# Leptogenesis, Neutrino Mixing Data and the Absolute Neutrino Mass Scale
^{1}^{1}1Compendium of [1] and [2] mostly based
on [3] with some new results in 3.4, 3.8 and 3.9.

\address
IFAE, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain

Recent developments in thermal leptogenesis are reviewed. Neutrino mixing data favor a simple picture where the matter-anti matter asymmetry is generated by the decays of the heavy RH neutrinos mildly close to thermal equilibrium and, remarkably, in the full non relativistic regime. This results into predictions of the final baryon asymmetry not depending on the initial conditions and with minimized theoretical uncertainties. After a short outline of a geometrical derivation of the asymmetry bound, we derive analytic bounds on the lightest RH neutrino mass and on the absolute neutrino mass scale. Neutrino masses larger than are not compatible with the minimal leptogenesis scenario. We discuss how the results get just slightly modified within the minimal supersymmetric standard model. In particular a conservative lower bound on the reheating temperature, , is obtained in the relevant effective neutrino mass range . We also comment on the existence of a ‘too-short-blanket problem’ in connection with the possibility of evading the neutrino mass upper bound.

## 1 Introduction

Cosmic rays and CMBR observations indicate that our observable Universe is baryon asymmetric [4]. Moreover the observation of the acoustic peaks in the power spectrum of CMBR [5], combined with large scale structures observations [6], provide a precise and robust measurement of such an asymmetry that can be expressed in terms of the baryon to photon number ratio at the recombination time,

(1) |

in very good agreement with the latest determination from (NACRE updated) Standard BBN and primordial Deuterium measurements that give [7]

(2) |

At the same time there is a growing evidence that an inflationary stage occurred during the early Universe. In this case this would have diluted any pre-existing initial asymmetry to a level many orders of magnitude below the measured value, thus requiring an explanation of the observed baryon asymmetry in terms of a dynamical generation, the aim of a model of baryogenesis that necessitates the accomplishment of the three famous Sakharov’s conditions: and violation, violation and departure from thermal equilibrium. Within the Standard Model all three conditions are fulfilled, yet the observed value is too large to be explained and therefore a successful model of baryogenesis requires some new physics ingredient. A host of models have been proposed since the first Sakharov idea [8]. Some examples of typologies of baryogenesis models are: Planck scale baryogenesis, baryogenesis from phase transitions, Affleck-Dine models, baryogenesis from black holes evaporation, models of spontaneous baryogenesis [9].

Even though leptogenesis [10] and GUT baryogenesis [11, 12] exhibit, from a particle physics point of view, substantial differences, they can be jointly regarded as two different examples belonging to the oldest class of models of baryogenesis from heavy particle decays. Such a classification privileges the thermodynamical aspect enlightening general properties that do not depend on the specific particle physics framework. We will thus discuss the kinetic theory of heavy particle decays in the first part, while in the second part we will see how leptogenesis is a specific remarkable example in which the new physics ingredient is provided by the seesaw mechanism and such that the observed baryon asymmetry is nicely related to neutrino mixing data.

## 2 Baryogenesis from heavy particle decays

### 2.1 Out-of-equilibrium decays

Let us consider a self-conjugate heavy () particle whose decays are asymmetric, in such a way that the decaying rate into particles, , is in general different from the decaying rate into anti-particles, , and such that the single decay process into particles (anti-particles) violate by a quantity (). The asymmetry parameter is then conveniently defined as

(3) |

For a joint discussion of baryogenesis () and leptogenesis () models, it is useful to introduce the quantity . The total decay rate is the product of the total decay width, , times the averaged dilation factor

(4) |

Sphaleron processes, while inter-converting and separately, leave unchanged [13] and for this reason the kinetic equations get much simpler if the evolution is tracked instead of the separate or evolution. Moreover it is convenient to use, as an independent variable, the quantity and to introduce the decay factor . Another useful choice is to track the number of particles, , and the amount of the asymmetry, , in a portion of comoving volume normalized in such a way to contain, averagely in ultra-relativistic thermal equilibrium, just one particle (i.e. ).

