Open Questions in Cosmology\chaptertitleLeptogenesis and Neutrino Masses in an Inflationary SUSY PatiSalam Model \webadresshttp://dx.doi.org/10.5772/b.bookid.chapterid
Abstract
We implement the mechanism of nonthermal leptogenesis in the framework of an inflationary model based on a supersymmetric (SUSY) PatiSalam Grand Unified Theory (GUT). In particular, we show that inflation is driven by a quartic potential associated with the Higgs fields involved in the spontaneous GUT symmetry breaking, in the presence of a nonminimal coupling of the inflaton field to gravity. The inflationary model relies on renormalizable superpotential terms and does not lead to overproduction of magnetic monopoles. It is largely independent of oneloop radiative corrections, and it can be consistent with current observational data on the inflationary observables, with the GUT symmetry breaking scale assuming its SUSY value. Nonthermal leptogenesis is realized by the outofequilibrium decay of the two lightest righthanded (RH) neutrinos, which are produced by the inflaton decay. Confronting our scenario with the current observational data on light neutrinos, the GUT prediction for the heaviest Dirac neutrino mass, the baryon asymmetry of the universe and the gravitino limit on the reheating temperature, we constrain the masses of the RH neutrinos in the range and the Dirac neutrino masses of the two first generations to values between and .
Keywords: Baryogenesis, Inflation, Grand Unified Theories
PACs Codes: 98.80.Cq, 11.30.Qc, 11.30.Er, 11.30.Pb, 12.60.Jv
1 Introduction
One of the most promising and wellmotivated mechanisms for the generation of the Baryon Asymmetry of the Universe (BAU) is via an initial generation of a lepton asymmetry, which can be subsequently converted to BAU through sphaleron effects – see e.g. Ref. (1; 2). NonThermal Leptogenesis (nTL) (3; 4) is a variant of this proposal, in which the necessitated departure from equilibrium is achieved by construction. Namely, the righthanded (RH) neutrinos, , whose decay produces the lepton asymmetry, are outofequilibrium at the onset, since their masses are larger than the reheating temperature. Such a setup can be achieved by the direct production of through the inflaton decay, which can also take place outofequilibrium. Therefore, such a leptogenesis paradigm largely depends on the inflationary stage, which it follows.
In a recent paper (5) – for similar attempts, see Ref. (6; 7; 8) –, we investigate an inflationary model where a Standard Model (SM) singlet component of the Higgs fields involved in the spontaneous breaking of a supersymmetric (SUSY) PatiSalam (PS) Grand Unified Theory (GUT) can produce inflation of chaotictype, named nonminimal Higgs Inflation (nMHI), since there is a relatively strong nonminimal coupling of the inflaton field to gravity (9; 10; 11; 12). This GUT provides a natural framework to implement our leptogenesis scenario, since the presence of the gauge symmetry predicts the existence of three . In its simplest realization this GUT leads to third family Yukawa unification (YU), and does not suffer from the doublettriplet splitting problem since both Higgs doublets are contained in a bidoublet other than the GUT scale Higgs fields. Although this GUT is not completely unified – as, e.g., a GUT based on the gauge symmetry group – it emerges in standard weakly coupled heterotic string models (13) and in recent Dbrane constructions (14).
The inflationary model relies on renormalizable superpotential terms and does not lead to overproduction of magnetic monopoles. It is largely independent of the oneloop radiative corrections (15), and it can become consistent with the fitting (16) of the sevenyear data of the Wilkinson Microwave Anisotropy Probe Satellite (WMAP7) combined with the baryonacoustic oscillation (BAO) and the measurement of the Hubble constant (). At the same time the GUT symmetry breaking scale attains its SUSY value and the problem of the Minimal SUSY SM (MSSM) is resolved via a PecceiQuinn (PQ) symmetry, solving also the strong CP problem. Inflation can be followed by nonthermal leptogenesis, compatible with the gravitino () limit (17; 18; 19) on the reheating temperature, leading to efficient baryogenesis. In Ref. (5) we connect nonthermal leptogenesis with neutrino data, implementing a twogeneration formulation of the seesaw (20; 21; 22) mechanism and imposing extra restrictions from the data on the light neutrino masses and the GUT symmetry on the heaviest Dirac neutrino mass. There we (5) assume that the mixing angle between the first and third generation, , vanishes. However, the most updated (23; 24) analyses of the low energy neutrino data suggest that nonzero values for are now preferred, while the zero value can be excluded at standard deviations. Therefore, a revision of our results, presented in Ref. (5), is worth pursuing.
