Soft superweak CP violation in a 3-3-1 model

Soft superweak CP violation in a 3-3-1 model

J. C. Montero,*V. Pleitez,† and O. Ravinez‡

Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900–Sa˜o Paulo, Sa˜o Paulo, Brazil

共Received 16 November 1998; published 8 September 1999兲

We show that it is possible to implement soft superweak C P violation in the context of a 3-3-1 model with only three triplets. All C P violation effects come from the exchange of singly and doubly charged scalars. We consider the implication of this mechanism in the quark and lepton sectors. In particular it is shown that, in this model, as in most of those which incorporate scalar mediated C P violation, it is possible to have large electric dipole moments for the muon and the tau lepton while keeping small those of the electron and neutron. The CKM mixing matrix is real at the tree level but gets a phase at the 1-up loop level.关S0556-2821共99兲04117-X兴 PACS number共s兲: 11.30.Er, 12.60.Cn

I. INTRODUCTION

The origin and the smallness of C P violation are still open questions. In the context of the electroweak standard model关1兴 the only source of CP violation in the quark sector is the surviving phase in the charged current coupled to the vector boson W⫾ 关2兴. This is called explicit or hard CP violation. On the other hand there is no C P violation in the lepton sector at lower orders since neutrinos are massless.

Although this is an interesting feature of the model it leaves open the question of why C P is so feebly violated. It is well known that C P is softly broken if it occurs through a dimensional coupling in the bare Lagrangian and/or sponta-neously. The possibility that C P nonconservation arises ex-clusively through Higgs boson exchange is a rather old one. It can be implemented in a spontaneous way as was proposed by Lee 关3兴 or, explicitly in the parameters of the scalar po-tential, as proposed by Weinberg关4,5兴. Of course, both pos-sibilities can be mixed. Since the works of Lee and Weinberg it has been known that in renormalizable gauge theories the violation of C P has the right strength if it occurs through the exchange of a Higgs boson of mass MH, i.e., it is propor-tional to G_Fm_f2/ M_H2 , where m_f is the fermion mass. Since then, there have been many realizations of that mechanism in extensions of the electroweak standard model. Even if we insist that the C P violation arises solely through the Higgs exchange we have several possibilities in the standard elec-troweak model with several doublets 关6兴.

Some years ago it was proposed a model based on the SU(3)C丢SU(3)L丢U(1)N gauge symmetry with exotic charged leptons关7兴. In this model since the scalar multiplets transform in a different way one from another the scalar potential is constrained in such a way that even with three triplets there is not spontaneous C P violations关8兴. Here we will show that it is possible to have soft C P violation be-cause the coupling constant of the trilinear term in the scalar potential is complex and the vacuum expectation value

共VEV兲 of the three triplets are also complex. Hence, CP is a symmetry in the full bare Lagrangian but in the trilinear term in the potential, in particular all Yukawa couplings are real at tree level. We will show that the model is a realization of a superweak C P violation 关9兴 in the sense that all flavor changing phenomena other than C P violation are accurately described by the real Cabbibo-Kobayashi-Maskawa 共CKM兲 matrix since C P violation is restricted to one operator of dimension three 关10兴.

If the condition that except in the trilinear term C P is a symmetry of the Lagrangian is assumed, we have verified that it is possible to choose the physical phases in such a way that the only places of C P violation in the Lagrangian den-sity are those related with the singly and doubly charged scalars which are present in the model.

This work is organized as follows. In Sec. II we review the Higgs sector of the model. We consider in particular the minimization of the scalar potential and we also give the definitions of the Goldstone and the physical scalar fields of the several charged sectors. In Sec. III we study the Yukawa interactions showing how the VEVs phases can be absorbed in the fermion fields in some places of the Lagrangian den-sity and that they only survive in the sector involving both simply and doubly charged scalar fields. In Sec. IV we show that there is not C P violation in the vector boson-fermion interactions. The phenomenology of the model is considered in Sec. V while our conclusions are in the last section.

II. A MODEL WITH THREE SCALAR TRIPLETS As we said before, here we will consider a model with SU(3)C丢SU(3)L丢U(1)N symmetry with both exotic quarks and charged leptons关7兴. In this model in order to give mass to all fermions it is necessary to introduce three scalar triplets. They transform as

␹⫽

冉

␹⫺ ␹⫺⫺ ␹0

冊

⬃共3,ⴚ1兲, ␳⫽

冉

␳⫹ ␳0 ␳⫹⫹

冊

⬃共3,1兲, 共1兲 ␩⫽

冉

␩0 ␩1⫺ ␩2⫹

冊

⬃共3,0兲. *Email address: [email protected]

†_{Email address: [email protected]}

‡_{Present address: Universidad Nacional de Ingenieria共UNI兲,} Fac-ultad de Ciencias, Avenida Tupac Amaru S/N apartado 31139, Lima, Peru. Email address: [email protected]

The more general scalar potential invariant under the gauge symmetry is V共␹,␩,␳兲⫽␮₁2␹†␹⫹␮₂2␩†␩⫹␮₃2␳†␳⫹共␣⑀i jk␹i␳j␩k⫹H.c.兲

⫹a1共␹†␹兲2⫹a2共␩†␩兲2⫹a3共␳†␳兲2⫹a4共␹†␹兲共␩†␩兲⫹a5共␹†␹兲共␳†␳兲⫹a6共␳†␳兲共␩†␩兲⫹a7共␹†␩兲共␩†␹兲

⫹a8共␹†␳兲共␳†␹兲⫹a9共␳†␩兲共␩†␳兲⫹关a10共␹†␩兲共␳†␩兲⫹H.c.兴. 共2兲

All terms of this potential but the a10term conserve the total

lepton number L共or L⫹B, where B is the baryonic number兲. The minimum of the potential must be studied after the shift-ing of the neutral components of the three scalar multiplets in Eq. 共1兲. Hence, we redefine the neutral components as follows: ␩0_→v␩

冑

2

冉

1⫹ X_␩0⫹iI_␩0 兩v_␩兩

冊

, ␳0→ v_␳

冑

2

冉

1⫹ X_␳0⫹iI_␳0 兩v␳兩

冊

, 共3兲 ␹0_→v␹

冑

2

冉

1⫹ X_␹0⫹iI_␹0 兩v␹兩

冊

, whereva⫽兩va兩 ei␪a with a⫽␩,␳,␹.

The condition that the first derivative of the potential in Eq.共2兲 is zero 共i.e., no linear terms in all neutral fields must survive兲 implies the following constraint equations:

a4 2 兩v␩兩 2_兩v ␹兩⫹ a5 2 兩v␳兩 2_兩v ␹兩⫹␮1 2_兩v ␹兩⫹a1兩v␹兩3 ⫹

_冑

1 2兩v_␹兩Re共␣v␹v␳v␩兲⫽0, 共4a兲 a2兩v␩兩3⫹ a4 2 兩v␹兩 2_兩v ␩兩⫹ a6 2 兩v␳兩 2_兩v ␩兩⫹␮2 2_兩v ␩兩 ⫹ 1

冑

2兩v_␩兩Re共␣v␹v␳v␩兲⫽0, 共4b兲 a6 2 兩v␩兩 2_兩v ␳兩⫹ a5 2 兩v␳兩兩v␹兩 2_⫹␮ 3 2_兩v ␳兩⫹a3兩v␳兩3 ⫹

_冑

1 2兩v_␳兩Re共␣v␹v␳v␩兲⫽0, 共4c兲 and, finally Im共␣v_␹v_␳v_␩兲⫽0. 共4d兲

Before considering how many physical phases will sur-vive in the potential共this must be done by taking at the same time the Yukawa interactions兲 we will study all scalar mass eigenstates in the model. Recall that we have to verify where the VEV’s phases appear in the several sectors of the La-grangian that is, in vertices and in mixing matrices. Hence, we will firstly write down explicitly the mass and mixing matrices of each charged sector of the model. Two of the

phases in Eqs. 共3兲 can be transformed away with a SU共3兲 transformation; whenever we do that we will mention it ex-plicitly and we will always choose␪_␩⫽␪_␳⫽0.

