Davide Meloni
email: davide.meloni@physik.uni-wuerzburg.de

A See-Saw S 4 model for fermion masses and mixings

We present a supersymmetric see-saw S 4 model giving rise to the most general neutrino mass matrix compatible with Tri-Bimaximal mixing. We adopt the S 4 ×Z 5 flavour symmetry, broken by suitable vacuum expectation values of a small number of flavon fields. We show that the vacuum alignment is a natural solution of the most general superpotential allowed by the flavour symmetry, without introducing any soft breaking terms. In the charged lepton sector, mass hierarchies are controlled by the spontaneous breaking of the flavour symmetry caused by the vevs of one doublet and one triplet flavon fields instead of using the Froggatt-Nielsen U (1) mechanism. The next to leading order corrections to both charged lepton mass matrix and flavon vevs generate corrections to the mixing angles as large as O(λ 2 C ). Applied to the quark sector, the symmetry group S 4 × Z 5 can give a leading order V CKM proportional to the identity as well as a matrix with O(1) coefficients in the Cabibbo 2 × 2 submatrix. Higher order corrections produce non vanishing entries in the other V CKM entries which are generically of O(λ 2 C ).

Introduction

The Tri-Bimaximal structure (TBM) [1] of the neutrino mixing matrix is remarkably in agreement with the experimental results in the neutrino sector [2]. Within 1-σ error, the values of the mixing angles can be approximated by their TBM values [3]:

tan 2 θ 23 = 1 tan 2 θ 12 = 1 2 sin θ 13 = 0 . ( 1
)
The simplified structure of the mixing matrix suggested the possibility to be explained using some discrete non-abelian groups, added to the Standard Model, and containing a triplet representations to fit the number of lepton families observed in Nature. The symmetry A 4 [4,5] (also in the context of Grand Unified theories [6]) emerged as a natural candidate because it is the smallest discrete group with triplet representation and it is sufficiently manageable to be broken differently in the charged and neutral lepton sectors, a necessary condition if we want to get a mixing matrix different from the identity.

In the context of see-saw models, it is interesting to observe that the TBM structure in A 4 is generally associated with a well defined relations among the complex eigenvalues of the light neutrino mass matrix:

1 m 3 = 1 m 1 - 2 m 2 , ( 2
)
implying that most A 4 models are quite predictive because of the reduced number of independent parameters. It should be stressed, however, that the realisation of the TBM strongly relies on the choice of symmetry breaking pattern. In fact, in the neutrino sector the group A 4 is usually broken into a subgroup generated by the matrices

U µ-τ =   1 0 0 0 0 1 0 1 0   and G =   -1 2 2 2 -1 2 2 2 -1   ,
which, in the basis where charged leptons are diagonal, leave invariant the most general neutrino mass matrix diagonalized by TBM:

m light =   x y y y x + v y -v y y -v x + v   .
(

) 3
It turns out that the representations of A 4 only contain G and the invariance under U µ-τ arises accidentally, as a consequence of the specific field content of the model. From this point of view, the group S 4 [7]- [9] arises as a natural candidate as a flavour group for neutrino mixing because one can find a suitable representation of it containing simultaneously the previous elements. It should also be noted that the extension of the A 4 symmetry from lepton to quarks seems to be complicated by the absence of doublet representations, whose use is suggested by the heaviness of the top quark, which S 4 possesses instead.

However, the problem of reproducing small mixing angles and strong mass hierarchy in the up-sector at the same time can only be partially alleviated by the bidimensional representations of S 4 because one or more fine tunings between the relevant Yukawas are invoked to correctly reproduce some of the m up /m down quark mass ratios. In this paper, we build a constrained see-saw S 4 -based model for fermion masses and mixing which, compared to models already existent in the literature, realises the most general neutrino mass matrix diagonalized by TBM at leading order (LO). This is obtained allowing the right-handed neutrinos to couple to singlet, doublet and triplet flavon fields. The light neutrino masses depend on six complex Yukawa parameters and the typical A 4 sum rule of eq.( 2) does not hold, leaving the model less predictive but more manageable. The mass hierarchy among charged leptons is obtained breaking the S 4 symmetry by the vevs of a doublet and triplet flavon fields, without invoking any Froggatt-Nielsen U (1) symmetry. The unwanted couplings are forbidden imposing an additional Z 5 symmetry to the model. The resulting S 4 × Z 5 symmetry is minimal from the point of view of the flavour symmetry and field content. We extend the S 4 × Z 5 symmetry to the quark sector including the left-handed components into triplet representations (and not into doublets, as usually done); we show that, even using a rigid structure like that proposed in this paper, some of the relevant features of the quark sector, like a good leading order V CKM and quark mass ratios, can still be accounted for. The paper is organized as follows: in Sect.2, we discuss the relevant feature of the S 4 symmetry and the structure of the model, presenting leading order results on neutrino as well as charged lepton mass matrices; in Sect.3 we compute the next to leading order corrections (NLO) to the vacuum alignment and the relevant higher order operators, both responsible for deviations from TBM mixing; in Sect.4 we discuss some phenomenological results obtained from our model with all Yakawas constrained to be O (1) and we also show that the whole model allows for acceptable leptogenesis parameters. Sect.5 is devoted to the quark sector whereas in Sect.6 we draw our conclusions.

The structure of the model

We introduce here the structure of the model which leads to TBM in first approximation. We recall that S 4 , the permutation group of 4 objects, can be generated by the two elements S and T obeying the relations (a "presentation" of the group):

S 4 = T 3 = 1, ST 2 S = T . ( 4
)
The action of the generators S and T can be assigned as follows:

(1234) → S (2341) 2 (1234) → T (2314)
and the 24 elements of the group, belonging to 5 conjugate classes, are:

C 1 : 1 C 2 : S 2 = (3412), T S 2 T 2 = (4321), S 2 T S 2 T 2 = (2143) C 3 : T, T 2 = (3124), S 2 T = (1423), S 2 T 2 = (2431), ST ST 2 = (4132) ST S = (4213), T S 2 = (4132), T 2 S 2 = (1342) C 4 : ST 2 = (1243), T 2 S = (4231), T ST = (1432) T ST S 2 = (3214), ST S 2 = (1324), S 2 T S = (2134) C 5 : S, T ST 2 = (2413), ST = (3142), T S = (3421), S 3 = (4123), S 3 T 2 = (4312)
The inequivalent irreducible representations of S 4 are 1 1 , 1 2 , 2 and 3. It is immediate to see that one-dimensional unitary representations are given by:

1 1 : S = 1 T = 1 1 2 : S = -1 T = 1 (5)
while the two-dimensional unitary representation, in a basis where the element T is diagonal, is given by:

T = ( ω 0 0 ω 2 ) , S = ( 0 1 1 0 ) . ( 6
)
Finally, the three-dimensional unitary representation is as follows:

3 1 : T =   1 0 0 0 ω 2 0 0 0 ω   , S = 1 3   -1 2ω 2ω 2 2ω 2ω 2 -1 2ω 2 -1 2ω   , ( 7
)
where ω = e 2πi/3 = (-1 + √ 3)/2, whereas in the 3 2 representation the generator T is the same but S is the opposite. It is useful to remind the product rules between the group representations:

1 1 ⊗ ξ = ξ 1 2 ⊗ 1 2 = 1 1 , 1 2 ⊗ 2 = 2, 1 2 ⊗ 3 i = 3 j (8) 2 ⊗ 2 = 1 1 ⊕ 1 2 ⊕ 2, 2 ⊗ 3 i = 3 1 ⊕ 3 2 , 3 i ⊗ 3 i = 1 1 ⊕ 2 ⊕ 3 1 ⊕ 3 2 , 3 1 ⊗ 3 2 = 1 2 ⊕ 2 ⊕ 3 2 ⊕ 3 2 ,
where the indeces i, j = 1, 2, with i = j and ξ indicates any other representation. The Clebsch-Gordan coefficients in the basis presented above are reported in Appendix A.

The general neutrino mass matrix of eq.( 3) can be obtained in the framework of the see-saw mechanism,

m light = -m T D m -1 M m D , (9)
where both Majorana (m M ) and Dirac (m D ) mass matrices are needed. They are derived from the most general lagrangian invariant under S 4 × Z 5 and containing fields in any of the S 4 representations, which are singlets, doublets and triplets. The group S 4 is broken by means of suitable vev's of Standard Model singlet fields (flavons), whose alignments have to guarantee the correct entries of the Majorana and Dirac mass matrices. Group theoretical considerations help in understanding the pattern of symmetry breaking needed to generate the wanted matrix; in the representation of App.A, the elements S 2 and (T ST S 2 ) leave invariant the m light of eq.( 3), that is

S 2 m light S 2 = m light (T ST S 2 ) m light (T ST S 2 ) = m light .
This means that the vevs of the flavon fields in the neutrino sector have to be invariant under the subgroup generated by them. This strong condition for the flavon alignment is realized, for the triplet representation, by the field configuration

ϕ S = v S (1, 1, 1) . ( 10
)
For the bidimensional representation, the same matrices should leave the vev of a doublet field invariant; if we choose

∆ = v ∆ (1, 1) (11)
these correspond to the first matrix of the groups C 4,5 in the doublet representation. Therefore, we have found a scalar field configuration which remains invariant under the action of the same matrices that leave m light invariant. In addition to the previous fields, we also include in the model a singlet flavon field ξ, with a non-vanishing vev. To realize the classical see-saw mechanism, we need to introduce right-handed neutrinos, which we assume to transform as a triplet representation of S 4 . To avoid large fine-tuning from terms of the form M ν c ν c , like those discussed in [5], we properly tune the ν c charge under Z 51 ; whatever this Z 5 charge is, if we want the singlet, doublet and triplet flavons to contribute to the neutrino mass matrix their charges have to be the same. We also attribute a Z 5 charge to the h u higgs boson in order to avoid the Weinberg operator O 5 = h u h u at the leading order [11], where is a triplet field of the SU (2) Standard Model lepton doublets. This also forbids a leading order Dirac mass term of the form (ν c ) h u , which is then generated at the O(1/Λ) through couplings with the same flavon fields ξ, ∆ and ϕ S . The Z 5 charges for the matter fields as well as for flavons and driving fields (discussed later) are reported in Tab.1, where we used the symbol ω = e 2πi/5 to indicate the Z 5 unit-charge.

