$\epsilon'/\epsilon$ from charged-Higgs-induced gluonic dipole operators

We study the effect of charged-Higgs-induced chromomagnetic operator, $Q_{8G}(-) \equiv \bar s \sigma^{\mu \nu} T^a \gamma_5 d G^a_{\mu \nu}$, on the Kaon direct CP violation $Re(\epsilon'/\epsilon)$. Using the matrix element $\langle \pi \pi | O_{8G}(-) | K^0 \rangle$ recently obtained by a large $N_c$ dual QCD approach, we find that if the Kobayashi-Maskawa phase is the origin of CP violation, the charged-Higgs-induced gluon penguin dipole operator in the type-III two-Higgs-doublet model can explain the measured $Re(\epsilon'/\epsilon)$ when the constraints from the relevant low energy flavor physics, such as $\Delta B(K)=2$, $B\to X_s \gamma$, and Kaon indirect CP violation parameter $\epsilon$, are included.

The potential missing contributions in the SM could be from long-distance (LD) final state interactions (FSIs). However, their effects have not yet been concluded, where the LD effects obtained in [13,34] cannot compensate for the insufficient Re(ǫ ′ /ǫ) SM , but the authors in [15] obtain Re(ǫ ′ /ǫ) SM = (15 ± 7) × 10 −4 when the short-distance (SD) and LD effects are combined.
We consider the charged-Higgs contributions to CMOs based on the following characteristics: (a) gluon-penguin effect can be enhanced by the top-quark mass m t , which arises from the top-quark Yukawa coupling and mass insertion in the propagator; (b) a large tan β enhancement from Yukawa couplings could occur in the gluon-penguin diagram, and (c) the CP violation phase can uniquely originate from the Kobaysahi-Maskawa (KM) phase of the Cabibbo-Kobayashi-Maskawa (CKM) matrix [37,38], where the same KM phase can be used to explain the Kaon indirect CP violation ǫ and CP asymmetries observed in the B-meson system. A charged-Higgs naturally exits in a two-Higgs-doublet model (2HDM), and its Yukawa couplings strongly depend on how the Higgs doublets couple to fermions. Before discussing a specific scenario for the charged-Higgs Yukawa couplings in the 2HDM, model-independent Yukawa couplings can be generally written as: where q ′ = d, s; v = v 2 1 + v 2 2 ≈ 246 GeV, and v 1(2) is the vacuum expectation value (VEV) of neutral Higgs field H 1 (H 2 ) in doublets; the top-quark related couplings are shown due to the m t enhancement and CP phase associated with V td , and C L(R) tq ′ denote the dimensionless couplings to the left(right)-handed down type quarks. As a result, the effective Hamiltonian for d → sg can be expressed as: where the Feynman diagram is sketched in Fig. 1, and the dimension-6 gluonic dipole oper- 8G are defined as: with g s being the strong gauge coupling constant and σ · G = σ µν T a G a µν . The dimesionless Wilson coefficients at the µ H ≡ µ H ± scale are obtained as: Since the electromagnetic dipole contributions are much smaller than chromomagnetic dipole, we ignore their contributions.
To estimate the hadronic matrix element for K 0 → ππ via the operators Q (′) 8G , we take the results obtained by a dual QCD approach as [34]: is the effective Wilson coefficient with mass dimension (−1) at the µ scale; f K(π) is the K(π)-meson decay constant, and Λ χ can be determined by: with f K /f π ≈ 1.193. Thus, the Kaon direct CP violation contributed by CMOs can be simply estimated as: where ω = ReA 2 /ReA 0 ≈ 1/22.46 denotes the ∆I = 1/2 rule, and A 0(2) denotes the K 0 → π + π − decay amplitude in the isospin I = 0(2) final state. With |ǫ| ≈ 2.228 ×10 −3 [39] and ReA 0 ≈ 33.22 × 10 −8 GeV, Eq. (10) can be expressed as: After formulating the necessary part for the calculation of Re(ǫ ′ /ǫ) 8G , in the following, in the 2HDM. In the literature, to avoid flavor-changing neutral currents (FCNCs) at the tree level, usually global symmetry is typically imposed in the 2HDM.
