# K 0 - K ¯ 0 Mixing in the Minimal Flavor-Violating Two-Higgs-Doublet Models

Authors: Natthawin Cho; Xin-Qiang Li; Fang Su; Xin Zhang
Journal: Advances in High Energy Physics
 (2017)

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## 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 theK0-K¯0 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 K0-K¯0 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 ΔmK and the CP-violating parameter ϵK 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.


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## Body

## 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 Z2 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 YU and YD. 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 SU(3)×SU(2)×U(1) representation of the second scalar doublet is fixed to be either 1,21/2 or 8,21/2 [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 W boson in the SM, indicating that the NP effects might be significant. For example, the charged-Higgs effects of these two models on the Bs0-B¯s0 mixing have been studied in [28]. In this paper, we shall explore the K0-K¯0 mixing within these two models and pursue possible differences between their effects. The general formula for K0-K¯0 mixing, including the charged-Higgs contributions, could be found, for example, in [33].Our paper is organized as follows. In Section2, we review briefly the 2HDMs under the MFV hypothesis and give the theoretical framework for the K0-K¯0 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 K0-K¯0 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, Φ1 and Φ2, and quarks as [24, 25](1)-LY=q¯L0Φ~1YUuR0+q¯L0Φ1YDdR0+q¯L0Φ~2aTRaY¯UuR0+q¯L0Φ2aTRaY¯DdR0+h.c.,where Φ~j=iσ2Φj∗j=1,2 with σ2 the Pauli matrix, and q¯L0, uR0, and dR0 are the quark fields given in the interaction basis. TR(a) is the SU(3) 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). YU,D and Y¯U,D are the Yukawa couplings and are generally complex 3×3 matrices in the quark flavor space.According to the MFV hypothesis, the transformation properties of the Yukawa coupling matricesYU,D and Y¯U,D under the quark flavor symmetry group SU3QL⊗SU3UR⊗SU3DR are required to be the same. This can be achieved by requiring Y¯U,D to be composed of pairs of YU,D [25]:(2)Y¯U=Au∗1+ϵu∗YUYU†+⋯YU,Y¯D=Ad1+ϵdYUYU†+⋯YD.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](3)LH+=g2mW∑i,j=13u¯iTRaAuimuiPL-AdimdjPRVijdjHa++h.c.,where Au,di are family-dependent Yukawa coupling constants [25, 28, 34]:(4)Au,di=Au,d1+ϵu,dmt2v2δi3,with v=Φ10=174 GeV. For simplicity, we consider only the family universal coupling case in which the family-dependent Yukawa couplings, Au,di, can be simplified to Au,di=Au,d.


### 2.2.K0-K¯0 Mixing

Both within the SM and in the 2HDMs with MFV, the neutral kaon mixing occurs via the box diagrams depicted in Figure1 (these Feynman diagrams are drawn with the LaTeX packageTikZ-Feynman [35]). As demonstrated in [36], the correction from the external momenta and quark masses is not negligible for the K0-K¯0 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].Figure 1
Box diagrams for theK0-K¯0 mixing in the unitary gauge both within the SM (a) and in the 2HDMs with MFV (b)–(d). Crossed diagrams, which are related to the original ones by interchanging the external lines, have also been taken into account.
(a) (b) (c) (d)Calculating the one-loop box diagrams and following the standard procedure of matching [36], we obtain the effective Hamiltonian responsible for the K0-K¯0 mixing:(5)Heff=GF2mW216π2CVLLμQVLL+CSLLμQSLL+CTLLμQTLL+h.c.,where GF is the Fermi coupling constant, mW is the W-boson mass, and Ciμ is the scale-dependent Wilson coefficients of the four-quark operators Qi, 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):(6)QVLL=s¯αγμ1-γ5dαs¯βγμ1-γ5dβ,QSLL=s¯α1-γ5dαs¯β1-γ5dβ,QTLL=s¯ασμν1-γ5dαs¯βσμν1-γ5dβ,with α and β being the color indices and σμν=1/2γμ,γν. Note that we include the QCD corrections only to the SM Wilson coefficient CVLL, but not to the NP ones. The hadronic matrix elements of these operators can be written as [37](7)K¯0∣QVLL∣K0=43mKFK2B1VLLμ,K¯0∣QSLL∣K0=-56RμmKFK2B1SLLμ,K¯0∣QTLL∣K0=-2RμmKFK2B2SLLμ,where mK is the kaon mass and FK the kaon decay constant. Bij(μ) is the scale-dependent bag parameters, and R(μ) is defined as [37](8)Rμ=mKmsμ+mdμ2.It should be noted that the SM and NP contributions to the Wilson coefficientsCi(μ) cannot be summed directly because they are given at different initial scales, μW for the SM and μH± 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 Bij(μ) are evaluated. The explicit expressions for these Wilson coefficients will be presented in Section 3.For theK0-K¯0 mixing, there exist two observables which can be calculated from the effective Hamiltonian given by (5) [37]:(9)ΔmK=2ReK0∣Heff∣K¯0,(10)ϵK=exp⁡iπ/42ΔmKIm⁡K0∣Heff∣K¯0.The above equations are the most general formulae for these two observables. It should be noted thatΔmK and ϵK receive both short-distance (SD) and long-distance (LD) contributions. With the LD contribution included, the mass difference ΔmK can be decomposed as [38](11)ΔmK=ΔmKSD+ΔmKLDππ+ΔmKLDη′,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](12)ΔmKLDππ=0.4ΔmKexp,ΔmKLDη′=-0.3ΔmKexp.We can see from (12) that the LD contribution to ΔmK 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 ΔmKLD 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 parameterϵK, with the LD contribution taken into account, is given by [40](13)ϵK=κϵeiϕϵ2Im⁡M12SDΔmKexp,where κϵ=0.942 [38], ϕϵ=43.525∘ [41], and M12SD=K0∣Heff∣K¯0. The LD contribution to ϵK has been included in the two phenomenological factors κϵ and ϕϵ. In the case with only the SD contribution, κϵ=1 and ϕϵ=π/4, and (13) goes back to (10).


