PHYSICAL REVIEW D 73, 016003 (2006) Soft CP violation in K-meson systems J. C. Montero,1,* C. C. Nishi,1,† V. Pleitez,1,‡ O. Ravinez,2,x and M. C. Rodriguez3,k 1Instituto de Fı́sica Teórica, Universidade Estadual Paulista, Rua Pamplona, 145 01405-900– São Paulo, Brazil 2Facultad de Ciencias, Universidad Nacional de Ingenieria UNI, Avenida Tupac Amaru S/N apartado 31139 Lima, Peru 3Departamento de Fı́sica, Fundação Universidade Federal do Rio Grande/FURG, Av. Itália, km 8, Campus Carreiros 96201-900, Rio Grande, RS, Brazil (Received 1 November 2005; published 17 January 2006) *Electronic †Electronic ‡Electronic xElectronic kElectronic 1550-7998=20 We consider a model with soft CP violation which accommodates the CP violation in the neutral kaons even if we assume that the Cabibbo-Kobayashi-Maskawa mixing matrix is real and the sources of CP violation are three complex vacuum expectation values and a trilinear coupling in the scalar potential. We show that for some reasonable values of the masses and other parameters the model allows us to explain all the observed CP violation processes in the K0- �K0 system. DOI: 10.1103/PhysRevD.73.016003 PACS numbers: 11.30.Er, 12.60.�i, 13.20.Eb I. INTRODUCTION Until some time ago, the only physical system in which the violation of the CP symmetry was observed was the neutral kaon system [1]. Besides, only the indirect CP violation described by the � parameter was measured in that system. Only recently has clear evidence for direct CP violation parametrized by the �0 parameter been observed in laboratory [2]. Moreover, the CP violation in the B-mesons system has been finally observed as well [3]. It is in fact very impressive that all of these observations are accommodated by the electroweak standard model with a complex Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix [4,5] when QCD effects are also included. In the context of that model, the only way to introduce CP violation is throughout its hard violation due to complex Yukawa couplings, which imply a surviving phase in the charged current coupled to the vector boson W� in the quark sector. In the neutral kaon system, despite the CKM phase being O�1�, the breakdown of that symmetry is naturally small because its effect involves the three quark families at the one loop level [6]. This is not the case of the B mesons where the three families are involved even at the tree level and the CP violating asymmetries are O�1� [7]. Notwithstanding, if new physics does exist at the TeV scale it may imply new sources of CP violation. In this context the question if the CKM matrix is complex be- comes nontrivial since at least part of the CP violation may come from the new physics sector [8]. For instance, even in the context of a model with SU�2�L �U�1�Y gauge sym- metry, we may have spontaneous CP violation through the complex vacuum expectation values (VEVs). This is the case of the two Higgs doublets extension of the standard model if we do not impose the suppression of flavor- address: montero@ift.unesp.br address: ccnishi@ift.unesp.br address: vicente@ift.unesp.br address: opereyra@uni.edu.pe address: mcrodriguez@fisica.furg.br 06=73(1)=016003(9)$23.00 016003 changing neutral currents (FCNCs), as in Ref. [9]. The CP violation may also arise throughout the exchange of charged scalars if there are at least three doublets and no FCNCs [10]. Truly soft CP violation may also arise throughout a complex dimensional coupling constant in the scalar potential and with no CKM phase [11]. In fact, all these mechanisms can be at work in multi-Higgs ex- tensions of the standard model [12]. Hence, in the absence of a general principle, all possible sources of CP violation must be considered in a given model. However, it is always interesting to see