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Universal Neutrino Mass Hierarchy and Cosmological Baryon Number Asymmetry

We conjecture that three light Majorana neutrinos and their right-handed counterparts may have a universal geometric mass hierarchy. Incorporating this phenomenological conjecture with the Fritzsch texture of lepton mass matrices in a simple seesaw mechanism, we show that it is possible to simultaneously account for current neutrino oscillation data and the cosmological baryon number asymmetry via leptogenesis.

Recent solar [1], atmospheric [2], reactor (KamLAND [3] and CHOOZ [4]) and accelerator (K2K [5]) neutrino oscillation experiments provide us with very compelling evidence that neutrinos have nonvanishing masses and their mixing involves two large angles (θ 12 ∼ 33 • and θ 23 ∼ 45 • ) and one small angle (θ 13 < 13 • ). These important results imply that the minimal standard electroweak model, in which neutrinos are massless Weyl particles, is actually incomplete. A very simple extension of the standard model is to include one right-handed neutrino in each of three lepton families , while the Lagrangian of electroweak interactions keeps invariant under SU(2) L × U(1) Y gauge transformation. In this case, the Lagrangian relevant for lepton masses can be written as where l L denotes the left-handed lepton doublet, E and N stand respectively for the righthanded charged lepton and Majorana neutrino singlets, and φ is the Higgs-boson weak isodoublet. After spontaneous gauge symmetry breaking, we obtain the charged lepton mass matrix M l ≡ Y l φ and the Dirac neutrino mass matrix M D ≡ Y ν φ with φ ≈ 174 GeV. The scale of M R may be considerably higher than φ , because right-handed neutrinos are SU(2) L singlets and their mass term is not subject to electroweak symmetry breaking. As a consequence, the effective (light and left-handed) neutrino mass matrix M ν can be derived from M D and M R via the well-known seesaw mechanism [6]: It becomes clear that the smallness of left-handed neutrino masses is attributed to the largeness of right-handed neutrino masses. The phenomenon of lepton flavor mixing, which has shown up in both solar and atmospheric neutrino oscillations, arises from a nontrivial mismatch between the diagonalizations of M l and M ν .
Note that lepton number violation induced by the third term of L lepton allows decays of the heavy Majorana neutrinos N i (for i = 1, 2, 3) to happen. Because the decay can occur at both tree and one-loop levels, their interference may lead to a CP -violating asymmetry ε i between the CP -conjugated N i → l + φ † and N i → l c + φ processes [7]. If the masses of N i are hierarchical (i.e., M 1 < M 2 < M 3 ), the interactions of N 1 can be in thermal equilibrium when N 2 and N 3 decay. The asymmetries ε 2 and ε 3 are therefore erased before N 1 decays, and only the asymmetry ε 1 produced by the out-of-equilibrium decay of N 1 survives. In the flavor basis where M R is diagonal and positive, we have [8] , and the condition M 2 1 ≪ M 2 2 ≪ M 2 3 has been taken. The point of leptogenesis is that ε 1 may result in a net lepton number asymmetry Y L ≡ (n L − nL)/s = dε 1 /g * , where s denotes the entropy of the early universe, g * = 106.75 is an effective number characterizing the relativistic degrees of freedom which contribute to s, and d measures the dilution effects induced by the lepton-number-violating wash-out processes. This lepton number asymmetry is eventually converted into a net baryon number asymmetry Y B via nonperturbative sphaleron processes [9]: Y B ≡ (n B − nB)/s ≈ −0.55Y L . Such an elegant mechanism may therefore interpret the cosmological matter-antimatter asymmetry, 7 × 10 −11 < ∼ Y B < ∼ 10 −10 , which is drawn from the recent WMAP observational data [10]. It is a highly nontrivial task to simultaneously account for the cosmological baryon number asymmetry and current neutrino oscillation data, because the textures of Y ν (or M D ), M R and M l are completely unknown. Although a number of ansätze for lepton mass matrices have been discussed in the literature [11], most of them remain rather preliminary and have little predictability.
The purpose of this paper is to propose a novel and viable phenomenological model of lepton mass matrices, from which both the cosmological baryon number asymmetry and neutrino mixing parameters can be calculated. Our main conjecture is that three light Majorana neutrinos and their heavy right-handed counterparts have a universal geometric mass hierarchy. Incorporating this conjecture with the Fritzsch texture of lepton mass matrices in a simple seesaw scenario, one may easily account for current neutrino oscillation data at the 3σ level. The leptogenesis mechanism allows us to obtain a very instructive result for the cosmological baryon number asymmetry Y B , which depends on a single CP -violating phase. We find that the observed value of Y B can set a strong constraint on the mass scale of three right-handed Majorana neutrinos.