## 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, Φ1 and Φ2, and quarks as [24, 25](1)-LY=q¯L0Φ~1YUuR0+q¯L0Φ1YDdR0+q¯L0Φ~2aTRaY¯UuR0+q¯L0Φ2aTRaY¯DdR0+h.c.,where Φ~j=iσ2Φj∗j=1,2 with σ2 the Pauli matrix, and q¯L0, uR0, and dR0 are the quark fields given in the interaction basis. TR(a) is the SU(3) 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). YU,D and Y¯U,D are the Yukawa couplings and are generally complex 3×3 matrices in the quark flavor space.According to the MFV hypothesis, the transformation properties of the Yukawa coupling matricesYU,D and Y¯U,D under the quark flavor symmetry group SU3QL⊗SU3UR⊗SU3DR are required to be the same. This can be achieved by requiring Y¯U,D to be composed of pairs of YU,D [25]:(2)Y¯U=Au∗1+ϵu∗YUYU†+⋯YU,Y¯D=Ad1+ϵdYUYU†+⋯YD.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](3)LH+=g2mW∑i,j=13u¯iTRaAuimuiPL-AdimdjPRVijdjHa++h.c.,where Au,di are family-dependent Yukawa coupling constants [25, 28, 34]:(4)Au,di=Au,d1+ϵu,dmt2v2δi3,with v=Φ10=174 GeV. For simplicity, we consider only the family universal coupling case in which the family-dependent Yukawa couplings, Au,di, can be simplified to Au,di=Au,d.


## 2.2.K0-K¯0 Mixing

Both within the SM and in the 2HDMs with MFV, the neutral kaon mixing occurs via the box diagrams depicted in Figure1 (these Feynman diagrams are drawn with the LaTeX packageTikZ-Feynman [35]). As demonstrated in [36], the correction from the external momenta and quark masses is not negligible for the K0-K¯0 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].Figure 1
Box diagrams for theK0-K¯0 mixing in the unitary gauge both within the SM (a) and in the 2HDMs with MFV (b)–(d). Crossed diagrams, which are related to the original ones by interchanging the external lines, have also been taken into account.
(a) (b) (c) (d)Calculating the one-loop box diagrams and following the standard procedure of matching [36], we obtain the effective Hamiltonian responsible for the K0-K¯0 mixing:(5)Heff=GF2mW216π2CVLLμQVLL+CSLLμQSLL+CTLLμQTLL+h.c.,where GF is the Fermi coupling constant, mW is the W-boson mass, and Ciμ is the scale-dependent Wilson coefficients of the four-quark operators Qi, 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):(6)QVLL=s¯αγμ1-γ5dαs¯βγμ1-γ5dβ,QSLL=s¯α1-γ5dαs¯β1-γ5dβ,QTLL=s¯ασμν1-γ5dαs¯βσμν1-γ5dβ,with α and β being the color indices and σμν=1/2γμ,γν. Note that we include the QCD corrections only to the SM Wilson coefficient CVLL, but not to the NP ones. The hadronic matrix elements of these operators can be written as [37](7)K¯0∣QVLL∣K0=43mKFK2B1VLLμ,K¯0∣QSLL∣K0=-56RμmKFK2B1SLLμ,K¯0∣QTLL∣K0=-2RμmKFK2B2SLLμ,where mK is the kaon mass and FK the kaon decay constant. Bij(μ) is the scale-dependent bag parameters, and R(μ) is defined as [37](8)Rμ=mKmsμ+mdμ2.It should be noted that the SM and NP contributions to the Wilson coefficientsCi(μ) cannot be summed directly because they are given at different initial scales, μW for the SM and μH± 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 Bij(μ) are evaluated. The explicit expressions for these Wilson coefficients will be presented in Section 3.For theK0-K¯0 mixing, there exist two observables which can be calculated from the effective Hamiltonian given by (5) [37]:(9)ΔmK=2ReK0∣Heff∣K¯0,(10)ϵK=exp⁡iπ/42ΔmKIm⁡K0∣Heff∣K¯0.The above equations are the most general formulae for these two observables. It should be noted thatΔmK and ϵK receive both short-distance (SD) and long-distance (LD) contributions. With the LD contribution included, the mass difference ΔmK can be decomposed as [38](11)ΔmK=ΔmKSD+ΔmKLDππ+ΔmKLDη′,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](12)ΔmKLDππ=0.4ΔmKexp,ΔmKLDη′=-0.3ΔmKexp.We can see from (12) that the LD contribution to ΔmK 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 ΔmKLD 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 parameterϵK, with the LD contribution taken into account, is given by [40](13)ϵK=κϵeiϕϵ2Im⁡M12SDΔmKexp,where κϵ=0.942 [38], ϕϵ=43.525∘ [41], and M12SD=K0∣Heff∣K¯0. The LD contribution to ϵK has been included in the two phenomenological factors κϵ and ϕϵ. In the case with only the SD contribution, κϵ=1 and ϕϵ=π/4, 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 Figure1(a). Without any QCD correction, they are given, respectively, as(14)CSMVLLμW=λc2S0xc+λt2S0xt+2λcλtS0xc,xt+xsλc2f1xc+λt2f1xt+2λcλtf1xc,xt,CSMSLLμW=xsλc2f2xc+λt2f2xt+2λcλtf2xc,xt,CSMTLLμW=xsλc2f3xc+λt2f3xt+2λcλtf3xc,xt,where xi=mi2(μ)/mW2, and S0 is the Inami-Lim function given by (A.1) [42]. Explicit expressions for the functions fi 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 CSMSLL and CSMTLL even in the SM case.The QCD corrections to the Wilson coefficients can be described by the factorsηcc, ηct, and ηtt, 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 get(15)K¯0∣Heff∣K0SMVLL=ζB^λc2ηccS0xc¯+λt2ηttS0xt¯+2λcλtηctS0xc¯,xt¯+PSMVLLxs,μWλc2f1xc,μW+λt2f1xt,μW+2λcλtf1xc,μW,xt,μW,K¯0∣Heff∣K0SMSLL=ζPSMSLLCSMSLLμW,K¯0∣Heff∣K0SMTLL=ζPSMTLLCSMTLLμW,where ζ=GF2mW2mKFK2/12π2, and xi¯≡mi(mi)/mW2 is the scale-independent mass ratio, whereas xi,μ≡mi(μ)/mW2 is the mass ratio at the scale μ. B^ is the RG independent bag parameter, and the factors Pi encode the RG evolution effects that are given, respectively, as [37](16)PSMVLL=η3GeVVLLSMB13GeV,PSMSLL=-58η113GeVSLLSMB23GeVeff-32η213GeVSLLSMB33GeVeff,PSMTLL=-58η123GeVSLLSMB23GeVeff-32η223GeVSLLSMB33GeVeff,where the effective bag parameters [Bi(3GeV)]eff are defined as [37](17)Bi3GeVeff≡R3GeVBi3GeV,with R(μ) defined in (8). The factors η and ηi,j are given by the formulae collected in [37] with(18)η4≡αs4μbαs43GeV,η5≡αs5μWαs5μb.