The lagrangian in the neutrino sector is then as follows:

W ν = 1 Λ ν c h u (y ν 1 ϕ S + y ν 2 ∆ + y ν 3 ξ) + ν c ν c (a ξ + b ϕ S + c ∆) , ( 12
)
Field ν c e c µ c τ c h d h u ϕ T η ∆ ϕ S ξ ϕ T 0 ϕ S 0 ∆ 0 ρ 0 S 4 3 3 1 1 2 1 1 1 1 1 1 1 1 3 1 2 2 3 1 1 1 3 1 3 1 2 1 1 Z 5 ω 2 1 ω 3 ω 2 ω 1 ω 2 ω 4 ω 4 ω ω ω ω 2 ω 3 ω 3 ω 2 U (1) R 1 1 1 1 1 0 0 0 0 0 0 0 2 2 2 2
Table 1: Transformation properties of leptons, electroweak Higgs doublets and flavons under S 4 × Z 5 and U (1) R .

where a, b, c and y ν i are complex Yukawa couplings. The Dirac mass matrix is obtained from the first term in eq.( 12) and it is given by:

m D = v u Λ   2 y ν 1 v S + y ν 3 u y ν 2 v ∆ -y ν 1 v S y ν 2 v ∆ -y ν 1 v S y ν 2 v ∆ -y ν 1 v S 2 y ν 1 v S + y ν 2 v ∆ y ν 3 u -y ν 1 v S y ν 2 v ∆ -y ν 1 v S y ν 3 u -y ν 1 v S 2 y ν 1 v S + y ν 2 v ∆   ( 13
)
where v u is the vacuum expectation value of the higgs field h u . The other terms give the Majorana mass matrix:

m M =   a u + 2 b v S -b v S + c v ∆ -b v S + c v ∆ -b v S + c v ∆ 2 b v S + c v ∆ a u -b v S -b v S + c v ∆ a u -b v S 2 b v S + c v ∆   (14)
whose eigenvalues are:

M 1 = a u + 3 b v S -c v ∆ M 2 = a u + 2 c v ∆ (15) M 3 = -a u + 3 b v S + c v ∆ .
Using eq.( 9), we can derive the light neutrino mass matrix, diagonalized by tri-bimaximal mixing, whose eigenvalues are:

m 1 = - ( v u Λ ) 2 (3 y ν 1 v S -y ν 2 v ∆ + y ν 3 u) 2 a u + 3 b v S -c v ∆ m 2 = - ( v u Λ ) 2 (2 y ν 2 v ∆ + y ν 3 u) 2 a u + 2 c v ∆ (16
)

m 3 = ( v u Λ ) 2 (3 y ν 1 v S + y ν 2 v ∆ -y ν 3 u) 2 a u -3 b v S -c v ∆ .
We see that the neutrino masses depend on six unrelated complex Yukawa parameters, which offer more freedom to tune mass differences and then recover the phenomenology associated to neutrino oscillation. Notice also that no sum rules can be found in this case among complex eigenvalues.

Charged leptons

It could be easier to work in a basis where the charged lepton mass matrix is diagonal.

In order to understand how this naturally arises in an S 4 -based model, we observe that a generic diagonal matrix m D l (with diagonal entries different to each other) is left invariant under the action of an element A of S 4 only if such an element is itself diagonal, with different phase factors at each diagonal entry, as it can be understood requiring that the relation

A † m D † l m D l A = m D † l m D l ( 17
)
is satisfied. The generator T , for example, is such an appropriate matrix. Since the charged lepton matrix is generated after spontaneous symmetry breaking, one could choose flavon fields with vevs invariant under the action of T . Instead, we prefer the choice

ϕ T = v T (0, 1, 0) (18)
which, even breaking completely the group S 4 , not only guarantees the diagonal form of the mass matrix but also generates the hierarchy among lepton families without introducing any addition U (1) F N symmetry2 . In the same way, we choose the vev for the doublet flavon η as:

η = v η (0, 1) ; (19)
the corresponding lagrangian in the charged lepton sector is as follows:

L = y τ Λ τ c ( ϕ T ) h d + y µ 1 Λ 2 µ c ( ϕ T ϕ T ) h d + y µ 2 Λ 2 µ c (η ϕ T ) h d + y e 1 Λ 3 e c [ϕ T (ϕ T ϕ T ) 2 ] 3 2 h d + y e 2 Λ 3 e c [ϕ T (ϕ T ϕ T ) 3 1 ] 3 2 h d + (20) y e 3 Λ 3 e c [η (ϕ T ϕ T ) 3 1 ] 3 2 h d + y e 4 Λ 3 e c [ϕ T (ηη) 2 ] 3 2 h d + y e 5 Λ 2 e c (∆ϕ S ) 3 2 h d .
It is interesting to observe that the last term in the lagrangian would be the dominant one and would drive the electron mass to a too large value3 . However, the leading order structures of the vacua in eqs.( 10)- (11) prevent this term to appear. We will show later that this is not the case when the next to leading order corrections to the flavon alignment are taken into account. After symmetry breaking, the mass matrix has the form:

m = v d v T Λ   1 Λ 2 [ v 2 T (y e 1 + 2 y e 2 ) -2 v T v η y e 3 + v 2 η y e 4 ] 0 0 0 1 Λ (2 y µ 1 v T + y µ 2 v η ) 0 0 0 y τ   , (21) where v d = h d .
To estimate the order of magnitude of v T and v η , we can use the experimental informations on the ratios of lepton masses. Assuming that the combinations of the y coefficients are of all of O( 1), one obtains:

( m µ m τ ) ∼ 2 ε T + ε η 0.06 ( m e m τ ) ∼ 3 ε 2 T -2 ε T ε η + ε 2 η 0.0003
where we introduced the small quantities

ε T = v T /Λ ε η = v η /Λ .
These relations are satisfied for:

(|ε T |, |ε η |) ∼ (0.017, 0.029). ( 22
)
so that we can roughly assume that both ε T and ε η are of the same order of magnitude,

ε ∼ O(λ 2 C ).

Superpotential and vacuum alignment

The most general driving superpotential w d invariant under S 4 × Z 5 with R = 2 is given by

w d = g 1 (ϕ S 0 ϕ S ϕ S ) + g 2 (ϕ S 0 ϕ S ) ξ + g 3 ϕ S 0 (ϕ S ∆) + g 4 ∆ 0 (∆∆) + g 5 ∆ 0 (ϕ S ϕ S ) + g 6 ∆ 0 ∆ξ + (23) h 1 ϕ T 0 (ϕ T ϕ T ) + h 2 ϕ T 0 (ηϕ T ) + r 1 ρ 0 (ϕ T ϕ T ) + r 2 ρ 0 (ηη) = w LO d (ϕ S 0 , ∆ 0 ) + w LO d (ϕ T 0 , ρ 0 )
where all possible contractions among flavon fields are understood. The equations which fix the components of the vevs of the various flavon fields are obtained solving a system of equations obtained deriving w d with respect to the component of the driving fields.

The charge assignment reported in Tab.1 allows to separate the equations into two sets of independent relations among the fields appearing in the neutrino sector and in the charged one, respectively. For the latter we have:

∂w d ∂ϕ T 01 = 2h 1 (ϕ T 2 1 -ϕ T 2 ϕ T 3 ) + h 2 (η 1 ϕ T 2 + η 2 ϕ T 3 ) = 0 ∂w d ∂ϕ T 02 = 2h 1 (ϕ T 2 2 -ϕ T 1 ϕ T 3 ) + h 2 (η 1 ϕ T 1 + η 2 ϕ T 2 ) = 0 (24) ∂w d ∂ϕ T 02 = 2h 1 (ϕ T 2 3 -ϕ T 1 ϕ T 2 ) + h 2 (η 1 ϕ T 3 + η 2 ϕ T 1 ) = 0 ∂w d ∂ρ 0 = r 1 (ϕ T 2 1 + 2 ϕ T 2 ϕ T 3 ) + 2 r 2 η 1 η 2 ,
whose solutions are:

η = v η (0, 1), ϕ T = v T (0, 1, 0) , v η = -2 ( h 1 h 2 ) v T . ( 25
)
It is important to observe that the last equation is crucial to avoid another solution of the form η = (1, -1), ϕ T = (1, 1, 1). We also recover the relation between the vevs v T and v η of eq.( 22). In the neutrino sector, the set of equations read as follows:

∂w d ∂ϕ S 01 = 2g 1 (ϕ S 2 1 -ϕ S 2 ϕ S 3 ) + g 2 ξ ϕ S 1 + g 3 (∆ 1 ϕ S 2 + ∆ 2 ϕ S 3 ) = 0 ∂w d ∂ϕ S 02 = 2g 1 (ϕ S 2 2 -ϕ S 1 ϕ S 3 ) + g 2 ξ ϕ S 3 + g 3 (∆ 1 ϕ S 1 + ∆ 2 ϕ S 2 ) = 0 ∂w d ∂ϕ S 03 = 2g 1 (ϕ S 2 3 -ϕ S 1 ϕ S 2 ) + g 2 ξ ϕ S 2 + g 3 (∆ 1 ϕ S 3 + ∆ 2 ϕ S 1 ) = 0 (26) ∂w d ∂∆ 01 = g 4 ∆ 2 1 + g 5 (ϕ S 2 3 + 2 ϕ S 1 ϕ S 2 ) + g 6 ∆ 2 ξ = 0 ∂w d ∂∆ 02 = g 4 ∆ 2 2 + g 5 (ϕ S 2 2 + 2 ϕ S 1 ϕ S 3 ) + g 6 ∆ 1 ξ = 0 .
The system is solved by:

ξ = u, ∆ = v ∆ (1, 1), ϕ S = v S (1, 1, 1) , ( 27
)
with the additional relations

v ∆ = - g 2 u 2 g 3 (28) v 2 S = ( 2g 2 g 3 g 6 -g 2 2 g 4 12g 5 g 2 3 ) u 2 (29)
and u undetermined. We have explicitly checked that the solutions of the vacuum alignment equations are unique; in fact, requiring for the generic field Φ LO to be shifted as Φ LO + δΦ, we found that the components δΦ are all in the same directions of the corresponding Φ LO ; thus, we do not need to introduce any soft term to drive the superportential into the wanted minimum.

Next to leading order

The discussion of the corrections to the previous results starts with a study of the next to leading order structure of the vacuum alignments of the flavon fields. It will turn out that such corrections will be enough to guarantee deviation from TBM at a level compatible with the recent experimental results.

Corrections to the vacuum alignment

The next to leading order terms mix the charged lepton and neutrino sectors in a nontrivial way. We want to find perturbations of (25,27) of the form

ϕ S = (v S + δv S 1 , v S + δv S 2 , v S + δv S 3 ) ϕ T = (δv T 1 , v T + δv T 2 , v T + δv T 3 ) ∆ = (v ∆ + δv ∆ 1 , v ∆ + δv ∆ 2 ) (30) η = (δv η 1 , v η + δv η 2 ) ξ = u + δu .
On a general ground, we have eleven unknowns but only nine equations (six from the two triplets, two from the doublet and one from the singlet); then, we expect two of the previous shifts to remain unconstrained. To better understand the output of such an analysis, we study the (ϕ S 0 , ∆ 0 ) and (ϕ T 0 , ρ 0 ) sectors separately.

