Vector-like quark interpretation for the CKM unitarity violation, excess in Higgs signal strength, and bottom quark forward-backward asymmetry

Due to a recent more precise evaluation of Vud and Vus, the unitarity condition of the first row in the Cabibbo-Kobayashi-Maskawa (CKM) matrix: |Vud|2 +|Vus|2 +|Vub|2 = 0.99798 ± 0.00038 now stands at a deviation more than 4σ from unity. Furthermore, a mild excess in the overall Higgs signal strength appears at about 2σ above the standard model (SM) prediction, as well as the long-lasting discrepancy in the forward-backward asymmetry AFBb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal{A}}_{\mathrm{FB}}^b $$\end{document} in Z→bb¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ Z\to b\overline{b} $$\end{document} at LEP. Motivated from the above three anomalies we investigate an extension of the SM with vector-like quarks (VLQs) associated with the down-quark sector, with the goal of alleviating the tension among these datasets. We perform global fits of the model under the constraints coming from the unitarity condition of the first row of the CKM matrix, the Z -pole observables AFBb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal{A}}_{\mathrm{FB}}^b $$\end{document}, Rb and Γhad, Electro-Weak precision observables ∆S and ∆T , B-meson observables Bd0−B¯d0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {B}_d^0-{\overline{B}}_d^0 $$\end{document} mixing, B+→ π+ℓ+ℓ− and B0→ μ+μ− and direct searches for VLQs at the Large Hadron Collider (LHC). Our results suggest that adding VLQs to the SM provides better agreement than the SM.


Introduction
The Standard Model (SM) particle content includes three families of fermions under the identical representation of the gauge symmetries SU(3) c × SU(2) L × U(1) Y . Each fermion family includes a quark sector (up-type and down-type quarks) and a lepton sector (charged leptons and a neutrino). The well-known quark mixing in crossing between the families is an indispensable ingredient in flavor physics. One can rotate the interaction eigenbasis to the mass eigenbasis in the quark sector through a unitary transformation, and it generates nonzero flavor mixings across the families in the charged-current interactions with the W boson. The quark mixing for the three generations in the SM can be generally parameterized by the 3×3 Cabibbo-Kobayashi-Maskawa (CKM) matrix V SM CKM [1,2]. Since V SM CKM is composed of two unitary matrices, unitarity of the CKM matrix shall be maintained. The existence of additional quarks beyond the three SM families shall extend the CKM matrix to a larger dimension. In such a case, the unitarity of original 3 by 3 submatrix will no longer hold.

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The recent updated measurements and analyses of V ud and V us are briefly outlined as follows. The most precise determination of |V ud | is extracted from the superallowed 0 + −0 + nuclear β decay measurements [3,4] |V ud | 2 = 0.97147 (20) 1 where ∆ V R accounts for short-distance radiative correction. Recently, according to the dispersion relation study with experimental data of neutrino-proton scattering, the inner radiative correction with reduced hadronic uncertainties ∆ V R = 0.02467 (22) was reported in ref. [5]. It significantly modified the value of |V ud | = 0.97370 (14) [4]. On the other hand, one can use various kaon decay channels to independently extract the values of |V us | and |V us /V ud |. Based on the analysis of semileptonic Kl3 decays [8] and the comparison between the kaon and pion inclusive radiative decay rates K → µν(γ) and π → µν(γ) [9], the values of |V us | = 0.22333(60) and |V us /V ud | = 0.23130(50) are obtained in ref. [4]. As a result, the matrix-element squared of the first row of V SM CKM |V ud | 2 + |V us | 2 + |V ub | 2 = 0.99798 ± 0.00038 , (1.2) which deviates from the unitarity by more than 4σ [4,5]. 1 If this deviation is further confirmed, it may invoke additional quarks to extend the CKM matrix. 2 After the final piece of the SM, Higgs boson, has been discovered in 2012 [13,14], the precise measurements of its properties become more and more important. The SM can fully predict the signal strengths of this 125 GeV scalar boson so that deviations from the SM predictions can help us to trace the footprint of new physics beyond the SM. Recently, the average on the Higgs-signal strengths from both ATLAS and CMS Collaborations indicated an excess at the level of 1.5σ. 3 If one looks more closely into each individual signal strength channel, one would find that mild 1σ excesses appear in the majority of channels. After taking into account of all available data from the Higgs measurements, the average of the 125 GeV Higgs signal strengths was obtained [17] µ Higgs = 1.10 ± 0.05 . (1.3) One simple extension of the SM with an SU(2) doublet of vector-like quarks (VLQs) with hypercharge −5/6 can be introduced to account for the excess by reducing the bottom 1 Reduction in the extracted value of V ud is due to the reduction of uncertainty in ∆ V R , which is made possible by a dispersion-relation based formulation of the γ −W box contribution to the neutron and nuclear beta decays [6]. However, the value is to be taken cautiously before jumping to a conclusion, because one has to include properly the quasielastic contribution from one-nucleon knock-out as well as advanced correction from two-nucleon knock-out. On the other hand, a recent proposal to study ∆ V R on lattice can be found in ref. [7]. 2 Another explanation for this deviation involves new physics in the neutrino sector with lepton-flavor

