Electroweak Resonant Leptogenesis in the Singlet Majoron Model
aa r X i v : . [ h e p - ph ] J un MAN/HEP/2008/09 arXiv:0805.1677
May 2008
Electroweak Resonant Leptogenesisin the Singlet Ma joron Model
Apostolos Pilaftsis
School of Physics and Astronomy, University of Manchester,Manchester M13 9PL, United Kingdom
ABSTRACT
We study resonant leptogenesis at the electroweak phase transition in the singlet Majoronmodel with right-handed neutrinos. We consider a scenario, where the SM gauge groupand the lepton number break down spontaneously during a second-order electroweak phasetransition. We calculate the flavour- and temperature-dependent leptonic asymmetries, byincluding the novel contributions from the transverse polarisations of the W ± and Z bosons.The required resummation of the gauge-dependent off-shell heavy-neutrino self-energies isconsistently treated within the gauge-invariant framework of the Pinch Technique. Takinginto consideration the freeze-out dynamics of sphalerons, we delineate the parameter spaceof the model that is compatible with successful electroweak resonant leptogenesis. Thephenomenological and astrophysical implications of the model are discussed. PACS numbers: 11.30.Er, 14.60.St, 98.80.Cq Introduction
Leptogenesis [1] provides an elegant framework to consistently address the observed BaryonAsymmetry in the Universe (BAU) [2] in minimal extensions of the Standard Model (SM) [3].According to the standard paradigm of leptogenesis, there exist heavy Majorana neutrinosof masses close to the Grand Unified Theory (GUT) scale M GUT ∼ that decay outof equilibrium and create a net excess of lepton number ( L ), which gets reprocessed intothe observed baryon number ( B ), through the ( B + L )-violating sphaleron interactions [4].The attractive feature of such a scenario is that the GUT-scale heavy Majorana neutrinoscould also explain the observed smallness in mass of the SM light neutrinos by means ofthe so-called seesaw mechanism [5].The original GUT-scale leptogenesis scenario, however, runs into certain difficulties,when one attempts to explain the flatness of the Universe and other cosmological data [2]within supergravity models of inflation. To avoid overproduction of gravitinos e G whose latedecays may ruin the successful predictions of Big Bang Nucleosynthesis (BBN), the reheattemperature T reh of the Universe should be lower than 10 –10 GeV, for m e G = 8–0.2 TeV [6].This implies that the heavy Majorana neutrinos should accordingly have masses as low as T reh < ∼ GeV, thereby rendering the relation of these particles with GUT-scale physicsless natural. On the other hand, it proves very difficult to directly probe the heavy-neutrinosector of such a model at high-energy colliders, e.g. at the LHC or ILC, or in any otherforeseeable experiment.A potentially interesting solution to the above problems may be obtained within theframework of resonant leptogenesis (RL) [7]. The key aspect of RL is that self-energy ef-fects dominate the leptonic asymmetries [8], when two heavy Majorana neutrinos happento have a small mass difference with respect to their actual masses. If this mass differencebecomes comparable to the heavy neutrino widths, a resonant enhancement of the leptonicasymmetries takes place that may reach values O (1) [7, 9]. An indispensable feature of RLmodels is that flavour effects due to the light-to-heavy neutrino Yukawa couplings [10] playa dramatic role and can modify the predictions for the BAU by many orders of magni-tude [11, 12]. Most importantly, these flavour effects enable the modelling [12] of minimalRL scenarios with electroweak-scale heavy Majorana neutrinos that could be tested at theLHC [13, 14] and in other non-accelerator experiments, while maintaining agreement withthe low-energy neutrino data. Many variants of RL have been proposed in the litera-ture [15, 16], including soft leptogenesis [17] and radiative leptogenesis [18].In spite of the many existing studies, leptogenesis models face in general a seriousrestriction concerning the origin of the required CP and L violation. If CP or L violation2ere due to the spontaneous symmetry breaking (SSB) of the SM gauge group, a net L asymmetry could only be generated during the electroweak phase transition (EWPT),provided the heavy Majorana neutrinos are not too heavy such that they have not alreadydecayed away while the Universe was expanding. ∗ In this paper we show how RL constitutes an interesting alternative to provide aviable solution to the above problem as well. For definiteness, we consider a minimalextension of the SM with right-handed neutrinos and a complex singlet field Σ. Themodel possesses a global lepton symmetry U(1) l which gets spontaneously broken throughthe vacuum expectation value (VEV) of Σ, giving rise to the usual ∆ L = 2 Majoranamasses. Because of the SSB of the U(1) l , the model predicts a true massless Goldstoneboson, the Majoron. Therefore, this scenario is called the singlet Majoron model in theliterature [20, 21]. Depending on the particular structure of the Higgs potential, the VEVof Σ may be related to the VEV of the SM Higgs doublet Φ. Such a relation, for example,arises if the bilinear operator Σ ∗ Σ is small or absent from the Higgs potential. In this case,the breaking of L occurs during the EWPT. For the model under study and given the LEPlimit [22] on the SM Higgs boson M H > ∼
