1 Introduction

# Vacuum Induced CP Violation Generating a Complex CKM Matrix with Controlled Scalar FCNC

## Abstract

We propose a viable minimal model with spontaneous CP violation in the framework of a Two Higgs Doublet Model. The model is based on a generalised Branco-Grimus-Lavoura model with a flavoured symmetry, under which two of the quark families are even and the third one is odd. The lagrangian respects CP invariance, but the vacuum has a CP violating phase, which is able to generate a complex CKM matrix, with the rephasing invariant strength of CP violation compatible with experiment. The question of scalar mediated flavour changing neutral couplings is carefully studied. In particular we point out a deep connection between the generation of a complex CKM matrix from a vacuum phase and the appearance of scalar FCNC. The scalar sector is presented in detail, showing that the new scalars are necessarily lighter than 1 TeV. A complete analysis of the model including the most relevant constraints is performed, showing that it is viable and that it has definite implications for the observation of New Physics signals in, for example, flavour changing Higgs decays or the discovery of the new scalars at the LHC. We give special emphasis to processes like , as well as , which are relevant for the LHC and the ILC.

CFTP/18-012

IFIC/18-nnn

Vacuum Induced CP Violation Generating a Complex CKM Matrix with Controlled Scalar FCNC

Miguel Nebot 1 Francisco J. Botella 2, Gustavo C. Branco 3,

Departamento de Física and Centro de Física Teórica de Partículas (CFTP),

Instituto Superior Técnico (IST), U. de Lisboa (UL),

Av. Rovisco Pais 1, P-1049-001 Lisboa, Portugal.

Departament de Fìsica Teòrica and Instituto de Física Corpuscular (IFIC),

Universitat de València – CSIC, E-46100 Valencia, Spain.

## 1 Introduction

The first model of spontaneous T and CP violation was proposed [1] by T.D. Lee in 1973 at a time when only two incomplete quark generations were known. The main motivation for Lee’s seminal work was to put the breaking of CP and T on the same footing as the breaking of gauge symmetry. In Lee’s model, the Lagrangian is CP and T invariant, but the vacuum violates these discrete symmetries. This was achieved through the introduction of two Higgs doublets, with vacuum expectation values with a relative phase which violates T and CP invariance. In Lee’s model, CP violation would arise solely from Higgs exchange, since at the time only two generations were known and therefore the CKM matrix was real. The general two Higgs Doublet Model (2HDM) [2, 3] has Scalar Flavour Changing Neutral Couplings (SFCNC) at tree level which need to be controlled in order to conform to the stringent experimental constraints. This can be achieved by imposing Natural Flavour Conservation (NFC) in the scalar sector, as suggested by Glashow and Weinberg (GW) [4]. Alternatively, it was suggested by Branco, Grimus and Lavoura (BGL) [5] that one may have 2HDM with tree level SFCNC but with their flavour structure only dependent on the CKM matrix .

BGL models have been extensively analysed in the literature [6, 7, 8, 9, 10, 11], and their phenomenological consequences have been studied, in particular in the context of LHC. Recently BGL models have been generalised [12] in the framework of 2HDM. Both the GW and the BGL schemes can be implemented through the introduction of extra symmetries in the 2HDM. On the other hand, it has been shown [13] that the introduction of these symmetries in the 2HDM prevents the generation of either spontaneous or explicit CP violation in the scalar sector, unless they are softly broken [14]. It was recently discussed [15] that for a scalar potential with an extra symmetry beyond gauge symmetry, there is an intriguing correlation between the capability of the potential to generate explicit and spontaneous CP violation.

In this paper we propose a realistic model of spontaneous CP violation in the framework of 2HDM. At this stage it is worth recalling the obstacles which have to be surmounted by any model of spontaneous CP violation:

• The scalar potential should be able to generate spontaneous CP breaking by a phase of the vacuum, denoted .

• The phase should be able to generate a complex CKM matrix, with the strength of CP violation compatible with experiment. Recall that the CKM matrix has to be complex even in the presence of New Physics [16].

• SFCNC effects should be under control so that they do not violate experimental bounds.

The paper is organised as follows. In the next section we present the structure of the model and specify the flavoured symmetry introduced. In the third section we show how a complex CKM matrix is generated from the vacuum phase. Section 4 contains a detailed analysis of the scalar potential with real couplings. In section 5 we derive the physical Yukawa couplings and the phenomenological analysis of the model is presented in section 6. Finally we present our conclusions in the last section.

