TeV Scale Leptogenesis, theta_13 And Doubly Charged Particles At LHC

We explore a realistic supersymmetric SU(2)_L \times SU(2)_R \times U(1)_B-L model spontaneously broken at around 10^12 GeV. The presence of D and F-flat directions gives rise to TeV mass doubly charged particles which can be found at the LHC. We implement TeV scale leptogenesis and employing both type I and II seesaw, the three light neutrinos are partially degenerate with masses in the 0.02-0.1 eV range. The effective mass parameter for neutrinoless double beta decay is 0.03-0.05 eV. We also find the interesting relation tan 2 \theta_13 ~ [\Delta m^2_{\odot} / \Delta m^2_{atm}] [sin 2 \theta_12 /tan 2 \theta_23]<~ 0.02.

It has been recognized for some time that spontaneously broken supersymmetric models with D and F -flat directions can lead to interesting phenomenological and cosmological consequences [1,2,3,4,5,6,7]. An intermediate symmetry breaking scale of order 10 8 −10 16 GeV is a characteristic feature of these models. Another important aspect is the appearance of thermal inflation typically involving about 10 or so e-foldings [2,4,6,7]. The entropy generation associated with thermal inflation has been exploited to try to resolve the gravitino [2] and moduli problem [8], and to suppress the primordial monopole number density to acceptable levels [5]. The entropy production does have an important drawback though. It will dilute , and in some cases completely wash away, any pre-existing baryon asymmetry.
This very depends on the magnitude of the intermediate scale M I [6].
In [9], with an intermediate scale of order 10 8 GeV, the observed baryon asymmetry was explained via resonant leptogenesis [10]. The relatively low intermediate scale causes a moderate amount of dilution of an initially large lepton asymmetry, such that the final baryon asymmetry is consistent with the observations. This paper is partly motivated by the desire to implement TeV scale leptogenesis in models with an intermediate scale that is higher, namely of order 10 12 GeV. (Scales significantly higher than this lead to a reheat temperature after thermal inflation that is too low for sphaleron transitions to be effective).
To be specific, we base our discussion on the supersymmetric version of the well known gauge group G 221 = SU(2) L × SU(2) R × U(1) B−L [11,12]. The presence of D and F -flat directions means that the 'flaton' fields φ, φ with vevs = M I ∼ 10 12 GeV, have an associated mass scale M s , the supersymmetry breaking scale, of the order of TeV. Being in the triplet representation of SU(2) R , these fields contain doubly charged particles which turn out to have masses of order M s . Hence, they should be found at the LHC. Thermal inflation is driven by φ, φ and after it is over, the flatons produce TeV mass right-handed neutrinos associated with the first two families, whose subsequent decay leads via leptogenesis to the observed baryon asymmetry. (Because it has mass of order M I , the third generation right-handed neutrino is not accessible at the TeV scale). Taking into account both type I [13] and type II [14] seesaw mechanism, the light neutrinos turn out to have partially degenerate [15]  Since G 221 is broken to the gauge group SU(2) L × U(1) Y with the vevs of φ (1, 3, −2) and φ (1,3,2), a discrete Z 2 symmetry remains unbroken [16] which is precisely 'matter' parity.
Consequently, the LSP is stable. To generate the scale M I via D and F -flat directions, we employ a discrete symmetry Z 4 × Z 8 which, among other things, prevents terms such as φφ from appearing in the superpotential.
The term proportional to κ can help resolve the MSSM µ problem (µ ∼ κ M I M *

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M I (see also [9]), and is expected to be of order few hundred GeV). With κ and γ 12 of comparable magnitudes, the dominant decay channel of φ is to first and second generation right-handed neutrinos, N 1 , N 2 , and this is useful in realizing TeV scale leptogenesis through N 1,2 decay. 1 With only one bidoublet higgs, there will be no CKM mixings. However, G 221 can be embedded in a bigger group such as SO(10), and it is then possible to induce non-zero CKM mixings through some additional 'matter fields' [17]. Another possibility is to include loop contributions in association with supersymmetry breaking terms [18]. To simplify our presentation we will not address these possibilities here. Generation of the lepton-mixing matrix will be discussed later in the paper.
