Supersymmetric Model of Neutrino Mass and Leptogenesis with String-Scale Unification

Adjoint supermultiplets (1,3,0) and (8,1,0) modify the evolution of gauge couplings. If the unification of gauge couplings occurs at the string scale, their masses are fixed at around $10^{13}$ GeV. This scale coincides with expected gaugino condensation scale in the hidden sector $M_{string}^{2/3} m^{1/3}_{3/2} \sim 10^{13}$ GeV. We show how neutrino masses arise in this unified model which naturally explain the present atmospheric and solar neutrino data. The out-of-equilibrium decay of the superfield (1,3,0) at $10^{13}$ GeV may also lead to a lepton asymmetry which then gets converted into the present observed baryon asymmetry of the Universe.

It is interesting to study the adjoint SU(2) triplet with no hypercharge T ≡ (1, 3, 0) having a mass approximately 10 13 GeV. In a supersymmetric theory, the lepton superfield L i ≡ (ν i , l i ), the Higgs superfield H 2 ≡ (h + 2 , h 0 2 ), and the superfield T ≡ (T + , T 0 , T − ) may be connected by the Yukawa coupling h i T L i H 2 T . As H 2 gets a nonzero vacuum expectation value (VEV) given by v 2 ≡ v sin β, where tan β ≡ h 0 2 / h 0 1 , the neutrino ν pairs up with the fermion T 0 to form a Dirac mass and because T has an allowed Majorana mass M 2 , a seesaw mass is generated for ν: [1] where the extra factor of 2 comes from the fact that T 0 couples to (νh 0 2 + lh + 2 )/ √ 2. The difference between this and the canonical seesaw mechanism [2] is the use of an SU(2) triplet instead of a singlet. This means that whereas the latter has negligible influence on the evolution of gauge couplings, the former changes it in a significant way. It is thus possible to have gauge coupling unification at the string scale [3] with M 2 ∼ 10 13 GeV as well as a realistic theory of neutrino mass and leptogenesis consistent with present atmospheric and solar neutrino experiments [4,5], as shown below.
One-loop string effects could lower the tree-level value of the string scale M string = g string M P lank somewhat, and one calculates [6] that the string unification scale is modified to M string = g string × 5.27 × 10 17 GeV ≃ 5.27 × 10 17 GeV.
Furthermore, string models having a G×G structure, when broken to the diagonal subgroup, naturally contain adjoint scalars with zero hypercharge. In this paper, we minimally extend the canonical supersymmetric standard model by including the superfields T ≡ (1, 3, 0), O ≡ (8, 1, 0) and S ≡ (1, 1, 0) [7]. We will show that if the unification of gauge couplings occurs at the string scale, two-loop renormalization-group equations (RGE) will fix the masses of T The dimensionless Yukawa couplings of this model in standard superfield notation is given by where we have introduced a singlet S ≡ (1, 1, 0), the utility of which will be explained later.
The one-loop coefficients b i and the two-loop coefficients b ij can be easily derived [7]. Also the effect of the Yukawa couplings on the running of the gauge couplings is brought in by the coefficients a ij . They are given by In the matrix a ik the index k refers to In the evolution equations we have generically used the notations Y = h 2 /4π and α = g 2 /4π.
As we know, we must also run the Yukawa couplings which are involved in the running of the gauge couplings. The RGE for a typical trilinear Yukawa term We now apply Eq. (7) to the Yukawa couplings of interest. We thus get the evolution equations for the extra Yukawa couplings h T , h S , h λ as well as their influence on the evolution of the other relevant Yukawa couplings. Here also we put t = ln µ/2π.
We assume that the unification is happening at the scale M X = 5.2 × 10 17 GeV [6] with the unified coupling of α X . We then use the two-loop RGE to evolve the couplings down to m Z .
In doing so, we must properly cross the thresholds M 2 and M 3 . Once we get the values of the couplings at m Z , we can numerically solve the set of quantities α X , M 2 , M 3 using as input Note that the running also depends on the Yukawa couplings present in our model. We must have the top quark mass at around 174 GeV. We keep the top-quark Yukawa coupling at its To first approximation, we let T 0 couple to (ν µ + ν τ )/ √ 2 and set m ν = 0.05 eV in Eq. (1) to account for the atmospheric neutrino data. For M 2 ≃ 3.7 × 10 13 GeV, this implies that To account for the solar neutrino data, we need another massive neutrino.
The couplings which are relevant for generating a lepton asymmetry of the Universe [12,15] in this scenario are contained in where we have considered one triplet T and one singlet S. Note that with only one triplet or There are one-loop vertex diagrams interfering with the tree-level decay diagrams of T and S, which will give rise to CP violation in these decays (see Fig. 2). This CP asymmetry will then generate a lepton asymmetry of the Universe. Unlike other models of leptogenesis where two or more heavy particles of the same type are used, there are no self-energy diagrams contributing to the CP asymmetry in this model.