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

The Wilson coefficients at the matching scaleμH±~mH± in the NP case are calculated from the box diagrams shown in Figures 1(b)–1(d), with the results given, respectively, as(19)CIIIVLLμH±=AuAu∗λc2f4xc,xH+xsg4xc,xH+λt2f4xt,xH+xsg4xt,xH+2λcλtf4xc,xt,xH+xsg4xc,xt,xH+AuAd∗xsλc2f5xc,xH+λt2f5xt,xH+2λcλtf5xc,xt,xH+Au2Au∗2xsλc2f6xc,xH+λt2f6xt,xH+2λcλtf6xc,xt,xH,CIIISLLμH±=xsAuAu∗λc2f7xc,xH+g7xc,xH+λt2f7xt,xH+g7xt,xH+2λcλtf7xc,xt,xH+g7xc,xt,xH+AuAd∗λc2f8xc,xH+λt2f8xt,xH+2λcλtf8xc,xt,xH+Au2Au∗2λc2f9xc,xH+λt2f9xt,xH+2λcλtf9xc,xt,xH+Au2Ad∗2λc2f10xc,xH+λt2f10xt,xH+2λcλtf10xc,xt,xH-Au2Au∗Ad∗λc2f10xc,xH+λt2f10xt,xH+2λcλtf10xc,xt,xH,CIIITLLμH±=0,CCVLLμH±=13AuAu∗λc2f4xc,xH+xsg4xc,xH+λt2f4xt,xH+xsg4xt,xH+2λcλtf4xc,xt,xH+xsg4xc,xt,xH+13AuAd∗xsλc2f5xc,xH+λt2f5xt,xH+2λcλtf5xc,xt,xH+1118Au2Au∗2xsλc2f6xc,xH+λt2f6xt,xH+2λcλtf6xc,xt,xH,CCSLLμH±=xs-512AuAu∗λc2f7xc,xH+g7xc,xH+λt2f7xt,xH+g7xt,xH+2λcλtf7xc,xt,xH+g7xc,xt,xH-512AuAd∗λc2f8xc,xH+λt2f8xt,xH+2λcλtf8xc,xt,xH-1972Au2Au∗2λc2f9xc,xH+λt2f9xt,xH+2λcλtf9xc,xt,xH-1972Au2Ad∗2λc2f10xc,xH+λt2f10xt,xH+2λcλtf10xc,xt,xH+1972Au2Au∗Ad∗λc2f10xc,xH+λt2f10xt,xH+2λcλtf10xc,xt,xH,CCTLLμH±=xs116AuAu∗λc2f7xc,xH+g7xc,xH+λt2f7xt,xH+g7xt,xH+2λcλtf7xc,xt,xH+g7xc,xt,xH+116AuAd∗λc2f8xc,xH+λt2f8xt,xH+2λcλtf8xc,xt,xH+796Au2Au∗2λc2f9xc,xH+λt2f9xt,xH+2λcλtf9xc,xt,xH+796Au2Ad∗2λc2f10xc,xH+λt2f10xt,xH+2λcλtf10xc,xt,xH-796Au2Au∗Ad∗λc2f10xc,xH+λt2f10xt,xH+2λcλtf10xc,xt,xH.Explicit expressions for the functionsfi introduced in the above equations are collected in the appendix. Note that the contribution to CTLL 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 as(20)K¯0∣Heff∣K0IIIVLL=ζPNPVLLCIIIVLLμt,K¯0∣Heff∣K0IIISLL=ζPNPSLLCIIISLLμt,K¯0∣Heff∣K0IIITLL=0,for type III, and(21)K¯0∣Heff∣K0CVLL=ζPNPVLLCCVLLμt,K¯0∣Heff∣K0CSLL=ζPNPSLLCCSLLμt,K¯0∣Heff∣K0CTLL=ζPNPTLLCCTLLμt,for type C 2HDM. The factors Pi are also similar to the SM case but with a different factor η5, which is now defined by(22)η5≡αs5μtαs5μb.Here the matching scale for the 2HDMs has been changed to μt~mt, because the evolution effect from μH±~mH± down to μt~mt 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 elementK¯0∣Heff∣K0, which can be written as(23)K¯0∣Heff∣K0i=K¯0∣Heff∣K0SMi+K¯0∣Heff∣K0NPi,where the superscript “i” labels the different four-quark operators.


