IMSc/2001/04/13
UWThPh200114
Ific/0118
The solar LMA neutrino oscillation solution
in the Zee model
Abstract
We examine the neutrino mass matrix in the version of Zee model where both Higgs doublets couple to the leptons. We show that in this case one can accommodate the large mixing angle (LMA) MSW solution of the solar neutrino problem, while avoiding maximal solar mixing and conflicts with constraints on lepton family numberviolating interactions. In the simplified scenario we consider, we have the neutrino mass spectrum characterized by and , where is the solar mixing angle.
1 Introduction
The results of the atmospheric [1] and solar neutrino experiments (for a recent presentation of the solar results see, e.g., Ref. [2]) indicate that neutrinos have nonzero masses. A suitable environment for obtaining Majorana neutrino masses is to extend the Higgs sector of the Standard Model [3]. Extending the Standard Model with a singly charged gauge singlet scalar and adding a second Higgs doublet allows to write down an explicitly lepton numberviolating interaction in the Higgs potential and leads to 1loop neutrino masses [4]. In the following we will only consider the lepton sector of the Zee model.
The Zee model is traded in two versions in the literature: the original model [4], which we will call general Zee model (GZM) in this paper, and a simpler version of the Zee model where only one of the two Higgs doublets couples in the lepton sector [5]. The latter version, which we will call restricted Zee model (RZM), can naturally be achieved with a discrete symmetry [6]; it has the advantage that the family lepton numberviolating interactions mediated by the couplings of the three physical neutral scalars are absent. The interesting point is that the RZM leads to a symmetric neutrino mass matrix with zeros in the diagonal^{4}^{4}4We always work in a basis where the charged lepton mass matrix is diagonal.; this mass matrix is called Zee mass matrix in the literature. One can easily check that all phases in the Zee mass matrix can be absorbed into the lefthanded neutrino fields and, therefore, in neutrino oscillations no CP violation is observable if the Zee mass matrix is the correct neutrino mass matrix.
The Zee mass matrix has a special feature: it has been shown [7, 8] that it allows to accommodate only bimaximal mixing, i.e., solar and atmospheric mixing angles are both very close to . While this is in excellent agreement with the atmospheric neutrino data, results from the solar neutrino experiments are not that well compatible with a solar mixing angle , though this value is also not excluded [2, 9]. The purpose of the present paper is to show that in the GZM it is no problem to accommodate the large mixing angle (LMA) MSW solution of the solar neutrino deficit, while at the same time all constraints on the additional couplings in the lepton sector stemming from the second Higgs doublet are respected. In the GZM also the diagonal elements of the neutrino mass matrix are nonzero in general. Since the GZM is a quite rich and intricate model, we restrict ourselves rather to an “existence proof” of the LMA MSW solution within the GZM instead of discussing the GZM in full generality.
Let us outline our procedure:

The general problem is still quite intricate, so we set one of these parameters of the mass matrix equal to zero and assume that the remaining ones are real. In this scenario we can relate in a simple way the three real parameters with the physical quantities , the solar mixing angle, , the solar masssquared difference, , the atmospheric masssquared difference and ( with are the neutrino masses) and it is possible to have the LMA angle solution of the solar neutrino problem^{5}^{5}5This means that is large but safely smaller than . (Section 3).

Now the GZM is brought into play. After a discussion of the neutrino mass matrix in this model (Section 4) we assume that all quantities in the model are real and we set to zero all but two of the additional Yukawa couplings present in the GZM; this has the purpose of avoiding as many family lepton numberviolating neutral scalar interactions a possible. We show that with these two additional coupling constants the restricted mass matrix mentioned in Point 2, which allows for the LMA MSW solution, can be accommodated (Section 4).

We complete our procedure by a numerical discussion of the parameters of our scenario and by estimates of the rates of (family) lepton numberviolating processes (Section 6).
We also review the features of the RZM as a limit of the GZM (Section 7) and summarize the results (Section 8). In the appendix we present the general formulas for the 1loop Majorana neutrino mass matrix induced by charged scalar loops.
