#
Strong thermal leptogenesis

and the

absolute neutrino mass scale

###### Abstract

We show that successful strong thermal leptogenesis, where the final asymmetry is independent of the initial conditions and in particular a large pre-existing asymmetry is efficiently washed-out, favours values of the lightest neutrino mass for normal ordering (NO) and for inverted ordering (IO) for models with orthogonal matrix entries respecting . We show analytically why lower values of require a higher level of fine tuning in the seesaw formula and/or in the flavoured decay parameters (in the electronic for NO, in the muonic for IO). We also show how this constraint exists thanks to the measured values of the neutrino mixing angles and could be tightened by a future determination of the Dirac phase. Our analysis also allows us to place a more stringent constraint for a specific model or class of models, such as -inspired models, and shows that some models cannot realise strong thermal leptogenesis for any value of . A scatter plot analysis fully supports the analytical results. We also briefly discuss the interplay with absolute neutrino mass scale experiments concluding that they will be able in the coming years to either corner strong thermal leptogenesis or find positive signals pointing to a non-vanishing . Since the constraint is much stronger for NO than for IO, it is very important that new data from planned neutrino oscillation experiments will be able to solve the ambiguity.

## 1 Introduction

The observed matter-antimatter asymmetry of the Universe is a long standing cosmological puzzle
calling for physics beyond the Standard Model (SM).
In terms of the baryon-to-photon number ratio the matter-antimatter asymmetry is today accurately and precisely measured by CMB observations. Recently the Planck collaboration found from CMB anisotropies plus lensing data
^{1}^{1}1More precisely the Planck collaboration finds
corresponding to .
[1]

(1) |

Leptogenesis [2] provides an attractive solution since it relies on a minimal and natural way to extend the SM incorporating neutrino masses and mixing discovered in neutrino oscillation experiments: the seesaw mechanism [3]. At the same time it should be noticed that leptogenesis also relies on the Brout-Englert-Higgs mechanism and, therefore, the recent discovery of the Higgs boson at the LHC nicely contributes to support the picture. On the other hand the non-observation of new physics at the LHC so far, places stronger constraints on low scale baryogenesis scenarios such as, for example, electroweak baryogenesis within the minimal supersymmetric standard model [4].

The prediction of the baryon asymmetry relies on some assumption on the initial conditions. A plausible and common one is that an inflationary stage before leptogenesis resets the initial conditions in the early Universe, enforcing vanishing values of the asymmetry and of the right-handed (RH) neutrino abundances prior to the onset of leptogenesis. However, it cannot be excluded, especially at the high temperatures required by a minimal scenario of leptogenesis [5], that other mechanisms, such as gravitational [6], GUT [7], Affleck-Dine baryogenesis [8], generate a large asymmetry at the end of inflation and/or prior to the onset of leptogenesis.

Since these mechanisms escape experimental probes, it would be certainly more attractive if the final asymmetry from leptogenesis were independent of the initial conditions. In this paper we show that, given the current low energy neutrino data, the possibility to enforce independence of the initial conditions in leptogenesis, so called strong thermal leptogenesis, barring quasi-degenerate RH neutrino masses and strong fine tuned cancellations in the flavoured decay parameters and in the seesaw formula, implies a lower bound on the absolute neutrino mass scale, more specifically on the lightest neutrino mass. Though this lower bound can be evaded allowing for fine tuned cancellations, most of the models require values of the lightest neutrino mass that will be tested during the coming years, especially in the case of NO.

The plan of the paper is the following. In Section 2 we introduce some basic notation and review current experimental information on low energy neutrino parameters. In Section 3 we briefly discuss strong thermal leptogenesis. In Section 4 we show the existence of a lower bound on the neutrino masses under certain conditions. We also present results from a scatter plot analysis confirming the existence of the lower bound and at the same time showing how the bulk of models require values of the lightest neutrino mass that can be potentially tested in future years mainly with cosmological observations. In Section 5 we draw the conclusions.