According to the different symmetry transformations, the 2HDM can be classified as type-I, type-II, lepton-specific, and flipped models, for which a detailed discussion can be found in [40]. The dimensionless parameters C L tq ′ and C R tq ′ in these models can be found as in [40]: with tan β = v 2 /v 1 . As a result, the direct CP violation can be estimated as: where we have used m s (m c ) ≈ 0.109 GeV, m d (m c ) ≈ 5.44 MeV [9], and m H ± = 300 GeV to estimate the numerical values. From the analysis, it can be clearly seen that the contributions of CMOs in the type-II and flipped models are independent of tan β, and the magnitude is two orders of magnitude smaller than the data. The situation in type-I and lepton-specific models is worse, where even the sign is opposite. It is obvious that it is necessary to look for another scenario to enhance Re(ǫ ′ /ǫ) in the 2HDM.
The most feasible scenario is the use of the generic 2HDM without imposing extra global symmetry, i.e. the type-III 2HDM, where the model can be successfully used to resolve anomalies that are indicated by the experiments, such as h → µτ , muon g − 2, R(D), and R(K ( * ) ) [41][42][43][44][45]. Although the type-III 2HDM has tree-level FCNCs, the flavorchanging effects, which involve light quarks, can be naturally suppressed when the Cheng-Sher ansatz [47] is applied. In order to understand the new characteristics of the charged-Higgs interactions, we write the H ± Yukawa couplings to the quarks in the type-III model as [41,42,45]: where the flavor indices are suppressed; V ≡ V u L V d † L stands for the CKM matrix, and X u,d are defined as: Here, V u,d L,R are the unitary matrices for diagonalizing the quark mass matrices, and Y u(d) 1 (2) is one of two Yukawa matrices, which consist of the up(down)-type quark mass matrix. When Y u(d) 1 (2) vanishes, the Yukawa couplings in the type-II model will be recovered. Thus, the new effects are from X u and X d , and they indeed dictate the tree-level FCNC effects. Using the Cheng-Sher ansatz, we can parametrize the X u,d as: In terms of above parametrizations, the new parameters χ u ij and χ d ij in general can be of O(1), and their magnitudes can be determined or constrained by the experimental data.
It is of interest to understand why the X u,d effects can play an important role in flavor physics. From Eqs. (15) and (17), the vertices of t R q ′ L H + can be expressed as: Due to m u /m t V uq ′ < m c /m t V cq ′ , we can simplify above equation as: Using |V ts(td) | ≈ 0.041(0.0088) and |V cs(cd) | ≈ 1(0.225), m c /m t V cq ′ /V tq ′ can be estimated to be ∼ 2.27 for q ′ = d and ∼ 2.14 for q ′ = s; that is, the second term can have an important effect. In addition, for a large tan β scheme, the vertex of t R q ′ R H ± is insensitive to tan β. Similarly, the vertex of t L q ′ H + can be simplified as: where we only retain the term with maximal CKM matrix element V tb ≈ 1. Intriguingly, the vertex of t L q ′ R H + does not have the V tq ′ suppression factor and its dependence on m q ′ is smeared by the square-root of m q ′ . Unlike the case of t R q ′ L H + , the t L q ′ R H + coupling is sensitive to tan β.
From Eqs. (15) and (17), we can write the relevant charged-Higgs couplings as: where the parameters ζ f tq ′ can be of O(1) and are defined as: According to the expressions in Eqs. (5) and (11), the Wilson coefficients for CMOs at the µ H scale in the type-III model can be written as: If the source of CP violation in the type-III 2HDM still originates from the KM phase, we find that the imaginary part of the effective Wilson coefficient C − 8G (µ H ) is only related to m s y H ± 8G and has a simple form as follows: In addition to the m H ± and tan β parameters, the new parameters for Re(ǫ ′ /ǫ) 8G involved in the type-III 2HDM are only χ d bs and χ u tt , where |χ d bs | could be of O(10 −2 ) due to the constraints from flavor physics, and |χ u tt | could be of O(1) [45,46]. In addition, because of the χ u tt term, the tan β enhancement factor from ζ d ts is retained, so the Re(ǫ ′ /ǫ) 8G can be significantly enhanced in a large tan β scheme. Note that although Eq. (24) is shown at the µ H scale, in our numerical analysis, we take the value at the µ = m c scale using the renormalization group (RG) running.
Before discussing the numerical analysis, we discuss the parameters that are involved as well as the theoretical and experimental inputs. In addition to Re(ǫ ′ /ǫ) 8G , the charged-Higgs has significant contributions on the low energy flavor physics, such as the ∆B q ′ = 2, ∆K = 2, B → X s γ processes, and the indirect CP violation ǫ [45,46]. Although the Re(ǫ ′ /ǫ) 8G related parameters are only tan β, m H ± , χ u t , and χ d bs , since these parameters and the other parameters are correlated in the mentioned flavor physics, we have to consider all parameters together when the strictly experimental constraints are taken into account.