A. Doubly charged scalars

In this sector we have the following mass matrix in the (␳⫹⫹, ␹⫹⫹)T _basis

冉

⫺a₂8兩v␹兩2⫹ A

冑

2兩v_␳兩2 ␣1*v␩*

冑

2 ⫺ a8 2 v␳v␹ ⫺a8 2 兩v␳兩 2_⫹ A

冑

2兩v_␹兩2

冊

共5兲

where we have defined A⬅Re(␣v_␹v_␩v_␳).

As expected, we have a doubly charged Goldstone boson G⫹⫹ and a physical doubly charged scalar Y⫹⫹

冉

␳⫹⫹ ␹⫹⫹

冊

⫽ 1 共兩v␳兩2_⫹兩v ␹兩2兲1/2

冉

兩v␳兩 ⫺兩v␹兩 e⫺i␪␹ 兩v␹兩 ei␪␹ 兩v␳兩

冊

⫻

冉

G ⫹⫹ Y⫹⫹

冊

, 共6兲

with the mass square of the Y⫹⫹ field given by

M_Y2_⫹⫹⫽ A

冑

2

冉

1 兩v␹兩2 ⫹ 1 兩v␳兩2

冊

⫺a8 2 共兩v␹兩 2_⫹兩v ␳兩2兲 . 共7兲

Notice that this mass is proportional to兩␣兩v_␹and兩v_␹兩2, here 兩␣兩 is an arbitrary mass scale while 兩v␹兩 is the VEV that is in control of the SU(3)Lsymmetry and so it is the largest mass scale of the model. So it means that the doubly charged scalar might be a heavy scalar.

B. Singly charged scalars

Next, for the simply charged scalar fields we have, in the (␩1⫹,␳⫹,␩2⫹,␹⫹)Tbasis,

冢

⫺a₂9兩v␳兩2⫹ A

冑

2兩v_␩兩2 ⫺ a9 2 v␳*v␩*⫹ ␣v_␹

冑

2 ⫺ a10 2 v␳*v␹* ⫺ a10 2 v␳*v␩ ⫺a9 2 兩v␩兩 2_⫹ A

冑

2兩v_␳兩2 ⫺ a10 2 v␹*v␩ ⫺ a10 2 v␩ 2 ⫺a7 2 兩v␹兩 2_⫹ A

冑

2兩v_␩兩2 ⫺ a7 2 v␹v␩⫹ ␣*v_␳*

冑

2 ⫺a7 2 兩v␩兩 2_⫹ A

冑

2兩v_␹兩2

冣

. 共8兲

Notice that if a10⫽0,␩1⫹and␳⫹decouple from␩2⫹and␹⫹.

Hence, the mass matrix in Eq. 共8兲 is reduced to two 2⫻2 mass matrices and we have two Goldstone bosons G₁⫹ and G₂⫹; and two massive fields Y₁⫹ and Y₂⫹,

冉

␩1⫹ ␳⫹

冊

⫽ 1 共兩v␩兩2⫹兩v␳兩2兲1/2

冉

⫺兩v_␩兩 兩v␳兩 兩v␳兩 兩v␩兩

冊

冉

G₁⫹ Y₁⫹

冊

, 共9兲 with m_Y 1 ⫹ 2 ⫽ A

冑

2

冉

1 兩v_␩兩2⫹ 1 兩v␳兩2

冊

⫺a9 2 共兩v␳兩 2_⫹兩v ␩兩2兲, 共10兲 and

冉

␩2⫹ ␹⫹

冊

⫽ 1 共兩v␩兩2⫹兩v␹兩2兲1/2

冉

⫺兩v_␩兩 兩v␹兩 ⫺兩v␹兩 兩v␩兩

冊

冉

G₂⫹ Y₂⫹

冊

, 共11兲 with m_Y 2 ⫹ 2 ⫽

_冑

A 2

冉

1 兩v␩兩2⫹ 1 兩v␹兩2

冊

⫺a₂7共兩v␹兩2⫹兩v␩兩2兲. 共12兲 There are 5 phases in the matrix in Eq. 共8兲, two of them can be transformed away with a SU共3兲 transformation 共here when we do that we will choose␪_␩⫽␪_␳⫽0). The other three phases can be absorbed by redefining three scalar fields. However these phases will appear in the Yukawa interactions or in the scalar potential in terms like a10␹⫹␩0␳⫺␩0. Hence,

there is C P violation in the propagators of the singly charged scalars if a10⫽” 0. Since we want CP softly broken we will

consider here a10⫽0 with the singly charged scalar given by

the expressions above.

C. Neutral scalars

Finally, in the neutral sector we have C P even fields, denoted here by H_i0, and C P odd fields, G_1,20 ,h0.

In the C P odd sector we have the following mass matrix:

冉

A

冑

2兩v_␩兩2 A

冑

2兩v_␩兩兩v_␳兩 A

冑

2兩v_␩兩兩v_␹兩 A

冑

2兩v_␳兩2 A

冑

2兩v_␹兩兩v_␳兩 A

冑

2兩v_␹兩2

冊

共13兲

in the (I_␩,I_␳,I_␹)Tbasis. There are in fact two neutral Gold-stone bosons G₁0 and G₂0 共as it must be since we have two massive neutral vector bosons Z and Z

⬘

) and a physical C P odd scalar field h0. Explicitly we have

冉

I_␩0 I_␳0 I_␹0

冊

⫽

冉

⫺兩v_N␩兩 1 ⫺兩v␳兩 2_兩v ␩兩 N₂ 兩v␳兩兩v␹兩 N₃ ⫺兩v_N␳兩 1 ⫺ 兩v␩兩2兩v␳兩 N2 兩v␩兩兩v␹兩 N3 0 兩v␹兩N1 2 N₂ 兩v_␩兩兩v␳兩 N3

冊

冉

G₁0 G₂0 h0

冊

, 共14兲 where N1⫽共兩v␩兩2⫹兩v␳兩2兲1/2, N2⫽共兩v␩兩2兩v␳兩4⫹兩v␩兩4兩v␳兩2⫹兩v␹兩2N1 4_兲1/2_{, 共15兲} and N₃⫽共兩v_␹兩2_兩v ␳兩2⫹兩v␹兩2兩v␩兩2⫹兩v␳兩2兩v␩兩2兲1/2, 共16兲 and with the mass of the h0 _{field given by}

M_h2⫽ A

冑

2

冉

1 兩v_␩兩2⫹ 1 兩v␹兩2 ⫹ 1 兩v␳兩2

冊

. 共17兲

冉

⫺2a2兩v␩兩2⫹ A

冑

2兩v_␩兩2 ⫺a6兩v␩兩兩v␳兩⫺ A

冑

2兩v_␩兩兩v_␳兩 ⫺a4兩v␩兩兩v␹兩⫺ A

冑

2兩v_␩兩兩v_␹兩 ⫺2a3兩v␳兩2⫹ A

冑

2兩v_␳兩2 ⫺a5兩v␳兩兩v␹兩⫺ A

冑

2兩v_␳兩兩v_␹兩 ⫺2a1兩v␹兩2⫹ A

冑

2兩v_␹兩2

冊

, 共18兲

in the (X_␩0,X_␳0,X_␹0)T basis. All those scalars are physical but we will not write the respective mass eigenstates 关12兴. It is enough to stress that X_a0⫽O_aiHH0_i where a⫽␩,␳,␹; i ⫽1,2,3 and OH_{is an orthogonal 3⫻3 matrix.}

III. YUKAWA INTERACTIONS

Here we will consider the most general Yukawa interac-tion with real coefficients which is invariant under the gauge symmetry. However, there are flavor changing neutral cur-rents through the neutral scalars and since these fields get nonzero VEVs as in Eq. 共3兲 it is not straightforward to see which phases can be absorbed by redefining the fermion fields or which of them, and where, survive in the Lagrang-ian density.

