{'fraction_non_alphanumeric': 0.045560684224816725, 'fraction_numerical': 0.02438229704045615, 'mean_word_length': 4.157098851862224, 'pattern_counts': {'":': 0, '<': 27, '<?xml version=': 0, '>': 19, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Neutrino masses, mixing and leptogenesis in a two Higgs doublet model for the third generation

We examine neutrino oscillations in a two Higgs doublet model (2HDM) in which the second doublet couples only to the third generation right-handed up-fermions, i.e., to t_R and to N_3 which is the heaviest right-handed Majorana neutrino. The inherently large tan(beta) of this model can naturally account for the large top mass and, based on a quark-lepton similarity ansatz, when embedded into a seesaw mechanism it can also account for the observed neutrino masses and mixing angles giving a very small \theta_{13}: -0.017<\theta_{13}<0.021 at 99% CL, and a very restrictive prediction for the atmospheric mixing angle: 42.9^0<\theta_{atm}<45.2^0 at 99% CL. The large value of tan(beta) also sets the mass scale of the heaviest right-handed Majorana neutrino N_3 and triggers successful leptogenesis through a CP-asymmetry in the decays of the N_1 (lightest right-handed Majorana) which is tan^2(beta) enhanced compared to the CP-asymmetry obtained in models for leptogenesis with one Higgs doublet or in the MSSM. This enhancement allows us to relax the lower bound on M_{N_1} and consequently also the lower bound on the reheating temperature of the early universe.

A monumental discovery of the past decade is that neutrinos have mass! This discovery is bound to have a significant impact on our understanding of the universe. It also presents us with a pressing challenge: unraveling of the mixing matrix in the lepton sector. When we recapitulate the analogous quark case we can easily grasp the importance and the difficulties of this new task. Indeed, an obvious but nonetheless crucial question is the relation of the mixing matrix in the lepton sector to that in the quark sector. Intensive experimental effort is underway and much more is being planned, round the globe, to address these issues of vital importance.
In this work we suggest that these impressive findings in the lepton sector are closely related to another remarkable finding of the 90's in the quark sector, namely that the top quark is enormously heavy compared to all the other quarks. Specifically, we will propose a simple, grounds-up approach which treats the 3rd generation neutrino in a completely analogous manner to the top quark. This reasoning leads us to a concrete framework for neutrino masses and mixings with considerable predictive power. For example, we find that the crucial mixing angle θ 13 , which is an important target of many neutrino experiments, is quite constrained, −0.017 < ∼ θ 13 < ∼ 0.021 at 99% CL, in our picture. Needless to say, these observations of neutrino oscillations [1], confirming the existence of massive neutrinos, implies that the Standard Model (SM) can no longer be regarded as a minimal theory that explains all observed phenomena in Particle Physics. Bearing this in mind, our objective here is to present a simple extension of the SM, which will serve as a low energy effective theory which captures the dominant features of phenomenology and models some underlying complicated dynamics of electroweak (EW) symmetry breaking at distances much shorter than the EW scale.
In order to explain the above neutrino mass differences without introducing extremely small and unnatural neutrino Dirac Yukawa couplings, the general practice is to add superheavy right handed Majorana neutrino fields, N i , and rely on the seesaw mechanism for generating sub-eV light neutrinos: where, in the minimal setup, L Y (N, L, φ) = Y ν ij L i HN j , with L i = (ν jL , ℓ jL ) and H is the only Higgs doublet that couples to neutrinos. Then, the light neutrino mass matrix is given by the see-saw mechanism as: with m D = vY ν and v =< H >.
In this work we wish to propose a 2HDM, which we will name the "2HDM for the 3rd generation" (3g2HDM).
The model relates the neutrino sector to the up-quark sector, and by giving the third generation right-handed neutrino similar status as the top quark it provides a natural framework for accommodating the observed pattern of neutrino masses and mixing angles. In addition, our 3g2HDM assumes a specific structure in the Yukawa sector which should be viewed as an effective low-energy parametrization of the underlying short distance theory. This concrete phenomenological framework for the Yukawa sector allows us to successfully address both the heaviness of the top-quark and the apparent hierarchical structure in the neutrino sector.
In particular, we extend the idea of the so called "2HDM for the top-quark" (t2HDM) [3] to the leptonic sector. In the t2HDM one assumes that φ 2 [the second Higgs doublet with a much larger vacuum expectation value (VEV)] couples only to the top-quark, while the other Higgs doublet φ 1 (with a much smaller VEV) couples to all the other fermions. The large mass hierarchy between the top quark and the other quarks is then viewed as a consequence of v 2 /v 1 ≡ tan β >> O(1), which, therefore, becomes the "working assumption" of the t2HDM.