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Yukawa coupling at about 6% from its SM value [17]. Since the h → bb mode takes up around 58% of the 125 GeV Higgs total decay width, the above extension can reduce the total Higgs width and universally raise the signal strengths by about 10% to fit the data. Finally, the measurement of the forward-backward asymmetry A b F B of the bottom quark at the Z 0 pole has exhibited a long-lasting −2.4σ deviation from the SM prediction [9]. Again, this anomaly can be reconciled by introducing an SU(2) doublet VLQs with hypercharge −5/6. The mixing between the isospin T 3 = 1/2 component of VLQs and the right-handed SM bottom quark with mixing angle sin θ R 0.2 can enhance the right-handed bottom quark coupling with Z boson. Meanwhile, the left-handed bottom quark coupling remains intact [17]. However, the mixing between VLQs and the SM bottom quark is under severe restrictions from other Z 0 -pole observables, for example, the Z hadronic decay width Γ had and the ratio of Z partial width into bb relative to the total hadronic width, R b , are both consistent with SM predictions. Earlier attempts in this direction can be found in refs. [18,19].
All the above three discrepancies can be explained with additional heavy quarks, which mix with the SM bottom quark. In order to guarantee the anomaly-free condition, one economical way is to introduce VLQs. The review of various types of VLQs can be found in ref. [20]. In this study, we need to modify both left-handed and right-handed downquark sectors in order to alleviate the above three anomalies. In general, both left-handed and right-handed mixing angles are generated and related to each other for each type of VLQs though one may be suppressed relative to another. It means that we need at least two types of VLQs to simultaneously explain these anomalies. We show that the minimal model requires coexistence of both doublet and singlet VLQs, B L,R and b L,R . This paper is organized as follows. In section 2, we first write down the general model and study the interactions between VLQs and SM particles, especially the modifications of couplings to W , Z, and h bosons. Then we boil down to the requirements of the minimal model. The various constraints from relevant experimental observables are discussed in section 3. In section 4, we perform the chi-square fitting and show numerical results, in particular we discuss the allowed parameter space that can explain all three anomalies. We summarize in section 5.

Standard model with extra vector-like quarks
In this work, a doublet and singlet of vector-like quarks (VLQs) are introduced: with hypercharges (Y /2) B L,R = −5/6 and (Y /2) b L,R = −1/3, respectively, under the SM U(1) Y symmetry. The upper component of the doublet and the singlets have the same quantum numbers as the SM down-type quarks, and thus they are allowed to mix with the SM down-type quarks if nontrivial Yukawa interactions exist among them. It was pointed out that the Yukawa interaction between B L and b R will induce a mixing between the JHEP05(2020)117 right-handed b R and b R , and so reduce the bottom Yukawa coupling. At the same time, it will increase the coupling of the Z boson to the right-handed b quark [17]. The reduction in the bottom Yukawa coupling gives rise to a decrease in the Higgs total decay width, and thus can help alleviate the overall Higgs signal-strength excess, while the increase in the Z coupling to the right-handed b quark can bring the prediction of the forward-backward asymmetry A b F B down to the experimental value. On the other hand, the mixing between b L and b L is suppressed due to the absence of Yukawa interaction between B R and b L , and so the modification of CKM matrix is negligible. However, the Higgs-induced Yukawa interaction between b L,R and the SM down quarks will give a larger left-handed mixing than the right-handed one. Thus, the non-negligible left-handed mixing can further modify the original 3 × 3 CKM matrix and the extra VLQs can extend the CKM matrix to 5 × 5 to restore the unitarity.

