Research Article  Open Access
Mixing in the Minimal FlavorViolating TwoHiggsDoublet Models
Abstract
The twoHiggsdoublet 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 lowenergy 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 colorsinglet or coloroctet, which are named as type III and type C 2HDM, respectively. In this paper, we study the chargedHiggs effects of these two models on the mixing, an ideal process to probe New Physics (NP) beyond the SM. Firstly, we perform a complete oneloop 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 uptodate 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 CPviolating 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 SMlike 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 socalled 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 CabibboKobayashiMaskawa (CKM) [8, 9] matrix, making it possible to explain the BAU [4].
It is known that, within the SM, the flavorchanging neutral current (FCNC) interactions are forbidden at tree level and are also highly suppressed at higher orders, due to the GlashowIliopoulosMaiani 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 righthanded 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 flavorviolating 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 colorsinglet or coloroctet. For convenience, they are referred as type III and type C model [25], respectively. Examples of the colorsinglet case include the aligned 2HDM (A2HDM) [26, 27] and the four types of 2HDM reviewed in [21, 22]. In the coloroctet case, the scalar spectrum contains one CPeven, colorsinglet Higgs boson (the usual SM one), and three coloroctet particles, one CPeven, one CPodd, and one electrically charged [24].
Although the scalarmediated flavorviolating interactions are protected by the MFV hypothesis, type III and type C models can still bring in many interesting phenomena in some lowenergy processes, especially due to the presence of a chargedHiggs boson [24, 25, 28–32]. The neutralmeson mixings are of particular interest in this respect, because the chargedHiggs 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 chargedHiggs 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 chargedHiggs 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 oneloop 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 colorsinglet or coloroctet). 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 masseigenstate basis, which are given by [25, 28]where are familydependent Yukawa coupling constants [25, 28, 34]:with GeV. For simplicity, we consider only the family universal coupling case in which the familydependent 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 TikZFeynman [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 strangequark momentum and mass to the second order; this is essential to guarantee the final result gaugeindependent [34].
(a)
(b)
(c)
(d)
Calculating the oneloop 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 scaledependent Wilson coefficients of the fourquark operators , which are defined, respectively, as follows (there are totally eight fourquark 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 scaledependent 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 shortdistance (SD) and longdistance (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 boldguess 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 CPviolating 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 InamiLim function given by (A.1) [42]. Explicit expressions for the functions can be found in the appendix. Note that when the external strangequark 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 nexttonexttoleading order [43–45] and are collected in [46]. Combining the renormalization group (RG) evolution with these QCD corrections, we getwhere , and is the scaleindependent 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 fourquark 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 twoloop approximation.
 
This value is calculated with the RunDec package [49] at the twoloop level in . 
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 strangequark 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 chargedHiggs 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 chargedHiggs 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 crosssections for the dijet, toppair, dijetpair, , 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 ManoharWise model [24], which is equivalent to type C 2HDM discussed here. Interestingly, they found that masses below 1 TeV have not been excluded for coloroctet scalars as is often claimed in the literature. For a variety of wellmotivated 2HDMs, the authors in [55] found that chargedHiggs 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 chargedHiggs 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