$\epsilon'_K/\epsilon_K$ and $K \to \pi \nu \bar\nu$ in a two-Higgs doublet model

The Kaon direct CP violation $Re(\epsilon'_K/\epsilon_K)$ below the experimental data of a $2\sigma$ in the standard model, which is calculated using the RBC-UKQCD lattice and a large $N_c$ dual QCD, indicates the necessity of a new physics effect. In order to resolve the insufficient $Re(\epsilon'_K/\epsilon_K)$, we study the charged-Higgs contributions in a generic two-Higgs-doublet model. If we assume that the origin of the CP-violation phase is uniquely from the Kobayashi-Maskawa (KM) phase when the constraints from the $B$- and $K$-meson mixings, $B\to X_s \gamma$, and Kaon indirect CP violating parameter $\epsilon_K$ are simultaneously taken into account, it is found that the Kaon direct CP violation through the charged-Higgs effects can reach $Re(\epsilon'_K/\epsilon_K)_{H^\pm}\sim 8 \times 10^{-4}$. Moreover, with the constrained values of the parameters, the branching ratios of the rare $K\to \pi \nu \bar\nu$ decays can be $BR(K^+\to \pi^+ \nu \bar\nu)\sim 14 \times 10^{-11}$ and $BR(K_L\to \pi^0 \nu \bar\nu)\sim 3.6 \times 10^{-11}$, where the results can be tested through the NA62 experiment at CERN and the KOTO experiment at J-PARC, respectively.

To pursue new physics contributions to the Re(ǫ ′ K /ǫ K ) and rare K decays, in this work, we investigate the influence of a charged-Higgs in a generic two-Higgs-doublet model (2HDM), i.e., the type-III 2HDM, where global symmetry is not imposed on the Yukawa sector.As a result, flavor changing neutral currents (FCNCs) in such models can be arisen at the tree level.To reasonably suppress the tree-level FCNCs for the purpose of satisfying the constraints from the B and K systems, such as ∆M B d ,Bs , B → X s γ, ∆M K , and ǫ K , we can adopt the so-called Cheng-Sher ansatz [54], where the neutral scalar-mediated flavorchanging effects are dictated by the square-root of the mass product of the involved flavors, denoted by √ m f i m f j /v.Thus, we can avoid extreme fine-tuning of the free parameters when they contribute to the rare K and B decays.
From a phenomenological viewpoint, the reasons why the charged-Higgs effects in the type-III 2HDM are interesting can be summarized as follows: firstly, Re(ǫ ′ K /ǫ K ) and K → πν ν in the SM are all dictated by the product of the CKM matrix elements V * ts and V td .The same CKM factor automatically appears in the charged-Higgs Yukawa couplings without introducing any new weak CP-violation phase; thus, we can avoid the strict limits from the time-dependent CP asymmetries in the B d and B s systems.Secondly, unlike the type-II 2HDM, where the charged-Higgs mass is bounded to be m H ± > 580 GeV via the B → X s γ decay [55,56], the charged-Higgs in the type-III model can be much lighter than that in the type-II model, due to the modification of the Yukawa couplings [56].Thirdly, a peculiar unsuppressed Yukawa coupling m c /m t V cq ′ /V tq ′ χ u * ct (see the later discussions), which originates from the FCNCs, also appears in the charged-Higgs couplings to the topquark and down-type quarks [56,57].The effects play a key role in enhancing Re(ǫ K /ǫ K ) and K → πν ν in this model.Fourthly, the charged-Higgs effects can naturally provide the lepton-flavor universality violation and can be used to resolve the puzzles in the semileptonic B decays, such as R(D), R(K ( * ) ), and large BR(B − u → τ ν) [56][57][58][59][60][61][62][63][64][65].Since the charged-Higgs effects have a strong correlation with different phenomena, the new free parameters are not constrained by only one physical observable.Therefore, the involved new parameters are strictly limited and cannot be arbitrarily free.It is found that when the constraints of ∆B = 2, ∆K = 2, B → X s γ, and ǫ K are simultaneously taken into account, the charged-Higgs contribution to the direct CP violation of K-meson can reach ( not including the SM contribution), and the BR for The paper is organized as follows: In Section II, we briefly review the charged-Higgs and neutral scalar Yukawa couplings to the fermions with the Cheng-Sher ansatz in the type-III 2HDM.In Section III, we formulate ∆M K , ǫ K , and Re(ǫ ′ K /ǫ K ) in the 2HDM.The charged-Higgs contributions to the rare K → πν ν decays are shown in Section IV.The detailed numerical analysis is shown in Section V, where the constraints from ∆M B d ,Bs , B → X s γ, ∆M K , and ǫ K are included.A conclusion is given in Section V.