Yukawa couplings and fermion masses
The generalized interactions between VLQs, SM quarks, and the Higgs doublet are expressed as where U, D represent the SM up-and down-quarks with i, j = 1, 2, 3 as the flavor indices, and superscript 0 indicates flavor eigenstates, for which the SM Yukawa matrix y u,d have been diagonalized. Note the implicit sum over the repeated indices in the above equation. The dual of Higgs field H ≡ iτ 2 H * carries Y /2 = −1/2, where τ 2 is the Pauli matrix. After the electroweak symmetry breaking (EWSB), H = (0, v/ √ 2) T , the mass matrix of the down-type quarks becomes Since both MM † and M † M are symmetric matrices, they can be diagonalized as

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where the mass eigenstates are related to the flavor eigenstates via the unitary matrices V R,L . Similarly, for the up-type quarks the mass eigenstates are related to the flavor eigenstates by Since the VLQs do not mix with up-type quarks, the up-type quark mass matrix remains the same as in SM. Due to the discrepancies between the mass matrix and Higgs interaction matrix, the Higgs couplings of down-type quarks will be modified from the SM Yukawa couplings, The coupling for b L b R h can be extracted out from the matrix element (Y) 33 , for example. Since we only introduce the vector-like quarks that can mix with the bottom quarks, the Higgs couplings to the up-type quarks will stay the same as the SM ones.

Modifications to the W couplings with SM quarks
The charged-current interactions via the W boson with the SM quarks and vector-like quarks are where P L,R = 1∓γ 5 2 . We define the 5 × 5 CKM matrix as

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Since the VLQs do not modify the up-quark sector, we simply extend the 3 × 3 matrix W L in eq. (2.9) to a 5 × 5 matrix. The exact parameterization of V 5×5 CKM will be shown in appendix A.
We further parameterize the charged current interactions in the following simple form [21], where q includes all SM quarks and VLQs. A L ij and A R ij are summarized as follows where α = 1 to 3, β = 1 to 5, and (U 1 , where Q f (T 3f ) is the electric charge (third component of isospin) of quarks, the gauge coupling g Z = g 2 / cos θ w , x w = sin 2 θ w is the sine-square of the Weinberg angle θ w . Again, the Z boson couplings to the SM up-type quarks are exactly the same as in the SM and are not modified by VLQs. We further parameterize the Z boson couplings with SM down-type quarks and VLQs in the following simple form [21], where X L ij and X R ij are summarized below, JHEP05(2020)117

Minimal models
In this subsection, we would like to narrow down to the most relevant couplings to the experimental anomalies. First, we consider non-zero couplings g B 3 , g b 1 , while M 1,2 are at TeV scale. According to ref. [17], the tensions of Higgs signal strength and A b FB can be alleviated by the g B 3 coupling from the doublet VLQ. Then the CKM unitarity violation mainly due to the |V ud | is relevant to g b 1 from the singlet VLQ. Other parameters in eq. (2.2) are set to zero. It simplifies the down-type quark mass matrix and V L,R as Here we have taken the liberty that the first two generations of the SM down-type quark masses are set at zero. If the couplings g .

(2.17)
According to eq. (2.7), the coupling for (h/v)b L b R is given by .

