Effects of Lightest Neutrino Mass in Leptogenesis
SISSA and INFN-Sezione di Trieste, Trieste I-34014, Italy
The effects of the lightest neutrino mass in “flavoured” leptogenesis are investigated in the case when the CP-violation necessary for the generation of the baryon asymmetry of the Universe is due exclusively to the Dirac and/or Majorana phases in the neutrino mixing matrix . The type I see-saw scenario with three heavy right-handed Majorana neutrinos having hierarchical spectrum is considered. The “orthogonal” parametrisation of the matrix of neutrino Yukawa couplings, which involves a complex orthogonal matrix , is employed. Results for light neutrino mass spectrum with normal and inverted ordering (hierarchy) are obtained. It is shown, in particular, that if the matrix is real and CP-conserving and the lightest neutrino mass in the case of inverted hierarchical spectrum lies the interval , the predicted baryon asymmetry can be larger by a factor of than the asymmetry corresponding to negligible . As consequence, we can have successful thermal leptogenesis for eV even if is real and the only source of CP-violation in leptogenesis is the Majorana and/or Dirac phase(s) in .
PACS numbers: 98.80.Cq, 14.60.Pq, 14.60.St
keywords: thermal leptogenesis, seesaw mechanism, lepton flavour effects
In the present article we continue to investigate the possible connection between leptogenesis [1, 2] (see also, e.g. [3, 4]) and the low energy CP-violation in the lepton (neutrino) sector (for earlier discussions see, e.g. [5, 6, 7, 8] and the references quoted therein). It was shown recently  that the CP-violation necessary for the generation of the observed baryon asymmetry of the Universe in the thermal leptogenesis scenario can be due exclusively to the Dirac and/or Majorana CP-violating phases in the Pontecorvo-Maki-Nakagawa-Sakata (PMNS)  neutrino mixing matrix, and thus can be directly related to the low energy CP-violation in the lepton sector (e.g. in neutrino oscillations, etc.). The analysis performed in  (see also [11, 12]) was stimulated by the progress made in the understanding of the importance of lepton flavour effects in leptogenesis [13, 14, 15, 16, 17, 18]. It led to the realisation that these effects can play crucial role in the leptogenesis scenario of baryon asymmetry generation [15, 16, 17]. It was noticed in , in particular, that “Scenarios in which while entail the possibility that the phases in the light neutrino mixing matrix are the only source of CP violation.”, and being respectively the individual lepton number and the total lepton number CP violating asymmetries.
As is well-known, the leptogenesis theory  is based on the see-saw mechanism of neutrino mass generation . The latter provides a natural explanation of the observed smallness of neutrino masses (see, e.g. [20, 21, 22]). An additional appealing feature of the see-saw scenario is that through the leptogenesis theory it allows to relate the generation and the smallness of neutrino masses with the generation of the baryon (matter-antimatter) asymmetry of the Universe, .
The non-supersymmetric version of the type I see-saw model with two or
three heavy right-handed (RH) Majorana neutrinos is the minimal scheme
in which leptogenesis can be implemented. In  the
analysis was performed within the simplest type I see-saw mechanism of
neutrino mass generation with three heavy RH Majorana neutrinos,
, . Taking into account the lepton flavour effects in
leptogenesis it was shown , in particular, that if the
heavy Majorana neutrinos have a hierarchical spectrum, i.e. if , being the mass of , the observed baryon
asymmetry can be produced even if the only source of
CP-violation is the Majorana and/or Dirac phase(s) in the PMNS
It should be noted that constructing a viable see-saw model which leads to real or purely imaginary matrix might encounter serious difficulties, as two recent attemps in this direction indicate [18, 25]. However, constructing such a model lies outside the scope of our study.
In the present article we investigate the effects of the lightest neutrino mass on “flavoured” (thermal) leptogenesis. We concentrate on the case when the CP-violation necessary for the generation of the observed baryon asymmetry of the Universe is due exclusively to the Dirac and/or Majorana CP-violating phases in the PMNS matrix . Our study is performed within the simplest type I see-saw scenario with three heavy RH Majorana neutrinos , . The latter are assumed to have a hierarchical mass spectrum, . Throughout the present study we employ the “orthogonal” parametrisation of the matrix of neutrino Yukawa couplings . As was already mentioned earlier, this parametrisation involves an orthogonal matrix , . Although, in general, the matrix can be complex, i.e. CP-violating, in the present work we are primarily interested in the possibility that conserves the CP-symmetry. We consider the two types of light neutrino mass spectrum allowed by the data (see, e.g. ): i) with normal ordering (), , and ii) with inverted ordering (), . The case of inverted hierarchical (IH) spectrum and real (and CP-conserving) matrix is investigated in detail. Results for the normal hierarchical (NH) spectrum are also presented.