3.1.1

The (ϕ S 0 , ∆ 0 ) sector This sector is responsible for the alignment of the fields ϕ S and ∆. At order O(1/Λ), the three-field terms entering the superpotential are combination of ϕ T and η; collecting these terms, we have:

δw d (ϕ S 0 ) = Σ 3 i=1 s i Λ ϕ S 0 (ϕ T ϕ T ϕ T ) i + Σ 5 i=4 s i Λ ϕ S 0 (ϕ T ηη) i + s 6 Λ ϕ S 0 (ϕ T ϕ T η) i (31) δw d (∆ 0 ) = Σ 2 i=1 δ i Λ ∆ 0 (ηηη) i + Σ 4 i=3 δ i Λ ∆ 0 (ϕ T ϕ T η) i + δ 5 Λ ∆ 0 (ϕ T ϕ T ϕ T ) i (32)
where, in both cases, the index i of the trilinear terms represents different S 4 contractions. These NLO corrections have to be evaluated with the leading order vev's in eq.( 25), so that they do not contain any of the unknowns under investigation. In particular, the structure of the vacua (25) produces

δw d (∆ 0 ) = 0 ;
and the total LO+NLO part of the superpotential responsible for the alignment of ϕ S and ∆ is given by:

w d (ϕ S 0 , ∆ 0 ) = w LO d (ϕ S 0 , ∆ 0 ) + δw d (ϕ S 0 ) .
Symmetry arguments allow to understand the structure of the solutions, whose detailed expressions can be found by explicitly solving the system of equations according to the procedure of Sect. (2.2). In fact, we see that, after symmetry breaking, the non-vanishing terms in δw d (ϕ S 0 ) are proportional to the vector (1, 0, 0), which is left invariant by the S 4 element (T ST S 2 ); this has a 2 ↔ 3 symmetry which forces the new vev of ϕ S to have the form (v S + δv S 1 , v S + δv S 2 , v S + δv S 2 ). At the same time, the shifts δv ∆ i should all be equal because of the vanishing NLO terms δw d (∆ 0 ). An explicit computation confirms these speculations and we can write the new vacua in the following form:

ϕ S = v S        1 + A S ( ε ε ) 2 ε 1 + B S ( ε ε ) 2 ε 1 + B S ( ε ε ) 2 ε        ∆ = v ∆   1 + A ∆ ( ε ε ) 2 ε 1 + A ∆ ( ε ε ) 2 ε   (33)
where the coefficients A S , B S and A ∆ are linear combinations of leading and next to leading order coefficients and, for the sake of simplicity, we have introduced the small parameter ε = ϕ S /Λ ∼ ∆ /Λ ∼ ξ /Λ. Notice that the three δv S i also depend on the undetermined parameter δu with identical coefficients, so that they can be readsorbed in the leading order result.

3.1.2

The (ϕ T 0 , ρ 0 ) sector This sector is responsible for the alignment of the fields ϕ T and η. At order O(1/Λ), the new terms in the superpotential are combinations of ϕ S , ∆ and ξ; we have:

δw d (ϕ T 0 ) = Σ 3 i=1 t i Λ ϕ T 0 (ϕ S ϕ S ϕ S ) i + Σ 5 i=4 t i Λ ϕ T 0 (ϕ S ∆∆) i + (34) t 6 Λ ϕ T 0 (ϕ S ϕ S ∆) + t 7 Λ ϕ T 0 (ϕ S ϕ S )ξ + t 8 Λ ϕ T 0 ϕ S ξ 2 + t 9 Λ ϕ T 0 (ϕ S ∆) ξ and δw d (ρ 0 ) = ρ 1 Λ ρ 0 (ϕ S ϕ S ϕ S ) + ρ 2 Λ ρ 0 (∆ϕ S ϕ S ) + ρ 3 Λ ρ 0 (ϕ S ϕ S ) ξ + (35) ρ 4 Λ ρ 0 (∆ ∆ ∆) + ρ 5 Λ ρ 0 (∆ ∆) ξ + ρ 5 Λ ρ 0 ξ 3 .
In this case, the corrections to ϕ T 0 do not conserve any of the S 4 basis elements and we expect that the components of ϕ T will point to different directions. Also the η 's vev is not preserved at the next to leading order and, consequently, the shifts δv η i are different from zero and different to each other. We choose to treat δv η 2 as the second undetermined parameters, so that the new vacua can be cast in the following form:

ϕ T = v T        A T ( ε ε ) 2 ε 1 + B T ( ε ε ) 2 ε C T ( ε ε ) 2 ε        η = v η   A η ( ε ε ) 2 ε 1 + B η δv η 2   . ( 36
)
Notice that a correction proportional to δv η 2 also appears in the second component of ϕ T in the form ϕ

T 2 = v T -(h 2 /2h 1 ) δv η 2 + O(ε ) which, using the last relation in eq.(25), gives ϕ T 2 = -(h 2 /2h 1 ) (v T + δv η 2 ) + O(ε )
, so that the shift δv η 2 can be readsorbed into a redefinition of v T .

Charged lepton mass matrix

The next to leading order corrections to the mass matrix of eq.( 21) come from the corrections to the vacuum alignments in eqs.( 33) and ( 36) and from higher dimensional operators, suppressed by a relative O(1/Λ) with respect to each of the terms quoted in eq.( 20). In the following we will study them separately.

Corrections from vacuum alignment

The main features of these corrections are related to the fact that all the vanishing entries in the matrix of eq.( 21) are suppressed by one additional O(1/Λ) factor compared to the diagonal entries, except for the first line, where the O(1/Λ 2 ) operator in eq.( 20) is non vanishing. As a result, we get the following charged lepton mass matrix at the NLO:

m = v d   a 1 ε 3 a 2 ε ε 2 -a 2 ε ε 2 b 1 ε 3 b 2 ε 2 b 3 ε 3 c 1 ε 3 ε c 2 ε 3 ε c 3 ε   (37)
where the coefficients a i , b i and c i can be easily reconstructed from eqs.( 20) and ( 36). The matrix m † m can be diagonalized by the unitary transformation

U =     1 ( b 1 b 2 ε 3 ε 2 ) * ( c 1 c 3 ε 3 ε 2 ) * -b 1 b 2 ε 3 ε 2 1 ( c 2 c 3 ε 3 ε 2 ) * -c 1 c 3 ε 3 ε 2 -c 2 c 3 ε 3 ε 2 1     . ( 38
)
so that, at the lowest order in ε , the coefficients of the electron row in eq.( 37) do not contribute. From the matrix in eq.( 38), we can compute the U PMNS mixing matrix from the lepton sector only: ) to fit the experimental data.

U PMNS = U † U T BM =      √ 2 3 + 1 √ 6 (a * + b * ) ( ε 3 ε 2 ) * 1 √ 3 -1 √ 3 (a * + b * ) ( ε 3 ε 2 ) * 1 √ 2 (a * -b * ) ( ε 3 ε 2 ) * - √ 1 6 + √ 2 3 a ( ε 3 ε 2 ) + 1 √ 6 c * ( ε 3 ε 2 ) * √ 1 3 + 1 √ 3 a ( ε 3 ε 2 ) -1 √ 3 c * ( ε 3 ε 2 ) * - √ 1 2 -c * ( ε 3 ε 2 ) * - √ 1 6 + 1 √ 6 (2b -c) ( ε 3 ε 2 ) √ 1 3 + (b + c) ( ε 3 ε 2 ) √ 1 2 - √ 1 2 c ( ε 3 ε 2 )      (

Corrections from higher dimensional operators

For the electron case, there are many of such operators, coming from the following contractions:

( ∆ 3 ϕ T ) ( ∆ 3 η ) ( ϕ 3 S ϕ T ) ( ϕ 3 S η ) ( ∆ 2 ϕ T ϕ S ) ( ∆ 2 ϕ T ξ ) ( ∆ 2 ϕ S η ) ( ∆ 2 ξη ) ( ϕ 2 S ϕ T ∆ ) ( ϕ 2 S ϕ T ξ ) ( 40
) ( ϕ 2 S ∆η ) ( ξ 2 ϕ T ∆ ) ( ξ 2 ϕ S η ) ( ξ 2 ϕ T ϕ S ) ( ξ 2 ∆ϕ S ) .
It is easy to understand that they contribute to the electron row by terms of order O(1/Λ 4 ), which are unimportant because they do not sizebly modify the diagonalizing matrix U . Also, the corrections to the τ row are of O(1/Λ 3 ), that is the coefficients b and c appearing in the U P M N S are modified at the next to next to leading order (NNLO) and can be safely neglected. For the muon case, the following higher dimensional operators modify the coefficient a appearing in eq.( 39) at the NLO :

L NLO = Σ 2 i=1 y µ N LO i Λ 3 µ c (ϕ S ϕ S ϕ S ) h d + Σ 4 i=3 y µ N LO i Λ 3 µ c (ϕ S ∆∆) h d + (41) y µ N LO 5 Λ 3 µ c ϕ S ξ 2 + y µ N LO 5 Λ 3 µ c (ϕ S ∆) ξ .
In conclusion, both types of corrections (from vacuum alignment and from higher dimensional operators) to the charged lepton mass matrix contribute to generate deviation from TBM at O(λ 2 C ).

Neutrinos

Also for neutrinos, we have to take into account corrections from both vacuum alignment and higher dimensional operators. In particular, the NLO corrections of the vevs of the flavon fields affect both Majorana and Dirac masses with the same pattern, as we can see from eq.( 12). Different higher order corrections are generated by higher order operators. These corrections turn out to be negligible compared with the NLO and of the same order of magnitude as those induced by the Weinberg operator (see later).