## 3.1. Wilson Coefficients within the SM

For the SM case, we calculate the Wilson coefficients from the box diagram shown in Figure1(a). Without any QCD correction, they are given, respectively, as(14)CSMVLLμW=λc2S0xc+λt2S0xt+2λcλtS0xc,xt+xsλc2f1xc+λt2f1xt+2λcλtf1xc,xt,CSMSLLμW=xsλc2f2xc+λt2f2xt+2λcλtf2xc,xt,CSMTLLμW=xsλc2f3xc+λt2f3xt+2λcλtf3xc,xt,where xi=mi2(μ)/mW2, and S0 is the Inami-Lim function given by (A.1) [42]. Explicit expressions for the functions fi 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 CSMSLL and CSMTLL even in the SM case.The QCD corrections to the Wilson coefficients can be described by the factorsηcc, ηct, and ηtt, 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 get(15)K¯0∣Heff∣K0SMVLL=ζB^λc2ηccS0xc¯+λt2ηttS0xt¯+2λcλtηctS0xc¯,xt¯+PSMVLLxs,μWλc2f1xc,μW+λt2f1xt,μW+2λcλtf1xc,μW,xt,μW,K¯0∣Heff∣K0SMSLL=ζPSMSLLCSMSLLμW,K¯0∣Heff∣K0SMTLL=ζPSMTLLCSMTLLμW,where ζ=GF2mW2mKFK2/12π2, and xi¯≡mi(mi)/mW2 is the scale-independent mass ratio, whereas xi,μ≡mi(μ)/mW2 is the mass ratio at the scale μ. B^ is the RG independent bag parameter, and the factors Pi encode the RG evolution effects that are given, respectively, as [37](16)PSMVLL=η3GeVVLLSMB13GeV,PSMSLL=-58η113GeVSLLSMB23GeVeff-32η213GeVSLLSMB33GeVeff,PSMTLL=-58η123GeVSLLSMB23GeVeff-32η223GeVSLLSMB33GeVeff,where the effective bag parameters [Bi(3GeV)]eff are defined as [37](17)Bi3GeVeff≡R3GeVBi3GeV,with R(μ) defined in (8). The factors η and ηi,j are given by the formulae collected in [37] with(18)η4≡αs4μbαs43GeV,η5≡αs5μWαs5μb.


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

The Wilson coefficients at the matching scaleμH±~mH± in the NP case are calculated from the box diagrams shown in Figures 1(b)–1(d), with the results given, respectively, as(19)CIIIVLLμH±=AuAu∗λc2f4xc,xH+xsg4xc,xH+λt2f4xt,xH+xsg4xt,xH+2λcλtf4xc,xt,xH+xsg4xc,xt,xH+AuAd∗xsλc2f5xc,xH+λt2f5xt,xH+2λcλtf5xc,xt,xH+Au2Au∗2xsλc2f6xc,xH+λt2f6xt,xH+2λcλtf6xc,xt,xH,CIIISLLμH±=xsAuAu∗λc2f7xc,xH+g7xc,xH+λt2f7xt,xH+g7xt,xH+2λcλtf7xc,xt,xH+g7xc,xt,xH+AuAd∗λc2f8xc,xH+λt2f8xt,xH+2λcλtf8xc,xt,xH+Au2Au∗2λc2f9xc,xH+λt2f9xt,xH+2λcλtf9xc,xt,xH+Au2Ad∗2λc2f10xc,xH+λt2f10xt,xH+2λcλtf10xc,xt,xH-Au2Au∗Ad∗λc2f10xc,xH+λt2f10xt,xH+2λcλtf10xc,xt,xH,CIIITLLμH±=0,CCVLLμH±=13AuAu∗λc2f4xc,xH+xsg4xc,xH+λt2f4xt,xH+xsg4xt,xH+2λcλtf4xc,xt,xH+xsg4xc,xt,xH+13AuAd∗xsλc2f5xc,xH+λt2f5xt,xH+2λcλtf5xc,xt,xH+1118Au2Au∗2xsλc2f6xc,xH+λt2f6xt,xH+2λcλtf6xc,xt,xH,CCSLLμH±=xs-512AuAu∗λc2f7xc,xH+g7xc,xH+λt2f7xt,xH+g7xt,xH+2λcλtf7xc,xt,xH+g7xc,xt,xH-512AuAd∗λc2f8xc,xH+λt2f8xt,xH+2λcλtf8xc,xt,xH-1972Au2Au∗2λc2f9xc,xH+λt2f9xt,xH+2λcλtf9xc,xt,xH-1972Au2Ad∗2λc2f10xc,xH+λt2f10xt,xH+2λcλtf10xc,xt,xH+1972Au2Au∗Ad∗λc2f10xc,xH+λt2f10xt,xH+2λcλtf10xc,xt,xH,CCTLLμH±=xs116AuAu∗λc2f7xc,xH+g7xc,xH+λt2f7xt,xH+g7xt,xH+2λcλtf7xc,xt,xH+g7xc,xt,xH+116AuAd∗λc2f8xc,xH+λt2f8xt,xH+2λcλtf8xc,xt,xH+796Au2Au∗2λc2f9xc,xH+λt2f9xt,xH+2λcλtf9xc,xt,xH+796Au2Ad∗2λc2f10xc,xH+λt2f10xt,xH+2λcλtf10xc,xt,xH-796Au2Au∗Ad∗λc2f10xc,xH+λt2f10xt,xH+2λcλtf10xc,xt,xH.Explicit expressions for the functionsfi introduced in the above equations are collected in the appendix. Note that the contribution to CTLL 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 as(20)K¯0∣Heff∣K0IIIVLL=ζPNPVLLCIIIVLLμt,K¯0∣Heff∣K0IIISLL=ζPNPSLLCIIISLLμt,K¯0∣Heff∣K0IIITLL=0,for type III, and(21)K¯0∣Heff∣K0CVLL=ζPNPVLLCCVLLμt,K¯0∣Heff∣K0CSLL=ζPNPSLLCCSLLμt,K¯0∣Heff∣K0CTLL=ζPNPTLLCCTLLμt,for type C 2HDM. The factors Pi are also similar to the SM case but with a different factor η5, which is now defined by(22)η5≡αs5μtαs5μb.Here the matching scale for the 2HDMs has been changed to μt~mt, because the evolution effect from μH±~mH± down to μt~mt 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 elementK¯0∣Heff∣K0, which can be written as(23)K¯0∣Heff∣K0i=K¯0∣Heff∣K0SMi+K¯0∣Heff∣K0NPi,where the superscript “i” labels the different four-quark operators.