The simplest case is when the life-time, , is much longer than the age of the Universe, , at , when the particles become non relativistic. In this way decays will occur when the temperature is much below the mass and the -production from inverse decays, or other possible processes, is Boltzmann suppressed. In this situation decays are the only relevant processes and the kinetic equations for the X-abundance and the asymmetry are particularly simple to be written,

(5) | |||||

(6) |

and solved,

(7) | |||||

(8) |

The solutions can be fully described just in terms of the decay parameter

(9) |

in terms of which . The dilation factor, averaged on the Boltzmann statistics, is simply approximated by [3]

(10) |

a useful simple expression that makes possible to solve analytically the integral in the Eq. (8), yielding the result

(11) |

In particular the final asymmetry is given by

(12) |

The baryon to photon ratio at recombination can then be obtained dividing by the number of photons at recombination (about thirty times the number of photons at the onset of decays) and taking into account that sphalerons will convert only a fraction of the asymmetry into a baryon asymmetry. In this way one can write:

(13) |

It is useful to introduce the efficiency factor defined as the ratio of the asymmetry produced from the decays, excluding the contribution from a possible initial quantity, to the asymmetry, i.e.

(14) |

In the case of out of equilibrium decays one has and the Eq. (7) can be re-casted as

(15) |

This definition is such that the final efficiency factor, , is equal to unity in the case of an initial thermal abundance with . In Fig. 1 we show two examples of out of equilibrium decays, for and , assuming an initial thermal abundance () and zero initial asymmetry (). The numerical results are compared with the analytic expression (cf. (11)).

The out-of-equilibrium picture is an efficient way to produce an asymmetry from decays. However it relies on the possibility that an initial abundance was thermalized by some unspecified mechanism at and that one can neglect a possible generated during or after inflation and before the onset of decays. Therefore, it is evident that this picture is plagued by a strong sensitivity to the initial conditions and hence it requires to be complemented with a model able to specify them, for example a detailed description of the inflationary stage.

### 2.2 Inverse decays

The out-of-equilibrium picture is strictly valid only in the limit .
If one defines as the value such that the life time
coincides with the age of the Universe () then, for ,
one has . Thus for the ’s will decay
for and the inverse decays have to be
taken into account. The kinetic equations
(5) and (6)
are then generalized in the following way [12, 14, 15, 16, 17]
^{2}^{2}2The equations (16) and (17) are actually not only
accounting for decays and inverse decays but also for the
real intermediate state contribution from
scattering processes. This term exactly cancels a non conserving
term from inverse decays that would otherwise lead to an
un-physical asymmetry generation in thermal equilibrium [18].

(16) | |||||

(17) |

In the equation for the second term accounts for the inverse decays that, remarkably, can now produce the ’s. On the other hand one can see that a new term appears in the second equation for the asymmetry too, a wash-out term that tends to destroy what is generated from the decays. This term is controlled by the (inverse decays) wash-out factor given by

(18) |

where is the number of baryons or leptons in the decay final state ( in the case of leptogenesis). Note that the decay parameter is still the only parameter in the equations and thus the solutions will still depend only on . They can be again worked out in an integral form [12]. In the case of the asymmetry one can write the final asymmetry as

(19) |

where now the efficiency factor is given by the integral

(20) |

In the limit the out-of-equilibrium case is recovered. In general one can see that the wash-out has the positive effect to damp a pre-existing asymmetry but also the negative one to damp the same asymmetry generated from decays, thus reducing the efficiency of the mechanism. A quantitative analysis is crucial and it is very useful to discuss separately the regime of strong wash out for and the regime of weak wash-out for .

### 2.3 Strong wash-out regime

The strong wash-out regime is characterized by the existence, for , of an interval such that and thus such that inverse decays are in equilibrium. Practically all the asymmetry produced at is washed-out including, remarkably, an initial one. Moreover the calculation of the residual asymmetry is made very simple by the possibility to use the close equilibrium approximation given by

(21) |

In this way the integral in the Eq. (20) can be easily evaluated [12, 3]. Indeed, this can be regarded as a Laplace integral, that means an integral of the form

(22) |

that receives a dominant contribution only from a small interval centered around a special value such that . In this way one can use the approximation of replacing with in the Eq. (20). With this approximation and assuming the integral can be easily solved obtaining

(23) |

For large and for
this expression coincides with that one found in [12]
^{3}^{3}3Note however that the definition (9) for
has to be used instead of ..
The calculation of the important quantity
proceeds from its definition, ,
approximately equivalent to the equation