In this section we will show that it is possible to choose that all Yukawa interactions which have a counterpart in the standard model are C P conserving. Hence, all C P violation effects arise from the singly and/or doubly charged scalar-fermion interactions. The vector gauge interactions with the fermions are also C P conserving including the singly and doubly charged bileptons as we will show in Sec. IV.

A. Quark-scalar interactions

First, let us consider the quark-scalar interactions. The quark multiplets are the following:

Q_1L

⬘

⫽

冉

u₁

⬘

d₁

⬘

J₁

⬘

冊

L ⬃共3,2 3兲, QmL

⬘

⫽

冉

d_m

⬘

u_m

⬘

J_m

⬘

冊

L ⬃共3*,⫺1 3兲, 共19a兲 and the respective right-handed components in singlets

U_␣R

⬘

⬃共3,1,2/3兲, D_␣R

⬘

⬃共3,1,⫺1/3兲, J_1R

⬘

⬃共3,1,5/3兲;

J_mR

⬘

⬃共3,1,⫺4/3兲, 共19b兲

where m⫽2,3 and␣⫽1,2,3.

With the quark multiplets in Eqs.共19兲 and the scalar ones in Eq.共1兲 we have the Yukawa terms

⫺LY⫽Q¯1L

⬘

兺

␣ 共G1␣U␣R

⬘

␩⫹G ˜_1␣D ␣R

⬘

␳兲 ⫹

兺

i Q¯_iL

⬘

兺

␣ 共Fi␣U␣R

⬘

␳*⫹F˜i␣D␣R

⬘

␩*兲 ⫹␭1Q¯1L

⬘

J1R

⬘

␹⫹

兺

i,m ␭im Q¯_iL

⬘

J_mR

⬘

␹*⫹H.c., 共20兲 here i⫽2,3 and we will assume that all coupling constants in Eq. 共20兲 are real. In the following subsections we will ana-lyze, case by case, all the charged sectors.

1. u-like sector–neutral scalar interactions

After the spontaneous symmetry breaking the VEVs of the neutral scalars are arbitrary complex numbers, as dis-cussed in Sec. II. Using the shifted neutral fields, given in Eq. 共3兲, from Eq. 共20兲 we obtain the interactions of the quarks with the neutral scalars.

In the u-like sector we have the interactions ⫺LY u_⫽u¯ 1L

⬘

兺

␣ G1␣U␣R

⬘

␩ 0_⫹

兺

m u ¯ mL

⬘

兺

␣ Fi␣U␣R

⬘

␳ 0_*⫹H.c. 共21兲 In this and in the following sections we will not use the freedom of choosing two phases equal to zero because we want to see how many phases can be absorbed in the fermion fields. Redefining the phases of the following fields 关11兴:

U_␣R

⬙

⬅ei␪␩_U␣R

⬘

_{, u} mL

⬙

⬅e⫺i(␪␩⫹␪␳)_u

mL

⬘

, 共22兲

the mass matrix of the u-like sector becomes real and can be diagonalized by real orthogonal matrices OL,R

u _{共recall that} ␣⫽1,2,3 and m⫽2,3): 共OL u_兲T_⌫u_O R u_⫽Mu_⫽diag共m u,mc,mt兲, 共23兲 with ⌫u_⫽ 1

冑

2

冉

兩v␩兩G11 兩v␩兩G12 兩v␩兩G13 兩v␳兩F21 兩v␳兩F22 兩v␳兩F23 兩v␳兩F31 兩v␳兩F32 兩v␳兩F33

冊

. 共24兲

Symmetry eigenstates 共singly and doubly primed fields兲 are related to the mass eigenstates fields u,c,t as follows:

冉

u1L

⬘

u_2L

⬙

u_3L

⬙

冊

⫽OL u

冉

u c t

冊

L ,

冉

U1R

⬙

U_2R

⬙

U_3R

⬙

冊

⫽OR u

冉

u c t

冊

R . 共25兲

The Yukawa interaction in Eq.共21兲 can be written as

⫺LY u_⫽U¯ LMuUR⫹ U¯LMuUR X_␳0⫺iI_␳0 兩v␳兩 ⫹U¯ L共OL u_兲T_⌬O L u MuUR

冉

X_␩0⫹iI_␩0 兩v_␩兩 ⫺ X_␳0⫺iI_␳0 兩v␳兩

冊

⫹H.c., 共26兲 with ⌬⬅

冉

1 0 0 0 0 0 0 0 0

冊

. 共27兲

Since there is no mixing among X’s and I’s we have no C P violation in this sector.

2. d-like sector–neutral scalar interactions

Similarly, in the d-like sector, the interaction with the neutral Higgs scalars are

⫺LY d_⫽d¯ 1L

⬘

兺

␣ G ˜ 1␣D␣R

⬘

␳0⫹

兺

m d ¯ mL

⬘

兺

␣ F ˜ m␣D␣R

⬘

␩0*⫹H.c. 共28兲 Making the following phase redefinition

D_␣R

⬙

⬅ei␪_␳D ␣R

⬘

, d_mL

⬙

⬅e⫺i(␪␳⫹␪␩)_d

mL

⬘

, 共29兲

we obtain a real mass matrix which can be diagonalized with an orthogonal transformation 共OL d_兲T_⌫d_O R d_⫽Md_⫽diag共m d,ms,mb兲, 共30兲 with ⌫d_⬅ 1

冑

2

冉

兩v␳兩G˜11 兩v␳兩G˜12 兩v␳兩G˜13 兩v_␩兩F˜21 兩v␩兩F˜22 兩v␩兩F˜23 兩v␩兩F˜31 兩v␩兩F˜32 兩v␩兩F˜33

冊

. 共31兲

The symmetry eigenstates 共singly and doubly primed fields兲 are related to the mass eigenstates 共unprimed fields兲 as follows:

冉

d_1L

⬘

d_2L

⬙

d_3L

⬙

冊

⫽OL d

冉

d s b

冊

L ,

冉

D_1R

⬙

D_2R

⬙

D_3R

⬙

冊

⫽OR d

冉

d s b

冊

R , 共32兲

and the interactions in Eq.共28兲 become

⫺LY d_⫽D¯ LMdDR⫹D¯LMdDR X_␩0⫺iI_␩0 兩v_␩兩 ⫹D¯ L共OL d_兲T_䉭O L d_Md_D R

冉

X_␳0⫹iI_␳0 兩v␳兩 ⫺ X_␩0⫺iI_␩0 兩v␩兩

冊

⫹H.c. 共33兲 Again, we see that if X’s and I’s do not mix among them we have not C P violation through the exchange of neutral fields. This is the case of the model with only three triplets in Sec. II.