The t2HDM has several other notable features, such as enhanced H + bc and H 0 cc couplings, new flavor changing H 0 tc and H 0 tu interactions and new CP phases. Thus, the t2HDM can influence CP-violation in B physics [4], flavor changing Z-decays [5] and the production of a Higgs in association with a c-jet or a b-jet in hadron colliders [6].
Following the idea of the t2HDM, we propose the following Yukawa interactions in the 3g2HDM: where N are right-handed Majorana neutrinos, Q and L are the usual quark and lepton doublets and such that, in both the quark and leptonic sectors, φ 2 couples only to the third generation right-handed upfermions. The Yukawa texture in (5) yields the following Dirac neutrino mass matrix (dropping the superscript ν): where t β ≡ tan β = v 2 /v 1 . Note that we have assumed the texture zeros (Y u,ν 1 ) 13 = 0 and (Y u,ν 2 ) 31 = 0, which gives a Dirac neutrino mass matrix in which the upper right and lower left entries vanish. This is required in order to simplify our discussion below and to avoid fine tuning in the neutrino Yukawa sector when confronted with the experimental limits on neutrino masses and mixings (in fact, it is sufficient to set only (Y ν 2 ) 13 → 0 in order to avoid fine tuning). The Dirac mass texture in (6) is similar to the one which arises in the minimal approach suggested in [7,8], that is, from the requirement of the most economic addition to the SM that can accommodate both neutrino oscillation data and the observed baryon asymmetry in the universe. Clearly, for such texture zeros to be natural, they have to be protected by underlying symmetries of the short distance theory. For example, SO(10) GUT theories dictate a symmetry between the up-quark and the Dirac neutrino mass matrices which, along with the empirically known properties of the upquark mass matrix, yields the texture zeros in (6), see e.g., [9]. In fact, as was shown in [10], there always exists a scalar sector such that any texture zero in the fermion mass matrices can be enforced by means of an abelian symmetry once the scalar sector is enlarged to the extent required. Another alternative and attractive framework for the zero entries in Y 1 and Y 2 was suggested in [8]: such texture zeros may easily result from a vanishingly small overlap between the wave functions of N and L with the corresponding indexes in extra-dimensional models. The above several possible ways to accommodate the desired texture zeros [in (6)] do not give us any further insight regarding the underlying reason for the structure of the Yukawa couplings in our model. Nonetheless, they do motivate us to study the implications and phenomenology of the 3g2HDM with the texture zeros assumed in (5).
We note also that, without loss of generality, further simplifications were taken in (5) that allow us to give a more transparent analytical derivation of the neutrino oscillations pattern in our 3g2HDM. In particular, we have chosen similar Yukawa couplings within each generation: In addition, a real parameter of O(1) (the parameter δ) was added to the (Y 1 ) 23 entry in (5). We emphasize that these simplifying assumptions have no effect on our main results below, aside from simplifying the analytical formulae given below.
In the basis where M N is diagonal (one can always choose a basis for the fields N such that M N is diagonal), we then obtain from the seesaw mechanism in (3): where and In the limit ω >> ǫ,ǭ, δ, the physical neutrino masses and mixing angles are then given by [11]: where we have defined the parameters: In the following, we will assume normal hierarchy for the light neutrinos: m 3 >> m 2 >> m 1 , therefore leading to m 3 ∼ ∆m 2 atm and m 2 ∼ ∆m 2 sol . Thus, setting for example 2m 0 ν ∼ ∆m 2 sol , it follows from the expression for m 3 in (10) that ω ∼ ∆m 2 atm /∆m 2 sol . Note that, using the definitions for ω in (8), for m 0 ν in (9) and the fact that m t ∼ ct β × (v 1 / √ 2), our model yields the following seesaw triple-relation between the heaviest light neutrino, the heaviest right-handed Majorana neutrino and the top quark: Moreover, from (9) with 2m 0 ν ∼ ∆m 2 sol we obtain the typical mass scale (M ) of the heavy right handed neutrinos: where, in the last equality, we have taken m t ∼ v 2 / √ 2 and t β ∼ O(10), which are the "working assumption" values within the 3g2HDM.
