Leptogenesis in GeV scale seesaw models
PPreprint typeset in JHEP style - HYPER VERSION
IFIC/15-44SISSA 35/2015/FISI
Leptogenesis in GeV-scale seesaw models
P. Hern´andez a , M. Kekic a , J. L´opez-Pav´on b , J. Racker a and N. Rius a a Instituto de F´ısica Corpuscular, Universidad de Valencia and CSIC,Edificio Institutos Investigaci´on, Apt. 22085, E-46071 Valencia, Spain b SISSA and INFN Sezione di Trieste, via Bonomea 265, 34136 Trieste Italy.
Abstract:
We revisit the production of leptonic asymmetries in minimal extensions of theStandard Model that can explain neutrino masses, involving extra singlets with Majoranamasses in the GeV scale. We study the quantum kinetic equations both analytically,via a perturbative expansion up to third order in the mixing angles, and numerically.The analytical solution allows us to identify the relevant CP invariants, and simplifiesthe exploration of the parameter space. We find that sizeable lepton asymmetries arecompatible with non-degenerate neutrino masses and measurable active-sterile mixings.
Keywords:
Beyond Standard Model, Cosmology of Theories beyond the SM, Neutrinophysics, CP violation. a r X i v : . [ h e p - ph ] A p r ontents
1. Introduction 12. Minimal Model of neutrino masses 2
3. Perturbative Solution of the Raffelt-Sigl equation 8
4. Numerical solution 205. Baryon asymmetry 216. Conclusions 26A. Results for the perturbative integrals 28
A.1 One dimensional integrals 28A.2 Two dimensional integrals 29A.3 Three dimensional integrals 30
B. Perturbative result for the invariants J W and I (3)2
1. Introduction
One of the interesting potential implications of (Majorana) neutrino masses is the gener-ation of a matter-antimatter asymmetry in the Universe. It has been demonstrated thatthe generation of sizeable leptonic asymmetries, leptogenesis, is generic in extensions of theStandard Model that can account for neutrino masses [1]. In particular two new ingredientsare essential for this mechanism to work: the existence of new weakly interacting particlesthat are not in thermal equilibrium sometime before the electroweak phase transition andthe existence of new sources of CP violation.Leptogenesis from the out-of-equilibrium decay of heavy Majorana fermions that ap-pear in type I seesaw models [1] has been extensively studied (for a comprehensive reviewsee e.g. [2]). The simplest version requires however relatively large Majorana masses > GeV [3, 4] (or > if flavour effects are included [5]), which imply that this scenario– 1 –ould be very difficult to test experimentally. It is possible to have sizeable asymmetriesfor smaller masses if a large degeneracy exists, through resonant leptogenesis [6].On the other hand, for Majorana masses in the GeV range, when the neutrino Yukawacouplings are small, another mechanism might be at work. In particular, the non-equilibriumcondition takes place not in the decay, but in the production of the heavy sterile neutrinos.The small Yukawa couplings imply that some of the species might never reach thermalequilibrium and a lepton asymmetry can be generated at production and seed the baryonasymmetry in the Universe. This mechanism was first proposed by Akhmedov, Rubakovand Smirnov (ARS) in their pioneering work [7] and pursued, with important refinementsin refs. [8, 9]. For a recent review and further references see [10]. In most of these works,the case of just two extra sterile species is considered, which is also the limiting case ofthe so-called ν MSM where there are three species, but one of them plays the role of warmdark matter (WDM) and is almost decoupled, having no impact in the generation of thelepton asymmetry. When the mechanism involves just two species, it has been found thatthe observed baryon asymmetry is only possible if the two states are highly degenerate inmass. This however was not the conclusion of the ARS paper.The purpose of this paper is to explore systematically the parameter space in the caseof three sterile species (which encompass the one with two neutrinos) as regards the baryonasymmetry, in particular we do not want to restrict the parameter space to have a WDMcandidate. The model has many free parameters (only 5 out of the 18 parameters arefixed by the measured light neutrino masses and mixings) and the exploration of the fullparameter space is challenging. Only with the help of approximate analytical solutions tothe kinetic equations this task is feasible. The analytical solutions furthermore allows usto identify the relevant CP invariants and to reach regions of parameter space where theequations become stiff and very difficult to deal with numerically.The paper is organised as follows. In section 2 we present the model, which is essen-tially a generic type I seesaw model, establish the notation and discuss on general groundswhat are the CP reparametrization and flavour invariants we expect to find in computingany CP violating quantity such as any putative lepton asymmetry. In section 3 we presentthe kinetic equations that describe the production of sterile neutrinos and solve them ana-lytically via a perturbative expansion in the mixing angles up to the third order. In section4 we compare the analytical and numerical solutions for several choices of the parame-ters, and identify the region of parameter space where the analytical solution accuratelydescribes the numerical one. In section 5 we use the analytical solutions and perform aMonte Carlo scan (using the software package MultiNest [11, 12]) to find regions of param-eter space that can reproduce the observed baryon asymmetry, and that are compatiblewith the measured neutrino masses and mixings. In section 6 we conclude.
2. Minimal Model of neutrino masses
We will concentrate on the arguably simplest model of neutrino masses that includes three– 2 –ight-handed singlets. The Lagrangian is given by: L = L SM − (cid:88) α,i ¯ L α Y αi ˜Φ N iR − (cid:88) i,j =1
12 ¯ N icR M ij N jR + h.c., where Y is a 3 × M a diagonal real matrix. The spectrum of thistheory has six massive Majorana neutrinos, and the mixing is described in terms of sixangles and six CP phases generically. One convenient parametrization for the problem athand is in terms of the eigenvalues of the yukawa and majorana mass matrices togetherwith two unitary matrices, V and W . In the basis where the Majorana mass is diagonal, M = Diag( M , M , M ), the neutrino Yukawa matrix is given by: Y ≡ V † Diag( y , y , y ) W. (2.1)Without loss of generality, using rephasing invariance, we can reduce the unitary matricesto the form : W = U ( θ , θ , θ , δ ) † Diag(1 , e iα , e iα ) ,V = Diag(1 , e iφ , e iφ ) U (¯ θ , ¯ θ , ¯ θ , ¯ δ ) , (2.2)where U ( θ , θ , θ , δ ) ≡ cos θ sin θ − sin θ cos θ
00 0 1 cos θ θ e − iδ − sin θ e iδ θ θ sin θ − sin θ cos θ . (2.3) Obviously not all the parameters are free, since this model must reproduce the lightneutrino masses, which approximately implies the seesaw relation: m ν (cid:39) − v Y M Y T , (2.4)where v = 246 GeV is the vev of the Higgs. On the other hand, the known neutrino massesand mixings do not give us enough information to determine the Majorana spectrum, noteven the absolute scale. Very strong constraints can be derived from neutrino oscillationexperiments for masses below the eV range [13, 14, 15, 16]. Cosmology can exclude a hugewindow below 100 MeV [17, 18, 19, 20, 21, 22], except maybe for one species that could belighter provided the lightest active neutrino mass is below (cid:46) × − eV [20, 21]. The GeVrange is interesting because an alternative mechanism for lepton asymmetry generationcould be at work [7, 8, 9]. Majorana neutrinos in this range are heavy enough to safelydecay before Big Bang Nucleosynthesis, while they are light enough that they might havenot reached thermal equilibrium by the time of the electroweak phase transition (EWPT),behaving as reservoirs of a putative lepton asymmetry.Our goal in this paper is to explore the full parameter space of this model allowed byneutrino masses, as regards leptogenesis. An essential condition will be that at least one Although we use the same notation for the mixing angles and phases of W as those in the usual PMNSmatrix, they should not be confused. Note the unconventional ordering of the 2 × U . – 3 –f the sterile neutrinos does not reach thermal equilibrium before the EWPT. This can beensured