In Table I we list the discrete charges of the various superfields. (For the cosmology of spontaneously broken discrete symmetries the reader is referred to [7].) The zero temperature effective scalar potential of φ (we use φ to also represent the scalar component of the superfield) along the D-flat direction with φ = φ † , is given by where µ 4 0 is introduced to ensure that at the minimum φ = M I , V (M I ) = 0, and −2M 2 s |φ| 2 is the soft supersymmetry breaking term with M s ∼ TeV. Here it is assumed that a positive supersymmetry breaking mass squared term generated at some superheavy scale can acquire a negative sign, via radiative corrections involving the superpotential coupling γ 33 L c 3 L c 3 φ, at a lower energy [19]. Minimization of the effective potential yields the intermediate scale, We will see shortly that M I ≃ 10 12 GeV and M * ≃ 5.5 × 10 13 GeV, for M s ≃ 5.5 TeV are compatible with TeV scale lepton asymmetry, with partial conversion of the latter via sphalerons into the observed baryon asymmetry.
For T > T c = 2/σM s the potential develops a minimum at φ = 0, with V (φ = 0) = µ 4 0 = 12 7 M 2 s M 2 I . For φ > T , the temperature-dependent term is exponentially suppressed and V (φ) develops another minimum at φ = M I . φ = 0 remains an absolute minimum for µ 0 T M I , but for T µ 0 , the true minimum at M I takes over. The dark energy density associated with the absolute minimum (10 −12 eV 4 ) is irrelevant for our purpose [21].
Due to the false vacuum energy density µ 4 0 the universe experiences roughly ln(µ 0 /T c ) ∼ 8 e-foldings of thermal inflation. The flaton has mass m φ of order 2 √ 6M s and it can decay into right-handed neutrinos (with mass M N ) via the superpotential coupling The decay width is given by where we have used Eq. (3) with m φ in terms of M s , and f φ = (1 − 4M 2 N /m 2 φ ) 3/2 . The other decay width, Γ φ→HH ∼ (κ 2 /8π)(M I /M * ) 10 m φ , is clearly suppressed compared to Γ φ . Therefore the final temperature (T f ) after thermal inflation can be expressed as To estimate T f we need to know γ 12 which can be estimated as follows.
From the superpotential in Eq. (1) the right-handed neutrino mass matrix M R is given by with real and positive eigenvalues |M 1 | = |M 2 | = |x| and |M 3 | = |M|. The phases of . Hence in this model we find the final temperature to be with M N /m φ ≃ 0.3. It is gratifying that T f is in a range where the electroweak sphalerons [22] are able to convert some fraction of the lepton asymmetry into baryon asymmetry [23], which sets an upper bound on M I of 10 12 GeV for M s ∼ few TeV. In Fig. (1 TeV corresponding to T f ≃ 90 GeV.
We now consider the case where N 1,2 are produced by the direct non-thermal decay of the flaton field φ. The ratio of the number density of right-handed neutrino, n N , to the entropy density s is given by where B r denotes the the branching ratio into the right-handed neutrino channel. The resulting total lepton asymmetry produced by the N 1 , N 2 decay is n L s = i n N s ǫ i , where ǫ i is the lepton asymmetry produced per ith right-handed neutrino decay.
Unlike thermal leptogenesis, there is no wash-out factor in this non-thermal scenario [24] corresponding to the lepton number violating 2-body scatterings mediated by right-handed neutrinos, as long as the light right-handed neutrino masses |M i | ≫ T f [25,26]. The washout factor is proportional to e −z , where z = M i /T f [27] and for z 10 it can be safely neglected.
The CP asymmetry ǫ i is given by [28] where and h = m ′ D v is the neutrino Yukawa coupling matrix in the basis where the right-handed neutrino mass matrix M R is diagonal with real and positive eigenvalues and v is the electroweak scale vev (≃ 174 GeV). This expression is valid in the limit where represents the decay width of the ith right-handed neutrino, and applies in our case.
The mass degeneracy between |M 1 | and |M 2 | can be broken by assuming the existence of new physics beyond M * (≃ 5.5 × 10 13 GeV) which does not respect the discrete symmetry.