(24) |

This is a transcendental algebraic equation and thus one cannot find an exact analytic solution (see [3] for an approximate procedure). However the expression

(25) |

provides quite a good fit that can be plugged into the Eq. (23) thus getting an analytic expression for the efficiency factor. At vary large this behaves as a power law . In Fig. 2 we compare the analytic solution for (cf. (23)) with the numerical solution (for ). One can see how for the agreement is quite good. Note that the Eq. (24) implies that for large values of K one has , that particular value of corresponding to the last moment when inverse decays are in equilibrium (). In this way almost all the asymmetry produced for is washed-out and most of the surviving asymmetry is produced in the period just around the inverse decays freeze out, simply because the abundance gets rapidly Boltzmann suppressed. An example of this picture is illustrated in Fig. 3 for (from [3]).

Instead of the abundance we plotted the deviation from the equilibrium value, the quantity . The deviation grows until the ’s decay at , when it reaches a maximum, and decrease afterwards when the abundance stays close to thermal equilibrium. Correspondingly the asymmetry grows for , reaching a maximum around , and then it is washed-out until it freezes at . The evolution of the asymmetry can induce the wrong impression that the residual asymmetry is some fraction of what was generated at and that one cannot relax the assumption without reducing considerably the final value of the asymmetry. Actually what is produced is also very quickly destroyed. A plot of the quantity , as defined in the Eq. (22) and shown in Fig 4 (from [3]), enlightens some interesting aspects.

This is the final asymmetry that was produced in a infinitesimal interval around . It is evident how just the asymmetry that was produced around survives and, for this reason, the temperature can be rightly identified as the temperature of baryogenesis for these models. It also means that in the strong wash out regime the final asymmetry was produced when the particles were fully non relativistic implying that the simple kinetic equations (16) and (17), employing the Boltzmann approximation, give actually accurate results and corrections from use of the exact quantum statistics can be safely neglected.

This is not the only nice feature of the strong wash-out regime. Since any asymmetry generated for gets washed-out, one can also rightly neglect any pre-existing initial asymmetry . At the same time the final asymmetry does not depend on the initial abundance.

In Fig. 5 we show how even starting from a zero abundance, the ’s are rapidly produced by inverse decays in a way that well before the number of decaying neutrinos is always equal to the thermal number [20]. The final asymmetry does not even depend on the initial temperature as far as this is higher than and thus if one relaxes the assumption to , the final efficiency factor gets just slightly reduced (for example for this is reduced approximately by ).

Summarizing we can say that that in the strong wash out regime the reduced efficiency is compensated by the remarkable fact that, for , the final asymmetry does not depend on the initial conditions and all non relativistic approximations work very well. These conclusions change quite drastically in the weak wash-out regime.

### 2.4 Weak wash-out regime

For one can see that rapidly tends to unity (cf. (25)). In Fig. 2 the analytic solution for the efficiency factor, Eq. (23), is compared with the numerical solution. It can appear surprising that, in the case of an initial thermal abundance, the agreement is excellent not only at large , but also at small , with some appreciable deviation just around . The reason is that when the wash-out processes get frozen, the efficiency factor depends only on the initial number of neutrinos and not on its derivative and thus the approximation Eq. (21) introduces a sensible error only in the transition regime .

The Eq. (23) can be easily generalized to any value of the initial abundance until one can neglect the ’s produced by inverse decays. More generally, one has to calculate such a contribution and it is convenient to consider the limit case of a zero initial abundance. The production lasts until , when the abundance is equal to the equilibrium value, such that

(26) |

At this time the number of decays equals the number of inverse decays. For decays can be neglected and the Eq. (16) becomes

(27) |

For one then simply finds

(28) |

In the weak wash out regime the equilibrium is reached very late, when neutrinos are already non relativistic and . In this way one can see that the number of reaches, at , a maximum value given by

(29) |

It is possible to interpolate between the two asymptotical regimes getting a global solution for any . For inverse decays can be neglected and the ’s decay out of equilibrium in a way that

(30) |

Let us now consider the evolution of the asymmetry calculating the efficiency factor. Its value can be conveniently decomposed as the sum of two contributions, a negative one, , generated at , and a positive one, , generated at . In the limit of zero wash-out we know that the final efficiency factor must vanish, since we are assuming an initial zero abundance. This implies that the negative and the positive contributions have to cancel each other. The effect of wash-out is to suppress the negative contribution more than the positive one, in a way that the cancellation is only partial. In the weak wash-out regime it is possible in first approximation to neglect completely the wash-out at . In this way it is easy to derive from the Eq. (20) the following expression for the final efficiency factor:

(31) |

One can see how it vanishes at the first order in and only at the second order one gets [20].