(2.18)
This gives rise to a reduction factor in the Higgs Yukawa coupling by C hbb ≡ c R 34 / 1 + (∆ 2 /M 2 1 ), and thus the enhancement of Higgs signal strengths. The modification of the CKM matrix is indicated by eq. (2.9). The first row of first three elements of V 5×5 CKM violates unitarity as However, the unitarity for the first row of V 5×5 CKM can be restored with the other two elements If s L 15 ∼ s L 34 , we anticipate the contribution from V ub will be dominant.

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Finally, from eq. (2.13) the Zbb couplings are modified as Since s R 34 enhances (g b ) R , it alleviates the tension between A b F B observation and SM prediction.
Second, we include one more non-zero coupling g b 3 . Then the mass matrix and unitary transformations matrices are (2.23) Here we diagonalize MM † via a 4-step block diagonalization procedure. We have used rotation matrices with the order of R(θ 15 ), R(θ 35 ), R(θ 34 ), and R(θ 45 ) to block diagonalize MM † in each step and finally V L and V R can be approximated by eq. (2.22). The mass of the bottom quark which is the same as eq. (2.18). The first three elements in the first row of V 5×5 CKM violate unitarity as

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Similarly, the unitarity in the fist row of V 5×5 CKM can be restore by the other two elements Once again, the contribution from V ub is the dominant one. Then the Zdd, Zbb, Zdb couplings are given by 4 The FCNC is generated from (g db ) L and shall be constrained by More details are shown in the following sections.

Z boson measurements
Once the d, s, b couplings to the Z boson are modified, we find that the following observables are modified: JHEP05(2020)117 1. Total hadronic width. At tree level, the change to the decay width into dd, ss, or bb is given by With this modification, the total hadronic width is changed to The R b is the fraction of hadronic width into bb, which is given by There is a large tension in the forward-backward asymmetry of b quark production at the Z resonance between the experimental measurement and the SM prediction, The couplings of fermions to the Z boson are basically given by T 3 − Qx w in the SM. For the electron it is simply It was pointed out in ref. [17] that the interaction term g B 3 B 0 L H b 0 R from the doublet vector-like quark B L,R is able to reconcile this tension.
For the second minimal model, where g B 3 , g b 1,2 are non-zero couplings, the modifications of (g b ) L and (g b ) R can be found from eq. (2.27). If we further assume s L 15 , s L Both s R 34 and s L 35 can reduce the the forward-backward asymmetry A b FB of the quark at Z-pole. They are good to fit the measured A b FB at a lower value from the SM prediction.

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On the other hand, s L 35 reduces R b but s R 34 increases R b . We can use both to maintain R b at the SM value. This is achieved in the leading order by Unfortunately, we will see from the Fit-2b in section 4 that the B-meson observables are too restrictive to fulfill this relation. Subsequently, mixing angles are chosen to fit the anomaly in A b FB .

125 GeV Higgs precision measurements
The data for the Higgs signal strengths for the combined 7 + 8 TeV data from ATLAS and CMS [22] and all the most updated 13 TeV data were summarized in ref. [23]. The overall average signal strength is µ Higgs = 1.10 ± 0.05 [23], which is moderately above the SM prediction. Using a total of 64 data points, the goodness of the SM description for the Higgs data stands at χ 2 /d.o.f. = 53.81/64, which gives a goodness of fit 0.814. A reduction in the total Higgs decay width can provide a better description of the Higgs data with χ 2 /d.o.f. = 51.44/63, corresponding to a goodness of fit 0.851 [23]. The p-value of the hypothesis of the single-parameter fit (∆Γ tot ) equals 0.12 when the SM is the null hypothesis. Although it is not significantly enough to say they are different, it may still give a hint that the single-parameter fit is indeed better than the SM. In this work, the reduction in the Higgs total width is achieved by a slight reduction in the RH bottom Yukawa coupling which can be found from the matrix element (Y) 33 in eq. (2.7) and predominately from the doublet vector-like bottom quark interaction term g B 3 B 0 L H b 0 R . 6

Electro-Weak Precision Observables(EWPOs)
The Electro-Weak Precision Observables (EWPOs) can be another important indirect constraint for the mixings and masses of the VLQs. The EWPOs can be represented by a set of oblique parameters S, T and U . We apply the data from Particles Data Group (PDG) 2018 review [9] with a fixed U = 0, and the best fits of S and T parameters are ∆S = 0.02 ± 0.07, ∆T = 0.06 ± 0.06. (3.6) where ∆S and ∆T are defined as We consider the 3σ allowed regions of ∆S and ∆T parameters in our fitting. 6 Once vector-like bottom quarks are heavier than 1 TeV, their contributions to gg → h and h → γγ are tiny. We will ignore these effects in our fitting.

