# \({B_s^0}\)–\({\bar{B}}_s^0\) mixing within minimal flavor-violating two-Higgs-doublet models

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## Abstract

In the “Higgs basis” for a generic 2HDM, only one scalar doublet gets a nonzero vacuum expectation value and, under the criterion of minimal flavor violation, the other one is fixed to be either color-singlet or color-octet, which are named as the type-III and type-C models, respectively. In this paper, the charged-Higgs effects of these two models on \(B_s^0\)–\({\bar{B}}_s^0\) mixing are studied. First of all, we perform a complete one-loop computation of the electro-weak corrections to the amplitudes of \(B_s^0\)–\({\bar{B}}_s^0\) mixing. Together with the up-to-date experimental measurements, a detailed phenomenological analysis is then performed in the cases of both real and complex Yukawa couplings of charged scalars to quarks. The spaces of model parameters allowed by the current experimental data on \(B_s^0\)–\({\bar{B}}_s^0\) mixing are obtained and the differences between type-III and type-C models are investigated, which is helpful to distinguish between these two models.

## Keywords

Large Hadron Collider Yukawa Coupling Standard Model Prediction Minimal Flavor Violation Standard Model Contribution## 1 Introduction

Thanks to the successful running of the Large Hadron Collider (LHC), particle physics has entered a new era, which is featured by the discovery of a new boson with a mass close to 125 GeV [1, 2]. Its measured properties are so far in good agreement with those of the Standard Model (SM) Higgs [3, 4, 5, 6, 7, 8, 9, 10, 11], suggesting that the electro-weak symmetry breaking (EWSB) is probably realized via the Higgs mechanism implemented through a single scalar doublet. It should be noted, however, that the EWSB is not necessarily induced by just a single scalar. Interestingly, many new physics (NP) scenarios are equipped with an extended scalar sector. The search for additional scalars is one of the important programs of the LHC experiments.

One of the extensions of SM scalar sector is the so-called two-Higgs-doublet model (2HDM) [11], in which a second scalar doublet is added to the SM field content. To avoid the experimental constraints on flavor-changing neutral-current (FCNC) transitions, which are forbidden at tree level in the SM due to the GIM mechanism [12], two different hypotheses, natural flavor conservation (NFC) [13] and minimal flavor violation (MFV) [14, 15, 16, 17], have been proposed.^{1} In the NFC hypothesis, depending on the \(Z_2\) charge assignments on the scalar doublets and fermions, there exist four types of 2HDM (type-I, II, X and Y) [23, 24]. In the MFV hypothesis, to control the flavor-violating interactions, all the scalar Yukawa couplings are assumed to be composed of the SM ones \(Y^U\) and \(Y^D\). In the “Higgs basis” [25], in which only one doublet gets a nonzero vacuum expectation value (VEV) and behaves as the SM one, the allowed \(SU(3)_C\otimes SU(2)_L\otimes U(1)_Y\) representation of the second scalar doublet that couples to quarks via Yukawa interactions is fixed to be either \((1,2)_{1/2}\) or \((8,2)_{1/2}\) [26], which implies that the second scalar doublet can be either color-singlet or color-octet. For convenience, they are referred to as type-III and type-C models [27], respectively. Examples of the former include the aligned 2HDM (A2HDM) [28] and the four types of 2HDM reviewed in Refs. [23, 24]. The scalar spectrum of the latter contains, besides a CP-even and color-singlet Higgs boson (the usual SM one), three color-octet particles, one CP-even, one CP-odd and one electrically charged [26].

Although the scalar-mediated flavor-violating interactions are protected by the MFV hypothesis, the type-III and type-C models still present very interesting phenomena in some low-energy processes, especially due to the presence of a charged Higgs boson [26, 27, 29]. In this paper, we shall study the \(B_s^0\)–\({\bar{B}}_s^0\) mixing within these two models and pursue possible differences between their effects. Since the charged Higgs contributes to the process at the same order as does the *W* boson in the SM, the NP effects might be significant.