3. Exotic quarks–neutral scalar interactions

In the sector involving exotic quarks we have

⫺LY J_⫽␭ 1¯J1L

⬘

J1R

⬘

␹ 0_⫹

兺

i,m ␭im J ¯ iL

⬘

J_mR

⬘

␹0*⫹H.c. 共34兲 Making the redefinition of the right-handed components of the exotic quarks

J_1R

⬙

⬅ei␪␹_J 1R

⬘

, J_mR

⬙

⬅e⫺i␪␹_J mR

⬘

, 共35兲 we have ⫺LY J_⫽␭ 1¯J1L

⬘

J1R

⬙

兩v␹兩

冑

2

冉

1⫹ X_␹0⫹iI_␹0 兩v␹兩

冊

⫹

兺

i,m ␭im J ¯ iL

⬘

J_mR

⬙

兩v␹兩

冑

2

冉

1⫹ X_␹0⫺iI_␹0 兩v␹兩

冊

⫹H.c. 共36兲 Notice that J₁

⬘

共or J₁

⬙

) does not mix with any other quarks but J_2,3

⬘

共or J_2,3

⬙

) mix between themselves since they have the same charge. Here we use, when necessary,

J⬅J₁

⬙

and 共J_L,R

⬙

兲m⫽共OL,R J _兲

mi共 jL,R兲i, 共37兲

OL,R J

are orthogonal 2⫻2 matrices; where J and ji and now with i⫽1,2 denote mass eigenstates, i.e., the mass eigen-states in the exotic quark sector will be denoted when nec-essary J for the charge 5/3 quark and j_1,2for the two charge ⫺4/3 quarks. Hence Eq. 共36兲 can be written in terms of mass eigenstates exotic fermions

⫺LY J_⫽m J¯JLJR

冉

1⫹ X_␹0⫹iI_␹0 兩v␹兩

冊

⫹ j¯LM J_j R

冉

1⫹ X_␹0⫺iI_␹0 兩v␹兩

冊

⫹H.c., 共38兲 where MJ⫽diag(mj₁,mj₂).

4. Singly charged scalar–quark interactions

⫺LY u⫺d_⫽

兺

␣ 共d¯1L

⬘

G1␣U␣R

⬘

␩1 ⫺_⫹u¯ 1L

⬘

G˜_1␣D_␣R

⬘

␳⫹兲 ⫹

兺

m,␣ 共d¯mL

⬘

Fi␣U_␣R

⬘

␳⫺⫹u¯mL

⬘

F˜i␣D_␣R

⬘

␩1⫹兲⫹H.c. 共39兲 Using the phase redefinition of Eqs.共22兲, 共29兲, and 共35兲 in Eq. 共39兲 we have ⫺LY u_⫺d⫽

冑

2 兩v␳兩D ¯_L共O L d_兲T_K 1OL u MuUR␳⫺ ⫹U¯_L共O L u_兲T_K 2OL d MdDR

冑

2 兩v_␩兩␩1⫹ ⫹D¯ L共OL d_兲T_K 1䉭OL u MuUR

冋

冑

2 兩v_␩兩␩1⫺⫺

冑

2 兩v␳兩␳ ⫺

册

⫹U¯ L共OL u_兲T_K 2䉭OL d MdDR

冋

冑

2 兩v␳兩␳ ⫹_⫺

冑

2 兩v_␩兩␩1⫹

册

⫹H.c., 共40兲 being

K1⫽diag共e⫺i␪␩,ei␪␳,ei␪␳兲 and K2⫽diag共e⫺i␪␳,ei␪␩,ei␪␩兲.

共41兲 Notice that if we consider the more general potential, i.e., with the a10term, we have, according to the Eq.共8兲 a general

mixing among all singly charged scalars. There is C P viola-tion through the exchange of a singly charged scalar in this case. However, when we consider the case when a₁₀⫽0 the mixing in that sector is given by Eqs.共9兲 and 共11兲 and there is no C P violation in the interaction of Eq.共40兲 if we also chose that␪_␩⫽␪_␳⫽0 by making a SU共3兲 transformation.

Finally, we have the interaction involving the exotic quarks, ⫺LY J⫺␹_⫽

兺

␣ J ¯ LG1␣OR␣␤ u _U ␤R␩2⫹e⫺i␪␩ ⫹mJ

冑

2 兩v␹兩 U ¯_L共O L u_兲T_⌬J R␹⫺e⫺i␪␹ ⫹_兩v

冑

2 ␹兩D ¯_L共O L d_兲T_⌬¯共O R J_兲T_M˜J_˜j R␹⫹e⫺i(␪␳⫹␪␩⫺␪␹) ⫹H.c., 共42兲

where we have already used Eqs.共22兲 and 共29兲 and defined

⌬¯⬅

冉

0 0 0 0 1 0 0 0 1

冊

, M˜⫽

冉

0 0 0 0 mj₁ 0 0 0 mj₂

冊

, j ˜_{R⫽diag共0,j1, j2兲. 共43兲} The phases of the fields u_1L

⬘

and d_1L

⬘

共and those of the J_1L

⬘

and J_mL

⬘

too兲 are, in principle, still free. However, we can not change the phases of these fields unless we change the phases in the Eqs. 共21兲 and 共28兲 关or also in Eqs.共40兲兴. Not-withstanding, the phases in those equations have already be-ing absorbed after the redefinition given in Eqs. 共22兲 and 共29兲, respectively. We can not absorb phases any more so we have C P violation through the exchange of the singly charged scalars which are coupled to the exotic and known quarks. Hence, we have shown that the only phase redefini-tion in the quark fields are those in Eqs.共22兲, 共29兲, and 共35兲. It means that the observable C P violating phase is Imv_␹共or

⫺Im␣). This will be also the case in the interactions with the doubly charged scalars.

5. Doubly charged scalar–quark interactions

The doubly charged Higgs scalars couple the known quarks with the exotic ones and by using Eqs. 共22兲 and 共29兲 we have ⫺LY J⫺␳_⫽

兺

␣ J ¯ LG˜1␣OR␣␤ d _D ␤R␳⫹⫹e⫺i␪␳⫹

兺

i,l,␣,␤ j ¯ Ll共OL J_兲 li T_F i␣OR␣␤ u _U ␤R␳⫺⫺e⫺i␪␩⫹H.c., 共44a兲 ␤⫽1, 2, 3, l⫽1, 2, and ⫺LY J⫺␹_⫽mJ

冑

2 兩v␹兩 D ¯ L共OL d_兲T_{⌬ J} R␹⫺⫺e⫺i␪␹⫹

冑

2 兩v␹兩U ¯ L¯⌬共OL u_兲T_O L J_M˜J_˜j R␹⫹⫹ei(␪␹⫺␪␳⫺␪␩)⫹H.c., 共44b兲

where we have omitted matrix indices in the last equation. From the same argument of the preceding subsections we can see that there is C P violation in the doubly charged Higgs exchange too. The fields ␳⫹⫹ and␹⫹⫹ are given in terms of the respective mass eigenstates in Eq. 共6兲. Fermion fields are all the mass eigenstates.

B. Lepton-scalar interactions

Now, let us consider the leptonic sector. The leptons are assigned to the following representations:

⌿aL⫽

冉

␯l_a l

⬘

_a⫺ E_a

⬘

⫹

冊

L ⬃共3,0兲; laR

⬘

⫺⬃共1,⫺1兲, 共45兲 E_aR

⬘

⫺⬃共1,⫹1兲, a⫽1,2,3.

The Yukawa interactions in this sector are ⫺LY

l_⫽G ab e _␺¯

aLlbR

⬘

⫺␳⫹Gab␺ ␺¯aLEbR

⬘

⫹␹⫹H.c. 共46兲 Next, we will consider each type of interactions as we did in the quark sectors.