115 GeV, the EWPT is expected to be secondorder and hence continuous from the symmetric phase to the broken one [23].We should now notice that all SM fermions and right-handed neutrinos have nochiral masses above the EWPT and therefore the generation of a net leptonic asymmetryis not possible. Consequently, in this model successful baryogenesis can result from RL atthe EWPT. Although the singlet Majoron model that we will be studying here violatesCP explicitly, the results of our analysis can straightforwardly apply to models with anextended Higgs sector that realise spontaneous CP violation at the electroweak scale.The paper is organised as follows: Section 2 presents the basic features of the singletMajoron model with right-handed neutrinos, including the interaction Lagrangians that arerelevant to the calculation of the leptonic asymmetries in Section 3. Moreover, in Section 3we consider the novel contributions to the leptonic asymmetries, coming from the transversepolarisations of the W ± and Z bosons. In the same context, the resummation of thegauge-dependent off-shell heavy-neutrino self-energies [24, 25] (which remains an essentialoperation in RL) is performed within the so-called Pinch Technique (PT) framework [26].In Section 4 we analyse the Boltzmann dynamics of the sphaleron effects on RL and presentpredictions for the BAU. Section 5 is devoted to the phenomenological and astrophysicalimplications of the singlet Majoron model. Finally, Section 6 contains our conclusions. ∗ An exception to this argument may result from a phase transition that is strongly first order. However,such a scenario is not feasible within the SM with singlet neutrinos [19] (see also our discussion below). The Singlet Ma joron Model
Here we describe the basic features of the singlet Majoron model [20, 21] augmented witha number n R of right-handed neutrinos ν αR (with α = 1 , , . . . , n R ) that will be relevantto our study. As mentioned in the introduction, the singlet Majoron model contains onecomplex singlet field Σ in addition to the SM Higgs doublet Φ. Although Σ is not chargedunder the SM gauge group SU(2) L ⊗ U(1) Y , it still carries a non-zero quantum numberunder the global lepton symmetry U(1) l . More explicitly, the scalar potential of the modelis given by − L V = m Φ † Φ + m Σ ∗ Σ + λ Φ † Φ) + λ Σ ∗ Σ) − δ Φ † Φ Σ ∗ Σ . (2.1)In order to minimise the potential (2.1), we first linearly decompose the scalar fields asfollows: Φ = G + v √ φ + iG √ , Σ = w √ σ + iJ √ . (2.2)Then, the extremal or tadpole conditions may easily be calculated by T φ ≡ − * ∂ L V ∂φ + = v m + λ Φ v − δ w ! = 0 , (2.3) T σ ≡ − * ∂ L V ∂σ + = w m + λ Σ w − δ v ! = 0 . (2.4)If m or m are negative, the tadpole conditions (2.3) and (2.4) imply that the groundstate of the scalar potential breaks spontaneously the local SU(2) L ⊗ U(1) Y and the globalU(1) l symmetries, through the non-zero VEVs v and w , respectively.Expanding the fields Φ and Σ about their VEVs, we obtain three would-be Goldstonebosons G ± and G , which become the longitudinal polarisations of W ± and Z bosons,and one true massless Goldstone boson J associated with the SSB of U(1) l . This masslessCP-odd field J is called the Majoron in the literature [20, 21]. In addition, there are twoCP-even Higgs fields H and S , whose masses are determined by the diagonalisation of themass matrix M = λ Φ v − δ vw − δ vw λ Σ w , (2.5)where M is defined in the weak basis ( φ , σ ). The Higgs mass eigenstates H and S arerelated to the states φ and σ , through the orthogonal transformation: φσ = c θ − s θ s θ c θ HS , (2.6)4ith t β = s β /c β = v/w and t θ = 2 δ t β λ Σ − λ Φ t β . (2.7)In the above we used the short-hand notation: s x ≡ sin x , c x ≡ cos x and t x ≡ tan x .Moreover, the squared mass eigenvalues of the CP-even H and S bosons may easily becalculated from M in (2.5) and are given by M H,S = v " λ Φ + λ Σ t − β ± r(cid:16) λ Φ − λ Σ t − β (cid:17) + δ t − β . (2.8)The requirement that M H,S be positive gives rise to the inequality conditions, λ Φ , Σ > , λ Φ λ Σ > δ , (2.9)for the quartic couplings of the potential. In this context, we note that if | m | ≪ ( δ/λ Φ ) | m | such that m can be completely neglected in the scalar potential, the VEV w of Σ is then entirely determined by the VEV v of Φ and the quartic couplings λ Σ and δ , viz. w = s δλ Σ v . (2.10)This is an interesting scenario, since the ratio t β = v/w = q λ Σ /δ does not strongly dependon the temperature T , as opposed to what happens to the VEVs v and w individually. Infact, as long as λ Φ , Σ , δ ≪
1, the thermally-corrected effective potential can be expanded, toa very good approximation, in powers of T /m . In such a high- T expansion, the quarticcouplings of L V turn out to be T -independent [27] and hence t β does not depend on T .We now turn our attention to the neutrino Yukawa sector of the model, which isnon-standard. After SSB, it is given in the unitary gauge by − L Y = φv ¯ ν iL ( m D ) iα ν αR + σ + iJ w ¯ ν CαR ( m M ) αβ ν βR + H.c. , (2.11)where summation over repeated indices is understood. Hereafter we use Latin indicesto label the left-handed neutrinos, e.g. ν iL , and Greek indices for the right-handed ones,e.g. ν αR . Observe that the spontaneous breaking of U(1) l generates lepton-number-violating∆ L = 2 Majorana masses ( m M ) αβ in addition to the lepton-number-preserving ∆ L = 0Dirac masses ( m D ) iα .The model under discussion predicts a number (3 + n R ) of Majorana neutrinos whichwe collectively denote by n I , with I = i , α . Their physical masses are obtained from thediagonalisation of the neutrino mass matrix M ν = m D m TD m M , (2.12)5y means of the unitary transformation U ν T M ν U ν = c M ν , where c M ν is a non-negativediagonal matrix. The neutrino mass eigenstates ( n I ) R and ( n I ) L are related to the states ν iL , ( ν iL ) C , ν αR and ( ν αR ) C through ν CL ν R I = U νIJ ( n J ) R , ν L ν CR I = U ν ∗ IJ ( n J ) L . (2.13)Assuming the seesaw hierarchy ( m D ) iα / ( m M ) αβ ≪