the potentialities of a given source to explain by itself all the present experimental data. This is not a trivial issue since, for instance, CP violation medi- ated by Higgs scalars in models without flavor changing neutral currents have been almost ruled out even by old data [13–17]. Among the interesting extensions of the standard model there are the models based on the SU�3�C � SU�3�L � U�1�X gauge symmetry called 3-3-1 models for short [18–20]. These models have shown to be very predictive not only because of the relation with the generation prob- lem, some representation content of these models allows three and only three families when the cancellation of anomalies and asymptotic freedom are used; they also give some insight about the observed value of the weak mixing angle [21]. The 3-3-1 models are also interesting context in which new theoretical ideas as extra dimensions [22] and the little Higgs mechanism can be implemented [23]. In the minimal 3-3-1 model [18] both mechanisms ofCP violation, hard [24] and spontaneous [25] have already been considered. In this paper we analyze softCP violation in the framework of the 3-3-1 model of Ref. [19] in which only three triplets are needed for breaking the gauge sym- metry appropriately and give mass to all fermions. Although it has been shown that in this model pure sponta- neous CP violation is not possible [25], we can still imple- ment soft CP violation if, besides the three scalar VEVs, a trilinear parameter in the scalar potential is allowed to be complex. In this case a physical phase survives violating -1 © 2006 The American Physical Society http://dx.doi.org/10.1103/PhysRevD.73.016003 MONTERO, NISHI, PLEITEZ, RAVINEZ, AND RODRIGUEZ PHYSICAL REVIEW D 73, 016003 (2006) the CP symmetry. This mechanism was developed in Ref. [26] but there a detailed analysis of the CP observ- ables in both kaons and B-mesons was not given. Here we will show that all the CP violating parameters in the neutral kaon system can be explained through this mecha- nism, leaving the case of the B-mesons for a forthcoming paper. The outline of this paper is as follows. In Sec. II we briefly review the model of Ref. [26] in which we will study a mechanism for soft CP violation. In Sec. III we review the usual parametrization of the CP violating pa- rameters of the neutral kaon system, � and �0, establishing what is in fact being calculated in the context of the present model. In Sec. IV we calculate �, and in Sec. V we do the same for �0. The possible values for those parameters in the context of our model are considered in Sec. VI, while our conclusions are in the last section. In the appendix we write some integrals appearing in box and penguin diagrams. II. THE MODEL Here we are mainly concerned with the doubly charged scalar and its Yukawa interactions with quarks since this is the only sector in which the soft CP violation arises in this model [26]. The interaction with the doubly charged vector boson will be considered when needed (Sec. V). As ex- pected, there is only a doubly charged would be Goldstone boson, G��, and a physical doubly charged scalar, Y��, defined by � ��� ��� � � 1 N jv�j �jv�je �i�� jv�je i�� jv�j !� G�� Y�� � ; (1) where N � �jv�j2 � jv�j2�1=2; the mass square of the Y�� field is given by m2 Y�� � A��� 2 p � 1 jv�j 2 � 1 jv�j 2 � � a8 2 �jv�j 2 � jv�j 2�; (2) where we have defined A � Re�fv�v�v�� with f a com- plex parameter in the trilinear term ��� of the scalar potential and a8 is the coupling of the quartic term ��y�� ��y�� in the scalar potential. For details and notation see Ref. [26]. Notice that since jv�j jv�j, it is ��� which is almost Y��. In Ref. [26] it was shown that all CP violation effects arise from the singly and/or doubly charged scalar-exotic