Tribimaximal Mixing, Leptogenesis, and theta13

We show that seesaw models based on flavor symmetries (such as A_4 and Z_7 X Z_3) which produce exact tribimaximal neutrino mixing, also imply a vanishing leptogenesis asymmetry. We show that higher order symmetry breaking corrections in these models can give a non-zero leptogenesis asymmetry and generically also give deviations from tribimaximal mixing and a non-zero theta13>~ 10^(-2)

There is an ongoing experimental program to measure or place an upper bound on θ 13 at the level of sin 2 2θ 13 ∼ 0.01 [3]. The ratio of the solar and atmospheric mass squared differences is r = ∆m 2 21 / ∆m 2 32 = (3.2±0.3)×10 −2 . Although the individual neutrino masses m i are not determined, the neutrino masses are known to be much smaller than the masses of all other standard model fermions from tritium endpoint, neutrinoless double beta decay and cosmological data. The smallness of neutrino masses can be naturally explained using the seesaw model [4], which extends the standard model by adding gauge singlet neutrinos. The singlet neutrinos N R of the seesaw model naturally have Majorana masses much larger than the electroweak scale, unlike the standard model fermions which acquire mass proportional to electroweak symmetry breaking. An interesting feature of the seesaw model is that CP -violating decays of heavy singlet neutrinos can produce a lepton asymmetry in the early universe, which is converted into a baryon asymmetry at the electroweak scale. This leptogenesis mechanism [5,6] provides a simple explanation for the observed baryon asymmetry of the universe.
The neutrino mixing matrix has two large angles (θ 12 , θ 23 ), and one small angle (θ 13 ). A particularly interesting ansatz for the mixing matrix is the tribimaximal matrix [7] with tan 2 θ 12 = 1/2, sin 2θ 23 = 1 and θ 13 = 0. The phase δ is undefined since θ 13 = 0. Eq. 4 can be easily extended to include non-vanishing Majorana phases α 1,2 , U T B → U T B diag(e iα1/2 , e iα2/2 , 1), which is the generalized form of tribimaximal mixing that we will consider in this work. The tribmaximal mixing matrix has been derived using models with discrete flavor symmetries. The models rely on the observation due to Ma [8] that a Majorana mass matrix of the form  is diagonalized by a mixing matrix with θ 13 = 0 and sin 2 2θ 23 = 1. If A + B = C + D, then tan 2 θ 12 = 1/2 and the mixing matrix is tribimaximal. The mixing matrix can have Majorana phases α 1,2 if A, B, C, D are complex. Particularly interesting are models based on the symmetries A 4 [8,9] and Z 7 ⋊ Z 3 [10]. These groups have a three-dimensional irreducible representation, and three inequivalent one-dimensional representations, so that the three generations of lepton doublets, charged leptons and singlet neutrinos can either transform as a 3, or as three inequivalent one-dimensional representations, which distinguish between the generations. It turns out that the seesaw models in the literature which derive exact tribimaximal mixing from a flavor symmetry do not allow for leptogenesis. In these models, the low-energy neutrino mass matrix generated by the seesaw mechanism has the Ma form, but the product of neutrino Yukawa coupling matrices Y † ν Y ν relevant to leptogenesis is proportional to the unit matrix, so the leptogenesis asymmetry parameter ǫ vanishes. This is true even if one considers the more general possibility of flavored leptogenesis [11,12]. The tribimaximal mixing models have complex parameters, and have CP violation. The low-energy PMNS matrix has CP violation through non-zero α 1,2 . The problem is that the symmetry breaking pattern which generates a tribimaximal PMNS mixing matrix does not allow for CP violation in the particular quantity Y † ν Y ν that is needed for leptogenesis. The models typically have higher order corrections from higher dimension operators which are second order in flavor symmetry breaking. If the small symmetry breaking parameter is η ≪ 1, we show that the leptogenesis asymmetry is order η 2 . One can obtain ǫ ∼ 10 −6 , which is the typical value necessary to obtain an adequate baryon asymmetry [6], with η ∼ 10 −2 . We show that the flavor symmetry breaking also leads to deviations from tribimaximal mixing at first order in η, so that θ 13 is typically non-zero, and larger than η ∼ 10 −2 .
