Constraints on the quark mixing matrix with vector-like quarks

While a chiral fourth generation of quarks is almost ruled out from the data on Higgs boson production and decay at the Large Hadron Collider, vector-like quarks are still a feasible option to extend the fermionic sector of the Standard Model. Such an extension does not suffer from any anomalies and easily passes the constraints coming from oblique electroweak parameters. We consider such minimal extensions with SU(2) singlet and doublet vector-like quarks that may mix with one, or at the most two, of the Standard Model quarks. Constraints on the new mixing angles and phases are obtained from several ∆B = 1 and ∆B = 2 processes.


Introduction
While all flavour observables, both CP-conserving and CP-violating, are more or less well explained by the 3 × 3 Cabibbo-Kobayashi-Maskawa (CKM) matrix, it is pertinent to investigate whether one can still accommodate one or more extra quarks. The new quarks can be chiral, like their Standard Model (SM) counterparts, or vector-like, where both left-and right-chiral components transform identically under the SU (2) of weak interaction. A chiral fourth generation [1] that gets its mass through the same Higgs mechanism as the other three generations do, is more or less ruled out from the data on Higgs boson production and decay at the Large Hadron Collider (LHC) [2][3][4], unless the scalar sector of the Standard Model (SM) is extended [5]. However, vector-like quarks (VQ) easily evade such a constraint. Moreover, by construction, inclusion of VQs still keeps the SM free from any gauge or mixed anomalies.
In this work, we would be interested to find out how much mixing any new VQ may have with the SM quarks. For simplicity, we consider only one charge + 2 3 VQ, called t , and/or only one charge − 1 3 VQ, called b . (Thus, SM gauge singlet VQs, a possible fermionic dark matter candidate, are outside our purview.) In other words, we would like to extend the CKM matrix by a row, or a column, or both, and try to find the bound on the new elements. Such studies have been performed earlier [6,7], and we will build on those studies, armed with more precise experimental data. Our main tool will be the loop-mediated ∆B = 1 (b → sγ, B s → µ + µ − ) or ∆B = 2 (B s -B s mixing) processes, and occasionally, the decay Z → bb. This is one of the reasons why we will not discuss the chiral fourth generation with more scalar fields, as those scalars affect these observables in a nontrivial way. A good example is the radiative b decay, b → sγ, which is affected by the charged scalar loop. Thus, the bounds on the new CKM elements are meaningful only if we have a complete control over the parameters of the scalar potential. As a spin-off, we will also show how anomalous top decays like t → cγ are affected.
If a chiral quark is extremely massive, say of the order of a TeV, the Yukawa coupling will be badly nonperturbative, and no loop calculations are trustworthy [8]. For VQs, one does not face such a problem, as they can have a mass term without symmetry breaking. No such heavy quarks have been observed, so for our analysis, we will keep their masses fixed at 2 TeV. This is just outside the LHC reach; however, in future, only the luminosity of LHC, and not the energy reach, will increase, so anything beyond 2 TeV should remain undiscovered. This increases the importance of such indirect detection studies.
The paper is organised as follows. In Section 2, we provide a short introduction to the VQ models as well as a brief summary of the SM expressions for various loop-mediated observables that are relevant for our study. We also enlist the data that will be used in our analysis. In Section 3, we recast these expressions in the framework of the extended SM with vector-like singlet or doublet quarks. In Section 4, we display our results for each of these models, followed by the summary and conclusion in Section 5.

The toolbox
The charged current Lagrangian in the SM involving the stationary quark fields looks like where i, j = 1, 2, 3 are the generation indices, P L = 1 2 (1 − γ 5 ), and V is the CKM matrix. By construction, V is unitary, and the magnitudes of seven of the elements, as determined from tree-level processes, are as follows: 1.013 ± 0.030 Table 1: Magnitudes of the directly measured CKM matrix elements, taken from Ref. [9].
The elements V td and V ts can only be determined from B 0 -B 0 or B s -B s mass difference, or indirectly from the unitarity of the CKM matrix. In our analysis, we will implicitly assume that the seven CKM elements shown in Table 1 are not affected in any significant way by the VQs.