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The general form of S parameter can be represented as [21,24,25] , M q i are the quark masses, and A L,R ij , X L,R ij are defined in eqs. (2.11) and (2.14) respectively. On the other hand, the functions inside S are The contributions from t and b quarks in the SM for the S parameter can be represented as Similarly, the general form of T parameter can be represented as [21,24,26] where the functions inside T are θ + (y 1 , y 2 ) = y 1 + y 2 − 2y 1 y 2 y 1 − y 2 log y 1 y 2 (3.12) The contributions from t and b quarks in the SM for the T parameter can be represented as in s-channel. The overall expression including the SM t-W box diagram and Z boson FCNC is [27] x where U 2 std−db is from the SM contribution of top-W box diagram, and −U db ≡ V * L35 V L15 from the Z boson FCNC induced by the singlet VLQ. On the other hand, the FCNC contribution from the doublet VLQ, V * L34 V L14 , is much smaller than that from the singlet VLQ, because the pattern of the mass matrix which suppresses the left-handed mixing angle for doublet VLQ with down and bottom quarks [17]. The prefactor was obtained by substituting the numerical values: the √ B B f B = 225 ± 9 MeV [9] from lattice calculation; the QCD correction η B = 0.55 [28]; the B d lifetime τ B d = 1.520(4) ps = 2.31 × 10 12 GeV −1 and mass m B d = 5.27963(15) GeV [9]; and Fermi constant G F . The expression for SM contribution is given by [29] where y t ≡ m 2 t /m 2 W and the loop function [29] f 2 (y) ≡ 1 − 3 4 Taking the most updated experimental values of |V tb | = 1.019 ± 0.025 and |V td | = (8.1 ± 0.5) × 10 −3 [9], the SM reproduces the central value of the current experimental measurement [9] x d | exp = 0.770 ± 0.004 . (3.17) However, the theoretical uncertainty is much larger than the experimental one. For conservative limit we require the new physics contribution to be less than the SM contribution, which implies that is much weaker than the constraints from B + → π + + − and B 0 → µ + µ − in the next two subsections. In addition, due to large theoretical uncertainties we do not use this data in our global analysis. On the other hand, the mixings between the second generation quarks and new VLQs are irrelevant in this study. In order to avoid the stringent constraints from the mixing of D 0 -D 0 , K 0 -K 0 , and B 0 s -B 0 s mesons, we suppress all the interaction terms between the second generation quarks and new VLQs for simplicity. 7 The more general study can be found in ref. [32].

(3.24)
In the following chi-square fitting, we combine both the experimental error and 30% theoretical uncertainty from the SM [30] to give conservative constraints.

The
operator also contributes to the B 0 → µ + µ − through the expression [31] Br where f B = 225 MeV. In our framework, the (g db ) R = 0 from eq. (2.27) guarantees no mixing among the right-handed d and b quarks and thus C 10 defined in ref. [31] is zero.