## 2 The Structure of the Model and the Flavoured symmetry

The Yukawa couplings in the 2HDM read

 LY=−¯Q0L(Γ1Φ1+Γ2Φ2)d0R−¯Q0L(Δ1~Φ1+Δ2~Φ2)u0R+H.c., (1)

with summation over generation indices understood and . We consider the following transformations to define the model:

 Φ1↦Φ1,Φ2↦−Φ2,Q0L3↦−Q0L3,Q0Lj↦Q0Lj,j=1,2, d0Rk↦d0Rk,u0Rk↦u0Rk,k=1,2,3. (2)

Invariance under eq. (2) gives the following form of the Yukawa coupling matrices:

 Γ1=⎛⎜⎝××××××000⎞⎟⎠, Γ2=⎛⎜⎝000000×××⎞⎟⎠, Δ1=⎛⎜⎝××××××000⎞⎟⎠, Δ2=⎛⎜⎝000000×××⎞⎟⎠. (3)

The symmetry assignment in eq. (2) and the Yukawa matrices in eq. (3) correspond to the generalised BGL models introduced in [12]. We impose CP invariance at the Lagrangian level, so we require the Yukawa couplings to be real:

 Γ∗j=Γj,Δ∗j=Δj. (4)

We write the scalar doublets in the “Higgs basis” [17, 18, 19] (see section 4 and appendix B for further details on the scalar sector)

 (H1H2)=Rβ(e−iθ1Φ1e−iθ2Φ2),withRβ=(−cβsβ−sβcβ), RTβ=R−1β. (5)

In this basis, only acquires a vacuum expectation value

 ⟨H1⟩=v√2(01),⟨H2⟩=(00). (6)

Equation (1) can then be rewritten as

 LY=−√2v¯Q0L(M0dH1+N0dH2)d0R−√2v¯Q0L(M0u~H1+N0u~H2)u0R+H.c., (7)

where the quark mass matrices , and the , matrices read

 M0d =veiθ1√2(cβΓ1+eiθsβΓ2), N0d =veiθ1√2(−sβΓ1+eiθcβΓ2), (8) M0u =ve−iθ1√2(cβΔ1+e−iθsβΔ2), N0u =ve−iθ1√2(−sβΔ1+e−iθcβΔ2), (9)

where is the relative phase among and . For simplicity, we remove the irrelevant global phases setting .
Notice that the matrices , can be written:

 N0d =tβM0d+eiθv√2(tβ+t−1β)sβΓ2=tβM0d−(tβ+t−1β)P3M0d, (10) N0u =tβM0u+e−iθv√2(tβ+t−1β)sβΔ2=tβM0u−(tβ+t−1β)P3M0u, (11)

where is the projector

 P3=⎛⎜⎝000000001⎞⎟⎠. (12)

## 3 Generation of a complex CKM matrix from the vacuum phase

In this section, we show how the vacuum phase is capable of generating a complex CKM matrix. As previously emphasized, this is a necessary requirement for the model to be consistent with experiment. Following eqs. (3)-(4) and (8)-(9), we write:

 M0d=⎛⎜⎝10001000eiθ⎞⎟⎠^M0d,M0u=⎛⎜⎝10001000e−iθ⎞⎟⎠^M0u, (13)

with and real. Then,

 M0dM0†d=⎛⎜⎝10001000eiθ⎞⎟⎠^M0d^M0Td⎛⎜⎝10001000e−iθ⎞⎟⎠ (14)

with real and symmetric, which is diagonalised with a real orthogonal transformation:

 OdTL^M0d^M0TdOdL=diag(m2di). (15)

Consequently, eq. (14) gives

 OdTL⎛⎜⎝10001000e−iθ⎞⎟⎠M0dM0†d⎛⎜⎝10001000eiθ⎞⎟⎠OdL=diag(m2di). (16)

That is, the diagonalisation of is accomplished with

 (17)

Similarly,

 (18)

Notice the important sign difference in between eqs. (17) and (18), which give the following CKM matrix ,

 V=OuTL⎛⎜⎝10001000ei2θ⎞⎟⎠OdL. (19)

Notice also that, if , is real, i.e. it does not generate CP violation. This can be understood through a careful analysis of the potential, which will be presented in section 4. The model we present here has spontaneous CP violation and thus a physical phase in the CKM matrix can only arise from . In section 4.1 we show that for the vacuum is CP invariant and no CP violation can be generated in this model. In particular CKM is necessarily real for this value of , as noticed in eq. (19).

