#### Abstract

The two-Higgs-doublet model (2HDM), as one of the simplest extensions of the Standard Model (SM), is obtained by adding another scalar doublet to the SM and is featured by a pair of charged Higgs, which could affect many low-energy processes. 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 type III and type C 2HDM, respectively. In this paper, we study the charged-Higgs effects of these two models on the mixing, an ideal process to probe New Physics (NP) beyond the SM. Firstly, we perform a complete one-loop computation of the box diagrams relevant to the mixing, keeping the mass and momentum of the external strange quark up to the second order. Together with the up-to-date theoretical inputs, we then give a detailed phenomenological analysis, in the cases of both real and complex Yukawa couplings of the charged Higgs to quarks. The parameter spaces allowed by the current experimental data on the mass difference and the CP-violating parameter are obtained and the differences between these two 2HDMs are investigated, which are helpful to distinguish them from each other from a phenomenological point of view.

#### 1. Introduction

The SM of particle physics has been proved to be successful because of its elegance and predictive capability. Almost all predictions in the SM are in good agreement with the experimental measurements, especially for the discovery of a Higgs boson with its mass around 125 GeV [1, 2]. The discovery of a SM-like Higgs boson suggests that the electroweak symmetry breaking (EWSB) is probably realized by the Higgs mechanism implemented via a single scalar doublet. However, the EWSB is not necessarily induced by just one scalar. It is interesting to note that many NP models are equipped with an extended scalar sector; for example, the minimal supersymmetric standard model requires at least two Higgs doublets [3]. Moreover, the SM does not provide enough sources of CP violation to generate the sufficient size of baryon asymmetry of the universe (BAU) [4–6].

One of the simplest extensions of the SM scalar sector is the so-called 2HDM [7], in which a second scalar doublet is added to the SM field content. The added scalar doublet can provide additional sources of CP violation besides that from the Cabibbo-Kobayashi-Maskawa (CKM) [8, 9] matrix, making it possible to explain the BAU [4].

It is known that, within the SM, the flavor-changing neutral current (FCNC) interactions are forbidden at tree level and are also highly suppressed at higher orders, due to the Glashow-Iliopoulos-Maiani mechanism [10]. To avoid the experimental constraints on the FCNCs, the natural flavor conservation (NFC) [11] and minimal flavor violation (MFV) [12–15] hypotheses have been proposed (the NFC and MFV hypotheses are not the only alternatives to avoid constraints from FCNCs; models with controlled FCNCs have also been addressed in the literature [16–20].). In the NFC hypothesis, the absence of dangerous FCNCs is guaranteed by limiting the number of scalar doublets coupling to a given type of right-handed fermion to be at most one. This can be explicitly achieved by applying a discrete symmetry to the two scalar doublets differently, leading to four types of 2HDM (usually named as types I, II, X, and Y) [21, 22], which have been studied extensively for many years. In the MFV hypothesis, to control the flavor-violating interactions, all the scalar Yukawa couplings are assumed to be composed of the SM ones and . In the “Higgs basis” [23], in which only one doublet gets a nonzero vacuum expectation value (VEV) and behaves the same as the SM one, the allowed representation of the second scalar doublet is fixed to be either or [24], which implies that the second scalar doublet can be either color-singlet or color-octet. For convenience, they are referred as type III and type C model [25], respectively. Examples of the color-singlet case include the aligned 2HDM (A2HDM) [26, 27] and the four types of 2HDM reviewed in [21, 22]. In the color-octet case, the scalar spectrum contains one CP-even, color-singlet Higgs boson (the usual SM one), and three color-octet particles, one CP-even, one CP-odd, and one electrically charged [24].

Although the scalar-mediated flavor-violating interactions are protected by the MFV hypothesis, type III and type C models can still bring in many interesting phenomena in some low-energy processes, especially due to the presence of a charged-Higgs boson [24, 25, 28–32]. The neutral-meson mixings are of particular interest in this respect, because the charged-Higgs contributions to these processes arise at the same order as does the boson in the SM, indicating that the NP effects might be significant. For example, the charged-Higgs effects of these two models on the mixing have been studied in [28]. In this paper, we shall explore the mixing within these two models and pursue possible differences between their effects. The general formula for mixing, including the charged-Higgs contributions, could be found, for example, in [33].

Our paper is organized as follows. In Section 2, we review briefly the 2HDMs under the MFV hypothesis and give the theoretical framework for the mixing. In Section 3, we perform a complete one-loop computation of the Wilson coefficients for the process within these two models. In Section 4, numerical results and discussions are presented in detail. Finally, our conclusions are made in Section 5. Explicit expressions for the loop functions appearing in the mixing are collected in the appendix.