### 2.5 Final efficiency factor: summary

Generalizing the procedure seen for the strong wash-out it is possible to find a global solution for valid for any . The calculation proceeds separately for and and the final results are given by

(32) |

and

(33) |

The function extends, approximately, the definition of to any value of

(34) |

The sum of the Eq.’s (33) and (32) is plotted, for , in Fig. 2 (short-dashed line) and compared with the numerical solution (solid line).

We can now outline some conclusions about a comparison between the weak and the strong wash-out regimes. A large efficiency in the weak wash-out regime relies on some unspecified mechanism that should have produced a large (thermal or non thermal) abundance before their decays. On the other hand the decrease of the efficiency at large in the strong wash-out regime is only (approximately) linear and not exponential [12]. This means that for moderately large values of a small loss in the efficiency would be compensated by a full thermal description such that the predicted asymmetry does not depend on the initial conditions, a nice situation that resembles closely the Standard Big Bang Nucleosynthesis scenario for the calculation of the primordial nuclear abundances.

## 3 Leptogenesis

Let us see now how the results that hold for generical baryogenesis models from heavy particle decays get specialized in the case of leptogenesis [10]. This is the cosmological consequence of the seesaw mechanism, explaining the lightness of the ordinary neutrinos through the existence of three new heavy RH neutrinos with masses respectively , much larger than the electroweak scale. The simple seesaw formula,

(35) |

relates the neutrino mixing matrix to the RH neutrino mass matrix and to the Dirac neutrino mass matrix generated by the Yukawa coupling matrix , where is the Higgs vacuum expectation value. Both light and heavy neutrinos are predicted to be Majorana neutrinos. All mass matrices are in general complex and this provides a natural source for the asymmetry while the new RH neutrinos are the natural candidates to play the role of the particles. In this case things are apparently more complicated since there are three of them. We will assume that the decays and inverse decays of the two heavier neutrino decays do not influence the value of the final asymmetry. This assumption holds for example either if the asymmetry produced by the two heavier RH neutrinos is negligible or if this is produced and then washed out by the inverse decays of the lightest (heavy RH). In this way we can straightforwardly apply the general picture of baryogenesis from decays to leptogenesis, with the ’s playing the role of the particles.

### 3.1 Decay parameter and neutrino masses

The total decay width is given by

(36) |

where the effective neutrino mass is defined as [15]

(37) |

It is then easy to see that the decay parameter is related to [21] by the following relation

(38) |

where the equilibrium neutrino mass can be written as

(39) |

with the quantity given by

(40) |

It is quite non trivial that the value of is close
to the neutrino mixing mass scales and we will show soon the relevance
of this result. For the moment note that the value of
is independent on the well known success of the seesaw mechanism
in explaining the atmospheric and solar neutrino mass scales and this is
why we wrote in a sort of seesaw-like form, introducing
the scale . Apart from the very generical consideration that
the logarithm of is expected to be close to the Planck scale,
this is not related to the grand unified scale, rather to the expansion rate
at the baryogenesis time
^{4}^{4}4It is then quite curious that the value
of is just the value of the supersymmetric unification scale..

Let us now assume that the simple decays plus inverse decays picture studied in the previous section is a good approximation of leptogenesis. It is then crucial to determine the value of the the effective neutrino mass , and thus, from the Eq. (38), the value of the decay parameter , in order to answer the important question whether leptogenesis lies in the strong or in the weak wash-out regime.

