In order to obtain the higher-dimensional operators that represent the effect of the VLQ in the energy scale between and the EW scale, we integrate out the VLQ fields in Eq. (1). At first, we perform tree-level matching at VLQ mass scale . In Fig. 1, we show the Feynman diagram (left figure) for the scattering of a quark / anti-quark pair into a Higgs pair , in which the VLQ is exchanged. We assume that the external particles have momenta much smaller than the mass of the VLQ. Then the amplitude of the left figure can be reproduced up to accuracy by computing the Feynman diagram of the right figure with the following low-energy effective Lagrangian [18,19,22–24]: (5) where .

The effective Lagrangian is written in terms of a dimension-six operator, and its coefficient is determined so that it reproduces the amplitude of the left figure in Fig. 1 within the precision of . By using the equations of motion derived from the SM Lagrangian, we can rewrite the effective Lagrangian of Eq. (5): (6)

Next we consider one-loop level matching between the full theory and the effective theory to obtain effective interactions that contribute to the radiative transition of the quarks. The procedure is as follows:

(i) We calculate the amplitudes of the Feynman diagrams for the decay of into and one of the gauge fields , , or at one-loop level (see the top figures in Fig. 2). These diagrams include the VLQ in the internal line. In this calculation, we renormalize the amplitudes with the scheme.

We can obtain the following effective Lagrangian at one-loop level: (7) where (8) (9) (10) (11)

To derive the above expressions, we use the equation of motion in the leading order of the expansion with respect to , namely that of the SM. In Eqs. (9)–(11), , , and are the field strength of , SU(2), and , respectively. The matching scale is typically taken to be the VLQ mass scale . The lepton doublet is denoted by . The effective Lagrangians , , and contain the effective operators that contribute to the decay of , , and , respectively. Since we use the equations of motion, the effective Lagrangians also contain the operators such as the four-Fermi operators that do not contribute to these processes.

Finally, the whole Lagrangian obtained by integrating out the VLQ fields is given as: (12) (13) where is given in Eq. (5) or Eq. (6), and is given in Eqs. (8)–(11). In Eq. (12), the kinetic term of the SM quark doublet is (14) (15) where the term comes from the first term in Eq. (8). To rewrite the kinetic term into a canonical form, we perform the following rescaling of : (16)

Then the kinetic term of the quark doublet becomes (17)

In terms of the rescaled fields introduced in Eq. (16), the Yukawa interactions in Eq. (13) are changed: (18) (19) where we redefine the SM Yukawa coupling as (20)

The rescaling of the field in Eq. (16) is absorbed into the Yukawa couplings. After the diagonalization of the mass matrices based on these couplings, it contributes to the CKM matrix as one-loop corrections. Since we only consider the charged current interaction in one-loop diagrams in the next section, these corrections lead to two-loop order effects and they are neglected.

The tree-level effective operator in Eq. (5) is also changed by the rescaling in Eq. (16): (21) where we redefine the Yukawa coupling between the SM quarks and the VLQ as (22)

As we will see in the next subsection, this redefinition of the Yukawa coupling in Eq. (22) adds corrections to the CKM matrix and the FCNC coupling. In the next section we will take into account only leading-order contributions in , and these corrections are neglected.

For the one-loop effective Lagrangian, the rescaling in Eq. (16) leads to two-loop order corrections and we can simply take in the one-loop effective Lagrangian.

In this subsection, we derive the Lagrangian for the broken phase of the SM gauge symmetry. We substitute the following forms for the Higgs doublet in the Lagrangian Eqs. (6)–(11): (23) (24)

Here we do not take into account the running effect from the VLQ mass scale to the EW scale for the coefficients in the effective interactions in Eqs. (6)–(11). For the effective Lagrangian in Eq. (6), we obtain (25) where the ellipsis represents the terms including more than four fields, represents , and denotes the cosine (sine) of the weak mixing angle . The mass matrices of the up-type and down-type quarks which correspond to in Eq. (13) are denoted by . Adding the tree-level effective Lagrangian in Eq. (25) to the SM Lagrangian in Eq. (13), the mass matrix of the down-type quarks changes into . We diagonalize this mass matrix. At first, we introduce unitary matrices and . These unitary matrices diagonalize the matrix : (26) where the prime indicates the mass basis of the SM. In this mass basis, the mass matrix of the down-type quarks changes into (27) where (28)

The mass matrix in Eq. (27) is not diagonal. In order to diagonalize the mass matrix including the contributions of , we introduce unitary matrices and , (29) where the double prime denotes the mass basis of the model with VLQ. The physical masses of the down-type quarks are denoted by . The mixing angles of these unitary matrices are of the order of . Hereafter we omit the double prime on the quark fields of the mass basis and denotes the in the right-hand side of Eq. (28). Finally, we obtain the following Lagrangian after the transformation in Eqs. (26) and (29): (30) where the ellipsis represents the terms that contain more than four fields. Each part of the Lagrangians is given below: (31) (32) (33) (34) (35) (36) (37)

In Eqs. (31)–(37), and denote the chiral projection operators, . The electromagnetic charges of up-type and down-type quarks are denoted by and , respectively. The CKM matrix is defined as (38)

The FCNCs arise from the non-diagonal matrix in the , , and interactions in Eqs. (34), (36), and (37). The matrix in the neutral currents is defined as follows: (39)

Using Eqs. (38) and (39), we obtain the relation between the CKM matrix and the matrix up to : (40)

Equation (40) shows that the unitarity of the CKM matrix for the three generations does not hold due to the deviation from the unit matrix of the matrix in Eq. (39). Taking the limit of , the unitarity relation is restored.