Our analysis is performed for negligible renormalisation group (RG) running of and of the parameters in the PMNS matrix from to . This possibility is realised (in the class of theories of interest) for sufficiently small values of the lightest neutrino mass [26, 27], e.g., for eV. The latter condition is fulfilled for the NH and IH neutrino mass spectra, as well as for spectrum with partial hierarchy (see, e.g. ). Under the indicated condition , and correspondingly and , and can be taken at the scale , at which the neutrino mixing parameters are measured.
Throughout the present work we use the standard parametrisation of the PMNS matrix:
where , , , is the Dirac CP-violating (CPV) phase and and are the two Majorana CPV phases [29, 30], . All our numerical results are obtained for the current best fit values of the solar and atmospheric neutrino oscillation parameters [31, 32, 33], , and , :
In certain cases the predictions for are very sensitive to the variation of and within their 95% C.L. allowed ranges:
2 Baryon Asymmetry from Low Energy CP-Violating Dirac and Majorana Phases in
Following  we perform the analysis in the framework of the simplest type I see-saw scenario. It includes the Lagrangian of the Standard Model (SM) with the addition of three heavy right-handed Majorana neutrinos () with masses and Yukawa couplings , . We will work in the basis in which i) the Yukawa couplings for the charged leptons are flavour-diagonal, and ii) the Majorana mass term of the RH neutrino fields is also diagonal. The heavy Majorana neutrinos are assumed to possess a hierarchical mass spectrum, .
In what follows we will use the well-known “orthogonal parametrisation“ of the matrix of neutrino Yukawa couplings :
where is, in general, a complex orthogonal matrix, , and are diagonal matrices formed by the masses of and of the light Majorana neutrinos , , , , , and GeV is the vacuum expectation value of the Higgs doublet field. We shall assume that the matrix has real and/or purely imaginary elements.
In the case of “hierarchical” heavy Majorana neutrinos , the CP-violating asymmetries, relevant for leptogenesis, are generated in out-of-equilibrium decays of the lightest one, . The asymmetry in the lepton flavour (lepton charge ) is given by [15, 16, 17]:
Thus, for real or purely imaginary elements of , .
There are three possible regimes of generation of the baryon asymmetry
in the leptogenesis scenario [15, 16, 17]. At
temperatures GeV the lepton flavours are
indistinguishable and the one flavour approximation is valid. The
relevant asymmetry is and in the case of interest (real or purely imaginary
CP-conserving ) no baryon asymmetry is produced. For
Boltzmann evolution of the asymmetry in the
flavour (lepton charge of the Universe) is
distinguishable from the evolution of the flavour (or
lepton charge ) asymmetry . This corresponds to the so-called “two-flavour
where the second expression corresponds to real and purely imaginary . Here is the number of relativistic degrees of freedom, , , is the “wash-out mass parameter” for the asymmetry in the lepton flavour [15, 16, 17],
and and are the efficiency factors for generation of the asymmetries and . The efficiency factors are well approximated by the expression :
At GeV, the three-flavour regime is realised and 
For real or purely imaginary of interest, , it proves convenient to cast the asymmetries in the form :
where we have used and , , . Note that real (purely imaginary) and purely imaginary (real) , , implies violation of CP-invariance by the matrix . In order for the CP-symmetry to be broken at low energies, we should have both and (see  for further details). Note also that if , , is real or purely imaginary, as the condition of CP-invariance requires , of the three quantities , and , relevant for our discussion, not more than two can be purely imaginary, i.e. if, for instance, and , then we will have .
3 Effects of Lightest Neutrino Mass: Real
We consider next the possible effects the lightest neutrino mass can have on (thermal) leptogenesis. We will assume that the latter takes place in the regime in which the lepton flavour effects are significant and that the CP-violation necessary for the generation of the baryon asymmetry is provided only by the Majorana or Dirac phases in the PMNS matrix . In the present Section we analyse the possibility of real elements , , of the matrix . The study will be performed both for light neutrino mass spectrum with normal and inverted ordering. We begin with the more interesting possibility of spectrum with inverted ordering (hierarchy).