1, the model predicts 3 light statesthat are identified with the observed light neutrinos ( n i ≡ ν i ), and a number n R of heavyMajorana neutrinos ( n α ≡ N α ) with masses of order ( m M ) αβ = ρ αβ w , where ρ αβ = ρ βα arethe Yukawa couplings of Σ to right-handed neutrinos.To obtain an accurate light and heavy neutrino mass spectrum within the context ofmodels of electroweak RL, it is important to go beyond the leading seesaw approximation.To this end, we need first to perform a block diagonalisation and cast M ν into the form: M ν → m ν m N . (2.14)This can be achieved by introducing the unitary matrix V [28]: V = ( + ξ ∗ ξ T ) − / ξ ∗ ( n R + ξ T ξ ∗ ) − / − ξ T ( + ξ ∗ ξ T ) − / ( n R + ξ T ξ ∗ ) − / , (2.15)where ξ is an arbitrary 3 × n R matrix. The expressions ( + ξ ∗ ξ T ) − / and ( n R + ξ T ξ ∗ ) − / are defined in terms of a Taylor series expansion about the N × N identity matrix N .These infinite series converge provided the norm || ξ || is much smaller than 1, where || ξ || ≡ q Tr( ξξ † ). This condition is naturally fulfilled within the seesaw framework [29]. Blockdiagonalisation of the matrix M ν given in (2.12) implies that the { } block element of V T M ν V vanishes, or equivalently that m D − ξ m M − ξ m TD ξ ∗ = 0 . (2.16)Equation (2.16) determines ξ in terms of m D and m M . It can be solved iteratively, withthe first iteration given by ξ = m D m − M − m D m − M m TD m ∗ D m ∗ − M m − M . (2.17)Note that the second term on the RHS of (2.17) is suppressed by the ratio of the light-to-heavy neutrino masses and can thus be safely neglected in numerical estimates. Uponblock diagonalisation, the block mass “eigen-matrices” are m N = (cid:16) n R + ξ † ξ (cid:17) − / (cid:16) m M + m TD ξ ∗ + ξ † m D (cid:17) (cid:16) n R + ξ T ξ ∗ (cid:17) − / , (2.18) m ν = − (cid:16) + ξξ † (cid:17) − / (cid:16) m D ξ T + ξm TD − ξm M ξ T (cid:17) (cid:16) + ξ ∗ ξ T (cid:17) − / = − ξ m N ξ T , (2.19)6here we used (2.16) to arrive at the last equality of (2.19). Keeping the leading orderterms in an expansion of m N in powers of m D m − M , we find that m N = m M + 12 (cid:16) m † D m − M m D + m TD m − M m ∗ D (cid:17) , m ν = − m D m − M m N m − M m TD . (2.20)These last expressions are used to calculate the light and heavy neutrino mass spectra ofthe RL scenarios discussed in Section 4.In order to calculate the leptonic asymmetries in the next section, we need to know theLagrangians that govern the interactions of the Majorana neutrinos n I and charged leptons l = e, µ, τ with: ( i ) the W ± and Z bosons; ( ii ) their respective would-be Goldstone bosons G ± and G ; ( iii ) the CP-odd Majoron particle J ; ( iv ) the CP-even Higgs fields H and S .In detail, these interaction Lagrangians are given by [21] L W ∓ = − g w √ W − µ ¯ l B lI γ µ P L n I + H.c. , (2.21) L Z = − g w θ w Z µ ¯ n I γ µ (cid:16) C IJ P L − C ∗ IJ P R (cid:17) n J , (2.22) L G ± = − g w √ M W G − ¯ l B lI (cid:16) m l P L − m I P R (cid:17) n I + H.c. , (2.23) L G = − i g w M W G ¯ n I " C IJ (cid:16) m I P L − m J P R (cid:17) + C ∗ IJ (cid:16) m J P L − m I P R (cid:17) n J , (2.24) L J = − i g w M W t β J ¯ n I " C IJ (cid:16) m I P L − m J P R (cid:17) + C ∗ IJ (cid:16) m J P L − m I P R (cid:17) + δ IJ m I γ n J , (2.25) L H = − g w M W ( c θ − s θ t β ) H ¯ n I " C IJ (cid:16) m I P L + m J P R (cid:17) + C ∗ IJ (cid:16) m J P L + m I P R (cid:17) − i t β t − θ − t β δ IJ m I γ n J , (2.26) L S = − g w M W ( s θ + c θ t β ) S ¯ n I " C IJ (cid:16) m I P L + m J P R (cid:17) + C ∗ IJ (cid:16) m J P L + m I P R (cid:17) + i t β t θ + t β δ IJ m I γ n J , (2.27)where P L,R = (1 ∓ γ ), g w is the SU(2) L gauge coupling of the SM and B lI = V llk U ν ∗ kI , C IJ = U νkI U ν ∗ kJ . (2.28)7n (2.28) V l is a 3-by-3 unitary matrix that occurs in the diagonalisation of the chargedlepton mass matrix M l . Without loss of generality, we assume throughout the presentstudy that M l is positive and diagonal, which implies that V l = . Finally, we commenton the limit of t β →