quark interactions. Notwithstanding, the CP violation in the singly charged scalar is avoided by assuming the total leptonic number L (or B� L, see below) conservation and, in this case, only two phases survive after the redefinition of the phases of all fermion fields in the model: a phase of the trilinear coupling constant f and the phase of a vacuum expectation value, say v�. Among these phases, actually only one survives because of the constraint equation 016003 Im �fv�v�v�� � 0; (3) which implies �� � ��f. Let us briefly recall the representation content of the model [26] with a little modification in the notation. In the quark sector we have QiL � �di; ui; ji� T L � �3; 3�;�1=3�; i � 1; 2 Q3L � �u3; d3; J� T L � �3; 3; 2=3�; U�R � �3; 1; 2=3�, D�R � �3; 1;�1=3�; � � 1; 2; 3, jiR � �3; 1;�4=3� and JR � �3; 1; 5=3�, and the Yukawa interac- tions are written as: �L � X i� QiL�Fi���U�R � ~Fi�D�R��� �Q3L�F3�U�R� � ~F3�D�R�� � X im �imQiLjmR�� � �3Q3LJR��H:c:; (4) where all couplings in the matrices F; ~F, and �’s are in principle complex. Although the fields in Eq. (4) are sym- metry eigenstates we have omitted a particular notation. Here we will assume that all the Yukawa couplings in Eq. (4) are real in such a way that we may be able to test to what extension only the phase �� can describe the CP violation parameters in the neutral kaon system, � and �0. In order to diagonalize the mass matrices coming from Eq. (4), we introduce real and orthogonal left- and right- handed mixing matrices defined as U0L�R� � Ou L�R�UL�R�; D0L�R� � Od L�R�DL�R�; (5) withU � �u; c; t�T etc.; the primed fields denote symmetry eigenstates and the unprimed ones mass eigenstates, being the Cabibbo-Kobayashi-Maskawa matrix defined as VCKM � OuT L Od L. In terms of the mass eigenstates the Lagrangian interac- tion involving exotic quarks, the known quarks, and doubly charged scalars is given by [26]: �LY � ��� 2 p �J � e�i�� jv�j N Md � jv�j R� e�i�� jv�j N mJ jv�j L � �Od L�3�d�Y �� � h:c:; (6) where N is the same parameter appearing in Eq. (1), i.e., N � �jv2 �j � jv2 �j� 1=2 and now, unlike Eq. (4), all fields are mass eigenstates, L � �1� �5�=2, R � �1� �5�=2, with mJ � �3jv�j= ��� 2 p . In writing the first term of Eq. (6) we have used ~F3� � ��� 2 p �Od LM dOdT R �3�=jv�j, where Md is the diagonal mass matrix in the d-quark sector and we have omitted the summation symbol in � so that d� � d; s; b. The Eq. (6) contains all CP violation in the quark sector once we have assumed that all the Yukawa couplings are real. Unlike in multi-Higgs extensions of the standard model [9–17] there is no Cabibbo suppression since in this model only one quark, J, contributes in the internal line, i.e., we have the replacement u; c; t;! J. -2 d s̄ JJ Y ++ g FIG. 1. Dominant CP violating penguin diagram contributing to the decay K0 ! . SOFT CP VIOLATION IN K-MESON SYSTEMS PHYSICAL REVIEW D 73, 016003 (2006) Notice that in Eq. (6) the suppression of the mixing angle in the sector of the doubly charged scalars [see Eq. (1)] has been written explicitly. We will use as illus- trative values jv�j 246 GeV and jv�j * 1 TeV. In this situation the CP violation in the neutral kaon system will impose constraints only upon the masses mJ, mY , and, in principle, on mU the mass of the doubly charged vector boson. Although Oj L has free parameters since the masses mj1;2 are not known, the exotic quarks j1;2; do not play any role in the CP violation phenomena of K mesons. We should mention that it was implicit in the model of Ref. [26] the conservation of the quantum number B� L defined in Refs. [19,20]. Only in this circumstance (or by introducing appropriately a Z2 symmetry) we can avoid terms like ���aL� c�bL� and �laL�cEbR, where �L, lR and ER denote the left-handed lepton