Introducing the vector-like quarks
Let us first introduce a charge − 1 3 VQ, b , which is an SU (2) singlet. The charged current Lagrangian now reads where U and D are the 3×3 and 4×4 basis transformation matrices. The indices can have the following values: i, j = 1, 2, 3, and k = 1, 2, 3, 4. Hence, the CKM matrix V ≡ U † D is a 3 × 4 matrix. Obviously, the SM part of V is no longer unitary, and we cannot use the unitarity condition to constrain its elements. The following points immediately become evident: • Such a 3 × 4 CKM matrix can be parametrized by 9 independent parameters, including 6 real angles and 3 phases. To see this, let us take U = 1, the identity matrix, so that V is just the top 3 × 4 block of D. The row unitarity conditions still hold, which gives 9 constraints, and one can remove 6 irrelevant phases through quark phase redefinition. Thus, one is left with 2 × 12 − 9 − 6 = 9 independent parameters.
• The tree-level Z-mediated flavour-changing neutral current (FCNC) processes will be present in the down sector, but not in the up sector. The vertex factor for d iL d jL Z tree-level interaction is while there is no such FCNC for the right-chiral quarks (the situation is exactly the opposite for vector-like doublet quarks). We use the standard abbreviation of s W ≡ sin θ W , c W ≡ cos θ W .
• Only if we consider the texture where U = 1 and hence D 3×4 = V , the FCNC vertex factor in Eq. (3) can be written as which can potentially be most significant for the bsZ vertex.
• Again, under such assumptions like U = 1, the 3 × 4 CKM matrix is a part of the full 4 × 4 D matrix, which is unitary, and we can use the row unitarity conditions as constraints, e.g., Similarly, one can introduce a charge + 2 3 singlet VQ, t , for which the CKM matrix V is 4 × 3. The FCNC will be present only for left-chiral up-type quarks, parametrized by a vertex factor analogous to that in Eq. (3): We will consider the following simplified models for singlet VQ (not all of them can be constrained from flavour observables): The structure forbids any tree-level FCNC. However, there will be new one-loop contributions to ZW W and γW W vertices, as well as to tt production 1 . As we will show, this model remains essentially unconstrained, except the bound on |V tb | coming from unitarity, see Eq. (5).

2.
Model VQ-S-D2: Similar to VQ-S-D1, except that both V tb and V cb are taken to be nonzero, while V ub is kept fixed at zero. One of these two new elements can contain a nontrivial phase. The tree-level bsZ interaction contributes to B s -B s mixing, b → sγ, and B s → µ + µ − . There will also be loop-mediated contributions to t → cγ [10]. To maximise the effects of the VQs, we will take U = 1, so that the FCNC coupling is given by Eq. (4).
3. Model VQ-S-U1: A mirror image of VQ-S-D1, with one singlet charge + 2 3 quark t , and V t d = V t s = 0 while V t b = 0. The CKM matrix is 4 × 3. The texture affects the triple gauge vertices at one-loop, as well as the decay Z → bb. Again, the most significant constraint on V t b comes from the column unitarity, analogous to Eq. (5), under possible assumptions about D.

4.
Model VQ-S-U2: Again, similar to VQ-S-U1, but keeping two nonzero mixing elements V t b and V t s . The same observables as for VQ-S-D2 are affected, but there is a major difference. For VQ-S-D2, processes like B s -B s mixing were affected by the tree-level bsZ vertex, while t → cγ was affected by one-loop processes (with the b quark running in the loop). For VQ-S-U2, it is just the opposite: t → cγ is affected by the tree-level tcZ interaction, while B-physics observables are affected by one-loop processes. For the VQ-S-U type models, we will consider U 3×4 = V † and D = 1; this maximises the FCNC effects.
Next, let us consider a degenerate (so that there are no constraints from oblique parameters) vectorlike doublet Q = (t b ) T , for which both left-and right-chiral components are in the fundamental representation of weak SU (2). Eq. (2) now extends to all the left-chiral quarks, i.e., all the indices i, j, k run from 1 to 4. Obviously, there is no tree-level FCNC in the left-chiral sector. In the right-chiral sector, the FCNC vertices, for the up-and down-quark sectors respectively, are where U R and D R are the basis transformation matrices for the right-chiral up-and down-type quarks respectively. The flavour-conserving vertices are exactly like the SM. As the CKM matrix does not contain any information about U R or D R matrices, one may either treat them as free parameters, or just set all the off-diagonal terms to zero (i.e., U R = D R = 1).
One may note that the 4 × 4 CKM matrix is unitary, so it may be parametrized by 6 real angles and 3 complex phases. We do not need to know the individual D L or U L matrices, and the analysis is similar to that for a sequential fourth generation [6]. The unitarity constraints can now be invoked, e.g., We will take U R = D R = 1, so that for doublet VQ, we only need to consider the loop-driven processes. Two models will be discussed, namely: 1. Model VQ-D-U2: Apart from V t b , only two other elements involving a VQ are taken to be nonzero: V t s and V t b . The b quark decays through a real (if kinematically allowed) or a virtual t . This texture affects processes like B s -B s mixing, b → sγ, B s → µ + µ − , and Z → bb. Another variant is to take V t d = 0 and V t s = 0; all the above processes are, in that case, replaced by their s → d analogues.
2. Model VQ-D-D2: The nonzero elements are taken to be V cb and V tb , apart from V t b . Such a choice affects top pair production, and decays like t → cγ.
Being loop-driven in the SM, the amplitudes for all the processes that we consider can be written in a generic way: where i runs over all the individual amplitudes, including higher order QCD and electroweak corrections, V i is some combination of the CKM elements, and F IL i (m) is the relevant Inami-Lim function, which depends on the quark (mostly top) and the gauge boson masses. All the other constants are clubbed in A i . For our analysis, we will take only the leading order effects coming from the VQs and try to find out the allowed region of the CKM elements vis-à-vis the VQ couplings.