#### 2. Theoretical Framework

##### 2.1. Yukawa Sector

Specifying to the “Higgs basis” [23], in which only one doublet gets a nonzero VEV, we can write the most general Lagrangian of Yukawa couplings between the two Higgs doublets, and , and quarks as [24, 25]where with the Pauli matrix, and , , and are the quark fields given in the interaction basis. is the color generator which determines the color nature of the second Higgs doublet (depending on which type of 2HDM we are considering, the second Higgs doublet can be either color-singlet or color-octet). and are the Yukawa couplings and are generally complex matrices in the quark flavor space.

According to the MFV hypothesis, the transformation properties of the Yukawa coupling matrices and under the quark flavor symmetry group are required to be the same. This can be achieved by requiring to be composed of pairs of [25]:

Transforming the Lagrangian in (1) from the interaction basis to the mass basis, one can obtain the Yukawa interactions of charged Higgs with quarks in the mass-eigenstate basis, which are given by [25, 28]where are family-dependent Yukawa coupling constants [25, 28, 34]:with GeV. For simplicity, we consider only the family universal coupling case in which the family-dependent Yukawa couplings, , can be simplified to .

##### 2.2. Mixing

Both within the SM and in the 2HDMs with MFV, the neutral kaon mixing occurs via the box diagrams depicted in Figure 1 (these Feynman diagrams are drawn with the LaTeX package* TikZ-Feynman* [35]). As demonstrated in [36], the correction from the external momenta and quark masses is not negligible for the mixing. Thus, unlike the traditional calculation performed in the limit of vanishing external momenta and external quark masses, we shall keep the external strange-quark momentum and mass to the second order; this is essential to guarantee the final result gauge-independent [34].

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Calculating the one-loop box diagrams and following the standard procedure of matching [36], we obtain the effective Hamiltonian responsible for the mixing:where is the Fermi coupling constant, is the -boson mass, and is the scale-dependent Wilson coefficients of the four-quark operators , which are defined, respectively, as follows (there are totally eight four-quark operators for the most general case [37], but we have written out only the operators that exist in our calculation):with and being the color indices and . Note that we include the QCD corrections only to the SM Wilson coefficient , but not to the NP ones. The hadronic matrix elements of these operators can be written as [37]where is the kaon mass and the kaon decay constant. is the scale-dependent bag parameters, and is defined as [37]

It should be noted that the SM and NP contributions to the Wilson coefficients cannot be summed directly because they are given at different initial scales, for the SM and for the 2HDM in particular. In order to sum these two contributions, they must be firstly run down to the lattice scale at which the bag parameters are evaluated. The explicit expressions for these Wilson coefficients will be presented in Section 3.

For the mixing, there exist two observables which can be calculated from the effective Hamiltonian given by (5) [37]:

The above equations are the most general formulae for these two observables. It should be noted that and receive both short-distance (SD) and long-distance (LD) contributions. With the LD contribution included, the mass difference can be decomposed as [38]where the SD part is derived from (9) with the effective Hamiltonian obtained from the box diagrams, while the two LD parts are estimated, respectively, as [38, 39]

We can see from (12) that the LD contribution to is about 10% of the experimental value. However, keeping in mind that this estimate is just a bold-guess based on an analysis at the leading chiral logarithm in the framework of chiral perturbation theory [38], we should note that the actual uncertainty on is quite huge (we thank Professor Antonio Pich for pointing out this to us). As the structure of LD contribution is still not well understood, we include this part only in the SM case but not in the NP one.

The formula for the CP-violating parameter , with the LD contribution taken into account, is given by [40]where [38], [41], and . The LD contribution to has been included in the two phenomenological factors and . In the case with only the SD contribution, and , and (13) goes back to (10).

#### 3. Analytic Calculation

##### 3.1. Wilson Coefficients within the SM

For the SM case, we calculate the Wilson coefficients from the box diagram shown in Figure 1(a). Without any QCD correction, they are given, respectively, aswhere , and is the Inami-Lim function given by (A.1) [42]. Explicit expressions for the functions can be found in the appendix. Note that when the external strange-quark momentum and mass are kept to the second order, we also get nonzero contributions to the Wilson coefficients and even in the SM case.