First of all, we conjecture that three light (left-handed) neutrinos have a geometric mass hierarchy at low energies: m 1 /m 2 = m 2 /m 3 ≡ r. Because of m 1 < m 2 or r < 1 [1], this geometric mass relation cannot be reconciled with the inverted neutrino mass hierarchy (i.e., m 3 < m 1 < m 2 ). Given the neutrino mass-squared differences ∆m 2 which have been determined from a global analysis of current neutrino oscillation data [12], it is easy to obtain where r = ∆m 2 21 /(∆m 2 31 − ∆m 2 21 ) lies in the range 0.122 < ∼ r < ∼ 0.270. A complete determination of the neutrino mass spectrum is therefore available. We stress that the geometric mass hierarchy of three light neutrinos can essentially be extrapolated up to the lowest mass scale of three heavy (right-handed) Majorana neutrinos. Taking the normal mass hierarchy M 1 < M 2 < M 3 , one may establish a simple relation for m i (with i = 1, 2, 3) between the electroweak scale µ = M Z and the seesaw mass scale µ = M 1 [13]: where measures the overall renormalization-group effect of the Higgs self-coupling, gauge coupling and top-quark Yukawa coupling from M Z to M 1 . Setting I α = 1 at M Z , we shall get I α > 1 at any energy scales higher than M Z . Because Eq. (4) is universally valid for three light neutrinos, we arrive at the geometric hierarchy m 1 (M 1 )/m 2 (M 1 ) ≈ m 2 (M 1 )/m 3 (M 1 ) to a good degree of accuracy at the seesaw scale.
We further conjecture that three heavy right-handed neutrinos have the same geometric mass hierarchy as three light neutrinos do: M 1 /M 2 = M 2 /M 3 = r. As a consequence of r ∼ 0.2, the condition M 2 1 ≪ M 2 2 ≪ M 2 3 holds to guarantee the validity of Eq. (2). One may then take advantage of baryogenesis via leptogenesis to fix the seesaw mass scale M 1 (or M 3 ). To do so, a specific texture of the Dirac neutrino coupling matrix Y ν must be assumed. We find that it is natural to assume three eigenvalues of Y ν , denoted as D i (for i = 1, 2, 3), to have the same geometric hierarchy as three heavy Majorana neutrinos do: D 1 /D 2 = D 2 /D 3 = r. Now the question is how to incorporate such a common geometric relation into the formula M ν ≈ Y ν M −1 R Y T ν φ 2 at the seesaw scale 1 . We consider a simple ansatz of lepton mass matrices to accommodate the conjectures made above. At the scale µ = M 1 , the textures of Y ν and M R are required to have a universal Fritzsch form: where λ = D or R, a = 1−r+r 2 , b = (1−r) r(1 + r 2 )/a and c = r r/a . It is easy to check that three eigenvalues of F λ have the geometric hierarchy r 2 : r : 1. The effective neutrino mass matrix M ν at low energies may also take the Fritzsch texture: where F ν has the same form as F λ defined in Eq. (5) but its two phases (denoted by φ ν and ϕ ν ) are free parameters. Since the running effects of three lepton flavor mixing angles and three CP -violating phases are negligibly small in the standard model with a normal neutrino mass hierarchy 2 , we extrapolate M ν (M Z ) up to the scale µ = M 1 in a good approximation: M ν (M 1 ) ≈ m 3 I α F ν , where I α ∼ 1.4 for M 1 ∼ 10 10 GeV [13]. It is then straightforward to obtain the seesaw relation Our result shows that the Fritzsch texture must be seesaw-invariant, if the eigenvalues of Y ν and M R share a universal geometric hierarchy and their phase parameters satisfy a very simple relation. Indeed, holds in this phenomenological scenario. We argue that the texture zeros of F λ could stem from an underlying horizontal flavor symmetry [16]. Although the dynamical reason for a universal geometric mass hierarchy of light and heavy Majorana neutrinos is completely unknown, we do have observed the approximate geometric hierarchy of up-and down-quark masses (i.e., m u /m c ∼ m c /m t and m d /m s ∼ m s /m b [17]). We proceed to take the charged-lepton Yukawa coupling matrix Y l to have a generic Fritzsch texture with no geometric mass hierarchy: whereã ≈ m τ / φ ,b ≈ √ m µ m τ / φ andc ≈ √ m e m µ / φ . Because of m e /m µ ∼ 0.005 and m µ /m τ ∼ 0.06 [17], the contribution of Y l (or M l ) to lepton flavor mixing is not expected to be very significant. A careful numerical analysis made in Ref. [18] has shown that the Fritzsch texture of lepton mass matrices is compatible with current neutrino oscillation data at the 3σ level. We find that the same conclusion can be drawn when the geometric mass hierarchy of three light Majorana neutrinos is taken. In this case, the allowed