Natural Peccei-Quinn symmetry in the 3-3-1 model with a minimal scalar sector J. C. Montero* and B. L. Sánchez-Vega† Instituto de Fı́sica Teórica—Universidade Estadual Paulista, R. Dr. Bento Teobaldo Ferraz 271, Barra Funda, São Paulo—SP, 01140-070, Brazil (Received 2 March 2011; revised manuscript received 17 August 2011; published 22 September 2011) In the framework of a 3-3-1 model with a minimal scalar sector we make a detailed study concerning the implementation of the Peccei-Quinn symmetry in order to solve the strong CP problem. For the original version of the model, with only two scalar triplets, we show that the entire Lagrangian is invariant under a Peccei-Quinn-like symmetry but no axion is produced since a Uð1Þ subgroup remains unbroken. Although in this case the strong CP problem can still be solved, the solution is largely disfavored since three quark states are left massless to all orders in perturbation theory. The addition of a third scalar triplet removes the massless quark states but the resulting axion is visible. In order to become realistic the model must be extended to account for massive quarks and an invisible axion. We show that the addition of a scalar singlet together with a ZN discrete gauge symmetry can successfully accomplish these tasks and protect the axion field against quantum gravitational effects. To make sure that the protecting discrete gauge symmetry is anomaly-free we use a discrete version of the Green-Schwarz mechanism. DOI: 10.1103/PhysRevD.84.055019 PACS numbers: 14.80.Va, 12.60.Cn, 12.60.Fr I. INTRODUCTION The standard model (SM) of the elementary particles physics successfully describes almost all of the phenome- nology of the strong, electromagnetic, and weak interac- tions. However, from the experimental point of view, the need to go to physics beyond the standard model comes from the neutrino masses and mixing, which are required to explain the solar and atmospheric neutrino data. On the other hand, from the theoretical point of view, the SM cannot be taken as the fundamental theory since some important contemporary questions, like the number of generations of quarks and leptons, do not have an answer in its context. Unfortunately we do not know what the physics beyond the SM should be. A likely scenario is that at the TeV scale physics will be described by models which, at least, give some insight into the unanswered questions of the SM. A way of introducing new physics is to enlarge the symmetry gauge group. For example, the gauge symmetry may be SUð3ÞC � SUð3ÞL �Uð1ÞX, instead of that of the SM. Models based on this gauge group have become known as 3-3-1 models [1–3]. Although the 3-3-1 models coincide with the SM at low energies, they explain some fundamental questions. This is the case of the number of generations cited above. In the 3-3-1 model framework, the number of generations must be three, or a multiple of three, in order to cancel anomalies. This is because the model is anomaly-free only if there is an equal number of triplets and antitriplets, including the color degrees of freedom. In this case, each generation is anomalous. The anomaly cancellation only occurs for the three, or multiple of three, generations together, and not generation by generation like in the SM. This provides, at least, a first step towards the understanding of the flavor question. Other interesting features of the 3-3-1 models concern the electric charge quantization and the vectorial character of the electromag- netic interaction [4,5]. These questions can be accommo- dated in the SM. However, in the 3-3-1 models these questions are related one to another and are independent of the nature of the neutrinos. In recent literature we find studies about the most differ- ent aspects of the 3-3-1 model phenomenology. Among others, a fundamental puzzling aspect is, Why is the CP nonconservation in the strong interactions so small [6,7]? The last question, quantified by the �� parameter of the effective QCD Lagrangian, is known as the strong CP problem. Several solutions based on different ideas have been proposed. According to the framework, they are based on unconventional dynamics [8], spontaneously bro- ken CP [9–11], and an additional chiral symmetry. In the framework of introducing an additional chiral symmetry, two suggestions have been made. If this symmetry is not broken, the symmetry is realized in the Wigner-Weyl manner and the only possible way of relating this unbroken chiral symmetry with flavor conserving gluons is to have at least one massless quark [12]. This suggestion is disfa- vored by standard current algebra analysis [13,14]. The second possibility is that the global Uð1Þ chiral symmetry, known as Uð1ÞPQ [15,16], is spontaneously broken down, which implies a Nambu-Goldstone boson (NG boson), currently known as the axion [17–19]. In this paper we consider the strong CP problem in the framework of a version of the 3-3-1 model in which the scalar sector is minimal [20]. This model has become known as the ‘‘economical 3-3-1 model.’’ The appealing feature of this 3-3-1 model is the natural existence of a *montero@ift.unesp.br †brucesan@ift.unesp.br PHYSICAL REVIEW D 84, 055019 (2011) 1550-7998=2011=84(5)=055019(10) 055019-1 � 2011 American Physical Society http://dx.doi.org/10.1103/PhysRevD.84.055019 Peccei-Quinn-like (PQ-like) Uð1Þ symmetry. To study the consequences of this symmetry in this model, we organize this paper as follows: in Sec. II we briefly describe the model, and in Sec. III we analyze the consequences of the natural PQ-like symmetry in the model and find that the symmetry is realized in the Wigner-Weyl manner implying three massless quarks, which disagrees with the standard current algebra analysis. Thus, we propose the introduction of two new scalar fields, � and �, in order to both give a solution to the massless quarks and implement the PQ mechanism. Since this mechanism needs the Uð1ÞPQ to be anomalous in order to solve the strong CP problem, it does not seem natural to impose this symmetry on the Lagrangian. However, it could be understood as being natural if it is a residual symmetry of a larger one which is not anomalous and spontaneously broken. Then, we consider a ZN discrete gauge symmetry to be a sym- metry of the Lagrangian. The discrete gauge anomalies are canceled by a discrete version of the Green-Schwarz mechanism. After this, two ZN symmetries, Z10 and Z11, which protect the axion against quantum gravity effects, are explicitly shown. Finally, our conclusions are given in Sec. IV. II. A BRIEF REVIEW OF THE ECONOMICAL 3-3-1 MODEL The different models based on a 3-3-1 gauge symmetry can be classified according to the electric charge operator Q ¼ T3 � bT8 þ X; (1) where T3 and T8 are the diagonal Gell-Mann matrices, X refers to the quantum number of the Uð1ÞX group, and b ¼ 1= ffiffiffi 3 p , ffiffiffi 3 p . The embedding b parameter defines the model. Here, we will consider the model with both b ¼ 1= ffiffiffi 3 p and the simplest scalar sector, which was pro- posed for the first time in Ref. [21]. It has become known in the literature as ‘‘economical 3-3-1 model.’’ This model had origin in a systematic study of all possible 3-3-1 models without exotic electric charges [22]. To give a brief review of the main features of this model, let us say that it has a fermionic matter content given by �aL ¼ ð�a;ea; ð�aRÞCÞTL�ð1;3;�1=3Þ; eaR�ð1;1;�1Þ; Q�L ¼ ðd�;u�;d0�ÞTL�ð3;3�;0Þ; Q3L ¼ ðu3;d3;u03ÞTL�ð3;3;1=3Þ; uaR�ð3;1;2=3Þ; u03R�ð3;1;2=3Þ; daR�ð3;1;�1=3Þ; d0�R�ð3;1;�1=3Þ; (2) where a ¼ 1, 2, 3, � ¼ 1, 2 (from now on Latin and Greek letters always take the values 1, 2, 3 and 1, 2, respectively), and the values in the parentheses denote quantum numbers based on the ðSUð3ÞC; SUð3ÞL; Uð1ÞXÞ factor, respectively. In this model the electric charges of the exotic quarks are the same as the usual ones, i.e., Qðd0�Þ ¼ �1=3 and Qðu03Þ ¼ 2=3. In the bosonic matter content there are only two scalar triplets, � and �: � ¼ ð�0; ��; �0 1ÞT � ð1; 3;�1=3Þ; � ¼ ð�þ; �0; �þ 1 Þ � ð1; 3; 2=3Þ: (3) These two scalars spontaneously break down the SUð3ÞL �Uð1ÞX gauge group. The vacuum expection val- ues (vevs) in this model satisfy the constraint V�0 � hRe�0i; V�0 � hRe�0i � V�0 1 � hRe�0 1i: With the quark, lepton, and scalar multiplets above we have the Yukawa interactions Ll Y ¼ Yab ��aLebR�þ Y0 ab� ijkð ��aLÞið�bLÞCj ð��Þk þ H:c:; (4) for leptons. Yab and Y 0 ab are arbitrary complex matrices and Y0 ab is also antisymmetric. Throughout the paper we use the convention that an addition over repeated indices is im- plied. The lepton masses are generated by the interactions in Eq. (4). The first term gives a general tree level mass matrix for the charged leptons [20]. However, for the neutrino mass generation, the interactions in the second term are not able to provide a realistic mass spectrum at the tree level. At least 1-loop corrections must be considered in order to obtain neutrino masses compatible with the solar and atmospheric neutrino data [23]. For quarks we have Lq Y ¼ G1 �Q3Lu 0 3R�þG2 � �Q�Ld 0 R� � þG3 a �Q3LdaR� þG4 �a �Q�LuaR� � þG5 a �Q3LuaR�þG6 �a �Q�LdaR� � þG7 � �Q3Ld 0 �R�þG8 � �Q�Lu 0 3R� � þ H:c:; (5) where Gi are arbitrary complex matrices. Notice that the Yukawa interactions given in Eqs. (4) and (5) are the most general allowed by the gauge symmetries. Here, we follow exactly Refs. [20,24]; i.e., no additional symmetries are imposed, contrary to what is done in Ref. [25] where a Z2 symmetry is imposed. The most general scalar potential invariant under the gauge symmetry is VH ¼ 2 �� y�þ 2 �� y�þ �1ð�y�Þ2 þ �2ð�y�Þ2 þ �3ð�y�Þð�y�Þ þ �4ð�y�Þð�y�Þ: (6) One of the main features of this model is that its scalar sector is the simplest possible. In principle, this should make the scalar potential analysis easier. A study of the stability of this scalar potential is presented in Ref. [26]. J. C. MONTERO AND B. L. SÁNCHEZ-VEGA PHYSICAL REVIEW D 84, 055019 (2011) 055019-2 III. Uð1ÞPQ SYMMETRY IN THE ECONOMICAL 3-3-1 MODEL A Uð1ÞPQ symmetry is global and chiral [15,16]; i.e., it treats the left- and right-handed parts of a Dirac field differently. Moreover, it must be both a symmetry of the entire Lagrangian and valid only at the classical level. In renormalizable theories, the key ingredient of theUð1ÞPQ is that it must be afflicted by a color anomaly; i.e., its asso- ciated current, jPQ , must obey @ jPQ � Ng2 16�2 G ~G; (7) being G ~G ¼ 1 2 � � �Gb �G b �, and Gb � is the color field strength tensor (b ¼ 1; . . . ; 8). N must not be zero. Now, we are going to prove that the economical 3-3-1 model entire Lagrangian is naturally invariant under a Uð1ÞPQ symmetry transformation. To do so, we search for how many Uð1Þ symmetries the model has. First of all, we write the relations that these symmetries must obey in order to keep