## 2 General set up

We assume a minimal model of leptogenesis where the SM Lagrangian is extended introducing three RH neutrinos with Yukawa couplings and a Majorana mass term . After spontaneous symmetry breaking the Higgs vev generates a Dirac neutrino mass term . In the seesaw limit the spectrum of neutrino masses splits into a set of three heavy neutrinos with masses , approximately equal to the eigenvalues of , and into a set of light neutrinos with masses given by the seesaw formula

(2) |

written in a basis where both the Majorana mass and the charged lepton mass matrices are diagonal, so that can be identified with the PMNS leptonic mixing matrix.

From neutrino oscillation experiments we know two mass squared differences, and . Neutrino masses can then be either NO, with and , or IO, with and . For example, in a recent global analysis [9], and analogously in [10, 11], it is found . and

In order to fix completely the three light neutrino masses, there is just one parameter left to be measured, the so called absolute neutrino mass scale. This can be conveniently parameterised in terms of the lightest neutrino mass . The most stringent upper bound on comes from cosmological observations. A conservative upper bound on the sum of the neutrino masses has been recently placed by the Planck collaboration [1]. Combining Planck and high- CMB anisotropies, WMAP polarisation and baryon acoustic oscillation data, it is found . When neutrino oscillation results are combined, this translates into an upper bound on the lightest neutrino mass,

(3) |

showing how cosmological observations start to corner quasi-degenerate neutrinos.

For NO the leptonic mixing matrix can be parameterised as

(4) |

() while for IO, within our convention of labelling light neutrino masses, the columns of the leptonic mixing matrix have to be permuted in a way that

(5) |

The mixing angles, respectively the reactor, the solar and the atmospheric one, are now constrained within the following () ranges [10] for NO and IO respectively,

(6) | |||||

It is interesting that current experimental data also start to put constraints on the Dirac phase and the following best fit values and errors are found for NO and IO respectively,

(7) |

while all values are still allowed at .
^{2}^{2}2It is also useful to give the constraints on the angles and on in degrees:

## 3 Strong thermal leptogenesis and the -dominated scenario

Within an unflavoured scenario and assuming, conservatively, that only the lightest RH neutrinos thermalise, the strong thermal condition translates quite straightforwardly into a condition on the lightest RH neutrino decay parameter , where is the expansion rate and is the total decay width. Given a pre-existing asymmetry , the relic value after the lightest RH neutrino wash-out is simply given by [2, 12]

(9) |

where we are indicating with the abundance of any (extensive) quantity in a co-moving volume containing one RH neutrino in ultra-relativistic thermal equilibrium (so that ). The relic value of the pre-existing asymmetry would then result in a contribution to given by , taking into account the dilution due to photon production and the sphaleron conversion coefficient.

Imposing , where is the contribution coming from leptogenesis, immediately yields the simple condition , with

(10) |

where is assumed to be large, meaning that . Since , where , the requirement is a sufficient (but not necessary) condition for strong thermal leptogenesis.

When flavour effects are considered, the possibility to satisfy both successful leptogenesis, , and strong thermal condition, , relies on much more restrictive conditions [13], due to the 3-dim flavour space and to the fact that the RH neutrino wash-out acts only along a specific flavour component [14].

It is then possible to show [13] that only in a -dominated scenario [15],
defined by having GeV and ,
so that the observed asymmetry is dominantly produced by the RH neutrinos,
with the additional requirements
^{3}^{3}3In this way the asymmetry production from
decays occurs in the two-flavour regime [14, 16].
and that the asymmetry is dominantly produced in the tauon flavour,
one can have successful strong thermal leptogenesis.

