5.2. Lepton-Number and Gravitino Abundances
The mechanism of nTL [
57,
58,
59] can be activated by the out-of-equilibrium decay of the
’s produced by the
decay, via the interactions in Equation (
45a). If
, the out-of-equilibrium condition [
52,
53] is automatically satisfied. Namely,
decay into (fermionic and bosonic components of)
and
via the tree-level couplings derived from the last term in the rhs of Equation (
3a). The resulting—see
Section 5.3—lepton-number asymmetry
(per
decay) after reheating can be partially converted via sphaleron effects into baryon-number asymmetry. In particular, the
B yield can be computed as
The numerical factor in the rhs of Equation (49a) comes from thesphaleron effects, whereas the one (
) in the rhs of Equation (49b) is due to the slightly different calculation [
105] of
—cf. Reference [
52,
53]. The validity of the formulae above requires that the
decay into a pair of
’s is kinematically allowed for at least one species of the
’s and also that there is no erasure of the produced
due to
mediated inverse decays and
scatterings [
106]. Theses prerequisites are ensured if we impose
Finally, the interpretation of BAU through nTL dictates [
56] at 95% c.l.
The
’s required for successful nTL must be compatible with constraints on the
abundance,
, at the onset of nucleosynthesis (BBN). Assuming that
is much heavier than the gauginos of MSSM,
is estimated to be [
62,
63,
64,
65,
66,
67]
where we take into account only the thermal
production. Non-thermal contributions to
[
74] are also possible but strongly dependent on the mechanism of soft SUSY breaking. Moreover, no precise computation of this contribution exists within HI adopting the simplest Polonyi model of SUSY breaking [
68,
69]. For these reasons, we here adopt the conservative estimation of
in Equation (
52). Nonetheless, it is notable that the non-thermal contribution to
in models with stabilizer field, as in our case, is significantly suppressed compared to the thermal one.
On the other hand,
is bounded from above in order to avoid spoiling the success of the BBN. For the typical case where
decays with a tiny hadronic branching ratio, we have [
64,
65,
66,
67]
The bounds above can be somehow relaxed in the case of a stable
—see e.g., Reference [
107,
108,
109,
110]. In a such case,
should be the LSP and has to be compatible with the data [
56] on the CDM abundance in the universe. To activate this scenario we need lower
’s than those obtained in
Section 4.2. As shown from Equation (
38b), this result can be achieved for lower
’s and/or larger
’s. Low
’s, implying large
r’s, generically help in this direction too.
Note, finally, that both Equations (49) and (
52) calculate the correct values of the
B and
abundances provided that no entropy production occurs for
. This fact can be achieved if the Polonyi-like field
z decays early enough without provoking a late episode of secondary reheating. A subsequent difficulty is the possible over-abundance of the CDM particles which are produced by the
z decay—see Reference [
111,
112,
113].
5.3. Lepton-Number Asymmetry and Neutrino Masses
As mentioned above, the decay of
, emerging from the
decay, can generate a lepton asymmetry,
, caused by the interference between the tree and one-loop decay diagrams, provided that a CP-violation occurs in
’s. The produced
can be expressed in terms of the Dirac mass matrix of
,
, defined in the
-basis, as follows [
114,
115,
116]:
where we take
, for large
and
The involved in Equation (
54a)
can be diagonalized if we define a basis—called
weak basis henceforth—in which the lepton Yukawa couplings and the
interactions are diagonal in the space of generations. In particular we have
where
U and
are
unitary matrices which relate
and
(in the
-basis) with the ones
and
in the weak basis as follows
Here, we write LH lepton superfields, i.e.,
doublet leptons, as row 3-vectors in family space and RH anti-lepton superfields, i.e.,
singlet anti-leptons, as column 3-vectors. Consequently, the combination
appeared in Equation (
54a) turns out to be a function just of
and
. Namely,
The connection of the leptogenesis scenario with the low energy neutrino data can be achieved through the seesaw formula, which gives the light-neutrino mass matrix
in terms of
and
. Working in the
-basis, we have
where
with
real and positive. Solving Equation (
55) wrt
and inserting the resulting expression in Equation (
58) we extract the mass matrix
which can be diagonalized by the unitary PMNS matrix satisfying
and parameterized as follows
where
,
and
,
and
are the CP-violating Dirac and Majorana phases.