## 4. Numerical Results and Discussions

### 4.1. Input Parameters and the SM Results

Firstly, we collect in Table1 the values of the relevant input parameters used throughout this paper, together with the experimental data on ΔmK and ϵK. For the bag parameters, we use the lattice results with Nf=2+1 flavors of dynamical quarks and evaluated at the renormalization scale 3 GeV [41, 48]. In addition, we have used theRunDec package [49] to obtain the running coupling constant and quark masses at different scales in the two-loop approximation.Table 1
Input parameters used throughout this paper, together with the experimental data.
Electroweak parameters [47] m W = 80.385 ( 15 ) GeV m Z = 91.1876 ( 21 ) GeV μ W = 80.385 GeV μ t = 163.427 GeVa G F = 1.1663787 ( 6 ) × 1 0 - 5 GeV−2  QCD coupling constant α s ( μ t ) = 0.1086 10 a α s ( m Z ) = 0.1182 12 [47] α s ( μ W ) = 0.1205 12 a α s ( μ b ) = 0.2243 45 a α s ( 3  G e V ) = 0.2521 - 0.0057 + 0.0058 a  Quark masses m d 2  G e V = 0.0047 - 0.0004 + 0.0005 GeV [47] m d ( 3  G e V ) = 0.0043 - 0.0004 + 0.0005 GeVa m s 2  G e V = 0.096 - 0.004 + 0.008 GeV [47] m s ( 3  G e V ) = 0.087 - 0.004 + 0.007 GeVa m s ( μ W ) = 0.057 - 0.002 + 0.005 GeVa m s ( μ t ) = 0.054 - 0.002 + 0.005 GeVa m c m c = 1.27 3 GeV [47] m c ( μ W ) = 0.660 ( 21 ) GeVa m c μ t = 0.623 20 GeVa m t p o l e = 173.21 ( 87 ) GeV [47] m t ( m t ) = 163.427 - 0.829 + 0.828 GeVa m t ( μ W ) = 173.276 - 1.586 + 1.590 GeVa  CKM matrix elements [47] λ = 0.22506 50 A = 0.811 26 ρ - = 0.124 - 0.018 + 0.019 η - = 0.356 11 ρ = 0.127 - 0.018 + 0.019 η = 0.365 11  Kaon mixing parameters m K = 0.497611 13 GeV [47] F K = 0.1562 ( 9 ) GeV [41] B ^ K = 0.7625 97 [41] B 1 ( 3  G e V ) = 0.519 ( 26 ) [48] B 2 ( 3  G e V ) = 0.525 ( 23 ) [48] B 3 ( 3  G e V ) = 0.360 ( 16 ) [48] B 4 ( 3  G e V ) = 0.981 ( 62 ) [48] B 5 ( 3  G e V ) = 0.751 ( 68 ) [48] η c c = 1.87 ( 76 ) [46] η t t = 0.5765 ( 65 ) [46] η c t = 0.496 ( 47 ) [46]  Experimental data [47] Δ m K e x p = 3.4839 59 × 1 0 - 15 GeV ϵ K e x p = 2.228 ( 11 ) × 1 0 - 3 aThis value is calculated with the RunDec package [49] at the two-loop level in αs.With the input parameters collected in Table1, we can now give the numerical results for ΔmK and ϵK 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 differenceΔmK without the corrections from the external strange-quark mass, xs, and from the LD contribution, agrees well with that obtained in [45].(ii)
The corrections fromxs to ΔmK and ϵK are 6.83% and −0.06%, respectively. Note that the correction to ΔmK is at the same order as that obtained in [36]. Moreover, the LD contributions to ΔmK and ϵK are 11.20% and −6%, respectively.(iii)
As thexs correction can be precisely calculated, we consider it both to ΔmK and to ϵK; especially, this correction is not too small for ΔmK. In addition, we include the LD contributions to ϵK but not to ΔmK, because the structure of LD contribution to ΔmK is still not well understood [38].Table 2
SM results forΔmK and ϵK with different corrections included and the ratios between these values and their experimental data. Here the column “None” denotes the results obtained with the external strange-quark momentum and mass ignored and without including the LD contribution.
Observables Corrections None With LD Withxs With LD andxs Δ m K S M   × 1 0 - 15  G e V 3.109 (1.258) 3.458 (1.258) 3.321 (1.258) 3.670 (1.258) Δ m K S M Δ m K e x p 89.24% 99.24% 95.34% 105.34% ϵ K S M   × 1 0 - 3 2.219-0.294+0.309 2.086-0.280+0.294 ⁡2.218-0.294+0.309 2.085-0.280+0.294 ϵKSMϵKexp 99.61% 93.63% 99.55% 93.58%