The threegeneration implementation of the seesaw mechanism is here adopted, following a bottomup approach, along the lines of Ref. (25; 26; 27; 28). In particular, we use as input parameters the low energy neutrino observables considering several schemes of neutrino masses. Using also the third generation Dirac neutrino mass predicted by the PS GUT, assuming a mild hierarchy for the two residual generations and imposing the restriction from BAU, we constrain the masses of ’s and the residual neutrino Dirac mass spectrum. Renormalization group effects (28; 29) are also incorporated in our analysis.
We present the basic ingredients of our model in Sec. 2. In Sec. 3 we describe the inflationary potential and derive the inflationary observables. In Sec. 4 we outline the mechanism of nonthermal leptogenesis, while in Sec. 5 we exhibit the relevant imposed constraints and restrict the parameters of our model. Our conclusions are summarized in Sec. 6. Throughout the text, we use natural units for Planck’s and Boltzmann’s constants and the speed of light (); the subscript of type denotes derivation with respect to (w.r.t) the field (e.g., ); charge conjugation is denoted by a star and stands for logarithm with basis .
2 The PatiSalam SUSY GUT Model
In this section, we present the particle content (Sec. 2.1), the structure of the superpotential and the Kähler potential(Sec. 2.2) and describe the SUSY limit (Sec. 2.3) of our model.
2.1 Particle Content
We focus on a SUSY PS GUT model described in detail in Ref. (30; 5). The representations and the transformation properties of the various superfields under , their decomposition under , as well as their extra global charges are presented in Table 1.
The th generation lefthanded (LH) quark and lepton superfields, , ( is a color index), and are accommodated in a superfield . The LH antiquark and antilepton superfields , , and are arranged in another superfield . The gauge symmetry can be spontaneously broken down to through v.e.vs which the superfields and acquire in the directions and . The model also contains a gauge singlet , which triggers the breaking of , as well as an 6plet , which splits into and under and gives (13) superheavy masses to and . In the simplest realization of this model (13; 30), the electroweak doublets and , which couple to the up and down quarks respectively, are exclusively contained in the bidoublet superfield .
Super  Represe  Trasfor  Decompo  Global  
fields  ntations  mations  sitions  Charges  
under  under  under  PQ  
Matter Superfields  
Higgs Superfields  
In addition to , the model possesses two global symmetries, namely a PQ and an R symmetry, as well as a discrete symmetry (‘matter parity’) under which , change sign. The last symmetry forbids undesirable mixings of and and/or and and ensures the stability of the lightest SUSY particle (LSP). The imposed R symmetry, , guarantees the linearity of the superpotential w.r.t the singlet . Finally the PQ symmetry, , assists us to generate the term of the MSSM. The PQ breaking occurs at an intermediate scale through the v.e.vs of , , and the term is generated via a nonrenormalizable coupling of and . Following Ref. (30), we introduce into the scheme quartic (nonrenormalizable) superpotential couplings of to , which generate intermediatescale masses for the and, thus, masses for the light neutrinos, , via the seesaw mechanism (20; 21; 22). Moreover, these couplings allow for the decay of the inflaton into , leading to a reheating temperature consistent with the constraint with more or less natural values of the parameters. As shown finally in Ref. (30), the proton turns out to be practically stable in this model.
2.2 Superpotential and Kähler Potential
The superpotential of our model splits into three parts:
(1) 
which are analyzed in the following.

is the part of which contains the usual terms – except for the term – of the MSSM, supplemented by Yukawa interactions among the lefthanded leptons and :
(2) Here and are the th generation doublet LH quark and lepton superfields respectively. Summation over repeated color and generation indices is assumed. Obviously the model predicts YU at since the fermion masses per family originate from a unique term of the PS GUT. It is shown (31; 32) that exact third family YU combined with nonuniversalities in the gaugino sector and/or the scalar sector can become consistent with a number of phenomenological and cosmological lowenergy requirements. On the other hand, it is expected on generic grounds that the predictions of this simple model for the fermion masses of the two lighter generations are not valid. Usually this difficulty can be avoided by introducing (33) an abelian symmetry which establishes a hierarchy between the flavor dependent couplings. Alternatively, the present model can be augmented (34) with other Higgs fields so that and are not exclusively contained in , but receive subdominant contributions from other representations too. As a consequence, a moderate violation of exact YU can be achieved, allowing for an acceptable lowenergy phenomenology, even with universal boundary conditions for the soft SUSY breaking terms. However, we prefer here to work with the simplest version of the PS model, using the prediction of the third family YU in order to determine the corresponding Dirac neutrino mass – see Sec. 5.1.