the entire Lagrangian invariant. From Eqs. (4)–(6) we obtain the following relations: �XQ3 þXu0 3R þX�¼0; �XQþXd0R �X�¼0; (8) �XQ3 þXuR þX�¼0; �XQþXdR �X�¼0; (9) �XQ3 þXdR þX�¼0; �XQþXuR �X�¼0; (10) �XQ3 þXd0R þX�¼0; �XQþXu03R �X�¼0; (11) � X� þ XeR þ X� ¼ 0; �2X� � X� ¼ 0; (12) where the notation Xc above is to be understood as the Uð1Þ charge of the c field. Solving the equations above, we find three independentUð1Þ symmetries. One of these is the Uð1ÞX gauge symmetry. The other two are the usual baryon number symmetry, Uð1ÞB, and a chiral symmetry acting on the quarks, Uð1ÞPQ. Thus, the model actually has a larger symmetry: SUð3ÞC � SUð3ÞL �Uð1ÞX �Uð1ÞB � Uð1ÞPQ. The two last symmetries are global. This is sum- marized in Table I. We can see that the Uð1ÞPQ chiral symmetry is afflicted by a color anomaly in the following way: APQ / �X� � 2X� ¼ �3; (13) where APQ is the coefficient of the ½SUð3ÞC�2Uð1ÞPQ anom- aly. Therefore, this chiral symmetry is a PQ-like symmetry. Also, notice that in this case the Uð1ÞPQ is an accidental symmetry; i.e., it follows from the gauge local symmetry plus renormalizability. In other words, the economical model naturally has a PQ symmetry. The naturalness of theUð1ÞPQ in the economical 3-3-1 model is a key point. In our understanding, since Uð1ÞPQ symmetry is anomalous its imposition is not sensible in the sense that in the absence of further constraints on very high energy physics we should expect all relevant and marginally relevant opera- tors that are forbidden only by this symmetry to appear in the effective Lagrangian with coefficient of order one, but if this symmetry follows from some other free anomaly symmetry, in our case from the gauge symmetry, all terms which violate it are then irrelevant in the renormalization group sense. Unfortunately, when � and � acquire vevs different from zero, a subgroup of Uð1ÞX �Uð1ÞPQ remains unbroken; i.e., the symmetry-breaking pattern is SUð3ÞL �Uð1ÞX �Uð1ÞPQ!h�iSUð2ÞL �Uð1ÞY �Uð1Þ0PQ !h�iUð1ÞQ �Uð1Þ00PQ; (14) whereUð1ÞQ is the electromagnetic symmetry. The SUð3ÞC and Uð1ÞB groups have been omitted in the expression above because these are both unbroken and irrelevant to the current analysis. An explicit expression of the Uð1Þ0PQ symmetry can be easily written as Uð1Þ0PQ � Uð1ÞPQ þ 3Uð1ÞX: (15) Also, note that Uð1Þ0PQ and Uð1Þ00PQ are PQ-like symmetries because these are chiral and afflicted by a color anomaly. As a consequence of the unbroken Uð1Þ00PQ chiral sym- metry [i.e.,Uð1Þ00PQ is realized in theWigner-Weyl manner], no axion appears in the scalar mass spectrum. Instead of that, some quarks remain massless after the spontaneous symmetry breaking, and these will remain massless to all orders of perturbation theory. To illustrate the preceding, we explicitly calculate the mass spectra of scalars and quarks. First, we calculate the scalar mass spectrum m2 H1;H2 ¼�1V 2 �0þðV2 �0þV2 �0 1 Þ�2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðV2 �0�1�ðV2 �0þV2 �0 1 Þ�2Þ2þðV2 �0þV2 �0 1 ÞV2 �0� 2 3 r ; (16) TABLE I. Assignment of quantum charges in the economical 3-3-1 model. Q�L Q3L (uaR, u 0 3R) (daR, d 0 �R) �aL eaR � � Uð1ÞX 0 1=3 2=3 �1=3 �1=3 �1 2=3 �1=3 Uð1ÞB 1=3 1=3 1=3 1=3 0 0 0 0 Uð1ÞPQ �1 1 0 0 �1=2 �3=2 1 1 NATURAL PECCEI-QUINN SYMMETRY IN THE 3-3-1 . . . PHYSICAL REVIEW D 84, 055019 (2011) 055019-3 m2 H 3 ¼ 1 2 ðV2 �0 þ V2 �0 þ V2 �0 1 Þ�4; (17) where V�0 , V�0 , V�0 1 are the vevs of �0, �0, �0 1, respectively. For simplicity, all the vevs have been assumed to be real. Additionally, there are exactly 8 NG bosons that will be- come the longitudinal components of the 8 gauge bosons [21]. The absence of one physical massless state (or axion) in the scalar spectrum shows that the Uð1Þ00PQ symmetry remains unbroken after the spontaneous symmetry breaking. On the other hand, in the quark spectra, there are three massless states, one in the up-quark sector and two in the down-quark sector. First, consider the up-quark mass matrix at the tree level which is written as �uLM ð0Þ u uR� 1ffiffiffi 2 p �uL G4 11V�0 G4 12V�0 G4 13V�0 G8 1V�0 G4 21V�0 G4 22V�0 G4 23V�0 G8 2V�0 G5 1V�0 G5 2V�0 G5 3V�0 G1V�0 G5 1V�0 1 G5 2V�0 1 G5 3V�0 1 G1V�0 1 2 66666664 3 77777775 uR; (18) where �uL � ð �u1L; �u2L; �u3L; �u03LÞ and uR � ðu1R; u2R; u3R; u 0 3RÞT . The third and fourth rows of the Mð0Þ u matrix are proportional; thus there is a massless up quark (we refer to this massless up quark simply as u) at the tree level. An analytical expression for this massless state can be given but it is useless for our analysis. Later we give arguments that the u quark remain massless to all orders of perturba- tion theory [27]. Similarly, the down-quark mass matrix at the tree level, Mð0Þ d , defined as 1ffiffi 2 p �dLM ð0Þ d dR, reads G6 11V�0 G6 12V�0 G6 13V�0 G2 11V�0 G2 12V�0 G6 21V�0 G6 22V�0 G6 23V�0 G2 21V�0 G2 22V�0 G3 1V�0 G3 2V�0 G3 3V�0 G7 1V�0 G7 2V�0 G6 11V�0 1 G6 12V�0 1 G6 13V�0 1 G2 11V�0 1 G2 12V�0 1 G6 21V�0 1 G6 22V�0 1 G6 23V�0 1 G2 21V�0 1 G2 22V�0 1 2 66666666664 3 77777777775 ; (19) where �dL � ð �d1L; �d2L; �d3L; �d01L; �d02LÞ and dR � ðd1R; d2R; d3R; d01R; d02RÞT . Since the first and fourth rows, and the second and fifth rows, are proportional to each other, the Mð0Þ d matrix has two eigenvalues equal to zero (we refer to these massless down quarks as d and s). Thus, the economical model has three massless quark states: one in the up-quark sector and two in the