LEPTONS
In this section, we summarize the Yukawa couplings of the neutral Higgses and charged-Higgs to the quarks and leptons in the generic 2HDM.The Yukawa couplings without imposing extra global symmetry can be in general written as: where all flavor indices are hidden; P R(L) = (1 ± γ 5 )/2; Q L and L L are the SU(2) L quark and lepton doublets, respectively; f R (f = U, D, ℓ) denotes the singlet fermion; Y f 1,2 are the 3 × 3 Yukawa matrices, and Hi = iτ 2 H * i .There are two CP-even scalars, one CPodd pseudoscalar, and two charged-Higgs particles in the 2HDM, and the relations between physical and weak states can be expressed as: where φ i (η i ) and η ± i denote the real (imaginary) parts of the neutral and charged components of H i , respectively; c α (s α ) = cos α(sin α), c β = cos β = v 1 /v, and s β = sin β = v 2 /v, v i are the vacuum expectation values (VEVs) of H i , and v = v 2 1 + v 2 2 ≈ 246 GeV.In this study, h is the SM-like Higgs while H, A, and H ± are new particles in the 2HDM.
Introducing the unitary matrices V f L,R to diagonalize the quark and lepton mass matrices, the Yukawa couplings of scalars H and A can then be obtained as: where m f is the diagonalized fermion mass matrix; s βα = sin(β − α), and X f s are defined as: We can also obtain the Higgs Yukawa couplings; however, it is found that the associated X f terms are always related to c αβ , which is strictly bound by the current precision Higgs data.
For simplicity, we take the alignment limit with c αβ = 0 in the following analysis.Thus, the Higgs couplings are the same as those in the SM.The charged-Higgs Yukawa couplings to fermions are found as: where L stands for the CKM matrix.Except the factor √ 2 and CKM matrix, the Yukawa couplings of charged Higgs are the same as those of pseudoscalar A.
From Eq. ( 9), the FCNCs at the tree level can be induced through the X f terms.To suppress the tree-induced ∆F = 2 (F = K, B d(s) , D) processes, we employ the Cheng-Sher ansatz [54] as: where χ f ij are the new free parameters.With the Cheng-Sher ansatz, the Yukawa couplings of scalars H and A to the down-type quarks can be straightforwardly obtained as: where the CKM matrix elements do not involve.
Since the charged-Higgs interactions are associated with the CKM matrix elements, the couplings involving the third generation quarks may not be small; therefore, for the K-meson decays, it is worth analyzing the charged-Higgs Yukawa couplings of the d-and s-quark to the top-quark, i.e., tdH + and tsH + .According to Eq. ( 11), the t R d L H + coupling can be written and simplified as where we have dropped the χ u * ut term because its coefficient is a factor of 4 smaller than the χ u * ct term.In addition to the m t enhancement, the effect associated with ct , which is in principle not suppressed.Intriguingly, the charged-Higgs coupling is comparable to the SM gauge coupling of (g/ √ 2)V td .Because can be approximated as: Although there is no V td suppression, because 66 GeV, the tsH + coupling can be similarly obtained as: The detailed analysis for the other charged-Higgs couplings can be found in [56].In sum, the charged-Higgs couplings to the d(s)-and top-quark in the type-III 2HDM can be formulated as: where the parameters ζ f ij are defined as: For the lepton sector, we use the flavor-conserving scheme with as a result, the Yukawa couplings of H ± to the leptons can be expressed as: The suppression factor m ℓ /v could be moderated using a large value of tan β.In this work, we use the interactions shown in Eqs. ( 13), (17), and (19) to study the influence on the K 0 − K0 mixing ∆M K , ǫ K , ǫ ′ K /ǫ K , and K → πν ν decays. III.