^{2}which are defined, respectively, by

Our paper is organized as follows. In Sect. 2, after a brief review of the 2HDMs under the MFV hypothesis, we perform a complete one-loop computation of the electro-weak corrections to the amplitudes of \(B_s^0\)–\({\bar{B}}_s^0\) mixing within the two models. In Sect. 3, the numerical results and discussions are presented in detail. Finally, our conclusions are made in Sect. 4. Explicit expressions for the loop functions appearing in \(B_s^0\)–\({\bar{B}}_s^0\) mixing are collected in the appendix.

## 2 Theoretical framework

### 2.1 Brief review of the 2HDMs under the MFV hypothesis

*g*is the \(SU(2)_L\) coupling constant,

*i*,

*j*the fermionic generation indices, and \(m_{u,d}\) the quark masses;

*V*denotes the Cabibbo–Kobayashi–Maskawa (CKM) matrix [33, 34], and \(P_{R,L}=\frac{1\pm \gamma _5}{2}\) are the right- and left-handed chirality projectors. The couplings \(A_{u,d}^i\) are generally family-dependent and read

Following the notation used in Ref. [27], we shall denote the model with the second scalar doublet being color-singlet and the one with the second scalar doublet color-octet as the type-III and the type-C model, respectively, both of which satisfy the principle of MFV. Their explicit contributions to the \(B_s^0\)–\({\bar{B}}_s^0\) mixing will be presented in the next subsection.

### 2.2 \(B_s^0\)–\({\bar{B}}_s^0\) mixing within the SM and the 2HDMs with MFV

^{3}For \(C^{\mathrm{SRR}}\) and \(C^{\mathrm{TRR}}\), on the other hand, in order to get a gauge-independent result, the external momenta of the heavy quarks inside the mesons should be taken into account, and the heavy-quark masses should be kept up to the second order; our results for these two coefficients differ from the ones presented in Refs. [36, 46, 47, 48, 50].

Values of the relevant input parameters throughout this paper

\(|V_{us}|=0.2253\pm 0.0008\), \(|V_{ub}|=0.00413\pm 0.00049\), \(|V_{cb}|=0.0411\pm 0.0013\), \(\gamma =(68.0^{+8.0}_{-8.5})^{\circ }\) [55] |

\({\bar{m}}_{s}(2\,\mathrm{GeV})=95 \pm 5\) MeV, \({\bar{m}}_{b}({\bar{m}}_{b})=4.18 \pm 0.03\) GeV, \(m_t=173.21\pm 0.87\) GeV [55] |

\(\frac{{\bar{m}}_s(\mu )}{{\bar{m}}_{u,d}(\mu )}=27.5 \pm 1.0\) [55], \(m_{b}^\mathrm{pow}=4.8^{+0.0}_{-0.2}\) GeV [44, 45] |

\(f_{B_s}=228 \pm 5\pm 6\) MeV, \(f_{B_s}\sqrt{B_1}=211\pm 5\pm 6\) MeV, \(f_{B_s}\sqrt{B_2}=195\pm 5\pm 5\) MeV, |

Numerical results for \(\Delta M_s\ [\mathrm{ps^{-1}}]\), \(\Delta \Gamma _s\ [\mathrm{ps^{-1}}]\), \(\phi ^{c{\bar{c}}s}_s\) and \(a_{sl}^s\ [\%]\) within the SM. The theoretical uncertainties are obtained by varying each input parameter listed in Table 1 within its respective allowed range and then adding the individual uncertainty in quadrature

\(\Delta M_s\) | \(\phi ^{c{\bar{c}}s}_s\) | \(\Delta \Gamma _s\) | \(a_{sl}^s\ [\%]\) | |
---|---|---|---|---|

Exp. | \(17.757\pm 0.021\) | \(-0.015\pm 0.035\) | \(0.081\pm 0.006\) | \(-0.75\pm 0.41\) |

SM | \(17.228^{+1.731}_{-1.672}\) | \(-0.043^{+0.006}_{-0.006}\) | \(0.082^{+0.009}_{-0.013}\) | \(0.0026^{+0.0004}_{-0.0004}\) |

## 3 Numerical results and discussions

We now proceed to the presentation of our numerical results and discussions. Values of the relevant input parameters used throughout this paper are summarized in Table 1. Our SM predictions for the observables of \(B_s^0\)–\({\bar{B}}_s^0\) mixing are given in the third row of Table 2, in which the experimental data averaged by the HFAG [30] are also listed in the second row for comparison. As mentioned already in the introduction section, there is no significant deviation between the SM predictions and the experimental data for the observables at the current level of precision, even though a slight disagreement appears for \(a_{sl}^s\). Therefore, these observables are expected to put strong constraints on the parameter spaces of 2HDMs with MFV.