down-quark sector. In other words, the economical 3-3-1 model has a remaining unbroken chiral symmetry, Uð1Þ00PQ, which allows us to transform uL ! ei�uL, dL ! ei�dL, sL ! ei�sL, leaving the Lagrangian invariant. This symmetry will protect these massless quarks to acquire mass at any level of perturba- tion theory [27]. At this point it is important to say that, since theUð1Þ00PQ symmetry is anomalous, these quarks will acquire mass only through QCD nonperturbative effects (for example, by instanton effects [28]). Although the quarks could acquire some mass through these nonpertur- bative processes, this is in conflict with both chiral QCD and lattice calculation where the ratio mu=md is 0:410 0:036 [13,14,29]. Before considering a possible solution to the problem mentioned above, for the sake of completeness, we find it important to say that in Ref. [20] one-loop contributions to the up-quark mass matrix were calculated, even though a subtle flaw makes these contributions not right. To dem- onstrate that, we exactly follow the same lines as in Ref. [20]. There, in Sec. 4, the authors consider, for sim- plicity, one-loop contributions to the submatrix Mð0Þ u3u 0 3 � 1ffiffiffi 2 p G5 3V�0 G1V�0 G5 3V�0 1 G1V�0 1 2 4 3 5; (20) where Mð0Þ u3u 0 3 is written in the base ðu3; u03Þ. The other two massive quark states, u1 and u2, which acquire mass at tree level [m1 ¼ G4 11V�0= ffiffiffi 2 p ,m2 ¼ G4 22V�0= ffiffiffi 2 p , see Eq. (27) in Ref. [20] ] are not important in the analysis. The matrix Eq. (20) mixes together the states u3 and u03. A combina- tion of them will be a massless quark and the orthogonal combination acquires a mass �V�0 1 . Now, the idea is to calculate the one-loop contributions coming from the Feynman diagrams in Fig. 1 to the up- quark mass submatrix defined in Eq. (20). Following Ref. [20], we get �u3L;u 0 3R ¼ �2iV�0V�0 1 �1Mu0 3 ðG1Þ2 Z d4p ð2�Þ4 p2 ðp2 �M2 u0 3 Þ2ðp2 �M2 �0Þðp2 �M2 �0 1 Þ � 2V�0V�0 1 �1Mu03ðG1Þ2IðM2 u03 ;M2 �0 ;M 2 �0 1 Þ; (21) where IðM2 u0 3 ;M2 �0 ;M 2 �0 1 Þ is defined as IðM2 u0 3 ;M2 �0 ;M 2 �0 1 Þ ��i Z d4p ð2�Þ4 p2 ðp2�M2 u03 Þ2ðp2�M2 �0Þðp2�M2 �0 1 Þ ; (22) and �u3L;u 0 3R is the one-loop contribution to the element ðMð0Þ u3u 0 3 Þ12 given by the Feynman diagram in Fig. 1(a). The value of the integral in Eq. (22) is not relevant in our analysis and thus it is not calculated. Now, �u3L;u3R is found in a similar way from the diagram in Fig. 1(b), �u3L;u3R ¼ �2iV�0V�0 1 �1Mu0 3 G5 3G 1 Z d4p ð2�Þ4 p2 ðp2 �M2 u0 3 Þ2ðp2 �M2 �0Þðp2 �M2 �0 1 Þ ¼ G5 3 G1 �u3L;u 0 3R : (23) J. C. MONTERO AND B. L. SÁNCHEZ-VEGA PHYSICAL REVIEW D 84, 055019 (2011) 055019-4 One-loop contributions to ðMð0Þ u3u 0 3 Þ21 and ðMð0Þ u3u 0 3 Þ22, found from the Feynman diagrams in 1(c) and 1(d), respectively, are also proportional to each other, i.e., �u0 3L ;u3R ¼ G5 3 G1 �u0 3L ;u0 3R : (24) Therefore, when considering simultaneously all the one- loop contributions above, the Mð0Þ u3u 0 3 becomes 1ffiffiffi 2 p G5 3 � V�0 þ �u3L;u 0 3R G1 � G1 � V�0 þ �u3L;u 0 3R G1 � G5 3 � V�0 1 þ �u0 3L ;u0 3R G1 � G1 � V�0 1 þ �u0 3L ;u0 3R G1 � 2 6664 3 7775: (25) This matrix still has a determinant equal to zero. In other words, we have shown that one combination of the up quarks still remains massless, as it should be. In the down-quark sector a similar analysis can be easily made. Thus, what makes the contributions to the up-quark and down-quark masses made in Ref. [20] not right is that those contributions were not considered simultaneously. To conclude, the 3-3-1 economical model has three massless quarks (one up quark and two down quarks) to all order of perturbation theory, which is in conflict with both chiral QCD and lattice calculation where the ratio mu=md is 0:410 0:036 [14]. Therefore, the economical model is not realistic and it must be modified to overcome that difficulty. One manner of doing that is introducing a new scalar triplet, �: � ¼ ð�0; ��; �0 1ÞT � ð1; 3;�1=3Þ: (26) When the scalar triplet, �, is introduced into the model, the Yukawa Lagrangian given in Eq. (5) has the following extra terms: Lq Y;extra ¼ G9 a �Q3LuaR�þG10 �a �Q�LdaR� � þG11 �Q3Lu 0 3R� þG12 � �Q�Ld 0 R� � þ H:c: (27) As can be seen from Eq. (5) and Eq. (27), the quark fields interact with different neutral scalar fields simultaneously. Hence, flavor-changing neutral currents (FCNCs) are, in general, induced. This characteristic is shared by most of multi-Higgs models [30]. In order to suppress the FCNC effects we must use some model dependent strategies, for instance, choosing an appropriate direction in the vev space, resorting to heavy scalars and/or small mixing FIG. 1. One-loop contributions to the up-quark mass matrix. NATURAL PECCEI-QUINN SYMMETRY IN THE 3-3-1 . . . PHYSICAL REVIEW D 84, 055019 (2011) 055019-5 angles in the quark and the scalar sectors, and considering adequate Yukawa coupling matrix textures [3,30–32]. In particular, in this model the exotic quarks have the same electric charge as the ordinary ones. This means that they can mix with the later ones and hence also induce FCNC. However, this kind of FCNC is suppressed when the vev which controls the exotic quark masses is taken much larger than the electroweak mass scale [3,32]. FCNC also occurs in models which have an extra neutral vector boson. They can be handled in a similar way. See, for example, [33]. Finally, from Eq. (4) we see that the lepton sector of the model is not