, is the part of which is relevant for the spontaneous breaking of and the generation of the term of the MSSM. It is given by
(3) where is the String scale. The scalar potential, which is generated by the first term in the RHS of Eq. (3), after gravitymediated SUSY breaking, is studied in Ref. (35; 30). For a suitable choice of parameters, the minimum lies at . Hence, the PQ symmetry breaking scale is of order . The term of the MSSM is generated from the second term of the RHS of Eq. (3) as follows:
(4) which is of the right magnitude if . Let us note that has an additional local minimum at , which is separated from the global PQ minimum by a sizable potential barrier, thus preventing transitions from the trivial to the PQ vacuum. Since this situation persists at all cosmic temperatures after reheating, we are obliged to assume that, after the termination of nMHI, the system emerges with the appropriate combination of initial conditions so that it is led (36) in the PQ vacuum.

, is the part of which is relevant for nMHI, the spontaneous breaking of and the generation of intermediate Majorana [superheavy] masses for [ and ]. It takes the form
(5) where is a superheavy mass scale related to – see Sec. 3.2 – and is the dual tensor of . The parameters and can be made positive by field redefinitions.
According to the general recipe (11; 12), the implementation of nMHI within SUGRA requires the adoption of a Kähler potential, , of the following type
(6) 
where is the reduced Planck scale and the complex scalar components of the superfields and are denoted by the same symbol. The coefficients and are taken real. From Eq. (6) we can infer that we adopt the standard quadratic nonminimal coupling for Higgsinflaton, which respects the gauge and global symmetries of the model. This nonminimal coupling of the Higgs fields to gravity is transparent in the Jordan frame. We also added the fifth term in the RHS of Eq. (6) in order to cure the tachyonic mass problem encountered in similar models (10; 11; 12) – see Sec. 3.1. In terms of the components of the various fields, in Eq. (6) reads
(7a)  
with  
(7b) 
and summation over the repeated Greek indices is implied.
2.3 The SUSY Limit
In the limit where tends to infinity, we can obtain the SUSY limit of the SUGRA potential. Assuming that the SM nonsinglet components vanish, the Fterm potential in this limit, , turns out to be
(8a)  
while the Dterm potential is  
(8b) 
Restricting ourselves to the Dflat direction , we find from that the SUSY vacuum lies at
(9) 
Therefore, leads to spontaneous breaking of . As we shall see in Sec. 3, the same superpotential, , gives rise to a stage of nMHI . Indeed, along the Dflat direction and , tends to a quartic potential, which can be employed in conjunction with in Eq. (6) for the realization of nMHI along the lines of Ref. (12).
It should be mentioned that soft SUSY breaking and instanton effects explicitly break to . The latter symmetry is spontaneously broken by and . This would lead to a domain wall problem if the PQ transition took place after nMHI. However, as we already mentioned above, is assumed already broken before or during nMHI. The final unbroken symmetry of the model is .
3 The Inflationary Scenario
Next we outline the salient features of our inflationary scenario (Sec. 3.1) and calculate a number of observable quantities in Sec. 3.2.