triplet, and the usual right-handed components for usual and exotic leptons. These interactions imply mixing among the left- and right-handed components of the usual charged leptons with the exotic ones [27]. The quartic term �y��y� in the scalar potential which would imply CP violation throughout the single charged scalar exchange is also avoided by imposing the B� L conservation. In fact, this model has the interesting feature that when a Z2 symmetry is imposed, the Peccei-Quinn U�1�, the total lepton num- ber, and the baryon number are all automatic symmetries of the classic Lagrangian [28]. III. CP VIOLATION IN THE NEUTRAL KAONS First of all let us say that in the present model there are tree level contributions to the mass difference �MK � 2ReM12 (where M12 � hK 0jH eff j �K0i=2mK). This is be- cause the existence of the flavor changing neutral currents in the model in both the scalar sector and in the couplings with the Z00. The H0’s contributions to �MK have been considered in Ref. [25]. For mH � 150 GeV the constraint coming from the experimental value of �MK implies �Od L�dd�O d L�ds & 0:01. There are also tree-level contribu- tions to �MK coming from the Z0 exchange which were considered in Ref. [18,29]. However, since there are 520 diagrams contributing to �MK, we will use in this work the experimental value for this parameter. In this vain a priori there is no constraints on the matrix elements of Od L. The definition for the relevant parameters in the neutral kaon system is the usual one [30–33]: �0 � ei� 2� 0� =2���� 2 p ReA2 ReA0 � ImA2 ReA2 � ImA0 ReA0 � ; � � ei =4��� 2 p � ImA0 ReA0 � ImM12 �MK � ; (7) We shall use the �I � 1=2 rule for the nonleptonic decays which implies that ReA0=ReA2 ’ 22:2 and that the phase 2 � 0 ’ � 4 is determined by hadronic parameters fol- lowing Ref. [34] and it is, therefore, model independent. 016003 The � parameter has been extensively measured and its value is reported to be [33] j�expj � �2:284� 0:014� 10�3: (8) More recently, the experimental status for the �0=� ratio has stressed the clear evidence for a nonzero value and, therefore, the existence of direct CP violation. The present world average (wa) is [33] j�0=�jwa � �1:67� 0:26� 10�3; (9) where the relative phase between � and �0 is negligible [35]. These values of j�j and j�0=�j imply j�0expj � 3:8 10�6: (10) On the other hand, we can approximate j�j � 1��� 2 p ��������ImM12 �MK ��������; (11a) j�0j � 1 22:2 ��� 2 p ��������ImA0 ReA0 ��������: (11b) In the prediction of �0=�, ReA0 and �MK are taken from experiments, whereas ImA0 and ImM12 are computed quantities [36]. The experimental values used in this work are ReA0 � 3:3 10�7 GeV and �MK � 3:5 10�15 GeV. Let us finally consider the condition with which we will calculate the parameters � and �0. The main �S � 1 con- tribution for the �0 parameter comes from the gluonic penguin diagram in Fig. 1 that exchanges a doubly charged scalar. The electroweak penguin is suppressed as in the SM and will not be considered. On the other hand the �S � 2 and CP violating parameter � has only contributions com- ing from box diagrams involving two doubly charged scalars Y�� [see Fig. 2(a)] and box diagrams involving one doubly charged scalar and one vector boson U�� [see Fig. 2(b)]. The relevant vertices for the calculations are -3 s d̄ d s̄ Y ++ Y ++ J J s d̄ d s̄ Y ++ U++ J J FIG. 2. (a) One of the box diagrams responsible for the transition �K0 ! K0 that involves the exchange of two doubly charged scalars Y��. (b) One of the box diagrams responsible for the transition �K0 ! K0 that involves the exchange of a doubly charged scalar Y�� and a doubly charged vector boson U��. MONTERO, NISHI, PLEITEZ, RAVINEZ, AND RODRIGUEZ PHYSICAL REVIEW D 73, 016003 (2006) given in Eq. (6) and we