afflicted by FCNC. The most general scalar potential invariant under the gauge symmetry, V ¼ VH þ VNH, has now the following extra terms: VH;extra¼ 2 �� y�þ�5ð�y�Þ2þ�y�½�6ð�y�Þþ�7ð�y�Þ� þ�8ð�y�Þð�y�Þþ�9ð�y�Þð�y�Þ; (28) and VNH¼ 2 4� y�þf�ijk�i�j�kþ�10ð�y�Þ2 þ�11ð�y�Þð�y�Þþ�12ð�y�Þð�y�Þ þ�13ð�y�Þð�y�Þþ�14ð�y�Þð�y�ÞþH:c: (29) Now, when the scalar triplets acquire vevs, it is straightfor- ward to see that the quark mass matrices do not have determinant equal to zero; thus all the quarks are massive. Additionally, as we will show below, there will be no accidental anomalous PQ-like symmetry. Returning to the question of the PQ symmetry, we note that due to these new terms in the Lagrangian the charges of the Uð1Þ symmetries must obey the following relations, �XQ3 þXuR þX�¼0; �XQ3 þXu0R þX�¼0; (30) �XQþXd0R �X�¼0; �XQþXdR �X�¼0; (31) X� þ X� þ X� ¼ 0; �X� þ X� ¼ 0; (32) besides the ones given in Eqs. (8)–(12). Solving Eqs. (8)–(12) and Eqs. (30)–(32) simultaneously, we find that there are only twoUð1Þ symmetries,Uð1ÞX andUð1ÞB. The assignment of quantum charges for these two Uð1Þ symmetries when � is included is shown in Table II. Thus, in this case, in contrast to the previous one, the Uð1ÞPQ is not allowed by the gauge symmetry. But, if the Lagrangian is slightly modified by imposing a Z2 symmetry such that � ! ��, u03R ! �u03R, d 0 R ! �d0 R and all the other fields being even under Z2, the trilinear term of the scalar potential, f�ijk�i�j�k, is eliminated. Consequently, the Uð1ÞPQ symmetry is automatically introduced. This can be seen by solving Eqs. (8)–(12) and Eqs. (30)–(32) with- out the equation X� þ X� þ X� ¼ 0: (33) Note that, in addition to the assignment of quantum charges given in Table I, the charge Uð1ÞPQ of the � triplet scalar is 1. Unfortunately, the axion that appears when the neutral components of the scalar triplets acquire vev is visible. This is easy to see as follows. In this model the � field is responsible for breaking the symmetry from SUð3ÞC � SUð3ÞL �Uð1ÞX to SUð3ÞC � SUð2ÞL �Uð1ÞY . Thus, to obtain an invisible axion, V�0 1 that breaks the PQ symmetry must be greater than 109 GeV. But, when � acquires a vev the combination Uð1Þ0PQ ¼ Uð1ÞPQ þ 3Uð1ÞX is not broken. Therefore, the new PQ symmetry is truly broken when the � field acquires a vev. As V�0 & 246 GeV, the axion induced is visible. A visible axion was long ago ruled out by experiments [34]. One usual way to resolve that problem is to introduce an electroweak scalar singlet, � [17,18]. Its role is to break the PQ symmetry at a scale much larger than the electro- weak scale. This field does not couple directly to quarks and leptons; however, it acquires a PQ charge by coupling to the scalar triplets. With the PQ charges given in Table I, the � scalar acquires a PQ charge by coupling to the �, �, � scalar triplets through the interaction term �PQ� ijk�i�j�k�: (34) From this coupling, the � field obtains a PQ charge of�3. Also, notice that this term is permitted provided the� field is odd under the Z2 symmetry, i.e., Z2ð�Þ ¼ ��. However, the Z2 and gauge symmetries do not prohibit some terms in the scalar potential violating the PQ sym- metry, such as �2, �4, �y��2, �y��2, �y��2, from appearing. Thus, the PQ symmetry should be imposed. Since the PQ symmetry is anomalous, it is somewhat awkward to do so. However, there is a way to overcome this difficulty. Consider that the entire Lagrangian is in- variant under a ZN discrete gauge symmetry [35], with N � 5, instead of a Z2 symmetry. The ZN charge assign- ment that allows the scalar potential to be naturally free of awkward terms violating the PQ symmetry must satisfy the following minimal conditions: TABLE II. Assignment of quantum charges when � is included. Q�L Q3L (uaR, u 0 3R) (daR, d 0 �R) �aL eaR � (�, �) Uð1ÞX 0 1=3 2=3 �1=3 �1=3 �1 2=3 �1=3 Uð1ÞB 1=3 1=3 1=3 1=3 0 0 0 0 J. C. MONTERO AND B. L. SÁNCHEZ-VEGA PHYSICAL REVIEW D 84, 055019 (2011) 055019-6 ZNð�Þ � ð0; N=2; N=3; N=4Þ; (35) ZNð�Þ þ ZNð�Þ þ ZNð�Þ � pN; (36) � ZNð�Þ þ ZNð�Þ ¼ rN; p; r 2 Z; (37) and, obviously, the other ones that leave the rest of the Lagrangian invariant under ZN. The �ZNð�Þ þ ZNð�Þ ¼ rN condition, with r 2 Z, is necessary to allow the terms in the scalar potential given in Eq. (29), except the trilinear f�ijk�i�j�k term and, thus, avoid the appearance of an additional dangerous massless scalar in the physical spectrum. In other words, with the conditions imposed by Eqs. (35)–(37) for this ZN discrete symmetry, none of Lagrangian terms, except the violating PQ terms, such as f�ijk�i�j�k, �2, �3, �4, etc., are prohibited from appearing. Furthermore, to stabilize the axion solution from quan- tum gravitational effects [36,37] we will make use of the ZN discrete symmetry with anomaly cancellation by a discrete version of the Green-Schwarz mechanism [38–41]. Quantum gravity effective operators, allowed by the gauge symmetry, of the form �N=MN�4 Pl can induce a nonzero �� given by �� ’ fNa �4 QCDM N�4 Pl : (38) From the neutron electric dipole moment (EDM) experi- mental data �� & 10�11 [29], and using fa � 1010 GeV, we find that N � 10, in order to keep the PQ solution stable. It means that effective operators with N < 10 must be for- bidden by the ZN symmetry. The neutron EDMwill also receive contributions that do not come from the �� ~GG term. Those which are similar to the SM contributions will pose no problems since they will have