Next we rewrite the one-loop level effective Lagrangian Eqs. (8)–(11) in terms of the mass basis defined in Eq. (29). Below we write the part of the dipole operators and omit the other parts of the effective Lagrangian: (41) (42) where the field strengths , , and are defined as (43) respectively. Note that the coefficient of the photon dipole operator with the down-type quarks is consistent with the case of the full theory calculation up to [25].

First we consider the process to calculate the branching ratio of the process in the next section. The diagrams that contribute to the effective Lagrangian up to are shown in Fig. 3.

The process occurs at tree level in the model with VLQ since there is the tree-level FCNC among the down-type quarks. In one-loop level, the contribution comes from the Feynman diagrams in Figs. 3(a) and (b) that are also present in the SM. However, these amplitudes include the additional contributions due to the violation of CKM unitarity. Since these contributions are suppressed by the loop factor compared with the contribution of the tree diagram in Fig. 3(c), we neglect the contribution from the violation of CKM unitarity in the computation of the process. Then the effective Lagrangian for is given as follows: (44) where , and denotes the fine structure constant of the electromagnetic interaction. The Inami–Lim function is [26,27] (45) where . The first term in Eq. (44) comes from the diagrams in Figs. 3(a) and (b) with the CKM unitarity relation for the SM , that is, the SM contribution. The second term of Eq. (44) comes from the diagram in Fig. 3, so this term is the new contribution in the model with VLQ.

Next we show the effective Lagrangian for the process in order to compute the mass difference of the meson in a later section. In Refs. [28–31], the process was computed up to and . The diagrams that contribute to the process are given in Fig. 4.

From the diagrams in Fig. 4, we obtain the effective Lagrangian for the process as [28–31]: (46) where (47) is the Inami–Lim function [26]. Also, is given in Eq. (45). The first term in Eq. (46) comes from the diagram in Fig. 4(a) with the CKM unitarity relation. The second term in Eq. (46) is obtained from the violation of CKM unitarity in the diagram in Fig. 4(a) in addition to the contribution from the diagram in Fig. 4(b). The CKM unitarity relation is used for the one-loop FCNC vertex in Fig. 4(b) since the contribution is neglected. The third term comes from the diagram in Fig. 4(c). Note that the effective Lagrangian in Eq. (46) contains only the Inami–Lim functions which are gauge-parameter independent [31].

In this section, we will make numerical calculations for the mass difference of the meson , the branching ratio of and the branching ratio of the inclusive radiative decay of the meson in the model with VLQ. In addition to these processes, we consider the constraint from Eq. (40). We will use the new physics parameters defined as (80) in the following computations.

The meson inclusive radiative decay is governed by the effective Hamiltonian at the -quark mass scale [34,36], (85) where and denote the four-Fermi operators and their Wilson coefficients, respectively. The effective operators and are given by (86) (87)

In the calculation of the branching ratio for , it is convenient to introduce the so-called “effective coefficients” [37,38]. The effective coefficient for the effective operator at the scale is given as [34,38,39] (88) where with and (89) (90)

In Eq. (88), the indices “(0)” mean the leading-order contributions. Since we do not take into account the running effect from the VLQ mass scale to the EW scale, we obtain the Wilson coefficients and at the EW scale (taken as ) as: (91) (92) where the Wilson coefficients, (93) (94) come from the SM contributions in Eq. (52). The Wilson coefficients (95) (96) are obtained from the VLQ contributions in Eqs. (41) and (42). The Wilson coefficients and are given as follows: (97) (98) where (99)

The Wilson coefficients with the suffix “uv” come from the effective Lagrangian in Eq. (53), whose origin is the violation of CKM unitarity. The Wilson coefficients with the suffix “NC” are obtained from the effective Lagrangian in Eq. (54) by taking the limits and . Here we neglect terms. The Wilson coefficients and can be obtained from the effective Lagrangian corresponding to the diagrams.