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

As can be seen clearly from Table2, there is no significant deviation between the SM predictions and the experimental data for ΔmK and ϵK, 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 Au,d, and the charged-Higgs mass mH±, in this paper. In the case of complex couplings, we can further choose Au and AuAd∗=AuAd∗eiθ as the independent variables, with θ being the relative phase between Au and Ad∗.The relevant model parameters are also constrained by the other processes. For the parameterAu, an upper bound can be obtained from the Z→bb¯ decay [25], while the parameter Ad 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 mH±>78.6 GeV (95% CL) has been set by the LEP experiment [50], which is obtained under the assumption that H± decays mainly into fermions without any specific Yukawa structure. In addition, direct searches for H± 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, tt¯bt¯, and bb¯bb¯ production at the LHC with the strongest available experimental limits from ATLAS or CMS at 8 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](24)Au∈0,3,Ad∈0,500,θ∈-π,π,mH±=100,250,500GeV.TakingmH±=500 GeV as a benchmark, we firstly explore the dependence of each Wilson coefficient evaluated at the matching scale μ=mH± or approximately at μ=mt on the other model parameters,(25)CIIIVLL×109=41.28-42.15iAu2+6.97-7.19iAu4+10-4·5.33-0.05iAuAd∗,CIIISLL×1015=3.28-3.29iAu2-0.09-0.09iAu4-308.73-25.25iAuAd∗+0.45-0.46iAu2AuAd∗-0.45-0.46iAuAd∗2,CCVLL×109=13.76-14.05iAu2+4.26-4.39iAu4+10-4·1.78-0.02iAuAd∗,CCSLL×1015=-1.37-1.37iAu2+0.02-0.02iAu4+128.64-10.52iAuAd∗-0.12-0.12iAu2AuAd∗+0.12-0.12iAuAd∗2,CCTLL×1016=2.05-2.06iAu2-0.06-0.06iAu4-192.96-15.78iAuAd∗+0.32-0.33iAu2AuAd∗-0.32-0.33iAuAd∗2.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 QVLL in both type III and type C 2HDM, due to the xs suppression in QSLL and QTLL. Furthermore, the coefficient of the AuAd∗ term in QVLL is quite small, being of order O10-4 compared to that of the Au terms.(ii)
Due to the color factor, the Wilson coefficientCVLL in type C is a little bit smaller than that in type III 2HDM, and the sign of CSLL in type C is also flipped relative to that in type III 2HDM.(iii)
There exists an extra operatorQTLL in type C 2HDM, and its Wilson coefficient CTLL differs from that of QSLL in sign.From the current experimental data onΔmK and ϵK, 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 parametersAu and Ad (also the relative phase θ for complex couplings) randomly within the ranges given by (24), with mH± fixed at 100, 250, and 500 GeV, respectively.(2)
With each set of values for the model parameters, we give the theoretical prediction forΔmK and ϵK, 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 the2σ 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.Figure 2
Allowed parameter spaces forAu and Ad in the case of real coupling for type III and type C 2HDMs, under the combined constraint from ΔmK and ϵK. The red, blue, and green regions are obtained with mH± fixed at 100, 250, and 500 GeV, respectively.
(a) (b)Figure 3
Allowed parameter spaces forAu, AuAd∗, and θ in the case of complex coupling for type III and type C 2HDMs, under the combined constraint from ΔmK and ϵK. The exclusion region in AuAd∗ comes from the constraint on Ad. The other captions are the same as in Figure 2.
(a) (b) (c) (d)From Figure2, we can make the following observations for the real coupling:(i)
In type III model, as shown in Figure2(a), the parameter Au is severely constrained due to the good agreement between the SM predictions and the experimental data, especially for ϵK; for example, the limit Au<0.7 is more stringent compared to that obtained in [34] with mH±=500 GeV. However, there is almost no constraints on Ad because of the smallness of the coefficient involving Ad, as mentioned earlier.(ii)
In type C model, we also get strong constraint forAu, but being looser than that in type III case, with the maximum value Au≈1. 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 Ad.(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 forAu is smaller with larger Ad for type III model, while the allowed range for Au in type C model can be larger with greater Ad. The reason is that the dominant contribution to the two observables ΔmK and ϵK comes from the operator QVLL, the Wilson coefficient of which in type C is smaller than that in type III model. Moreover, the cancellation between the Wilson coefficients QSLL and QTLL 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 betweenAu and AuAd∗, especially when mH±=100 GeV, as shown in Figure 3(a). It is also found from Figure 3(c) that the large values of AuAd∗ are allowed at θ≈±π/2, due to the cancellation between the complex terms,(ii)
in type C model, as shown in Figures3(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 AuAd∗ in the AuAd∗-θ plane occur around θ≈0 and ±π, 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 fromK0-K¯0 mixing, it is still hard to distinguish them from each other, especially for the real coupling case or for small Ad. 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.