approximately the same values and will give dSM-CKMn � 10�32 e cm [42], i.e., 6 to 7 orders of magni- tude smaller than the experimental limit [43]. The other contributions, which are specific to the present 3-3-1 model, like the 1-loop contribution due to the exchange of a charged scalar (��), can be used to constrain the still free parameters of the model, in order to be consistent with the experimental neutron EDM data [43,44]. Wewill return to this point later. Before introducing the ZN symmetry to stabilize the PQ mechanism, we calculate the axion state. With the intro- duction of the scalar singlet �, the scalar potential gains the following extra terms: V�;extra ¼ � 2 �� y�þ ��ð�y�Þ2 þ �15ð�y�Þð�y�Þ þ �16ð�y�Þð�y�Þ þ �17ð�y�Þð�y�Þ: (39) Now, to calculate the eigenstate of the axion field, we write the fields as � ¼ �þ 1ffiffi 2 p ðV�0 þ Re�0 þ iIm�0Þ �þþ 0 BB@ 1 CCA; � ¼ 1ffiffi 2 p ðV�0 þ Re�0 þ iIm�0Þ �� 1ffiffi 2 p ðV�0 1 þ Re�0 1 þ iIm�0 1Þ 0 BB@ 1 CCA; � ¼ 1ffiffi 2 p ðV�0 þ Re�0 þ iIm�0Þ �� 1ffiffi 2 p ðV�0 1 þ Re�0 1 þ iIm�0 1Þ 0 BB@ 1 CCA; � ¼ 1ffiffiffi 2 p ðV� þ Re�þ iIm�Þ: (40) The axion field must be isolated from the eight NG bosons that are absorbed by the gauge bosons in the unitary gauge. This is fundamental to do a correct phenomenological study of the axion properties. By following standard pro- cedures, the axion field, aðxÞ, is determined to be aðxÞ ¼ 1 fa � V2� V�0 Im�0 � V�0 1 Im�0 þ V�0 Im�0 1 þ V�0 1 Im�0 � V�0 Im�0 1 � � V2� V2 �0 þ V2þ V2� � V�Im� � ; (41) where V2� � V�0V�0 1 � V�0 1 V�0 ; (42) V2þ � V2 �0 þ V2 �0 1 þ V2 �0 þ V2 �0 1 ; (43) and fa is the normalization constant given by fa � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� V2� V�0 � 2 þ V2þ þ � V2� V2 �0 þ V2þ V2� � 2 V2 � vuut : (44) Note that in the limit V� � V�0 , V�0 1 , V�0 , V�0 1 aðxÞ ’�Im�þ � V2� V2 �0 þV2þ V2� ��1 V�1 � � V2� V�0 Im�0�V�0 1 Im�0 þV�0 Im�0 1þV�0 1 Im�0�V�0 Im�0 1 � ; (45) i.e., the axion is primarily composed of the Im� field. As is well-known, to make the invisible axion compatible with astrophysical and cosmological considerations, the axion decay constant, fa, must be in the range 109 GeV fa 1012 GeV. Now, returning to the stabilization of the axion by the ZN symmetry, let us put that in a short way. If the ZN symmetry that survives at low energies was part of an ‘‘anomalous’’ Uð1ÞA gauge symmetry, the ZN charges of the fermions in the low energy theory must satisfy nontrivial conditions: The anomaly coefficients for the full theory are given by NATURAL PECCEI-QUINN SYMMETRY IN THE 3-3-1 . . . PHYSICAL REVIEW D 84, 055019 (2011) 055019-7 the coefficients for the low energy sector, in our case A3C � ½SUð3ÞC�2Uð1ÞA and A3L � ½SUð3ÞL�2Uð1ÞA, plus an integer multiple of N=2 [45,46], i.e., A3C þ pN=2 k3C ¼ A3L þ rN=2 k3L ¼ �GS; (46) with p and r being integers. The k3C and k3L are the levels of the Kac-Moody algebra for the SUð3ÞC and SUð3ÞL, respectively. In the present case these are positive integers. Finally, the �GS is a constant that is not specified by the low energy theory alone. Other anomalies such as ½Uð1ÞA�3, ½Uð1ÞA�2Uð1ÞX do not give useful low energy constraints because these depend on some arbitrary choices concern- ingUð1ÞA [47]. This is why these do not appear in Eq. (46). Now, to identify that anomalous Uð1ÞA symmetry, it is helpful to write it as a linear combination of the Uð1ÞPQ and the Uð1ÞB symmetries, i.e., Uð1ÞA ¼ �½Uð1ÞPQ þ Uð1ÞB�; (47) where � is a normalization constant used to make the Uð1ÞA charges integer numbers. With the charges given in Table I, it is straightforward to calculate the anomaly coefficients A3C and A3L, A3C ¼ � 3 2 �; A3L ¼ � � 9 4 þ 3 2 � �: (48) Thus, the parameter that satisfies the condition given in Eq. (46) is ¼ 1 3 � �3 k3L k3C þ 9 2 þ N � � k3L k3C p� r �� : (49) Taking the simplest possibility for the parameters k3C and k3L, i.e., k3C ¼ k3L, the parameter becomes ¼ 1 3 � 3 2 þ N � ðp� rÞ � : (50) Recalling that to stabilize the axion from the quantum gravity corrections we needN > 10, we show two possible solutions with N ¼ 10 and 11. The corresponding charge assignment of these two discrete subgroups of the Uð1ÞA symmetry is given in Table III. Also, it is important to remember that those charges are defined mod N. It can be explicitly verified that the charges in Table III satisfy Eq. (46), as it should be, since Z10 and Z11 are discrete subgroups of Uð1ÞA, which is anomaly-free by the Green-Schwarz mechanism. At this point, an important remark is in order. In its most general form, this model possesses other CP-violating sources apart from the strong CP-violating �� term, which can give additional contributions to the electric dipole moment of the neutron. The reason is that not all phases can be absorbed into the quark and lepton field definitions. Therefore, it is necessary to estimate if these additional contributions do not require tuning the model parameters at the same order of the �� parameter. Then, let us compute a representative case: the up-quark electric dipole moment, deu. One dominant diagram contributing to deu is derived from the one given in Fig. 1(b), when an external photon line is attached. To compute the resultant diagram, we need to know the mixing of the scalar fields, Cij, coming from the diagonalization of the scalar mass matrix. However, we will consider Cij �Oð1Þ, which is the worst case. Standard calculations lead to deujm�mu0 ;m� � eG5 