Semi-aligned two Higgs doublet model

In the left-right symmetric model based on $SU(2)_L\times SU(2)_R\times U(1)_{B-L}$ gauge symmetry, there appear heavy neutral scalar particles mediating quark flavor changing neutral currents (FCNCs) at tree level. We consider a situation where such FCNCs give the only sign of the left-right model while $W_R$ gauge boson is decoupled, and name it"semi-aligned two Higgs doublet model"because the model resembles a two Higgs doublet model with mildly-aligned Yukawa couplings to quarks. We predict a correlation among processes induced by quark FCNCs in the model, and argue that future precise calculation of meson-antimeson mixings and CP violation therein may hint at the semi-aligned two Higgs doublet model and the left-right model behind it.


I. INTRODUCTION
The left-right symmetric model [1] based on SU (2) L ×SU (2) R ×U (1) B−L gauge symmetry and the left-right parity (symmetry under the Lorentzian parity transformation accompanied by the exchange of SU (2) L and SU (2) R ) is a well-motivated extension of the Standard Model (SM). In the model, the chiral nature of the SM is beautifully attributed to spontaneous breaking of SU (2) R × U (1) B−L gauge symmetry. More importantly, the model offers a new look into the strong CP problem; since the left-right parity demands the Yukawa couplings to be Hermitian and forbids gluon θ term at tree level, if one could find a symmetry-based reason that the two vacuum expectation values (VEVs) of the SU (2) L × SU (2) R bi-fundamental scalar are both made real, the strong CP problem would be solved. Put another way, the original strong CP problem, which is about a miraculous cancellation between the θ term in QCD and the quark mass phases in the theory of electroweak symmetry breaking, is simplified to the issue of why the scalar potential does not break CP spontaneously. Thus, the strong CP problem becomes more tractable, although the left-right model by itself does not solve it.
The first experimental hint of the left-right model would probably come in the form of quark flavor changing neutral currents (FCNCs) mediated by heavy neutral scalar particles at tree level, because the SU (2) L × SU (2) R bi-fundamental scalar necessarily has two unaligned Yukawa couplings both contributing to up and down-type quark masses to accommodate the Cabibbo-Kobayashi-Maskawa (CKM) matrix, and the resultant quark FCNCs and CP violation are efficiently searched for through meson-antimeson mixings [2][3][4][5]. On the other hand, recent studies have centered on the possibility of a direct measurement of W R gauge boson at the LHC, by elaborating a scalar potential where SU (2) R × U (1) B−L breaking VEV v R is below ∼5 TeV (hence W R mass being several TeV), while the heavy scalar particles have masses above ∼20 TeV to evade the bounds on FCNCs [2,3]. Such a scalar potential contains O(1) quartic couplings that quickly become non-perturbative along renormalization group (RG) evolutions [6,7], and even if this is circumvented, there is no theoretical reason that favors having specially heavy scalar particles. Conversely, if W R gauge boson has a mass similar to or larger than the heavy scalar particles (e.g. (W R mass) (heavy scalar mass)= 100 TeV, so that the model evades the constraints from meson-antimeson mixings), then W R gauge boson leaves no direct or indirect experimental signature and it is the heavy scalar particles that allow us to probe the left-right model through FCNCs they mediate 1 .
In this paper, therefore, we pose the following question: If quark FCNCs mediated by the heavy neutral scalars give the only sign of the left-right model, can we test the model?
To answer this, we extract the bi-fundamental scalar part of the model, assuming that W R gauge boson is decoupled from phenomenology, and systematically study its indirect signatures and their correlation. We coin the term "semi-aligned two Higgs doublet model (semi-aligned 2HDM)" to describe the bi-fundamental scalar part, in light of the fact that the bi-fundamental scalar contains two SU (2) L doublet scalars and their quark Yukawa couplings are mildly aligned reflecting the smallness of CKM mixing angles.
