Scalar triplet flavored leptogenesis: a systematic approach

Type-II seesaw is a simple scenario in which Majorana neutrino masses are generated by the exchange of a heavy scalar electroweak triplet. When endowed with additional heavy fields, such as right-handed neutrinos or extra triplets, it also provides a compelling framework for baryogenesis via leptogenesis. We derive in this context the full network of Boltzmann equations for studying leptogenesis in the flavored regime. To this end we determine the relations which hold among the chemical potentials of the various particle species in the thermal bath. This takes into account the SM Yukawa interactions of both leptons and quarks as well as sphaleron processes which, depending on the temperature, may be classified as faster or slower than the Hubble rate. We find that when leptogenesis is enabled by the presence of an extra triplet, lepton flavor effects allow the production of the $B-L$ asymmetry through lepton number conserving CP asymmetries. This scenario becomes dominant as soon as the triplets couple more to leptons than to SM scalar doublets. In this case, the way the $B-L$ asymmetry is created through flavor effects is novel: instead of invoking the effect of $L$-violating inverse decays faster than the Hubble rate, it involves the effect of $L$-violating inverse decays slower than the Hubble rate. We also analyze the more general situation where lepton number violating CP asymmetries are present and actively participate in the generation of the $B-L$ asymmetry, pointing out that as long as $L$-violating triplet decays are still in thermal equilibrium when the triplet gauge scattering processes decouple, flavor effects can be striking, allowing to avoid all washout suppression effects from seesaw interactions. In this case the amount of $B-L$ asymmetry produced is limited only by a universal gauge suppression effect, which nevertheless goes away for large triplet decay rates.


Introduction
Non-vanishing neutrino masses [1,2,3], and the cosmic asymmetry between baryons and anti-baryons [4,5], constitute two well-established experimental facts which have particularly well demonstrated that physical degrees of freedom beyond the standard model (SM) must be at work at certain unknown energy scale. Although a large number of scenarios capable of accounting for these experimental facts exist, arguably the treelevel seesaw models, type-I [6], type-II [7] and type-III [8], are of special interest: they constitute the simplest frameworks, are well theoretically motivated and, through the leptogenesis mechanism, provide a common explanation for both puzzles (for reviews see [9,10] and [11]).
The type-II leptogenesis scenario [11,12,13,14,15], in which the baryon asymmetry is generated from the decay of one or several scalar triplets, is more intricate than the standard scenario based in the type-I seesaw. First of all, while leptogenesis in type-I seesaw is driven by right-handed (RH) neutrinos which do not couple to gauge bosons, in type-II seesaw (as well as in type-III seesaw) the states which dynamically generate the B − L asymmetry do have electroweak interactions. Since at high temperatures gauge reactions are much more faster than the Universe Hubble expansion, one may be tempted to believe that gauge couplings constitute a non-circumventable obstacle which unavoidably imply the inviability of leptogenesis in these scenarios. This however is not the case [11,13,15,16,17,18]. Once the temperature of the heat bath reaches the mass of the decaying triplet, gauge reactions-being doubly Boltzmann suppressedrapidly decouple and the dynamics becomes dominated by Yukawa reactions which then operate to a large extent as in the type-I seesaw case. Secondly, while a single scalar triplet suffices for fitting neutrino masses and mixing angles, leptogenesis requires extra degrees of freedom (e.g. in the form of extra heavier triplets or RH neutrinos). Thirdly, since the scalar triplet is not a self-conjugated particle as a RH neutrino is, a scalar triplet/anti-triplet asymmetry develops [15,11], thus calling for an additional Boltzmann equation accounting for the new asymmetry populating the heat bath. As a result, while the SM scalar asymmetry in the type-I case is fully determined by the evolution of the B − L asymmetry, here it is determined in addition by the evolution of the triplet scalar asymmetry.
In this paper we aim to study the generation of the B − L asymmetry arising from the CP violating and out-of-equilibrium decays of a scalar triplet, taking into account in a systematic way any relevant effect that a SM interaction could have at a given temperature. This includes the flavor effects of the charged lepton Yukawa couplings 1 and the "spectator" effects of the quark Yukawa couplings (in particular the role of the top Yukawa reaction) and the sphalerons processes. To this end, we will first derive the full network of flavored Boltzmann equations and then will consider the redistribution of the B/3 − L i asymmetries in the heat bath, which in turn requires considering the conservation laws and chemical equilibrium conditions implied by slow and fast reactions.