Direct searches for the vector-like bottom quarks
The vector-like bottom quarks can be pair produced by QCD processes or singly produced via a t-channel Z boson exchange at hadron colliders. Assuming that the new vectorlike bottom quarks can only decay to SM particles, there are three possible decay modes: The searches for pair production of vectorlike bottom quarks only depend on their masses, decay patterns, and branching ratios. According to ref. [36], the ATLAS Collaboration has published their combined searches for pair production of vector-like bottom quarks with the above three decay modes. The SU(2) singlet vector-like bottom quark b is excluded for masses below 1.22 TeV, and the SU(2) doublet vector-like bottom quark B = (b −1/3 , p −4/3 ) T is excluded for masses below 1.14 TeV. Other recent searches for pair production of vector-like bottom quarks from CMS Collaboration can be found in refs. [37,38], and those constraints are similar to ref. [36].
On the other hand, the searches for single production of vector-like bottom quarks depend not only on their masses, but also on their mixing with SM down-type quarks. Recently, the ATLAS Collaboration has published their searches for single production of vector-like bottom quark with decays into a Higgs boson and a b quark, followed by H → γγ in ref. [39]. Again, this constraint is roughly the same as the above ones. Similarly, the searches for pair production and single production of vector-like quark p with electric charge −4/3 can be found in refs. [40,41]. A lower mass limit about 1.30 TeV at 95% confidence level is set on the p . In order to escape the constraints from these direct searches at the LHC, we can increase m b , m p , and m b to be above the lower bounds of the mass constraints. Therefore, we safely set their masses at 1.5 TeV in the analysis.

Fitting
Five data sets are considered in our analysis. Totally, we used 75 data points: 64 from 125 GeV Higgs signal strengths; four from CKM; three from A b FB , R EXP b , Γ had each; two from ∆S, ∆T ; and two from Br(B + → π + + − ) and Br(B 0 → µ + µ − ). They are summarized in table 1.
The SM CKM matrix is parameterized using the Wolfenstein parameters [9] quoted from the global fit [9]. The SM values of |V SM us |, |V SM us /V SM ud |, |V SM ud |, and |V ub | are listed in table 1, and the uncertainties from global fit in SM are included in our chisquare analysis. In fact, the SM does not fit well to the above datasets, as it gives a total χ 2 (SM)/d.o.f. = 88.946/75, which is translated into a goodness of fit only 0.130. Note that during the parameter scan, the unitarity condition of i=d,s,b,b ,b" |V ui | 2 = 1 is always held from our analytical parameterization. The unitary violation only happens on i=d,s,b |V ui | 2 .
According to the minimal model of additional VLQs with various options on the parameters in subsection 2.4, we perform several fittings to investigate if these models can provide better explanations for the data. Without loss of generality we fix the VLQs mass at 1.5 TeV, which is above the current VLQs mass lower bounds from ATLAS and CMS searches [36,39,[41][42][43][44].
For Fit-1, keeping g b 3 = 0 can guarantees the flavor-changing coupling (g db ) L from eq. (2.27) to be zero. Therefore the constraints from B 0 d -B = 0.789. Comparing with the SM fit Fit-1 has a p-value of 2.5 × 10 −6 against the SM null hypothesis. It is shown in both table 2 and figure 1 that the best-fit points prefer a non-zero value of g B 3 = ±1.177 and g b 1 = ±0.335 at a level more than 2.5σ and 4σ from zero, respectively. Furthermore, the bottom-quark Yukawa coupling deviates from the SM prediction by more than 2σ, and the best-fit points give C hbb = 0.98, which is about 2% smaller than the SM value. It helps to enhance the overall Higgs signal strengths. In fact, the Higgs signal-strength dataset prefers bottom Yukawa coupling 6% smaller than the SM value [17]. Since the R EXP b was quite precisely measured and consistent with the SM prediction, the deviation of the bottom-Yukawa coupling cannot exceed more than a couple of percent. From the (V L15 , V R34 ) panel of figure 1, since V L15 s L 15 ∝ g b 1 and V R34 s R 34 ∝ g B 3 , it does not show correlation between g B 3 and g b 1 . In the (V R34 , ∆S) and (V R34 , ∆T ) panels, they show that the best-fit regions are consistent with the oblique parameters from electroweak precision measurements.
In Fit-2, both couplings g b 1 and g b 3 can vary from zero. In this case, according to eq. (2.27), flavor-changing coupling (g db ) L is induced and therefore is constrained B + → π + + − and B 0 → µ + µ − (B 0 d -B 0 d mixing is not included in any of the fits.) In figure 2 for Fit-2a, which has not included these flavor-changing constraints in the global fit, it allows both couplings g b 1 and g b 3 to significantly deviate from zero. Indeed, we see that the best-fit points prefer g B 3 = ±1.651 and g b 3 = ±0.614, and (s R 34 ) 2 5(s L 35 ) 2 are correlated in (V L35 , V R34 ) panel. This is in accordance with our discussion at end of subsection 3.2, where the VLQs contributions to R b cancel among themselves, meanwhile A b F B anomaly is explained by (g b ) L . Since the VLQs contributions to R b are canceled, the bottom-Yukawa coupling now is allowed to deviate from the SM by more than 6%, and the best-fit points give C hbb = 0.96, which deviates form the SM prediction by more than 3σ. Hence, Fit-2a can further lower the minimal chi-square than Fit-1, and gives mixing, B + → π + + − and B 0 → µ + µ − , which will restrict simultaneously large non-zero values of g b 1 and g b 3 . In order to study the effects from those B physics constraints, we further include both B + → π + + − and B 0 → µ + µ − in the Fit-2b.
In figure 3 for Fit-2b, we can understand how the constraints from B + → π + + − and B 0 → µ + µ − affect the allowed parameter region. In the (g b 3 , ∆χ 2 ) panel, the coupling g b 3 is restricted to be small within 3σ, more precisely, it requires |g b 3 | ≤ 0.076. Since g b 3 is restricted close to zero, the best-fit points and the corresponding C hbb of Fit-2b overlap with Fit-1. In the same panel, we can observe there are two local minima at g b 3 ±0.6 at 4σ, which is correlated to g b 1 0 in (g b 1 , ∆χ 2 ) panel. From the (U db , ∆χ 2 ) panel, we know that the flavor constraints from B + → π + + − is more stringent than B 0 d -B 0 d mixing due to more precise theoretical uncertainty in the former. Around the minimum, we can identify the two-tine fork shape structure, and it is due to the interference between JHEP05(2020)117 VLQs and SM contributions for B + → π + + − from eq. (3.20). Finally, comparing with B + → π + + − , the B 0 → µ + µ − gives similar but weaker constraint on (g db ) L . We can also find in table 2 that both the values of Br(B + → π + + − ) and Br(B 0 → µ + µ − ) in Fit-2b are largely reduced by three orders of magnitude compared with Fit-2a. On the other hand, we observe that the value of Br(B + → π + + − ) in Fit-2b is indeed closer to the measurement from LHCb in eq. (3.24) than the SM prediction in eq. (3.23), because the central value in eq. (3.23) is more than 1σ larger than the central value in eq. (3.24). Once both theoretical and experimental uncertainties are reduced in the future with almost the same central value in Br(B + → π + + − ), it will be a smoking-gun signature for adding VLQs to the SM.
For discovery prospects of the doublet+singlet VLQs, there are some signatures which can be searched for at the LHC. The VLQs can be pair produced via QCD processes, such JHEP05(2020)117  Here, we assume the mass degeneracy of b and p from the doublet VLQ to avoid the decay mode b → p W or p → b W . Even though there is slight mass splitting between b and p of order O(10) GeV due to the mixing effect, the decay p → b W or b → p W can only give very soft leptons or jets, which are very difficult for detection at the LHC. The decay branching ratios of VLQs, for example, from the best-fit points for Fit-1 and Fit-2b from table 2 are