3.1 Light Neutrino Mass Spectrum with Inverted Ordering
The case of inverted hierarchical (IH) neutrino mass spectrum, , , is of particular
interest since, as was already mentioned in the Introduction, for real
, , IH spectrum and negligible lightest neutrino mass
, it is impossible to generate the observed baryon
asymmetry in the regime of
“flavoured” leptogenesis , i.e. for , if the only source of CP violation are the
Majorana and/or Dirac phases in the PMNS matrix. For and real , the terms proportional to in the
expressions for the asymmetries and wash-out mass
parameters , , will be negligible if
, or if and , . The main reason for the indicated negative result
lies in the fact that if , or and , the lepton asymmetries are suppressed by the
factor , while
, and the resulting baryon asymmetry is too
In what follows we will analyse the generation of the baryon asymmetry for real , , when is non-negligible. We will assume that is produced in the two-flavour regime, GeV. Under these conditions the terms in will be dominant provided 
This inequality can be fulfilled if , or , and if is sufficiently large. The neutrino mass spectrum will be of the IH type if still obeys . The latter condition can be satisfied for having a value . Our general analysis will be performed for values of from the interval .
Consider the simple possibility of . We will present later results of a general analysis, performed without setting to 0. For the asymmetry of interest is given by:
The two relevant wash-out mass parameters are given by:
The orthogonality of the matrix implies that , which in the case under consideration reduces to . It is not difficult to show that for and satisfying this constraint, the maximum of the function , and therefore of the asymmetry , takes place for
At the maximum we get
The second approximate equalities in eqs. (22) and (23) correspond to IH spectrum, i.e. to . Thus, the maximum of the asymmetry thus found i) is not suppressed by the factor , and ii) practically does not depend on in the case of IH spectrum. Given the fact that
, where we have used , and , and that , we find the absolute upper bound on the baryon asymmetry in the case of IH spectrum and real matrix (real ):
This upper bound allows to determine the minimal value of for which it is possible to reproduce the observed value of lying in the interval for IH spectrum, real and :
The values of , for which is maximal, can differ, in general, from those maximising due to the dependence of the wash-out mass parameters and of the corresponding efficiency factors on . However, this difference, when it is present, does not exceed 30%, as our calculations show, and is not significant. At the same time the discussion of the wash-out effects for the maximal is rather straightforward and allows to understand in a rather simple way the specific features of the generation of in the case under discussion. For these reasons in our discussion of the wash-out mass parameters we will use maximising . All our major numerical results and most of the figures are obtained for maximising .
For (), which maximises the ratio and the asymmetry , the relevant wash-out mass parameters are given by:
Equations (24), (27) and (28)
suggest that in the case of IH spectrum with non-negligible ,
, the generated baryon asymmetry
can be strongly enhanced in comparison with the asymmetry
produced if . The enhancement can be by a factor of . Indeed, the maximum of the asymmetry (with
respect to ), eq. (23), does not contain the
and its magnitude is not controlled by , but rather by
. At the same time, the wash-out mass parameters
and , eqs. (27) and
(28), are determined by . The
latter in the case under discussion can take values as large as eV. The efficiency factors
and , which
enter into the expression for the baryon asymmetry, eq. (8),
have a maximal value when eV (weak wash-out regime). Given the
range of values of for IH spectrum extends to eV, one can always find a value of in this range such
that or take a value maximising
or , and
qualitative discussion suggests that there always exists an interval
of values of for which the baryon asymmetry is produced in the
weak wash-out regime. On the basis of the above considerations one
can expect that we can have successful leptogenesis for a
non-negligible in the case of IH spectrum even if the requisite
CP-violation is provided by the Majorana or Dirac phase(s) in the PMNS
matrix. This is confirmed by the detailed (analytic and numerical)
analysis we have performed,
the results of which are described below.
A. Leptogenesis due to Majorana
We will assume first that the Dirac phase has a CP-conserving value, . For , we have and correspondingly , where and we have used the best fit values of and , and the limit . For we get: . The terms proportional to have a subdominant effect on the magnitude of the calculated and .
It is easy to check that the asymmetry and the wash-out mass parameters remain invariant with respect to the change , . Thus, the baryon asymmetry satisfies the following relation: . Therefore, unless otherwise stated, we will consider the case of in what follows.
The absolute maximum of the asymmetry with respect to
is not obtained for for which
We are interested primarily in the dependence of on . As increases from the value of eV up to eV, in the case of under discussion the maximal possible for a given increases monotonically, starting from a value which for GeV is much smaller than the observed one, . At approximately eV, we have for GeV. As increases beyond eV, for a given continues to increase until it reaches a maximum. This maximum occurs for such that eV and is maximal, , while is considerably smaller. As can be shown, for , it always takes place at . For , and , the maximum of in question is located at eV. It corresponds to the CP-asymmetry being predominantly in the flavour. As increases further, and correspondingly , rapidly decrease. At certain value of , typically lying in the interval eV, one has
and goes through a deep
minimum: one can have even . This minimum of
corresponds to a partial or complete cancellation between the
asymmetries in the flavour and in the flavour