3jG1j sin� 48�2 mu0 m2 � KðrÞ; (51) where K ðrÞ ¼ 1 2r � 1 r2 þ 1 r3 lnð1þ rÞ; (52) with r ¼ m2 u0 m2 � � 1; mu0 and m� are the exotic quark and scalar masses, respectively; G5 3 and G1 are the Yukawa couplings given in Eq. (5); and sin� is the sine of the CP-violating phase � related to the complex parameter G1. Also, we have taken the limit m � m� and m � mu0 , with m the up-quark mass. Furthermore, to give numerical results, it is interesting to consider mu0 � m� in Eq. (51), since these two exotic particles obtain mass from the same vev, V�0 1 . In this case, we have G5 3jG1j sin� � 1 TeV m� � & 2:1 10�6; (53) since Kð0Þ ¼ 1=3. To obtain the bound in Eq. (53), we have used den � 4 3d e d � 1 3 d e u � OðdeuÞ< 0:29 10�25 e � cm [48]. Now, for instance, let us assume that the CP-violating phase is such that sin�� 10�2 and m� � 1 TeV. In this case the parameters G5 3 � 10�2 and jG1j � 10�2 satisfy the upper bound given in Eq. (53). In the general case, i.e.,m� � mu0 , it can be shown that when m� >mu0 the limit on the couplings is softer than the one given in Eq. (53). Hence, since the order of the model parameters differs from �� & 10�11 by several order of magnitude, a solution to the strong CP problem, as the one presented above, is required. TABLE III. The charge assignments for Z10 and Z11 that stabilize the axion, for � ¼ 6. Q�L Q3L (uaR, u 0 3R) (daR, d 0 �R) �aL eaR � (�, �) � Z10 þ5 þ7 þ1 þ1 þ7 þ1 þ6 þ6 þ2 Z11 þ6 þ7 þ1 þ1 þ8 þ2 þ6 þ6 þ4 J. C. MONTERO AND B. L. SÁNCHEZ-VEGA PHYSICAL REVIEW D 84, 055019 (2011) 055019-8 IV. CONCLUSIONS In this paper we have shown a detailed and compre- hensive study concerning the implementation of the PQ symmetry into a 3-3-1 model in order to solve the strong CP problem. We have considered a version of the 3-3-1 model in which the scalar sector is minimal. In its original form this version has only two scalar triplets (�, �) and it is found that the model presents an auto- matic PQ-like symmetry. However, for this scalar con- tent, there is a Uð1Þ subgroup of Uð1ÞX �Uð1ÞPQ that remains unbroken and hence no axion field, aðxÞ, arises. Therefore, the strong CP problem cannot be solved by the dynamical properties of the axion field. However, as we have shown in the text, the problem can be solved due to the appearance of three massless quark states. We show explicitly that those massless quark states remain massless to all orders in perturbation theory. This solu- tion is disfavored since results from lattice and current algebra do not point in that direction. When the model is slightly extended by the addition of a third scalar triplet �, with the same quantum numbers as �, we do not have massless quarks anymore but we cannot implement a PQ symmetry in a natural way. The trilinear term in the scalar potential forbids this symmetry. We can resort to a Z2 symmetry to remove the trilinear term. In this case, we can define a PQ symmetry and an axion field appears in the physical scalar spectrum. Unfortunately this axion is visible since it is related to the V�0 energy scale, which is of the order of the electroweak scale. Therefore, the model must be extended. We have succeeded in imple- menting a stable PQ mechanism by introducing a � scalar singlet and a ZN discrete gauge symmetry. The introduction of the � scalar makes the axion invisible provided V� * 109 GeV, i.e., aðxÞ ’ Im�. On the an- other hand, the ZN protects the axion against quantum gravity effects because both it is anomaly-free, as it was shown by using a discrete version of the Green-Schwarz mechanism, and it forbids all effective operators of the form ��N=MN�4 Pl , with N < 10, which could destabilize the PQ mechanism. ACKNOWLEDGMENTS B. L. Sánchez-Vega was supported by CAPES. [1] F. Pisano and V. Pleitez, Phys. Rev. D 46, 410 (1992). [2] P. H. Frampton, Phys. Rev. Lett. 69, 2889 (1992). [3] R. Foot, H.N. Long, and T. A. Tran, Phys. Rev. D 50, R34 (1994). [4] C. A. de S. Pires and O. P. Ravinez, Phys. Rev. D 58, 035008 (1998). [5] C. A. de S. Pires, Phys. Rev. D 60, 075013 (1999). [6] Palash B. Pal, Phys. Rev. D 52, 1659 (1995). [7] Alex G. Dias and V. Pleitez, Phys. Rev. D 69, 077702 (2004). [8] S. Y. Khlebnikov and M. E. Shaposhnikov, Phys. Lett. B 203, 121 (1988). [9] S.M. Barr, Phys. Rev. D 30, 1805 (1984). [10] S.M. Barr, Phys. Rev. D 34, 1567 (1986). [11] A. E. Nelson, Phys. Lett. 136B, 387 (1984). [12] D. B. Kaplan and A.V. Manohar, Phys. Rev. Lett. 56, 2004 (1986). [13] J. Gasser and H. Leutwyler, Phys. Rep. 87, 77 (1982). [14] D. R. Nelson, G. T. Fleming, and G.W. Kilcup, Phys. Rev. Lett. 90, 021601 (2003). [15] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38, 1440 (1977). [16] R. D. Peccei and H. R. Quinn, Phys. Rev. D 16, 1791 (1977). [17] Jihn E. Kim, Phys. Rev. Lett. 43, 103 (1979). [18] M. Dine, W. Fischler, and M. Srednicki, Phys. Lett. 104B, 199 (1981). [19] A. P. Zhitnitskii, Yad. Fiz. 31, 497 (1980) [Sov. J. Nucl. Phys.31, 260 (1980)]. [20] P. V. Dong, Tr. T. Huong, D. T. Huong, and H.N. Long, Phys. Rev. D 74, 053003 (2006). [21] William A. Ponce, Yithsbey Giraldo, and L.A. Sánchez, Phys. Rev. D 67, 075001 (2003). [22] William A. Ponce, J. B Flórez, and L.A. Sánchez, Int. J. Mod. Phys. A 17, 643 (2002). [23] P. V. Dong, H. N. Long, and D.V. Soa, Phys. Rev. D 75, 073006 (2007). [24] P. V. Dong, H. N. Long, D. T. Nhung, and D.V. Soa, Phys. Rev. D 73, 035004 (2006). [25] D. A. Gutiérrez, W.A. Ponce, and L.A