- (i)
In both the type-III and the type-C models, the charged-Higgs contributions to \(\mathcal{A}^{\mathrm{VLL}}\) (Eqs. (26) and (29)) depend only on the Yukawa coupling parameter \(A_u\) via \(|A_u|\), and hence are always constructive to the SM one. For a value \(|A_u|\sim \mathcal{O}(1)\), the type-III contribution could be comparable with the SM one, while the type-C model provides a relatively smaller correction.

- (ii)
Comparing Eqs. (27)–(28) with (26) (for the type-III model) and Eqs. (30)–(31) with (29) (for the type-C model), one can see that the NP contributions to \(\mathcal{A}^{\mathrm{SRR}}\) and \(\mathcal{A}^{\mathrm{TRR}}\) are much smaller than to \(\mathcal{A}^{\mathrm{VLL}}\), especially when \(|A_d|\sim |A_u|\). This is because the Wilson coefficients \(C^{\mathrm{SRR}}(\mu _W)\) and \(C^{\mathrm{TRR}}(\mu _W)\) are always suppressed by the factor \(x_b\) with respect to \(C^{\mathrm{VLL}}(\mu _W)\), both within the SM and in the 2HDMs with MFV.

- (iii)
In the case with large complex values of \(A_dA_u^{*}\), however, the charged-Higgs contributions to \(\mathcal{A}^{\mathrm{SRR}}\) and \(\mathcal{A}^{\mathrm{TRR}}\) could provide a large imaginary part to the off-diagonal mass matrix element \(M_{12}^s\), which may result in a significant correction to the CP-violating observables, such as \(\phi _s\) and \(\phi _s^{c{\bar{c}}s}\).

- (iv)
Different from the type-C model, the type-III contribution to \(\mathcal{A}^{\mathrm{TRR}}\) is induced only by the RG evolution effect, and is numerically much smaller. There are, however, cancelations between the charged-Higgs contributions to \(\mathcal{A}^{\mathrm{SRR}}\) and \(\mathcal{A}^{\mathrm{TRR}}\) in the type-C model.

- (i)
In the type-III model, as shown in Fig. 2a, the module of Yukawa coupling parameter \(A_u\) is severely constrained by the good agreement between the SM prediction and the experimental data for \(\Delta M_s\); for instance \(|A_u|<1\) is obtained with \(m_{H^{\pm }}=500~\hbox {GeV}\). There are, however, almost no constraints on the coupling \(A_d\), because the contribution involving it is negligible with respect to the one involving only \(A_u\).

- (ii)
In the type-C model, because the charged-Higgs contribution to \(\mathcal{A}^{\mathrm{VLL}}\) is relatively small and large cancelation effects exist between the terms involving \(A_d\) and \(A_u\), the allowed values of \(A_d\) and \(A_u\) could be large simultaneously, with either the same or the opposite signs, as shown by the four “legs” in Fig. 2b.

- (iii)
Besides the “legs” in Fig. 2b, the difference between the two models is also featured by the different shapes of the allowed parameter spaces. The current data on \(B_s^0\)–\({\bar{B}}_s^0\) mixing generally puts a stronger constraint on the type-III model; for instance, with the assumption \(|A_d|\sim |A_u|\) and choosing \(m_{H^{\pm }}=500~\hbox {GeV}\), the upper bound \(|A_u|\sim 1.5\) obtained in type-C model is obviously looser than the one \(|A_u|\sim 1\) in type-III model.