In Fig. 1 we give a scatter plot of the allowed ranges in the θ 13 − θ 23 and θ 13 − θ 12 planes, i.e., subject to the 3σ limits listed in (1). This is done by randomly varying the three input parameters ǫ, r and δ (requiring ω >> ǫ,ǭ, δ) with a sample of 3 · 10 6 points. We see that the 3g2HDM predicts θ 13 to lie within (in radians) −0.05 < ∼ θ 13 < ∼ 0.035 and the atmospheric mixing angle θ 23 to be at most a few degrees away from maximal. In fact, a minimum χ 2 analysis with respect to θ 13 and θ 23 yields: with the best fitted value at θ 13 ∼ −0.011 and θ 23 ∼ 43.9 0 . Notice that this is a rather restrictive prediction for the atmospheric mixing angle since the experimentally allowed 3σ range for θ 23 spans over about 20 0 , i.e., 35 0 < ∼ θ exp To better understand what is enforcing maximal atmospheric mixing and a very small θ 13 (as is evident from Fig. 1), we note that from (10) we have: since by definition r ≥ 1. Therefore, for values of ǫ, δ and r of O(1), we have: as is typical to grand unified (or quark-lepton unified) theories [13]. Furthermore, from (15) we have (taking δ < −1, see discussion below and Fig. 2): as can be seen in Fig. 1.
We see that case I (corresponding to the best fitted value for r [12]) is compatible with neutrino oscillation data for 0.4 < ∼ ǫ < ∼ 0.7 with −1.7 < ∼ δ < ∼ − 1, while the allowed As an example, in table I we give the neutrino mixing angles and masses in cases I and II, for some specific values of the allowed parameter space.
Let us now examine the consequences of our model on the mass spectrum of the heavy Majorana neutrinos, taking into account the above constraints coming from neutrino oscillation. For definiteness, we will consider case I (i.e., r ∼ 1, which we find as the best fitted value with respect to oscillation data), and based on quarklepton similarity (perhaps motivated by GUT scenarios, see e.g., [9]), we will take the following Ansatz for the neutrino Dirac Yukawa couplings: Indeed, this choice follows from the assumption that the Dirac mass matrices of the neutrinos and up-quarks exhibit a similar hierarchical structure. This Ansatz is, therefore, natural in the 3g2HDM under consideration, since this model is constructed in order to simultaneously explain the large top mass and the hierarchical structure of the neutrino masses, by relating the quark sector to the leptonic sector. That is, taking v 1 ∼ O(10) and t β ∼ O(10), this Ansatz follows from the up-quark sector; if one assumes the quark-lepton similarity: It is then easy to extract the mass spectrum of the heavy Majorana neutrinos in our Ansatz, subject to the constraints given in (1). In particular, from (12) we have: Furthermore, we find that our Ansatz (19) yields the following masses for N 2 and N 1 [for r ∼ 1 and for ǫ ∼ O(1) in accord with the constrains coming from neutrino oscillations, see Fig. 2]: which, for M ∼ 10 13 GeV [see (13)], gives M N2 ∼ 10 11 GeV and M N1 >> 10 7 GeV. Let us now examine how the 3g2HDM with our quarklepton similarity Ansatz in (19) and with r ∼ 1 and ǫ ∼ O(1) fits into the mechanism of leptogenesis [16]. Leptogenesis is an additional attractive feature, associated with the seesaw mechanism, since it allows the possibility of generating the observed baryon excess in the universe from a lepton asymmetry, driven by decays of the heavy Majorana neutrinos [14,15,16]. In particular, a CP-asymmetry, ǫ Ni , in the decay N i → ℓφ j can generate the lepton asymmetry (see e.g., [14,15]): where Y Ni = n Ni /s, n Ni being the number density of N i and s the entropy, with Y Ni (T >> M Ni ) = 135ζ(3)/(4π 4 g * ) and g * being the effective number of spin-degrees of freedom in thermal equilibrium. Also, η is the "washout" parameter (efficiency factor) that measures the amount of deviation from the out-of-equilibrium condition at the time of the N i decay. The lepton asymmetry n L /s can then be converted into a baryon asymmetry through nonperturbative spheleron processes. The conversion factor is [17]: n B /s = −(8N G + 4N H )/(22N G + 13N H ) × n L /s, where N G and N H are the number of generations and Higgs doublets, respectively. Thus, in our case, i.e., N G = 3, N H = 2 and g * > ∼ 100, we obtain: As seen from (20) and (21), our 3g2HDM can lead to a hierarchical mass spectrum for the heavy Majorana neutrinos, M N1 << M N2 << M N3 , within a large portion of the allowed parameter space. In particular, imposing M N1 << M N2 , from (21) and with M ∼ 10 13 GeV, we get: M N1 ∼ 10 8 − 10 