assuming a large hierarchy in the yukawas [7]: y (cid:28) y , y . (2.5)It is mandatory, however, to have an accurate analytical description, since the uncon-strained parameter space is huge. We will solve the quantum kinetic equations in a per-turbative expansion in the mixings in the next section. Since the lepton asymmetry isnecessarily a CP-odd observable, on general grounds we can derive what are the expecta-tions in terms of weak-basis CP invariants. In [23], weak basis (WB) invariants sensitive to the CP violating phases which appear inleptogenesis, within the type I seesaw model, were derived. All of them should vanish ifCP is conserved, and conversely the non-vanishing of any of these invariants signals CPviolation. They must be invariant under the basis transformations: (cid:96) L → W L (cid:96) L ,N R → W R N R . (2.6)Defining h ≡ Y † Y , and H M ≡ M † M , a subset of the invariants can be written as: I ≡ ImTr[ hH M M ∗ h ∗ M ] , (2.7) I ≡ ImTr[ hH M M ∗ h ∗ M ] , (2.8) I ≡ ImTr[ hH M M ∗ h ∗ M H M ] . (2.9)Since the I i are WB invariants, we can evaluate them in any basis. In the WB where thesterile neutrino mass matrix M is real and diagonal, one obtains: I = M M ∆ M Im( h ) + M M ∆ M Im( h ) + M M ∆ M Im( h ) , (2.10) I = M M ( M − M )Im( h ) + M M ( M − M )Im( h )+ M M ( M − M )Im( h ) , (2.11) I = M M ∆ M Im( h ) + M M ∆ M Im( h ) + M M ∆ M Im( h ) , (2.12)where ∆ M ij ≡ M i − M j and, using the parametrization of eq. (2.1)Im( h ij ) = Im[( Y † Y ) ij ] = (cid:88) α,β y α y β Im[ W ∗ αi W ∗ βi W αj W βj ] . (2.13)It is explicit in the above expression that such unflavoured invariants depend only on theCP phases of the sterile neutrino sector, which are encoded in the unitary matrix W : oneDirac-type phase, δ and two Majorana-type phases α , α . Not surprisingly, these invariantsare the relevant ones in unflavoured leptogenesis, i.e., in the conventional computation ofthe CP asymmetry generated by heavy Majorana neutrino decay neglecting flavour effects.– 4 –he combinations of W matrix elements which appear in Im( h ij ) can be expressed interms of the rephasing invariants defined in [24] as follows:Im[ W ∗ αi W ∗ βi W αj W βj ] = Im[ W αi W ∗ βi W ∗ αj W βj ( W αj W ∗ αi ) ] | W αi W αj | . (2.14)Notice that J W ≡ ± Im[ W αi W ∗ βi W ∗ αj W βj ] is the Jarlskog invariant for the matrix W ,while the quantities Im[( W αj W ∗ αi ) ] determine the Majorana phases, α , . When consid-ering processes, such as heavy neutrino oscillations, where the Majorana nature does notplay a role, only the Dirac phase δ will be relevant and therefore we expect to find just theJarlskog invariant of the matrix W.Since there are six independent CP-violating phases, it is possible to construct threemore independent WB invariants, which would complete the description of CP violationin the leptonic sector. One simple choice are those invariants obtained from I i under thechange of the matrix h by ¯ h ≡ Y † h (cid:96) Y , with h (cid:96) = λ (cid:96) λ † (cid:96) , being λ (cid:96) the charged lepton Yukawacouplings, i.e., ¯ I = ImTr[ Y † h (cid:96) Y H M M ∗ Y T h ∗ (cid:96) Y ∗ M ] , (2.15)and analogously for ¯ I , ¯ I . The corresponding CP odd invariants are Im(¯ h ij ), which in thebasis where also the charged lepton Yukawa matrix is real and diagonal can be written as:Im(¯ h ij ) = (cid:88) α,β λ α λ β Im[ Y ∗ αi Y αj Y βj Y ∗ βi ] . (2.16)The lepton number (L) violating part of the flavoured CP asymmetries in leptogenesisdepends on the above combinations [25]: (cid:15) (cid:54) Liα = (cid:88) β,j Im[ Y ∗ αi Y αj Y βj Y ∗ βi ] ˜ f ( M i , M j ) , (2.17)where ˜ f is an arbitrary function. Upon substitution of the neutrino Yukawa couplings asgiven in eq. (2.1) can be written as: (cid:15) (cid:54) Liα = (cid:88) j (cid:88) β,δ,σ y β y δ y σ Im[ W ∗ βi V βα V ∗ δα W δj W ∗ σi W σj ] ˜ f ( M i , M j ) . (2.18)These asymmetries contain the additional rephasing invariants of the form Im[ W ∗ βi V βα V ∗ δα W δj ],which depend on the phases in the matrix V (¯ δ, φ , φ ), showing that the flavoured CP asym-metries of leptogenesis are also sensitive to the CP phases in the V leptonic mixing matrix,besides those in W .Alternatively, we choose to construct the WB invariants which will appear when theMajorana character of the sterile neutrinos is not relevant, i.e., L-conserving ones. These– 5 –re given by:¯ I (cid:48) ≡ ImTr[ hH M ¯ hH M ]= M M ∆ M Im( h ¯ h ) + M M ∆ M Im( h ¯ h )+ M M ∆ M Im( h ¯ h ) , (2.19)¯ I (cid:48) ≡ ImTr[ hH M ¯ hH M ]= M M ( M − M )Im( h ¯ h ) + M M ( M − M )Im( h ¯ h )+ M M ( M − M )Im( h ¯ h ) , (2.20)¯ I (cid:48) ≡ ImTr[ hH M ¯ hH M ]= M M ∆ M Im( h ¯ h ) + M M ∆ M Im( h ¯ h )+ M M ∆ M Im( h ¯ h ) , (2.21)where Im( h ij ¯ h ji ) = (cid:88) α,β λ α Im[ Y αi Y ∗ αj Y βj Y ∗ βi ] . (2.22)The L-conserving CP asymmetry in leptogenesis via heavy neutrino decay, as well asthe CP asymmetries encountered in leptogenesis through sterile neutrino oscillations, aresensitive to the above combinations of Yukawa couplings [25]: (cid:15) Liα = (cid:88) j,β Im[ Y αi Y ∗ αj Y βj Y ∗ βi ] f ( M i , M j ) , (2.23)where f is an arbitrary function, and can be written in terms of the rephasing invariantsas: (cid:15) Liα = − (cid:88) j (cid:88) β,δ,σ y β y δ y σ Im[ W ∗ βi V βα V ∗ δα W δj W σi W ∗ σj ] f ( M i , M j ) . (2.24)Notice that the crucial difference between the L -violating and the L -conserving CP asym-metries is that in (cid:15) Liα the combination of W matrix elements is such that all dependenceon the Majorana phases α , disappears, as expected.In the approximation of neglecting y (cid:28) y , y , we obtain that Im[ Y αi Y ∗ αj ( Y † Y ) ij ] = (cid:80) β Im[ Y αi Y ∗ αj Y βj Y ∗ βi ] reduces toIm[ Y αi Y ∗ αj ( Y † Y ) ij ] = y y ( | V α | − | V α | )Im[ W ∗ i W j W ∗ j W i ]+ y y (cid:8)(cid:2) y | W i | − y | W i | (cid:3) Im[ W ∗ j V α V ∗ α W j ]+ (cid:2) y | W j | − y | W j | (cid:3) Im[ W ∗ i V α V ∗ α W i ] (cid:9) , (2.25)so in principle we expect that the lepton asymmetry will depend on ten CP invariants,namely Im[ W ∗ i V α V ∗ α W i ], with i = 1 , , α = 1 , , J W .However, they are not all independent. In ref. [24] it has been shown that in theminimal seesaw there are only six independent CP invariants that can be made out of thematrices V, W . Two of them correspond to the Majorana phases of W , α , , which aswe have argued before will not contribute in the limit of small sterile neutrino Majoranamasses that we are considering. Other two are the equivalent of the Jarlskog invariants– 6 –or the matrices V, W and therefore determine the Dirac phases, ¯ δ, δ , respectively. Thelast two are of the form Im[ W ∗ i V α V ∗ α W i ], for two reference values of i, α , that fix theadditional phases φ , . Moreover, it can be shown that since we are neglecting the Yukawacoupling y , the phase φ of the matrix V does not appear in eq. (2.25), thus we are leftwith only three independent invariants.The unitarity of the mixing matrices V, W implies that (cid:88) α V α V ∗ α = 0 , (2.26) (cid:88) i W ∗ i W i = 0 , (2.27)which allows to write the invariants Im[ W ∗ i V α V ∗ α W i ] for α = 2 in terms of those with α = 1 ,
3, and the invariants for i = 2 in terms of the corresponding ones with i = 1 ,
3. Byexploiting the identitiesIm[ W ∗ i V β V ∗ β W i ] = Im[( W ∗ i V α V ∗ α W i )( V ∗ β V α V ∗ α V β )] | V α V α | . (2.28)we can write for instance one of the invariants with β = 3 in terms of the invariant with α = 1 and the Jarlskog invariant for V , Im[ V ∗ β V α V ∗ α V β ] = ± J V .It is simpler, though, to write the results in terms of the following four invariants, evenif only three are independent, expanded up to 3rd order in the small mixing angles θ ij , ¯ θ ij : I (2)1 = − Im[ W ∗ V V ∗ W ] (cid:39) θ ¯ θ sin φ ,I (3)1 = Im[ W ∗ V V ∗ W ] (cid:39) θ ¯ θ ¯ θ sin(¯ δ + φ ) ,I (3)2 = Im[ W ∗ V V ∗ W ] (cid:39) ¯ θ θ θ sin( δ − φ ) ,J W = − Im[ W ∗ W W ∗ W ] (cid:39) θ θ θ sin δ. (2.29)A generic expectation for the CP-asymmetry relevant for leptogenesis is∆ CP = (cid:88) α,k | Y αk | ∆ α , (2.30)with ∆ α = (cid:88) i (cid:15) Liα = (cid:88) i,j Im[ Y αi Y ∗ αj ( Y † Y ) ij ] f ( M i , M j ) . (2.31)Since the CP rephasing invariants are at least second order in the angles, we just need totake the diagonal elements in ∆ CP , to keep the result up to 3rd order. Then, in the limit y = 0, we get:∆ CP = y y ( y − y ) (cid:88) i,j Im[ W ∗ i W j W ∗ j W i ] f ( M i , M j )+ y y (cid:16) ( y − y ) (cid:110) I (2)1 [ g ( M ) − g ( M )] + I (3)2 [ g ( M ) − g ( M )] (cid:111) (2.32) − y I (3)1 [ g ( M ) − g ( M )] (cid:17) , – 7 –here g ( M i ) ≡ y [ f ( M , M i ) − f ( M i , M )] − y [ f ( M , M i ) − f ( M i , M )] . (2.33)From the above definition of g ( M i ), it immediately follows that g ( M ) − g ( M ) =( y − y )[ f ( M , M ) − f ( M , M )], so ∆ CP simplifies to∆ CP = y y ( y − y ) (cid:88) i,j Im[ W ∗ i W j W ∗ j W i ] f ( M i , M j )+ y y ( y − y ) (cid:110)(cid:104) ( y − y ) I (2)1 − y I (3)1 (cid:105) [ f ( M , M ) − f ( M , M )]+ I (3)2 [ g ( M ) − g ( M )] (cid:111) . (2.34)We will see in the next section that this is precisely the yukawa and mixing angledependence we will find when solving the kinetic equations, which is a strong crosscheckof the result.