2 Neutrino mixing and the mass matrix
The Majorana neutrino mass matrix is diagonalized with a unitary matrix by
(1) 
With the assumptions mentioned in the introduction in Point 1, we can write the matrix as
(2) 
The phases in suggest the definitions
(3) 
Then can be expressed by and the parameters of in the following way:
(4) 
with and (see, e.g., Refs. [7, 11]). Thus, the mass matrix has the structure
(5) 
Consequently, for having and an atmospheric mixing angle of exactly the following conditions on are necessary:
Condition 1:  (6a)  
Condition 2:  (6b)  
Condition 3:  (6c) 
Finally, with the parameterization (5) the complex masses are found as
(7a)  
(7b)  
(7c) 
3 A simplified mass matrix
Having discussed the general form of the mass matrix which leads to the mixing matrix (2), we now investigate the consequences of the following simplifying assumptions:
(8) 
In the next section, this scenario will be reproduced in the framework of the Zee model. With the reality assumptions, the quantities (3) are identical with the neutrinos masses apart from possible signs. The experimentally accessible quantities are expressed as
(9a)  
(9b)  
(9c)  
(9d) 
by the parameters , , . We have chosen as representative of the absolute neutrino mass values, since it is simply given by Eq. (9d). Without loss of generality we will adopt henceforth the following conventions: , , . It follows from the last relation and from Eq. (7c) that is positive. Note that in Eq. (9c) no absolute value of the righthand side of the equation is necessary, because it must be positive. The argument goes as follows. Suppose that . Then it follows that . Therefore, is positive, which allows to derive the inequality from Eq. (9b). This is a contradiction to the values of the masssquared differences, fitted from the data [2].
In Eqs. (9a), (9b), (9c) and (9d), four physical quantities are expressed by three parameters. Therefore, a consistency condition exists, which is given by
(10) 
The signs and occurring in this equation are and . In the context of the Zee model we will finally need the relations
(11a)  
(11b) 
Looking at the consistency condition (10) and assuming that is of the order of or smaller, we obtain , which amounts to bimaximal mixing. Since we want to show that the Zee model allows to avoid bimaximal mixing we concentrate on
(12) 
With this assumption it is easy to obtain an approximate expression for . One can check that for one arrives again at bimaximal mixing. Using , we have and we easily calculate
(13) 
Using this equation the condition (12) implies
(14) 
Equation (13) together with and , allows to estimate from Eq. (11a) that
(15) 
From Eqs. (9a) and (9b) and the convention , we can express the masses and as
(16a)  
(16b) 
Inspection of Eqs. (7a) and (7b) reveals that our conventions fix the signs of : . Then, with Eq. (15), an estimate of the neutrino masses which neglects the solar masssquared difference is given by
(17) 
This equation tells us that , at least in the regime of large solar mixing.
4 Neutrino masses in the general Zee model
In the previous section we have discussed the mass matrix determined by Eqs. (5) and (8) without reference to any specific model of neutrino masses. Now we introduce the Zee model [4] and discuss the neutrino mass matrix in the case that both scalar doublets of the Zee model couple in the lepton section. The Yukawa Lagrangian is given by
(18) 
where is an antisymmetric matrix [4]. The mass matrix of the charged leptons arises at tree level through
(19) 
with GeV. The physical charged scalar fields , with masses , , respectively, and the wouldbe Goldstone boson are obtained by the unitary transformation [4, 6, 12]
(20) 
As anticipated in the introduction, we assume to be in a basis where the charged lepton mass matrix is diagonal, i.e., , where the hat symbolizes that this mass matrix is diagonal. In this basis we have
(21) 
In our parameterization the flavourchanging Higgs couplings are given by the offdiagonal elements of . Furthermore, offdiagonal elements of the 22 unitary matrix are present because of the vacuum expectation value of the lepton numberviolating term
(22) 
in the Higgs potential.
The physical charged Higgses couple to the leptons in the following way:
(23)  
With the formulas in the appendix we obtain [4]
(24) 
with
(25) 
The infinity in Eq. (24) cancels [4] because . Defining
(26) 
from Eq. (24) we obtain the final result [4, 12]
(27)  
with () and . Note that for the function simplifies to
(28) 
For the product of elements of the chargedscalar mixing matrix , we obtain the relation
(29) 
It shows explicitly that the Majorana neutrino masses are proportional to the coupling in the Higgs potential (22).