10 GeV. With such an hierarchical mass spectrum, only the CP-asymmetry produced by the decay of N 1 survives. In our model N 1 can only decay to the "light" doublet φ 1 , as can be seen from (4) and (5). Therefore, the CPasymmetry ǫ N1 is similar to the one obtained in single-Higgs models in the limit M N1 << M N2 , M N3 (see e.g., [15]): where Y ν 1 is the Yukawa coupling of N 1 to φ 1 given in (5). Note that, since N 3 does not couple to φ 1 in our model, it does not contribute to the sum in (24) above. Therefore, ǫ N1 is generated only from the i = 2 term in (24), giving: where the CP-phases arise from the possible complex entries in Y ν 1 : a = |a|e iθa and b = |b|e iθ b . Thus, using (13) and ǫ ∼ b 2 /ǫ M2 (as implied by a fit of the 3g2HDM to neutrino oscillation data, i.e., r ∼ 1), we obtain: It is interesting to note that the above CP-asymmetry is a factor of ∼ t 2 β larger than the CP-asymmetry obtained in models with one Higgs doublet or for that matter in SUSY-like models where the condensate of a single Higgs is responsible for generating the Dirac neutrino mass term. Thus, in our 3g2HDM the enhancement factor is of O(100) since t β ∼ O (10). To see that we can rewrite ǫ N1 in (25) as follows: where in our model ǫ ∼ 0.5 and ω ∼ 5 are fixed by oscillation data and ǫ max N1 is the maximum of the CPasymmetry in models with one Higgs doublet (see e.g., [15]): with the atmospheric neutrino mass m atm = m 3 given in (10) and v = v 2 1 + v 2 2 = 246 GeV. As mentioned above, this enhancement is possible due to the presence of the second Higgs doublet, since in this case the light neutrino mass spectrum is determined by v 1 (the VEV of the lighter Higgs field) and not by the EW-scale VEV v (as in the usual scenario with one Higgs doublet), and the hierarchy m atm >> m sol is dictated by the large tan β. In this way, the CP-asymmetry no longer suffers from a direct relation to the Dirac neutrino mass term and is, therefore, not directly constraint by oscillation data. This observation was also made by Fukuyama and Okada in [18] who have investigated leptogenesis within a generic two Higgs doublet model. As will be shown below, this enhancement of the CP-asymmetry allows us to reproduce the observed baryon asymmetry in the universe with M N1 as low as 10 9 GeV, which in turn relaxes the lower bound on the reheating temperature in the early universe to T RH > ∼ 2 · 10 8 GeV, [15]. The efficiency factor, η, which measures the amount of lepton asymmetry left from the CP-asymmetry ǫ N1 , is calculated by solving the appropriate Boltzmann equations.
Summarizing, we have presented here a simple extension to the Standard Model, in a grounds-up approach, which leads us to a framework for neutrino masses and mixings with considerable predictive power. This can clearly have significant impact on on-going as well as planned experiments. To briefly recapitulate, we have constructed a 2HDM in which the second doublet, with a much larger VEV, couples only to the third generation right-handed up-fermions (the top-quark and the 3rd generation right-handed neutrino) and the other doublet couples to all other fermions. Thus, this model is a possible effective low energy parametrization of an underlying short distance theory which envisions a close relation between quark dynamics and neutrino physics. The key parameter of this 2HDM is tan β which is assumed to be of O(10) in order to naturally accommodate a large mass for the top quark. We have shown that the large value of tan β in this model is directly responsible for successfully reproducing the observed neutrino oscillation data, predicting a very small θ 13 : −0.017 < ∼ θ 13 < ∼ 0.021 at 99% CL, a very restricted allowed range for the atmospheric mixing angle: 43 0 < ∼ θ 23 < ∼ 45 0 at 99% CL, as well as successfully reproducing the observed baryon asymmetry of the universe through leptogenesis. In particular, in our model, the CP-asymmetry in the N 1 decays that drives leptogenesis is larger by a factor of O(tan 2 β) ∼ O(100) than the CP-asymmetry obtained in models for leptogenesis with one Higgs doublet or in the MSSM. This enhancement allows to relax the lower bound on M N1 and accordingly also the lower bound on the reheating temperature of the early universe.
S.B.S thanks the hospitality of the theory group in Brookhaven National Laboratory where part of this study was performed. This work was supported in part by US DOE Contract Nos. DE-FG02-94ER40817 (ISU) and DE-AC02-98CH10886 (BNL).