Corrections to the Majorana and Dirac mass matrices

The relevant contributions come from the NLO vacuum alignments. In particular, eq.( 14) is modified only by ϕ S at the NLO because the other two vevs, namely those of ∆ and ξ entering in the lagrangian in eq.( 12), are aligned along the LO direction. However, the shifts in ϕ S are not enough to introduce independent corrections to all the (six) elements of the Majorana and Dirac mass matrices; in fact, the explicit structure of the NLO corrections are as follows:

δm M = b ( ε ε ) 2 ε v S Ŝ (42) δm D = v u ( ε ε ) ε 2 y ν 1 Ŝ (43)
where the common matrix Ŝ is:

Ŝ =   2 A S -B S -B S -B S 2 B S -A S -B S -A S 2 B S   . ( 44
)
With these results, we can build the light neutrino mass matrix; since the NLO matrix elements have now complicated expressions in terms of the leading order Yakawas of eq.( 12) and of the correction coefficients A S and B S , we prefer to summarize its structure as follows:

m light = ε 2   x y y y x + v y -v y y -v x + v   + ε 3   x y y y z z y z z   , ( 45
)
where the relation z+z = x +y implies that m light is not diagonalized by U T BM . However, it is easy to show that the almost symmetric structure of the ε 3 contribution prevents to generate corrections to the TBM values of θ 13 and θ 23 because the diagonalization of m light is achieved by a matrix U ν = U T BM + δU , where δU has a vanishing last column. If we want to obtain a non-vanishing θ 13 , we need to distinguish the elements ( 12) from ( 13) and/or ( 22) from (33) in eq.( 44). Without taking into account the corrections from the charged lepton sector, this can be accomplished only at the NNLO. In fact, a Z 5 singlet built with the ν c ν c bilinear requires a charge ω for the accompaining fields, and this can be achieved with at least three fields, which induce terms of O(1/Λ 2 ). There are more than twenty of these terms, generated by the following contractions:

( ∆ 2 ϕ T ) ( ∆ 2 η ) ( ϕ 2 S ϕ T ) ( ϕ 2 S η ) ( ξ 2 ϕ T ) ( ξ 2 η ) (ϕ T ∆ϕ S ) (ϕ T ∆ξ) (ϕ T ϕ S ξ) (∆ηϕ S ) (∆ηξ) (ϕ S ηξ) .
Similarly, the Dirac mass matrix receives corrections at the same NNLO relative to the leading terms due to the fact that the term (ν c ) h u has a total Z 5 = ω 4 . Beside the previous operators, we also have to consider effective terms of the form llh u h u ; given its charge assignment (Z 5 = ω 4 ), the Weinberg operator arises at O(1/Λ 2 ) with the insertion of one flavon field, as for the Dirac and Majorana terms:

W eff ν = 1 Λ 2 (α 1 ϕ S + α 2 ∆ + α 3 ξ) ( h u h u ) . ( 46
)
After spontaneous symmetry breaking, W eff ν generates terms which are of order:

m W ∼ v 2 u Φ Λ 2 ∼ ε ( v u Λ ) v u . ( 47
)
Compared to the NLO corrections of the Dirac mass term, eq.( 43), we see that:

m W /δm D ∼ v u /Λ ε << 1 ;
then this type of operators is more suppressed with respect to the Dirac mass corrections (and even more if compared with the Majorana mass terms) and can be safely neglected.

In conclusion, from the neutrino sector only, the following mixing matrix arises:

U ν =      √ 2 3 - √ 2 3 (x +y -z-z ) * 9y * ( ε 3 ε 2 ) * 1 √ 3 + 2 (x +y -z-z ) 9 √ 3y ( ε 3 ε 2 ) 0 - √ 1 6 - √ 2 3 (x +y -z-z ) * 9y * ( ε 3 ε 2 ) * √ 1 3 -(x +y -z-z ) 9 √ 3y ( ε 3 ε 2 ) - √ 1 2 - √ 1 6 - √ 2 3 (x +y -z-z ) * 9y * ( ε 3 ε 2 ) * √ 1 3 -(x +y -z-z ) 9 √ 3y ( ε 3 ε 2 ) √ 1 2      (48)

Mixing angles at the NLO

From the previous discussions, it clearly appears that the NLO corrections to θ 13 and θ 23 come only from U whereas θ 12 is modified by both U and U ν . They are as follows:

s 13 = |U e3 | = 1 √ 2 (A -B) * ( ε 3 ε 2 ) * s 12 = |U e2 | √ 1 -|U e3 | 2 = 1 √ 3 - 1 √ 3 (A + B) * ( ε 3 ε 2 ) * - √ 2α ( ε 3 ε 2 ) ( 49
)
s 23 = |U µ3 | √ 1 -|U e3 | 2 = 1 √ 2 + C * ( ε 3 ε 2 ) *
where we used the short-hand notation

α = - √ 2 3 (x + y -z -z ) 9y .
In particular, the first relation can be used to put a bound on |ε |; in fact, given the maximum allowed value for θ 13 and assuming that |A -B| ∼ O( 1) we get:

|ε | [ √ 2 λ 4 C θ max 13 ] 1/3 (50)
which is close to (2 λ 2 C ) for θ max 13 ∼ 10 o .

A bit of phenomenology

The model we have presented has a huge parameter space, made by the six complex Yukawa couplings y ν i and a, b and c. It is clear, and we have checked this numerically, that the various phenomenological constraints, coming for example from the smallness of the parameter r = ∆m 2 sol /|∆m 2 atm |, are easily satisfied, for both type on neutrino mass hierarchies. It seems then more interesting to ask whether the model can still give an acceptable phenomenology in some particular cases, like for example allowing all the Yukawa couplings to be of O( 1) and the flavon vevs ϕ S , ∆ and ξ to be of the same order of magnitude 4 . In this case, we do not expect any huge hierarchies among the heavy neutrinos, see eq.( 15). Our numerical study aims to predict some interesting physical quantities only imposing the following 3-σ experimental constraints [13]: [16] (upper solid lines) and from [16] + [17] (lower solid lines). The vertical line in both panels is the future sensitivity of 0.2 eV on m lightest from the KATRIN experiment [14]. 10) for the normal ordering and O( 10 -2 ) for the inverted one. The ratio tends to a degenerate spectrum in both cases for |m lightest | ∼ 10 -1 eV which, however, is disfavoured in our model. In the right panel we show the sum of the light neutrino masses as a function of the lightest neutrino mass m lightest . The vertical line denotes the future sensitivity of 0.2 eV on of |m lightest | from the KATRIN experiment [14], and the horizontal lines are the cosmological bounds [15] at 0.60 eV, obtained combining the data from ref. [16], and at 0.19 eV, corresponding to all the previous data combined to the small scale primordial spectrum from Lyman-alpha (Lyα) forest clouds [17]. We see that our model predicts Σ m i too similar for both hierarchies to be distinguished using the current cosmological information on the sum of the neutrino masses; however, such a discrimination could be possible if some improvements on these bounds would be achieved in the near future. Finally, we present in Fig.

∆m 2 sol > 0 |∆m 2 atm | = 2.41 ± 0.34 × 10 -3 eV 2 (
as
( 2) the predictions for the values of the effective mass |m ee | as a function of the lightest neutrino mass, for both normal and inverted hierarchy. We also show the future sensitivity of the KATRIN (vertical solid line) and of CUORE [18] (horizontal solid line at 15 meV) experiments. The main feature of the analysis is that a large set of points falls into the region of |m lightest | around 10 -2 eV but, given the still large number of parameters of the model, many values of |m ee | can be obtained for both hierarchies among the experimental allowed ranges. On the other hand, the region above this value (the regime of degenerate spectra) is strongly disfavoured.

Leptogenesis

The formal description of the asymmetry parameters can be done in the general context of arbitrary Dirac mass matrix because the final expressions are quite compact and transparent. The asymmetry parameters are defined as follows:

i = 1 8π( Ŷ Ŷ † ) ii ∑ j =i Im { [ ( Ŷ Ŷ † ) ij ] 2 } f ( |M j | 2 |M i | 2 ) ( 52
)
where the hat matrices are Yukawa matrices evaluated in the basis in which the Majorana mass matrix is diagonal and M i are the Majorana masses. For supersymmetric theories, the f -function is given by:

f (x) = - √ x [ 2 x -1 + log ( 1 + x x )] . ( 53
)
Defining Ω as the unitary matrix which diagonalizes the Majorana mass matrix, the LO Yukawa matrix in this basis is given by

v u Ŷ = v u Ω T Y ν = Ω T m D and the product Ŷ Ŷ † reads: Ŷ Ŷ † = Ω T Y ν Y † ν Ω * . ( 54
)
At LO, Ω = U T BM and the product Ŷ Ŷ † is a diagonal matrix: the i parameters are all vanishing [12]. At the next to leading order, one has to take into account the corrections to the Yukawa matrix as well as to the Majorana mass matrix, which reflects in a different structure of both the Ω and Y ν matrices in such a way that:

v u Y ν = ( m D + v u ( ε ε ) ε 2 y ν 1 Ŝ) = v u (Y LO + δY ) (55) Ω = U T BM U φ + δΩ
where δΩ has a structure similar to the corrections δU computed in Sect. (3.3.1) and δY is of O(λ 4 C ) compared to its leading order result. This means that the correction to the matrix product Ŷ Ŷ † is given by:

δ( Ŷ Ŷ † ) = (δΩ) T Y LO Y † LO Ω * + Ω T Y LO Y † LO (δΩ) * + Ω T δ(Y ν Y † ν ) Ω * .
The first two terms do not contribute to the i parameters because they are complex conjugate of each other and do not contain any imaginary part; then the only contribution arises from the last term. In the basis in which the charged leptons are diagonal and considering ε as a real variable for simplicity, one easily obtains:

δ( Ŷ Ŷ † ) =   σ 1 ε 3 √ 2 e i(φ 1 -φ 2 ) σ 2 ε 3 0 √ 2 e -i(φ 1 -φ 2 ) σ 3 ε 3 0 0 0 0 σ 4 ε 3   ( 56
)
where the σ i coefficients are complicated functions of y ν i , A S and B S . Then, at leading order, the parameters are given by:

1 = ( ε ε ) 2 ε 4 (A S -B S ) 2 4π [y ν 1 (y ν 2 + 2y ν 3 ) + 3|y ν 1 | 2 ] 2 f ( |M 2 | 2 |M 1 | 2 ) sin [2(φ 1 -φ 2 )] 9|y ν 1 | 2 + |y ν 2 | 2 + |y ν 3 | 2 -2(y ν 2 y ν 3 ) + 3y ν 1 y ν 2 -3y ν 1 y ν 3 2 = ( ε ε ) 2 ε 4 (A S -B S ) 2 4π [y ν 1 (y ν 2 + 2y ν 3 ) + 3|y ν 1 | 2 ] 2 f ( |M 1 | 2 |M 2 | 2 ) sin [2(φ 1 -φ 2 )] 4|y ν 2 | 2 + |y ν 3 | 2 + 4(y ν 2 y ν 3 ) (57) 3 = 0.
A relevant feature of the model is that 3 always vanishes; however, we can see that, barring possible fine-tunings in the parameters and/or suppressions or enhancements due to f (|x| 2 ) sin [2∆φ], 1,2 can be of the right order of magnitude to fulfill the experimental requirements for a successful leptogenesis because