In the GZM considered here, there are family lepton numberviolating processes induced by the charged and the neutral scalar interactions. Experimental bounds constrain the coupling matrices and .
5 The simplified mass matrix within the Zee model
In this section, our aim is to reproduce the neutrino mass matrix defined by Eqs. (5) and (8) within the GZM. In order to save the amount of writing we introduce the notation
(30) 
Both constants and are of the order of 1 GeV, resulting from dividing the electroweak scale by :
(31) 
Since the offdiagonal elements of introduce flavourchanging neutral interactions, we adopt the philosophy to set to zero as many of them as possible. As we will see, it turns out that
(32) 
and all other elements of being equal to zero, is sufficiently general to avoid bimaximal mixing, which necessarily happens for [7]. It can easily be checked that with this assumption Condition 2 (6b) is fulfilled by having . Furthermore, we assume that all quantities we deal with are real: , , the elements of the matrix , and and . Thus we identify with .
6 Numerical estimates
Let us now estimate the values of the coupling constants. For definiteness we take eV and GeV. Furthermore, we need the values of and . Defining and assuming that^{6}^{6}6This avoids some finetuning for . Note, however, that is also possible. and , we obtain
(34a)  
(34b)  
(34c)  
(34d) 
These equations serve to see the orders of magnitude and any effects of are neglected. As can be seen from Eq. (15), a considerable amount of finetuning is involved in order to reproduce the solar masssquared difference.
Now we concentrate on the LMA MSW solution of the solar neutrino problem, where is in the first octant. With Eq. (9a) it follows that . In this case a representative value of the mixing angle is given by the best fit value () of Ref. [2], with the corresponding masssquared difference eV. We note that in this case we have , which is similar to the case of the Zee mass matrix [7]. On the other hand, in our scenario we have , whereas for the Zee mass matrix the relation holds [7]. As far as is concerned, with we can have .
Due to our assumption (32), flavourchanging neutral scalar interactions at the tree level are very constrained. Among the charged lepton decays we only have with . A generous estimate of the branching ratio of this decay for is obtained by
(35) 
where we have taken ( GeV) and is a generic neutral Higgs mass. At the 1loop level, neutral Higgs exchange also induces the decay . Making again an estimate, we obtain [13, 14]
(36) 
Both estimates are well compatible with the experimental upper bounds on these branching ratios of the order of [15]. More detailed discussions of these decays are found in Refs. [14, 16]. We have used and in Eqs. (35) and (36).
Also the charged scalars participate in various charged lepton decays as intermediate particles. Numerous decays of the type proceed at tree level via the Lagrangian (23). According to the couplings in this Lagrangian we can distinguish between , and vertices, and we can have decay amplitudes with all possible combinations of these vertices, except , which is forbidden due to the restricted form of , Eq. (32). E.g., to the Standard Model amplitude of ordinary muon decay there is an amplitude with couplings and another one with ; another example is with couplings , which is lepton number violating. The branching ratios of all these decays are negligible because of the smallness of the coupling constants and and the ratios . Also negligible are radiative decays induced by charged Higgs loops [13]. In this case one always has two vertices in the loop graph, except in the case of the amplitude for , where there are two contributions, proportional to and . Recent reviews of the restrictions on the coupling constants are found in Refs. [17, 18].
Scalar contributions to the anomalous magnetic moments of the electron and muon involving [18] and couplings are totally negligible because these constants are too small. A contribution from the couplings to the electron magnetic moment coming from exchange is proportional to [19] and is thus zero in view of .
In our scenario we have . The matrix element is identical with the effective neutrino mass probed in neutrinoless doublebeta decay. Therefore, this decay is allowed and the effective neutrino mass is given by
(37) 
This represents an order of magnitude which is accessible in future experiments (for recent reviews see, e.g., Ref. [20]).