Assuming this scale, M G , to be near the GUT scale 2 , an additional term in the superpotential such as W ξ = ηL c 2 L c 2 φ(φφ/M 2 G ) can provide a suitable splitting (ξ = ηM I (M I /M G ) 2 ), which can lead to the desired lepton asymmetry. To estimate the latter, let us assume that in the basis with M R given by Eq. (6), the Dirac mass matrix m D can be diagonalized by a bi-unitary transformation which leads to where U R = U RD U RR . We will consider m diag diag(m e , m µ , m τ ) tan β which is possible within a left-right framework [29,30]. The diagonal entries are taken to be real and positive. Notice that the left-handed rotation U L is not present in Eq. (13). From Eq. (1), m D is given by where violating supersymmetry breaking contributions [31].) The deviation of U L from identity matrix is parameterized by a small angle θ 13 L proportional to ε 1 /m D 3 10 −2 .
Substituting U RR from Eq. (7) into Eq. (13), we find In the limit of degenerate neutrinos y → 1 and thus f (y) ≃ 2/(1 − y). From Eq. (10) we then have where we have used Eq. (15). Recall that the mass degeneracy between |M 1 | , |M 2 | is removed by the introduction of the ξ term 3 with the superpotential W ξ . 3 The introduction of new contributions like W ξ and additional terms in m D (to be considered later) which are important for light neutrino mixings, will not have much impact on Eqs. (15) and (16), and the results for lepton asymmetry will remain unaffected.
Before discussing the magnitudes of the parameters involved in Eq. (19) in order to be consistent with the observed n B /s, let us first consider the light neutrino masses and related issues. From solar, atmospheric and terrestrial neutrino data (at 95% C.L.) [37], we have Our next task is to make sure that the light neutrino mass matrix m ν is consistent with Eq. (20). The lepton mixing matrix [38] is given by U P M N S = U L U ν , where U L arises from the charged lepton sector, and U ν comes from the diagonalization of m ν , namely Since θ 13 L 10 −2 , the bilarge mixings must arise from U ν . The structure of m D given in Eq. (14) must be modified to generate appropriate atmospheric and solar 4 In this limit, sin(α 1 − α 2 ) ≃ |ξ| |x| sin(δx−δ ξ ) 1− |ξ| 2 |x| 2 cos 2 (δx−δ ξ ) . 5 Here, the final temperature is just below the electroweak crossover scale, and we follow [36] to estimate the approximate conversion factor relating lepton and baryon asymmetries as 36/111 ≃ 0.324. We thank the referee for raising this point.  0 (to leading order). The deviation of U RD from the identity matrix is parameterized by θ 13 R ∼ ε 2 /m D 3 ≪ 1.
Including the type II seesaw contribution to the neutrino mass matrix from the induced vev of ∆ L , we have 6 where, for simplicity, a ∆ = 2 p ∆ L ≃ 2 p (cd/|a| 2 )(v 2 /M I )(M I /M * ) 4 is taken to be real.
Note that the terms proportional to ξ (with η ∼ 10 −2 , |ξ| ∼ 100 GeV) are accompanied by factors ε i ε j /x 2 , i, j = 2, 3, and can be safely ignored. As for the lepton asymmetry, only the relative phase between ξ and x will be important.
To obtain the mass eigenstates, we first rotate m ν by U ′ = U 23 U 13 (where U ij denotes the rotation matrix in the ij sector and we will ignore CP violation here) and express the effective mass matrix m ν = U ′T m ν U ′ in the new basis as where and c ij = cos θ ij and s ij = sin θ ij . Furthermore, Note that for 2m D 2 ε 2 /x ≫ m 2 D 3 /M I , 23 mixing can be maximized. An approximate diagonalization of m ν is achieved by focusing on the 12 block of m ν and noting that θ 13 is relatively small (see Eq. (20)). With ρ 1 ≪ (a ∆ − Λ + ), to a good approximation the third state in Eq. (22) decouples. The upper left 2 × 2 block of m ν is readily diagonalized and the resulting mass eigenvalues are with Barring cancellation between a ∆ and Λ − a large but non-maximal mixing angle θ 12 is possible. The light neutrinos turn out to be partially degenerate (|m ν 1 | ∼ |m ν 2 | ∼ |m ν 3 | ∆m 2 atm ).
