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Escape from washing out of baryon number in a two-zero-texture general Zee model compatible with the large mixing angle MSW solution

We propose a two-zero-texture general Zee model, compatible with the large mixing angle Mikheyev-Smirnov-Wolfenstein solution. The washing out of the baryon number does not occur in this model for an adequate parameter range. We check the consistency of a model with the constraints coming from flavor changing neutral current processes, the recent cosmic microwave background observation, and the Z-burst scenario.


I. INTRODUCTION
problem in the RZM, and complete baryon number washing out does not occur [27].
In the present paper, we focus on the original Zee model in which both of the two Higgs doublets couple to the ordinary matter fields. We call this model the general Zee model (GZM) in order to clearly distinguish it from the RZM [15]. Although this model was proposed earlier, it has not been extensively discussed, as we mentioned above. We shed light on the GZM once again and find not only that the two phenomenological problems above can be avoided [58] but also that the GZM can be consistent with experiments involving FCNCs. This model is attractive in the sense that the extension from the RZM, of which properties have been extensively discussed, is minimal. Although the extension is minimal, many undetermined parameters arise and the investigation in the full parameter region is very difficult. We then constrain the structure of the neutrino mass matrix obtained from the Zee model. Two constraints for the neutrino mass matrix have been proposed: imposing a texture to have two independent zeros in the base, where the masses of the charged leptons are diagonalized [31], and requiring the mass matrix to have a vanishing determinant [32].
While the vanishing determinant constraint is good in that the condition does not depend on the weak base we take, and it is suitable for an investigation into the Zee model with a doubly charged Higgs singlet [33,34], it does not fit the structure of the mass matrix obtained from the Zee model considered here. We therefore employ the two-independent-zero-texture constraint, of which phenomenological aspects were investigated in Refs. [35,36,37]. The suitable two-zero-texture mass matrix for the Zee model considered here is the case C defined in Ref. [31], The main aim of the present paper is to search for the simplest GZM with this structure of the mass matrix, which is compatible with the LMA MSW solution of the solar neutrino problem and free from the baryon number washing out [59]. It turns out that such a GZM can be constructed by minimal extension from the RZM, i.e., by adding only one coupling constant to the RZM. We also examine whether this model is allowed from the viewpoint of experiments: µ(τ ) decay rate, µ(τ ) → eγ [15] and neutrinoless double beta decay [39].
We briefly consider the constraints from the recent CMB observation [4,5,6,7] and the possibility of the Z-burst scenario in this model [40,41].
The outline of this paper is as follows: In Sec. II, we first analyze the structure of the neutrino mass matrix that is compatible with SNO/S-K and CHOOZ data. We next derive the Majorana neutrino mass matrix that is induced through the radiative correction in the GZM. The parameters of the GZM are determined by comparing the structure of the neutrino mass matrix obtained from the experiments with that obtained from the GZM.
In Sec. III, we investigate the cosmological implication of the GZM. From the viewpoint of baryogenesis, there must be some combinations of the lepton numbers that avoid the washing out of the primordial baryon number. We first examine what combinations of lepton numbers are conserved in each type of GZM. We next obtain a range of parameters in which the baryon number is not washed out. We finally calculate the resulting lepton and baryon number. Section IV is devoted to a summary.