Our analysis starts by realizing that the flavor and CP-violating couplings of the heavy scalar particles are uniquely determined as follows: Since the search for neutron electric dipole moment (EDM) has put a severe bound on the strong CP phase, we may concentrate on the limit with a vanishing spontaneous CP phase for the bi-fundamental scalar VEVs [9] 2 . In this limit, the quark mass matrices are Hermitian and the mixing matrix for righthanded quarks is identical with the CKM matrix, which allows one to express the couplings of the heavy scalar particles to quarks in terms of the SM quark masses and CKM matrix, without free parameters. The amplitudes for various FCNC processes are then calculated as functions of just one free parameter, that is, the nearly degenerate mass of the heavy scalar particles, and we thus predict a correlation among various FCNC processes. We will show that indirect CP violation in kaon system, Re , the B 0 d mass splitting, ∆M B d , and the B 0 s mass splitting, ∆M Bs , are the most sensitive probes for the semi-aligned 2HDM, and if uncertainties in their calculation are reduced and the prediction including the contributions of heavy scalar particles converges to the experimental values for some unique value of the heavy scalar mass, it is evidence for the semi-aligned 2HDM and the left-right model behind it.
A novelty of our study compared to previous works [2,3] is that we concentrate on the limit with decoupled W R gauge boson, which enables us to compute all amplitudes with only one free parameter and investigate their correlation. Also, we pay attention to the 1 For a collider study of the heavy scalar particles in the left-right model, see Ref. [8]. 2 In this paper, we focus on phenomenological consequences of a vanishing spontaneous CP phase and do not discuss its theoretical origin. For attempts to derive the vanishing spontaneous CP phase in the framework of the left-right model, see Ref. [10].
fact that new physics contributions can distort the determination of the CKM matrix. To avoid this, when we derive the current bound on the semi-aligned 2HDM, we refit the CKM matrix, attempting to fit the experimental data with SM+new physics contributions and using a tension in the fitting to constrain the model. When we make a prediction for FCNC processes, we assume that the CKM matrix is determined beforehand in a way unaffected by new physics contributions. This paper is organized as follows: In Section II, we describe the semi-aligned 2HDM induced from the left-right model, and calculate the mass and couplings of the heavy scalar particles. In Section III, we review the procedure for computing amplitudes for ∆F = 2 processes. In Section IV, we derive the current bound on the heavy neutral scalar mass.
Section V presents our main results, which are Re , ∆M B d and ∆M Bs expressed in terms of one parameter. Section VI summarizes the paper.

II. MODEL
We start from the left-right symmetric model based on SU (3) C × SU (2) L × SU (2) R × U (1) B−L gauge symmetry [1] and the left-right parity, whose full expression is in Appendix A.
The VEV of the SU (2) R triplet scalar, v R , breaks SU (2) R ×U (1) B−L into hypercharge U (1) Y and also breaks the left-right parity. In this paper, we focus on a limit where v R is much larger than the electroweak scale, and at the same time, the quartic couplings between two SU (2) R triplets and two bi-fundamentals are much smaller than 1, namely, where α 1 , α 2R , α 2I , α 3 are defined in the Lagrangian of Appendix A. In the limit with Eq. (1), the low-energy theory at scales below v R (we call it the semi-aligned 2HDM) contains the SM fermions+three right-handed neutrinos+the bi-fundamental scalar and pos- However, the Yukawa couplings and self-couplings of the bi-fundamental scalar respect global SU (2) R symmetry and the left-right parity, which are remnants of the left-right model, and hence have highly constrained structures. In Table I, we summarize the fields in the semi-aligned 2HDM and their Here, the SM right-handed fermions form doublets of global SU (2) R symmetry, q R and R . The bifundamental scalar Φ is expressed as a 2 × 2 matrix transforming under a SU (2) L × SU (2) R gauge transformation as which is then decomposed into two SU (2) L doublet scalars with hypercharge Y = ±1/2, H u and H d , as Φ = (iσ 2 H * u , iσ 2 H * d ). The Lagrangian of the semi-aligned 2HDM is given by where m 2 1 , m 2 2R , m 2 2I , m 2 3 , λ 1 , λ 2 , λ 3 , λ 4 are all real, and the Yukawa coupling matrices are Hermitian, Notice that m 2 2I softly breaks the left-right parity, and m 2 3 softly breaks SU (2) R symmetry. Both result from the spontaneous left-right symmetry breaking and are proportional to v 2 R .