In the -dominated scenario the contribution to the asymmetry from leptogenesis can be calculated as the sum of the three (charged lepton) flavoured asymmetries , [17, 18, 19]

(11) | |||||

where and . As we will show soon, the strong thermal condition implies and, therefore, in this case the contribution to the asymmetry from leptogenesis simply reduces to

(12) |

The baryon-to-photon number ratio from leptogenesis can then be simply calculated as . The flavoured decay parameters are defined as

(13) |

The ’s and the ’s can be regarded as the zero temperature limit of the flavoured decay rates into leptons, , and anti-leptons, in a three-flavoured regime, where lepton quantum states can be treated as an incoherent mixture of the three flavour components. They are related to the total decay widths by , with . The efficiency factors can be calculated using [20, 21]

(14) |

This is the expression for an initial thermal abundance but, since we
will impose the strong thermal leptogenesis condition, this will automatically select
the region of the space of parameters where there is no dependence on
the initial conditions anyway.
^{4}^{4}4Moreover in this case this analytical expression approximates the numerical result
with an error below .

Within the -dominated scenario the flavoured asymmetries, defined as , can be calculated in the hierarchical limit simply using [22]

(15) |

with and .

In the orthogonal parameterisation the neutrino Dirac mass matrix, in the basis where both charged lepton and RH neutrino mass matrices are diagonal, can be written as , where is an orthogonal matrix encoding the information on the properties of the RH neutrinos [23]. This parameterisation is quite convenient in order to easily account for the experimental low energy neutrino information. Barring strong cancellations in the seesaw formula, one typically expects . More generally, we will impose a condition , studying the dependence of the results on . In the orthogonal parametrisation the flavoured decay parameters can be calculated as

(16) |

The quantity can also be expressed in the orthogonal parameterisation,

(17) |

Now, we have finally to impose the strong thermal condition, and to this extent we need to calculate the relic value of the pre-existing asymmetry distinguishing two different cases.

### 3.1 Case

In the case the heaviest RH neutrino either, for , is not thermalised or it cannot in general wash-out completely the pre-existing asymmetry, as requested by the strong thermal leptogenesis condition. This is because the wash-out would occur in the one-flavour regime and, for a generic pre-existing asymmetry, the component orthogonal to the -flavour direction would survive. Therefore, without any loss of generality, we can simply neglect its presence. The relic value of the pre-existing asymmetry can then be calculated as [24] , with

(18) | |||||

In this expression
^{5}^{5}5Notice that in the limit
one has . Notice also that this expression
incorporates flavour projection [14]
and exponential suppression of the parallel components,
two effects that have been both confirmed within a
density matrix approach [19].
the quantities and
are the fractions of the pre-existing asymmetry in the tauon and
components respectively, where is the -orthogonal flavour component of the leptons
produced by decays, while
is the fraction
of -asymmetry that is first washed-out by the inverse processes in the tauon-orthogonal plane and
then by the inverse processes.

The terms , and , with , take into account the possibility of different flavour compositions of the pre-existing leptons and anti-leptons. This would lead to initial values of the pre-existing asymmetries that are not necessarily just a fraction of . The presence of these terms depends on the specific mechanism that produced the pre-existing asymmetry. For example in leptogenesis itself they are in general present, they are the so called phantom terms. However, this indefiniteness has just a very small effect on the results. If the -terms are not present, then in principle very special flavour configurations with could also lead to a wash-out of the pre-existing asymmetries without the need to impose . We will comment on this possibility but for the time being we will assume that these terms are present. In this case the condition of successful strong thermal leptogenesis translates into the straightforward set of conditions

(19) |

These conditions guarantee a washout of the electron and muon asymmetries,
only possible in the three-flavoured regime at GeV, and at the same time also a wash-out of the
tauon asymmetry in the two-flavoured regime. The latter is still compatible with a
generation of a sizeable tauon asymmetry from decays. This is the only possibility [13].
It should be noticed that in the -dominated scenario the existence of the heaviest RH neutrino
is necessary in order to have an interference of tree level decays with one-loop decay graphs
containing virtual yielding sufficiently large .
Therefore, within the -dominated scenario, where by definition
, one has a phenomenological reason to have at least
three RH neutrino species [15].
^{6}^{6}6
In the limit , when decouples and a two RH neutrino scenario is effectively recovered with ,
one has (cf. eq. (15)). In this limit the only possibility to realise successful leptogenesis
is to have sizeable asymmetries from the interference terms with the lightest RH neutrinos that we neglected
when we wrote eq. (15).
These terms are and
successful leptogenesis necessarily requires in the end a lower bound
[25]. However, then in this case
the -produced asymmetry not only cannot be neglected but typically dominates on the -produced asymmetry
and moreover, more importantly for us, strong thermal leptogenesis cannot be realised [13].
This well illustrates that in the -dominated scenario,
the presence of a (coupled) is necessary for successful leptogenesis.