It is always possible to work in a basis in which the heavy neutrino mass matrix is diagonal, such that . Moreover one can also simultaneously diagonalize the light neutrino mass matrix by mean of the unitary MNS matrix , such that

(41) |

In this way the seesaw formula (35) gets specialized in the following way:

(42) |

This expression can be also re-casted as an orthogonality condition,

(43) |

for the matrix defined as [22]

(44) |

and whose matrix elements are then simply given by

(45) |

where . The matrix is fully determined by three complex parameters. Four of them are needed to fix the three first column entries , and , particularly important for leptogenesis. This because if one inverts the relation (44), in a way to get an expression of in terms of , and plugs it into the effective neutrino mass definition (cf. (37)), then one easily gets [23]

(46) |

From the orthogonality of it follows that . This is the only fully model independent restriction on . For configurations such that

(47) |

one has . Models with rely on the possibility of strong phase cancellations.

Neutrino mixing data provide two important pieces of information on the neutrino mass spectrum. In the case of normal hierarchy one has and . In the case of inverted hierarchy and . The third, still undetermined, independent information, the absolute neutrino mass scale, can be conveniently expressed in terms of the lightest neutrino mass . The two heavier neutrino masses are then given, for normal (inverted) hierarchy, by

(48) | |||||

(49) |

where we defined . The latest measurements give [24]

(50) |

and for solar neutrinos [25]

(51) |

from which it follows that

(52) |

These relations imply that for neutrinos are quasi-degenerate (), whereas for they are hierarchical ().

For fully hierarchical neutrinos () there is practically no restriction on . However the case requires and . This situation cannot be excluded [17] but, because of the observed large mixing angles in the mixing matrix , it relies on a fine tuning between the and matrix elements (cf. (44)), such that the off-diagonal terms are very small. This qualitative and general argument is supported by different investigations on specific models or classes of models for which typically one finds [26]. Therefore, in the case of normal hierarchy one has that the favored range for the value is given by , that in terms of the decay parameter (cf. (38)) gets translated into the range

(53) |

while for inverted hierarchy the situation is even simpler since
and .
One thus arrives to the interesting conclusion that
neutrino mixing data favor leptogenesis to lie in a mildly strong wash out regime,
strong enough to benefit from the advantages we discussed, independence on the initial conditions
plus minimal theoretical uncertainties, but not too much to result in an untenable
efficiency loss. This conclusion derives because both the
two independent experimental quantities, and , are about
ten times and so now one can better appreciate the nice matching
of the theoretical quantity with the experimental data
^{5}^{5}5Note that this is also a consequence of the recent exclusion of the low solution
in the solar neutrino data, that would have implied ..
In the range a good fit of the
final efficiency factor (cf. Eq. (23)) is given by the power law

(54) |

shown in Fig. 2 (dot-dashed line). These conclusions hold under the assumption that leptogenesis is well approximated by the simple decays plus inverse decays picture and we have now to verify whether they are drastically modified or just corrected by the account of scatterings and processes.

### 3.2 Scatterings

The ’s can also be destroyed or produced in scatterings involving the top quark. These are mediated by the Higgs and can occur in the s channel, like , or in the t channel, like . The account of these processes modify the kinetic equations (16) and (17) in the following way:

(55) | |||||

(56) |

Note that scatterings have two effects: they contribute both to the neutrino production (the function) and to the wash-out (the function).

The first one is important in the weak wash-out regime. As one can see from the Eq.’s (55) and (56), the production of the ’s from the function is not associated to a production of the asymmetry, simply because these processes do not violate . In Fig. 5 (from [3]) we show an example of production for

(), comparing the case when scatterings are included with the case when they are neglected. It can be seen how at the number of neutrinos is approximately doubled while the final asymmetry is two orders of magnitude larger. The reason is that the neutrino production from the scatterings is not associated to a production of a negative asymmetry. On the other hand all produced neutrinos yield a positive contribution when they decay. The expression (31) for the final efficiency factor in the weak wash-out regime gets thus modified in the following way at the first order in K

(57) |

If scatterings are switched off the negative and the positive contribution cancel at the first order like we saw already. If the positive term is enhanced while the negative one remains unchanged and in this way the sum does not vanish any more. Hence this effect makes more efficient the asymmetry production at small , without having to assume an initial thermal abundance. There is however a drawback. The final result is quite sensitive to the theoretical assumptions. The scattering cross section depends on the ratio, , of the Higgs mass to the RH neutrino mass. The case depicted in Fig. 5 is for . For smaller values of this ratio the result does not change much. However it has been recently pointed out [27] that the Higgs mass is better described by its thermal mass such that . The relevant values of for neutrino production are and so the ratio . Such an high value has the effect to suppress heavily the term and the suppression is made even stronger by the account of the running of the top Yukawa coupling at high temperature. In this way the simple decays plus inverse decays picture is practically recovered.