Before proceeding further, we first review the standard seesaw scenario for neutrino mixing and leptogenesis. The lepton mass terms in the seesaw theory are (following Ref. [13]): where i, j are flavor indices, L = (e L , ν L ) are the lepton doublets, E R are charged lepton SU (2) L singlets with non-vanishing hypercharge, N R are gauge-singlet fermion fields, and φ is the Higgs It is convenient to pick a basis in which M and Y E are diagonal, real, and non-negative, In this basis, the only freedom to redefine Y ν is given by a diagonal U L rephasing. This rephasing can be used to eliminate three phases in Y ν , so the 3 × 3 complex matrix Y ν contains 9 real and 6 imaginary physical parameters [13].
The singlet Majorana mass matrix M is not proportional to the weak scale v, and is naturally much larger than v in unified theories. The Lagrangian Eq. (6) leads to three heavy neutrinos with masses M i which are dominantly N R , and three light neutrinos with masses of order v 2 /M which are dominantly ν L . Integrating out the heavy right-handed neutrinos leads to the dimension-five operator in the effective theory below M , with When the Higgs field gets a vacuum expecation value, this generates a Majorana mass matrix m = −(v 2 /2)c 5 for the light neutrinos. By definition, the PMNS ma- where the light neutrino masses m i are real and nonnegative. The masses m i and the PMNS matrix U are sufficient to describe neutrino physics at energies below M , and are the observables accessible in low-energy neutrino experiments. The PMNS matrix, however, does not give complete information about the mixing structure of the seesaw Lagrangian. The general form of Y ν consistent with U and m i in a basis where Y E and M are diagonal is [14] v and O is a complex orthogonal matrix OO T = 1. Note that in general O is not unitary. Low-energy physics fixes U and m, but leaves O and M undetermined. At dimension six [13], there is flavor mixing in the light neutrino kinetic terms after electroweak spontaneous symmetry breaking, with Measuring c 6 in addition to U and m completely determines the parameters of the high-energy seesaw theory if the number of generations of heavy neutrinos is equal to the number of generations of standard model fermions.
Out-of-equilibrium decays in the early universe of N Ri to lepton and Higgs doublets produces lepton asymmetries. In a basis where M is diagonal and real, the lepton asymmetry parameters are [5,6,15,16] where For almost degenerate neutrinos, The ǫ i depend on the heavy neutrino masses, and the product using the form Eq. (11). One can have non-zero asym- If leptogenesis takes place at temperatures below about 10 12 GeV, then decoherence effects due to Yukawa interactions of the charged leptons are important, and the flavor of the charged leptons produced in the decays N Ri → ℓ α + φ,l α + φ † are relevant. This scenario is referred to as flavored leptogenesis 2 [11,12], and the lepton asymmetry depends on the asymmetry parameters [11,12,15,16] The flavor independent asymmetry parameter Eq. (14) is given by ǫ i = α=e,µ,τ ǫ (α) i . U does not cancel in the combination Y † ν jα (Y ν ) αi in Eq. (18). There have been several studies of leptogenesis which assume that the PMNS matrix has tribimaximal form [17,18]. It was shown that mass matrices can be constructed in the seesaw Lagrangian which produce a large enough lepton asymmetry for the leptogenesis mechanism to lead to the baryon asymmetry of the universe. However, this construction implicitly assumes that the tribimaximal form of the PMNS matrix is a lowenergy accident, rather than a consequence of an underlying symmetry, as in the examples of Ref. [8,9,10].
References [8,9,10] obtain the tribimaximal structure using a broken discrete flavor symmetry with a specific symmetry breaking pattern generated by the expectation values of scalar fields φ i . The symmetry breaking structure that leads to tribimaximal mixing in these models gives no leptogenesis. If one includes additional symmetry breaking terms of higher order in φ , which exist via higher dimensional operators in the field theory, then one can avoid the leptogenesis problem. The higher dimension operators also lead to deviations from tribimaximal mixing. One can have non-zero leptogenesis while retaining the exact tribimaximal structure only if the higher dimension operators are tuned so that they do not perturb the PMNS matrix, in which case the exact tribimaximal form is accidental.