3. Perturbative Solution of the Raffelt-Sigl equation
Our starting point is the Raffelt-Sigl formulation [26] of the kinetic equations that describethe production of sterile neutrinos in the early Universe. The density matrix is the expec-tation value of the one-particle number operator for momentum k : ρ N ( k ) for neutrinos,and ¯ ρ N ( k ) for antineutrinos. We will assume that only sterile neutrinos and the leptondoublets are out of chemical equilibrium, but assume that all the particles are in kineticequilibrium, using Maxwell-Boltzmann statistics: ρ a ( k ) = A a ρ eq ( k ) , A a = e µ a ; ρ ¯ a ( k ) = A ¯ a ρ eq ( k ) , A ¯ a = e − µ a , (3.1)where ρ eq ( k ) ≡ e − k /T , with k = | k | , and µ a denotes the chemical potential normalised bythe temperature. We will furthermore neglect spectator processes and the washout inducedby the asymmetries in all the fields other than the sterile neutrinos and lepton doublets.We expect this approximation to give uncertainties of O (1) which for our purpose is goodenough [27].In [7], only the asymmetry in the sterile sector was considered, neglecting the feedbackof the leptonic chemical potentials. In this case, the equations get the standard form˙ ρ N = − i (cid:2) H, ρ N (cid:3) − (cid:8) Γ , ρ N − ρ eq (cid:9) , (3.2)and the analogous for ¯ ρ N with H → H ∗ , where H is the Hamiltonian (we neglect matterpotentials for the time being but we will include them later on) H ≡ W ∆ W † , ∆ ≡ Diag (cid:16) , ∆ M k , ∆ M k (cid:17) . (3.3)– 8 – is the rate of production/annihilation of sterile neutrinos in the plasma, which is diagonalin the basis that diagonalises the neutrino Yukawa’s:Γ = Diag(Γ , Γ , , Γ i ∝ y i , (3.4)where we assume y = 0. In deriving eq.(3.2) it is assumed that the particles involvedin the production/annihilation of the sterile neutrinos are in full equilibrium (all chemicalpotentials vanish), and that kinematical effects of neutrino masses are negligible.Note that only the matrix W appears in these equations and therefore any CP asym-metry generated can only be proportional to the invariant J W which depends at third orderon the mixing angles of W .In [8] it was correctly pointed out that the asymmetries in the sterile sector will bemodified by the leptonic chemical potentials that will be generated as soon as sterile neu-trinos start to be produced. Including the evolution of the leptonic chemical potentials hastwo important consequences: new sources of CP violation become relevant and washouteffects are effective. Leptons are fastly interacting through electroweak interactions in theplasma and therefore it is a good approximation to assume they are in kinetic equilibrium.An important question is what is the flavour structure of these chemical potentials.For T (cid:46) GeV the Yukawa interactions of the tau and muon are very fast, which impliesthat µ will be diagonal in the basis that diagonalises the charged lepton Yukawa matrix,since no other interaction changing flavour is in equilibrium before the heavy neutrinosare produced. Note however that this is not the basis where the neutrino Yukawas arediagonal, the two are related by the mixing matrix V . As a result, when the evolutionof the lepton chemical potentials is taken into account, the CP phases of the matrix V become relevant.Adapting the derivation of [26] to this situation, we find that the evolution of theCP-even and CP-odd parts of the neutrino densities: ρ ± ≡ ρ N ± ¯ ρ N and the lepton chemicalpotentials , µ α , to linear order in µ α , ρ − , satisfy in this case:˙ ρ + = − i [ H re , ρ + ] + [ H im , ρ − ] − γ aN + γ bN { Y † Y, ρ + − ρ eq } + iγ bN Im[ Y † µY ] ρ eq + i γ aN (cid:8) Im[ Y † µY ] , ρ + (cid:9) , ˙ ρ − = − i [ H re , ρ − ] + [ H im , ρ + ] − γ aN + γ bN (cid:8) Y † Y, ρ − (cid:9) + γ bN Re[ Y † µY ] ρ eq + γ aN (cid:8) Re[ Y † µY ] , ρ + (cid:9) , ˙ µ α = − µ α (cid:16) γ bν Tr[
Y Y † I α ] + γ aν Tr (cid:2) Re[ Y † I α Y ] r + (cid:3)(cid:17) +( γ aν + γ bν ) (cid:16) Tr (cid:2) Re[ Y † I α Y ] r − (cid:3) + i Tr (cid:2) Im[ Y † I α Y ] r + (cid:3)(cid:17) , (3.5)where H re ≡ Re[ H ], H im ≡ Im[ H ], I α is the projector on flavour α and γ a,bN , γ a,bν are therates of production/annihilation of a sterile neutrino or a lepton doublet neglecting allmasses, after factorizing the flavour structure in the Yukawas, γ a ( b ) N ( ν ) ≡ k (cid:88) i (cid:90) p , p , p ρ eq ( p ) |M ( a ( b )) N ( ν ) ,i | (2 π ) δ ( k + p − p − p ) , (3.6)– 9 – L α Φ N L α Φ a b Figure 1: a, b topologies for annihilation/production of sterile neutrinos where k is the momentum of the N or ν and a ( b ) refer to the s-channel (t,u-channels)depicted in figure 1. In topology a the lepton and sterile neutrino are both in the initial orfinal state, while topology b corresponds to those diagrams where one is in the initial andother in the final state. Finally r ± ≡ (cid:80) i (cid:82) p , p , p ρ ± ( p ) |M ( a ) νi | (2 π ) δ ( k + p − p − p ) (cid:80) i (cid:82) p , p , p ρ eq ( p ) |M ( a ) νi | (2 π ) δ ( k + p − p − p ) . (3.7)A similar derivation can be found in [28] and we agree with their findings.These equations reduce to those in eq. (3.2) in the limit µ → i = y i ( γ aN + γ bN ).Most previous studies have assumed that the rates are dominated by the top quarkscatterings. In this case, the rates are given (in the Boltzman approximation) by thewell-known result [29, 30] γ bN,Q = 2 γ aN,Q = 2 γ bν,Q = 4 γ aν,Q = 316 π y t T k . (3.8)The factor of 2 difference between the rates of the N and the ν is due to the fact that thelepton is a doublet and the sterile neutrino is a singlet. Note that there is a non-linearterm of the form O ( µρ + ), as first noted in [28]. More recently in [31], the equations havebeen written in terms of the µ B − L α / chemical potentials, however not all the chemicalpotentials (e.g. higgs and top quark) have been included. A full treatment including allchemical potentials will be postponed for a future work, but we expect that including thesespectator effects will change the results by factors of O (1).In [30, 32], it has been pointed out that the scattering processes ¯ LN ↔ W H get astrong enhancement from hard thermal loops and are actually the dominant scatterings.The results of [30, 32] however do not include the chemical potentials of spectators, so itis not clear how to include them consistently in the above equations. We will neglect theseeffects in the following. Note however that the lepton flavour structure of these and of thetop quark scatterings is the same. – 10 –t is easy to see also that total lepton number is conserved as it should:2 (cid:88) α ˙ µ α + Tr[ ˙ r − ] = 0 . (3.9)Two approximations are often used in solving these equations: 1) assume that the mo-mentum dependence of ρ ± follows that of ρ eq , i.e. kinetic equilibrium for the sterile states,which implies r ± = ρ ± /ρ eq are constants and the integro-differential equations become justdifferential equations, 2) neglect the k dependence of the rates by approximating (cid:104) k − (cid:105) (cid:39) T − . (3.10)The effect of these approximations has been studied numerically in [28] and the results donot differ too much. We will therefore adopt both approximations that simplify consider-ably the perturbative treatment. We are going to solve these equations perturbing in the mixing angles up to third order.We first consider the simpler case, neglecting leptonic chemical potentials and consideringin turn the evolution in a static Universe and in the expanding case.