II. THE GZM MASS MATRIX COMPATIBLE WITH THE LMA-MSW SOLU-TION
In the present section, we review the structure of the neutrino mass matrix arising in the GZM and determine the parameters of the GZM using the neutrino oscillation data. We further examine whether the GZM is consistent with the constraints from other phenomenological experiments.
A. Neutrino mass matrix that satisfies SNO/S-K and CHOOZ experimental data The Majorana neutrino mass term is written as where M ν is a complex valued symmetric matrix. Here, the weak eigenstate ν L is taken to diagonalize the mass matrix of the charged leptons. Using the Maki-Nakagawa-Sakata (MNS) matrix U, we can diagonalize M ν toM ν , where ν m L is in the neutrino mass eigenstate. We write a diagonal matrix with a caret hereafter. The Lagrangian density is written in this base, We assume that M ν is real and the MNS matrix is orthogonal for simplicity. Under these assumptions, we can parametrize the orthogonal matrix U as where c ij ≡ cos θ ij and s ij ≡ sin θ ij .
We summarize the current results of neutrino experiments to make clear what type of neutrino mass matrix is allowed. The S-K experiment shows that there exist a mass squared difference ∆ a and a mixing angle θ atm in order to explain the atmospheric neutrino oscillation [1,42,43] of 1.6 × 10 −3 eV 2 < ∆ a < 4.0 × 10 −3 eV 2 , 0.88 < sin 2 2θ atm ≤ 1.0 (90% C.L.), with the best fit values ∆ a = 2.5 × 10 −3 eV 2 and sin 2 2θ atm = 1.
with the best fit values, ∆ s = 6.9×10 −5 eV 2 and tan 2 θ ⊙ = 0.46 [46]. This analysis concludes with three remarks: first, the original LMA region is narrowed and is separated into two islands (8); second, the LMA is the unique solution to 4σ level; thirdly, the maximal solar neutrino mixing angle is excluded at the 3σ level (9). These data are rewritten using the mass squared difference ratio R and the deviation of the solar neutrino angle from π/4, θ ⊙ ≡ π/4 − ǫ 3 , as with the best fit values R = 2.8 × 10 −2 and ǫ 3 = 0.19. The CHOOZ experiment shows the upper bounds for the mixing angle θ 13 [47] to be Since we assume that the mass matrix M ν is real, we can generally parametrize it as Using the orthogonal matrix the first matrix M 0 ν in Eq. (13) is diagonalized as follows [15]: According to the data (7) and (11), the deviations of the angles θ 23 and θ 13 from the values θ 23 = π/4 and θ 13 = 0 are small, and the influence of η and ξ is expected to be treated as a small perturbation. Defining these deviations as ǫ 1 ≡ π/4 − θ 23 and ǫ 2 ≡ θ 13 , we find that they should lie in the range 0 < ǫ 1 < 0.18 and 0 < ǫ 2 < 0.16. We obtain the following relations within the first order approximation: Judging from the range of ǫ 1 and ǫ 2 , we find that η and ξ are much smaller than the elements of M 0 ν . We therefore assume that η and ξ are zero in the present paper. We further assume c = 0 for simplicity. In this case, the structure of the mass matrix belongs to the type-C texture with two independent zeros, defined in Ref. [31], and the phenomenology is discussed in Refs. [35,36,37]. We obtain the three mass eigenvalues and ǫ 3 as follows: Since we identify ∆ a and ∆ s as ∆ a ≡ |m 2 It is shown in Ref. [21] that a must have an appropriate magnitude obtained from the experimental data mentioned above. This can be understood as follows. If we set a = 0, we obtain a simple relation between tan 2ǫ 3 and R, On the other hand, R and tan 2ǫ 3 must satisfy from the data (10). We therefore find that a must have an appropriate magnitude because the conditions (21) and (22) are incompatible. Similarly, we can show that d must also have an appropriate magnitude. From Eqs. (19), (20), and (10), we find that mild fine-tuning |a + d| ≫ |a − d| is needed to fit the model with the LMA MSW solution.
Because the difference a − d (≡ ǫ) is small in comparison with b, a, d, it can be treated as a small perturbation. We obtain the following values for b, a, d, and ǫ: for the best fit values ∆ a = 2.5 × 10 −3 eV 2 , ǫ 3 = 0.19, and R = 2.8 × 10 −2 .

B. GZM neutrino mass matrix
In the present section, we review the GZM [13,15]. The interactions of the GZM are the following: Here, h − is the Zee singlet, l L is the lepton doublet in a weak eigenstate, Φ 1 and Φ 2 are the two Higgs doublets, Γ 1 and Γ 2 are two Yukawa coupling matrices, f is an antisymmetric coupling matrix, λ is a cubic coupling constant, and a, b are indices of the SU(2) L doublets.
We write the vacuum expectation value (VEV) of each Higgs doublet as φ 0 . Rotating Φ 1 and Φ 2 as follows: where tan β = v 1 /v 2 , we make only one of the two Higgs doublets have the VEV, i.e., 2 is the would-be Nambu-Goldstone boson, the diagonalization of the mass matrix for the remaining physical charged Higgs fields using the orthogonal transformation where the rotational angle α and the two mass eigenvalues M 1 , M 2 are determined as Using Φ ′ 1 and Φ ′ 2 in Eq. (30), the Lagrangian density (27) is written as Since only φ ′ 0 2 has the VEV, using the biunitary transformation The matrix (sin whereM l−m is the mass matrix of the charged leptons. The Lagrangian density (27) is written as where . Since we estimate the interaction rates in the early Universe in Sec. III, we write down the Lagrangian density in the symmetric phase (v = 0) in addition to that in the symmetry broken phase (v = 246GeV) where the neutrinos have the Majorana masses.