As the dominant contributions to ρ 1,2 come from the second term in their expressions, we have |ρ 2 | ≃ |ρ 1 | cot θ 23 . Using Eq. (28) we find tan 2θ 13 ≃ sin 2θ 12 tan 2θ 23 • The 13 mixing angle is well below the upper limit allowed by experiments. This is due to the fact that ∆m 2 ⊙ depends upon |ρ 2 | and in turn on tan 2θ 13 . Higher values of tan θ 13 near the experimental upper limit cannot reproduce the appropriate ∆m 2 ⊙ . With the parameters in Table II, the mass-squared differences are ∆m 2 ⊙ ∼ 7.6 × 10 −5 eV 2 and ∆m 2 atm ∼ 2 × 10 −3 eV 2 . In Fig. (2) the allowed region for ε 2 , ε 3 is shown for fixed M s and M I .
• For simplicity we have taken Y 13 associated with ε 1 to be 10 −1 and the relation (31) holds to a good accuracy. We have checked numerically that Y 13 ∼ O(1) would not change anything except that somewhat higher values of ε 3 (∼ 3.3 × 10 −8 GeV) are allowed.
• Finally, θ 13 L induces a tiny correction to θ 12 and θ 23 . θ 13 receives a correction of order θ 13 L c 23 which could be significant for Y 13 of order unity, i.e. sin θ eff 13 ≃ sin θ 13 − (ε 1 /m D 3 )c 23 . Even for this case the prediction for θ 13 remains unaltered.
From Eq. (26) we find that the light neutrino masses are of the same order, close to 0.02−0.1 eV. Figs. (3) and (4) display the range of allowed values for m ν 1 and m ν 3 . Following [15] these are partially degenerate neutrinos. This range of neutrino masses is below the so-called quasi-degenerate case. Furthermore, the effective mass parameter in neutrinoless double beta decay (which is the ee element (≡ a ∆ ) of the neutrino mass matrix) [41] is estimated to be of order 0.03 − 0.05 eV, corresponding to M I = 10 12 GeV and M s = 5.5 TeV.
We have checked that renormalization effects [42] do not alter our conclusions in any significant way. The estimated splitting between N 1 and N 2 due to running from M * to M 1,2 is of order m 2 D 2 4π 2 v 2 ln( M * 10 4 GeV )M 2 , which is much smaller than the contribution arising from the term proportional to ξ. Furthermore, the running in m D can be absorbed through a rescaling of m D i .
With the specified range of parameters involved, we are now in a position to calculate n L /s from Eq. (19). Table III presents a sample value of the phase involved in n L /s which is required to produce correct amount of lepton asymmetry. All other parameters are taken from  An important feature of our model is the existence of TeV scale doubly charged particles [43]. Writing and letting φ 0 = M I + η/ √ 2 and φ 0 = M I + η/ √ 2 (η = η * is the real flaton field), the mass-squared matrix for the doubly charged particle is where Eq. (3)  to pseudo-Goldstone bosons. Finally, we note that in addition to the full MSSM spectrum 7 The contributions from the D terms turn out to be quite small.
of fields, the model contains a new singly charged field of mass √ 2M s ≃ 8 TeV.
There is yet another singly charged field with mass of order M I , well beyond the reach of LHC.
In summary, we have presented a realistic supersymmetric model with gauge symmetry SU(2) L × SU (2)  GeV, while the mass of the third right-handed neutrino is M 3 ∼ 10 12 GeV. The physics of neutrino oscillations requires both type I and type II seesaw, and the three light neutrinos turn out to be partially degenerate with masses around 0.02 − 0.1 eV. This is close to the value of the mass parameter associated with neutrinoless double beta decay [41]. An important test of the model is the presence of doubly charged particles that should be found at the LHC. Another important feature is the prediction sin θ 13 0.01. It would be of some interest to extend the discussion to larger gauge groups such as SU (3) 3 [29,30,45] and SU(4) c × SU(2) L × SU (2)