1,2 ∼ λ 8 C ∼ 6 × 10 -6 . ( 58
)

The quark sector

It is well known that the extension of a flavour symmetry from the neutrino sector to the quark one is highly non trivial due to the "non-trigonometric" structure of the quark mixing matrix V CKM . The common way to introduce quarks in these kind of models is to put the two SU (2) doublets of left-handed quarks into a doublet representation of S 4 and assign the heaviest ones (top and bottom quarks) into singlets, in such a way to easily maintain the hierarchy among mass eigenstates. Better results are obtained if other extra symmetries, i.e. Froggatt-Nielsen or extra Z N , are introduced in order to further suppress the unwanted couplings; this is particularly true for the m up /m top mass ratio, which otherwise tends to be larger than the experimental counterpart. It is clear that such scenarios are more flexible than the simple S 4 × Z 5 illustrated in this paper; however, it is important to stress that it is still possible to get a satisfactory description of the quark sector without invoking any other extra symmetries and using the triplet representation of S 4 instead of the doublet ones. In the down-quark sector the simplest choice is to copy the coupling allowed in the charged lepton sector because the hierarchy between the d, s and b quark masses is quite similar to that existing for e, µ and τ leptons. On the other hand, the mass hierarchy in the up-quark sector does not follow such a simple prescription and we can arrange the S 4 × Z 5 charge assignment in several ways. One possibility is summarized in Tab. (2), where Q is a triplet of SU (2) left-handed doublets. The corresponding leading order lagrangian is the following:

Field Q d c s c b c u c c c t c S 4 3 1 1 2 1 1 1 1 1 2 1 1 1 1 Z 5 1 ω 3 ω 2 ω ω 4 1 ω 4 U (1) R 1 1 1 1 1 1 1
Table 2: Transformation properties of quarks under S 4 × Z 5 and U (1) R .

L = y b Λ b c (Qϕ T ) h d + y s 1 Λ 2 s c (Q ϕ T ϕ T ) h d + y s 2 Λ 2 s c Q (η ϕ T ) h d + y d 1 Λ 3 d c Q [ϕ T (ϕ T ϕ T ) 2 ] 3 2 h d + y d 2 Λ 3 d c Q [ϕ T (ϕ T ϕ T ) 3 1 ] 3 2 h d + y d 3 Λ 3 d c Q [η (ϕ T ϕ T ) 3 1 ] 3 2 h d + y d 4 Λ 3 d c Q [ϕ T (ηη) 2 ] 3 2 h d + y d 5 Λ 2 d c Q (∆ϕ S ) 3 2 h d + y t Λ t c (Qϕ T ) h u + (59) y c 1 Λ 2 c c (Q ϕ T ϕ T ) h u + y c 2 Λ 2 c c Q (η ϕ T ) h u + y u 1 Λ 3 u c Q [∆(ϕ T ϕ T ) 3 1 ] 3 2 h u + y u 2 Λ 3 u c Q [ϕ S (ϕ T ϕ T ) 2 ] 3 2 h u + y u 3 Λ 3 u c Q [ϕ S (ϕ T ϕ T ) 3 1 ] 3 2 h u + y u 4 Λ 3 u c Q [ϕ S (ηη) 2 ] 3 2 h u + y u 5 Λ 3 u c Q [η (ϕ T ϕ S ) 3 1 ] 3 2 h u + y u 6 Λ 3 u c Q [η (ϕ T ϕ S ) 3 2 ] 3 2 h u .
This lagrangian gives rise to a diagonal mass matrix for the down-type quarks and an almost diagonal mass matrix for the up-type quarks (all the up entries are filled with terms of the same order ε 2 ε ). This picture slightly changes when including the next to leading order effects. As for the case of charged leptons, in the down sector the relevant NLO corrections come from the corrections to the vacuum alignment of the flavon fields, which are all of relative O(1/Λ) with respect to their leading order counterparts, whereas the higher order operators give corrections of the same size only for the s-quark, via couplings like (ϕ 3 S ), (ϕ S ∆ 2 ), (ϕ S ξ 2 ), (ϕ S ∆ξ) , (

and are much more suppressed for the down and bottom quarks. Similarly, for the uptype quarks the mass matrix is modified by the NLO structure of vacuum alignment of the flavon fields and, for the c-quark, also by the same higher order operators modifying the s-quark entries in eq.( 60). All in all, the mass matrices for both type of quarks are given by:

m down = v d   a d 1 ε 3 a d 2 ε ε 2 -a d 2 ε ε 2 b d 1 ε 3 b d 2 ε 2 b d 3 ε 3 c d 1 ε 3 ε c d 2 ε 3 ε c d 3 ε   m up = v u   a u 1 ε 2 ε a u 2 ε 2 ε a u 3 ε 2 ε b u 1 ε 3 b u 2 ε 2 b u 3 ε 3 c u 1 ε 3 ε c u 2 ε 3 ε c u 3 ε   (61)
with the following mass eigenvalues:

m d = v d [ a d 1 ε 3 + ( c d 1 c d 3 - b d 1 b d 3 ) a d 2 ε ε 3 ] m s = v d b d 2 ε 2 + O(ε 4 ) (62) m b = v d c d 3 ε + O(ε 4 )
and

m u = v u [ a u 1 ε 2 ε + ( b u 1 a u 2 b u 2 - a u 3 c u 1 c u 3 ) ε 4 ] m c = v u b u 2 ε 2 + O(ε 4 ) (63) m t = v u c u 3 ε + O(ε 4 ) .
The previous mass matrices can be diagonalized by unitary matrices is such a way that:

(m † u m u ) diag = U † u L (m † u m u ) U u L (m † d m d ) diag = U † d L (m † d m d ) U d L ,
where:

U d L =      1 ( b d 1 b d 2 ε 3 ε 2 ) * ( c d 1 c d 3 ε 3 ε 2 ) * - b d 1 b d 2 ε 3 ε 2 1 ( c d 2 c d 3 ε 3 ε 2 ) * - c d 1 c d 3 ε 3 ε 2 - c d 2 c d 3 ε 3 ε 2 1      , U u L =      1 ( b u 1 b u 2 ε 3 ε 2 ) * ( c u 1 c u 3 ε 3 ε 2 ) * - b u 1 b u 2 ε 3 ε 2 1 ( c u 2 c u 3 ε 3 ε 2 ) * - c u 1 c u 3 ε 3 ε 2 - c u 2 c u 3 ε 3 ε 2 1     (64)
The resulting V CKM is given by:

V CKM = U † u L U d L =      1 [( b d 1 b d 2 - b u 1 b u 2 ) ε 3 ε 2 ] * [( c d 1 c d 3 - c u 1 c u 3 ) ε 3 ε 2 ] * - ( b d 1 b d 2 - b u 1 b u 2 ) ε 3 ε 2 1 [( c d 2 c d 3 - c u 2 c u 3 ) ε 3 ε 2 ] * - ( c d 1 c d 3 - c u 1 c u 3 ) ε 3 ε 2 - ( c d 2 c d 3 - c u 2 c u 3 ) ε 3 ε 2 1      . ( 65
)
As for other flavour models, the matching of the V CKM and the quark mass ratios to their experimental values requires some fine-tunings between the Yukawas. As anticipated, all the experimental mass ratios in the down sector are easily reproduced for the natural values

a d 1 , b d 2 , c d 3 ∼ O(1) because m d /m s ∼ m s /m b ∼ ε ∼ λ 2 C .
A moderate hierarchy is also present in the up sector and it mainly depends on the parameter ε , which we estimated in Sect. (3.3.2) to be not larger than some units of λ 2 C . This same parameter appears in the off-diagonal entries of V CKM with the same power in each entries. This is essentially the reason why it is difficult to explain at the same time the mass hierarchy in the up-type quark sector and the off-diagonal values of the quark mixing matrix. For example, one can treat ε and tan β as free parameters of the model. In that case, one can use tha ratio m u /m c to fix the value of ε and m u /m d for that of tan β; assuming for all Yukawas the natural O( 1) value, we get:

m u m c = ε m u m d = ( ε ε ) tan β (66) and then ε = ( m u m c ) exp tan β = λ 2 C ( m u m d ) exp / ( m u m c ) exp ∼ 12 . ( 67
)
It is easy to verify that the previous results are enough to accomodate all the independent mass ratios that can be built from six different quarks. However, ε turns out to be very small and, having assumed all Yukawas of O( 1), it is really difficult to enhance the offdiagonal elements of V CKM to their experimental values. The other possibility is to preserve

ε ∼ O(λ 2 C ), requiring that [( b d 1 b d 2 - b u 1 b u 2 )] ∼ O(1/λ C ) [( c d 1 c d 3 - c u 1 c u 3 )] ∼ O(λ C ) [( c d 2 c d 3 - c u 2 c u 3 ) ] ∼ O(1) . ( 68
)
This fine-tuning is a condition on the individual Yukawas, which can be chosen to be:

c d 2 ∼ b u 1 ∼ O(1) c d 1 ∼ O(λ C ) (69) b d 1 ∼ O(1/λ C ) ,
providing that b u 2 , c u 3 1 to fit the charm and top quark masses. Although this situation is not completely satisfactory, it illustrates a way to account for many experimental informations in the quark sector using a simple and unconstrained S 4 × Z 5 model, the prize to be paid being a fine-tuning in four of the Yukawa couplings appearing in the lagrangian.

Notice that it is not more difficult to modify the charge assignment proposed in Tab.( 2) in such a way to obtain at LO a V CKM matrix with entries of O(1) in the Cabibbo 2 × 2 submatrix. This can be accomplished slightly changing only the Z 5 charge of the quark c. In fact, giving the assignment c ∼ (1 1 , ω 3 ) under (S 4 , Z 5 ), the c-quark part of lagrangian is now:

L c = y c 1 Λ 2 c c (Q ϕ T ∆) h u + y c 2 Λ 2 c c Q (ϕ T ϕ S ) h u + y c 3 Λ 2 c c Q ϕ T ξ h u + y c 4 Λ 2 c c Q (ϕ S η) h u (70)
and the c-quark entries in the mass matrix are modified according to

(c c , u) : b u 1 ε ε (c c , c) : b u 2 ε ε (c c , t) : b u 3 ε ε . (71)
Correspondingly, the u and c-quark masses are:

m u = v u ( a u 1 - b u 1 a u 2 b u 2 ) ε 2 ε m c = v u b u 2 ε ε (72)

Conclusions

We have presented and discussed an S 4 model for TB mixing (of the see-saw type) and quark mixing which, in spite of being based on a most economical flavour symmetry and field content, it is still phenomenologically viable. In the neutrino sector, we realized the most general neutrino mass matrix diagonalized at LO by TB mixing. At the NLO, all mixing angles receive corrections at higher orders at the level of O(λ 2 C ) and, in particular, the elusive θ 13 is predicted to be within the sensitivity of the experiments which are now in preparation and will take data in the near future. Also, we have shown that the leptogenesis parameters 1,2 can be of the right order of magnitude to satisfy the requirements needed to reproduce the observed asymmetry and that an acceptable phenomenology is still obtainable even with all the Yukawa couplings at O (1).