Sánchez, Int. J. Mod. Phys. A 21, 2217 (2006). [26] Yithsbey Giraldo, William A. Ponce, and Luis A. Sánchez, Eur. Phys. J. C 63, 461 (2009). [27] T. Banks, Y. Nir, and N. Seiberg, in Yukawa Couplings and the Origins Of Mass, edited by P. Ramond (International Press, Boston, 1996), ISBN 1-57146-025-X. [28] G. ’t Hooft, Phys. Rev. Lett. 37, 8 (1976). [29] J. Kim and G. Carosi, Rev. Mod. Phys. 82, 557 (2010). [30] T. P. Cheng and Marc Sher, Phys. Rev. D 35, 3484 (1987). [31] J. C. Montero, F. Pisano, and V. Pleitez, Phys. Rev. D 47, 2918 (1993). [32] P. Langacker and D. London, Phys. Rev. D 38, 886 (1988). [33] R. H. Benavides, Y. Giraldo, and W.A. Ponce, Phys. Rev. D 80, 113009 (2009). [34] W.A. Bardeen, R. D. Peccei, and T. Yanagida, Nucl. Phys. B279, 401 (1987). [35] Lawrence M. Krauss and Frank Wilczek, Phys. Rev. Lett. 62, 1221 (1989). NATURAL PECCEI-QUINN SYMMETRY IN THE 3-3-1 . . . PHYSICAL REVIEW D 84, 055019 (2011) 055019-9 http://dx.doi.org/10.1103/PhysRevD.46.410 http://dx.doi.org/10.1103/PhysRevLett.69.2889 http://dx.doi.org/10.1103/PhysRevD.50.R34 http://dx.doi.org/10.1103/PhysRevD.50.R34 http://dx.doi.org/10.1103/PhysRevD.58.035008 http://dx.doi.org/10.1103/PhysRevD.58.035008 http://dx.doi.org/10.1103/PhysRevD.60.075013 http://dx.doi.org/10.1103/PhysRevD.52.1659 http://dx.doi.org/10.1103/PhysRevD.69.077702 http://dx.doi.org/10.1103/PhysRevD.69.077702 http://dx.doi.org/10.1016/0370-2693(88)91582-1 http://dx.doi.org/10.1016/0370-2693(88)91582-1 http://dx.doi.org/10.1103/PhysRevD.30.1805 http://dx.doi.org/10.1103/PhysRevD.34.1567 http://dx.doi.org/10.1016/0370-2693(84)92025-2 http://dx.doi.org/10.1103/PhysRevLett.56.2004 http://dx.doi.org/10.1103/PhysRevLett.56.2004 http://dx.doi.org/10.1016/0370-1573(82)90035-7 http://dx.doi.org/10.1016/0370-1573(82)90035-7 http://dx.doi.org/10.1103/PhysRevLett.90.021601 http://dx.doi.org/10.1103/PhysRevLett.90.021601 http://dx.doi.org/10.1103/PhysRevLett.38.1440 http://dx.doi.org/10.1103/PhysRevLett.38.1440 http://dx.doi.org/10.1103/PhysRevD.16.1791 http://dx.doi.org/10.1103/PhysRevD.16.1791 http://dx.doi.org/10.1103/PhysRevLett.43.103 http://dx.doi.org/10.1016/0370-2693(81)90590-6 http://dx.doi.org/10.1016/0370-2693(81)90590-6 http://dx.doi.org/10.1103/PhysRevD.74.053003 http://dx.doi.org/10.1103/PhysRevD.67.075001 http://dx.doi.org/10.1142/S0217751X02005815 http://dx.doi.org/10.1142/S0217751X02005815 http://dx.doi.org/10.1103/PhysRevD.75.073006 http://dx.doi.org/10.1103/PhysRevD.75.073006 http://dx.doi.org/10.1103/PhysRevD.73.035004 http://dx.doi.org/10.1103/PhysRevD.73.035004 http://dx.doi.org/10.1142/S0217751X06029442 http://dx.doi.org/10.1142/S0217751X06029442 http://dx.doi.org/10.1140/epjc/s10052-009-1101-4 http://dx.doi.org/10.1103/PhysRevLett.37.8 http://dx.doi.org/10.1103/RevModPhys.82.557 http://dx.doi.org/10.1103/PhysRevD.35.3484 http://dx.doi.org/10.1103/PhysRevD.35.3484 http://dx.doi.org/10.1103/PhysRevD.47.2918 http://dx.doi.org/10.1103/PhysRevD.47.2918 http://dx.doi.org/10.1103/PhysRevD.38.886 http://dx.doi.org/10.1103/PhysRevD.80.113009 http://dx.doi.org/10.1103/PhysRevD.80.113009 http://dx.doi.org/10.1016/0550-3213(87)90003-4 http://dx.doi.org/10.1016/0550-3213(87)90003-4 http://dx.doi.org/10.1103/PhysRevLett.62.1221 http://dx.doi.org/10.1103/PhysRevLett.62.1221 [36] M. Kamionkowski and J. March-Russell, Phys. Lett. B 282, 137 (1992). [37] R. Holman, Stephen D.H. Hsu, T.W. Kephart, Edward W. Kolb, R. Watkins, and L.M. Widrow, Phys. Lett. B 282, 132 (1992). [38] M.B. Green and J. H. Schwarz, Phys. Lett. 149B, 117 (1984). [39] M.B. Green and J. H. Schwarz, Nucl. Phys. B255, 93 (1985). [40] M. Green, J. Schwarz, and P. West, Nucl. Phys. B254, 327 (1985). [41] K. S. Babu, Ilia Gogoladze, and Kai Wang, Phys. Lett. B 560, 214 (2003). [42] B. H. J. McKellar, S. R. Choudhury, X.G. He, and S. Pakvasa, Phys. Lett. B 197, 556 (1987). [43] Jen-Chieh Peng, Mod. Phys. Lett. A 23, 1397 (2008). [44] J. C. Montero, V. Pleitez, and O. Ravinez, Phys. Rev. D 60, 076003 (1999). [45] Luis E. Ibáñez, Nucl. Phys. B398, 301 (1993). [46] K. S. Babu, Ilia Gogoladze, and Kai Wang, Nucl. Phys. B660, 322 (2003). [47] T. Banks and M. Dine, Phys. Rev. D 45, 1424 (1992). [48] K. Nakamura et al. (Particle Data Group), J. Phys. G 37, 075 021 (2010). J. C. MONTERO AND B. L. SÁNCHEZ-VEGA PHYSICAL REVIEW D 84, 055019 (2011) 055019-10 http://dx.doi.org/10.1016/0370-2693(92)90492-M http://dx.doi.org/10.1016/0370-2693(92)90492-M http://dx.doi.org/10.1016/0370-2693(92)90491-L http://dx.doi.org/10.1016/0370-2693(92)90491-L http://dx.doi.org/10.1016/0370-2693(84)91565-X http://dx.doi.org/10.1016/0370-2693(84)91565-X http://dx.doi.org/10.1016/0550-3213(85)90130-0 http://dx.doi.org/10.1016/0550-3213(85)90130-0 http://dx.doi.org/10.1016/0550-3213(85)90222-6 http://dx.doi.org/10.1016/0550-3213(85)90222-6 http://dx.doi.org/10.1016/S0370-2693(03)00411-8 http://dx.doi.org/10.1016/S0370-2693(03)00411-8 http://dx.doi.org/10.1016/0370-2693(87)91055-0 http://dx.doi.org/10.1142/S0217732308027771 http://dx.doi.org/10.1142/S0217732308027771 http://dx.doi.org/10.1103/PhysRevD.60.076003 http://dx.doi.org/10.1103/PhysRevD.60.076003 http://dx.doi.org/10.1016/0550-3213(93)90111-2 http://dx.doi.org/10.1016/S0550-3213(03)00258-X http://dx.doi.org/10.1016/S0550-3213(03)00258-X http://dx.doi.org/10.1103/PhysRevD.45.1424 http://dx.doi.org/10.1103/PhysRevD.45.1424 http://dx.doi.org/10.1088/0954-3899/37/7A/075021 http://dx.doi.org/10.1088/0954-3899/37/7A/075021