It is also straightforward to observe that and are real and symmetric, and are thus diagonalised with real orthogonal matrices and ,

 OdTRM0†dM0dOdR=diag(m2di),OuTRM0†uM0uOuR=diag(m2ui), (20)

such that the bi-diagonalisation of and reads

 Md=diag(mdi)=Ud†LM0dOdR,Mu=diag(mui)=Uu†LM0uOuR. (21)

Following eq. (10),

 Nd≡Ud†LN0dOdR=tβUd†LM0dOdR−(tβ+t−1β)Ud†LP3M0dOdR=tβMd−(tβ+t−1β)Ud†LP3UdLMd, (22)

with the projector in eq. (12) and in eq. (17). In the last term of eq. (22),

 Ud†LP3UdL=OdTLP3OdL, (23)

that is, in eq. (22) is real. Introducing a real unit vector and a complex unit vector with components

 ^r[d]j≡[OdL]3j,^n[d]j≡[UdL]3j=eiθ^r[d]j, (24)

one has, for in eq. (23),

 [Ud†LP3UdL]ij=^n∗[d]i^n[d]j=^r[d]i^r[d]j. (25)

Similarly, for we have

 Nu≡Uu†LN0uOuR=tβMu−(tβ+t−1β)Uu†LP3UuLMu, (26)

with

 Uu†LP3UuL=OuTLP3OuL, (27)

and

 ^r[u]j≡[OuL]3j^n[u]j≡[UuL]3j=e−iθ^r[u]j,[Uu†LP3UuL]ij=^n∗[u]i^n[u]j=^r[u]i^r[u]j. (28)

Like , is real; and have the form:

 [Nd]ij =tβδijmdi−(tβ+t−1β)^n∗[d]i^n[d]jmdj, (29) ij =tβδijmui−(tβ+t−1β)^n∗[u]i^n[u]jmuj. (30)

Since , the complex unitary vectors and are not independent:

 ^n[d]i=^n[u]jVji,^n[u]i=V∗ij^n[d]j. (31)

It is interesting to notice that the 2HDM scenario studied in [20], where the soft breaking of a symmetry is the source of CP violation, shares some interesting properties with the present one: there, the CKM matrix can also be factorised in terms of real orthogonal rotations and a diagonal matrix containing the CP violating depence; the tree level SFCNC are also real in that phase convention. Other aspects of the model like the structure of the Yukawa couplings as well as the scalar sector to be discussed in section 4 are, however, completely different.
In the rest of this section, we analyse in detail the generation of a complex CKM matrix from the vacuum phase . The couplings of the physical scalars to the fermions are discussed in section 5, after the discussion of the scalar sector in 4.

It is clear that is necessary in order to have an irreducibly complex CKM matrix. However, one has to verify that one can indeed obtain a realistic CKM matrix, one that it is in agreement with the experimental constraints on the moduli (in particular of the moduli of the first and second rows), and on the CP violating phase (the only one accessible through tree level processes alone). Concerning CP violation, one can alternatively analyse that the unique (up to a sign) imaginary part of a rephasing invariant quartet (, ) has the correct size . Starting with eq. (19), one can compute that imaginary part. For the task, it is convenient to trade and for the real unit vector in eq. (24) and the real orthogonal matrix :

 ^r[d]j=[OdL]3j,R≡OuTLOdL. (32)

Then, one can rewrite

 V=OuTL[1+(ei2θ−1)P3]OdL⇒Vij=Rij+(ei2θ−1)Sij, (33)

and we introduce to allow for compact expressions:

 Sij≡[OuTLP3OdL]ij=^r[u]i^r[d]j=3∑k=1Rik^r[d]k^r[d]j=3∑k=1^r[u]i^r[u]kRkj. (34)

The real and imaginary parts of are4

 Re(Vij)=Rij−2s2θSij,Im(Vij)=s2θSij. (35)

Notice that, although eq. (35) is not rephasing invariant, this poses no problem when considering rephasing invariant quartets. With eq. (35), one can obtain:

 Im(Vi1j1V∗i1j2Vi2j2V∗i2j1)=sin2θ{4s2θ(Ri1j1Si2j1Ri2j2Si1j2−Si1j1Ri2j1Si2j2Ri1j2)+4s2θ(Si1j1Si2j1Si2j2Ri1j2−Si1j1Si2j1Ri2j2Si1j2+Si1j1Ri2j1Si2j2Si1j2−Ri1j1Si2j1Si2j2Si1j2)+Si1j1Ri2j1Ri2j2Ri1j2−Ri1j1Si2j1Ri2j2Ri1j2+Ri1j1Ri2j1Si2j2Ri1j2−Ri1j1Ri2j1Ri2j2Si1j2}. (36)

Although eq. (36) is not very illuminating, one can nevertheless illustrate that realistic values of can be obtained even in cases with less parametric freedom, as done in subsection 3.3 below. The general case is analysed in subsection 3.4. Before addressing those questions, we discuss two important aspects that deserve attention in the next two subsections: (i) the number of independent parameters and the most convenient choice for them, (ii) the fact that in this model, if tree level SFCNC were completely absent in one quark sector, then the CKM matrix would not be CP violating. One encounters again a deep connection [21] between the complexity of CKM and SFCNC, in the context of models with spontaneous CP violation.

### 3.1 Parameters

The CKM matrix requires 4 physical parameters, while the tree level SFCNC require 2, since is a unit real vector. One can parametrise in terms of two angles , :

 ^r[d]=(sinθdcosφd,sinθdsinφd,cosθd), (37)

as shown in Figure 15.

The orthogonal matrix requires 3 real parameters; together with and , these 6 parameters match the parameters necessary to describe (4 parameters) and the products , (2 parameters). However, in terms of and , there are a priori 3+3 real parameters; together with , 7 parameters in all. This apparent mismatch can be readily understood: a common redefinition

 OdL↦(cosαsinα0−sinαcosα0001)OdL,OuL↦(cosαsinα0−sinαcosα0001)OuL, (38)

leaves , , and unchanged, effectively removing one parameter from the , , parameter count. Consequently, it is convenient to adopt a parametrisation of of the form

 R=⎛⎜⎝1000cα3sα30−sα3cα3⎞⎟⎠⎛⎜⎝cα20sα2010−sα20cα2⎞⎟⎠⎛⎜⎝cα1sα10−sα1cα10001⎞⎟⎠=⎛⎜ ⎜⎝cα1cα2sα1cα2sα2−sα1cα3−cα1sα2sα3cα1cα3−sα1sα2sα3cα2sα3sα1sα3−cα1sα2cα3−cα1sα3−sα1sα2cα3cα2cα3⎞⎟ ⎟⎠, (39)

and a parametrisation of of the form6

 (40)

where is readily identified in the third row and the redundant , as in eq. (38), can be set to . One can then concentrate on in order to reproduce a realistic CKM matrix.

### 3.2 SFCNC and CP Violation in CKM

In eqs. (29)–(30), tree level SFCNC are a priori present in both the up and the down quark sectors and controlled by . Therefore, if has a vanishing component, SFCNC in that sector ( or ) do only appear in one type of transition (the one not involving that component). If had two vanishing components (then the remaining one equals ), there would not be SFCNC in that sector: interstingly, in this model, having no tree level SFCNC in one quark sector is incompatible with a CP violating CKM matrix. This can be readily checked by noticing that, in that case, in eq. (34), the matrix with entries has only a non vanishing row (column), corresponding to the absence of tree level SFCNC in the up (down) sector, for which . Then, with and , in eq. (36) all terms except two out of the last four automatically vanish, and those two terms appear with opposite sign, giving . As illustrated below, in subsection 3.4, this implies that a lower bound on the size of the second largest component in should exist, that is a lower bound on the intensity of some SFCNC in both quark sectors. Appendix A completes the discussion of the interplay in this model among flavour non-conservation and CP violation in the CKM matrix.