will use the unitary gauge in our calculations. In other renormalizable R� gauges we must to take into account the would be Goldstone contributions and notice that, according to Eq. (1), the component of ��� � O�1�G��. The hadronic matrix elements will be taken from litera- ture and whenever possible we also take, for the reasons we expose at the beginning of this section, from the experi- mental data or as free parameters. One of the features of this model is that there is no GIM mechanism since the only CP violation source comes from the vertices involv- ing a d-type quark, an exotic quark, and a single doubly charged scalar. 0 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5 3 FIG. 3. Using Eq. (28) and (29) we studied the x-dependence of C2 ds (left scale) and sin2�� (right scale) on z, respectively, with z defined by 10�z � mY=mJ � ��� x p . We have used PL � �1=2�PL�BM� and BL ! 3BL�VI�, where BM indicates the value of PL in the bag model, and VI means the vacuum insertion value of BL. We have also used N � 0:7 TeV and mU=mJ � 1. Notice that sin2�� does not depend on N. IV. DIRECT CP VIOLATION The dominant contributions to the �0 parameter come from the penguin diagram showed in Fig. 1 [32,37]. The part of the Lagrangian that takes into account this ampli- tude is obtained from Eq. (6) and the corresponding imagi- nary effective interaction is given by ImL�0 � gs 16 2N2 Cdsms � �s� � �a 2 � L� md ms R � d � Ga � 1 2 �h�x� � xh0�x�� sin2��; (12) where we have defined Cds � �Od L�3d�O d L�3s, and Ga � in the context of the effective interactions is just Ga � � @ Ga � � @�Ga , x � m2 Y=m 2 J and the function h�x� is given in the appendix, and the prime denotes first derivative. Neglecting the �; Z contributions, i.e., the amplitudes with I � 2, and using the values for the other parameters given above, Eq. (11b) leads to j�0j � 1��� 2 p 1 22:2 jImA0j jReA0j � 9:6 104 jImA0j 1 GeV ; (13) where we have used ReA0 � 3:3 10�7GeV�1, with, jImA0j � ��� 3 p gs 16 2 ms N2 Cds ��������1 2 �h�x� � xh0�x�� �������� ��������PL �md ms PR ��������sin2��: (14) 016003 We can write j�0j as follows: j�0j j�0expj � CdsA�x� sin2��; (15) A�x� � ��� 3 p gs �4 �2 ms j�0expjN 2 ��������PL � ms md PR �������� ��������1 2 �h�x� � xh0�x�� ��������9:6 104 1 GeV ; (16) where we have defined the matrix elements PL � h �I � 0�j � �s� �L �a 2 d � Ga �jK 0i; PR � h �I � 0�j � �s� �R �a 2 d � Ga �jK0i: (17) Using the bag model (BM) it has been obtained that PL � �0:5GeV2 [15]. The other term in Eq. (14) with the matrix element PR is negligible [even if jPRj � O�jPLj�] -4 SOFT CP VIOLATION IN K-MESON SYSTEMS PHYSICAL REVIEW D 73, 016003 (2006) since it has a md factor. We will also use the following values: mK � 498 MeV, md=ms � 1=20, ms � 120MeV, and �s � 0:2. The function jh�x� � xh0�x�j has its maxi- mum equal to one at x � 0. Both PL and PR matrix elements can be considered as free parameters, for instance in Fig. 3 we use PL � �1=2�PL�BM�. Of course, there is also a solution if we use the bag model value of PL. V. INDIRECT CP VIOLATION The contributing diagrams for the � parameter are of two types, one with the exchange of two Y�� and the other with 016003 one U�� and one Y��. They are shown in the Figs. 2(a) and 2(b), respectively. The imaginary part for this class of diagrams has been derived in Refs. [16,17]. The Higgs scalar-quark interaction is given in Eq. (6) and the gauge boson-quark Lagrangian interaction is LW � � g��� 2 p �J�Od L�3�� Ld�U �� �H:c: (18) The contributions to the effective Lagrangian of dia- grams like that shown in Fig. 2(a) are given by ImLYY � � C2 ds �4 �2 2m2 K N2 m2 s N2 �� sin4�� m2 K � ��sLd�2 � m2 d m2 s ��sRd�2 �� g0�x� � jv�j 4jv�j sin2�� �� �s� Li@ $ d �s � L� md ms R � d � � 5g0�x� � 3 2 xg00�x� � � �s � L� md ms R � i@ $ d �s� Ld � g0�x� � 3 2 xg00�x� ��� ; (19) where g0�x� is given in the appendix. On the other hand, the contributions to the effective Lagrangian of diagrams like that shown in Fig. 2(b) are given by ImLUY � � C2 ds �4 �2 2m2 K N2 m2 s N2 � g2 2 N2 4m2 J � sin2�� msm2 K �� �s� �� � L� md ms R � i@ $ d � ��s��Ld�E1�x;y� � �s��Li@ $ d �s� �� � L� md ms R � dE2�x;y� � � �s � L� md ms R � i@ $ d��s� Ld� � �s� Li@ $ d �s � L� md ms R � d � E3�x;y� � � i@ � �s� �� � L� md ms R � d � �s��Ld� i@ � �s��Ld� �s� �� � L� md ms � d � E4�x;y� � � i@ � �s � L� md ms R � d � �s� Ld� i@ � �s� Ld� �s � L� md ms R � d � E5�x;y� � ; (20) where y � m2 U=m 2 J and the functions E1;2;3;4;5 are defined in the appendix. Taking into account both contributions in Eqs. (19) and (20) and using ImM12 � Imh �K0jL��0�jK 0i 2mK ; (21) we obtain ImM12 � � C2 ds �4 �2 m2 K N2 f2 K 2N2 � 1� md ms � �2 � 5 6 sin4�� � 1� m2 d m2 s � g0�x� � jv�j 2jv�j sin2�� � 5 12 � 5g0�x� � 3 2 g00�x� � � 1 3 � 1� 1 4 � ms �md mK � 2 �� g0�x� � 3 2 xg00�x� �� � g2 2 N2 2m2 J sin2�� � 2 3 �E01�x; y� � E3�x; y�� � 2 3 � ms �md mK � 2 �E1�x; y� � E4�x; y�� � 1 12 � 1� � ms �md mK � 2 � �E2�x; y� � E4�x; y�� �� : (22) Thus, we can calculate j�j from Eq. (11a) using fK � 161:8 MeV and �MK � 3:5 10�15 GeV [33]. We have used the vacuum insertion (VI) approximation, and obtained: -5 MONTERO, NISHI, PLEITEZ, RAVINEZ, AND RODRIGUEZ PHYSICAL REVIEW D 73, 016003 (2006) BL � h �K0j��sL�R�d�2jK0i � � 5 12 m4 Kf 2 K �ms �md� 2 ; h �K0j��s� R�L�d�2jK0i ’ 2 3 f2 Km 2 Kh �K0j �sRd�sLdjK0i � 1 2 m4 Kf 2 K �ms �md� 2 � 1 12 m2 Kf 2 K; h �K0j �s� Ld�sL�i@ �djK0i � � 5 6 mdm 4 Kf 2 K �ms �md� 2 ; h �K0j �s� Ld��i@ � �sLdjK0i � 1 3 msm 4 Kf 2 K �ms �md� 2 � 1 12 msm2 Kf 2 K; h �K0j��i@ ��s��Ld�s�mu��LdjK0i � � 2 3 msm 4 Kf 2 K �ms �md� 2 : (23) We have verified that the main contribution to the box diagrams in Eqs. (19) comes from the matrix element denoted by BL. Thus, in order not to be restricted to the VI approximation, BL can be considered a free parameter and, for instance in Fig. 3, we have used BL � 3BL�VI�, but there is also a solution using the VI value of BL. VI. FITTING THE EXPERIMENTAL VALUES In order to compare the prediction of the model with the experimental data for the CP violation in the neutral kaon system we use Eqs. (13)–(17) for j�0j and rewrite Eq. (22) for j�j as j�j j�expj � C2 dsB�x� � 1 2 sin4�� � b�x; y� sin2�� � : (24) where B�x� � 1 �4 �2j�expj m2 K N2 f2 K 2N2 ��� 2 p mK �MK � 1� md ms � �2 � 1� m2 d m2 s � 5 6 g0�x� � 1:34 � 1 TeV N � 4 g0�x� (25) b�x;y�� 6 5 � 1� m2 d m2 s � �1 1 g0�x� � jv�j 4jv�j � 5 12 � 5g0�x�� 3 2 xg00�x� � � 1 3 � 1� 1 4 � ms�md mK � 2 �� g0�x�� 3 2 xg00�x� �� � g2 2 N2 4m2 J � 2 3 � E2�x;y��E4�x;y� � � 2 3 � ms�md mK � 2 � E1�x;y��E4�x;y� � � 1 12 � 1� � ms�md mK � 2 � �E3�x;y��E5�x;y�� �� : (26) 016003 Next we use the constraints��������� 0�Cds; x; ��� �0exp ��������� 1; ����������Cds; x; y; ����exp ��������� 1: (27) Notice that the above conditions are the strongest since we are not considering the experimental error. After some algebraic manipulations the constraints in Eqs. (27) imply C2 ds � D4�x� D2�x� � b2�x;y� A2�x� � b�x;y�A�x� B�x� 1; (28) and Cds sin�� � 1 A�x� ; (29) where we have defined D2�x� � 1 A2�x� � A2�x� B2�x� ; (30) with A�x� defined in Eq. (16), and B�x� and b�x; y� were defined in Eqs. (25) and (26) , respectively. It is interesting to note that Cds sin2�� � 1 A�0� � 0:072 � N 1 TeV � 2 ; (31) where we have used the value of the parameters as dis- cussed below Eq. (17). We have study numerically Eq. (28) and (29) and veri- fied that they are sensible to the values of the matrix elements PL in Eq. (17) and BL defined in Eq. (23). The curves in Fig. 3 are curves of compatibility with experimental data