0. It is easy to see from (2.7) that this limit leads to t θ → S and J decouple from matter; only the Higgs field H couples to Majorana neutrinosand to the rest of the SM fermions (cf. [13]). In this section we calculate the leptonic asymmetries produced by the decays of the heavyMajorana neutrinos during a second-order EWPT. The novel aspect of such a calculationis that, in stark contrast to the conventional leptogenesis scenario, the W ± and Z bosonsalso contribute to the decays and leptonic asymmetries of the heavy Majorana neutrinos.This fact raises new issues related to the gauge invariance of off-shell Green functions whichare here addressed within the so-called Pinch Technique (PT) framework [26].Since sphalerons act on the left-handed SM fermions converting an excess in leptonsinto that of baryons, we only need to consider the decays of the heavy Majorana neutrinos N α into the left-handed charged leptons l − L and light neutrinos ν lL . In detail, we haveto calculate the partial decay width of the heavy Majorana neutrino N α into a particularlepton flavour l ,Γ lN α = Γ( N α → l − L W + , G + ) + Γ( N α → ν lL Z, G, J, H, S ) . (3.1)To compute Γ lN α , it proves more convenient to first calculate the absorptive part Σ abs αβ ( p )of the heavy Majorana-neutrino self-energy transition N β → N α in the Feynman–’t Hooftgauge ξ = 1, where p µ is the 4-momentum carried by N α,β . The Feynman–’t Hooft gauge isnot a simple choice of gauge, but the result obtained in the gauge-independent frameworkof the PT [26], within which issues of analyticity, unitarity and CPT invariance can self-consistently be addressed [24, 25].Neglecting the small charged-lepton and light-neutrino masses, Σ abs αβ ( p ) acquires thesimple spinorial structure:Σ abs αβ ( p ) = A αβ ( s ) p P L + A ∗ αβ ( s ) p P R , (3.2)where s = p is the squared Lorentz-invariant mass associated to the self-energy transi-tion N β → N α . Considering the Feynman graphs shown in Fig. 1 and the interaction8 β l − L , ν lL N α W + , Z (a) N β l − L , ν lL N α G + , G, J, H, S (b)Figure 1: Feynman graphs that determine the 1-loop absorptive part A αβ ( s ) of the heavyMajorana-neutrino self-energy Σ αβ ( p ) . Lagrangians (2.21)–(2.27), the absorptive transition amplitudes A αβ ( s ) are calculated tobe A αβ ( s ) = α w X l = e,µ,τ ( B ∗ lα B lβ " − M W s ! θ ( s − M W ) + 2 M Z M W − M Z s ! θ ( s − M Z ) + m N α m N β M W B lα B ∗ lβ " − M W s ! θ ( s − M W ) + − M Z s ! θ ( s − M Z ) + t β θ ( s )+ ( c θ − s θ t β ) − M H s ! θ ( s − M H ) + ( s θ + c θ t β ) − M S s ! θ ( s − M S ) , (3.3)where α w = g w / (4 π ) is the SU(2) L fine-structure constant and θ ( x ) is the usual stepfunction: θ ( x ) = 1 for x >
0, whilst θ ( x ) = 0 if x ≤
0. In the calculation of A αβ ( s ),we used the fact that B lα = C ν l α + O ( C ν l α ), which is an excellent approximation in thephysical charged-lepton mass basis.We should bear in mind that all masses involved on the RHS of (3.3) depend onthe temperature T , through the T -dependent VEVs v ( T ) and w ( T ) related to the Higgsdoublet Φ and the complex singlet Σ, respectively [cf. (4.8) and (4.17)]. In the symmetricphase of the theory, i.e. for temperatures above the electroweak phase transition, theseVEVs vanish and the absorptive transition amplitude becomes A αβ ( s ) = α w m TD m ∗ D ) α β M W t β ! . (3.4)Note that this last formula is only valid in the weak basis in which the Majorana massmatrix m M is diagonal.To account for unstable-particle-mixing effects between heavy Majorana neutrinos,we follow [7, 9] and define the resummed effective couplings B lα and their CP-conjugateones B clα related to the vertices W − l L N α and W + ( l L ) C N α , respectively. For a symmetric9odel with 3 left-handed and 3 right-handed neutrinos, the effective couplings B lα exhibitthe same analytic dependence on the absorptive transition amplitudes A αβ as the one foundin [9]: † B lα = B lα − i X β,γ =1 | ε αβγ | B lβ (3.5) × m α ( m α A αβ + m β A βα ) − iR αγ h m α A γβ ( m α A αγ + m γ A γα ) + m β A βγ ( m α A γα + m γ A αγ ) i m α − m β + 2 i m α A ββ + 2 i Im R αγ (cid:16) m α | A βγ | + m β m γ Re A βγ (cid:17) , where all transition amplitudes A αβ , A βγ etc are evaluated at s = m N α ≡ m α and R αβ = m α m α − m β + 2 i m α A ββ ( m α ) . (3.6)Moreover, | ε αβγ | is the modulus of the usual Levi–Civita anti-symmetric tensor. The re-spective CP-conjugate effective couplings B cli are easily obtained from (3.5) by replacingthe ordinary W − -boson couplings B lα and A αβ ( s ) by their complex conjugates. In thedecoupling limit of m N ≫ m N , , we recover the analytic results known for a model with2 right-handed neutrinos [7, 9], where the effective couplings B l , are given by B l = B l − i B l m N (cid:16) m N A ( m N ) + m N A ( m N ) (cid:17) m N − m N + 2 im N A ( m N ) , (3.7) B l = B l − i B l m N (cid:16) m N A ( m N ) + m N A ( m N ) (cid:17) m N − m N + 2 im N A ( m N ) . (3.8)In all our results, we neglect the 1-loop corrections to the vertices W ± l L N α , Zν lL N α etc,whose absorptive parts are numerically insignificant in leptogenesis, but essential otherwiseto ensure gauge invariance and unitarity within the PT framework [25].In terms of the resummed effective couplings B lα and B clα and the absorptive transitionamplitudes A αβ ( s ), the partial decay widths Γ lN α and their CP-conjugates Γ lN α are now givenby Γ lN α = m N α A αα ( m N α ; B lα ) , Γ lN α = m N α A αα ( m N α ; B clα ) , (3.9)where the dependence of the absorptive transition amplitudes on B lα and B clα has explicitlybeen indicated. Note that no summation over the individual charged leptons and lightneutrinos running in the loop should be performed when calculating Γ lN α and Γ lN α using (3.3) † Here we eliminate a typo that occurred in [9], where R αγ in the numerator of the fraction needs bemultiplied with − i . δ lN α = ∆Γ lN α Γ N α = | B lα | − | B clα | P l = e,µ,τ (cid:16) | B lα | + | B clα | (cid:17) , (3.10)with Γ N α = X l = e,µ,τ (cid:16) Γ lN α + Γ lN α (cid:17) , ∆Γ lN α = Γ lN α − Γ lN α . (3.11)Notice that both Γ N α and δ lN i do in general