1 Introduction
 SISSA 60/2007/EP arXiv:0709.0413

Effects of Lightest Neutrino Mass in Leptogenesis

[0.3cm]

E. Molinaro1, S. T. Petcov2, T. Shindou34 and Y. Takanishi5

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

## 1 Introduction

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 [9] 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) [10] 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 [9] (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 [16], 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 [1] is based on the see-saw mechanism of neutrino mass generation [19]. 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 [9] 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 [9], 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 matrix6 . Let us recall that in the case of hierarchical spectrum of the heavy Majorana neutrinos, the lepton flavour effects can be significant in leptogenesis provided the mass of the lightest one satisfies the constraint [15, 16, 17] (see also [23]): . In this case the predicted value of the baryon asymmetry depends explicitly (i.e. directly) on and on the CP-violating phases in . The results quoted above were demonstrated to hold both for normal hierarchical (NH) and inverted hierarchical (IH) spectrum of masses of the light Majorana neutrinos (see, e.g. [20]). In both these cases they were obtained for negligible lightest neutrino mass and CP-conserving elements of the orthogonal matrix , present in the “orthogonal” parametrisation [24] of the matrix of neutrino Yukawa couplings. The CP-invariance constraints imply [9] that the matrix could conserve the CP-symmetry if its elements are real or purely imaginary. As was demonstrated in [9], for NH spectrum and negligible lightest neutrino mass one can have successful thermal leptogenesis with real . In contrast, in the case of IH spectrum and negligible lightest neutrino mass (), the requisite baryon asymmetry was found to be produced for CP-conserving matrix only if certain elements of are purely imaginary: for real the baryon asymmetry is strongly suppressed [8] and leptogenesis cannot be successful for (i.e. in the regime in which the lepton flavour effects are significant). It was suggested in [9] that the observed value of can be reproduced for in the case of IH spectrum and real (CP-conserving) elements of if the lightest neutrino mass is non-negligible, having a value in the interval , where is the mass squared difference responsible for the solar neutrino oscillations, and are the masses of the two additional light Majorana neutrinos. In this case we still would have since , being the mass squared difference associated with the dominant atmospheric neutrino oscillations.

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 [24]. 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. [20]): 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. [28]). 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:

 U=⎛⎜⎝c12c13s12c13s13e−iδ−s12c23−c12s23s13eiδc12c23−s12s23s13eiδs23c13s12s23−c12c23s13eiδ−c12s23−s12c23s13eiδc23c13⎞⎟⎠\leavevmode\nobreak diag(1,eiα212,eiα312) (1)

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 , :

 Δm2⊙=Δm221=8.0×10−5 eV2,\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak sin2θ12=0.30, (2) |Δm2A|=|Δm231(32)|=2.5×10−3 eV2,\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak sin22θ23=1. (3)

In certain cases the predictions for are very sensitive to the variation of and within their 95% C.L. allowed ranges:

 0.26≤sin2θ12≤0.36,\leavevmode\nobreak 0.36≤sin2θ23≤0.64,\leavevmode\nobreak \leavevmode\nobreak 95%\leavevmode\nobreak \leavevmode\nobreak C.L. (4)

We also use the current upper limit on the CHOOZ mixing angle [34, 31, 32]:

 sin2θ13<0.025\leavevmode\nobreak (0.041),\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak 95%\leavevmode\nobreak (99.73%)\leavevmode\nobreak C.L. (5)

## 2 Baryon Asymmetry from Low Energy CP-Violating Dirac and Majorana Phases in UPMNS

Following [9] 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 [24]:

 λ=1v√MR√mU†, (6)

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]:

 ϵl = −3M116πv2Im(∑jkm1/2jm3/2kU∗ljUlkR1jR1k)∑imi|R1i|2. (7)

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 , the 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 regime”7. At smaller temperatures, GeV, the evolution of the flavour (lepton charge ) and of also become distinguishable (three-flavour regime). The produced baryon asymmetry is a sum of the relevant flavour asymmetries, each weighted by the corresponding efficiency factor accounting for the wash-out processes.