With these tools at hand, and in order to illustrate how does scalar triplet flavored leptogenesis works, we will analyze two scenarios. (i) A scenario where the extra degrees of freedom correspond to additional scalar triplets, with the lepton number conserving CP flavored asymmetries naturally dominating the generation of the B −L asymmetry (purely flavored leptogenesis (PFL) scenario [20,21,22,23]); (ii) general triplet leptogenesis models involving lepton number violating CP asymmetries stemming from the presence of any seesaw state heavier than the decaying scalar triplet (RHNs, fermion or extra scalar electroweak triplets).
The paper is organized as follows. In Sec. 2 we fix our notation, discuss tree-level triplet decays, neutrino mass generation and the different CP asymmetries. In Sec. 3 we derive the network of flavored and unflavored Boltzmann equations, discuss chemical equilibration and analytical solutions to the flavored Boltzmann equations. In Secs. 4 and 5 we study scenarios (i) and (ii). Finally in Sec. 6 we present our conclusions. In Appx. A we present useful formulae.

Generalities
As regards the CP asymmetries, the details of scalar triplet leptogenesis strongly depend on the extra beyond SM degrees of freedom. As already pointed out, here we aim to analyze two generic scenarios: (i) models featuring several scalar triplets, or in other words extended pure type-II seesaw models, focusing on the cases where the B − L asymmetry production is dominated by the lepton conserving CP asymmetries; (ii) models involving a scalar triplet (minimal type-II seesaw) and heavier seesaw states, more specifically focusing on the effects of the lepton number violating asymmetries. The latter are particularly relevant for models where the generation of a B − L asymmetry becomes possible due to the interplay between type-I and type-II seesaws, scenarios arising in many well-motivated gauge extensions of the SM.

Interactions and tree-level triplet decays
The new interactions induced by extending both the scalar and fermion sectors of the SM with n ∆ scalar SU (2) triplets (∆ α ) and n R RH neutrinos (N α ) can be written, in the basis in which the RH neutrino Majorana and charged lepton mass matrices are diagonal, according to with T L = (ν L , e L ) and φ T = (φ + , (v + h 0 + iφ 0 3 )/ √ 2) the leptons and scalar boson SU (2) doublets,φ = iτ 2 φ * , τ T = (τ 1 , τ 2 , τ 3 ) (with τ i the 2 × 2 Pauli matrices) and the scalar ∆ α triplets given in the SU (2) fundamental representation i.e. ∆ α = (∆ 1 α , ∆ 2 α , ∆ 3 α ). Here Y α and λ are 3 × 3 and n R × 3 Yukawa matrices in flavor space and C is the charge conjugation matrix. Throughout the text we will be denoting lepton flavors e, µ, τ with Latin indices i, j, k . . . while RH neutrinos and scalar triplets with Greek labels α, β, . . . . The covariant derivative in (2) reads where T are the dimension three representations of the SU (2) generators. In our notation the SU (2) components of the fundamental scalar triplet representation have not all well defined electric charges, electric charge eigenstates are instead given by with the different components reading as The scalar interactions in (2) induce non-vanishing triplet vacuum expectation values which can be calculated from the minimization of the scalar potential: ∆ α = v ∆α µ ∆α v 2 /2m 2 ∆α . Both Lagrangians in (1) and (2), involving lepton number violating sources (from the coexistence of λ and M N and of Y and µ ∆ ), induce tree-level light neutrino Majorana masses through the standard type-I (assuming v λ · M −1 N 1) and type-II seesaw mechanisms. The structure of the full neutrino mass matrix will of course depend on whether a single or both mechanisms intervene. Since here we will be dealing with scenarios determined by either the setup of Eq. (2) or an interplay between (1) and (2), in what follows we write the effective neutrino mass matrix in each case, namely The light neutrino mass spectrum is thus derived from these matrices by diagonalization through the leptonic mixing matrix U = U (θ 23 )U (θ 13 , δ)U (θ 12 )P , with θ ij being the neutrino mixing angles, δ the Dirac CP phase andP = diag(1, e −iϕ 1 , e −iϕ 2 ) containing the Majorana CP phases. Regardless of the scenario considered, we are interested in the B − L asymmetry generated in triplet decays. Generating a sufficiently large B − L asymmetry, that after sphaleron reconversion matches the observed baryon asymmetry, requires certain balance between production and washout. Production is controlled by the CP violating asymmetry ( ∆α ) which structure is determined by the details (interactions) of the corresponding scenario, but which in any case arises via the interference of the tree-level decay and its one-loop corrections, as required by the unitarity of the scattering matrix [24].