We start with eq. (3.2) and assume y = 0. In this case, neither H nor Γ depend on time.Defining ρ Nij /ρ eq ≡ a ij + ib ij and taking into account the hermiticity of ρ N we change thematrix equation into a vector equation: r ≡ ( a , a , a , b , a , b , a , b , a ) . (3.11)At 0-th order the system of equations of eq. (3.2) can be rewritten as˙ r (0) = A r (0) + h , (3.12)with h ≡ (Γ ρ eq , Γ ρ eq , , .... , (3.13)and the matrix A is constant and has a block structure: A ≡ ( A I ) × A II ) ×
00 0 0 , (3.14) A I ≡ − Γ − Γ − Γ +Γ − ∆ ∆ − Γ +Γ , A II ≡ − Γ / − ∆ − Γ / − Γ ∆ − ∆ − (∆ − ∆ ) − Γ . (3.15) – 11 –he matrix can be easily diagonalised and exponentiated so the general solution to theequation is r (0) ( t ) = e A t (cid:90) t dx e − A x h . (3.16)At the next order we have to keep O ( θ ij ) in the Hamiltonian and translate the matrixform into the vector form: − i [ H (1) , ρ (0) ( t )] → A r (0) . (3.17)The equation for the first order correction to the density is˙ r (1) = A r (1) + A r (0) ( t ) . (3.18)The solution at this order is therefore r (1) ( t ) = e A t (cid:90) t dxe − A x A r (0) ( x ) . (3.19)We can iterate this procedure to get the correction at order n :˙ r ( n ) ( t ) = A r ( n ) ( t ) + n − (cid:88) i =1 A i r ( n − i ) ( t ) , (3.20)with solution r ( n ) ( t ) = e A t (cid:90) t dxe − A x n − (cid:88) i =1 A i r ( n − i ) ( x ) . (3.21)We can define the evolution operator U ( t, x ) ≡ e A t e − A x , (3.22)so that the solution can be written as r ( n ) ( t ) = (cid:90) t dx U ( t, x ) n − (cid:88) i =1 A i r ( n − i ) ( x ) . (3.23)As a first estimate of the leptonic asymmetry that can be generated, we are interested in∆ ρ since this is the sector that will never reach equilibrium (in the absence of mixing)and therefore can act as reservoir of the leptonic asymmetry until the electroweak phasetransition [7].One can easily compute the solution of the eq. (3.21) up to order n = 3, which is thefirst order that gives a non-vanishing result, as expected from general considerations onCP invariants. The result at finite t is not particularly illuminating but the limit t → ∞ is rather simple:lim t →∞ ∆ ρ ρ eq ≡ lim t →∞ ρ N − ¯ ρ N ρ eq = 2 J W (Γ − Γ )∆ ∆ (∆ − ∆ ) (cid:104) ∆ + Γ (cid:105) (cid:104) (∆ − ∆ ) + Γ (cid:105) . (3.24)– 12 – .01 0.1 1 10 10002. (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) t (cid:68)Ρ Ρ eq
100 500 1000 50001 (cid:180) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) t (cid:68)Ρ Ρ eq Figure 2:
Comparison of numerical (blue) and perturbative (red) solution for ∆ ρ as a functionof time, in the case with no expansion of the Universe. The two curves are indistinguishable (leftplot) until large times (right plot): the vertical line on the lower plot corresponds to ( θ Γ ) − ,while those on the upper one correspond to Γ − and Γ − respectively. A few comments are in order. We have not assumed any expansion in Γ i in this expression,only in the mixing angles. According to general theorems the equations should reach astationary solution if all the eigenvalues of the matrix A + A + A + ... are real andnegative. However, because Γ = 0, one of the eigenvalues of A vanishes and it is liftedonly at second order in perturbation theory, ∼ θ i Γ i , therefore we expect the perturbativeexpansion should break down for t ∼ θ i Γ i , which is the time scale of equilibration of thethird state. On the other hand, if θ is small, the perturbative solution should be accuratefor times t ≥ Γ − . Indeed this is precisely what we find comparing the perturbative andnumerical solutions in figure 2.The result is proportional to J W which is the only CP rephasing invariant that canappear in this case. The result vanishes if any two of the masses or the yukawa’s aredegenerate, since the CP phase would be unphysical in this case. Let us turn now to the realistic case of an expanding Universe. As usual, we will considerthe evolution as a function of the scale factor x ≡ a , in such a way that the Raffelt-Siglequation becomes ddt → xH u ( x ) ∂∂x ρ ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) fixed y = − i [ H ( x, y ) , ρ ( x, y )] − { Γ( x ) , ρ ( x, y ) − ρ eq ( y ) } , (3.25)where H u ( x ) is the Hubble parameter, H u = (cid:113) π g ∗ ( T )45 T M Planck , and y ≡ pT . Assuming forsimplicity a radiation dominated Universe with constant number of degrees of freedom,during the sterile evolution time we can assume xT = constant that we can fix to be one.Therefore the scaling of the different terms is H ( x, y ) ≡ xW ∆ M y W † , Γ i ( x ) ≡ c i x , xH u ( x ) ≡ M ∗ P x , (3.26)– 13 –here M ∗ P ≡ M Planck (cid:113) π g ∗ ( T ) and g ∗ ( T ) is the number of relativistic degrees of freedomin the plasma during the sterile evolution.Therefore the equation as function of x is:˙ ρ = − ix [ W ∆ W † , ρ ] − { γ, ρ − ρ eq } , (3.27)where we have defined ∆ ij ≡ ∆ M ij y M ∗ P , γ i ≡ c i M ∗ P . (3.28)The perturbative expansion works as in section 3.2.1, but now all the A n ( x ) are x -dependent: A n ( x ) with n ≥ x , while A ( x ) contains terms that scalewith x and others that do not depend on x . Fortunately, there is an important sim-plification in that A ( x ) can be diagonalised by an x -independent matrix, therefore thepath-ordered exponential can be easily evaluated. The result can be written in the sameform of eq. (3.23), with the evolution operator given by U ( t, r ) = e (cid:82) t A ( x ) dx e − (cid:82) r A ( y ) dy . (3.29)At third order in the mixings, after algebraic simplifications and partial integrations,the result can be given in terms of integrals of the form J n ( α , β , .., α n , β n , t ) ≡ (cid:90) t dx e iα x + β x (cid:90) x dx e iα x + β x .. (cid:90) x n − dx n e iα n x n + β n x n , (3.30)where α i are combinations of ∆ ij and β i are combinations of γ i . Up to third order in theperturbative expansion only integrals with n ≤ γ i (cid:28) | ∆ ij | (1 / , the integrands are highly oscillatoryand hard to deal with numerically. To evaluate the integrals, we separate the integrationinterval [0 , t ] = [0 , t ] + [ t , t ] with t such that t | ∆ ij | / (cid:29) t γ i (cid:28) J n ( α , β , .., α n , β n , t ) = J n ( α , β , ...α n , β n , t ) + ∆ J n ( α , β , ...α n , β n , t , t ) . (3.31)To solve the integrals up to t we can safely Taylor expand in β i (which results in anexpansion in γ , / | ∆ ij | ) and write the integrals in terms of simpler integrals of the form: J n k ( α , .., α n , t ) ≡ (cid:90) t dx x k e i α x (cid:90) x dx x k e i α x .. (cid:90) x n − dx n x k n n e i αnx n , (3.32)up to third order in the β i expansion we just need integrals with n + (cid:80) i k i ≤