FORMULATIONS OF ∆M
To study the new physics contributions to ∆M K and ǫ K , we follow the notations in [67] and write the effective Hamiltonian for ∆S = 2 as: where V i CKM are the involved CKM matrix elements; C i (µ) are the Wilson coefficients at the µ scale, and the relevant operators Q i are given as: The operators In the type-III 2HDM, the ∆S = 2 process can arise from the H/A-mediated tree FCNCs and the H ± -mediated box diagrams, for which the representative Feynman diagrams are sketched in Fig. 1.According to the interactions in Eq. ( 13), the H/A-induced effective Hamiltonian can be expressed as: where the subscript S denotes the scalar and pseudoscalar contributions.Clearly, no CKM matrix elements are involved in the tree FCNCs.Since the involved operators are Q SLL,SRR 1 and Q LR 2 , the corresponding Wilson coefficients at the µ S scale are obtained as: with x q = m 2 q /m 2 W .It can be seen from r S that although the H/A effects are suppressed by m d m s /m 2 W , due to the tan β enhancement, the ∆M K through the intermediates of H and A becomes sizable.We then can use the measured ∆M K to bound the parameters χ = 0 and that C LR S2 is a real parameter; under this condition, ǫ K that has arisen from the neutral scalars will be suppressed.
From the charged-Higgs interactions in Eq. ( 17), we find that with the exception of Q SLL 2 , the W ± -H ± , G ± -H ± , and H ± -H ± box diagrams can induce all operators shown in Eq. ( 21); the associated CKM matrix element factor is V i CKM = (V * ts V td ) 2 , and the Wilson coefficients at the µ = m H ± scale can be expressed as: where the subscript H ± denotes the charged-Higgs contributions, y q = m 2 q /m 2 H ± , and the loop integral functions are defined as: We note through box diagrams that the couplings of sbH(A) and dbH(A) can induce ∆S = 2; however, because the involved quark in the loop is the bottom-quark, the effects should be much smaller than those from the top-quark loop.Here, we ignore their contributions.
To obtain ∆M K and ǫ K , we define the hadronic matrix element of K-K mixing to be: Accordingly, the K-meson mixing parameter and indirect CP violating parameter can be obtained as: where we have ignored the small contribution of ImA 0 /ReA 0 from K → ππ in ǫ K .Since ∆M K is experimentally measured well, we will directly take the ∆M K data for the denominator of ǫ K .It has been found that the short-distance SM result on ∆M K can explain the data by ∼ 70%, and the long-distance effects may contribute another 20 − 30% with a large degree of uncertainty [77].In this work, we take ∆M exp K as an input to bound the new physics effects; using the constrained parameters, we then study the implications on the other phenomena.
To estimate the M 12 defined in Eq. ( 26), we need to run the Wilson coefficients from a higher scale to a lower scale using the renormalization group (RG) equation.In addition, we also need the hadronic matrix elements of K0 |Q i |K 0 .In order to obtain this information, we adopt the results shown in [67], where the RG and nonperturbative QCD effects have been included.Accordingly, the ∆S = 2 matrix element can be expressed as: where the Wilson coefficients C χ F i are taken at the m t scale with F = S(H ± ), and the values of P χ i at µ = 2 GeV are [67]: It can be seen that the values of P χ i , which are related to the scalar operators, are one to two orders of magnitude larger than the value of P V LL

1
, where the enhancement factor is from the factor m 2 K /(m s + m d ) 2 .The similar enhancement factor in the B-meson system is just slightly larger than one.Although the new physics scale is dictated by µ S(H ± ) (µ S(H ± ) > m t ), as indicted in [67], the RG running of the Wilson coefficients from µ S(H ± ) to m t is necessary only when µ S(H ± ) > 4m t .To estimate the new physics effects, we will take µ S(H ± ) 800 GeV and ignore the running effect between µ S(H ± ) and m t scale.In Eq. ( 28), we have explicitly shown the CKM factor to be V CKM = 1 for F = S and V CKM = (V * ts V td ) 2 for F = H ± .

B.