- (i)
In the type-III model, as shown in Fig. 3a and c, large values of \(|A_u|\) and \(|A_u^*A_d|\) are still allowed around \(\theta \sim \pm 90^{\circ }\), which makes it different from the case of real couplings. This is due to the fact that large cancelation effects appear among the charged-Higgs contributions when \(\theta \sim \pm 90^{\circ }\), which can also be seen from Eqs. (27) and (28). Moreover, as shown in Fig. 3a, an approximately linear relationship is observed between \(|A_u^*A_d|\) and \(|A_u|\) when \(|A_u|\gtrsim 0.5\).

- (ii)
As shown in Fig. 3b and d, similar observations could also be made in the type-C model, except for the fact that the constraints on the model parameters are now much looser. In addition, the cancelation effects among the charged-Higgs contributions occur around \(\theta \sim 0^{\circ }\) and \({\pm }180^{\circ }\), which is different from that observed in the type-III model.

As a final comment, it should be noted that the same analysis could also be applied to the \(B_{d}^0\)–\({\bar{B}}_d^0\) mixing, which is another important related low-energy process. The charged-Higgs effect on it can be obtained from that on the \(B_{s}^0\)–\({\bar{B}}_s^0\) mixing, with the replacement \(s\rightarrow d\) throughout the theoretical formulas presented in Sect. 2.2. However, we find that the bounds on the model parameters derive from the \(B_{d}^0\)–\({\bar{B}}_d^0\) mixing are quite similar to the ones from the \(B_{s}^0\)–\({\bar{B}}_s^0\) mixing, and no further information on the model parameters at all could be obtained from the former. Therefore, the constraints from \(B_{d}^0\)–\({\bar{B}}_d^0\) mixing will not be shown here.

## 4 Conclusion

In this paper, we have calculated the one-loop electro-weak corrections to the \(B_s^0\)–\({\bar{B}}_s^0\) mixing within the type-III and type-C 2HDMs with MFV, in which the second scalar doublet is fixed to be color-singlet and color-octet, respectively. It is noted that, in order to get a gauge-independent result, the external momenta of the heavy quarks inside the mesons should be taken into account, and the heavy-quark masses should be kept up to the second order.

- (i)
While the type-C model gives a nonzero contribution to the Wilson coefficient \(C^{\mathrm{TRR}}_{C}(\mu _W)\), the type-III contribution to the amplitude \(\mathcal{A}^{\mathrm{TRR}}\) is induced only by RG evolution effect.

- (ii)
In the case of real couplings, the allowed spaces of the Yukawa coupling parameters \(A_u\) and \(A_d\) in the two models are obviously different, as shown in Fig. 2.

- (iii)
In the case of complex couplings, due to the cancelation effects among the charged-Higgs contributions, large values of \(|A_u|\) and \(|A_d|\) are still allowed around \(\theta \sim \pm 90^{\circ }\) in the type-III and around \(\theta \sim 0^{\circ },\pm 180^{\circ }\) in the type-C model, which is shown in Fig. 3.

## Footnotes

- 1.
- 2.
The phase \(\phi _s^{c{\bar{c}}s}\) appears in tree-dominated \(b\rightarrow c{\bar{c}}s\) \(B_s\) decays, such as \(B_s\rightarrow J/\psi \phi \), and is generally different from \(\phi _s\) unless the terms proportional to \(V_{cb}V_{cs}^{*}V_{ub}V_{us}^{*}\) and \((V_{ub}V_{us}^{*})^2\) in \(\Gamma ^s_{12}\) are neglected [32].

- 3.
There are two typos in Eq. (26) of Ref. [50]: a global factor 2 should be added to the term proportional to \(|\eta _U|^2\) and 1 / 2 to the term proportional to \(|\eta _U|^4\).

## Notes

### Acknowledgments

We thank Antonio Pich and Martin Jung for useful discussions and cross-checks for the box Feynman diagrams. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11475055, 11435003, 11105043 and 11005032). Q. Chang is also supported by the Foundation for the Author of National Excellent Doctoral Dissertation of People’s Republic of China (Grant No. 201317) and the Program for Science and Technology Innovation Talents in Universities of Henan Province (No. 14HASTIT036). X. Q. Li is also supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, by the Open Project Program of SKLTP, ITP, CAS, China (No. Y4KF081CJ1), and by the self-determined research funds of CCNU from the colleges’ basic research and operation of MOE (CCNU15A02037).

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