On the other hand in [28, 27] it has been noticed how scatterings involving gauge bosons should also be included. These scatterings yield an additional contribution to the S function such that the final result is between a situation where scatterings are neglected and one where scatterings involving top quark and small are taken into account. The conclusion is that in the weak-wash out regime the theoretical uncertainties are such that it seems that any result between the simple decays plus inverse decays picture, for which or a behavior (cf. Eq. (57) ) cannot be firmly excluded at the moment. These large theoretical uncertainties, represented in Fig. 6 with the short-dashed region, are in addition to the model dependence in the description of the initial conditions.

In the strong wash-out regime all difficulties get considerably reduced. The theoretical uncertainties in the description od scatterings can change the final efficiency factor no more than and this is clearly shown in Fig. 6, where at large the (thin solid line) range shrinks considerably compared to the (short-dashed line) range at small . This because the thermal abundance limit is saturated at anyway and therefore the number of decaying neutrinos does not depend on the function. A residual source of uncertainty is still present because of the scattering contribution, , to the wash-out. The effect of this term is however small, for the simple reason that in the strong wash out regime the surviving asymmetry is produced sharply around and at such low temperatures inverse decays are dominant compared to scatterings. The conclusion is that in the strong wash-out regime the simple decays plus inverse decays picture does not get modified by scatterings within the theoretical uncertainties. It has been also pointed out [29] that an accurate description of the dynamics of sphalerons in converting the lepton number into a baryon number is expected to lead to a suppression of the final asymmetry of a factor and since this is currently neglected it gives an additional contribution to the theoretical uncertainties. Taking into account all these effects, an expression for the final efficiency factor in the strong wash out regime that accounts for the theoretical uncertainties is given by the power law [3]

(58) |

The central value corresponds to the curve represented in Fig. 5 with circles
(more precisely this is obtained for a power law ),
while the range that is spanned by the error, corresponds approximately
to the thin solid line area. The upper values of this range is the power law
(54), that well describes the simple decays plus inverse decays
picture where the wash-out from scatterings is neglected
^{6}^{6}6The result obtained in [27] corresponds to this situation because
at the Higgs thermal mass suppresses the wash-out from scatterings
involving the top quark while the contribution from scatterings involving gauge bosons
is negligible..

### 3.3 processes

There is another important contribution to the wash-out term arising from the processes like and mediated by the RH neutrinos. In the non relativistic regime this contribution tends simply to [30]

(59) |

with , and dominates on the other Boltzmann suppressed wash-out terms arising from inverse decays and scatterings. A well known problem is that at temperatures one has to be sure that the cross section of processes does not double count the on-shell contribution already accounted by inverse decays followed by decays to a final state with opposite lepton number (i.e. ). In [17] it has been found that the subtraction procedure usually employed in the previous literature gives arise to a washout term that is very well approximated by the asymptotical non-relativistic limit Eq. (59) plus a term that is just half the washout from inverse decays. In [27] this second term has been shown to be spurious and to disappear when a proper subtraction procedure is employed. This result has been confirmed in [3]. Therefore the effect of the processes is entirely well approximated by its non-relativistic limit. It is easy to see that for , this term can be neglected. Thus, for sufficiently small neutrino masses and in the strong wash-out regime, we can conclude that leptogenesis is well approximated by a simple decays plus inverse decays picture.

### 3.4 asymmetry and seesaw geometry

So far we concentrated on the kinetic theory of leptogenesis and we have seen how neutrino mixing data favor a very simple regime in which predictions are model independent and theoretical uncertainties are minimized. We have now to answer the crucial question whether the resulting final asymmetry can explain the measured CMB value (cf. (1)). The thermodynamical point of view, i.e. the efficiency factor, is not enough to answer this question, since one needs to know the value of the asymmetry too. This is a specific leptogenesis issue that concerns what can be called the seesaw geometry.