Having seen that the LMA MSW solution of the solar neutrino problem can be accommodated in our scenario, we now proceed to the small mixing angle (SMA) MSW solution. In this case we take as illustration the best fit value of Ref. [2], (), which has a corresponding eV. From Eq. (34a) we see that now becomes relatively large and barely compatible with the requirement that the Zee boson does not have an effect on the muon decay rate so that it does not destroy the agreement in electroweak precision tests [17]. Thus in our simple scenario we cannot incorporate safely the SMA solution.
7 The limit
To make contact with Refs. [7, 8], we explore the effect of . This means that all diagonal elements of the neutrino mass matrix are zero [4] and, in our notation, we have , i.e., and (see (Eq. (11a)). These relations and inspection of the consistency condition (10) leads to
(38) 
From the last relation we read off that is 45 for all practical purposes. The consequences for the coupling matrix are
(39) 
The first of these two relations is obtained by taking the ratio of Eq. (33b) and Eq. (33a), where is taken into account. In the second relation, Eq. (11b) has been used. The results (38) and (39) agree with those of Refs. [7, 8].
8 Summary
In this paper we have discussed neutrino masses and mixing in the general Zee model, where both Higgs doublets couple in the lepton sector. In this endeavour, we were motivated by the result that for , where denotes the Yukawa coupling matrix of the second Higgs doublet, the Zee mass matrix leads to bimaximal mixing. It is true that a mixing angle of is perfect for the description of the atmospheric neutrino data, but it does not represent a very good fit for the solar neutrino data. The general Zee model is a rather rich and intricate model. Therefore, in order to simplify the analysis, we have assumed that all quantities appearing in the neutrino mass matrix are real and that in the second Yukawa coupling matrix , which is set to zero usually, only two elements, the and elements, are nonzero. We have shown that this is sufficient to accommodate solutions of the solar neutrino problem with a large mixing angle instead of maximal mixing; on the other hand, in the atmospheric sector maximal mixing remains. At the same time, all dangerous processes induced by the new couplings are sufficiently suppressed. Actually, we could have even set the element of equal to zero and used a single nonzero element in this coupling matrix. However, the element represents a possibility to achieve a further suppression of the potentially more dangerous element. As in the case , the neutrino mass is the smallest neutrino mass, and the resulting mass spectrum is rather of the type which is called “inverted hierarchy”. However, whereas for one has , now is of the same order of magnitude as the other two masses.
Numerous finetunings are involved in our scenario. Implementing the condition (6a) and setting all but two elements of equal to zero represent rather severe finetunings. These procedures were useful for exactly having and an atmospheric mixing angle of , and for avoiding a dangerous class of lepton family numberviolating interactions. In order to reproduce the neutrino masses within our scenario, we need and a quite drastic finetuning between and in the mass matrix (5), in order to accommodate the solar masssquared difference. Of course, the masssquared difference eV of the quasivacuum oscillation solution [2] would require a stronger finetuning than eV of the large mixing angle solution. Examination of the Zee model with a general and with small deviations of the neutrino mass matrix from the form (5) should reveal how much our finetunings could be relaxed. In the present work we have confined ourselves to prove that it is possible to reproduce the LMA MSW solution within the Zee model.
In conclusion, even in the case that bimaximal mixing is ruled out, the Zee model will remain a viable and interesting scenario in order to accommodate the neutrino oscillation solutions of the solar and atmospheric neutrino problems.
Acknowledgements
This work was supported by Spanish DGICYT under grant PB980693, by the European Commission TMR network HPRNCT200000148 and the ESF network 86. T.S. is supported by the Marie Curie Training Site Program for Particle Physics Beyond the Standard Model, contract number HPMT200000124.
Appendix A Charged scalar exchange and the 1loop neutrino mass
We proceed from the general Lagrangian
(A1) 
where the are 33 coupling matrices in the case of 3 families and denote chiral charged fermion fields. The neutrino Majorana mass Lagrangian is defined by
(A2) 
With the scalar interactions (A1) we obtain
(A3) 
where we have defined
We have used dimensional regularization; thus, with being the number of spacetime dimensions, is Euler’s constant and is an arbitrary mass scale.
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