1. Lepton-neutral scalar interactions

In this sector the relation between the symmetry eigen-state fields (E_a

⬘

, l_aL

⬘

) and the mass eigenstate fields (li ⫽e,␮,␶; Ei⫽E1,E2,E3) is obtained through orthogonal

matrices共note that in the leptonic sector i⫽1,2,3) laL

⬘

⫽OLai

e

liL, laR

⬘

⫽ORai e

liR,

E_aL

⬘

⫽O_LaiE E_iL, E_aR

⬘

⫽O_RaiE E_iR. 共47兲 The respective mass matrices are defined as follows:

Me⫽兩v␳兩

冑

2 共OL e_兲T_Ge _O R e , ME_⫽兩v␹兩

冑

2 共OL E_兲T_GE_O R E_{, 共48兲} with Me_⫽diag(m e,m␮,m␶) and ME⫽diag(mE₁,mE₂,mE₃). Notice that the exotic lepton masses are proportional to the larger VEV,v_␹. We see that there is no C P violation in this sector too.

In terms of the physical lepton fields we have

⫺LY l_{⫽ l¯} eLMeleR⫹E¯LMEER⫹ l¯eLMeleR X_␳0⫹iI_␳0 兩v␳兩 ⫹E¯LMEER X_␹0⫹iI_␹0 兩v␹兩 ⫹H.c., 共49兲

and we have redefined the right-handed lepton components l_iR

⬙

⫽ei␪␳liR, EiR

⬙

⫽e

i␪_␹E

iR, 共50兲

but however we have omitted the double primed in Eq.共49兲.

2. Lepton-singly charged scalar interactions

The interactions involving singly charged scalars are

⫺LY␯⫺l,E⫽

冑

2 兩v␳兩 ¯ M␯L

⬘

e_l R␳⫹ ⫹_兩v

冑

2 ␹兩 ␯L

⬘

¯_KME_E R␹⫺e⫺i(␪␹⫺␪␳)⫹H.c., 共51兲 where we have defined

␯L

⬘

¯ ⫽¯␯LOL e e⫺i␪␳_{, 共52兲} andK⬅(OL e )TOL E

. We see that even if we choose␪_␳⫽0 the phase ␪_␹ survives and we have C P violation by the singly charged scalar exchanging, i.e., in the neutrino-exotic lepton vertex.

3. Lepton-doubly charged scalar interactions

The interactions with the doubly charged scalars in the lepton sector are given by

⫺LY l_⫺E⫽

冑

2 兩v␳兩E ¯_LKT_Me_l R␳⫹⫹e⫺i␪␳ ⫹_兩v

冑

2 ␹兩lL ¯_KME_E R␹⫺⫺e⫺i␪␹⫹H.c. 共53兲

As in the previous case we have C P violation in the doubly charged scalar sector even if we choose ␪_␳⫽0.

IV. GAUGE INTERACTIONS

Next, we will verify in what conditions all phase redefi-nitions that have been done in the previous section do not appear in the vector boson-fermion interactions.

A. Quark-vector boson interactions

With the redefinition of the phases in Eqs. 共22兲 and 共29兲 the mixing matrix in the charged currents coupled to the W⫾ is real LY W_⫺q⫽⫺ g

冑

2U ¯_L␥u_V CKMDLW_␮⫹⫹H.c., 共54兲

with the CKM matrix defined as VCKM⫽(OL u

)TO_Ld. So we have no C P violations in this sector. Similarly, we have in the charged currents coupled to the vector bileptons V⫹and U⫺⫺, LY V⫺q_⫽⫺ g

冑

2

冉

J ¯ 1L

⬘

␥␮_u 1L

⬘

⫺

兺

i,m d ¯ iL

⬙

␥␮_J mL

⬘

冊

V_␮⫹e⫺i(␪␳⫹␪␩) ⫹H.c. 共55兲 and

LY U⫺q_⫽⫺ g

冑

2

冉

J ¯ 1L

⬘

␥␮_d 1L

⬘

⫺

兺

i,m u ¯ iL

⬙

␥␮_J mL

⬘

冊

U_␮⫺⫺e⫺i(␪␳⫹␪␩) ⫹H.c. 共56兲

However, we always can choose v_␩⫽v_␳⫽0 by using a

SU共3兲 transformation. Hence, we will not have CP violation in the bilepton sector.

B. Lepton-vector boson interactions

The charged current interactions with the vector bosons in the leptonic sector are

Ll CC_⫽⫺ g

冑

2

兺

i 共␯ ¯ iL␥␮liLW␮⫹⫹␺¯iL␥␮␯iLV␮⫹ ⫹␺¯_iL␥␮_liLU ␮ ⫹⫹_{⫹H.c.兲, 共57兲}

where all fields are still symmetry eigenstates 共but we have omitted the prime兲. Then, the charged current interactions in terms of the physical basis, using Eq. 共47兲, is given by

Ll CC_⫽⫺ g

冑

2共␯ ¯ L␥␮lLW␮⫹⫹␺¯L␥␮KT␯LV␮⫹ei␪␳ ⫺␺¯_L␥␮_KlLU ␮ ⫹⫹_{⫹H.c.兲, 共58兲} where K⬅(OL e

)TOL␺ and we have used the redefinition of the neutrino fields in Eq.共52兲. Although a phase appears we can always choose it as being zero by an SU共3兲 transforma-tion.

We see from Eq.共58兲 that for massless neutrinos we have no mixing in the charged current coupled to W_␮⫹ but we still have mixing in the charged currents coupled to V_␮⫹ and U_␮⫹⫹. That is, if neutrinos are massless we can always choose¯␯_L

⬘

⬅¯␯LOL

e

. However, the charged currents coupled to V_␮⫹ and U_␮⫹⫹ are not diagonal in flavor space since the mixing matrixK survives. Thus, there is not CP violation in this sector if we use the freedom to choose ␪_␳⫽0.

The pure gauge boson interactions also conserve C P if the SU(3)L gauge and U(1)N vector bosons Wa with a ⫽1, . . . 8, and B, respectively transform as

共W_␮1

,W_␮2W_␮3 ,W_␮4,W_␮5,W_␮6 ,W_␮7,W_␮8,B_␮兲→ C P

⫺共W1␮_,⫺W2␮_,W3␮_,⫺W4␮_,W5␮_,⫺W6␮_,W7␮_,W8␮_,B␮_{兲. 共59兲}

It means that the physical fields transform as

共W␮⫹,V␮⫹,U⫹⫹␮ ,A␮,Z␮,Z␮

⬘

兲→ C P

⫺共W␮⫺_,⫺V␮⫺_,⫺U␮⫺⫺_,A␮_,Z␮_,Z

_⬘

␮_{兲. 共60兲}

We have shown that if all term, except the trilinear term in the scalar potential conserve C P this symmetry will be broken when the neutral scalars gain a complex VEV.

V. PHENOMENOLOGICAL CONSEQUENCES As it is well known, the violation of the C P symmetry was discovered in 1964 in the K0_⫺K¯0 _{system 关13兴. Up to}

now, it is only in this particular system in which C P violat-ing effects has been seen. If the source of the C P violation is the weak interactions we expect also to see its effects in B decays. However, only a general discussion is presented here concerning the mesons case 关14兴. On the other hand a de-tailed study of the electric dipole moments of the neutron and charged leptons is shown.

A. Quark sector

1. CP violation in the neutral meson systems

In Fig. 1 we show the tree level contributions to the mass difference ⌬MK⫽2ReM12 共where M12⫽

具

K0兩Heff兩K¯0

典

).

These diagrams exist because of the flavor changing neutral currents in Eq. 共33兲 and in the couplings with the Z

⬘

0. The H0’s contributions to ⌬M_K have been considered in Ref.

关15兴. For mH⬃150 GeV the constraint coming from the ex-perimental value of ⌬MK implies (OL

d₎

11(OL d₎

12ⱗ0.01.

There are also tree level contributions to⌬MKcoming from the Z

⬘

exchange which were considered in Ref.关16兴. Simi-larly for the mass difference of Bd

0_⫺B¯ d 0 , Bs 0_⫺B¯ s 0 , and D0 ⫺D¯0 _{systems. However, in the present model, the C P}

vio-lating parameters like ␧K have only contributions coming from box diagrams involving two doubly charged scalars as can be seen from Fig. 2. The direct C P violation parameter ␧K

⬘

has contributions at the 1-loop level too, as is shown in Fig. 3. There are penguin contributions only at the 2-loop level as we will see later on.