values of r are restricted to the range 0.215 < ∼ r < ∼ 0.270. Now let us turn to the cosmological baryon number asymmetry. To calculate the CPviolating asymmetry ε 1 in Eq. (2), we need to figure out the unitary matrix U R which is defined to diagonalize M R . The result is where a = 1 − r + r 2 has been given below Eq. (5). After a straightforward calculation, we arrive at where ω ≡ (φ D − φ R )/2 and m 3 has been given in Eq. (3). Note that ε 1 does not depend on the phase difference (ϕ D − ϕ R ), because the latter is cancelled out in (Y † ν Y ν ) due to the phase relation in Eq. (6). At the seesaw scale, the effective neutrino mass parameter where m 1 has been given in Eq. (3). Ifm 1 lies in the range 10 −2 eV < ∼m1 < ∼ 1 eV (the so-called strong washout regime [19]), one may estimate the dilution factor d by using the approximate formula d ≈ 0.02 × (0.01 eV/m 1 ) 1.1 given in Ref. [19] (11) presented in Ref. [20]. These two simple power laws lead respectively to d ≈ 2.0 × 10 −2 and 3.4 × 10 −2 atm 1 ≈ 0.01 eV, or d ≈ 1.3 × 10 −4 and 1.6 × 10 −4 atm 1 ≈ 1 eV. Therefore their difference is rather insignificant, less than a factor of 2 in the chosen region ofm 1 . Given appropriate inputs of ω and M 1 , the observed baryon number asymmetry Y B can then be reproduced from this simple leptogenesis scenario.
For the purpose of illustration, we typically take ∆m 2 31 = 2.5 × 10 −3 eV 2 , r = 0.25 and m H = 144 GeV (Higgs masss) to evaluate Y B . Restricting the output of Y B to the generous range 7 × 10 −11 < ∼ Y B < ∼ 10 −10 [10], we arrive at the allowed region of ω and M 1 as shown in FIG. 1. Note that regions (A) and (B) are obtained by using the approximate formulas of d given in Refs. [19] and [20], respectively. One can see that the lower bound of M 1 is about 3.0 × 10 10 GeV for region (A) or 1.7 × 10 10 GeV for region (B). In both cases, the lower limit of ω is about 11.4 • . When M 1 is much larger than 10 12 GeV, the value of ω approaches 90 • . This result is a straightfoward consequence of Y B ∝ ε 1 ∝ M 1 sin 3 ω cos ω. We have noticed that our numerical outputs are insensitive to the Higgs mass m H , which slightly affects the running function I α . The effective neutrino massm 1 is found to monotonically increase with ω, from 0.01 eV to 0.16 eV in the interval ω ∈ [11.4 • , 90 • ). Hence the lepton-numberviolating wash-out processes (measured by d) are quite efficient in our scenario.
It is worth mentioning that the above calculation of Y B can simply be extended to the minimal supersymmetric standard model. The sizes of ε 1 and g * in the supersymmetric case are twice as large as in the standard model [8], thus the two effects tend to cancel in the estimate of Y L . For mild values of tan β (from 10 to 50, for example), the supersymmetric running function I α is essentially comparable in magnitude with its standard model counterpart [13]. Therefore, the numerical result shown in FIG. 1 is roughly valid for the minimal supersymmetric standard model.
It is also worth remarking that the CP -violating asymmetry ε 1 has no direct connection with the low-energy CP violation in neutrino oscillations. The reason is simply that the former is governed by the phase difference (φ D − φ R ), while the latter is associated with the phase differences (φ ν − φ l ) and (ϕ ν − ϕ l ). This feature is certainly dependent on the phase choice of lepton mass matrices. If Y l in Eq. (7) is taken to be real (i.e., φ l = ϕ l = 0 or π), for instance, CP violation at low energies turn out to rely on the phase parameters φ ν = 2φ D − φ R and ϕ ν = 2ϕ D − ϕ R . In this case, there appears an indirect connection between CP violation at high scales and that at low scales.
In summary, we have proposed a simple but viable seesaw model of lepton mass matrices based on the phenomenological conjecture that left-and right-handed Majorana neutrinos have a universal geometric mass hierarchy. The cosmological baryon number asymmetry and current neutrino oscillation data can simultaneously be interpreted in this model with very few free parameters. A test of our conjecture and the model itself is possible at low energies, provided more accurate neutrino data are accumulated in the near future.
This work was supported in part by the National Natural Science Foundation of China. Note that regions (A) and (B) are obtained, respectively, by using the approximate formulas of d given in Refs. [19] and [20].