In the charged lepton sector, the mass hierarchy is determined by the S 4 × Z 5 flavour symmetry itself without invoking a Froggatt-Nielsen U (1) symmetry. The NLO corrections to the mass matrix turn out to be relevant for a θ 13 = 0.

In the quark sector, we have discussed in detail to which extent the model can reproduce the data on masses and mixing, emphasizing where it is successful and the reasons for its failures. Two different LO examples for V CKM have been given, one proportional to the identity matrix and the other with O(1) elements in the 2 × 2 sector; both realizations need almost the same amount of fine-tuning to reproduce the off-diagonal entries at the NLO.

Acknowledgements

I am grateful to Guido Altarelli for very stimulating discussions. I also wish to thank Luca Merlo for some interesting comments. This work was supported by the Deutsche Forschungs-gemeinschaft, contract WI 2639/2-1.

A The Group S 4

We adopt the following convention for the generators S and T , according to [9] S 4 = T 3 = (ST 2 ) 2 = 1 1 (74)

In the different representations, they can be written as reported in Tab.( 3):

rep 1 1 1 2 2 3 1 3 2 S 1 -1 ( 0 1 1 0 ) 1 3   -1 2ω 2ω 2 2ω 2ω 2 -1 2ω 2 -1 2ω   1 3   1 -2ω -2ω 2 -2ω -2ω 2 1 -2ω 2 1 -2ω   T 1 1 ( ω 0 0 ω 2 )   1 0 0 0 ω 2 0 0 0 ω     1 0 0 0 ω 2 0 0 0 ω  
S 2 = 1 3   -1 2 2 2 -1 2 2 2 -1   T S 2 T 2 = 1 3   -1 2ω 2ω 2 2ω 2 -1 2ω 2ω 2ω 2 -1   S 2 T S 2 T 2 = 1 3   -1 2ω 2 2ω 2ω -1 2ω 2 2ω 2 2ω -1   • C 3 : T =   1 0 0 0 ω 2 0 0 0 ω   T 2 =   1 0 0 0 ω 0 0 0 ω 2   S 2 T = 1 3   -1 2ω 2 2ω 2 -ω 2 2ω 2 2ω 2 -ω   S 2 T 2 = 1 3   -1 2ω 2ω 2 2 -ω 2ω 2 2 2ω -ω 2   ST ST 2 = 1 3   -1 2 2 2ω 2 -ω 2 2ω 2 2ω 2ω -ω   ST S = 1 3   -1 2ω 2 2ω 2ω 2 -ω 2 2ω 2 -ω 2   T S 2 = 1 3   -1 2ω 2ω 2 2ω -ω 2 2 2ω 2 2 -ω   T 2 S 2 = 1 3   -1 2 2 2ω -ω 2ω 2ω 2 2ω 2 -ω 2   • C 4 : ST 2 = 1 3   -1 2ω 2 2ω 2ω 2 -ω 2 2ω 2 -ω 2   T 2 S = 1 3   -1 2ω 2ω 2 2ω 2 2 -ω 2ω -ω
  -1 2ω 2ω 2 2ω 2ω 2 -1 2ω 2 -1 2ω   T ST 2 = 1 3   -1 2ω 2 2ω 2 2ω 2 -ω 2 -ω 2 2ω   ST = 1 3   -1 2 2 2ω 2ω -ω 2ω 2 -ω 2 2ω 2   T S = 1 3   -1 2ω 2ω 2 2 2ω -ω 2 2 -ω 2ω 2   S 3 = 1 3   -1 2ω 2 2ω 2ω 2 2ω -1 2ω -1 2ω 2   S 3 T 3 = 1 3   -1 2 2 2ω 2 2ω 2 -ω 2 2ω -ω 2ω  
For the 3-dimensional representation 3 2 , the matrices representing the elements of the group can be obtained from the list for the representation 3 1 in the following way: for C 1,2,3 are the same, while for C 4,5 are the opposite. In the previous basis, the Clebsch-Gordan coefficients are as follows (α i indicates the elements of the first representation of the product and β i the second one):

1 1 ⊗ η = η ⊗ 1 1 = η with η any representation 1 2 ⊗ 1 2 = 1 1 ∼ αβ 1 2 ⊗ 2 = 2 ∼ ( αβ 1 -αβ 2
)

1 2 ⊗ 3 1 = 3 2 ∼   αβ 1 αβ 2 αβ 3   1 2 ⊗ 3 2 = 3 1 ∼   αβ 1 αβ 2 αβ 3   .
The multiplication rules with the 2-dimensional representation are the following:

2 ⊗ 2 = 1 1 ⊕ 1 2 ⊕ 2 with              1 1 ∼ α 1 β 2 + α 2 β 1 1 2 ∼ α 1 β 2 -α 2 β 1 2 ∼ ( α 2 β 2 α 1 β 1 ) 2 ⊗ 3 1 = 3 1 ⊕ 3 2 with                    3 1 ∼   α 1 β 2 + α 2 β 3 α 1 β 3 + α 2 β 1 α 1 β 1 + α 2 β 2   3 2 ∼   α 1 β 2 -α 2 β 3 α 1 β 3 -α 2 β 1 α 1 β 1 -α 2 β 2   2 ⊗ 3 2 = 3 1 ⊕ 3 2 with                    3 1 ∼   α 1 β 2 -α 2 β 3 α 1 β 3 -α 2 β 1 α 1 β 1 -α 2 β 2   3 2 ∼   α 1 β 2 + α 2 β 3 α 1 β 3 + α 2 β 1 α 1 β 1 + α 2 β 2  
The multiplication rules with the 3-dimensional representations are the following:

3 1 ⊗ 3 1 = 3 2 ⊗ 3 2 = 1 1 ⊕ 2 ⊕ 3 1 ⊕ 3 2 with                                      1 1 ∼ α 1 β 1 + α 2 β 3 + α 3 β 2 2 ∼ ( α 2 β 2 + α 1 β 3 + α 3 β 1 α 3 β 3 + α 1 β 2 + α 2 β 1
)

3 1 ∼   2α 1 β 1 -α 2 β 3 -α 3 β 2 2α 3 β 3 -α 1 β 2 -α 2 β 1 2α 2 β 2 -α 1 β 3 -α 3 β 1   3 2 ∼   α 2 β 3 -α 3 β 2 α 1 β 2 -α 2 β 1 α 3 β 1 -α 1 β 3   3 1 ⊗ 3 2 = 1 2 ⊕ 2 ⊕ 3 1 ⊕ 3 2 with                                      1 2 ∼ α 1 β 1 + α 2 β 3 + α 3 β 2 2 ∼ ( α 2 β 2 + α 1 β 3 + α 3 β 1 -α 3 β 3 -α 1 β 2 -α 2 β 1
)

3 1 ∼   α 2 β 3 -α 3 β 2 α 1 β 2 -α 2 β 1 α 3 β 1 -α 1 β 3   3 2 ∼   2α 1 β 1 -α 2 β 3 -α 3 β 2 2α 3 β 3 -α 1 β 2 -α 2 β 1 2α 2 β 2 -α 1 β 3 -α 3 β 1  

  39) where a = b 1 /b 2 , b = c 1 /c 3 and c = c 2 /c 3 . Then, any entries of U TBM get corrected by terms of O(ε 3 /ε 2 ) which can be as large as O(λ 2 C

Figure 1 :

 1 Figure 1: Neutrino mass spectrum and sum of neutrino masses, as predicted from the model with all Yukawa couplings of O(1) and the flavon vevs of the same order of magnitude. Left panel: behaviour of the ratio |m 3 |/|m 2 | as a function of the lightest neutrino mass |m lightest |, for both neutrino hierarchies. Right panel: sum of the light neutrino masses as a function of |m lightest |. Also shown the bound from[16] (upper solid lines) and from[16] +[17] (lower solid lines). The vertical line in both panels is the future sensitivity of 0.2 eV on m lightest from the KATRIN experiment[14].

Figure 2 :

 2 Figure 2: |m ee | as a function of the lightest neutrino mass, for both normal and invertedThe filled regions correspond to the possible values of |m ee | in the limit of exact tri-bimaximal mixing, with mass differences computed at the central values in eq.(51). The horizontal and vertical lines are the future expected bounds on |m ee | and |m lightest | from theCUORE and KATRIN experiments, respectively.

C 4 :

 4 ST 2 , T 2 S, T ST T ST S 2 , ST S 2 , S 2 T S C 5 : S, T ST 2 , ST, T S, S 3 , S 3 T 2 .The explicit expression of the elements in the 2-dimensional representation is:

  a function of the lightest neutrino mass |m lightest |, which is |m 1 |(|m 3 |) for the normal (inverted) hierarchy. We clearly see that the largest hierarchies, obtained at the smallest allowed |m lightest |, are at the level of O(

Table 3 :

 3 Generators S and T in different representations.The 24 elements of the group belong to five conjugacy classesC 1 : 1 C 2 : S 2 , T S 2 T 2 , S 2 T S 2 T 2 C 3 : T, T 2 , S 2 T, S 2 T 2 , ST ST 2 ST S, T S 2 , T 2 S 2

Similar considerations, but using two discrete groups, have been discussed in[10].

This works in the same way as described in[5] noticing that, like in the case of A 4 , (0, 1, 0) 2 = (0, 0, 1) and (0, 1, 0)

= (1, 0, 0).3 This is because this term explicitly breaks the residual symmetries needed to generate the correct hierarchies between the charged lepton masses.

In the numerical simulation, this means that we allow the absolute values of the Yukawas to be in the interval [1/2, 3/2], while no restriction whatsoever has been imposed on the flavon vevs a part from being all equal within a factor of 10; also the large scale Λ is left free.

and the V CKM has the following structure:

where, for simplicity, we introduced the short-hand notation K = √ (b u 1 ) 2 + (b u 2 ) 2 and considered real vevs and Yukawa couplings. In this case, one can reproduce at the same time the correct values for the (11) and ( 22) entries of V CKM as well as the mass ratio m u /m c imposing b u 2 ∼ O(λ 2 C ); playing with ε and the other Yukawas one can also reproduce the off-diagonal entries, but one or more fine-tunings in the coefficients in front of ε 3 /ε 2 ratio are obviously needed.