Following a bottom-up approach, along the lines of References [
54,
55,
106,
117], we can find
via Equation (
60b) adopting the normal or inverted hierarchical scheme of neutrino masses. In particular,
’s can be determined via the relations
where the neutrino mass-squared differences
and
are listed in
Table 5 and computed by the solar, atmospheric, accelerator and reactor neutrino experiments. We also arrange there the inputs on the mixing angles
and on the CP-violating Dirac phase,
, for normal [inverted] neutrino mass hierarchy [
70]—see also Reference [
71,
72]. Moreover, the sum of
’s is bounded from above by the current data [
56], as follows
Taking also
as input parameters we can construct the complex symmetric matrix
see Equation (
60a)—from which we can extract
as follows
Note that
is a
complex, hermitian matrix and can be diagonalized numerically so as to determine the elements of
and the
’s. We then compute
through Equation (
57) and the
’s through Equation (
54a).
5.4. Results
The success of our inflationary scenario can be judged, if, in addition to the constraints of
Section 3.3, it can become consistent with the post-inflationary requirements mentioned in
Section 5.2 and
Section 5.3. More specifically, the quantities which have to be confronted with observations are
and
which depend on
,
,
and
’s —see Equations (49) and (
52). As shown in Equation (
41),
is a function of
n and
whereas
in Equation (
48) depend on
,
y and the masses of the
’s into which
decays. Throughout our computation we fix
which is a representative value. Also, when we employ
and
we take
which allows for a quite broad available
margin. As regards the
masses, we follow the bottom-up approach described in
Section 5.3, according to which we find the
’s by using as inputs the
’s, a reference mass of the
’s –
for NO
’s, or
for IO
’s –, the two Majorana phases
and
of the PMNS matrix, and the best-fit values, listed in
Table 5, for the low energy parameters of neutrino physics. In our numerical code, we also estimate, following Reference [
118], the RG evolved values of the latter parameters at the scale of nTL,
, by considering the MSSM with
as an effective theory between
and the soft SUSY breaking scale,
. We evaluate the
’s at
, and we neglect any possible running of the
’s and
’s. The so obtained
’s clearly correspond to the scale
.
We start the exposition of our results arranging in
Table 6 some representative values of the parameters which yield
and
compatible with Equations (
51) and (
53), respectively. We set
in accordance with Equations (18a) and (18b). Also, we select the
value in Equation (39b) which ensures central
n and
r in Equations (
1) and (31a). We obtain
and
for
or
and
for
or
. Although such uncertainties from the choice of
K’s do not cause any essential alteration of the final outputs, we mention just for definiteness that we take
or
throughout. We consider NO (cases A and B), almost degenerate (cases C, D and E) and IO (cases F and G)
’s. In all cases, the current limit of Equation (
62) is safely met—in the case D this limit is almost saturated. We observe that with NO or IO
’s, the resulting
and
are of the same order of magnitude, whereas these are more strongly hierarchical with degenerate
’s. In all cases, the upper bounds in Equation (
45b) is preserved thanks to the third term adopted in the rhs of Equation (
3b)—cf. Reference [
12,
13]. We also remark that
decays mostly into
’s—see cases A–D. From the cases E–G, where the decay of
into
is unblocked, we notice that, besides case E, the channel
yields the dominant contribution to the calculation
from Equation (49), since
. We observe, however, that
(
is constantly negligible) and so the ratios
introduce a considerable reduction in the derivation of
. This reduction could have been eluded, if we had adopted—as in References [
12,
13,
117]—the resolution of the
problem proposed in Reference [
119] since then, the decay mode in Equation (
46a) would have disappeared. This proposal, though, is based on the introduction of a Peccei-Quinn symmetry, and so the massless during HI axion generates possibly CDM isocurvature perturbation which is severely restricted by the
Planck results [
56]. In
Table 6 we also display, for comparison, the
B yield with
or without
taking into account the renormalization group running of the low energy neutrino data. We observe that the two results are in most cases close to each other with the largest discrepancies encountered in cases C, E and F. Shown are also the values of
, the majority of which are close to
, and the corresponding
’s, which are consistent with Equation (
53) for
. These values are in nice agreement with the ones needed for the solution of the
problem of MSSM—see, e.g.,
Figure 3 and
Table 4.