Symmetric phase (v = 0)
The interaction Lagrangian density is as follows: Here, all leptons are massless and f m (≡ V L f V T L ) is antisymmetric.

Symmetry broken phase(v = 246 GeV)
The interaction Lagrangian density is as follows: Here, the Higgs fields S − 1 , S − 2 have masses M 1 , M 2 , respectively. The fields φ ′ + 2 and N 2 are the would-be Nambu-Goldstone bosons.
FIG . 1: The diagrams that are used to calculate the radiatively induced neutrino mass matrix. We use s α ≡ sin α, c α ≡ cos α, and t α ≡ tan α and so on in this figure.

C. Matching the GZM with the experiments
We obtained the relation between the value of a and the experimental data in Sec. II A, On the other hand, we calculated the radiatively induced a in the GZM in Sec. II B, where B = v/ √ 2 cos β. From Eq. (47) and Eq. (48), we find that one of γ eτ 2 and γ eµ 2 must be nonzero at least. Because we prefer a minimal extension here, we assume that the other elements of Γ m 2 are zero. From now on, we further assume that f and Γ m 2 are real. Here, we match the neutrino mass matrix in the GZM (46) with that determined from the experimental data (23)−(25) as follows In this model, there are nine parameters (M 1 , M 2 , α, β, f eµ , f eτ , f µτ , γ eτ 2 , γ eµ 2 ). From Eqs. (49) − (51), we parametrize four parameters (f eµ , f eτ , f µτ , γ eτ 2 ) using the other five parameters (M 1 , M 2 , α, β, γ eµ 2 ) as follows We consider the simplest two cases, (1)γ eτ 2 = 0, γ eµ 2 = 0 and (2) γ eτ 2 = 0, γ eµ 2 = 0. In these cases, we have the following relations.

τ (µ) decay
Since the leptonic flavor changing neutral current does not exist in the standard model at all, stringent constraints are imposed on their decay amplitudes [48]. There is, however, τ (µ) decay through the FCNCs at the tree level in the GZM. These constraints are satisfied for the parameter set (56) and (57) as follows for β ≃ π/4: where G F is the Fermi coupling constant and M ′ 0 1 > ∼ 100 GeV. Since the coupling constants f eµ , f eτ , and f µτ are much smaller than γ eτ 2 and γ eµ 2 , the contributions to the branching ratios purely from f are negligible [49,50,51].

τ (µ) → e γ
Since these processes violate the lepton flavor, they do not exist in the standard model at all. On the other hand, these processes become possible in the GZM through the exchange of the charged and neutral Higgs bosons. Since they are rare events, stringent constraints are imposed on the amplitudes [48]. These constraints are safely satisfied for the parameter set (56) and (57) as follows for β ≃ π/4: 3 × 10 −10 ≪ 2.7 × 10 −6 (experiment [48]), Again, since the coupling constants f eµ , f eτ , and f µτ are much smaller than γ eτ 2 and γ eµ 2 , the contributions to the branching ratios purely from f are negligible [49,50,51].

0νββ
The observation of the neutrinoless double beta decay (0νββ) gives evidence that the neutrinos have Majorana masses. The study of this process is a crucial test of the validity of our model, since neutrinos have Majorana masses, the interaction of the Zee singlet h − violates lepton number, and the Yukawa coupling Γ m 2 is flavor changing. As was discussed by Schechter and Valle [52] in the theory with an exotic doubly charged scalar, the extension of the Higgs sector possibly provides a new source of the decay.
We can, in fact, think of a process with the exchange of our exotic scalar, i.e., h − [see The value obtained in Sec. II A is consistent with this upper bound as follows: Since the Yukawa coupling Γ m 2 is flavor changing, another source of decay exists as shown in Fig. 2(b), where m µµ or m τ τ contributes, but not m ee . This diagram, however, does not exist from reasoning similar to the above; namely, the amplitude of the diagram is also proportional to γ µe 2 or γ τ e 2 , which does not exist. Thus the ordinary process due to m ee discussed above is the unique source of double beta decay. If the sensitivity of the experiment is improved by a factor of 5 or so, the decay should be seen if our model is valid. Both processes fail to exist because of the vanishing Yukawa coupling γ µe 2 = 0. In both diagrams ν µ may be replaced by ν τ .