Re(ǫ ′ K /ǫ K ) from the charged-Higgs induced QCD and electroweak penguins Using isospin decomposition, the decay amplitudes for K → ππ can be written as [68]: where A 0(2) denotes the isospin I = 0(2) amplitude; δ 0(2) is the strong phase, and the measurement is δ 0 − δ 2 = (47.7 ± 1.5) • .In terms of the isospin amplitudes, the direct CP violating parameter in K system can be written as [14]: where a = 1.017 [69] and Ωeff = (14.8± 8.0) × 10 −2 [14] include the isospin breaking corrections and the correction of ∆I = 5/2, and With the normalizations of A 0,2 used in [14] , the experimental values of ReA 0 and ReA 2 should be taken as: Although the uncertainty of the predicted ReA 0 in the SM is somewhat large, the results of ReA 2 obtained by the dual QCD approach [77] and the RBC-UKQCD collaboration [5][6][7] were consistent with the experimental measurement.Thus, we can use the (ReA 2 ) exp to limit the new physics effects.Then, the explanation of the measured Re(ǫ ′ K /ǫ K ) will rely on the new physics effects that contributes to ImA 0 and ImA 2 .
From Eq. ( 13), the couplings qqH(A) with Cheng-Sher ansatz indeed are suppressed by m d(u) tan β/v ∼ 10 −3 (tan β/50).If we take m H(A) to be heavier, the effects will be further suppressed.Thus, in the following analysis, we neglect the neutral scalar boson contributions to the K → ππ processes.According to the results in [56], the couplings ud(s)H ± as compared with the SM are small; therefore, we also drop the tree-level charged-Higgs contributions to K → ππ.Accordingly, the main contributions to the ǫ ′ K are from the top-quark loop QCD and electroweak penguins.Since the induced operators are similar to the SM, in order to consider the RG running of the Wilson coefficients, we thus write the effective Hamiltonian for ∆S = 1 in the form of the SM as [78]: where τ = −V * ts V td /(V * ts V td ), and z i (µ) and y i (µ) are the Wilson coefficients at the µ scale.The effective operators for (V − A) ⊗ (V − A) are given as: where α, β are color indices; the color indices in qq ′ are suppressed, and (qq ′ ) V ±A = qγ µ (1 ± γ 5 )q ′ .For the QCD penguin operators, they are: where q in the sum includes u, d, s, c, and b quarks.For the electroweak penguins, the effective operators are: where e q is the q-quark electric charge.The H ± -mediated Wilson coefficients can be expressed as [78]: y 8 (µ H ) = y 10 (µ H ) = 0, and the functions D H , C H , and E H are given by [72,73]: To calculate ǫ ′ K , in addition to the Wilson coefficients, we need the hadronic matrix elements of the effective operators.The Q 1,2 matrix elements can be obtained from ReA 0 and ReA 2 through the parametrizations [14]: where we ignore the small imaginary part in V * us V ud ; the Wilson coefficients and the matrix elements of the effective operators are taken at the µ = m c scale; the subscripts of the brackets denote the isospin states I = 0 and I = 2; .2120, and q = z + Q + 0 /(z − Q − 0 ).In the isospin limit, the hadronic matrix elements of Q 4,9,10 can be related to Q +,− [70].Therefore, we show the matrix elements for isospin I = 0 as [14,70]: for isospin I = 2, they are given as: where h = 3/2; the small matrix elements for Q 3,5,7 are neglected; = 0.76 ± 0.05, and B (1/2) 8 = 1.0 ± 0.2 [23], which are extracted from the lattice calculations [8,9].
Using the introduced hadronic matrix elements and Eq. ( 31), the direct CP violating parameter via the charged-Higgs contributions can be expressed as: where λ t = V * ts V td , a ∆I i are defined as [14]: ReA 0 , a We note that the Wilson coefficients in Eq. ( 44) should be taken at the µ = m c scale through the RG running.

IV. CHARGED-HIGGS ON THE K → πν ν DECAYS
To investigate the new physics contributions to the rare K decays, we adopt the parametrizations shown in [24] as: where λ is the Wolfenstein parameter; ∆ EM = −0.003;P c (X) = 0.404 ± 0.024 denotes the charm-quark contribution [23,74,75]; ] combines the new physics contributions and the SM result of X SM L (K) = 1.481 ± 0.009 [24], and the values of κ +,L are given as: Here, X L (K) and X R (K) denote the contributions from the left-handed and right-handed quark currents, respectively.
