1. Leptogenesis beyond the limit of hierarchical heavy neutrino masses
IMPRS Seminar
Leptogenesis beyond the limit of hierarchical heavy neutrino masses
Reference: hep-ph/
Steve Blanchet, in collaboration with Pasquale Di Bari
Max-Planck-Institut für Physik, Munich
June 9, 2006

2. Introduction
Leptogenesis was first proposed by Fukugita & Yanagida in 1986 as a way to solve one of the most outstanding problems in modern cosmology, namely the baryon asymmetry of the Universe (BAU):
At T~1010 GeV:
L generated
But
in equilibrium
B-L and hence B are generated!
violate B+L
conserve B-L

3. Introduction
With the growing evidence for non-zero neutrino masses since 1998, leptogenesis became very attractive, because it bears in itself (see-saw !) a natural way of giving not only mass to neutrinos, but also very small ones.
Under the assumption of hierarchical heavy neutrino masses , one typically obtains the N1-dominated scenario.
The other extreme case which has been extensively studied is when two RH neutrinos are close to degenerate, : this is the so-called resonant leptogenesis scenario.
Here we want to go beyond these two limiting cases and in particular show the border between them.

4. The model
Leptogenesis is based on the following extension of the SM Lagrangian:
Crucial for the production of a lepton asymmetry!
Yukawa coupling
Majorana mass term
where and
are the Higgs and left-handed lepton doublets, respectively, and
This extension is clearly acceptable on grounds of gauge invariance and renormalizability, and is minimal in its particle content (here: 3 new particles).

5. The model
The masses of the singlet neutrinos are essentially free parameters, and thus can be taken to be very large
See-saw!
After spontaneous symmetry breaking, the vev of the Higgs leads to a Dirac mass term . The see-saw assumes so that the neutrino mass term can be block-diagonalized as:
1st order
3 heavy Majorana neutrinos, mass
Natural estimate :
After diagonalization: 3 light Majorana neutrinos, mass
Constraint: experiment give two different mass scales:

6. Leptogenesis
In order to produce a baryon asymmetry in the Early Universe, one needs to fulfill three conditions [Sakharov, 1967] :
Baryon number violation
2. C and CP violation
3. Departure from thermal equilibrium
In thermal leptogenesis:
Satisfied thanks to the electroweak sphaleron processes (SM violates B+L!)
2. C is maximally violated in the neutrino sector of the SM, and CP is violated in the decay of the heavy neutrinos:
CP asymmetry parameter
3. The decay of the heavy neutrinos will be out-of-equilibrium at some point:
``decay parameter´´

7. Leptogenesis
Two very different scenarios can occur in the framework of leptogenesis :
The weak wash-out scenario for
The strong wash-out scenario for
The fundamental Boltzmann equations for leptogenesis are the following:
where D denotes decay, S scattering, W the wash-out (inverse decays and scatterings) and all number densities NX are calculated in a portion of comoving volume containing 1 RH neutrino in ultra-relativistic thermal equilibrium.
Sphalerons conserve B-L !
CP violation
Out-of-equilibrium condition

8. Leptogenesis
The strong wash-out scenario is favoured both from a theoretical and an experimental perspective: it allows the model to be self-contained (no dependence on the initial conditions) and the two mass scales coming from neutrino experiments typically imply
The pure particle physics part (CP asymmetry) can always be disconnected from the pure thermodynamical part (processes occurring in the Early Universe) so that it is convenient to write the solution in the form:
where is the efficiency factor, approximately given by the number of Ni that decay out-of-equilibrium.

9. Leptogenesis The final baryon asymmetry is given by
where , and should be compared to the measured value of the BAU (e.g. WMAP+SDSS):
Assuming a hierarchical heavy neutrino spectrum,
one typically has the so-called N1-dominated scenario, where both production and wash-out from the heavier two RH neutrinos can be neglected. Then, one has
which only depends on a limited set of parameters!

10. The N1-dominated scenario
For our purposes, it is sufficient to consider the simplified picture with only decays and inverse decays. The fundamental Boltzmann equations then read
The efficiency factor is analytically given by
Solving these two equations for one case of weak wash-out (K=0.01) and one case of strong wash-out (K=10), both with initial thermal abundance, the following very different dynamics results:

11. WEAK WASH-OUT
STRONG WASH-OUT

12. WEAK WASH-OUT
STRONG WASH-OUT

13. Lower bounds on M1 and Treh
In leptogenesis, the different models can be parametrized using the see-saw orthogonal matrix:
PMNS mixing matrix:
Low energy parameters
so that
High energy parameters
For a given mass of N1, the CP asymmetry in the HL is bounded from above :
1 (<1) for m1=0 (> 0)
``Leptogenesis phase´´

14. Lower bounds on M1 and Treh
Recalling that this clearly implies both a lower bound on M1 and on the reheating temperature of the Universe (highest temperature after inflation).
Let us give two examples with m1=0:
1) Maximal phase 2)
5x109
2x109

15. Beyond the hierarchical limit
Assume a partial hierarchy
which implies ,
so that
Assume moreover that the two lightest right-handed neutrinos are in the strong wash-out, i.e. K1, K2 >> 1, we are then in the so-called strong-strong wash-out scenario.
the wash-outs sum!
In this case,

16. Dynamics beyond the HL
Let us choose the case to illustrate how the dynamics is modified beyond the HL.
In the HL:

17. Dynamics beyond the HL
Let us now allow the masses to become close, so that we are not in the HL anymore: 1 2 3

18. Dynamics beyond the HL There are two main effects:1
The two efficiency factors can sum. 2
The wash-out from N2 acts on N1 as well ! 3
The final efficiency factor is reduced.
What about the final B-L asymmetry? Is it also reduced?
Yes… But there is a competing effect in the CP asymmetry, which gets enhanced when the degeneracy is increased. The latter will even become dominant.
When exactly?

19. Lower bounds on M1 and Treh
a) Maximal Phase:

20. Lower bounds on M1 and Treh
b) Very heavy N3:
Here both

21. Conclusions
We have seen that it is possible to study in some detail leptogenesis beyond the usual hierarchical limit.
Degeneracies of the order 10-2 and below can relax the bounds
Useful to evade the gravitino problem in SUGRA scenarios and to have a quark-like Dirac neutrino mass matrix.
Are these degeneracies theoretically well-motivated?
In going beyond the HL, there is a greater model dependence that has to be handled (see-saw orthogonal matrix!)
We have seen that leptogenesis seems to favour normal hierarchies over inverted ones.