## 4.1. Input Parameters and the SM Results

Firstly, we collect in Table1 the values of the relevant input parameters used throughout this paper, together with the experimental data on ΔmK and ϵK. For the bag parameters, we use the lattice results with Nf=2+1 flavors of dynamical quarks and evaluated at the renormalization scale 3 GeV [41, 48]. In addition, we have used theRunDec package [49] to obtain the running coupling constant and quark masses at different scales in the two-loop approximation.Table 1
Input parameters used throughout this paper, together with the experimental data.
Electroweak parameters [47] m W = 80.385 ( 15 ) GeV m Z = 91.1876 ( 21 ) GeV μ W = 80.385 GeV μ t = 163.427 GeVa G F = 1.1663787 ( 6 ) × 1 0 - 5 GeV−2  QCD coupling constant α s ( μ t ) = 0.1086 10 a α s ( m Z ) = 0.1182 12 [47] α s ( μ W ) = 0.1205 12 a α s ( μ b ) = 0.2243 45 a α s ( 3  G e V ) = 0.2521 - 0.0057 + 0.0058 a  Quark masses m d 2  G e V = 0.0047 - 0.0004 + 0.0005 GeV [47] m d ( 3  G e V ) = 0.0043 - 0.0004 + 0.0005 GeVa m s 2  G e V = 0.096 - 0.004 + 0.008 GeV [47] m s ( 3  G e V ) = 0.087 - 0.004 + 0.007 GeVa m s ( μ W ) = 0.057 - 0.002 + 0.005 GeVa m s ( μ t ) = 0.054 - 0.002 + 0.005 GeVa m c m c = 1.27 3 GeV [47] m c ( μ W ) = 0.660 ( 21 ) GeVa m c μ t = 0.623 20 GeVa m t p o l e = 173.21 ( 87 ) GeV [47] m t ( m t ) = 163.427 - 0.829 + 0.828 GeVa m t ( μ W ) = 173.276 - 1.586 + 1.590 GeVa  CKM matrix elements [47] λ = 0.22506 50 A = 0.811 26 ρ - = 0.124 - 0.018 + 0.019 η - = 0.356 11 ρ = 0.127 - 0.018 + 0.019 η = 0.365 11  Kaon mixing parameters m K = 0.497611 13 GeV [47] F K = 0.1562 ( 9 ) GeV [41] B ^ K = 0.7625 97 [41] B 1 ( 3  G e V ) = 0.519 ( 26 ) [48] B 2 ( 3  G e V ) = 0.525 ( 23 ) [48] B 3 ( 3  G e V ) = 0.360 ( 16 ) [48] B 4 ( 3  G e V ) = 0.981 ( 62 ) [48] B 5 ( 3  G e V ) = 0.751 ( 68 ) [48] η c c = 1.87 ( 76 ) [46] η t t = 0.5765 ( 65 ) [46] η c t = 0.496 ( 47 ) [46]  Experimental data [47] Δ m K e x p = 3.4839 59 × 1 0 - 15 GeV ϵ K e x p = 2.228 ( 11 ) × 1 0 - 3 aThis value is calculated with the RunDec package [49] at the two-loop level in αs.With the input parameters collected in Table1, we can now give the numerical results for ΔmK and ϵK 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 differenceΔmK without the corrections from the external strange-quark mass, xs, and from the LD contribution, agrees well with that obtained in [45].(ii)
The corrections fromxs to ΔmK and ϵK are 6.83% and −0.06%, respectively. Note that the correction to ΔmK is at the same order as that obtained in [36]. Moreover, the LD contributions to ΔmK and ϵK are 11.20% and −6%, respectively.(iii)
As thexs correction can be precisely calculated, we consider it both to ΔmK and to ϵK; especially, this correction is not too small for ΔmK. In addition, we include the LD contributions to ϵK but not to ΔmK, because the structure of LD contribution to ΔmK is still not well understood [38].Table 2
SM results forΔmK and ϵK with different corrections included and the ratios between these values and their experimental data. Here the column “None” denotes the results obtained with the external strange-quark momentum and mass ignored and without including the LD contribution.
Observables Corrections None With LD Withxs With LD andxs Δ m K S M   × 1 0 - 15  G e V 3.109 (1.258) 3.458 (1.258) 3.321 (1.258) 3.670 (1.258) Δ m K S M Δ m K e x p 89.24% 99.24% 95.34% 105.34% ϵ K S M   × 1 0 - 3 2.219-0.294+0.309 2.086-0.280+0.294 ⁡2.218-0.294+0.309 2.085-0.280+0.294 ϵKSMϵKexp 99.61% 93.63% 99.55% 93.58%