3. We canuse the relation ddx [ F n ( x )] = x n e i αx (3.33)– 14 –ith F n ( α, x ) = − n − ( − iα ) − n Γ (cid:20) n , − iαx (cid:21) , (3.34)to evaluate immediately the one-dimensional integrals in terms of incomplete Γ functions.The integrals in the range [ t , t ] can be approximated by the large t behaviour of the J n ( α, t ) functions, after resumming the Taylor series in β i . Further details are presentedin appendix A.The finite t dependence of the asymmetry ∆ ρ is rather complicated, but the asymp-totic value is non-zero and rather simple:lim t →∞ ∆ ρ ρ eq = − J W γ γ ( γ − γ ) lim t →∞ Im[ J (∆ − ∆ , − ∆ , ∆ , t ) + J (∆ − ∆ , ∆ , − ∆ , t )+ J (∆ , − ∆ , ∆ − ∆ , t ) + J (∆ , ∆ − ∆ , − ∆ , t )] . (3.35)This can be simplified tolim t →∞ ∆ ρ ρ eq = − J W γ γ ( γ − γ )(∆ ∆ ∆ ) / Im (cid:20) I (cid:18) ∆ ∆ , − ∆ ∆ (cid:19) + I (cid:18) − ∆ ∆ , − ∆ ∆ (cid:19)(cid:21) , (3.36)where I (cid:18) ∆ ∆ , ∆ ∆ (cid:19) ≡ (∆ ∆ ∆ ) / (cid:90) ∞ dxe i ∆1 x J (∆ , x ) J (∆ , x ) . (3.37)Comparing eq. (3.36) and eq. (3.24) we see that in the expanding case the asymmetryis cubic in γ i and not linear. Note that the dependence on the yukawa’s is precisely thatexpected from a flavour invariant CP asymmetry. In fact this is effectively the situationin the expanding case, because the asymmetry is generated at times t (cid:28) γ − i and thedependence in the yukawa’s in this regime is therefore perturbative. This is in contrastwith the non-expanding case, where the asymmetry evolves all the way till t ∼ γ − i . Tounderstand the reason behind this different behavior, it is useful to recall the definition of∆ ij from eq. (3.28). Then, we see that ∆ ij x (cid:29) M ij / (4 T ) (cid:29) T /M ∗ P = H u ( T ),therefore in this regime the sterile neutrino oscillations are much faster than the Hubbleparameter and no asymmetry is produced anymore, since oscillations are averaged out.Thus in the expanding Universe the generation of the asymmetry occurs at x ∼ | ∆ ij | − / (cid:28) γ − i .Until now we have neglected the matter potentials, however given the suppression inthree powers of γ of the leading result, there are corrections of same order coming fromthe potentials, and in fact they are numerically more important.The equation including the potentials in the basis with diagonal neutrino Yukawas is:˙ ρ = − ix [ W ∆ W † , ρ ] − i [ v, ρ ] − { γ, ρ − ρ eq } , (3.38)– 15 –here v ij = y i M ∗ P δ ij ≡ v i δ ij . (3.39)The result for the asymmetry including the potentials is given by:lim t →∞ ∆ ρ ρ eq = J W lim t →∞ Re [ z J (∆ − ∆ , − ∆ , ∆ , t ) + z J (∆ − ∆ , ∆ , − ∆ , t )+ z J (∆ , ∆ − ∆ , − ∆ , t ) + z J (∆ , − ∆ , ∆ − ∆ , t )] . (3.40)with z ≡ γ γ ∆ v + γ v ∆ γ + i (cid:18) γ γ ∆ γ − γ v ∆ v (cid:19) ,z ≡ (cid:104) γ v − γ v + i (cid:16) γ γ v v (cid:17)(cid:105) ∆ γ ,z ≡ − γ γ ∆ v − γ v ∆ γ + i (cid:18) γ γ ∆ γ − γ v ∆ v (cid:19) . (3.41)and ∆ v ≡ v − v and ∆ γ ≡ ( γ − γ ).The leading terms O ( v γ ) at asymptotic times t (cid:29) γ − , are:lim t →∞ ∆ ρ ρ eq = 9 y t π J W y y ( y − y ) M ∗ P | ∆ M ∆ M ∆ M | / κ, (3.42)where κ ≡ | ∆ ∆ ∆ | / Im (cid:2) J (∆ − ∆ , − ∆ , ∆ , t ) − J (∆ − ∆ , ∆ , − ∆ , t ) − J (∆ , ∆ − ∆ , − ∆ , t ) + J (∆ , − ∆ , ∆ − ∆ , t ) (cid:3) (3.43)depends only on the ratios of mass differences and/or the ordering of the states. This resultis parametrically the same as the result of [7] if we neglect the dependence of κ on the massdifferences and has the dependence on the yukawas expected from eq. (2.34).Considering the naive seesaw scaling y i ∼ m ν M i v , for m ν ∼ M i ∼ ∆ M ij ∼ M , leads tolim t →∞ ∆ ρ ρ eq ∼ × − J W (cid:16) m ν (cid:17) (cid:18) M
10 GeV (cid:19) . (3.44)The asymmetry is highly sensitive to the light neutrino mass. Note that we have pushed thevalue to the limit, a light neutrino mass in the less constrained 0 . ∼
10 -100 GeV lepton number violatingtransitions via the Majorana mass could washout further the asymmetry, an effect thatrequires a refinement of the formulation to be taken into account.– 16 – .3 Lepton asymmetries in the active sector
The asymmetry generated ignoring the µ evolution depends only on the Dirac-type phase, δ , appearing in W as we have seen. However when the evolution of the leptonic chemicalpotentials is included, other phases contribute to the total lepton asymmetry. We willperform a perturbative expansion to third order in the mixings of both V and W matrices.The result at finite t (cid:28) θ i (¯ θ i ) γ − i can be written in the form:Tr[ µ ]( t ) = (cid:88) I CP I CP A I CP ( t ) (3.45)where all the four CP invariants appear, I CP = (cid:110) J W , I (2)1 , I (3)1 , I (3)2 (cid:111) , given in eqs. (2.29).At finite t , the result for the functions A I CP is well approximated by A I (2)1 ( t ) = y y ( y − y ) (cid:18) − γ N ¯ γ N (cid:19) γ N G ( t ) ,A I (3)1 ( t ) = − y y ( y − y ) (cid:18) − γ N ¯ γ N (cid:19) γ N G ( t ) ,A I (3)2 ( t ) = y y (cid:18) − γ N ¯ γ N (cid:19) γ N G ( t ) ,A J W ( t ) = γ γ (cid:18) − γ N ¯ γ N (cid:19) G ( t ) − γ N γ N G ( t ) . (3.46)where γ N ≡ γ aN + γ bN and ¯ γ N ≡ γ aN +3 γ bN , while G ( t ) ≡ (cid:0) e − ¯ γ t − e − ¯ γ t (cid:1) Re [ iJ (∆ , − ∆ , t ) + 2∆ v J (∆ , − ∆ , t )]+ 12 (cid:88) k =1 ( − k e − ¯ γ k t Re [ J (∆ , − ∆ , t ) ( − v + i (2¯ γ k − γ − γ ))] , (3.47)and G ( t ) = G ( t ) | ¯ γ =0 , (3.48)where we have defined ¯ γ i ≡ y i ¯ γ N and ∆ v ≡ v − v , and the result for G ( t ) , G ( t ) , G ( t )are lengthier and reported in the appendix B. These results would get modified for γ i t (cid:29) γ i ( v i ) / ∆ / when∆ / t (cid:29)
1, while the terms γ i ( v i ) t are resumed.In figure 3 we plot the functions A I (2)1 ( t ) and A I (3)1 ( t ), which depend only on oneneutrino mass difference. We show two physical situations: one with very degenerateneutrinos and the other with no strong degeneracies.These two invariants are the only ones relevant for the scenario that has been consideredin most previous studies, where it has been assumed that only two sterile neutrinos have arole in generating the lepton asymmetry (see for instance [33] for a very recent analysis).– 17 – (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Log (cid:64) t (cid:144) t EW (cid:68) A I (cid:72) t (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Log (cid:64) t (cid:144) t EW (cid:68) A I (cid:72) t (cid:76) Figure 3:
Functions A I (2)1 ( t ) (left) and A I (3)1 ( t ) (right) assuming the rates are dominated by topquark scattering, and taking y / √ y = 10 − , for two choices of ∆ M = 1GeV (dashed)and ∆ M = 10 − GeV (solid). t EW is the electroweak phase transition time, corresponding to T EW (cid:39) This is the situation in the limit of complete decoupling of N , ensured by the condition θ i = 0, implying that only the invariants I (2)1 and I (3)1 survive. In [8] an approximateanalytical solution was obtained, expanding in the yukawa’s, under the assumption that | ∆ | − / (cid:28) t EW (cid:28) ¯ γ − i . In this limit, the result of eqs. (3.46) and (3.47) can be simplifiedtoTr[ µ ]( t EW ) (cid:39) − (cid:0) ( y − y (cid:1) I (2)1 − y I (3)1 ) y y ( y − y ) (cid:18) − ¯ γ N γ N (cid:19) γ N Im[ J (∆ , − ∆ , ∞ )] T EW . (3.49)Comparing with eq. (2.34), we see that the dependence on the yukawa’s is again thatexpected from a flavour invariant CP asymmetry. UsingIm[ J (∆ , − ∆ , ∞ )] = − (cid:18) (cid:19) / π / Γ[ − /
6] sign(∆ ) | ∆ | / , (3.50)and ¯ γ N = γ N , and assuming the naive seesaw relations y = 2 √ ∆ sol M v , y = 2 √ ∆ atm M v we find: Tr[ µ ]( t EW ) (cid:39) − ( I (2)1 − I (3)1 ) (cid:113) M M / GeV / (cid:18) M | ∆ M | (cid:19) / , (3.51)while for y = y / − (that would correspond to light neutrino masses in the eVrange and heavy ones in the GeV range) we would haveTr[ µ ]( t EW ) (cid:39) × − I (2)1 − I (3)1 | ∆ M (GeV ) | / . (3.52)Even if the CP invariants are of O (1), the asymmetry is too small unless there is a significantdegeneracy between the two states [8]. It is important however to realise that the naiveseesaw scaling is too naive and a full exploration of parameter space is necessary.– 18 – (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Log (cid:64) t (cid:144) t EW (cid:68) A I (cid:72) t (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Log (cid:64) t (cid:144) t EW (cid:68) A J c kg (cid:72) t (cid:76) Figure 4:
Functions A I (3)2 ( t ) (left) and A J W ( t ) (right) assuming the rates are dominated bytop quark scattering, and taking y / √ y = 10 − , for three choices of [∆ M , ∆ M ] =[1 , , [10 − ,
2] and [10 − , × − ] in GeV (dashed, dotted and solid). t EW is the electroweakphase transition time, corresponding to T EW (cid:39) In figure 4 we plot the functions A I (3)2 ( t ) and A J W ( t ). They depend on the two neutrinomass differences, so we show three examples here: one in which there are no degeneracies,one where there are two almost degenerate states, and the case where the three states arealmost degenerate. As in the previous case we see a large enhancement when only one ofthe mass differences is small and a further enhancement when the two are small comparedto the absolute scale. In the case of A J W we find that there is a significant difference inthe regime ∆ / ij t (cid:28) A J W ( t ) truncated to the terms of O ( y i ). As we will seein the next section, the latter is much closer to the numerical result. The reason for thisdifference is that at small times, ∆ / t (cid:28)
1, only some terms of order O ( y i ) are kept ineqs. (3.46), while there is a strong cancellation if all had been included. Note however thatthis effect is only important at times where the asymmetry is suppressed and seems toaffect only A J W .It is interesting to note that even though the dependence on the yukawas of the func-tions A I CP ( t ) is different (fourth or sixth order), the maxima for all cases are roughly of thesame order of magnitude. Note, however, that in the limit t (cid:29) γ − i , only the contributionof two invariants, J W and I (3)1 , survive:lim t →∞ Tr[ µ ]( t ) (cid:39) − γ N γ N lim t →∞ (cid:20) ∆ ρ ( t ) ρ eq (cid:21) eq . (3 . − / π / / Γ (cid:2) − (cid:3) I (3)1 y y ( y − y ) | ∆ | / (cid:18) − γ N ¯ γ N (cid:19) γ N , (3.53)where we kept only the leading terms O ( y ) proportional to I (3)1 and we have used theresult of eq. (3.50).The first term in this expression corresponds to the expectation of [7], ie. the finalasymmetry is proportional to that stored in the third sterile state, eq. (3.40), while thesecond term was missing in the simplified treatment of [7]. Note that they depend ondifferent CP invariants. – 19 – - - - - - - - Log tt ew Tr @ m D I H L - - - - - - - - Log tt ew Tr @ m D I H L Figure 5:
Left: full numerical solution (solid blue) and numerical solution neglecting non-linearterms (dotted green) for case 1, normalised to the invariant I (2)1 , compared with the prediction, A I (2)1 ( t ) (dashed red). Right: same for case 2 normalised to the invariant I (3)1 compared to A I (3)1 ( t ).The parameters are the same as in figure 3 for the degenerate case.
4. Numerical solution
In order to check the accuracy of the analytical solutions presented in the previous sec-tion, we have solved the differential equations numerically. As shown in [28], the momen-tum dependence does not change significantly the results so we will consider the average-momentum approximation.In figures 5-6 we compare the analytical and numerical solutions for the functions A I CP ( t ) in the highly degenerate case (the values of the mixing angles are of O (10 − )) . Inorder to isolate the appropriate invariant we make the following choices: • Case 1: θ i = ¯ θ i = 0 isolates I (2)1 , • Case 2: θ i = ¯ θ = 0 isolates I (3)1 , • Case 3: θ = ¯ θ i = 0 isolates I (3)2 , • Case 4: ¯ θ ij = 0 isolates I J W .The numerical results normalised by the corresponding CP invariant are shown togetherwith the predictions of the previous section. In the case of J W , we plotted the function A J W keeping only the terms of O ( y ) that is more accurate at small t and the full functionat large t . The agreement in all cases is quite good. The differences observed at large t come from the non-linear terms in the equations. We also show the numerical results ofthe equations without them and find a very good agreement also at large t . Note that theapproximation works well in the regime γt (cid:29)
1, that is in the strong washout regime ofthe fast modes.Numerically it is very hard to go to regimes where the ratios γ/ | ∆ | / become verysmall, since the system becomes stiff. On the other hand, there is no reason why theperturbative solution is not accurate in such regime. We will therefore assume this to be– 20 – - - - - - - - Log tt ew Tr @ m D I H L - - - - - - - - Log tt ew Tr @ m D J ckg Figure 6:
Left: full numerical solution (solid blue) and numerical solution neglecting non-linearterms (dotted green) for case 3, normalised to the invariant I (3)2 , compared with the prediction, A I (3)2 ( t ) (dashed red). Right: same for case 4 normalised to the invariant J W compared to A J W ( t ).The parameters are the same as in figure 4 for the double degenerate case. the case in the following section and use the perturbative solution to perform a scan ofparameter space.