The QCD corrections to the Wilson coefficients can be described by the factors , , and , which have been calculated up to the next-to-next-to-leading order [43–45] and are collected in [46]. Combining the renormalization group (RG) evolution with these QCD corrections, we getwhere , and is the scale-independent mass ratio, whereas is the mass ratio at the scale . is the RG independent bag parameter, and the factors encode the RG evolution effects that are given, respectively, as [37]where the effective bag parameters are defined as [37]with defined in (8). The factors and are given by the formulae collected in [37] with

##### 3.2. Wilson Coefficients in the 2HDMs with MFV

The Wilson coefficients at the matching scale in the NP case are calculated from the box diagrams shown in Figures 1(b)–1(d), with the results given, respectively, as

Explicit expressions for the functions introduced in the above equations are collected in the appendix. Note that the contribution to is zero for type III but is not for type C 2HDM. With the RG evolution effect included, the final result is similar to the SM case and can be written asfor type III, andfor type C 2HDM. The factors are also similar to the SM case but with a different factor , which is now defined byHere the matching scale for the 2HDMs has been changed to , because the evolution effect from down to is quite small and can be safely neglected.

After performing the proper RG evolution, we can then sum directly both the SM and NP contributions to the matrix element , which can be written aswhere the superscript “” labels the different four-quark operators.

#### 4. Numerical Results and Discussions

##### 4.1. Input Parameters and the SM Results

Firstly, we collect in Table 1 the values of the relevant input parameters used throughout this paper, together with the experimental data on and . For the bag parameters, we use the lattice results with flavors of dynamical quarks and evaluated at the renormalization scale 3 GeV [41, 48]. In addition, we have used the* RunDec* package [49] to obtain the running coupling constant and quark masses at different scales in the two-loop approximation.

With the input parameters collected in Table 1, we can now give the numerical results for and in the SM case, which are listed in Table 2. We make the following comments on the SM results:(i)Our result for the mass difference without the corrections from the external strange-quark mass, , and from the LD contribution, agrees well with that obtained in [45].(ii)The corrections from to and are 6.83% and −0.06%, respectively. Note that the correction to is at the same order as that obtained in [36]. Moreover, the LD contributions to and are 11.20% and −6%, respectively.(iii)As the correction can be precisely calculated, we consider it both to and to ; especially, this correction is not too small for . In addition, we include the LD contributions to but not to , because the structure of LD contribution to is still not well understood [38].

##### 4.2. Results in the 2HDMs with MFV

As can be seen clearly from Table 2, there is no significant deviation between the SM predictions and the experimental data for and , especially for the latter. Therefore, these two observables are expected to put strong constraints on the parameter spaces of type III and type C 2HDMs, which are both featured by the three parameters, the two Yukawa couplings , and the charged-Higgs mass , in this paper. In the case of complex couplings, we can further choose and as the independent variables, with being the relative phase between and .

The relevant model parameters are also constrained by the other processes. For the parameter , an upper bound can be obtained from the decay [25], while the parameter is much less constrained phenomenologically [25, 28]. However, the perturbativity of the theory requires that these couplings cannot be too large. As for the charged-Higgs mass, the lower bound GeV (95% CL) has been set by the LEP experiment [50], which is obtained under the assumption that decays mainly into fermions without any specific Yukawa structure. In addition, direct searches for are also performed by the Tevatron [51], ATLAS [52], and CMS [53] experiments, among which most constraints depend strongly on the underling Yukawa structures. Recently, by comparing the cross-sections for the dijet, top-pair, dijet-pair, , and production at the LHC with the strongest available experimental limits from ATLAS or CMS at or 13 TeV, Hayreter, and Valencia [54] has extracted constraints on the parameter space of the Manohar-Wise model [24], which is equivalent to type C 2HDM discussed here. Interestingly, they found that masses below 1 TeV have not been excluded for color-octet scalars as is often claimed in the literature. For a variety of well-motivated 2HDMs, the authors in [55] found that charged-Higgs bosons as light as 75 GeV can still be compatible with all the results from direct charged and neutral Higgs boson searches at LEP and the LHC, as well as the most recent constraints from flavor physics, although this implies severely suppressed charged-Higgs couplings to all fermions. Thus, based on the above observations, we generate randomly numerical points for the model parameters as [34]

Taking GeV as a benchmark, we firstly explore the dependence of each Wilson coefficient evaluated at the matching scale or approximately at on the other model parameters,From the above numerical results, we can make the following observations: (i)The dominant contribution to the effective Hamiltonian given by (5) comes from the operator in both type III and type C 2HDM, due to the suppression in and . Furthermore, the coefficient of the term in is quite small, being of order compared to that of the terms.(ii)Due to the color factor, the Wilson coefficient in type C is a little bit smaller than that in type III 2HDM, and the sign of in type C is also flipped relative to that in type III 2HDM.(iii)There exists an extra operator in type C 2HDM, and its Wilson coefficient differs from that of in sign.