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and for Fit-2a, The above relation BR(b → Zb) BR(b → hb) comes from the equivalence theorem, in which the longitudinal mode of gauge bosons behaves like the Goldstone boson in the limit M b ,b ,p m Z,h . Therefore, one clear signature at the LHC from pair produced where X could be either h or Z. Such charged lepton pair(s) plus jets searches have been performed at the 13 TeV LHC [37,44]. Here we roughly estimate the current sensitivity on the lower mass limit of b . The event rate with at least one charged lepton pair is where = 0.0028 taking into account the branching ratios of b and Z → + − . Then using L = 36.1 fb −1 and requiring N < 2, we obtain By adopting the VLQ pair production cross section, the above upper limit translates into the lower mass limit of M b 1.1 TeV. Other decay modes of VLQs from pair production have been searched for by ATLAS and CMS Collaborations in refs. [36,38,42,43]. The lower mass limits of VLQs are around 1 TeV from these searches. Single VLQ production via the electroweak interaction, which depends on the size of mixing between VLQ and SM quark, was investigated in refs. [41,45]. We emphasize the predicted g B 3 and g b 1 values in table II all give s R 34 g B 3 v/( √ 2M 1 ) 0.14 and s L 15 0.04, that can be measured from the single VLQ productions via Zb(W u) fusion and ready to be tested in the near future. For example, the single p produced from the W b fusion has been studied by the ATLAS [41]. Assuming BR(p → W b) = 100% and varying |s R 34 | between 0.17 and 0.55, the lower mass limit of p can be set from 800 to 1800 GeV.
A distinctive signature of our proposed model from other phenomenological models is the singlet VLQ decay mode b → W − u. On the other hand, most of experimental searches at the LHC were focused on the mixing between VLQs and the third generation quarks. Hence, we stress the searches for the mixing between VLQs and the first generation quarks are also well-motivated in this work. The sizeable or dominant BR(b → W − u) can be a distinguishable feature of our scenario.