5. Baryon asymmetry
The observed baryon asymmetry is usually quoted in terms of the abundance, which isthe number-density asymmetry of baryons normalised by the entropy. After Planck thisquantity is known to per cent precision [34]: Y exp B (cid:39) . × − . (5.1)The lepton asymmetries in the left-handed (LH) leptons generated in the production ofthe sterile neutrinos are efficiently transferred via sphaleron processes [35] to the baryons.The baryon asymmetry is given by Y B = 2879 Y B − L . (5.2)Since we have neglected spectator processes in the transport equations, the B − L asymme-try is related to the chemical potentials computed in the previous sections by the relation Y B − L = − π g ∗ Tr[ µ ] , (5.3)where g ∗ = 106 .
75 (which ignores the contribution to the entropy of the sterile states).Our estimate for the baryon asymmetry is therefore Y B (cid:39) × − Tr[ µ ( t )] | t EW . (5.4)We have performed a first scan of the full parameter space of the model. Given thetheoretical uncertainties mentioned in different sections of the paper, we have considered– 21 –s interesting the points that can explain the baryon asymmetry within a factor of 5. Forthis we have used the analytical solutions, even though in some regions of parameter spacethey will not be precise, since they are based on a perturbative expansion on the mixingangles of the matrices V and W . We have considered however a few cases where the anglesare not small and we find that the analytical solutions differ from the numerical ones onlyin some global numerical factor of a few , but the time dependence is very similar.Even with an analytical expression the exploration of the large parameter space is achallenge. We have used the package Multinest [11, 12] to perform a scan on the Casas-Ibarra parameters [36], where the Yukawa matrix is written as Y = − iU ∗ PMNS √ m light R ( z ij ) T √ M √ v . (5.5) m light is a diagonal matrix of the light neutrino masses and R is a complex orthogonalmatrix that depends on three complex angles z ij . We fix the light neutrino masses andmixings to the present best fit points in the global analysis of neutrino oscillation data ofref. [37] and leave as free parameters: three complex angles, the three phases of the PMNSmatrix, the lightest neutrino mass as well as the heavy Majorana masses that are allowedto vary in the range M i ∈ [0 . , | log | Y B ( t EW ) /Y exp B || (in the range ≤ .
5) and the MultiNest algorithm is optimised to sample properly when there are severalmaxima. For the determination of Y B we use the analytical results of the previous sections,for which the CP invariants are computed directly from the matrix elements of the V, W matrices that can be easily calculated by diagonalising the Yukawa mass matrix obtainedin the Casas-Ibarra parametrization. Since the mechanism to work requires that at leastone of the modes does not get to equilibrium before the electroweak phase transition werestrict the search to the range where one of the yukawa eigenvalues, y , is much smallerthan the others and the following conditions are satisfied y ≤ . y , y ] , (cid:88) i =1 , Γ i (cid:0) | V i | + | W i | (cid:1) ≤ . H u ( T EW ) . (5.6)Furthermore, since the kinetic equations neglect lepton number violating effects in therates, we impose additionally the constraint (cid:18) M i T EW (cid:19) Γ i (cid:28) H u ( T EW ) . (5.7)We first consider a case where one of the sterile neutrinos is effectively decoupled frombaryon number generation, that we can assume to be N . This can be achieved with thechoice of parameters: m = 0 , z i = 0 , R ( z ij ) → R ( z ij )( P ) , (5.8)for the IH(NH), where P is the 123 →
312 permutation matrix (only necessary for theNH). With this choice, only the terms corresponding to the CP invariants I (2)1 and I (3)1 contribute. This case is the one that has been considered in most previous works on the– 22 – igure 7: Points on the plane ∆ M = M − M versus M for which Y B > / × Y exp B (blue), Y B > Y exp B (green) and Y B > × Y exp B (red) for NH (left) and IH (right), with only two sterileneutrino species. subject [8, 9, 38, 28, 31], where the number of parameters is reduced to six: only onecomplex angle, two PMNS CP phases and two Majorana neutrino masses are relevant.It is believed that a large degeneracy of the two sterile neutrinos is needed to obtain thecorrect baryon asymmetry. In figure 7 we show the result of the scan under the conditionsof eq. (5.8) on the plane ∆ M ≡ M − M versus M for normal and inverted orderingsof the light neutrinos. The different colours correspond to values of Y B > / , , × Y exp B (blue,green,red). Successful leptogenesis is possible in a larger range of parameter spacefor IH than for NH. In the range shown our results agree reasonably well with those inref. [39] for the IH, while the range for NH looks a bit smaller. We see that there are asignificant number of points for which the degeneracy is mild for the IH. We have analysedmore carefully some of these points by solving the full numerical equations. We find thateven though these points correspond to cases where the angles in V, W are not small, theanalytical and numerical solution agree very well and have the same t dependence as shownin figure 8. Note that the numerical solution is difficult at large times for non-degeneratesolutions and the standard methods that we use fail. An optimised numerical methodis needed to solve the stiffness problem and this will be studied elsewhere. It is veryinteresting to correlate the baryon asymmetry with observables that could be in principlemeasured such as the Dirac CP phase of the PMNS matrix, the amplitude of neutrinolessdouble beta decay or the active-sterile mixings that control the probability for the heavysterile states to be observed in accelerators or in rare decays of heavy mesons. The effectivemass entering the 0 νββ decay is given by m ββ = (cid:88) i =1 U ei m i + (cid:88) i =1 U e ( i +3) M i M νββ ( M i ) M νββ (0) , (5.9)where M νββ are the Nuclear Matrix Elements (NMEs) defined in [40] . The first term The results for the NMEs computation in the interacting shell model [41, 42] are available in AppendixA of [40] – 23 – - - - - - - - - - - tt ew Tr @ m D Figure 8:
Comparison of the analytical (red-dashed) and numerical (blue-solid) solution for oneof the points with mild degeneracy and Y B ≥ Y exp B , corresponding to log ( M ( GeV )) = 0 . (∆ M ( GeV )) = − .
92 and yukawa couplings y = 1 . × − , y = 9 . × − . corresponds to the standard light neutrino contribution and the second is the contributionfrom the heavy states. U ei with i ≥ M but the one for M is almost identical.We see that most of the parameter space for successfull baryogenesis is not excluded bypresent constraints and that the active-sterile mixings tend to be larger for the IH. Asizeable region in the range of the GeV could be explored in the future experiment SHiPin the case of the IH and by LBNE near detectors. It is interesting to note that the lessdegenerate solutions can not have very small active-sterile mixing, as shown in figure 10,where we plot the points on the plane (cid:15) deg ≡ | M − M | / ( M + M ) versus the active-sterilemixing in the electron flavour. The degeneracy can be lifted to some extent at the expenseof larger yukawa couplings which also imply larger mixings.We have looked for direct correlations of the baryon asymmetry with the phases of thePMNS matrix. We have found that the distribution on the Dirac phase and the Majoranaphase are flat. This is due to the fact that the complex angle can provide the necessary CPviolation, even if the PMNS phases would vanish. The same is true for the effective mass ofneutrinoless double beta decay, which depends on the Majorana phase. A dedicated scanis needed to quantify how the putative measurement of various observables could constrainthe lepton asymmetry. This will be done elsewhere.In the general case, N is also relevant and the main difference with respect to theprevious situation is that there is a significantly enlarged parameter space where degeneracyis not necessary. This was already found in refs. [46] for some points of parameter space.– 24 – igure 9: Points on the plane | U e | (left), | U µ | (middle), | U τ | (right) versus M for which Y B is in the range [1 / − × Y exp B (blue) and [1 − × Y exp B (green) for NH (up) and IH (down), withonly two sterile neutrino species. The red bands are the present constraints [45], the solid blackline shows the reach of the SHiP experiment [43] and the solid red line is the reach of LBNE neardetector [44]. Figure 10:
Points on the plane (cid:15) deg = | M − M | M + M versus | U e | for which the asymmetry is in therange [1 / , × Y exp B in the range explored for IH. In figure 12 we show the points on the plane (∆ M , M ) for the general case. The active-sterile mixings are shown in figure 13. These mixings can be larger in this case, speciallyin the case of the NH. The SHiP prospects are therefore more promising in this context.As in the N = 2 case there is no direct connection between the asymmetry and the PMNSCP phases. On the other hand, the lightest neutrino mass is non-zero in this case, but therequirement that one yukawa needs to be significantly smaller than the others, eq. (5.6),– 25 – (GeV) m log-18 -17 -16 -15 -14 -13 -12 -11 -10020406080100120140160 Figure 11:
Distribution of m for points that satisfy Y B > Y exp B for the NH. Figure 12:
Points on the plane ∆ M = M − M versus M for which Y B > / × Y exp B (blue), Y B > Y exp B (green) and Y B > × Y exp B (red) for NH (left) and IH (right), in the general case withthree neutrinos. implies that the lightest neutrino mass must be small. In figure 11 we show the distributionof this quantity for those points that satisfy Y B ≥ Y exp B in the case of NH (the IH beingvery similar).