FIG. 1. Tree level contributions to⌬MK.

Although we will not make here a detailed calculation of ␧K and␧K

⬘

and the respective parameters in the Bd

0_⫺B¯

d

0

and B_s0⫺B¯_s0 systems we can notice that in principle the model can give values for these C P violating parameters which are in accord with data since they depend on different mixing matrices O_L,Ru,d. From Eqs. 共44a兲 and 共44b兲 we see that ␧_K comes from diagrams like those shown in Figs. 2共a兲 and 2共b兲. 共There are two other diagrams as the ones in Figs. 2 but with the lines of Y and J interchanged.兲 There are diagrams similar to those in Figs. 2 but with one of the scalar bileptons being changed by a vector bilepton and with no mass inser-tion. These contributions are less suppressed by the mixing angles than by the vector bilepton mass.

Similar diagrams do exist for the other B mesons. The coefficient of the amplitude produced by diagrams like that in Fig. 2 are proportional to the several mixing matrix ele-ments and Yukawa couplings. For instance, for the neutral K system, from Fig. 2共a兲, up to a sin(4␪_␣) factor the amplitude is proportional to G1␣共OR d_兲␣1共O L d_兲 21G1␤共OR d_兲␤1共O L d_兲 11, 共61a兲

while for B mesons, similar diagrams to that in Fig. 2共a兲 imply that the amplitudes are proportional to

G₁␣共O_Rd兲␣1共O_Ld兲₃₁G₁␤共O_Rd兲␤1共O_Ld兲₁₁, 共61b兲 for the neutral Bd system, and

G1␣共OR d_兲 ␣2共OL d_兲 31G1␤共OR d_兲 ␤3共OL d_兲 21, 共61c兲

for the neutral Bs system. We see from Eqs. 共61兲 that the orthogonality condition implies that if we have chosen two of the ⑀K,B_d,B_s parameters the third one is fixed. A similar

analysis follows from Fig. 2共b兲 and the equivalent diagrams for the B systems.

In the D mesons case the mixing matrices involved are different since in these models the left-handed mixing matrix

OL u

survives in different places of the Lagrangian so it is not natural to set them equal to zero共see below兲. Hence, we see that all C P phenomenology in the meson systems can be accommodated, in principle, in the present model.

Concerning the direct C P violating parameters⑀_K

⬘

its con-tribution for the neutral kaons comes from diagrams like the one in Fig. 3共and the one with the line of Ys and j1,2

inter-changed兲. The vertices are given in Eqs. 共42兲 and 共44b兲 and we see that there is a GIM-like cancellation between the contributions of j1 and j2. It means that the suppression of

⑀K

⬘

does not give necessarily a strong constraint on the masses of j1,2. There are similar diagrams with the Y⫺

sub-stituted by a V⫺ vector bilepton and mass insertions in the exotic quarks lines. More details will be given elsewhere 关14兴 however since the contributions given in Fig. 3 depend on the masses of the j1,2exotic quarks it seems no problem

to produce the value

Re

冉

⑀K

⬘

⑀K

冊

⫽0.00200⫾0.00030 共stat兲⫾0.00026 共syst兲

⫾0.00019 共MC stat兲 共62兲

obtained recently by the KTeV collaboration at Fermilab 关17,18兴. Hence in this model we have naturally that 0⫽” 兩⑀

⬘

兩 Ⰶ兩⑀兩.

As in the case of the⑀ parameters, once we have chosen the appropriate value of the mixing matrix elements 共and Yukawa couplings兲 for explaining the observed value in the K0⫺K¯0system, we can still choose the⑀related to the Bdor to the Bssystems but not for both at the same time. The third one has to have a small value of ⑀

⬘

due to the orthogonality of the mixing matrix.

2. Electric dipole moment

It is well known that the discovery of a nonzero electric dipole moment 共EDM兲 for the neutron 共or another elemen-tary nondegenerate system兲 would be a direct evidence for both C P and T violation. The current experimental upper bound is 关19兴

兩dn兩⬍6.3⫻10⫺26e cm. 共63兲

In the standard model the neutron EDM arises at the three-loop level and for this reason it is very small, ⬃10⫺34e cm 关20兴.

In the present model we can calculate the EDM of the d and u quarks at the one-loop level. In principle the contribu-tions are those shown in Figs. 4 and 5. However, we can see from the interactions in Eqs.共40兲, 共42兲, 共44a兲, and 共44b兲 with the phase convention␪_␩⫽␪_␳⫽0 and the coefficient a10⫽0,

that neither the diagram in Fig. 4共a兲 nor that in 4共b兲

contrib-FIG. 2. Some of the box diagram contributions to␧ and M12.

ute to the EDM of the quark d, thus we have dd⫽0, at this level of approximation. For the u-like quarks we have not only the diagram in Fig. 5共a兲 but also those like the one in Fig. 5共b兲. Here we will show only the contributions from Fig. 5共a兲 and its related diagram with the photon attached to the internal fermion line共recall that␪_␹⫽⫺␪_␣):

du⫽e

冑

2G1␣共OR u_兲 ␣1共OL u_兲 11mJ 2_兩v ␩兩 32␲2_共兩v ␹兩2⫹兩v␩兩2兲mY⫹ 2 ⫻

冋

F⫹共mu,mJ兲⫹ 5 3F⫺共mu,mJ兲

册

sin␪␣, 共64兲 where F_⫾共mu,mJ兲⫽⫺ m_Y2_⫹ 2m_u2ln m_Y2_⫹ m_J2 ⫹ m_Y2_⫹ 2m_u2⌬u 共mY⫹ 2 _⫾m u 2_⫺m J 2_兲 ⫻ln

冋

mJ 2_⫹m Y⫹ 2 _⫺m u 2_⫹⌬ u m_J2⫹m_Y2_⫹⫺m_u2⫺⌬_u

册

, 共65兲 and ⌬u 2_⫽关共m Y⫹⫹mJ兲2⫺mu 2_兴关共m Y⫹⫺mJ兲2⫺mu 2_{兴, 共66兲}

the 5/3 factor is because of the charge of the quark-J. The measured parameter is the EDM of the neutron which in the quark model is given in terms of the constituent quarks’s EDM: dn⫽ 4 3 dd⫺ 1 3du⫽⫺ 1 3 du ⯝⫺0.486⫻10⫺21_G ␣1共OR u_兲 1␣共OL u_兲 11 sin␪␣e cm, 共67兲 where we have made the approximation兩v_␩兩⬇兩v_␳兩 in order to use the doubly charged vector bilepton mass M_U, given by M_U2⫽(g2/4)(兩v_␹兩2⫹兩v_␳兩2), instead of兩v_␹兩2⫹兩v_␩兩2in Eq. 共64兲; and GF/

冑

2⫽g2/8MW

2

. We have used MU⫽300 GeV, MY⫹⫽100 GeV; mJ⫽50 GeV and 兩v␩兩⫽100 GeV. How-ever, the value of du 共or dn) is not sensible to the masses of the exotic particles m_Y⫹, M_U and m_J, at least with M_U lesser than 10 TeV. It means that the factor G1␣(OR

u

)_␣1(O_Lu)11, for any value of sin␪␣, have to be

in-voked in order to obtain an EDM of the neutron compatible with the data in Eq. 共63兲. This is not in conflict with the CKM mixing matrix in the charged current coupled to the vector boson W⫾ since the later is defined as V_CKM ⫽(OL

u )TOL

d

. Since in Eq. 共67兲 only the matrix element (O_R,Lu )₁₁ related to the u-like quarks appears and, with the phase convention used here, the mixing matrices related to the d-like quarks do not enter at all in this sort of models and we cannot use V_CKM⫽O_Ld as is usually done in the standard electroweak model. Notice that the mass of the quark u does not enter in the EDM. This is because the interaction in Eq. 共42兲 are not proportional to the mass of the u-like quarks.