A perturbative calculation from the interference between tree level and vertex plus self energy one-loop diagrams yields [31]

(60) |

The function , describing the vertex contribution, is given by

(61) |

while the function , describing the self-energy contribution, is given by

(62) |

In the limit , corresponding to have a mild RH neutrinos mass hierarchy with , one has

(63) |

In this limit and barring strong phase cancellations [32] the expression (60) simplifies into [34]

(64) |

Replacing with (cf. (45) ) one then gets [35]

(65) |

where we introduced the convenient quantity

(66) |

The final asymmetry is proportional to the product of the asymmetry times the final efficiency factor that, in the simplified decays plus inverse decays picture, depends only on the effective neutrino mass . The expression (65) shows that the asymmetry depends on the three complex numbers and thus it introduces a model dependence in the prediction of the final asymmetry that one was hoping to have removed in the calculation of the final efficiency factor. It is however possible to maximize the absolute value of the asymmetry respect to the ‘geometrical’ parameters , thus finding a non trivial maximum depending only on , and . One can then define an effective leptogenesis phase such that the expression (66) can be re-casted in the following way

(67) |

The maximum of the absolute value of the asymmetry and of the function are thus realized for those particular geometrical configurations, corresponding to some ’s values, such that . A general procedure for the calculation of and is presented in [36]. Here we just sketch some general features and describe two particularly interesting limit cases.

If one represents the three in the complex plane, the orthogonality condition fixes the sum of the three to start from the origin and to end up onto the real axis at the point Re, as shown in Fig. 8 for a generic configuration (solid line arrows).

Using the orthogonality condition, defining and using the definition of (cf. 46), this can be re-casted as

(68) |

with . The absolute value of has to be maximized for constant. In general one always finds that [35, 39]

(69) |

with , and .

An interesting limit case is that of fully hierarchical neutrinos for . In this case and there is no global suppression. Moreover one has and thus, for any change configuration such that and are constant, the quantity is also constant while can be arbitrarily modified. Hence is maximized for configurations such that . It is then easy to see that it is further maximized for and , corresponding to the configuration shown in Fig.8 with dotted line arrows. In this case one has very simply . Therefore, the case corresponds, for a fixed , to an absolute maximum of the asymmetry given by [37, 35] (cf. (65) and (66))

(70) |

Note that with this last definition of , together with the expressions (67) and (69), the Eq. (65) for the asymmetry can be re-casted like

(71) |

showing the sequence of different maximization steps.

In the quasi-degenerate limit
the expression (46) for becomes simply .
Thus the condition is equivalent to select all those configurations
for which is constant.
Hence it is straightforward to conclude that is maximum for a configuration
such that and , shown in Fig. 8 with dashed line arrows.
Using that in the quasi degenerate limit
, one obtains
^{7}^{7}7This limit expression has been first shown in
[32] using the approximation . Here we derived it
in a more general way.

(72) |

Note that in both the two limit cases the maximum asymmetry is obtained for configurations such that . It is possible to show that this result holds in general, for any value of [36].

Therefore, for maximal asymmetry (), one can still express all predictions in terms just of and . In particular it is possible to express the CMB constraints, for all neutrino models, just in terms of these three parameters. For specific models it can happen of course that and the constraints get, in general, more restrictive.

### 3.5 CMB bound

From the Eq.’s (13) and (19) one obtains for the predicted baryon to photon number ratio

(73) |

where the quantity is defined as

(74) |

In the Standard Model case one has , while the number of photons at recombination, assuming a standard thermal history, is given by

(75) |

and thus .

The maximum baryon asymmetry is defined like the asymmetry corresponding to the maximum asymmetry

(76) |

The CMB bound is then simply equivalent to require

(77) |

and therefore will yield constraints on the space of the three parameters , and .

### 3.6 Lower bounds on the lightest RH neutrino mass and on the reheating temperature

We have seen that the absolute maximum of the asymmetry is obtained for . For the function suppresses the asymmetry [35]. Furthermore the wash-out term gets enhanced when the absolute neutrino mass scale increases (cf. (59)). Therefore, the maximum baryon asymmetry is maximal when . In this case the allowed region in the space of the parameters and and compatible with the CMB constraint is maximum [17] and one finds an interesting lower bound on the value [35, 17] just plugging the expression (70) into the CMB constraint (cf. (77))