according to the constraints in Eq. (27). The dashed curve shows all the allowed values for sin2�� while the continue curve shows the allowed values for Cds as a function of x. However, these values are not indepen- dent from one another if we want to satisfy both constraints at the same time. The compatibility with the experimental data is obtained by drawing a vertical line for a given value of z. For instance using z � 2 (i.e.,mJ � 100mY) we found sin2�� � 0:15 and C2 ds � 0:6, for z � 1 we obtain sin2�� � 0:25 and C2 ds � 0:3. Notice that from Fig. 3 we see that we have solution in the range 2:5 � & �� & 22:5 � . -6 SOFT CP VIOLATION IN K-MESON SYSTEMS PHYSICAL REVIEW D 73, 016003 (2006) VII. CONCLUSIONS AND DISCUSSIONS The study showed that the 3-3-1 model considered here can account for the direct and indirect CP violation present in the K0 � �K0 system for sensible values of the unknown parameters. Within the approximations used, N & 1TeV there are infinitely many possible values for Cds, �� and m2 Y=m 2 J allowed by the experimental data. Although they are not all independent and the constraint jCdsj< 1 implies a very small upper bound for the ratio m2 Y=m 2 J. Such bound becomes smaller as N becomes greater. Thus very large values of N leads to unrealistically small values of the ratio. Notice also that the constraints used in Eq. (27) are very strong. However, weaker constraints arise if a detailed analysis which take into account the experimental error in both � and �0 is done. Of course, it is clear that in this case there will exist a solution as well. The model implies also some contributions to the neu- tron electric dipole moment (EDM) as in Ref. [26] dn ’ 4:9 10�22 �X � G�1�O u R�1��O u L�11 sin�� � e cm; (33) and we see that a value compatible with the experimental bound of [38] jdnj< 6:3 10�26 e cm �90%CL�; (34) is obtained for practically any value of the phase ��, ifP �G�1�O u R�1��O u L�11 � 10�5. The EDM of the charged leptons also produces results compatible with the experi- mental limit for a large range of the parameters of the model. In addition this model allows magnetic dipole mo- ments for massive neutrinos in the range 10�13–10�11 B almost independently of the neutrino mass [39], which is near the experimental upper limit for the electron neutrino magnetic moment [40] e < 10�11 B�90%CL�: (35) Moreover, as in the standard model the lepton charge asymmetry in the Kl3 decay, L, which has the experimen- tal value (the weighted average of � � and �e� [33]) L � �3:27� 0:12� 10�3, is also automatically fitted in the present model because jA�K0 ! �e��e�j � jA� �K0 ! �e� ��e�j is still valid. Recent analysis on CP violation indicates that the phase of the CKM matrix, which is O�1�, is the dominant con- tribution to the CP violation in both K and B mesons so, new phases coming from physics beyond the standard model must be small perturbations. The CKM mechanism is also at work in the present model but we switch it off in order to study the possibilities of the extra phase of the model. Concerning the K meson and EDM for elementary particles it seems that the model does well. Presently we are working out the case of B decays; if the model is not able to fit these data it implies that CKM phase must be 016003 switched on. It is still possible that new phases may be at work if decays based on b! s gluonic dominated transi- tion really need new physics [7]. Either way, the extra phase in the model could be important for other CP viola- tion parameters like the EDM or if new CP violation observables in B-mesons will not be fitted by the CMK mechanism. Finally, some remarks concerning the masses of the extra particles in 3-3-1 models. First, let us consider the Z0 vector boson it contributes to the �MK at the