III. COSMOLOGICAL IMPLICATIONS OF THE GZM
In Sec. II, we obtained the values for the coupling constants (f eµ , f eτ , f µτ , γ eτ 2 , γ eµ 2 ) suggested by the neutrino oscillation data. In the present section, we investigate whether the primordial baryon number is washed out or not for these values. If there are some leptonic conserved numbers, the primordial baryon number remains finite even though the sphaleron process partially washes out the baryon number. We therefore consider what leptonic numbers are effectively conserved at a high temperature. We then calculate what amount of the primordial baryon number remains in the case that there exist some conserved numbers. We also briefly consider the constraints from the recent CMB observation [4,5,6,7] and the possibility of the Z-burst scenario in this model [40,41].

A. Conserved number for each coupling
It is convenient to define the following global U(1) charges as for an investigation into the washing out of the baryon number. These quantities are conserved numbers in the standard model. We, however, have coupling constants which violate these conserved numbers in our model. Each coupling conserves two of the three independent numbers which are shown in the Table I using a linear combination of L eµ , L eτ , and L µτ .

B. Interaction rates induced by each coupling
We here estimate the interaction rates induced by the coupling constants f and γ 2 .

L µτ violating process induced by the coupling constant f µτ
We consider the h − decay process and the inverse decay process induced by the coupling constant f µτ . The interaction rates for these processes are estimated at a temperature T > ∼ M h [27,56], where M h is the mass of the Zee singlet and E ≃ 2.7 T (boson) is the averaged energy.
Similar results are obtained for the processes induced by the coupling constants f eµ and f eτ .