∆B = 2 processes: B q -B q mixing
The mass difference ∆M q (q = d, s) for the B q -B q system is given by where x t = m 2 t /M 2 W , and the Inami-Lim function is where we neglect the charm-and up-quark contributions. We will focus only on the q = s case. All other relevant quantities are shown in Table 2. This leads to x t = 4.60 ± 0.04 , S 0 (x t ) = 2.52 ± 0.21 , The phase coming from the mixing in the B s -B s system is given by where M s 12 is the absorptive part of the (12)-element of the Hamiltonian matrix, and the phase comes from V 2 ts , whose value is found to be [9] 2.3 ∆B = 1 processes The radiative decay b → sγ has a branching ratio of [11] Br(b → sγ) = (3.32 ± 0.15) × 10 −4 , E γ > 1.6 GeV (15) with a corresponding SM prediction of [12,13] Br To reduce the theoretical uncertainties, one generally uses the ratio of the decay rates: At leading logarithmic order, the ratio can be written as where is the phase space factor in semileptonic b-decays. The effective Wilson coefficient C where The values of C 2 (M W ) ∼ 1, a i and h i can be found in Ref. [14]. C The other parameters entering in Eq. (20) are given in Table 3. A similar analysis follows for b → dγ, with the suitable replacement of s → d.  Table 3: Values of different parameters required for evaluating ∆B = 1 processes taken from Ref. [9], and F Bs from Ref. [15].

Parameter
whereas the SM prediction is [16,17] Br The expression for the branching ratio is given by where τ B sH is the lifetime of the heavier B s eigenstate 2 . The Wilson coefficient C A , evaluated at the regularisation scale µ b , gets contribution from both W -mediated box and Z-mediated penguin amplitudes: and each of the C A s can be written as indicating the leading order (LO) terms and the higher-order QCD corrections; results up to n = 2 can be found in Ref. [17]. The LO contribution is given by the Inami-Lim function The higher-order contributions can be parametrized by with η Y = 0.905.

t → cγ
The partial decay width for the radiative decay t → cγ is given by where A γ and B γ are respectively the vector and axial form factors, whose expressions may be found in Ref. [10]. The three-point Passarino-Veltman C-functions involved in the form factors are numerically evaluated with LoopTools [18]. The SM prediction for Br(t → cγ) is approximately 4 × 10 −14 .

Z → bb
The ratio R b , defined as with R q ≡ Γ(Z → bb)/Γ(Z → qq), involves two non-universal corrections, namely, δ b tQCD , an O(α 2 s ) correction originating from doublets with large mass splitting, and δ b , that represents the additional contribution to the Zbb vertex due to non-zero top quark mass [19]: Using the expressions given in Ref. [19] for R c , R s , R u and δ b tQCD , one gets R b ≈ 0.217.

Modifications with vector-like quarks
With singlet or doublet VQs introduced, the processes discussed above get modified. Table 4 provides a quick display, which we elaborate in this Section. The model VQ-S-D1 cannot be constrained from any of these observables, except the row unitarity shown in Eq. (5). This, in conjunction with the fact that ∆M d and ∆M s are not affected by the VQ so that the SM estimates for |V td | and |V ts | are still valid, leads to a tiny value of |V tb |. and In Eq. (35), the first term inside the parenthesis gives the SM box contribution, the second term comes from the amplitude with a tree-level FCNC coupling on one side and the SM loop on the other, while the last term is a pure Z-mediated tree-level FCNC contribution. We neglect the contributions due to charm and up quarks, and take η Bs Z = 0.57 [7].