Φ develops a VEV to break SU (2) L × U (1) Y symmetry. Through a SU (2) L plus σ 3 part of SU (2) R symmetry transformation 3 , the VEV is made into the following form, with one VEV having a CP phase α: The mass matrices for up-type quarks, M u , down-type quarks, M d , charged leptons, M e , are given by The spontaneous CP phase α has already been severely constrained by the search for neutron EDM. Since W R is decoupled and the coupling of scalar particles to up and down quarks is Yukawa-suppressed, perturbative corrections to neutron EDM are negligible, and the experimental bound is directly translated into a bound on arg det(M u M d ) and hence on α. In the limit of neglecting the quark flavor mixing (but no assumptions are made on β or α), we obtain, from Eqs. (7,8), the following formula: The current experimental bound [11] roughly gives where it should be reminded that the QCD θ term is prohibited at tree level by the left-right parity. α is thus constrained to be much below 1, and based on this fact, we fix α = 0 in the rest of the paper 4 .
The physical scalar particles after the electroweak symmetry breaking are 5 a charged scalar, H ± , a CP-odd scalar, A, a lighter CP-even scalar which we identify with the SM Higgs particle, h, and a heavier CP-even scalar, H. The H ± and A masses read The H and h masses expanded to the order of O(v 2 /|m 2 3 |) are found to be The bi-fundamental scalar Φ = (iσ 2 H * u , iσ 2 H * d ) can be decomposed into the physical scalar particles H ± , A, H, h and Nambu-Goldstone bosons, G ± , G 0 , in the following way: where γ is the mixing angle of the CP-even scalars satisfying Since the A and H masses are experimentally constrained to be above ∼10 TeV, we work in h , in which case the masses of H ± , A, H and the CP-even scalar mixing angle γ satisfy 5 Since α = 0, CP is not broken spontaneously and the CP-even and odd scalar particles can be defined.

Equation (76) induces Yukawa couplings for quarks and H ± , H and A, given by
where Eq. (19) has been used. We diagonalize the quark mass matrices by the rotation, to obtain where m D u and m D d are diagonalized mass matrices for up and down-type quarks, respectively, and s f = ±1(f = u, c, t, d, s, b), which reflects sign uncertainty of the mass eigenvalues. Note that the left and right-handed quarks are rotated with the same unitary matrix, since α = 0 and the mass matrices are Hermitian. Accordingly, the Yukawa couplings become where we have defined the CKM matrix, V , as We find that the following part in Eq. (25) induces FCNCs at tree level by the exchange of heavy neutral scalars H, A, with the strength controlled by the CKM matrix multiplied by quark masses: Flavor violation is suppressed by off-diagonal components of the CKM matrix, and possibly by light quark masses, which is a characteristic property of the model.
Since the Yukawa couplings always appear in combination with the factor 1/ cos 2β, hereafter we redefine the heavy scalar masses as 6 As a reference, we present the absolute values of the flavor-violating part of the Yukawa For down − type : The off-diagonal components in the above matrices indicate the strength of FCNCs mediated by the neutral scalars. Here, the CKM matrix components are obtained from the Wolfenstein parameters reported by the CKMfitter, which read [12], We comment in passing that the Yukawa couplings for leptons and H ± , H and A are obtained by a simple replacement: M D being the neutrino Dirac mass involved in the seesaw mechanism. Due to our ignorance of the seesaw scale, we cannot predict the strength of the flavor-violating couplings for charged leptons.

