Advances in High Energy Physics

Volume 2017, Article ID 2863647, 15 pages

https://doi.org/10.1155/2017/2863647

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

Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOE), Central China Normal University, Wuhan, Hubei 430079, China

Correspondence should be addressed to Fang Su; nc.ca.pti@gnafus

Received 14 May 2017; Accepted 10 July 2017; Published 29 August 2017

Academic Editor: Alexey A. Petrov

Copyright © 2017 Natthawin Cho et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

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

#### 1. Introduction

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

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

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

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

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

#### 2. Theoretical Framework

##### 2.1. Yukawa Sector

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

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

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

##### 2.2. Mixing

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