In the two-flavour regime, GeV, the baryon asymmetry8 predicted in the case of interest is given by [17] (see also [9]):

 YB ≅ −1237g∗(ϵ2η(417589˜m2)+ϵτη(390589˜mτ)) (8) = −1237g∗ϵτ(η(390589˜mτ)−η(417589˜m2)),

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],

 ˜ml = ∣∣ ∣∣∑kR1km1/2kU∗lk∣∣ ∣∣2,\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak l=e,μ,τ, (9)

and and are the efficiency factors for generation of the asymmetries and . The efficiency factors are well approximated by the expression [17]:

 η(X)≅(8.25×10−3eVX+(X2×10−4eV)1.16)−1. (10)

At GeV, the three-flavour regime is realised and [17]

 YB≅−1237g∗(ϵeη(151179˜me)+ϵμη(344537˜mμ)+ϵτη(344537˜mτ)). (11)

For real or purely imaginary of interest, , it proves convenient to cast the asymmetries in the form [9]:

 ϵl=−3M116πv2∑k∑j>k√mkmj(mj−mk)ρkj|R1kR1j|Im(U∗lkUlj)∑imi|R1i|2,\leavevmode\nobreak if\leavevmode\nobreak Im(R1kR1j)=0, (12) ϵl=−3M116πv2∑k∑j>k√mkmj(mj+mk)ρkj|R1kR1j|Re(U∗lkUlj)∑imi|R1i|2,\leavevmode\nobreak if\leavevmode\nobreak Re(R1kR1j)=0. (13)

where we have used and , , . Note that real (purely imaginary) and purely imaginary (real) , , implies violation of CP-invariance by the matrix [9]. In order for the CP-symmetry to be broken at low energies, we should have both and (see [9] for further details). Note also that if , , is real or purely imaginary, as the condition of CP-invariance requires [9], 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 R1j

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 [9], 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 small 9.

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 [9]

 2⎛⎜ ⎜⎝m3√Δm2⊙⎞⎟ ⎟⎠12(Δm2AΔm2⊙)34|R13|∣∣R11(12)∣∣≫1. (14)

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:

 ϵτ≅−3M116πv2√m3m2(1−m3m2)ρ23rIm(U∗τ2Uτ3), (15)

where

 m2=√m23+|Δm2A|, (16) r=|R13R12||R12|2+m3m2|R13|2, (17)

and

 Im(U∗τ2Uτ3)=−c23c13Im(ei(α31−α21)/2(c12s23+s12c23s13e−iδ)). (18)

The two relevant wash-out mass parameters are given by:

 ˜mτ = m2R212|Uτ2|2+m3R213|Uτ3|2+2√m2m3ρ23|R12R13|Re(U∗τ2Uτ3), (19) ˜m2 ≡ ˜me+˜mμ=m2R212+m3R213−˜mτ, (20)

where .

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

 R212=m3m3+m2,\leavevmode\nobreak \leavevmode\nobreak R213=m2m3+m2,\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak R212

At the maximum we get

 max(|r|)=12(m2m3)12≅12⎛⎜ ⎜⎝√|Δm2A|m3⎞⎟ ⎟⎠12, (22)

and

 |ϵτ|≅3M132πv2(m2−m3)∣∣Im(U∗τ2Uτ3)∣∣≅3M132πv2√|Δm2A|∣∣Im(U∗τ2Uτ3)∣∣. (23)

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

 |ϵτ|≅5.0×10−8m2−m3√|Δm2A|⎛⎜ ⎜⎝√|Δm2A|0.05\leavevmode\nobreak eV⎞⎟ ⎟⎠(M1109\leavevmode\nobreak GeV)∣∣Im(U∗τ2Uτ3)∣∣\leavevmode\nobreak , (24)

, 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 ):

 |YB|to0.0pt<%$∼$4.8×10−12⎛⎜ ⎜⎝√|Δm2A|0.05\leavevmode\nobreak eV⎞⎟ ⎟⎠(M1109\leavevmode\nobreak GeV)\leavevmode\nobreak . (25)

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 :

 M1to0.0pt>∼1.7×1010\leavevmode\nobreak GeV\leavevmode\nobreak . (26)

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:

 ˜mτ = m2m3m3+m2[\leavevmode\nobreak |Uτ2|2+|Uτ3|2+2ρ23Re(U∗τ2Uτ3)], (27) ˜m2 = 2m2m3m3+m2−˜mτ. (28)

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 suppression factor 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 . This 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 CP-Violation in

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 is maximal10, but rather for having a value in the interval if , or in the interval if . The maximal value of at is smaller at least by a factor of than the value of at its absolute maximum (see further Fig. 3). As can be easily shown, for there is a rather strong mutual compensation between the asymmetries in the lepton charges and owing to the fact that, due to , and have relatively close values and . Actually, in certain cases one can even have , and thus , for lying in the interval (see further Fig. 3). Similar cancellation can occur for at . Obviously, we have for .

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 flavour11. In our example of , and , the indicated minimum of occurs at eV. As increases further,