b → sγ
With a charge − 1 3 VQ b introducing FCNC couplings in the down-quark sector, the radiative decay b → sγ involves more penguin amplitudes, where the canonical W -propagator is replaced by either the Z boson, or the Higgs boson h [21]. The additional contributions to the Wilson coefficients C 7γ and C 8G in Eqs. (22) and (23) are given by where X sb X b b ≈ −X sb (assuming |D 44 | 2 ≈ 1), and with y i = (m i /M Z ) 2 and (w i = m i /M h ) 2 [21]. Three other Wilson coefficients, zero in the SM, receive non-zero contribution in this scenario: At the scale µ = 5 GeV, one gets In the SM, C 2 (M W ) = 1 because of the unitarity of the CKM matrix; here, it is better to use The contribution due to tree level Z exchange to Eq. (26) is given by [22] Br

t → cγ
The loop mediated contributions of b to the form factors in Eq. (31) can be easily incorporated as where i indicates the contribution coming from various diagrams [10].

VQ-S-U1
With an additional up-type singlet quark t and only V t b = 0, a similar constraint as for VQ-S-D1 comes from the column unitarity |V ib | 2 = 1. However, there is one decay channel where t can affect. This is the modification of the Zbb vertex; the loop corrections due to t and t modify the decay width Γ(Z → bb) and thus R b . As all the effects due to large top quark mass are contained in the vertex correction factor δ b , the mixing of top quark with vector quarks modifies the function F in Eq. (33), which is taken into account by making the substitution F → F + F 2 [7,23], where In addition to diagrams involving top quarks in the loop, there are triangle diagrams involving t s in the loop, or t and t . Therefore, δ b is modified as where the t-t contribution is given by the last term [7], with Since, t mixes only with third generation in this model, we have Z tt = V * tb V t b .

VQ-S-U2
With the VQ t in the loop, Eq. (10) is modified as whereS In our analysis, we will take all QCD correction factors to be equal: and neglect the contribution coming from the up and the charm quarks.

b → sγ
Similarly, for the radiative decay with the t quark loop, the additional contribution to the Wilson coefficients in Eqs. (22) and (23) can be written as where In presence of t , the branching ratio for B s → µ + µ − is given by where Again, we use the approximation η t ≈ η t = η Y for our estimates.

t → cγ
The introduction of t induces new FCNC vertices in the up-quark sector. Thus, there will be new loop amplitudes with FCNC Z-vertices. The flavour diagonal couplings also get modified according to Eq. (6). The contribution of the new diagrams to the form factors in Eq. (31) is given in Ref. [10].

Z → bb
The contribution of the new singlet to the vertex correction factor for Z → bb is described in Section 3.2. However, as t mixes with two generations in this model, from Eq. (45) we have

VQ-D-U2
In this model, the processes B s -B s mixing, b → sγ and B s → µ + µ − receive an additional contribution from t in the loop similar to the model VQ-S-U2, and the corresponding expressions in Section 3.3 can be used. However, since there is no tree level FCNC in the left-chiral as well as the right-chiral sector (our choice of U R = 1 ensures this), the process t → cγ does not receive any contribution from the new quarks. For Z → bb, the vertex correction factor of Eq. (33) can be written as Apart from these processes, one must also consider the constraints coming from the unitarity of the 4 × 4 CKM matrix, as in Eq. (8).

VQ-D-D2
Here, as all elements of the fourth row except V t b are taken to be zero, the only affected process is t → cγ, originating from the contribution of b at one loop. There are no effects due to t , as FCNC couplings are absent. The contribution to the form factors are, therefore, same as that shown in Section 3.1.