6. Conclusions
We have studied the mechanism of leptogenesis in a low-scale seesaw model that is arguablythe simplest extension of the Standard Model that can account for neutrino masses. ForMajorana neutrino masses in the GeV range, sizeable lepton asymmetries can be generatedin the production of these states some of which never reach thermal equilibrium before theelectroweak phase transition. Lepton asymmetries are efficiently transferred to baryonsvia sphaleron processes. This mechanism was proposed in [7, 8] and studied in many– 26 – igure 13:
Points on the plane | U e | (left), | U µ | (middle), | U τ | (right) versus M for which the Y B is in the range [1 / − × Y exp B (blue) and [1 − × Y exp B (green) for NH (up) and IH (down),with three sterile species. The red bands are the present constraints, the solid black line shows thereach of the SHiP experiment [43] and the solid red line is the reach of LBNE near detector [44]. works, but a full exploration of parameter space in the general case of three neutrinosis lacking. To this aim we have developed an accurate analytical approximation to thequantum kinetic equations which works both in the weak and strong washout regimes ofthe fast modes (there is always a slow mode that does not reach thermal equilibrium beforethe EW phase transition). It relies on a perturbative expansion in the mixing angles of thetwo unitary matrices that diagonalise the Yukawa matrix. This analytical approximationallows us to identify the relevant CP invariants, and explore with confidence the regimeof non-degenerate neutrino masses which is very challenging from the numerical point ofview. We have used this analytical solution to scan the full parameter space using theMultiNest package to identify the regions where the baryon asymmetry is within an orderof magnitude of the experimental value. We have performed first a scan in the simplersetting where one of the sterile neutrino decouples, which reduces the parameter space,and is the approximation that has been considered in most previous works on the subject,for example in the so-called ν MSM. Although baryon asymmetries tend to be larger in thecase of highly degenerate neutrinos, we find solutions with a very mild degeneracy thatalso correlate with a larger active-sterile mixing. These non-degenerate solutions appearfor an inverted ordering of the light neutrinos. On the other hand we do not observe adirect correlation with other observables, such as the PMNS CP phases nor the neutrinolessdouble beta decay amplitude.We have also performed a scan in the full parameter space, with the only requirementthat one of the yukawa matrix eigenvalues is very small, and that one mode will not reachequilibrium before the electroweak transition, for the washout not to be complete. The– 27 –ain difference with the simpler case of two neutrinos is that the parameter space withsuccessfull baryogenesis is significantly enlarged, in particular as regards non-degeneratespectra. Also the active-sterile mixings can reach larger values, particularly in the normalhierarchy case, improving the chances of future experiments such as SHiP or LBNE to findthe GeV sterile neutrinos. There is much less difference in this case between normal andinverted neutrino orderings and also no direct correlation with the PMNS phases. On theother hand, the requirement of a small yukawa eigenvalue implies that the lightest neutrinomass cannot be large.A number of refinements are needed to improve the precision of the determination ofthe baryon asymmetry. First a more precise determination of the scattering rates of thesterile neutrinos is required. Most previous studies, and this one, have included only top-quark scatterings, but it has been pointed out recently that gauge scatterings are also veryimportant. A correct treatment of these processes in the kinetic equations is necessary.Also the kinetic equations neglect effects of O (( M i /T ) ). Such effects are not so smallfor masses in the GeV near the electroweak phase transition and their effect should bequantified. Finally, spectator processes and the asymmetries of fields other than the sterileneutrinos and LH leptons have not been taken into account in the kinetic equations. Aproper treatment could easily bring corrections of O (1). Finally, a more ambitious scanof parameter space should define more accurately the limits of eq. (5.6) for successfullbaryogenesis. These effects will be studied in the future. Acknowledgments
We wish to thank R. Ruiz de Austri, J. Mart´ın-Albo and J.M. Mart´ı for their help withthe used software, and D. Bodeker, V. Domcke, M. Drewes and M. Laine for useful dis-cussions. This work was partially supported by grants FPA2011-29678, FPA2014-57816-P,PROMETEOII/2014/050, CUP (CSD2008-00037), ITN INVISIBLES (Marie Curie Ac-tions, PITN-GA-2011-289442), the INFN program on Theoretical Astroparticle Physics(TASP) and the grant 2012CPPYP7 (
Theoretical Astroparticle Physics ) under the pro-gram PRIN 2012 funded by the Italian Ministry of Education, University and Research(MIUR). PH acknowledges the support of the Aspen Center for Physics (National ScienceFoundation grant PHY-1066293), where this work was completed. MK thanks Fermilabfor hosting her while part of this work was done.
A. Results for the perturbative integrals
A.1 One dimensional integrals
We just need them up to O ( β/α ) : J ( α , β , t ) (cid:39) J ( α , t ) + β J ( α , t ) + β J ( α , t ) + ∆ J ( α , β , t, t ) , (A.1)– 28 –ith∆ J ( α , β , t, t ) = (cid:88) n β n n ! J n ( α , t, t ) (cid:39) i (cid:88) n β n n ! e i α t α t − n − e i α t α t − n + O ( t − , t − )= i e i α t + β t α t − e i α t + β t α t . (A.2)We can factor out the α dependence and define: J ( α, t ) = 1 | α | / (cid:16) Re (cid:2) J (1 , t | α | / ) (cid:3) + i sign( α )Im (cid:2) J (1 , t | α | / ) (cid:3)(cid:17) . (A.3) A.2 Two dimensional integrals
We just need them up to O ( β/α ): J ( α , β , α , β , t ) (cid:39) J ( α , α , t ) + β J ( α , α , t ) + β J ( α , α , t )+ ∆ J ( α , β , α , β , t, t ) , (A.4)where if (cid:80) i α i (cid:54) = 0:∆ J ( α , β , α , β , t, t ) = J ( α , β , t ) + i e i α t + β t α t ∆ J ( α , β , t, t ) − i i e i (cid:80) i αit + (cid:80) i β i t α (cid:80) i α i t − i e i (cid:80) i αit + (cid:80) i β i t α (cid:80) i α i t , (A.5)and for those terms where (cid:80) i α i = 0∆ J ( α , β , α , β , t, t ) = J ( α , β , t ) + i e i α t + β t α t ∆ J ( α , β , t, t ) − iα (cid:32)(cid:90) tt e (cid:80) i β i x x (cid:33) . (A.6)We can factorize the α -dependence: J ( − α, α, t ) = 1 | α | / (cid:16) Re (cid:2) J ( − , , t | α | / ) (cid:3) + i sign( α )Im (cid:2) J ( − , , t | α | / ) (cid:3)(cid:17) ,J ( − α, α, t ) = 1 | α | (cid:16) Re (cid:2) J ( − , , t | α | / ) (cid:3) + i sign( α )Im (cid:2) J ( − , , t | α | / ) (cid:3)(cid:17) ,J ( − α, α, t ) = 1 | α | (cid:16) Re (cid:2) J ( − , , t | α | / ) (cid:3) + i sign( α )Im (cid:2) J ( − , , t | α | / ) (cid:3)(cid:17) , (A.7)and reduce the integrals to the basic ones.– 29 – .3 Three dimensional integrals We need the integrals up to O ( β/α ) in this case. We can use the relation: J ( α , α , α , t ) = J ( α , t ) J ( α , α , t ) − (cid:90) t dx e iα x J ( α , x ) J ( α , x ) . 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