Another interesting class of models is based on D 4 [19], S 3 [20], and µ ↔ τ symmetry [21]. In these models, symmetry relates entries with µ ↔ τ , and the low-energy neutrino mass matrix has the Ma form Eq. (5) but without the restriction A + B = C + D, so that tan 2 θ 12 is not fixed to be 1/2. These models typically give non-zero leptogenesis. However, if one wants the exact tribimaximal form with tan 2 θ 12 = 1/2 without any accidential fine tunings, then the leptogenesis asymmetry also vanishes. To be specific, in the D 4 model of Grimus and Lavoura [19], where all the parameters can be complex. This example has complex entries in Y † ν Y ν in the basis in which M is diagonal as long as a = b, but in general tan 2 θ 12 = 1/2. Requiring that c 5 (Eq. (9)) have the Ma form Eq. (5) with A + B = C + D for exact tribimaximal mixing leads to the constraint a 2 M 2 = abM χ + b 2 M 1 . To satisfy this relation requires an accidental fine-tuning between the Majorana mass matrix M and the Dirac matrix Y ν , which are independent objects. One could obtain the constraint more naturally by assuming an additional symmetry of the underlying theory (as happens in the A 4 model) that restricts Y ν and M separately by a = b and M 2 = M χ + M 1 . But then the leptogenesis asymmetry vanishes.
In the remainder of the paper, we will use the specific seesaw implementation of A 4 symmetry given by Altarelli and Feruglio [9] to illustrate our point about the incompatibility of tribimaximal mixing derived from an exact flavor symmetry with leptogenesis. A 4 is the symmetry ω 2 ω 2 ω 2 ω ω 2 ω 2 1 ω 2 U (1)R 1 1 1 1 1 0 0 0 group of the tetrahedron, or the group of even permutations on four objects, and has order 12. It has three inequivalent one dimensional representations, 1, 1 ′ , 1 ′′ and a three dimensional representation 3.
Altarelli and Feruglio use a supersymmetric theory with an A 4 ⊗ Z 3 discrete flavor symmetry. In addition to the standard model and singlet neutrino multiplets, they have scalar fields φ S and φ T which transform as A 4 triplets, and ξ which transforms as an A 4 singlet. These scalars develop vacuum expectation values, There also are additional fields needed to construct a superpotential, which are not important for our analysis. The A 4 ⊗ Z 3 representations of the relevant fields are given in Table I  We first summarize the seesaw model of Altarelli and Feruglio. The superpotential terms x A ξN c N c , x B φ S N c N c , and yN c LH u , where x A , x B , y are coupling constants, generate the matrices The leading contribution to the charged lepton masses is from the superpotential terms y e e + (φ T L) 1 H d /Λ, y µ µ + (φ T L) 1 ′ H d /Λ, y τ τ + (φ T L) 1 ′′ H d /Λ which are higher dimension operators 3 suppressed by one power of 3 (φ T L) 1,1 ′ ,1 ′′ denotes that φ T and L are combined to form the A 4 representations 1, 1 ′ , 1 ′′ .

Λ,
It is necessary to introduce higher dimension operators to get non-zero charged lepton masses. The Yukawa coupling y τ is of order η, so η 10 −2 to get a large enough τ mass. It is simple to verify that Eqs. (20,21) lead to tribimaximal mixing, with heavy neutrinos of mass and light neutrinos of mass m i = |yv u | 2 /M i [9]. Let 2φ 1,2,3 be the phases of 2x A u + 2x B v S , 2x A u, −2x A u + 2x B v S , respectively, and φ y be the phase of y. 4 Then the PMNS matrix is U T B e iΦ with Φ = diag(φ 1 − φ y , φ 2 − φ y , φ 3 − φ y ), which can be converted by a phase redefinition into Φ = diag(φ 1 − φ 3 , φ 2 − φ 3 , 0), which is the standard form with only two Majorana phases. Since the three M i are given in terms of two complex numbers 2x A u and 2x B v S , it is not possible to have arbitrary values for M i . For the case of normal hierarchy, m 1 < m 2 < m 3 , and ∆m 2 21 ≪ ∆m 2 32 , so that 2x A u ≈ 2x B v S (which requires a fine-tuning at the level of 1/r ∼ 30 between the two terms), M 1 ≈ 2M 2 , and φ 1 ≈ φ 2 . The known neutrino mass differences then imply that M 3 ≈ 4r/3 M 2 ≈ M 2 /5. Equivalently, m 1 = m 2 /2 = r/3 m 3 , so that m 1 = 5.