depend on the temperature T , through the T -dependent masses, during a second-order electroweak phase transition. More details onthis issue will be presented in the next section. In this section we present the relevant Boltzmann equations (BEs) that will enable us toevaluate the lepton-to-photon and baryon-to-photon ratios, η L l and η B , during a second-order EWPT. In our numerical estimates, we only include the dominant collision termsrelated to the 1 ↔ N α . Wealso neglect chemical potential contributions from the right-handed charged leptons andquarks [3]. A complete account of the aforementioned subdominant effects may be givenelsewhere.To start with, we first write down the BEs that govern the photon normalised numberdensities η N α and η ∆ L l for the heavy Majorana neutrinos N α and the left-handed leptons l L , ν lL , respectively: dη N α dz = z D N α H ( T c ) − η N α η eq N α ! , (4.1) dη ∆ L l dz = z D N α H ( T c ) " η N α η eq N α − ! δ lN α − B lN α η ∆ L l . (4.2)Although our conventions and notations follow those of [12], there are several key differencespertinent to our EWPT scenario that need to be stressed here. Specifically, we express the T -dependence of the BEs (4.1) and (4.2) in terms of the dimensionless parameter z : z = T c T , (4.3)where T c is the critical temperature of the EWPT to be determined below [cf. (4.6)].The parameter H ( T c ) ≈ × T c /M P is the Hubble constant at T = T c , where M P =11 . × GeV is the Planck mass. The parameter B lN α denotes the branching fraction ofthe decays of the heavy Majorana neutrino N α into a particular lepton flavour l , i.e. B lN α =(Γ lN α + Γ lN α ) / Γ N α . Moreover, η eq N α is the equilibrium number density of the heavy neutrino N α , normalised to the number density of photons n γ = 2 T /π : η eq N α = m N α ( T )2 T K m N α ( T ) T ! , (4.4)where K n ( x ) is the n th-order modified Bessel function [30]. Finally, D N α is the T -dependentcollision term related to the decay and inverse decay of the heavy Majorana neutrino N α : D N α = Γ N α ( T ) n γ g N α Z d p N α (2 π ) m N α ( T ) E N α ( T ) e − E Nα ( T ) /T = m N α ( T )2 T Γ N α ( T ) K m N α ( T ) T ! , (4.5)where E N α ( T ) = [ | p N α | + m N α ( T ) ] / and g N α = 2 is the number of helicities of N α .Our next step is to include the effect of the ( B + L )-violating sphalerons [4] on thelepton-number densities produced by the decays of N α during the EWPT. In particular,our interest is to implement the temperature dependence of the rate of B + L violation justbelow the critical temperature T c , where T c is given by [31] T c = v
12 + 3 g w λ Φ + g ′ λ Φ + h t λ Φ ! − / . (4.6)In the above, g ′ is the U(1) Y gauge coupling and h t is the top-quark Yukawa coupling.We should notice that Φ-Σ mixing effects have been omitted in (4.6), which is a goodapproximation for scenarios with δ/λ Φ ≪ B + L )-violating sphaleron transitions canbe obtained for temperatures satisfying the double inequality M W ( T ) ≪ T ≪ M W ( T ) α w , (4.7)where α w = g w / π is the SU(2) L fine structure constant, M W ( T ) = g w v ( T ) / T -dependent W -boson mass and v ( T ) = v − T T c ! / (4.8)is the T -dependent VEV of the Higgs field. In detail, the rate of B + L violation per unitvolume is [32] γ ∆( B + L ) = ω − π N tr ( N V ) rot (cid:18) α w T π (cid:19) α − e − E sp /T κ . (4.9)12 Φ /g w ω − N rot N tr κ A × M W Table 1:
Values of the parameters occurring in (4.9) for λ Φ /g w = 0 . , which correspondsto a SM Higgs-boson mass of 120 GeV when δ = 0 . Given the double inequality (4.7), this last expression is valid for temperatures T < ∼ T c .Following the notation of [32], the parameters ω − , N tr and N rot that occur in (4.9) arefunctions of λ Φ /g w , V rot = 8 π and α = α w T / [2 M W ( T )]. The quantity E sp is the T -dependent energy of the sphaleron and is determined by E sp = A M W ( T ) α w , (4.10)where A is a function of λ Φ /g w and is O (1), for values of phenomenological interest. Thedependence of the parameter κ on λ Φ /g w has been calculated in [32, 33], and the results ofthose studies are summarised in Table 1, for λ Φ /g w = 0 . M H of 120 GeV in the vanishing limit of a Φ-Σ mixing.Since the SM Higgs-boson mass is M H > ∼
115 GeV, it can be shown [23] that the EWPTin the SM is not first order, but continuous from v ( T c ) = 0 to v , without bubble nucleationand the formation of large spatial inhomogeneities in particle densities. Therefore, weuse the formalism developed in [12], where the ( B + L )-violating sphaleron dynamics isdescribed in terms of spatially independent B - and L -number densities η B and η L j . Moreexplicitly, the BEs of interest to us are [12]: dη B dz = − z Γ ∆( B + L ) H ( T c ) (cid:20) η B + 2851 η L + v ( T ) T (cid:18) η B + 16187 η L (cid:19) (cid:21) , (4.11) dη L i dz = dη ∆ L i dz + 13 dη B dz , (4.12)where η L = P l = e,µ,τ η L l is the total lepton asymmetry andΓ ∆( B + L ) = 1683132 T + 51 T v ( T ) γ ∆( B + L ) . (4.13)We observe that in the limit Γ ∆( B + L ) /H ( T c ) → ∞ and for T > T c , the conversion ofthe lepton-to-photon ratio η L to the baryon-to-photon ratio η B is given by the knownrelation [34, 35]: η B = − η L . (4.14)13ikewise, when 1 < ∼ z < ∼ . κ = 1, it is Γ ∆( B + L ) /H ( T c ) ≫ η B is then related to the total lepton-to-photon ratio η L by η B = − v ( T ) T ! v ( T ) T ! − η L . (4.15)For z > ∼ .