Processes induced by the coupling constant γ eτ 2
We next estimate the interaction rates for the processes induced by the coupling constant γ eτ 2 . We estimate the interaction rates of φ ′ − 1 decay and the inverse decay. The interaction rate for the process φ where M ′ + 1 is the mass of the field φ ′ − 1 . Similar results are obtained for the processes induced by the coupling constant γ eµ 2 .
C. Out-of-equilibrium condition to avoid the baryon number washing out In the present section, we consider which processes discussed in the previous section were out of equilibrium in the early Universe [61]. We will consider what numbers are conserved in the case where these processes are out of equilibrium in the next section. The condition that the processes are out of equilibrium is given by where H is the Hubble parameter, H = 1.66 √ g * T 2 /M P l ≃ 1.47 × 10 −18 (T 2 /GeV) with M P l being the Planck mass. Here, g * is the total degrees of freedom of effectively massless particles. In the GZM, we adopt the value g * = g SM * + g h − * + g Φ 2 * ≃ 112.75.
1. Coupling constants f eµ , f eτ , and f µτ are out of equilibrium f µτ process. We examine the condition that the process h − ↔ µ − ν τ is out of equilibrium at T = M h . We apply the condition (65) to the rate (63), We obtain the condition for T = M h , 2.9 × 10 6 tan 4 β sin 4 2α where we use the following relations: At the higher temperatures, T > M h , the out-of-equilibrium condition Eq. (67) is more easily satisfied. At lower temperatures, T < M h , the number density of the Zee singlet h − in the thermal bath is so small that the L µτ -violating processes are negligible. It is therefore sufficient to consider the out-of-equilibrium condition at T ≃ M h . We set β = π/4 for simplicity hereafter. The inequality (68) is written for this value of β, From Eq. (71), we find that this condition is easily satisfied for the angle sin 2α ∼ 1. We then fix the value of α to π/4. In this case, the inequality (71) is simplified as where we parametrize as x ≡ M 1 /100 GeV, y ≡ M 2 /100 GeV (x, y ≥ 1, x ≥ y). From this, we find that this process is out of equilibrium for M 1 ≥ 310 GeV, when we fix the value of f eτ process. The out-of-equilibrium condition for the f eτ process is obtained just as the condition for the f µτ process. We examine the condition that the process h − ↔ e − ν τ is out of equilibrium at T = M h . We apply the condition (65) to the rate (63), In the case α = π/4, the inequality (74) is simplified as From this, we find that this process is out of equilibrium for M 1 ≥ 500 GeV, when we fix the value of M 2 to 100 GeV.
f eµ process. The out-of-equilibrium condition for the f eµ process is obtained in a similar manner to that in the f µτ process. The condition Γ(h − ↔ e − ν µ ) < H is written as From this, we find that this process is out of equilibrium for M 1 ≥ 1.2 × 10 6 GeV, when we fix the value of α to π/4 and the value of M 2 to 100 GeV. Since we recognize that this mass range for M 1 is quite large and somewhat unnatural, we consider that this process is in equilibrium at T = M h .
2. Coupling constants γ eτ 2 and γ eµ 2 are out of equilibrium We consider which processes induced from the coupling constants γ eτ 2 and γ eµ 2 are out of equilibrium for each case considered in Sec. II C: γ eτ 2 = 0,γ eµ 2 = 0 and γ eτ 2 = 0,γ eµ 2 = 0. γ eτ 2 = 0 and γ eµ 2 = 0. We examine the condition that the process φ where we fix the value of α to π/4. From this, we find that this process is out of equilibrium for M 1 ≥ 1.2 × 10 11 GeV, when we fix the value of M 2 to 100 GeV. We consider that this process is in equilibrium at T = M h for the same reason mentioned above.
γ eτ 2 = 0 and γ eµ 2 = 0. We examine the condition that the process φ ′ − 1 ↔ (ν e L ) c µ − R is out of equilibrium as follows: where we fix the value of α to π/4. From this, we find that this process is out of equilibrium for M 1 ≥ 4.0 × 10 8 GeV, when we fix the value of M 2 to 100 GeV. We consider that this process is also in equilibrium at T = M h .
1. γ eτ 2 = 0, γ eµ 2 = 0 Since the processes induced by γ eτ 2 and f eµ are in equilibrium from the previous section, the only possible conserved quantity is This quantity is conserved if the process induced by f eτ is out of equilibrium [Eq. (74)].
The region in which this quantity is conserved is shown in Fig. 3. This conserved number is converted to the baryon number through the sphaleron process and the final baryon (B eq f ) and lepton number (L eq f ) are calculated as where (B/3 + L eτ ) i is the primordial value generated at the temperature much higher than T = M h . The region shown in Fig. 3 is therefore the allowed region for avoiding the washing out of the primordial baryon number in this case.
2. γ eτ 2 = 0, γ eµ 2 = 0 Since the processes induced by γ eµ 2 and f eµ are in equilibrium, the only possible conserved quantity in this case is This quantity is conserved if the process induced from f eτ and f µτ is out of equilibrium [Eq.
(74) and Eq. (71)]. The region in which this quantity is conserved is shown in Fig. 3. This conserved number is converted to the baryon number through the sphaleron process and the final baryon (B eq f ) and lepton number (L eq f ) are calculated as B eq f = 60 185 where (B/3 − L τ ) i is a primordial value generated at the temperature much higher than T = M h . The region shown in FIG. 3 is therefore the allowed region for avoiding the washing out of the primordial baryon number in this case, also.
This value is consistent with that obtained in Sec. II A, If the ultrahigh energy cosmic rays with energy beyond the Greisen-Zatsepin-Kuzmin limit are explained in the Z-burst scenario, the heaviest neutrino mass should lie in the range [40,41,57] 0.08 eV < |m ν (heaviest)| < 1.3 eV (68% C.L.).
The value obtained in Sec. II A, is near this range, and the ultrahigh energy cosmic rays might be explained in the Z-burst scenario in our model.

IV. SUMMARY
We propose a simple GZM which is compatible with all the neutrino oscillation data for the LMA MSW solution of the solar neutrino problem, and is free from the baryon number washing out. Our extension from the RZM is minimal, i.e., we add only one coupling constant γ eτ 2 or γ eµ 2 in addition to those in the RZM. To avoid the baryon number washing out, the masses M 1 and M 2 must be in the region shown in Fig. 3. The final baryon and lepton numbers are calculated for the two cases (1)γ eτ 2 = 0, γ eµ 2 = 0 and (2) γ eτ 2 = 0, γ eµ 2 = 0: (1) γ eτ 2 = 0, γ eµ 2 = 0 : B eq f = 60 563 (2) γ eτ 2 = 0, γ eµ 2 = 0 : B eq f = 60 185 In order to explain today's baryon number, it is therefore necessary that the quantity (B/3+ L eτ ) or (B/3 − L τ ) is generated in the early Universe or the baryon number should be produced at lower temperatures T ≪ M h . We check that the predicted lepton flavor violating processes are not in conflict with the phenomenological constraints for τ (µ) decay, τ (µ) → eγ, and 0νββ decay.