III. ∆F = 2 AMPLITUDES
We give formulae for ∆F = 2 amplitudes to analyze flavor observables in the quark sector. In particular, mass differences for K, B 0 d and B 0 s , and a CP violating observable in kaon system are given. The effective Hamiltonian contributing to ∆S = 2 processes is given as follows, particles are degenerate, one can obtain a further simplified Hamiltonian. Including the SM contribution, we can write, where we follow the notation of Ref. [13]. In Eq. (37), we do not include operators, Q SLL 1 , Q SLL 2 and other chirality-flipped ones given in Ref. [13], since they do not arise from the FCNH diagram. Furthermore, Q SLL 1 and Q SLL 2 are decoupled from the mixing with the other operators so that we omit these contributions. The Wilson coefficients and the operators in Eqs. (36, 37) are given as, where P R(L) = (1±γ 5 )/2 denotes chirality projection operators while α and β represent color indices. In Eq. (38), C VLL 1 stands for the contribution within the SM, and the Inami-Lim function [14] is given as, where NLO QCD correction factors within the SM, (η 1 , η 2 , η 3 ), have been calculated in Ref. [15]. To be precise, one should multiply Eq. (40) by an overall factor which accounts renormalization scale of lattice QCD calculation.
As for the ∆B = 2 processes, we only take account of the contribution of internal top quarks, and the corresponding NLO QCD correction is obtained through the method in Ref. [13]. The formulae for an anomalous dimension matrix including two-loop contribution are given in Ref. [16]. As remarked in the literature [13], this renormalization group effect drastically enhances C LR 2 while it does not significantly change C LR 1 . In our analysis, new world averages of the QCD scale obtained by PDG [17] are used.
The matrix elements of the ∆F = 2 transition are parametrized as, where q = d, s. In this normalization, kaon decay constant is given as f K = 156.1 MeV.
The matrix elements in Eqs. (43-48) are written in terms of bag parameters, which represent the deviation from vacuum saturation approximation. For these parameter, we use the data which are calculated by the ETM collaboration [18,19]. Their results are obtained in MS scheme, and extracted from the result in the supersymmetric basis. The correspondence between bag parameters in the operator basis in Eq. (39) and ones in the supersymmetric basis is given in Ref. [13].
The mass differences of neutral meson system are obtained as follows, where M (q) 12 is divided into the SM part and the new physics part, Moreover, indirect CP violation in kaon system is characterized by

IV. CURRENT BOUND ON THE MODEL
In this section, we obtain the bound on mass of heavy Higgs through flavor observables.
First, sin 2β eff , which represents CP violation of interference in B 0 d −B 0 d mixing and B 0 d → J/ψK S , is analyzed. As discussed later, the bounds on Higgs mass are determined by p-values of CKM fitting.
In Table II, input data which appear in the numerical analysis are summarized.
A. sin 2β eff measured in B 0 d → J/ψK S decay Throughout this paper, we assume that direct CP violation in B 0 d → J/ψK S decay is negligible. Within this approximation, the time-dependent CP asymmetry in B 0 d → J/ψK S decay is given as follows, where q/p is a mixing parameter [21] in B 0 In the semi-aligned 2HDM, the decay amplitude in Eq. (55) does not deviate from the SM prediction, since a diagram of the charged scalar exchange gives rise to minor modification due to smallness of Yukawa couplings. We do not take account of such negligible contribution for the decay amplitude. Furthermore, the correction coming from penguin pollution in the SM is also small [22] because of the suppression for the CKM and the loop factor, and hence we ignore this effect. Meanwhile, the mixing parameter for B 0 d system, q/p, and one for kaon system are modified due to the diagrams for FCNH exchange. Hence, the parameter in Eq. (55) is given as [23,24], sin 2β eff = S J/ψK S = sin 2β + arg 1 + where the definitions of M (d) SM 12 and M (d) FCNC 12 are given in Eqs. (51, 52). Thus, the experimental observable deviates from sin 2β due to modification of the mixing parameters.