In fact, the matrices O_Lu,d will appear in the neutral cur-rents coupled to the extra neutral vector boson Z

⬘

0. We have

LZ⬘⫽⫺ g 2cW关U ¯_L␥␮_O L u Y_LU共Z

⬘

兲O_LuUL ⫹U¯ R␥␮OR u_Y R U_共Z

_⬘

_兲O R u_U R⫹D¯L␥␮OL d_Y L D_共Z

_⬘

_兲O L d_D L ⫹D¯_R␥␮_O R d Y_RD共Z

⬘

兲O_RdDR兴Z_␮

⬘

, 共68兲 with the matrices

Y_LU共Z

⬘

兲⫽Y_LD共Z

⬘

兲⫽⫺ 1

冑

3h共sW兲

冉

1 0 0 0 2s_W⫺1 0 0 0 2s_W⫺1

冊

共69兲 and Y_RU共Z

⬘

兲⫽⫺ 4sW

冑

3h共sW兲

冉

1 0 0 0 1 0 0 0 1

冊

, 共70兲

FIG. 4. Possible diagram contributing to the EDM of the quarks

d.

FIG. 5. Possible diagram contributing to the EDM of the quarks

u.

YR D_共Z

_⬘

_兲⫽ 2sW

冑

3h共s_W兲

冉

1 0 0 0 1 0 0 0 1

冊

, where h(sW)⫽(1⫺4sW

2₎1/2_{. Notice that the right-handed}

neutral currents remain diagonal, but it is not so for the left-handed ones. Here the matrices O_Lu and O_Ld survive in a different form than they appear in the VCKM matrix. These

matrices O_Lu, O_Ld, and V_CKM have to be only determined from experiment since we cannot set none of them to be diagonal in a natural way.

B. Lepton sector

Beside the neutron EDM, other ones which are experi-mentally well studied共alas not definitively yet兲 is the EDM of the electron and the muon. It is interesting that the EDM of the electron (d_e) in the standard model is rather small, of the order of magnitude of 2⫻10⫺38e cm关21兴. However, the value of decould be even smaller, of the order of 10⫺41e cm if there is a cancellation of the three-loop diagrams关22兴 even if QCD corrections are included关23兴. On the other hand, the experimental upper limit is ⭐4⫻10⫺27e cm关24兴.

Hence, a measurement of a nonzero large electron EDM 共and other elementary particles like muon, tau, and neutron, see below兲 would indicate new physics beyond the standard model.

Although we could have C P violation in the singly charged scalar sector共if a10⫽” 0), there would be no

contri-bution to the EDM of the leptons at the one-loop level since neutrinos are massless. If neutrinos remain massless the con-tribution to the electric dipole moment will arise at the three-loop level共this also happens in multi-Higgs extensions of the standard model 关25兴兲. This is not the case for diagrams in-volving doubly charged scalars and the known leptons and the exotic ones 共see Fig. 6兲. It is always possible to choose the doubly charged scalars those fields which carry this phase, i.e., the C P violation in the leptonic sector occurs

only through the exchange of both exotic leptons and of doubly charged scalar fields.

The Yukawa couplings of leptons with the doubly charged scalars are given in Eq. 共53兲 where the scalar fields are still symmetry eigenstates. In fact, there is one Goldstone boson G⫹⫹ and a physical one Y⫹⫹; we denote its mass by mY⫹⫹.共In this model there is not lepton number violation in the interactions with the neutral scalars.兲 As we have shown in Secs. III B and IV B it is not possible to absorb all phases in the complete lepton Lagrangian density. Since neutrinos are considered massless here, the only contributions to the leptonic EDM arise from the doubly charged scalars as shown in Fig. 6. From this we obtain

dl⫽⫺ eml 32␲2m_Y2⫹⫹

冑

2 M_W2GFOl sin共2␪␣兲, l⫽e,␮,␶; 共71兲 where we have defined

Ol⫽

兺

j Kl j 2 4mEj 2 M_U2 关F⫹共ml,mEj兲⫹F⫺共ml,mEj兲兴, 共72兲 the matrix K has been introduced in Eqs. 共51兲 and 共53兲;

F_⫾共m_l,m_E j兲⫽⫺ m_Y2_⫹⫹ 2m_l2 ln m_Y2_⫹⫹ m_E j 2 ⫹mY⫹⫹ 2 2m_l2⌬l 共mY⫹⫹ 2 _⫾m l 2_⫺m E_j 2 _兲 ⫻ln

冋

mEj 2 _⫹m Y⫹⫹ 2 ⫺ml 2_⫹⌬ l m_E j 2 _⫹m Y⫹⫹ 2 _⫺m l 2_⫺⌬ l

册

, 共73兲 and ⌬l 2_⫽关共m Y⫹⫹⫹mE_i兲2⫺ml 2_兴关共m Y⫹⫹⫺mE_i兲2⫺ml 2_兴. 共74兲 For nondegenerate heavy leptons the mixing angles re-main in Eq.共72兲. In the following, for the sake of simplicity, we will assume that mE₁⬍mE₂⬍mE₃ and these heavy lep-tons couple mainly with electron, muon, and tau respec-tively. For instance, the contribution of E1 to the electron

EDM, using mE₁⫽50 GeV and mY⫹⫹⫽100 GeV 关26兴 is given by de⬇10⫺25

冉

M_W2 M_U2

冊

Ke1 2 sin共2␪_␣兲e cm. 共75兲

Assuming MU⫽300 GeV and the factor with the mixing angles K_e12 ⬍10⫺1 we obtain de⬇10⫺27e cm for any value of␪_␣, which is compatible with the experimental upper limit

FIG. 6. Diagrams contributing to the EDM of the charged lep-tons.

of 10⫺27e cm 关24兴. However, if we have used MU⫽3000 GeV we obtain a de compatible with the experimental data for K_e12 ⬇1.

For the muon we have an experimental EDM’s upper lim-its of d_␮⬍10⫺19e cm关27兴. With m_E

2⫽500 GeV we obtain

d_␮⬇10⫺23K_␮22 sin共2␪_␣兲e cm. 共76兲 It means a constraint in K_␮22 ⬇1 and any ␪_␣ we obtain a value smaller than the experimental upper limit.

For the tau lepton a limit of 10⫺17e cm is derived from ⌫(Z→␶⫹_␶⫺_{) 关28,29兴. On the other hand, analyzing the} pro-cess e⫹e⫺→␶␶␥, the L3 Collaboration has obtained the value d_␶⫽(0.0⫾1.5⫾1.3)⫻10⫺16e cm 关30兴. In the present model assuming m_E

3⫽3000 GeV we obtain

d_␶⬇10⫺21K_␶12 sin共2␪␣兲e cm. 共77兲 It means that even ifK_␶1⬇1 for any value of␪_␣give a value smaller that the experimental limit. It means that, the tau lepton has not a EDM of the order of the present experimen-tal upper limit, however it still may induce anomalous cou-plings of the Z boson to fermions which could be seen in tau-charm factories 关31兴.

In this model there is no rare decays such as ␮→3e at tree level. However, the same loop diagrams that contribute for the EDM of the leptons imply transition magnetic and electric moments, like ␮→e␥. However, it will constrain only the matrix elements关(O_Le)TO_LE兴_␮12 .