7, sphaleron effects get sharply out of equilibrium and η B freezes out. To accountfor the T -dependent ( B + L )-violating sphaleron effects, our numerical estimates will bebased on the BEs (4.1), (4.2), (4.11) and (4.12).In the singlet Majoron model, the restoration of the global symmetry U(1) l will occurfor temperatures above a critical temperature T lc that could in general differ from T c of theSM gauge group given in (4.6). For example, in the absence of a doublet-singlet mixing,the critical temperature related to the SSB of U(1) l is [27] T lc = − m λ Σ . (4.16)Consequently, the T -dependence of w ( T ) for T < T lc will be analogous to v ( T ) in (4.8), i.e. w ( T ) = w − T ( T lc ) ! / . (4.17)However, if m vanishes, the singlet VEV w ( T ) and the doublet VEV v ( T ) will be relatedby an expression very analogous to (2.10), namely w ( T ) ≈ t − β v ( T ) . (4.18)As was mentioned after (2.10), the above relation becomes exact in a high- T expansion ofthe thermally corrected effective potential. Such an expansion is a very good approximationto the level of a few % for perturbatively small quartic couplings [27]. As a consequence, theSM gauge group and the global lepton symmetry U(1) l will both break down spontaneouslyvia the same second-order electroweak phase transition, with T lc = T c . Even though thefocus of the paper will be on this class of scenarios, we will comment on possible differencesfor models with T lc = T c .If T lc = T c , the heavy neutrino masses m N α , the gauge-boson masses M W,Z and theHiggs masses M H,S all scale with the same T -dependent factor, (1 − T /T c ) / , for temper-atures T < T c of our interest. Hence, the T -dependence drops out exactly in the expres-sion (3.3) of the absorptive transition amplitudes A αβ ( m N γ ), and likewise in the leptonicasymmetries δ lN α and the branching fractions B lN α . However, as can be seen from (4.5), thecollision terms D N α exhibit a non-trivial T -dependence that needs be carefully implementedin the BEs. 14or our numerical estimates of the BAU, we consider the 3-generation flavour scenarioof the RL model discussed in [11, 12]. Specifically, the Majorana sector is assumed to beapproximately SO(3) symmetric, m M = m N + ∆ M S , (4.19)where ∆ M S are small SO(3)-breaking terms that are of order m † D m D /m N as these arenaturally expected from (2.20). Plugging (4.19) into (2.20), we find that, to leading orderin ∆ M S , the heavy neutrino mass matrix m N deviates from m N by an amount δ m N = ∆ M S + 12 m N (cid:16) m † D m D + m TD m ∗ D (cid:17) . (4.20)It is interesting to observe that possible renormalisation-group (RG) running effects froma high-energy scale M X , e.g. GUT scale, down to m N will induce a negative contributionto δ m N [18], i.e.( δ m N ) RG = − α w π m N M W (cid:16) m † D m D + m TD m ∗ D (cid:17) ln M X m N ! . (4.21)For M X = M GUT ∼ and m N = 80–150 GeV, the RG-induced terms are typicallysmaller by a factor ∼ . m D ) iα , which in our case possess an approximate U(1)-symmetric flavour pattern [11]: m D = v √ a e − iπ/ a e iπ/ b e − iπ/ b e iπ/ c e − iπ/ c e iπ/ + δm D , (4.22)where the 3-by-3 matrix δm D , δm D = v √ ε e ε µ ε τ , (4.23)violates the U(1) symmetry by small terms of order of the electron mass m e . Instead, theU(1)-symmetric Yukawa couplings a and b can be as large as the τ -lepton Yukawa coupling m τ /v , i.e. of order 10 − –10 − . For successful RL, it was found [11, 12] that the parameter c needs to be taken of the order of the electron Yukawa coupling m e /v . It is important to15 iggs λ Φ λ Σ δ tan β M H [GeV] M S [GeV] Sector λ Φ 115 λ Φ / √ Neutrino ( δ m N ) m N ( δ m N ) m N ( δ m N ) m N ( δ m N ) m N ( δ m N ) m N ( δ m N ) m N Sector − − − − × − × − (6 . − . i ) 5 . × − × − a b c ε e ε µ ε τ × − − × q m N
100 GeV × q m N
100 GeV × q ∆ m N
100 GeV × q ∆ m N
100 GeV × q ∆ m N
100 GeV
Table 2:
Complete set of the theoretical parameters used for the singlet Majoron model,where ∆ m N = 2( δ m N ) + i [( δ m N ) − ( δ m N ) ] . stress here that the approximate flavour symmetries SO(3) and U(1) ensure the stabilityof the light- and heavy-neutrino sector under loop corrections [11, 13, 36].For our numerical analysis, we fully specify in Table 2 the values of the theoreticalparameters for the Higgs and neutrino sectors. The only parameter that we allow to varyis the heavy Majorana mass scale m N . For 50 GeV < ∼ m N < ∼
200 GeV, the choice ofparameters in Table 2 leads to an inverted hierarchical light-neutrino spectrum with thefollowing squared mass differences and mixing angles: m ν − m ν = (7 . . × − eV , m ν − m ν = 2 . × − eV , sin θ = 0 . , sin θ = 0 . , sin θ = 0 .
047 (4.24)and m ν = 0. The spectrum is compatible with the light-neutrino data at the 3 σ confidencelevel (CL) [37].In Fig. 2 we present numerical estimates of the lepton-flavour asymmetries η L e,µ,τ and the baryon asymmetry η B as functions of z = T c /T , for a typical electroweak RLscenario with m N = 100 GeV. As initial conditions at T = T c ≈
133 GeV, we take η in N α = 1 for the heavy neutrino number densities and vanishing lepton-to-photon andbaryon-to-photon ratios, i.e. η in L e,µ,τ = 0 and η in B = 0. The thermal in-equilibrium condition η in N α = 1 is expected, since the heavy neutrinos N , , have no chiral masses when T >T c and get rapidly thermalised by the sizeable light-to-heavy neutrino Yukawa couplings16 - - - - - - z = T c T Η X m N =
100 GeV Η B obs Η B Η L e = Η L Μ -Η L Τ -Η L Figure 2:
Numerical estimates of η B (solid), η L τ (dash-dotted), η L e = η L µ (dotted) and η L (dashed) as functions of z = T c /T , for a model with m N =
100 GeV, and η in N α = 1 .The model parameters are given in Table 2. The horizontal grey line corresponds to theobserved baryon-to-photon ratio η obs B = 1 . × − , after evolving the latter back to thehigher temperature T = T c / . √ m D ) iα /v > ∼ − . As can be seen from Fig. 2, a net baryon asymmetry η B is generatedby a non-zero τ -lepton asymmetry η L τ . This L τ -excess is created before sphalerons sharplyfreeze out, i.e. for temperatures T > ∼ T sph ≈
78 GeV ( z < ∼ . < ∼ T < ∼
133 GeV,where a leptonic asymmetry can be converted into the observed BAU for our scenarioswith spontaneous lepton-number violation at the electroweak scale.Figure 3 exhibits the dependence of the baryon-to-photon ratio η B on z = T c /T for different values of the heavy Majorana mass scale m N . We notice that the lighterthe heavy neutrinos are, the smaller the created baryon asymmetry is. For example, forheavy-neutrino masses m N ∼
80 (50) GeV, η B falls short almost by one order (two orders)of magnitude with respect to the observed BAU η obs B . This is a generic feature of our17 - - - - - - z = T c T Η B Η B obs
120 GeV100 GeV80 GeV50 GeV
Figure 3:
Numerical estimates of η B versus z = T c /T for m N = 120 GeV (dashed),100 GeV (solid), 80 GeV (dash-dotted), 50 GeV (dotted). The meaning of the horizontalgrey line is the same as in Fig. 2. electroweak RL scenarios based on large wash-out effects due to the relatively large Dirac-neutrino Yukawa couplings ( m D ) iα /v . If the heavy neutrinos have masses m N <
90 GeV,their number densities will start decreasing for
T < m N , potentially creating a net leptonasymmetry that can be converted into η obs B . However, this should happen above the freeze-out temperature T sph ≈
78 GeV of sphalerons. Thus, successful electroweak RL requiresthat m N > T sph . ‡ ‡ Recently, a different leptogenesis scenario with m N ≪ T sph was studied in [38], where the BAU isgenerated by sterile-neutrino oscillations. Such a realisation relies on the assumption that the oscillatingsterile neutrinos start evolving from a coherent state and retain their coherent nature within the thermalplasma of the expanding Universe. In the singlet Majoron model we have been studying here however, t -channel 2 ↔ JJ ↔ ν CαR ν αR , that occur before the EWPT ( T > ∼ T c ) arestrong, with rates O [ ρ αα T / (8 π )] ≫ H ( T ), for Higgs-singlet Yukawa couplings ρ αα ∼
1. They can thereforelead to rapid thermalization and loss of coherence of the massless right-handed neutrinos. Shortly afterthe EWPT, for z = T c /T > ∼ .