so that there is no leptogenesis (see Eq. (14)). There is also no flavored leptogenesis (see Eq. (18)). The combina- contain one factor of Y † ν Y ν in which the charged lepton index has been summed over. Equation (22) implies that Y † ν Y ν is diagonal and proportional to the unit matrix; thus ǫ (α) i vanish since they contain a factor of the off-diagonal elements i = j of Y † ν Y ν . While we have used a specific A 4 model, this conclusion is common to all models in the literature which generate exact tribimaximal mixing from a symmetry. (Note that in the case of the studied A 4 model, the assignment of three-dimensional irreducible representations to the singlet neutrinos and to the lepton doublets forces the Y ν matrix to take the given form.) There are corrections to Eq. (20,21) from operators suppressed by higher powers of 1/Λ. The leading correction to Y ν is from x C N c (Lφ T ) 3S H u /Λ and x D N c (Lφ T ) 3A H u /Λ where 3 S,A are the triplets in the symmetric and antisymmetric product 3 ⊗ 3, and gives The leading corrections to M are from the terms They give a correction to M of the form The (3 S , 3 S ), (3 S , 3 A ) and (1, 1) terms can be absorbed in redefinitions of x A,B,E . The leading corrections to Y E are from (e + L) 3 Since the vacuum expectation value of φ 2 T 3S is proportional to φ T , these corrections can be absorbed into a redefinition of y e,µ,τ in Eq. (21).
There can be direct higher dimension operator contributions to c 5 , which are not generated by the seesaw mechanism [9]. These arise from operators such as LLH u H u ξ 2 /Λ 3 , LLH u H u ξφ S /Λ 3 , LLH u H u φ 2 S /Λ 3 , and produce neutrino mass terms of order v 2 u V 2 /Λ 3 . They are of order V 3 /Λ 3 relative to the seesaw generated mass terms of order v 2 u /V , and can be neglected in our analysis.
The corrections Eq. (23,24) lead to deviations from exact tribimaximal mixing, and from the diagonal form Eq. (22), which can be computed in perturbation theory. It is convenient to go to a basis where the lowest order matrices Eq. (20,21), which we now denote by Y † ν0 and M † 0 , are diagonal. Let up to first order in η = V /Λ, where φ ij = φ i − φ j . S cancels since Y † ν0 Y ν0 ∝ 1. It can be seen from Eq. (26) that Im{[(Y † ν Y ν ) ij ] 2 } is non-zero and order η 2 in the basis where M is diagonal.
For the normal hierarchy, the lightest right-handed neutrino is M 3 , and leptogenesis is governed by ǫ 3 , i.e. by Y † ν Y ν 3i , i = 1, 2, which are non-zero, and complex. Hence Eq. (26) leads to non-zero leptogenesis for the case of normal hierarchy. Using Eq. (14), the lowest order values M 1 ≈ 2M 2 , φ 1 ≈ φ 2 , and the asymptotic form for f (x) for normal hierarchy gives combining the M 2 and M 1 terms. For the inverted hierarchy, M 2 is the lightest right-handed neutrino, and ǫ 2 controls leptogenesis, Re 2x C v T y * 3Λ 2 sin 2φ 21 . (28) In the second term, we have used Eq. (16) for f (M 2 1 /M 2 2 ). The PMNS matrix is U = U T B e iΦ0 U 1 , where U 1 is defined so that U T 1 c 5 U 1 is diagonal, with c 5 ≡ e −2iφy Y * ν M * −1 Y † ν . Writing U 1 = exp iX with X hermitian, and expanding X = δ (1) X + . . ., we can solve for X to first order in V /Λ, The light-neutrino mass shifts are δm i /m i ∼ O (V /Λ) due to the higher dimension operators.