From the current experimental data on and , one can constrain the model parameters and even distinguish the two scenarios of 2HDM with MFV. To get the plots for the allowed parameter spaces, we do as follows:(1)We scan the Yukawa coupling parameters and (also the relative phase for complex couplings) randomly within the ranges given by (24), with fixed at 100, 250, and 500 GeV, respectively.(2)With each set of values for the model parameters, we give the theoretical prediction for and , together with the corresponding uncertainty resulted from the input parameters listed in Table 1. The method of calculating the theoretical uncertainty is the same as in [34].(3)We select the points which lead to the theoretical predictions overlapping with the range of the experimental data.The final allowed spaces for the model parameters are shown in Figure 2 for the real coupling and in Figure 3 for the complex coupling case, respectively.

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From Figure 2, we can make the following observations for the real coupling:(i)In type III model, as shown in Figure 2(a), the parameter is severely constrained due to the good agreement between the SM predictions and the experimental data, especially for ; for example, the limit is more stringent compared to that obtained in [34] with GeV. However, there is almost no constraints on because of the smallness of the coefficient involving , as mentioned earlier.(ii)In type C model, we also get strong constraint for , but being looser than that in type III case, with the maximum value . The wider allowed range in type C model comes from the additional color factor. Similar to that observed in type III model, there is also almost no constraint on .(iii)The patterns of the allowed parameter spaces of these two models are different, looking like “convex lens” for type III and like “concave lens” for type C model. This means that the allowed range for is smaller with larger for type III model, while the allowed range for in type C model can be larger with greater . The reason is that the dominant contribution to the two observables and comes from the operator , the Wilson coefficient of which in type C is smaller than that in type III model. Moreover, the cancellation between the Wilson coefficients and in type C model also reduces their contribution to the observables.For the complex coupling, on the other hand, the results shown in Figure 3 imply that (i)in type III model, there exists a strong correlation between and , especially when GeV, as shown in Figure 3(a). It is also found from Figure 3(c) that the large values of are allowed at , due to the cancellation between the complex terms,(ii)in type C model, as shown in Figures 3(b) and 3(d), similar observations can also be made, except for the fact that the constraints on the couplings are now a little bit looser than that in type III model. What makes difference from type III model is that larger values of in the plane occur around and , which are resulted from the cancellation between the complex terms.

From the above discussions, one can conclude that although type III and type C model present some significantly different behaviors under the experimental constraints from mixing, it is still hard to distinguish them from each other, especially for the real coupling case or for small . This is due to the significant uncertainties of both the theoretical predictions and the experimental data. Therefore, more refined theoretical and experimental efforts are needed for a much clearer phenomenological picture.

#### 5. Conclusion

In this paper, we have performed a complete one-loop computation of the box diagrams for the mixing, both within the SM and in type III and type C 2HDMs. It is noted that, in order to get a gauge-independent result, the external strange-quark momentum and mass should be taken into account, which has been kept up to the second order.

Combining the latest experimental data on the mixing, we then performed a detailed phenomenological analysis of the charged-Higgs effects on this process. Our main conclusions can be summarized as follows:(i)The operator appears already at the matching scale in type C model, while its appearance in type III model is induced by the RG evolution effect from the high-down to the low-energy scale.(ii)We get strong constraint on the Yukawa coupling parameter in both the real and the complex coupling case, being even stronger than that obtained in [34], while there is almost no constraint on the other Yukawa coupling .(iii)The allowed parameter spaces for and in the case of real coupling are similar for both types of models, with a wider range in type C model. If we extend the range, however, the allowed region for will be smaller in type III and larger in type C model, behaving like “convex lens” and “concave lens,” respectively.(iv)In the case of complex coupling, the strong correlation between and is observed, especially for GeV in type III model. The relative phase between and , , allows the large values of at in type III and and in type C model. This is due to the cancellation effect between the complex terms in the Wilson coefficient .

Although these two types of models present some significantly different behaviors under the experimental constraints from mixing, it is still hard to distinguish one from the other, especially for the real coupling case or for small . We need more refined theoretical and experimental efforts for a much clearer phenomenological picture.

#### Appendix

#### Basic Functions for Mixing

In this appendix, we collect the relevant functions during the calculation of the Wilson coefficients for mixing. Note that the notations and are applied to each function listed below.

The Inami-Lim function is given by [42]The functions introduced in Section 3 are given explicitly as