Discussion
We have advocated an extension of the SM with vector-like quarks, including a doublet and a singlet, in aim of alleviating a few experimental anomalies. An urgent one is a severe JHEP05(2020)117 unitarity violation in the first row of the CKM matrix standing at a level more than 4σ due to a recent more precise evaluation of V ud and V us . Another one is the long-lasting discrepancy in the forward-backward asymmetry A b FB in Z → bb at LEP. Furthermore, a mild excess in the overall Higgs signal strength appears at about 2σ above the standard model (SM) prediction, In this work, we have performed global fits of the model under the constraints coming from the unitarity condition of the first row of the CKM matrix, the Z-pole observables A b FB , R b and Γ had , Electro-Weak precision observables ∆S and ∆T , B-meson observables B 0 d -B 0 d mixing, B + → π + + − and B 0 → µ + µ − , and direct searches for VLQs at the LHC. We found that the extension with a VLQ doublet and a singlet can improve the fitting to the datasets, especially the improvement to the unitarity condition of the first row of the CKM matrix with two additional entries in the first row.
We offer the following comments before closing.
1. By extending the CKM matrix to 5 × 5 with the extra VLQs, the unitarity condition in the first row is fully restored.
2. Without taking into account the B-meson constraints the best-fit (see Fit-2a) can allow the bottom-Yukawa coupling to decrease by about 6%, which can then adequately explain the 2σ excess in the Higgs signal strength. At the same time, it can also account for the A b FB without upsetting R b due to a nontrivial cancellation between two contributions. However, the resulting branching ratios for B + → π + + − and B 0 → µ + µ − become exceedingly large above the experimental values.
3. However, including the B-meson constraints the allowed parameter space in g b 3 is restricted to be very small due to the presence of the FCNC in Z-b-d.

4.
Last but not least, the extra 5 physical CP phases in V 5×5 CKM matrix can be a trigger for electroweak baryogenesis. In order to generate the strong first-order electroweak phase transition, one needs to add an extra singlet complex scalar [47,48]. On the other hand, adding extra Z boson as in the ref. [31] would be possible to cancel the FCNC contributions from VLQs. Therefore, a gauge U(1) extension of our minimal model with a singlet complex scalar may simultaneously alleviate the constraints from B meson observables and explain the matter-antimatter asymmetry of the Universe. However, this extension is beyond the scope of this work and we would like to study this possibility in the future.
We first parameterize the original 3 × 3 CKM matrix in the usual form with s ij = sinθ ij and c ij = cosθ ij [46]. Then we can further parameterize the full 5 × 5 CKM matrix based on V 3×3 CKM as

JHEP05(2020)117
Notice that there is some freedom to arrange the positions of extra 5 CP phases in those matrices. We assign there is no CP phase in the rotation matrices of θ 34 and θ 35 in this study. On the other hand, since we don't involve the vector-like up-type quarks t , t inside the model, only the measurable 3 × 5 sub-matrix of V 5×5 CKM is corresponding for our study here.
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