3.1 Structure of the Inflationary Potential
At treelevel the Einstein Frame (EF) SUGRA potential, , is given by (11)
(10a)  
where is the unified gauge coupling constant and the summation is applied over the generators of the PS gauge group – see Ref. (5). Also, we have  
(10b) 
The ’s are given in Eq. (7b). If we parameterize the SM singlet components of and by
(11) 
we can easily deduce that a Dflat direction occurs at
(12) 
Along this direction, the Dterms in Eq. (10a) – and, also, in Eq. (8b) – vanish, and so takes the form
(13) 
with
(14) 
From Eq. (13), we can verify that for and , takes a form suitable for the realization of nMHI, since it develops a plateau – see also Sec. 3.2. The (almost) constant potential energy density and the corresponding Hubble parameter (along the trajectory in Eq. (12)) are given by
(15) 
We next proceed to check the stability of the trajectory in Eq. (12) w.r.t the fluctuations of the various fields. To this end, we expand them in real and imaginary parts as follows
(16) 
Notice that the field can be rotated to the real axis via a suitable R transformation. Along the trajectory in Eq. (12) we find
(17) 
To canonically normalize the fields and , we first diagonalize the matrix . This can be achieved via a similarity transformation involving an orthogonal matrix as follows:
(18) 
Utilizing , the kinetic terms of the various fileds can be brought into the following form
(19) 
where , and and the dot denotes derivation w.r.t the cosmic time, . In the last line, we introduce the EF canonically normalized fields, and , which can be obtained as follows – cf. Ref. (11; 12; 37; 5):
(20) 
Taking into account the approximate expressions for , and the slowroll parameters , which are displayed in Sec. 3.2, we can verify that, during a stage of slowroll inflation, since , and since . On the other hand, we can show that , since the quantity , involved in relating to , turns out to be negligibly small compared with . Indeed, the ’s acquire effective masses – see below – and therefore enter a phase of oscillations about with decreasing amplitude. Neglecting the oscillatory part of the relevant solutions, we find
(21) 
where represents the initial amplitude of the oscillations, and we assume . Taking into account the approximate expressions for and the slowroll parameter in Sec. 3.2, we find
(22) 
Having defined the canonically normalized scalar fields, we can proceed in investigating the stability of the inflationary trajectory of Eq. (12). To this end, we expand in Eq. (10a) to quadratic order in the fluctuations around the direction of Eq. (12), as described in detail in Ref. (5). In Table 2 we list the eigenvalues of the masssquared matrices
(23) 
involved in the expansion of . We arrange our findings into three groups: the SM singlet sector, , the sector with the and the fields which are related with the broken generators of and the sector with the and the fields. Upon diagonalization of the relevant matrices we obtain the following mass eigenstates:
(24) 
As we observe from the relevant eigenvalues, no instability – as the one found in Ref. (37) – arises in the spectrum. In particular, it is evident that assists us to achieve – in accordance with the results of Ref. (12). Moreover, the Dterm contributions to and – proportional to the gauge coupling constant – ensure the positivity of these masses squared. Finally the masses that the scalars acquire from the second and third term of the RHS of Eq. (5) lead to the positivity of for of order unity. We have also numerically verified that the masses of the various scalars remain greater than the Hubble parameter during the last efoldings of nMHI, and so any inflationary perturbations of the fields other than the inflaton are safely eliminated.
Fields  Masses Squared  Eigenstates 
The – – Sector  
2 real scalars  
1 complex scalar  
The – () and – Sectors  
real scalars  
The – and – () Sectors  
real scalars  
The Goldstone bosons, associated with the modes and with and , are not exactly massless since – contrary to the situation of Ref. (30) where the direction with non vanishing minimizes the potential. These masses turn out to be . On the contrary, the angular parametrization in Eq. (11) assists us to isolate the massless mode , in agreement with the analysis of Ref. (11). Employing the wellknown ColemanWeinberg formula (15), we can compute the oneloop radiative corrections to the potential in our model. However, these have no significant effect on the inflationary dynamics and predictions, since the slope of the inflationary path is generated at the classical level – see the expressions for and below.
3.2 The Inflationary Observables
Based on the potential of Eq. (13) and keeping in mind that the EF canonically inflaton is related to via Eq. (20), we can proceed to the analysis of nMHI in the EF, employing the standard slowroll approximation. Namely, a stage of slowroll nMHI is determined by the condition – see e.g. Ref. (38; 39):
where
(25a)  
and  
(25b) 
are the slowroll parameters and – see Sec. 4.1. Here we employ Eq. (15) and the following approximate relations:
(26) 
The numerical computation reveals that nMHI terminates due to the violation of the criterion at a value of equal to , which is calculated to be
(27) 
The number of efoldings, , that the scale suffers during nMHI can be calculated through the relation:
(28) 
where is the value of when crosses the inflationary horizon. Given that , we can write as a function of as follows
(29) 
The power spectrum of the curvature perturbations generated by at the pivot scale is estimated as follows
(30) 
Since the scalars listed in Table 2 are massive enough during nMHI, can be identified with its central observational value – see Sec. 5 – with almost constant . The resulting relation reveals that is to be proportional to . Indeed we find