### 3.3 A simple example

As a simplified example of how a realistic CKM matrix can be obtained, consider a scenario with

 ^r[d]=(cosφd, sinφd, 0). (41)

Then, for , , eq. (36) reduces to

 Im(V11V∗12V22V∗21)=12(R11R21+R12R22)(R12R21−R11R22)sin2φdsin2θ. (42)

Since the rows of form a complete orthonormal set of 3-vectors,

 R11R21+R12R22=−R13R23,R12R21−R11R22=−R33, (43)

and eq. (45) is further reduced to

 Im(V11V∗12V22V∗21)=12R13R23R33sin2φdsin2θ. (44)

With in eq. (39), eq. (42) gives

 Im(V11V∗12V22V∗21)=18cosα2sin2α2sin2α3sin2φdsin2θ. (45)

A complete example of this type which reproduces correctly the CKM matrix, is given by:

 θ=π/8, (46)
 R=⎛⎜⎝1−7×10−60−3.746×10−3−1.536×10−4−1+8.41×10−4−0.041−3.743×10−30.041−1+8.48×10−4⎞⎟⎠, (47)

and

 OdL=⎛⎜⎝00−10.95090.309600.3096−0.95090⎞⎟⎠. (48)

The parameters underlying eqs. (47)–(48) have the values

 R: α1=0, α2=−3.746×10−3, α3=0.041−π, OdL: θd=π/2, φd=−1.2561, α=0. (49)

For , has been chosen for simplicity, since in this scenario it does not enter eq. (45). With the previous values, one can easily check that

 |Vus|=0.2253, |Vub|=3.75×10−3, |Vcb|=0.041, % Im(V11V∗12V22V∗21)=3.195×10−5. (50)

Concerning the discussion in subsection 3.2, this example shows that, although the complete absence of tree level FCNC in one sector is incompatible with a CP violating CKM matrix, this incompatibility does not extend to the case of tree level SFCNC circumscribed to only one type of transition (in this example, transitions).

### 3.4 General case

The previous example illustrates that the CKM matrix can be adequately reproduced even in a restricted scenario where one has less number of free parameters. For the general case one can explore with a simple numerical analysis the regions of parameter space where the CKM matrix is in agreement with data, that is moduli in the first two rows and the phase agree with experimental results [22].
Figure 2(a) shows the region of the plane vs. which can yield a good CKM matrix. It is to be noticed that (i) regions rather close to , with , are allowed and require , while (ii) for , allowed regions require . In any case, is a necessary requirement, as expected, since there is no CP violation in that limit.
Figures 2(d)2(f) also show the allowed regions in the vs. plane7, separating in 2(d), in 2(e), and in 2(f).
Following the discussion in subsection 3.2, one can sort the components of according to their size:

 |^r[d]Min|≤|^r[d]Mid|≤|^r[d]Max|. (51)

Incompatibility with a CP violating CKM matrix implies that , while the simple example in subsection 3.3 shows that is allowed. Figure 2(b) shows the allowed region of vs. confirming that it is necessary that , while figure 2(c) corresponds similarly to : both figures illustrate that, in this model, necessarily, there is at least some minimal presence of tree SFCNC8.

In conclusion, it is clear that the first requirement on the model, i.e. that it can reproduce the observed CKM matrix, can be fulfilled.

## 4 The scalar potential with real couplings

We consider the 2HDM with CP invariance and impose the symmetry of eq. (2) which is only softly broken by a term. All couplings are real, so that CP holds at the Lagrangian level. The scalar potential can be written:

 V(Φ1,Φ2)=μ211Φ†1Φ1+μ222Φ†2Φ2+μ212(Φ†1Φ2+Φ†2Φ1)+λ1(Φ†1Φ1)2+λ2(Φ†2Φ2)2+2λ3(Φ†1Φ1)(Φ†2Φ2)+2λ4(Φ†1Φ2)(Φ†2Φ1)+λ5[(Φ†1Φ2)2+(Φ†2Φ1)2]. (52)

The vacuum expectation values are

 (53)

and break electroweak symmetry spontaneously. As anticipated in section 2, we use , , , and , with , .

### 4.1 Minimization

The minimization conditions for are

 ∂V∂θ=−v1v2sinθ(μ212+2λ5v1v2cosθ)=0, (54) ∂V∂v1=μ211v1+λ1v31+(λ3+λ4)v1v22+v2(μ212cosθ+λ5v1v2cos2θ)=0, (55) ∂V∂v2=μ222v2+λ2v32+(λ3+λ4)v21v2+v1(μ212cosθ+λ5v1v2cos2θ)=0. (56)

In order to have spontaneous CP violation, we consider a solution of eqs. (54)–(56) with . From eq. (54) one obtains

 cosθ=−μ2122λ