## 4.2. Results in the 2HDMs with MFV

As can be seen clearly from Table2, there is no significant deviation between the SM predictions and the experimental data for ΔmK and ϵK, 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 Au,d, and the charged-Higgs mass mH±, in this paper. In the case of complex couplings, we can further choose Au and AuAd∗=AuAd∗eiθ as the independent variables, with θ being the relative phase between Au and Ad∗.The relevant model parameters are also constrained by the other processes. For the parameterAu, an upper bound can be obtained from the Z→bb¯ decay [25], while the parameter Ad 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 mH±>78.6 GeV (95% CL) has been set by the LEP experiment [50], which is obtained under the assumption that H± decays mainly into fermions without any specific Yukawa structure. In addition, direct searches for H± 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, tt¯bt¯, and bb¯bb¯ production at the LHC with the strongest available experimental limits from ATLAS or CMS at 8 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](24)Au∈0,3,Ad∈0,500,θ∈-π,π,mH±=100,250,500GeV.TakingmH±=500 GeV as a benchmark, we firstly explore the dependence of each Wilson coefficient evaluated at the matching scale μ=mH± or approximately at μ=mt on the other model parameters,(25)CIIIVLL×109=41.28-42.15iAu2+6.97-7.19iAu4+10-4·5.33-0.05iAuAd∗,CIIISLL×1015=3.28-3.29iAu2-0.09-0.09iAu4-308.73-25.25iAuAd∗+0.45-0.46iAu2AuAd∗-0.45-0.46iAuAd∗2,CCVLL×109=13.76-14.05iAu2+4.26-4.39iAu4+10-4·1.78-0.02iAuAd∗,CCSLL×1015=-1.37-1.37iAu2+0.02-0.02iAu4+128.64-10.52iAuAd∗-0.12-0.12iAu2AuAd∗+0.12-0.12iAuAd∗2,CCTLL×1016=2.05-2.06iAu2-0.06-0.06iAu4-192.96-15.78iAuAd∗+0.32-0.33iAu2AuAd∗-0.32-0.33iAuAd∗2.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 QVLL in both type III and type C 2HDM, due to the xs suppression in QSLL and QTLL. Furthermore, the coefficient of the AuAd∗ term in QVLL is quite small, being of order O10-4 compared to that of the Au terms.(ii)
Due to the color factor, the Wilson coefficientCVLL in type C is a little bit smaller than that in type III 2HDM, and the sign of CSLL in type C is also flipped relative to that in type III 2HDM.(iii)
There exists an extra operatorQTLL in type C 2HDM, and its Wilson coefficient CTLL differs from that of QSLL in sign.From the current experimental data onΔmK and ϵK, 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 parametersAu and Ad (also the relative phase θ for complex couplings) randomly within the ranges given by (24), with mH± fixed at 100, 250, and 500 GeV, respectively.(2)
With each set of values for the model parameters, we give the theoretical prediction forΔmK and ϵK, 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 the2σ 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.Figure 2
Allowed parameter spaces forAu and Ad in the case of real coupling for type III and type C 2HDMs, under the combined constraint from ΔmK and ϵK. The red, blue, and green regions are obtained with mH± fixed at 100, 250, and 500 GeV, respectively.
(a) (b)Figure 3
Allowed parameter spaces forAu, AuAd∗, and θ in the case of complex coupling for type III and type C 2HDMs, under the combined constraint from ΔmK and ϵK. The exclusion region in AuAd∗ comes from the constraint on Ad. The other captions are the same as in Figure 2.
(a) (b) (c) (d)From Figure2, we can make the following observations for the real coupling:(i)
In type III model, as shown in Figure2(a), the parameter Au is severely constrained due to the good agreement between the SM predictions and the experimental data, especially for ϵK; for example, the limit Au<0.7 is more stringent compared to that obtained in [34] with mH±=500 GeV. However, there is almost no constraints on Ad because of the smallness of the coefficient involving Ad, as mentioned earlier.(ii)
In type C model, we also get strong constraint forAu, but being looser than that in type III case, with the maximum value Au≈1. 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 Ad.(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 forAu is smaller with larger Ad for type III model, while the allowed range for Au in type C model can be larger with greater Ad. The reason is that the dominant contribution to the two observables ΔmK and ϵK comes from the operator QVLL, the Wilson coefficient of which in type C is smaller than that in type III model. Moreover, the cancellation between the Wilson coefficients QSLL and QTLL 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 betweenAu and AuAd∗, especially when mH±=100 GeV, as shown in Figure 3(a). It is also found from Figure 3(c) that the large values of AuAd∗ are allowed at θ≈±π/2, due to the cancellation between the complex terms,(ii)
in type C model, as shown in Figures3(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 AuAd∗ in the AuAd∗-θ plane occur around θ≈0 and ±π, 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 fromK0-K¯0 mixing, it is still hard to distinguish them from each other, especially for the real coupling case or for small Ad. 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 theK0-K¯0 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 theK0-K¯0 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 operatorQTLL 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 parameterAu 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 Ad.(iii)
The allowed parameter spaces forAu and Ad 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 Ad range, however, the allowed region for Au 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 betweenAu and AuAd∗ is observed, especially for mH±=100 GeV in type III model. The relative phase between Au and Ad, θ, allows the large values of AuAd∗ at θ≈±π/2 in type III and θ≈0 and ±π in type C model. This is due to the cancellation effect between the complex terms in the Wilson coefficient CVLL.Although these two types of models present some significantly different behaviors under the experimental constraints fromK0-K¯0 mixing, it is still hard to distinguish one from the other, especially for the real coupling case or for small Ad. We need more refined theoretical and experimental efforts for a much clearer phenomenological picture.



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