In Figure IV.1, in order to illustrate the FCNH mass dependence of sin 2β eff , we plot, where for the CKM matrix components, we used the values in Eq.
This factor is the same as the box diagram in the SM up to its sign. Therefore,  For completeness, we show the Wolfenstein parametrization of the CKM matrix [25], where terms of O(λ 4 ) are ignored. In Eq. (60), (ρ, η) are redefined in terms of phase convention independent parameters, (ρ,η) [26], In addition, angles which constitute the unitarity triangle are defined as, where the other angle, β, is given in Eq. (57). We note that (α, β, γ) can be written in terms of the Wolfenstein parameters, (λ, A,ρ,η).
In carrying out the analysis, the following facts are considered: • Measurements of |V ud |, |V us |, |V ub |, |V cd |, |V cs | and |V cb | are unchanged in the presence of FCNH. This is because these absolute values are determined through semi-leptonic decays, while the model predicts minor correction for semi-leptonic processes.
• Angle α, which is measured in time-dependent processes for B → ππ, B → πρ and • Measurement of angle γ is carried out in B ± → DK ± and B 0 → D ( * )± π ∓ decays.
These tree-level processes are not significantly modified by the FCNH exchange.
In the fitting, we include uncertainties of the Wolfenstein parameters [12], the bag parameters [18,19] and NLO QCD correction factors for the ∆S = 2 process denoted by η 1 , η 2 and η 3 [15]. As for the ∆B = 2 processes, uncertainty in short-distance QCD correction factors is not considered because they are more precisely determined than those for the ∆S = 2 process. We use the central value of this QCD correction calculated through the method of Ref. [13].
In the CKM fitting, a statistic, In this section, we present the prediction for observables to illustrate the pattern of deviation in the model. For this purpose, the Wolfenstein parameters are estimated from observables which are not affected by FCNH. Hence, we consider the following statistic, Through the minimization of Eq. (67), one can extract (λ, A,ρ,η).
The Belle II experiment announces that the expected integrated luminosity is 50 ab −1 in five years of running. Motivated by this, we consider Case I and Case II described below.
• Case I: The errors of |V ub | and γ are reduced by 1/3 and 1/7, respectively, without changing where a correlation matrix for (λ, A,ρ,η) is  In the following, we comment on other flavor-violating observables. Since a charged scalar exchange alters b → cl − ν decay rate at tree level, this model might be able to address the anomaly in R D ( * ) . However, this is not the case because the absolute value of the bcH − coupling is small up to √ , which does not exceed the SM bcW − coupling, |V cb | ∼ O(10 −2 ).
We examine the correction to b → ssd decay. In the SM, this proceeds via the box diagram and is highly suppressed by the Glashow-Iliopoulos-Maiani mechanism as Br SM [b → ssd] = O(10 −12 ) [28,29]. For the corresponding exclusive mode, experimental searches have been performed [30] and recently, the LHCb collaboration has reported Br[B − → K − K − π + ] < 1.1 × 10 −8 (90% CL) [31]. In Ref. [32], the FCNH contribution to the decay width is calculated in the Type-III 2HDM and is found to be Using the above formula, we verify that the correction to b → ssd decay in the semi-aligned    and SU (2) R gauge groups are interchanged and the fields transform as We write the bi-fundamental scalar Φ and triplet scalars ∆ L , ∆ R as 2 × 2 matrices that transform under a SU (2) L × SU (2) R gauge transformation as Φ → e iτ a θ a L Φe −iτ a θ a R , ∆ L → e iτ a θ a L ∆ L e −iτ a θ a L , ∆ R → e iτ a θ a R ∆ R e −iτ a θ a R , θ a L , θ a R : gauge parameters, τ a ≡ σ a /2.
Through a SU (2) R × U (1) B−L symmetry transformation, one can set the VEV of ∆ R in the following form: Through a subsequent SU (2) L and σ 3 part of SU (2) R symmetry transformation, one can which gives rise to a phase for the ∆ R VEV, but this can be negated by a U (1) B−L symmetry transformation.