The neutral scalar bosons H and A do not couple to neutrinos; therefore, the rare K → πν ν decays can be generated by the Z-mediated electroweak penguins and the W ± -and H ± -mediated box diagrams, for which the representative Feynman diagrams are shown in Fig. 2. Since the dominant H ± contributions are from the left-handed quark currents, we only show the X L (K) results in the following analysis.Using the H ± Yukawa couplings in Eq. ( 17), the Z-penguin contribution can be obtained as: FIG. 2: Sketched Feynman diagrams for the K → πνν process.
where g u L is the Z-boson coupling to the left-handed up-type quarks and is given as g u L = 1/2 − 2 sin 2 θ W /3 with sin θ W ≈ 0.23, and J 1 (y t ) is the loop integral function.
According to the intermediated states in the loops, there are three types of box diagrams contributing to the d → sν ν process: W ± H ± , G ± H ± , and H ± H ± .Their results are respectively shown as follows.For the W ± H ± diagrams, the result is obtained as: J 2 (y t , y W ) = I 2 (y t , y W ) − I 2 (y t , 0) , where x ℓ = m 2 ℓ /m 2 W , and the function J 2 is the loop integration.The result of G ± H ± diagrams is given as: It can be seen that except the loop functions, X G ± H ± L,Box and X W ± H ± L,Box have the common factor from the charged-Higgs effect.The pure H ± -loop contribution to the box diagram can be The values of the CKM matrix elements are taken as follows: where Re(V * ts V td ) ≈ −3.3 × 10 −4 and Im(V * ts V td ) ≈ 1.4 × 10 −4 are taken to be the same as those used in [14].The particle masses used to estimate the numerical values are given as: ) is taken.Intriguingly, the small θ CP may not be a fine-tuning result in the case of m H = m A .If the Yukawa matrices Y f i in Eq. ( 8) are hermitian matrices, due to 10) also being hermitian, we can obtain χ d * ds = χ d sd and θ CP = 0. Hence, a small θ CP can be ascribed to a slight break in a hermitian Yukawa matrix.
Next, we analyze the charged-Higgs loop contributions to ∆M K and ǫ K .According to Eqs. ( 17) and ( 18), the relevant Yukawa couplings are χ u tt,ct and χ d bd,bs .However, the same parameters also contribute to ∆M B d,s and B → X s γ, where the former can arise from the tree H/A-mediated and charged-Higgs-mediated box diagrams, and the latter is from the H ± -penguin loop diagrams [56].Thus, we have to constrain the free parameters by taking the ∆M B d ,Bs and B → X s γ data into account.To scan the parameters, we set the ranges of the scanned parameters to be: ǫ ′ K /ǫ K can be significantly enhanced only by the light H ± ; therefore, we set m H ± around 200 GeV.
In order to consider the constraints from the B-meson decays, we use the formulae and results obtained in [56].To understand the influence of B and K systems on the parameters, we show the constraints with and without the ∆M K and ǫ K constraints.Thus, for the ∆M B d ,Bs and B → X s γ constraints only, we respectively show the allowed ranges of χ u tt and χ u ct and the allowed ranges of χ d bs and χ d bd in Fig. 4(a) and (b), where the sampling data points for the scan are 5 • 10 6 .When the ∆M K and ǫ K constraints are included, the corresponding situations are shown in Fig. 5(a) and (b), respectively.From the plots, it can be clearly seen that K-meson data can further constrain the free parameters.
In this subsection, we analyze the charged-Higgs effect on Re(ǫ ′ K /ǫ K ) H ± in detail.To estimate Re(ǫ ′ K /ǫ K ) H ± , we need to run the Wilson coefficients from the µ H scale to the τ /m 2 W ζ ℓ τ t β , where although there is a t β enhancement factor, their contributions are still small and negligible.The Wilson coefficient from the H ± H ± box diagram can be enhanced through the (ζ ℓ τ t β ) 2 factor; however, its sign is opposite to that of the SM, so that it has a destructive effect on the SM results.Thus, we cannot rely on H ± H ± to enhance BR(K + → π + ν ν) and BR(K L → π 0 ν ν).Hence, the main charged-Higgs effect on the d → sν ν process is derived from the Z-penguin diagram.