Results
Let us now discuss the bounds on the elements of the quark mixing matrix, in the presence of one or more VQs. To be on the conservative side, we take the theoretical uncertainties as well as experimental error margins at 2σ, i.e., 95% confidence level (assuming the uncertainties to be Gaussian in nature). The benchmark mass for all the VQs is taken to be 2 TeV, just outside the LHC detection limit.
The one-loop ∆B = 1 and 2 processes, namely, b → sγ, B s → µ + µ − , and B s -B s mixing, put bounds on V ts , X sb (Eq. (36)), and λ t sb ≡ V * t s V t b . In view of the importance of QCD corrections, and uncertainties associated with fixing µ eff for the radiative decay b → sγ, we just compare the branching ratios in the VQ model vis-à-vis the SM. This is an approximation on the universality of the higherorder effects for the SM quarks and the VQs, but at least holds for the QCD corrections. Thus, we have where C (0)eff 7γSM (µ) is given by Eq. (20). Similarly, to analyse the bound on V t b from the process Z → bb, we take the ratio as the dominant contribution from the VQs appears in the vertex correction factor of Eq. (33).
The new mixing matrix elements may contain nontrivial phases. For our analysis, we parametrize For the complex phase, we use the measurement of β s in the B s -B s system: with β s = (1.1 ± 1.6) × 10 −2 .
In a similar vein, the bounds on V cb , V tb , and Z ct can be obtained from t → cγ. However, this is not yet observed and only an upper limit, orders of magnitude above the SM prediction, exists [9]. Thus, no stringent bounds on these elements can be obtained from t → cγ; rather, constraints from unitarity seem more promising.
The SM estimates and experimental numbers for the observables have been listed in Table 5. Note that the SM numbers for b → sγ or B s → µ + µ − involve V ts which, in turn, is obtained from the B s -B s system. We discuss the bounds on the magnitude and phase of the new mixing matrix elements for different VQ models, and summarise our results in Table 6. For all the VQ types, we generate several models, specified by the new mixing matrix elements, and see if they pass all the experimental constraints. This produces a scatter plot for the allowed parameter space.

VQ-S-D1
The only constraint comes from Eq. (5), assuming U = 1:  which results in a fast decay b → tW .

VQ-S-D2
The ∆B = 1 and ∆B = 2 processes put bound on X sb as well as |V ts |, which are shown in Figure 1.
Combined constraints from all the three processes, namely, b → sγ, B s → µ + µ − , and B s -B s mixing, yield |X sb | < 1.2 × 10 −3 and 0.036 < |V ts | < 0.045. One may note the role the phase θ, associated with X sb , plays in determining the allowed parameter space for both |X sb | and |V ts |. On the other hand, constraints from the row unitarity of the 3 × 4 CKM matrix, Eq. (5), give |V cb | ≤ 0.15 and |V tb | ≤ 0.30. Using the above bounds, the maximum branching ratio for t → cγ comes out to be ∼ O(10 −8 ), still well below the current reach.

VQ-S-U1
The only process where this model contributes is Z → bb, and gives a bound on V t b . The minimum value of Z tt = |V td | 2 +|V ts | 2 +|V tb | 2 , which is a measure of the deviation from unitarity of the third row of the CKM matrix, comes out to be Z tt ≈ 0.91. From Eqs. (33), (46), and (58), we obtain |V t b | < 0.047. The column unitarity produces a worse bound: |V t b | < 0.30.

VQ-S-U2 and VQ-D-U2
Most of the affected processes and the bounds obtained are identical for these two models, so we lump them together. In Fig. 2, we show the bounds on λ t sb and |V ts | arising from B s -B s mixing, b → sγ and B s → µ + µ − . The hollow bell-shaped plot, the right-hand panel of Fig. 2, is interesting. Large values of λ t sb near δ = π indicate a destructive interference with the SM, with the VQ amplitude about twice in magnitude compared to the SM one.

VQ-D-D2
The treatment for the only affected process, namely, t → cγ, is identical to that of VQ-S-D2.
A brief summary of our results is shown in Table 6.

Summary
In this paper, we have considered several models with vector-like quarks, including SU (2) singlet and doublet representations, and of charges + 1 3 and/or − 2 3 , so that they can mix with their SM counterparts. To make our life simple, we assume that they mix only with the third generation quarks, or at the most, with the second and the third generation quarks. This introduces new complex elements in the  Table 6: Constraints on the quark mixing matrix elements and their combinations in presence of vector-like quarks. A dash indicates that these observables are not affected in the model, whereas entries marked with a dagger indicate that they have been obtained from unitarity constraints. Note that we have taken V t d , V ub = 0 for all the models.
expanded CKM matrix. At the same time, the 3×3 SM block of the full CKM matrix no longer remains unitary.
We use the low-energy observables, namely, B s -B s mixing, the decays b → sγ and B s → µ + µ − , and the partial decay width R b , to constrain these new elements. Not all models can be constrained from these observables, and further extensions (like mixing with the first generation quarks) will bring in other observables. Our constraints have been discussed in the previous Section, which are consistent with other similar studies in the literature. We have also found that the width of the anomalous top decay t → cγ may be significantly enhanced in the presence of such VQs, by a few orders of magnitude compared to the SM, but still remains well below the present LHC reach. The parameter space will naturally get more squeezed with new data from LHCb and Belle-II.