Assuming M 1 >> M 2 , the lepton asymmetry is generated by the decay of the triplet superfield T . The singlet S enters in the loop diagram to give CP violation. The amount of asymmetry thus generated is given by where the factor ζ comes from the overlap between the neutrino states which couple to T and S. In this scenario, the triplet superfield T has gauge interactions, which will bring its number density to equilibrium through the interaction T + T → W → L +L. However, the decay and the inverse decay of T will be faster if we take h 2 From our RGE analysis we get α 2 (M 2 ) ∼ 1/25. Since an asymmetry is only generated by a departure from equilibrium, interactions faster than expantion rate of the universe will bear an additional suppression factor in the asymmetry they generate. This factor can be estimated by numerically solving the full set of Boltzmann equations. We borrow the result from Ref. [13] that when Γ/H is 5, the supression factor is 0.02 when it is 1000 the supression factor can be as large as 8 × 10 −6 . Here we make a rough estimate for our case by taking the Yukawa interaction and neglecting the gauge interaction and use an approximate supression factor, where H = 1.7 √ g * T 2 /M P l (with g * the number of relativistic degrees of freedom) is the Hubble expansion parameter and Γ = h 2 T M 2 /8π is the decay width of T . From our choice of numerical values for the neutrino mass matrix, we get an asymmetry where δ is the relative phase between h S and h T . Using M 2 = 3.7 × 10 13 GeV, ζ ∼ 0.05, and δ ∼ 0.01, we then get a lepton asymmetry ǫ L ∼ 10 −10 as required. A numerical solution of the Boltzmann equations can give errors introduced in parameters δ and ζ of this simple estimate. We plan to report this analysis in a future publication. In this case, the amount of lepton asymmetry is directly related to the neutrino masses valid for atmospheric and neutrino oscillations as well as the intermediate scale required for string-scale unification.
The scale of supersymmetry breaking in the hidden sector (for a particular choice of the hidden sector fields) in this scenario may also be 10 13 GeV, hence this particular intermediate mass scale allows us to have a consistent description of string-scale unification, neutrino mass, leptogenesis, as well as supersymmetry breaking.
If M 2 >> M 1 , it will be the decay of the singlet S which generates the lepton asymmetry.
In this case the singlet does not have any gauge interactions but its Yukawa interaction will be similar to that of the triplet in the previous case. Hence the amount of lepton asymmetry is again similar, except that the roles of M 1 and M 2 are reversed. Finally, this lepton asymmetry gets converted into the present baryon asymmetry of the Universe from the action of the B + L violating electroweak sphalerons [14], in analogy with the canonical leptogensis decay of heavy right-handed neutrinos [15].  Fig. 1. Such a heavy gravitino should decay otherwise it will over-close the universe. Now let us say that the gravitino decays predominantly to photon and photino. Upper bound on the reheating temperature depends on the mass of gravitino. We see from Figure (17) of the reference [16] that for m 3/2 more than 5 TeV, reheating temperature upper-bound is more than 10 13 GeV. Furthermore the produced photon may further produce hadrons [17].
In that case we get from Figure (14) of reference [18] that for m 3/2 more than 200 TeV, the reheating upper-bound is more than 10 13 GeV. In both these cases our scenario is consistent with post inflationary reheating. Infact note that from RGE analysis we get values of M I which actually gives m 3/2 in the 200 TeV range in a natural way. However the dominant decay mode of the gravitino may not be photon and photino. The case where the gravitino decays to a neutrino and sneutrino, when it is kinematically allowed, has been studied in [19].
In this case neutrinos and sneutrinos produce photons in cascade those interact and change predicted abundance of the light elements which may differ from the observed values of the abundance of light elements. In this case the upper bound on the reheating temperature is tighter, which is around 10 12 GeV. This intermediate scale will produce a smaller gravitino mass unless the string scale is lowered. In this case the present scenario would be in trouble.
Also, one must remember that there is experimental uncertainty in the determination of the abundance of light elements themselves such as the primordial fraction of 4 He [22]. Finally in various supersymmetric extensions of the standard model one can have light axinos in the KeV range and gravitino decays to axino. In such scenarios the upper bound on the reheating increases to 10 15 GeV [20].
A valid case can be made for larger soft masses of 100 TeV range as it is good for suppressing supersymmetric flavor changing neutran current and CP violation [21] problems.
In any case both supersymmetry and neutrino mass are physics beyond the standard model. it has been shown that the unification of gauge couplings occur at 5.2 × 10 17 GeV. In this paper we have shown that this scenario can lead to a neutrino mass matrix which produces (∆m 2 ) atm = 2.5 × 10 −3 eV 2 , sin 2 2θ atm = 1, and (∆m 2 ) sol = 4.9 × 10 −5 eV 2 , sin 2 2θ sol = 0.92. This is in good agreement with atmospheric and solar neutrino data. Furthermore, there are two ways that the triplet superfield T may decay to L and H 2 , and in one of which the singlet S resides in a loop. The interference of these decay amplitudes allows for the CP violation needed for leptogenesis. We have shown that after we take into account the suppression in the generated lepton asymmetry due an approximate equilibrium condition between the forward and inverse decays of T , the final lepton asymmetry emerges in the range ǫ L ∼ 10 −10 as required.
This work was supported in part by the U. S. Department of Energy under Grant No. DE-FG03-94ER40837. BB thanks Probir Roy for communications on gravitino decay modes.