Therefore, the involved parameters and their taken ranges are shown as: where we choose χ u tt and χ u ct to be opposite in sign because the resulted Re(ǫ ′ /ǫ) from the gluon and electroweak penguins can reach O(10 −4 ) [46]. In addition to the charged-Higgs effects, the neutral scalars H and A in the type-III model can also contribute to B q ′ mixing via the tree-level FCNCs. To include their effects, we fix m H = m A = 600 GeV.
The experimental inputs used to bound the free parameters are taken as [39]: Since ǫ in the SM fits well with the experimental data [48], for new physics effects, we thus use [19]: For the ∆K = 2 process, we take a combination of the short-distance (SD) and long-distance (LD) effects in the SM as ∆M SM K (SD+LD) = (0.80±0.10)∆M exp K [49]; therefore, the allowed new physics contribution to ∆M K should satisfy: The values of the CKM matrix elements are taken as: 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 [9]. The particle masses used to estimate the numerical values are given as: To consider the constraints from ∆M B ′ q , B → X s γ, ∆M K , and ǫ, we employ the results obtained in [45,46]. Using the experimental and theoretical inputs mentioned earlier, the bounds on χ u tt -χ u ct and χ u tt -χ d bs are shown in Fig. 2(a) and (b), where we use 5 × 10 6 sampling points to scan the parameters. From the results, we see that |χ u tt | 0.8 and |χ d bs | 0.06 are allowed. Using the constrained parameters, we can estimate the Kaon direct CP violation derived from the gluon and electroweak penguin operators. According to the result obtained in [46], the Re(ǫ ′ /ǫ) 4F via the penguin four-fermion operators as a function of m H ± is given in Fig. 3, where the electroweak penguin operator Q 8 dominates. From the result, it can be seen that the penguin four-fermion operator contribution becomes Re(ǫ ′ /ǫ) 4F < 10 −4 when m H ± 250 GeV.
In the following, we discuss the CMO contribution to Re(ǫ ′ /ǫ). Since only y H ± 8G (µ H ) contributes to the Kaon direct CP violation in this model, the associated Wilson coefficient at the µ = m c scale can be obtained as: where C 2 (m W ) ≈ 1 is the Wilson coefficient of the Q 2 = (sc) V −A (cd) V −A operator; for the new physics effects, we only use the leading-order QCD anomalous-dimension matrix for the operators Q 1−6 , O 7γ , and Q 8G [48], and µ H = 300 GeV is fixed. Using Eqs. (12) and (24), we show Re(ǫ ′ /ǫ) 8G in units of 10 −4 as a function of χ d bs -χ u tt in Fig. 4(a), where tan β = 30 and m H ± = 300 GeV are fixed. To obtain a positive Re(ǫ ′ /ǫ) 8G , χ d bs and χ u tt have to be the same sign. We show the correlations of Re(ǫ ′ /ǫ) 8G to χ u tt -tan β and χ d bs -tan β in plots Fig. 4(b) and (c), respectively, where we only show the positive χ u tt and χ d bs because the results for the case of χ u(d) tt(bs) < 0 are the same. The correlation of Re(ǫ ′ /ǫ) 8G to m H ± and tan β is shown in Fig. 4(d). From these results, it can be clearly seen that with the constrained parameter In summary, we studied the Kaon direction CP violation Re(ǫ ′ /ǫ), which arises from the charged-Higgs-induced chromomagnetic operator. If we assumed that the Kobayashi-Maskawa phase is the unique origin of CP violation, it was found that in addition to the tan β parameter, the new parameters χ u tt of O(1) and χ d bs of O(10 −2 ) in the type-III 2HDM play an important role in the contribution of the gluon dipole, where the former is a flavor-conserving coupling, and the latter is directly associated with the tree-level FCNCs. Although the contributions of the charged-Higgs-induced gluon and electroweak penguin operators to Re(ǫ ′ /ǫ) can be ∼ 8 ×10 −4 when m H ± ∼ 200 GeV, it was found that the charged-Higgs-induced chromomagnetic operator can explain the observed Re(ǫ ′ /ǫ) with wider parameter spaces, even in the case of m H ± > 250 GeV.