Tree-level triplet decays involve leptonic and scalar final states. The leptonic partial decay widths, depending on the lepton flavor composition of the final states, involve extra factors of 1/2 which avoid overcounting: where Q stands for the electric charges of the different SU (2) triplet components, . On the other hand, scalar triplet decay modes can be written according to so the total decay width, after summing over lepton flavors, can be written as where the "neutrino mass-like" parameterm ∆α is defined as with B α and B α φ standing for the ∆ α triplet decay branching ratios to lepton and scalar final states: where of course the relation B α + B α φ = 1 holds. As can be seen directly from Eqs. (10) and (11), for fixedm ∆α and m ∆α , Γ Tot ∆α exhibits a minimum at B α = B α φ = 1/2. Thus, the farther we are from B α = B α φ = 1/2, the faster the scalar triplet decays.

CP asymmetries in triplet decays
As already pointed out, the one-loop corrections to the tree-level decay depend on the details of the corresponding model. In purely triplet models, that is to say models entirely determined by the Lagrangian in (2), the corrections to the leptonic tree-level decay mode involve only wave-function type corrections [12]. The CP asymmetry follows from Figure 1: Tree-level and one-loop Feynman diagrams responsible for the flavored CP asymmetry i ∆α in the pure type-II seesaw scenario.
the interference between the tree-level and wave-function corrections shown in Fig. 1, it therefore consists of two pieces: a lepton number and flavor violating one (scalar loops) and a purely flavor violating part (lepton loops). The total flavored CP asymmetry in ∆ α decays can then be written as where the two pieces read with and y = (Γ Tot ∆ β /m ∆ β ) 2 . Note that the CP asymmetry in Eq. (14) is in-line with what has been found in [19]. This piece, which we refer to as purely flavored CP violating asymmetry, satisfies the total lepton number conservation constraint and so the total CP asymmetry can consequently be written as In terms of triplet decay observables the total flavored asymmetries can be recasted according to Figure 2: Tree-level and one-loop Feynman diagrams accounting for the flavored CP asymmetry i ∆α in scenarios featuring type-I and type-II interplay.
If flavor effects are operative, that is to say if leptogenesis takes place below 10 13 GeV, the purely flavored CP asymmetry in (15) will play a role in the generation of the B − L asymmetry. These asymmetries, conserving total lepton number, involve only the Y α Yukawa couplings and not the lepton number violating parameter µ ∆α . Hence, as also noted in Ref. [19], they are not necessarily suppressed by the smallness of the neutrino masses. As can be seen by comparing (14) and (15), when the condition is satisfied, the purely flavored CP asymmetry overshadows the lepton number violating piece, therefore leading to a regime where leptogenesis is entirely driven by flavor dynamics. In terms of scalar triplet interactions, this means that a purely flavored scalar triplet leptogenesis scenario naturally emerges whenever the triplets couple substantially less to SM scalars than to leptons, B α B α φ for at least one value of α. Note that although PFL scenarios in type-I seesaw can be defined as well, they differ significantly from the purely flavored scalar triplet leptogenesis scenario in that the latter just require suppressed lepton number violation in a single triplet generation i.e. suppression of lepton number breaking interactions in the full Lagrangian is not mandatory, as can be seen by noting that condition (20) can be satisfied even if µ ∆α /m ∆α Y α for a single value of α.