1, it is Γ N , /H ∼ –10 and Γ N /H ∼ N , , . m N > T sph for scenarioswith T lc = T c . This condition will still be valid, as long as T lc > T sph . However, for scenarioswith T lc < ∼ T c , the predicted BAU η B will sensitively depend on the initial values η in L e,µ,τ and η in B at T = T c . Instead, if T lc ≫ T c and m N > ∼
90 GeV, the predictions for the BAU willremain almost unaffected, even if η in B ∼ η obs B at T = 10 T c [12]. It is interesting to discuss the implications of the singlet Majoron model for astrophysicsand low-energy phenomenology. To quantify the effects of heavy Majorana neutrinos, wedefine the new-physics parametersΩ ll ′ = δ ll ′ − B ∗ lk B l ′ k = B ∗ lα B l ′ α , (5.1)where l, l ′ = e, µ, τ . Evidently, in the absence of light-to-heavy neutrino mixings, theparameters Ω ll ′ vanish. LEP and low-energy electroweak data put severe limits on thediagonal parameters Ω ll [39]:Ω ee ≤ . , Ω µµ ≤ . , Ω ττ ≤ . , (5.2)at the 90% CL. On the other hand, lepton-flavour-violating (LFV) decays, such as µ → eγ [40], µ → eee , τ → eγ , τ → eee , µ → e conversion in nuclei [41, 42] and Z → ll ′ [43], constrain the off-diagonal parameters Ω ll ′ , with l = l ′ . The derived constraintsstrongly depend on the heavy neutrino masses m N α and the size of the Dirac masses ( m D ) lα .However, for models relevant to leptogenesis, with ( m D ) lα ≪ M W [41], we obtain thefollowing limits: | Ω eµ | < ∼ . , | Ω eτ | < ∼ . , | Ω µτ | < ∼ . , (5.3)including the recent BaBar data on LFV τ decays [44].The predictions for LFV decays in models of resonant leptogenesis has been exten-sively discussed in [12]. Since our results obtained in Section 4 agree well with this earlieranalysis, we will not repeat the details of this study here. Here, we only reiterate the factthat successful electroweak RL requires that m N > ∼
100 GeV. This latter constraint givesrise to the following upper limits:Ω ee < ∼ . × − , | Ω eµ | < ∼ . × − , Ω µµ < ∼ . × − , (5.4)whereas all remaining parameters Ω ll ′ are O (10 − ) and so unobservably small. All theselimits are deduced by using the model parameters of Table 2.19 − N α N β l l ′ J (a) G − N α N β l l ′ J (b) ZN α N β l, q l, qJ (c)Figure 4: Loop-induced couplings of the Majoron to charged leptons l, l ′ and quarks q . In the singlet Majoron model under study, there are additional LFV decays for themuon and the tau-lepton that involve the Majoron, i.e. µ → J e , τ → J e and τ → J µ . Asshown in Fig. 4, these LFV decays are induced by heavy Majorana neutrinos at the 1-looplevel. Detailed analytic expressions for the loop-induced couplings
J ll ′ and J qq , where q isa quark, may be found in [21]. To leading order in Ω ll ′ , the prediction for the LFV decay l − → l ′− J is R ( l → l ′ J ) ≡ Γ( l − → l ′− J )Γ( l − → l ′− ν l ¯ ν l ′ ) = 3 α w π t β | Ω ll ′ | M W m l λ N (1 − λ N ) λ N − λ N ! , (5.5)where λ N = m N /M W . For λ N = 1, the prediction for the observable R ( l → l ′ J ) takes onthe simpler form: R ( l → l ′ J ) = 3 α w π t β | Ω ll ′ | M W m l . (5.6)The requirement for successful electroweak RL, i.e. m N > ∼
100 GeV, gets translated intothe following upper bounds: R ( µ → eJ ) < ∼ . × − , R ( τ → eJ ) < ∼ . × − , R ( τ → µJ ) < ∼ . × − . (5.7)On the experimental side, however, the following upper limits are quoted: R ( µ → eJ ) ≤ . × − , at 90% CL [45]; R ( τ → eJ ) ≤ . × − , at 95% CL [46]; (5.8) R ( τ → µJ ) ≤ . × − , at 95% CL [46] . It is interesting to remark that the predicted value for R ( µ → eJ ) is close to the presentexperimental sensitivity, whereas the other decay modes turn out to be very suppressed forthe given RL model with inverted light-neutrino hierarchy. Had we chosen a model withnormal hierarchy, the decay rates R ( τ → eJ ) and R ( τ → µJ ) would have been enhanced20y a factor ∼ , but they will still be rather small O (10 − ) to be observed; the predictionsgenerally lie 4 orders of magnitude below the current experimental upper bounds.Useful constraints on the parameters of the theory are obtained from astrophysics aswell [47]. Specifically, observational evidence of cooling rates of white dwarfs implies thatthe interaction of the Majoron to electrons, g Jee J ¯ eiγ e , should be sufficiently weak and thecoupling g Jee must obey the approximate upper bound [48]: | g Jee | < ∼ − . (5.9)The above limit gets further consolidated by considerations of the helium ignition processin red giants, leading to the excluded range: 3 × − < ∼ | g Jee | < ∼ × − . To leadingorder in Ω ll , the loop-induced coupling g Jee , is given by [21]: g Jee = g w α w π m e M W t β λ N " Ω ee λ N − λ N λ N − λ N ! + 12 X l = e,µ,τ Ω ll . (5.10)If λ N ≫
