom search induced constraints [58] improve but are still weaker than the ones coming from mono-jet searches [76].
Then we consider the case with m χ 2 > mK/2. The largest cross-section that may be obtained when mK = 115 GeV is for m χ 2 58 GeV. We get off-shellK produced jχ 2χ2 cross-section to be ∼ 0.01 pb for g D = 0.36 and sin α = −0.03, where we only cut on p j T > 250 GeV. This is safe from the constraints at 13TeV LHC [76] that cross-section after all cuts should be smaller than 0.57 pb. When m χ 2 is larger, the limits are even weaker due to smaller cross-section. We also check the process jjχ 2χ 2 with our benchmark setup and found it is even safer from Ref. [58]. For m χ 2 > mK/2, it is in general safe from the limits, due to small heavy quark PDF, 3-body phase space and off-shell suppression.

B. Majorana Dark Matter
If the mass of dark matter has contribution from a Majorana mass, the interactions between χ 2 and other particles are listed in Eq. (76). In this section, we will discuss the phenomenology of such Majorana DM, including constraints from dark matter relic abundance, direct detection, indirect detection and collider searches.

DM annihilation
The dominant annihilation process of χ 2χ 2 → ff is also mediated byK, after considering the mixing angle and Yukawa coupling suppression. The annihilation cross section is From the annihilation formula in Eq. (81), we can find out there are two contributions, one is p-wave suppressed which is proportional to (s − 4m 2 case, the vector coupling becomes γ µ γ 5 which induces spin dependent (SD) interaction, or velocity (momentum transfer) suppressed SI interaction. Therefore, the cross-section of vector mediated processes is very small and can be ignored. There are vector coupling between DM η 2 and its excited state η 1 , but it will be irrelevant if mass splitting is larger than O(100) keV. For the scalar part, if M m m χ 2 , s χ 2 = −m χ 2 /M m is very tiny so we can ignore the cross-section. Therefore, there are no constraints from direct detection. In the small splitting case M m m χ 2 , s χ 2 ∼ c χ 2 ∼ 1/ √ 2, then the coupling between χ 2 and Higgs are similar as in the Dirac case, and hence the scattering cross-section between χ 2 and nucleon is With the scattering cross-section, we give the constraints on m H 0 3 -m χ 2 plane, which are shown as the green area in Fig. 6. If m H 0 3 is large enough, the cross-section will be very tiny.

DM search at LHC
As previously discussed, the search for dark matter at LHC is dominantly via the interaction between dark matter andK. The branching ratio of BR(K) → χ 2χ 2 goes from 1/7 to 0 when m χ 2 < mK/2 for M m m χ 2 . Since the cross-section of jK and jjK at 13 TeV LHC does not change with respect to the Dirac case, if we consider the case m χ 2 < mK/2, then the constraints on invisible decay branching ratio is the same as the Dirac case. After combining the constraints from ATLAS and CMS, we can still make use of the limit BR(K) → χ 2χ 2 < 0.14 leading to a constraint on m χ 2 8.5GeV for M m m χ 2 . We show the LHC constraints as a gray area in Fig. 6, which does exclude low mass DM benchmark.
If m χ 2 > mK/2, for the search of jet+MET at ATLAS, we compare our cross-section ∼ 0.01 pb after cut p j T > 250 GeV to the constraint at 13 TeV LHC which is 0.57 pb [76]. It shows that off-shellK is very safe from limits from mono-jet searches. We also check the process jjχ 2χ 2 with our benchmark setup and found it is even safer from the constraints obtained in Ref. [58].

VI. CONCLUSIONS
In this article we have studied a gauge extension of the SM that allows to explained the Cancellation of anomalies in this model leads to the presence of a charged, vector-like lepton singlet state, as well as a vector-like neutral state that serves as a good DM candidate.
If it is a pure Dirac fermion, we can obtain the right relic abundance when its mass is around 14 or 236 GeV. The indirect detection for DM annihilation induced gamma-rays rules out the low mass DM benchmark, but keeps the high mass benchmark intact. The direct detection excludes the heavy Dirac χ 2 benchmarks, unless a 10% fine tuning in H 0 3 mass is applied. If χ 2 is split into two Majorana fermions, as is naturally the case, the direct detection constraint is easily evaded for large enough m H 0 3 > 400 GeV. We also can get the right relic abundance for a mass m χ 2 ∼ 22 or 142 GeV. There are no indirect detection limits because the annihilation cross-section at low temperatures is highly suppressed. The LHC searches does not rule out the both DM benchmark points, but is marginal for low mass DM benchmark point. In this section, we list a more detailed expression for CP-even Higgs mass and mixing matrix. The mass eigenvalues for CP-even Higgs in small cot β ≡ v 1 /v 2 and λ 6 expansion are given below, The mixing matrix in Eq. (29) are given in the more detailed expressions below, where the U matrix is approximate anti-symmetric that U 21 ∼ −U 12 , U 31 ∼ −U 13 and U 23 ∼ −U 32 .