VI. CONCLUSIONS

The model of C P violation of this work seems like an admixture of both spontaneous breaking as in Lee’s two dou-blets model 关3兴 and the charged-Higgs-boson exchange of Weinberg’s three doublets one 关4兴. However, some differ-ences are important to be pointed out. In a pure charged-scalar-exchange where the phases are in the coupling con-stants of the scalar potential it was shown that there is also C P violation through the exchange of neutral Higgs bosons 关32兴. This is correct in models with SU(2)L丢U(1)Y symme-try with several scalar doublets, that is, with all of them having the same quantum number. For this reason a more general mixing among the scalar fields of the same charge is possible. In the present model all triplets have different U(1)N charges and this constrains their interaction terms. Hence, there is not mixing among the real and imaginary part of the neutral scalar fields even with three scalar triplets as can be seen from Eqs. 共13兲 and 共18兲. If we had considered a10⫽” 0 in the scalar potential in Eq. 共2兲 we would have CP

violation in the propagator of single charged scalars like in the Weinberg model; however even in this case since the a10

term in the scalar potential does not contribute to the mass matrix of the neutral Higgs there is not mixing among C P-even and C P-odd neutral scalars. We see that in this model the C P violation comes only from charged scalars exchange.

There are other interesting features of this model:共a兲 No-tice that since we have no C P violation in the neutral scalar

sector the Weinberg’s three-gluon operators involving neu-tral Higgs boson exchanges do not contribute to the EDM of the neutron at all 关33兴. There is also no contribution to the EDM of the electron and neutron coming from the diagrams of Ref.关34兴 which involve CP violation in the propagator of the neutral Higgs bosons and it does not depend on the phase convention since in Eqs. 共26兲 and 共33兲 phases do not appear at all;共b兲 it was suggested in Ref. 关34兴 that similar contribu-tions to the ones given in Refs. 关33,34兴 but mediated by charged scalar could exist. In the present model two loop contribution to the EDMs as those in Ref. 关35兴 do exist but with top quark substituted by an exotic quark or lepton. However, when they involve a single charged scalars the W⫺ is substituted by V⫺and when they involve a doubly charged scalars W⫺ is substituted by an U⫺⫺; hence they are sup-pressed with respect to the one-loop level contributions by a factor ( MW/ MV)2 in the first case or by ( MW/ MU)2 in the second one for both the quark u and charged leptons; how-ever, these are the main contributions to the EDM of the d-quark since for this quark the one-loop contribution vanish at one-level as discussed above; 共c兲 the same diagrams 共but without the photon兲 that induce a EDM for fermions will induce phases in the mixing matrices. For instance, radiative corrections at one-loop level will induce phases in the mix-ing matrix in the charge 2/3 sector. However, for the charge ⫺1/3 sector the phases will be induced only at two-loop level. The usual Cabibbo-Kobayashi-Maskawa matrix is de-fined as VCKM⫽(OL

u

)TO_Ld then we can see that phases in this matrix are induced at 1-loop level. Moreover, since this type of diagrams are the same which would contribute to the pen-guin diagram, say of the ⑀_K

⬘

parameter, hence there is also penguin contributions to the C P violating parameters how-ever these are 2-loop contributions. Hence, it means that al-though arg detM vanishes at tree level, it has contributions for the charge 2/3 sector at the one-loop order, and for the charge⫺1/3 sector at the two-loop level; 共iv兲 if we assume that C P is also conserved at the tree level in the pure QCD part then ␪QCD⫽0 at the tree level and¯␪⫽␪QCD⫹arg detM

will be finite and calculable. Of course, this is not such a natural solution to the ␪-vacua problem but at least it is in the same foot than the assumption that all Yukawa couplings are real at the tree level. In the 3-3-1 model with a scalar sextet which is needed in order to give to the charged lepton a mass, it was shown by Pal关36兴 that a Peccei-Quinn sym-metry arise naturally if the trilinear␩␳␹ is drops out in the scalar potential. However in the present model no sextet has been added, so there is no such a global symmetry and the referred trilinear is necessary in order to have the correct VEVs pattern in Eqs.共3兲.

Notice that in the u-quark EDM diagrams in Fig. 5 there is a vertex that does not depend on any of the fermion mass because in the model there is flavor violating neutral cur-rents. For this reason the EDM is proportional to the VEV

v_␩. This feature is model dependent, for instance, in the model of Ref.关37兴 the electron’s EDM is proportional to the tau lepton mass because none of the Yuakawa interaction is proportional to the electron mass. In our case in the lepton sector the EDM’s are inversely proportional to the lepton

masses, as can be seen from Eqs. 共71兲, 共72兲, and 共73兲. This happens because in our model: 共i兲 only one of the vertex in Figs. 6 depend on the known lepton masses; 共ii兲 there is a mass insertion of the exotic lepton; and共iii兲 the other vertex which is proportional to the exotic lepton mass. Hence we have a factor m_lm_E

j

2 _{. Notice that if the contributions in Fig.}

6共a兲 would involve a neutral scalar the lepton in the internal line could be a known lepton, say, the same that in the ex-ternal line, and in the case in the numerator of the expression for d_l it will appear m_lm_E

J

2 _→m

l

3 _{as is usually the case in}

models where CP violation arises in the neutral scalar sector as in the first paper in Ref. 关3兴. On the other hand, the same mass dependence appear in the model of the second refer-ence of Ref.关3兴. All these possibilities occur in scalar medi-ated C P violation since in this case an explicit Higgs-fermion coupling flip the chirality. However, it does not mean that we have a EDM singularity for massless fermions since a careful study of the EDM expressions shows that we still have a zero EDM when the fermion masses go to zero. For instance, in the case m_YⰇm_E,m_l the F_⫹⫹F_⫺ function defined in Eq. 共73兲 behaves as ln(m_Y2/m_E2)⫺2; on the other hand for mEⰇmY,ml we have F_⫹⫹F_⫺⬀2mY

2_/m

E

2

⫹(mY

2

/m_E2)ln(m_Y2/m_E2); hence in both cases dl⬀ml.

This model of C P violation seems also like a particular realization of the soft superweak model proposed recently by some authors 关35,38兴. In that model the violation of CP is due to the couplings of the usual left-handed doublets to a heavy sector. In our model, the heavy sector corresponds to

the exotic quarks J, ji, the exotic leptons Ej and the scalars Y⫹ and Y⫹⫹; the only complex numbers are the phase ␪_␣ appearing in the trilinear term in the scalar potential and the complex VEVs, eventually we have only one phase, say␪_␹, after using the SU共3兲 freedom to eliminate two phases and the constrain equation in Eq. 共4d兲. As it was emphasize in Ref.关35兴 the calculation of the electron EDM due to solely to CP violation in the charged Higgs sector has not been study in literature共up to their work兲. Our model it is perhaps, the first renormalizable electroweak model in which the CP vio-lation arise, at the tree level, only through that sector 共the models in Refs.关35,38兴 are effective ones兲.

With three triplets and one sextet which are needed in the model of Ref. 关39兴 it is possible to have truly spontaneous violation of the C P symmetry 关8兴. In this case, the minimi-zation condition of the scalar potential implies more compli-cated constraint equations on the imaginary part of the neu-tral scalars so that two of the phases of the VEV survive in the Lagrangian density. The phenomenology of this model has been studied in Ref. 关15兴.

ACKNOWLEDGMENTS

This work was supported by Fundac¸ao de Amparo a` Pes-quisa do Estado de Sao Paulo共FAPESP兲, Conselho Nacional de Cieˆncia e Tecnologia共CNPq兲, and by Programa de Apoio a Nu´cleos de Exceleˆncia共PRONEX兲. One of us 共V.P.兲 grate-fully acknowledges useful discussions with C. O. Escobar.

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