### 3.2 Case

If , then the heaviest RH neutrinos can contribute to wash-out the tauon component together with the next-to-lightest RH neutrinos . In this way, for the relic value of the pre-existing asymmetry, one obtains ()

(20) | |||||

where we defined and . The inclusion of the -washout relaxes the condition to . In this way one can have strong thermal leptogenesis with lower values of and so the condition of successful leptogenesis can be more easily satisfied. Therefore, in this case the constraints from successful strong thermal leptogenesis could potentially get relaxed.

## 4 Lower bound on neutrino masses

In this Section we show finally that the strong thermal condition implies, for sufficiently large pre-existing asymmetries and barring fine tuned conditions on the values of the flavour decay parameters and in the seesaw formula, the existence of a lower bound on the lightest neutrino mass and, more generally, a strong reduction of the accessible region of parameters for .

The main point is that the conditions and can be satisfied simultaneously only for sufficiently large values of .

### 4.1 Case

Let us start discussing the more significant case , when, as already pointed out, the wash-out can be neglected. The cases of NO and IO need also to be discussed separately. Let us start from NO.

#### 4.1.1 NO neutrino masses

We want to show that the conditions and can be satisfied simultaneously, without fine-tuned conditions, only if is sufficiently large. Let us start by analysing . The general eq. (16) for the ’s specialises into

(21) |

From this expression, anticipating that the lower bound falls into a range of values so that we can approximate and , we can write

(22) |

where is some generic phase. If we now insert this expression into the expressions for and , we can impose ()

(23) |

where we defined and such that

(24) |

From this condition one obtains a lower bound on (),

(25) |

when , where we defined

(26) |

Because of the smallness of the reactor mixing angle there are two consequences: the first is that the maximum is found for and the second is that, imposing , both the two terms in proportional to are suppressed and in this way there is indeed a lower bound for a sufficiently small value of .

In the left panel of Fig. 1 we have conservatively taken
and plotted at
for as a function of the Dirac phase .
^{7}^{7}7We used Gaussian ranges for the mixing angles within as in eq. (6),
except for the atmospheric mixing angle for which we used a Gaussian distribution
, i.e. centred on the maximal mixing value since on this angle results
are still unstable depending on the analysis. We have also used, in the scatter plot analysis as well,
,
corresponding to a flavour blind pre-existing asymmetry. Notice in any case that
results depend only logarithmically on these parameters, so they are insensitive to a precise choice.

At we find (top right panel) while for we obtain , showing how a future determination of the Dirac phase could tighten the lower bound. The lower bound becomes more stringent for and we find . On the other hand for the lower bound gets relaxed and we obtain . For one can easily verify that the condition is not verified and there is no lower bound on .

In order to verify the existence of the lower bound, to test the validity of the analytic estimation and to show in more detail the level of fine tuning involved in order to saturate the lower bound, we performed a scatter plot analysis in the space of the 13 parameters (, 6 in , 6 in ) for . The results are shown in Fig. 1. for three values of (respectively the red, green and blue points). One can see that for the minimum values of in the left panel at different values of are much higher than the analytic estimation (one has to compare the red points with the red solid line). The reason is due to the fact that the lower bound is saturated for very special choices of such that are as close as possible to the maximum value but at the same time not to suppress too much the asymmetry needed to have successful leptogenesis. This is confirmed by Fig. 2 where in the three panels we have plotted , and for . We have made a focused search (by fine-tuning the parameters) managing to find a point (the red diamond) where is very close to the lower bound. For this point gets considerably reduced since it corresponds to a situation where the term in the flavoured decay parameters becomes negligible and the strong thermal condition is satisfied for a very special condition, basically the eq. (22) when the terms are neglected in the right-hand side and become maximal, that leads to a asymmetry suppression.