In our numerical calculation, we use the NLO expression for the branching ratio given as [40] (100) where and are non-perturbative corrections for the semi-leptonic and radiative meson decay rates, respectively. The quantity at NLO is summarized in Ref. [40] as (101)(1-\delta)E_\gamma^{\rm max}}}{\Gamma[b\rightarrow X_ce\overline{\nu_e}]} = \frac{\left|\lambda_{sb}^t\right|^2}{\left|V_\mathrm{CKM}^{cb}\right|^2} \frac{6\alpha_{em}}{\pi g(z)}F(z) \left\{|D|^2 + A(\delta)\right\}\! . \label{Eq:RquarkNLO} \end{align} ]]>

The function with corresponds to the phase space factor for the semi-leptonic decay. The function contains the NLO correction for the semi-leptonic decay and the difference between the pole mass and the mass of the -quark. The is the lower cut on the photon energy in the bremsstrahlung correction: (102)(1-\delta)E_\gamma^{\rm max} \equiv (1-\delta)\frac{m_b}{2} . \end{align} ]]>

The term originates from the bremsstrahlung corrections and the virtual corrections [40–43], (103) where the functions can be found in Ref. [40]. The term in Eq. (101) is constituted by the NLO Wilson coefficient for and the virtual corrections for [40–42]. Here, is defined as (104) where , , and can be found in Ref. [40].

In our numerical calculation, we take , 1.6$]]> GeV, and neglect the correction to the new physics contributions. Therefore the Wilson coefficients and are only included in the first term in Eq. (104).

The violation of CKM unitarity is shown in Eq. (40). For , we obtain the following relation: (105)

This relation can be rewritten as follows: (106) where we define (107)

The relation in Eq. (105) leads to a quadrilateral in the complex plane, as shown in Fig. 7.

In the following numerical analyses, we obtain constraints on FCNC couplings by using the current experimental data of rare decays, and . We also take account of the quadrilateral constraint Eq. (106) and Fig. 7. The values of the input parameters used in the numerical analyses are shown in Table 1.

At first we analyze the branching ratio of the process by using the expression in Eq. (83). Note that the branching ratio depends on the new physics parameters and . We equate in Eq. (83) with the experimental value, (108)

As the experimental value, we adopt the branching ratio measured by LHCb [45]: (109)

In Fig. 8, we show the dependence of on the absolute value of the FCNC coupling . The different colors of the dots in the scattered plots represent the different ranges for the value of the new physics parameter . The colored region satisfies the quadrangle constraint Eq. (106) with . Note that the expression of the branching ratio and quadrilateral constraint depend on through its cosine, and the plotted regions do not depend on the sign of . The experimentally allowed range of the branching ratio in Eq. (109) is shown as the gray shaded region. The horizontal solid line corresponds to the central value of the experimental branching ratio in Eq. (109). In Fig. 8, as approaches zero, comes close to the SM prediction [35], (110)

As increases from zero to , decreases for , while it increases for . As becomes larger, increases regardless of the range of since the third term in Eq. (84) is dominant. The dependence on for smaller can be also understood from Eq. (84), since the coefficient of the term linear in is proportional to .

Next we analyze the branching ratio of the process by using the expression in Eq. (100). Here we denote the branching ratio in the model with VLQ as . The new physics parameters and are included in the Wilson coefficients in Eqs. (91) and (92). In order to obtain constraints on and , we take account of the current average [46], (111) of experimental data [47–53]. In Fig. 9, we show the dependence of on . The different colors of the dots in the scattered plots represent the different ranges for the value of the new physics parameter . The colored region satisfies the quadrilateral constraint Eq. (106) with . The experimentally allowed range of the branching ratio in Eq. (111) is shown as the gray shaded region. The horizontal solid line corresponds to the central value of the experimental branching ratio in Eq. (111). In Fig. 9, as approaches zero, the value of comes close to that of the SM prediction at NNLO accuracy [54], (112)

We note that the number of light-blue dots is much less than that of the red ones, since the quadrilateral constraint for is tighter than that for . For the smaller , the filled regions with colored dots are almost the same as each other. Thus, depends on weakly compared with .

In the left figure of Fig. 10, we show the region allowed by the experimental data for the parameters and . The blue dots satisfy both the constraint from and the quadrilateral constraint Eq. (106) with . The green dots satisfy both the constraint from and the quadrilateral constraint. The values of and in the region where the blue and green region overlap satisfy all three constraints. The blue region has the shape of a ring. The region inside the ring is excluded because and in this region lead to predictions of smaller than the experimental value. One finds that the stringent constraint on the parameters and comes from .

Using the definition of in Eq. (39), we obtain a constraint on the VLQ mass and the product of the Yukawa coupling . This result is shown in the right figure of Fig. 10, where we use GeV [44]. Since the constraint on and from is stronger than that from (see Fig. 10 left), we show the region with blue dots where satisfy the constraint from and the quadrilateral constraint Eq. (106). One finds that the lower limit on the VLQ mass is around TeV for .

Finally, we show the violation of CKM unitarity on the complex plane in Fig. 11. The definition of each side is the same as that of Fig. 7. In order to obtain Fig. 11, we choose and and use the central values of the CKM matrix elements in Table 1. The side for is connected with the real axis at . The left figure is the case of , while the right figure is that of . One can see that the side for can be as large as that for , and the sign of affects the value of the angle in Fig. 7.