With the input parameter values, the BRs for the K + → π + ν ν and K L → π 0 ν ν processes in the SM can be estimated to be BR(K + → π + ν ν) ≈ 8.8 × 10 −11 and BR(K L → π 0 ν ν) ≈ 2.9 × 10 −11 .Using the parameter values, which are constrained by the B-meson and Kmeson data, we calculate the charged-Higgs contributions to K + → π + ν ν and K L → π 0 ν ν, where the BRs (in units of 10 −11 ) as a function of χ u ct are shown in Fig. 8(a) and (b).In order to suppress the contribution from the H ± H ± diagram, we take χ ℓ τ = 1.From the plots, we clearly see that BR(K + → π + ν ν) can be enhanced to ∼ 14 × 10 11 while BR(K + → π + ν ν) is enhanced to ∼ 3.6 × 10 −11 .Since the CP violating source in the charged-Higgs loop is the same as that of the SM, the K L → π 0 ν ν enhancement is limited.Although the charged-Higgs cannot enhance K + → π + ν ν by a factor of 2, it can increase the SM result by 60%.

VI. CONCLUSION
We comprehensively studied the Re(ǫ ′ K /ǫ K ) and the rare K + (K L ) → π + (π 0 )ν ν decays in the type-III 2HDM, where the Cheng-Sher ansatz was applied, and the main CP-violation phase was still from the CKM matrix element V td when the Wolfenstein parametrization was taken.We used |∆M K | < 0.2∆M exp K and |ǫ H ± K | < 0.4 × 10 −3 to bound the free parameters.The charged-Higgs related parameters, which contribute to ∆M K and ǫ K , also contribute to ∆M B d ,Bs and B → X s γ processes.When the constraints from the K and B systems are satisfied, we found that it is possible to obtain (ǫ ′ K /ǫ K ) H ± ∼ 8 × 10 −4 in the generic 2HDM, where the dominant effective operator is from the electroweak penguin Q 8 .
The dominant contribution to the rare K → πν ν decays in the type-III 2HDM is the H ± -loop Z-penguin diagram.With the same set of constrained parameters, we found that K L → π 0 ν ν can be slightly enhanced to BR(K L → π 0 ν ν) ∼ 3.6 × 10 −11 , whereas K + → π + ν ν can enhanced to be BR(K + → π + ν ν) ∼ 14 × 10 −11 .Although the BRs of the rare K → πν ν decays cannot be enhanced by one order of magnitude in the type-III 2HDM, the results are still located within the detection level in the KOTO experiment at J-PARC and the NA62 experiment at CERN.
d sd and χ d ds .If we take m H = m A and χ d sd = χ d ds , it can be seen that C SLL S1 = C SRR S1

m
K ≈ 0.489 GeV , m B d ≈ 5.28 GeV , m Bs ≈ 5.37 GeV , m W ≈ 80.385 GeV , m t ≈ 165 GeV , m c ≈ 1.3 GeV , m s (m c ) ≈ 0.109 GeV , m d (m c ) ≈ 5.44 MeV .(58) B. Constraints from ∆F = 2, B → X s γ, and ǫ K Since the free parameters for the tree-induced ∆K = 2 are different from that are boxinduced, we analyze them separately.According to Eq. (23), in addition to the t β parameter, the main parameters in the H/A-mediated M 12 are χ d * ds χ d sd = |χ d * ds χ d sd |e −iθ CP , where θ CP is the weak CP-violation phase.Using Eq. (28) and the taken input values, ∆M K (solid) and ǫ K (dashed) as a function of |χ d * ds χ d sd | (in units of 10 −4 ) and θ CP are shown in Fig. 3, where we only show the range of θ f = [0, π] and fix t β = 30.It is seen that the typical value of |χ d ds,sd | constrained by the K − K mixing is ∼ 7 × 10 −3 .Because the ǫ K and ∆M K both arise from the same complex parameter χ d * ds χ d sd , to obtain ǫ K of O(10 −3 ), the CP-violation phase θ CP inevitably has to be of O(10 −2 ) away from zero or π when the |χ d * ds χ d sd | of O(10 −5

FIG. 4 :FIG. 5 :
FIG. 4: Constraints from ∆M B d ,Bs and B → X s γ, where the plots (a) and (b) denote the allowed ranges of χ u tt and χ u ct and the allowed ranges of χ d bs and χ d bd , respectively.The number of sample points used for the scan is 5 • 10 6 .