We have also performed a scatter plot letting the mixing angles to vary within the whole range of physical values with no experimental constraints. In the right panel of Fig. 1 we show the results in the plane . One can see how the smallness of is crucial for the existence of the lower bound. This can be well understood analytically considering that in the expression for there are two terms (cf. eq. (26)).

In Fig. 3 we also show the results for the values of the three () and for , the four relevant flavoured decay parameters, for . First of all one can see how the values of the flavoured decay parameters respect the strong thermal conditions eq. (19). However, the most important plot is that one for , showing how for values the maximum value of gets considerably reduced until it falls below , indicated by the horizontal dashed line for , at the lower bound value (very closely realised by the red diamond point). It is also clear that already below the possibility to realise strong thermal leptogenesis requires a high fine tuning in the parameters since in this case for large asymmetries and not too unreasonably high values of .

This is well illustrated in Fig. 4 where we plotted the distribution of the values from the scatter plots for and for .

One can see that there is a clear peak around . One can also see that the distributions rapidly tend to zero when . For example, for our benchmark value and for , it can be noticed how more than of points falls for values (the value quoted in the abstract). Even for one still has that the of points satisfying successful strong thermal leptogenesis is found for . It is also interesting to notice how this constraint gets only slightly relaxed for lower values of the pre-existing asymmetry. Only for one obtains that of points fall at . For , not shown in the plots, this would decrease at (untestable) values . This provides another example of how, more generally, leptogenesis neutrino mass bounds tend to disappear in the limit [26]. It should be however said how large values of imply high cancellations in the see-saw formula such that the lightness of LH neutrinos becomes a combined effect of these cancellations with the the see-saw mechanism and they are typically not realised in models embedding a genuine minimal type I see-saw mechanism.

Clearly the results on the distributions in Fig. 4 depend on the orthogonal matrix parameterisation that
we used in order to generate the points on the scatter plots but they provide quite a useful indication
of the level of fine tuning required to satisfy successful strong thermal leptogenesis
for values of the lightest neutrino mass below . In any case it is fully
explained by our analytical discussion and
by the plot of the maximum of values that is independent of the specific parameterisation.
We also double checked the results producing scatter plots for two different parameterisations.
In a first case we used the usual parameterisation of
the orthogonal matrix in terms of complex rotations described by three complex Euler angles,
that, however, has the drawback not to be flavour blind. In a second case
we used a parameterisation based on the isomorphism between the group of complex orthogonal
matrices and the Lorentz group. We did not find any appreciable difference.
^{8}^{8}8As a technical detail it is probably worth to stress that
for the first time we have randomly generated complex orthogonal matrices
(about 10 million of points for both parameterisations) within the whole 6-dim parameter space, without any restriction
(except for the bound ).

#### 4.1.2 IO neutrino masses

Let us now discuss the case of IO. The analytical procedure we have discussed for NO can be repeated in the IO case and one finds the same expression eq. (25) for the lower bound on where, however, one has to replace and .

The replacement tends to push all values to much higher values and this is indeed what happens for . If one considers again the quantity (cf. eq. (26)) it is possible to check that this time one has always for . On the other hand this time the value of has to be fine tuned in order to be greater than . The reason is that for IO there is now a cancellation in the quantity that suppresses though not as strongly as in the NO case. Indeed one finds now that , the condition for the existence of the lower bound, holds only for . This implies that the lower bound on for IO is much looser than for the NO case. This result is again confirmed by a scatter plot analysis. The results are shown in Fig. 5 directly in the form of the distribution of probabilities for .

One can see how this time there is no lower bound for and we could obtain points satisfying successful strong thermal leptogenesis with arbitrarily small .

However, the fact that is just slightly higher than (this time ) still implies that one has to fine tune the parameters in the orthogonal matrix in order to maximise , and this still acts in a way that in the limit the density of points drops quickly. For example one can see th