L M Robledo
email: luis.robledo@uam.es

Remarks on the use of projected densities in the density dependent part of Skyrme or Gogny functionals

Keywords: numbers: 21, 60, Jz, 24, 10, Cn

I discuss the inadequacy of the "projected density" prescription to be used in density dependent forces/functionals when calculations beyond mean field are pursued. The case of calculations aimed at the symmetry restoration of mean fields obtained with effective realistic forces of the Skyrme or Gogny type is considered in detail. It is shown that at least for the restoration of spatial symmetries like rotations, translations or parity the above prescription yields catastrophic results for the energy that drive the intrinsic wave function to configurations with infinite deformation, preventing thereby its use both in projection after and before variation.

Introduction

Given the enormous success of phenomenological density dependent forces/functionals like the different variants of Skyrme, Gogny or relativistic mean fields in describing nuclear properties with mean field wave function [1], it was expected that going beyond mean field both restoring broken symmetries or considering configuration mixing would pave the way for a more satisfactory understanding of nuclear structure both in the qualitative and quantitative sides. Early attempts with the Skyrme functional [2] applying the Generator Coordinate Method (GCM) produced very encouraging results. Soon after, fully microscopic parity projection was implemented with the Gogny force [3] for octupole deformed states in a consistent framework and the comparison with other results as well as with experiment was encouraging. After that, many GCM and symmetry restoration calculations both with Skyrme and Gogny were reported. Here, I will only mention a few like the restoration of particle number symmetry that was tackled in [4] with the Gogny force. In this reference it was realized that a consistent inclusion of the direct, exchange and pairing terms of the interaction was needed for a divergent free evaluation of the energy. This was a real problem for functionals of the Skyrme type were the exchange and pairing fields of different terms of the functional were usually neglected (probably the most disappointing situation was the one of the Coulomb potential, as the consideration of its exchange and pairing fields increased by orders of magnitude the computational cost of the Skyrme type calculations). Recently, a regularization procedure to solve this problem and also to address the inherent self-energy and selfpairing problem of any energy density functional was proposed [5] and applied to the case of particle number projection (PNP) in [6]. Although the regularization procedure of [5] is basis-dependent and the impact of this arbitrariness has still to be checked, the results obtained so far [6] are very promising.

Another problem that plagues beyond mean field calculations has to do with the prescription to be used for the density dependent part of the "Hamiltonian"/functional (quotes will be used through the paper to stress the dependency of the "Hamiltonian" operator with the intrinsic wave function through the density dependent part of the interaction). The first prescription used [2] was inspired by the three body force origin of some Skyrme functionals and is referred to as the "mixed density" prescription (see below). Although attempts to use other prescriptions are not uncommon [7,8,9,10] it was assumed that the consistency requirements summarized in [11] favored the use of the "mixed density" prescription. However, as pointed out in [12,13] the "mixed density" prescription has a serious and in general insurmountable drawback when the density dependent part of the interaction depended upon non-integer powers of the density. As the mixed density is in general a complex quantity, non integer powers of it are not uniquely defined in the complex plane unless a "branch cut" is introduced and the Riemann sheet where the root lives is specified. In many cases, the mixed density is a real quantity and such a problem can be circumvented, but in general (including the most complex symmetry restoration of triaxial deformed states) it is complex and the lack of a physically guided rule to choose the roots yields to projected energies that depend on external artifacts like the integration contours entering the definition of the projector operators [12,13]. The only two known ways out to this problem in the general case are either to consider forces/functionals depending upon integer powers of the density (see below for a recently proposed functional) or to consider other prescriptions producing real densities. In the latter case, the most promising candidate is the "projected density" prescription [8,9,10] that, in addition, has some other appealing properties to be discussed below. In this paper, I am going to show by means of an example concerning parity projection that the "projected density" prescription yields to unphysical energies that favor infinitely deformed intrinsic states with diverging energies. This catastrophic behavior can also be extended to other spatial symmetry restorations like angular momentum projection or translational invariance.

I would like to emphasize that the present discussion concerns the inadequacy of using on the same footing symmetrized densities and intrinsic ones to obtain the energy associated to the strongly repulsive density dependent interaction, which is characteristic of the nuclear Energy Density Functional (EDF). The present discussion has little to do with the solution [14] in terms of symmetrized density matrices of the symmetry dilemma of the Kohn-Sham Density Functional Theory (DFT). The symmetry dilemma refers to the use of KS-DFT density functionals which break in some circumstances the symmetries of the interaction and lead to Kohn-Sham wave functions that, in contrast to the full interacting wave function, cannot be assigned to an irreducible representation of the symmetry group of the interaction.

The plan of the paper is as follows: in the next section, the "mixed density" prescription will be briefly discussed as a way to justify the introduction of the projected density prescription to be discussed in Section 3. In Section 4 I will try to convey what the open questions are regarding this issue as well as some thoughts about possible ways out. Conclusions will be given at the end of the paper.

Mixed density prescription

In the evaluation of the matrix element of a density dependent "Hamiltonian" between different states |ϕ and |ϕ a prescription for the density dependent part is needed. A common choice, proposed in [2] is the "mixed density" prescription that uses the overlap

ρ Mixed ( r) = ϕ|ρ( r)|ϕ / ϕ|ϕ (1)
as the density of the density dependent term of the "Hamiltonian". In this definition ρ( r) = A i=1 δ( r -r i ) is the density operator. This prescription is based on the expressions obtained with three body forces in spin saturated systems. The use of this prescription makes the density dependent "Hamiltonian" to break the symmetries broken by the density and also makes it a complex (non-hermitian) quantity. One might think, after considering these two properties, that the mixed density prescription should be sentenced to oblivion. However, as was discussed in detail in Ref. [15], the energy, which is the physical quantity to be computed out of the above mentioned matrix elements, is a well defined quantity, i.e. it is invariant under the broken symmetries and it is a real quantity. To understand these facts let us consider briefly the case of rotational invariance. Let us now consider an intrinsic wave function |ϕ that represents a deformed state and therefore is not characterized by the angular momentum quantum numbers. For standard (density independent) Hamiltonians the energy is independent of the orientation

ϕ| R+ (Ω) Ĥ R(Ω)|ϕ = ϕ| Ĥ|ϕ (2)
as a consequence of the scalar nature of the Hamiltonian, namely R(Ω) Ĥ = Ĥ R(Ω) for all Euler angles Ω. For density dependent "Hamiltonians" Ĥ[ρ |ϕ ] it was shown in Ref [15] what the scalar property for overlaps below holds

ϕ| R(Ω) Ĥ[ R ρ]|ϕ = ϕ| Ĥ[ρ R ] R(Ω)|ϕ (3)
where the shorthand notation

ρ R = ϕ|ρ R(Ω)|ϕ / ϕ| R(Ω)|ϕ and R ρ = ϕ| R(Ω)ρ|ϕ / ϕ| R(Ω)|ϕ
is used. This property allows to extend Eq. ( 2) to the case of density dependent "Hamiltonians" showing that the ensuing energy is orientation independent.

Concerning the reality of the energy, one has to realize that whenever a term like ϕ| Ĥ[ ϕ ρ ϕ ]|ϕ appears in the expression of the energy, the symmetric term ϕ | Ĥ[ ϕ ρ ϕ ]|ϕ will also be present and weighted with the complex conjugate weight. It is not difficult to show [15] that the last term is the complex conjugate of the first and therefore the contribution of the two add up to a real quantity. This analysis is kind of a näive one as the density dependence in the Hamiltonian usually comes through a non integer power of the (complex) density what forces the introduction of "branch cuts" in the complex plane and to specify in which Riemann sheet the roots are defined. As a consequence of this difficulty it was shown in Ref, [13] (see also [12] for an early consideration of this issue) that the energy was not independent of the specific way it was computed. To be more specific, in the case of PNP with density dependent "Hamiltonians" consisting of a density dependent term with a density raised to a non integer power, the projected energy was dependent upon the integration path chosen in the complex plane to perform the integrations needed in the particle number procedure. This is a real challenge to existing theoretical models as most of the Gogny, Skyrme or relativistic models contain non-integer powers of the density in their definition. Recently, an exception has shown up in the market in the form of the so called Barcelona-Catania-Paris (BCP) functionals [16,17] which are only defined in terms of integer powers of the density and are therefore promising candidates to incorporate the self-energy renormalization techniques of [5] to produce well defined quantities in calculations beyond the mean field.

A possibility that also comes up to solve the complex mixed density dilemma is to use the so called "projected density" prescriptions discussed next.

Projected density prescription

The projected density prescription was postulated for Particle Number Projection (PNP) [8,9] and Angular Momentum Projection (AMP) [10] to solve, in one hand , the problems associated with the complex nature of the mixed density prescription and, in the other, to give back to the density dependent part of the "Hamiltonian" the nice property of preserving fundamental symmetries like rotational invariance. In this way, the density dependent part of the "Hamiltonian" could be looked at as an effective part of a real Hamiltonian and not as a device to generate a non trivial density dependence in the energy functional [10].

In the case of PNP the particle number projected density

ρ N ( r) = φ| P + N ρ( r) PN |φ φ| P + N PN |φ (4)
was considered in [8,9] and it was even proved [9] that this prescription as well as the mixed density prescription produced the same expressions for the Lipkin Nogami parameter in the strong deformation limit. This has been the prescription used in all particle number projected calculations with the Gogny force. In the rotational symmetry case the rotational invariant density corresponding to the J = 0 projected state ρ J=0 (r) was postulated [10] to be used in the density dependent part of the Hamiltonian.

The "projected density" prescription looked reasonable and the only argument found against it in the PNP case [11] was in the context of the GCM where, under some assumptions, it is possible to derive the Random Phase Approximation (RPA) as a small amplitude approximation of the GCM [18]. This idea can be extended to the case of density dependent interactions [11,[START_REF] Robledo | The GCM-RPA connection for density dependent forces[END_REF] with the "mixed density" prescription to produce the RPA with the same matrix elements that appear in the standard derivation of the RPA for density dependent forces [START_REF] Blaizot | [END_REF]. However, this is not the case for the "projected density" prescription or other prescriptions postulated in the literature [7]. It has also to be mentioned that the "projected density" prescription produces in the context of the PNP and in the strong deformation limit an approximate expression for the projected energy (Kamlah expansion, see [START_REF] Ring | The nuclear many body problem[END_REF] for a detailed discussion) including a parameter, to be identified later with the Lagrange multiplier of the particle number constraint, that includes the rearrangement term characteristic of mean field theories with density dependent forces [8,9,11]. This property do not hold, however, for spatial symmetries like rotations as the leading term of the Kamlah expansion for the projected density is not the intrinsic density as in the case of PNP but some kind of symmetry preserving average (see below for the parity symmetry). Therefore, in the general case the density dependent interaction of the underlying mean field is different from the density dependent interaction to be used in projection (i.e. both densities are "decoupled" in a sense to make clear below) suggesting the possibility to use the "projected density" prescription also at the mean field level. That would make the Hartree-Fock-Bogoliubov (HFB) equations more involved due to the new rearrangement The blue dashed increasing curve corresponds to the intrinsic HFB energy. The red full line is for the projected energy to positive parity using the "mixed density" prescription. The dotted green down-sloping curve is for a calculation of the HFB energy where the intrinsic density of the density dependent interaction is replaced by the projected density to positive parity. Finally, the purple full line down-sloping curve is for the projected energy corresponding to positive parity and with the parity projected density in the density dependent part of the interaction. The result was obtained with the Gogny D1S force. On the right hand side, I plot the same quantities as before but for the heavy actinide and octupole deformed nucleus 224 Ra.

term but no fundamental problems were foreseen. As I will show below this possibility is also doomed to fail.