The gauge symmetry considered here does not predict any particular Yukawa unification pattern and so, the
’s are free parameters. For the sake of comparison, however, we mention that the simplest realization of a SUSY Left-Right [Pati-Salam] GUT predicts [
117,
120]
[
], where
are the masses of the up-type quarks and we ignore any possible mixing between generations—these predictions may be eluded though in more realistic implementations of these models as in References [
117,
120]. Taking into account the SUSY threshold corrections [
91] in the context of MSSM with universal gaugino masses and
, these predictions are translated as follows
Comparing these values with those listed in
Table 6, we remark that our model is not compatible with any pattern of large hierarchy between the
’s, especially in the two lighter generations, since and
. On the other hand,
is of the order of
in cases A–E whereas
only in case A. This arrangement can be understood, if we take into account that
and
separately influence the derivation of
and
correspondingly—see, e.g., References [
12,
13,
106]. Consequently, the displayed
’s assist us to obtain the
’s required by Equation (
51).
In order to investigate the robustness of the conclusions inferred from
Table 6, we examine also how the central value of
in Equation (
51) can be achieved by varying
as a function of
and
in
Figure 4a,b respectively. Since the range of
in Equation (
51) is very narrow, the
c.l. width of these contours is negligible. The convention adopted for these lines is also described in each plot. In particular, we use solid, dashed, dot-dashed, double dot-dashed and dotted line when the inputs—i.e.,
,
,
,
and
—correspond to the cases B, C, E, D and G of
Table 6, respectively. In both graphs we employ
or
with
and
.
In
Figure 4a we fix
to the value used in
Table 6. Increasing
above its value shown in
Table 6 the ratio
gets lower and an increase of
—and consequently on
—is required to keep
at an acceptable level. As a byproduct,
and
increase too and jeopardize the fulfillment of Equation (
53). Actually, along the depicted contours in
Figure 4a, we obtain
whereas the resulting
’s [
’s] vary in the ranges (0.8–3)
, (0.4–2)
and (2.9–3.1)
, [(4–6)
,
and (3–4)
] for the inputs of cases B, D and G respectively. Finally,
remains close to its values presented in the corresponding cases of
Table 6. At the upper [lower] termination points of the contours, we obtain
lower [upper] that the value in Equation (
51).
In
Figure 4b we fix
and vary
in the allowed range indicated in
Figure 2a. Only some segments from that range fulfill the post-inflationary requirements, despite the fact that the Majorana phases in
Table 6 are selected so as to maximize somehow the relevant
margin. Namely, as inferred by the numbers indicated on the curves in the
plane, we find that
may vary in the ranges (0.008–0.499), (0.025–0.47) and (0.06–0.4) for the inputs of cases B, C and E respectively. The lower limit on these curves comes from the fact that
is larger than the expectations in Equation (
51). At the other end, Equation (50b) is violated and, therefore, washout effects start becoming significant. At these upper termination points of the contours, we obtain
of the order
or
and so, we expect that the constraint of Equation (
53) will cut any possible extension of the curves beyond these termination points that could survive the possible washout of
. As induced by Equations (
41) and (
42),
increases with
and so, an enhancement of
’s and similarly of
’s is required so that
meets Equation (
51). The enhancement of
becomes sharp until the point at which the decay channel of
into
’s rendered kinematically allowed.
Compared to the findings of the same analysis in other inflationary settings [
12,
13,
54,
55], the present scenario is advantageous since
is allowed to reach lower values. Recall—see
Section 5.1—that the constant value of
obtained in the papers above represents here the upper bound of
which is approached when
tends to its maximal value in Equation (
15). In practice, this fact offers us the flexibility to reduce
and
at a level compatible with
values as light as
which are excluded elsewhere. On the other hand,
increases when
decreases and can be kept in accordance with the expectations due to variation of
and
. As a bottom line, nTL not only is a realistic possibility within our models but also it can be comfortably reconciled with the
constraint.