tree level so that there is a constraint over the quantity [41,42] �Od L�3d�O d L�3s MZ MZ0 ; (36) which must be of the order of 10�4 to have compatibility with the measured �MK. This can be achieved with MZ0 � 4 TeV if we assume a Fritzsch-structure Od Lij � �������������� mj=mi q or, since there is no a priori reason for Od L having the Fritzsch-structure, it is possible that the product of the mixing angles saturates the value 10�4 [41], in this case Z0 can have a mass near the electroweak scale. However, in 3-3-1 models there are flavor-changing neutral currents in the scalar sector implying new contributions to �MK which are of the form �Od L�d3�d3��OR��s MZ MH (37) that involve the mass of the scalarMH, the unknown matrix elements Od R, and also the Yukawa coupling �d, so their contributions to �MK can have opposite sing relative to that of the Z0 contribution. This calculation has not been done in literature, where only the latter contribution has been taken into account [41,42]. The model has also dou- bly charged scalars that are important in the present CP violating mechanism. The lower limit for the mass of doubly charged scalars is a little bit above 100 GeV [43]. Concerning the doubly charged vector boson, if they have masses above 500 GeV they can be found (if they really do exist) by measuring left-right asymmetries in lepton-lepton scattering [44]. Fermion pair production at LEP, and lepton flavor violating of the charged leptons suggest a low bound of 750 GeV for the U�� mass [45]. In e�e�; e�, �� colliders the detection of bileptons with masses between 500 GeV and 1 TeV [46] is favored, while if their masses are of the order of & 1 TeV they could be also observed at hadron colliders like LHC [47]. Muonium-antimuonium transitions would imply a lower bound of 850 GeV on the masses of the doubly charged gauge bileptons, U�� [48]. However, this bound depends on assumptions on the mix- ing matrix in the lepton charged currents coupled to U�� and also it does not take into account that there are in the model doubly charged scalar bileptons which also contrib- ute to that transition [49]. The muonium fine structure only implies mU=g > 215 GeV assuming only the vector bilep- ton contributions [50]. Concerning the exotic quark -7 MONTERO, NISHI, PLEITEZ, RAVINEZ, AND RODRIGUEZ PHYSICAL REVIEW D 73, 016003 (2006) masses, there is no lower limit for their masses but if they are in the range of 200–600 GeV they may be discovered at the LHC [51]. A search for free stable color triplets quarks has been carried out in a p �p collider at an energy of 1.8 GeV, excluding these particles in the range 50– 139 GeV, 50–116 GeV, and 50–140 GeV for the electric charges of �1, 2=3, and 4=3, respectively [52]. We can conclude that the masses for the extra degrees of freedom which distinguish 3-3-1 models with respect to the stan- dard model may be accessible at the energies of the col- liders of the next generations. ACKNOWLEDGMENTS This work was partially supported by CNPq under the processes 141874/03-1 (CCN), 305185/03-9 (JCM) and 306087/88-0 (VP) and by FAPESP (MCR and OR). APPENDIX: INTEGRALS g0�x; y� � � 1 x� y �� x x� 1 � 2 lnx� � y y� 1 � 2 lny � 1 x� 1 � 1 y� 1 � (A1) g1�x; y� � � 1 x� y � x �x� 1�2 lnx� y �y� 1�2 lny � 1 x� 1 � 1 y� 1 � (A2) 016003 g0�x� � lim y!x g0�x; y� � � 2 �x� 1�2 � x� 1 �x� 1�3 lnx (A3) h�x� � � x �x� 1�2 lnx� 1 x� 1 ; h0�x� � 2� 2x� �1� x� lnx �x� 1�3 ; (A4) E1�x; y� � � 5 2 � 1 4 �x@x � y@y� � g0�x; y� � 5 8y g1�x; y� (A5) E2�x; y� � � 1 2 � 1 4 �x@x � y@y� � g0�x; y� � 1 8y g1�x; y� (A6) E3�x; y� � 3 4y �1� x@x � y@y�g1�x; y�: (A7) E4�x; y� � 1 4 � �x@x � y@y�g0�x; y� � 1 2y g1�x; y� � (A8) E5�x; y� � � 1 4y �3� x@x � y@y�g1�x; y� (A9) [1] J. 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