In order to show the catastrophic consequences of having in the density dependent part of the "Hamiltonian" a quantity that is decoupled from the intrinsic density I will consider the case of parity projection of octupole deformed intrinsic states. The Gogny [START_REF] Dechargé | [END_REF] force D1S [23] will be used for the calculations and I will proceed along the same lines as our early study of parity projection with the Gogny force of Refs. [3,24]. The parity projection operator to parity π = ±1 is given by

Pπ = 1 √ 2 (I + π Π) ( 5
)
where Π is the parity operator. The projected energy is given by

E π = φ| Ĥ Pπ |φ φ| Pπ |φ = φ| Ĥ|φ + π φ| Ĥ Π|φ φ|φ + π φ| Π|φ (6)
The definite expression for the projected energy depends upon the prescription used for the density dependent part of the "Hamiltonian". For the "mixed density" prescription the standard intrinsic density ρ( r) has to be used in the density dependent part of the Hamiltonian for the evaluation of its mean value (first term in the numerator of Eq. ( 6)

) and the density θ( r) = φ|ρ( r) Π|φ φ| Π|φ in the evaluation of φ| Ĥ Π|φ / φ| Π|φ . In the case of the "projected density" prescription the density

ρ π=+1 PROJ ( r) = φ| P + π ρ( r) Pπ |φ φ| P + π Pπ |φ = ρ( r) + π φ| Π|φ θ( r)
1 + π φ| Π|φ (7) will be used in the density dependent part of the "Hamiltonian". In this expression the averaged matrices ρ( r) = 1 2 (ρ( r) + ρ(-r)) and θ( r) = 1 2 (θ( r) + θ(-r)) have been introduced. In the strong deformation limit (corresponding to φ| Π|φ → 0) the projected density goes to the average density ρ( r) instead of the intrinsic one. This is the fact that is at the heart of the problems encountered below for the projected energy.

The positive parity projected energy E + has been computed using the above formulas and prescriptions for given intrinsic configurations with different octupole moments and in different nuclei. A light nucleus 36 Ar has been chosen as an example of a system with a density which is mainly surface and also as a system where the intrinsic ground state is reflection symmetric (i.e. positive parity in this case). On the other hand, the heavy nucleus 224 Ra has been chosen as an example of an octupole deformed nucleus and with a density which is mostly volume. The results obtained are shown in Fig. 1 as a function of the octupole moment Q 3 . Several energies are computed: the mean field energy computed in the standard way E, the mean field energy computed with the parity projected density in the density dependent "Hamiltonian" E PD (the modified HFB framework discussed above), the positive parity projected energy computed with the "mixed density" prescription E + MD and finally the positive parity projected energy computed with the "projected density" prescription E + PD . First, it is observed the typical parabolic behavior of E as a function of Q 3 for a non octupole deformed nucleus like 36 Ar as well as the presence of a minimum for the case of the octupole deformed 224 Ra case. The positive parity projected energy E + MD obtained with the "mixed density" prescription has a behavior close to the mean field energy being lower than it in all the cases (see [3]) and quickly converging to the mean field energy for increasing values of the octupole moment. This is a consequence of the behavior of the mean value of the parity that quickly decreases to zero as the octupole moment increases reaching the strong deformation limit. On the other hand, the two corresponding energies computed with the "projected density" prescription show a very pronounced decrease of tens of MeV that keep going as the octupole moment is increased. In these two cases, the configuration minimizing the corresponding energies would be an infinitely deformed octupole configuration with energies going to minus infinity. Obviously this is a catastrophic situation that will lead the system to unphysical energies and intrinsic configurations. Therefore the "projected density" prescription, at least for spatial symmetries (see below), can be categorized as unphysical and its use has to be avoided. Concerning the particle number projected density prescription widely used with the Gogny force, the reasonable results obtained so far clearly demonstrate that no catastrophe is taking place in that case. However, and from a fundamental (as opposed to phenomenological) point of view, the use of a prescription which yields to catastrophic results when used in other contexts should also be avoided.

In order to understand the catastrophic results obtained it is convenient to analyze the structure of the density dependent energy which is given in the case of the Gogny force by the following integral

ϕ| ĤDD |ϕ = t 3 2 τ,τ (1 + x 0 2 -(x 0 + 1 2 )δ τ,τ ) d 3 rρ τ ρ τ ρ α (8)
where ρ τ are the intrinsic densities with isospin τ . The replacement of the standard ρ α ( r) by ρ α Proj ( r) produces a "mismatch" in the integrand because the new density ρ Proj does not have the same shape as ρ as can be seen in Fig. 2 where contour plots of the two densities are plotted for the nucleus 224 Ra at an octupole moment Q 3 = 4b 3/2 (which corresponds to the ground state of this octupole deformed nucleus). As ρ Proj is reflection invariant it must have "humps" at each side of the nucleus whereas the intrinsic density only has a "hump" at one of the sides (which one depends on the sign of the octupole moment). The two "humps" of the projected density will have half the density of the one of the intrinsic density as the projected density is roughly given (strong deformation limit) by the average density ρ( r) = 1 2 (ρ( r) + ρ(-r)) (see Eq. ( 6)). As a consequence of the mismatch of the different factors, the integrand corresponding to the "projected density" prescription is slightly reduced as compared to the integrand with the intrinsic density in the density dependent term. As a consequence, the value of the integral is also reduced producing a reduction of the (repulsive) density dependent energy that is magnified by the large values of the density dependent strength parameter t 3 ( 1562.22 MeV fm 4 , 1609.46 MeV fm 4 and 1350.0 MeV fm 4 for Gogny D1M [26], D1N [25] and D1S [23], respectively).

The mismatch between the projected density and the intrinsic density will also be very pronounced in the case of quadrupole deformation and associated to rotations in space as well as in the case of spatial translations. It is therefore evident that those two cases will also be plagued with the catastrophic decrease of the projected energy for infinitely deformed configurations. Therefore in the case of angular momentum projection and restoration of translational invariance the "projected density" prescription is doomed to badly fail.

Open problems

From the results presented here it is clear that, at least for spatial symmetries, the only possibility to get rid of all the different problems encountered in the extension of the density dependent forces paradigm to calculations going beyond mean field is to solely consider forces/functionals depending upon integer powers of the density. Apart from some scarce versions of Skyrme, there is a recently proposed functional with this property, the Barcelona-Catania-Paris (BCP) functional [16,17] (see also an article in this issue by Baldo et al concerning this functional [27]) which is based on a fit to a realistic nuclear matter equation of state followed by a Local Density Approximation map to produce a functional for finite nuclei. Additional finite range surface terms, spin-orbit, Coulomb and density dependent pairing are ingredients of this functional. Given the good performance of BCP at the mean field level [16,17,27], this functional can become a serious contender for beyond mean field studies in the spirit of the Energy Density Functional (EDF).

Another alternative to get rid of the above problems would be to use approaches based on the Gaussian Overlap Approximation (GOA) [START_REF] Ring | The nuclear many body problem[END_REF] or its more complete accurate and satisfactory version (the Topological GOA, see for instance [28] for a recent treatment) to perform the symmetry restorations and/or configuration mixing calculations (in the GCM spirit). This possibility has been explicitly suggested in [29] in the framework of the Bohr Hamiltonian method but the philosophy behind was also implicit in early applications of the method [30]. The reason why this proposal will cure the problems discussed before is because the residual interaction is defined as second derivatives of the energy (where the density dependent term is well defined) and therefore are free from the above problems of divergences and complex valued quantities. Obviously, these ideas can be generalized to a framework where overlaps are computed exactly but the residual interaction is defined as a second derivative in a way that closely resembles the RPA for density dependent forces/functionals. Work along this direction is in progress [START_REF] Robledo | The GCM-RPA connection for density dependent forces[END_REF].

Conclusions

I have explicitly shown with an example using parity projection that the "projected density" prescription, advocated by some authors to overcome some difficulties encountered in the application of the "mixed density" prescription in applications beyond the mean field, leads to catastrophic consequences for the projected energy as this prescription favors a system where the reflection symmetry violation is maximal and the projected energy in unbounded by below. This consideration can also be extended to other symmetry restorations involving spatial symmetry transformations like rotations and translations. The result is the consequence of the mismatch between the shape of the intrinsic and the projected densities and the strongly repulsive character of the density dependent term.

Figure 1 .

 1 Figure 1. (Color online) On the left hand side various energies for the nucleus36 Ar at a fixed value of β 2 of 0.23 are plotted as a function of the octupole moment Q 3 . The blue dashed increasing curve corresponds to the intrinsic HFB energy. The red full line is for the projected energy to positive parity using the "mixed density" prescription. The dotted green down-sloping curve is for a calculation of the HFB energy where the intrinsic density of the density dependent interaction is replaced by the projected density to positive parity. Finally, the purple full line down-sloping curve is for the projected energy corresponding to positive parity and with the parity projected density in the density dependent part of the interaction. The result was obtained with the Gogny D1S force. On the right hand side, I plot the same quantities as before but for the heavy actinide and octupole deformed nucleus 224 Ra.

Figure 2 .

 2 Figure 2.On the left hand side, the contour plots of the intrinsic density of the nucleus224 Ra with an octupole moment of 4 b 3/2 (roughly corresponding to is ground state) are depicted. The contour lines start at ρ = 0.02fm -3 for the outer contour and end at ρ = 0.14fm -3 for the inner one. On the right hand side, contour plots of the positive parity projected density corresponding to the intrinsic wave function of the left hand side plot. The contours are the same as in the left hand side plot.

Acknowledgments

Work supported in part by the MICINN (Spain) contract FPA2007-66069 and the Consolider-Ingenio 2010 program CPAN (Spain) contract CSD2007-00042.