1, the expression for the coupling g Jee simplifies to g Jee = g w α w π m e M W t β λ N (cid:16) Ω µµ + Ω ττ − Ω ee (cid:17) , (5.11)whilst for λ N = 1 g Jee becomes g Jee = g w α w π m e M W t β (cid:16) Ω µµ + Ω ττ (cid:17) . (5.12)Given the limits (5.4) for successful RL, we can estimate that g Jee < ∼ − . × − , (5.13)which passes comfortably the astrophysical constraint given in (5.9).Useful astrophysical constraints may also be obtained from considerations of the cool-ing rate of neutron stars [47]. Neutron stars will loose energy by Majoron emission throughthe interaction: g J N N J N iγ N , where N is a nucleon, specifically a neutron. The obser-vational limit on g J N N is [49] g J N N < ∼ − . (5.14)On the other hand, the theoretical prediction for g Jqq at the quark level is g Jqq = g w α w π m q M W t β λ N (cid:16) Ω ee + Ω µµ + Ω ττ (cid:17) . (5.15)From naive dimensional analysis arguments, one expects that g J N N ∼ ( m N /m q ) g Jqq . Inthis way, one may estimate that g J N N ≈ × − , (5.16)21fter taking into consideration the limits stated in (5.4).Cosmic microwave background (CMB) data and BBN put stringent limits on the max-imum number of weakly-interacting relativistic degrees of freedom, such as light neutrinosand Majorons [50, 51]. In particular, the allowed range obtained for the effective number N ν of left-handed neutrino species is N ν = 2 . +0 . − . at the 68% CL [51]. The upper boundon N ν may naively be translated into an upper limit on ∆ N ν = N ν − .
61 of extraeffective neutrino species beyond the 3 SM left-handed neutrinos. The singlet Majoroncontributes ∆ N ν = ( × ) / ≈ . T J isequal to the corresponding one T ν of the neutrinos. Although this result does not poseby itself a serious limitation on the singlet Majoron model, it can be estimated, however,that T J ≫ T ν ≈ J to ∆ N ν becomes even more suppressed.Specifically, the freeze-out temperature T J is determined when the annihilation rate ofMajorons through the process J J → νν becomes smaller than the Hubble expansion rate H ( T ) of the Universe. The annihilation process J J → νν is mediated by the H and S bosons in the s -channel and by the heavy neutrinos N , , in the t -channel. Consideringthe latter reactions only, one may naively estimate that T J T ν ∼ G F m N Ω ee t β ! / ∼ – 10 . (5.17)A similar value for T J /T ν is obtained if the S, H -boson exchange processes are used forthe model parameters of Table 2. Thus, the freeze-out temperature T J lies in the range0 . g ∗ ( T J ) ≈
66. Then, theactual contribution of the Majoron to ∆ N ν is reduced with respect to the T J = T ν caseby a factor ( g ∗ ( T ν ) /g ∗ ( T J )) / ≈ .
016 to the value ∆ N ν ≈ . J and singlet scalars S may also give rise to interestingcollider phenomenology [52] through the singlet-doublet mixing parameter δ in the scalarpotential (2.1). However, since δ ≪ λ Φ (cf. Table 2), the singlet Majoron scenario understudy predicts a rather small mixing angle s θ ≈ − .
1. The production cross section of S ,via the process e + e − → ZS , is then suppressed with respect to the SM one by a factor s θ ≈ .
01. Moreover, the so-produced Higgs singlets may decay quasi-invisibly into a pairof Majorons J , which makes difficult to fully rule out such a scenario by LEP2 data or atthe LHC. Future high-energy e + e − colliders of higher luminosity will severely constrain theallowed parameter space of this singlet Majoron model.22 Conclusions
The origin of CP violation in nature still remains an open physics question. If CP violationoriginates from the SSB of the SM gauge group, the original scenario [1] of GUT-scaleleptogenesis will be excluded. Similar will be the fate of all high-scale leptogenesis models,if the source of lepton-number violation is due to the SSB of a global U(1) l symmetry at theelectroweak scale. In this paper we have shown how resonant leptogenesis at the EWPTconstitutes a realistic alternative for successful baryogenesis in models with spontaneouslepton-number violation. Specifically, we have considered a minimal extension of the SM,the singlet Majoron model, which includes right-handed neutrinos and a complex singletfield that carries a non-zero lepton number. Depending on the form of the scalar potential,the lepton number can get broken spontaneously through the VEV of the SM Higgs doublet.Taking into consideration the Boltzmann dynamics of sphaleron effects, we have analysedthe BAU for different values of the Majorana mass scale m N within the context of abenchmark scenario whose model parameters are given in Table 2. The generic constraintfrom having successful electroweak RL is that m N > ∼ T sph , where T sph ≈
78 GeV is thefreeze-out temperature of the sphalerons.The singlet Majoron model predicts a massless Goldstone particle, the Majoron J .The Majoron can be produced via the LFV decays, µ → J e , τ → J µ and τ → J e .Considering the constraints from successful electroweak RL and the astrophysical limitsderived from the cooling rate of neutron stars, we have found that the decay mode µ → J e is the most promising channel, with sizeable branching fraction that can be looked for inthe next-round low-energy experiments.The predictions obtained for the BAU in this study are limited by the approximationsthat are inherent in the calculation of the non-perturbative sphaleron dynamics. Thepredicted values should be regarded as order-of-magnitude estimates, since the ( B + L )-violating sphaleron transitions crucially depend on the parameter κ that varies by a factorof 10 or so. It would therefore be very valuable to go beyond the current approximationmethods and improve the computation of the out-of-equilibrium sphaleron dynamics duringa second-order electroweak phase transition. Acknowledgements
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