We now turn to the case where the new states beyond the scalar triplet are RH neutrinos. In these scenarios the tree-level triplet decay involves only a vertex one-loop correction as shown in Fig. 2. The interference between the tree and one-loop level diagrams leads to the following CP asymmetry [14,15]: Here the triplet generation index, being superfluous, has been dropped. In contrast to what has been found in the previous case, the resulting flavored CP asymmetry violates lepton flavor as well as lepton number. So, unless a specific (and somehow arbitrary) flavor alignment is assumed, so that i i ∆ = 0, in these "hybrid" schemes PFL scenarios are not definable.
In the hierarchical case, m ∆ M Nα , the flavored CP asymmetry can be recasted in terms of triplet decay observables, namely with M ν ∆ and M ν N given by Eqs. (6) and (7).

Boltzmann equations
In general, the evolution equations of particle asymmetries in the early Universe couple all particles species and thus involve a large number of reactions. However, a simplification is possible given that for specific temperature regimes different reactions have different timescales. Any reaction occurring in the heat bath will necessarily fall in one of the following categories 2 : (I) Reactions which at a given temperature T 0 are much slower than the Hubble Universe expansion rate H(T 0 ): Γ SR H(T 0 ).
(II) Reactions which at a given temperature T 0 are much faster than the Hubble Universe expansion rate H(T 0 ): Γ FR H(T 0 ).
(III) Reactions which at a given temperature T 0 are comparable to the Hubble Universe expansion rate H(T ): Γ CR ∼ H(T 0 ).
At T 0 , reactions falling in category I basically have not taken place, so they are of no relevance in the actual problem. The parameters responsible for such reactions can then be put to zero at the Lagrangian level, leading to the corresponding early Universe effective Lagrangian which involves new global symmetries implying new conservation laws [25]. In contrast, the reactions in II at T 0 have occurred so often that the particles involved attain thermodynamic equilibrium and so are subject to chemical equilibrium constraints, which enforce relations among the different particles chemical potentials (the particle asymmetries) 3 . These chemical equilibrium conditions, when coupled with the constraints implied by the conservation laws of the early Universe effective Lagrangian, allow to express the particle asymmetries of all the species in the thermal bath in terms of quasi-conserved charge asymmetries, the asymmetries related with charges that are only (slowly) broken by the reactions in III. Finally, reactions of type III are not fast enough to equilibrate the distributions of the intervening particles, and so they have to be accounted for via Boltzmann equations, which dictate the evolution of the quasi-conserved charge asymmetries and therefore of all the asymmetries in the heat bath. Note that for reactions of category I one has nevertheless to be cautious before dropping them from the Boltzmann equations. A well-known example, relevant in some cases for the dark matter abundance, is the freeze-in regime, i.e. slow production of dark matter particles from an out-of-equilibrium process. Further on, in Sec. 4, we will see that in the PFL scenario a relatively similar effect, from very slow triplet interactions, can be relevant or even crucial, i.e. dominates the whole baryogenesis process (due to the fact that an additional asymmetry, the scalar triplet asymmetry, populates the heat bath, thus implying an additional Boltzmann equation).
The comparable-to-the-expansion triplet decays induce a B − L (B/3 − L i in the lepton flavored regime) and φ asymmetries. Although stemming from the same state and occurring at the same stage, these asymmetries follow different behaviors and so have to be treated in different ways. Let us discuss this in more detail. In the absence of triplet interactions, the full Lagrangian possesses a U (1) B−L symmetry no matter what the value of T 0 is. The B − L charge asymmetry is therefore only affected by slow washouts (in that sense is a quasi-conserved charge) which implies that it is not entirely washed away and spreads all over the thermal bath feeding all the SM particle asymmetries. In the same token, the φ asymmetry is partially transferred to asymmetries in SM fermions through Yukawa interactions (those which at T 0 are in thermal equilibrium). However, while the evolution of the B −L asymmetry is analyzed with the corresponding Boltzmann equations, the analysis of the evolution of the φ asymmetry does not require a Boltzmann equation: its evolution is determined by the chemical equilibrium conditions enforced by the reactions that at T 0 are in thermal equilibrium.
The network of Boltzmann equations for scalar triplet leptogenesis, no matter whether lepton flavor effects are active or not, corresponds to a system of coupled differential equations accounting for the temperature evolution of the triplet density Σ α = ∆ α + ∆ † α , the triplet asymmetry ∆ ∆α = ∆ α −∆ † α and the B−L (B/3−L i in the lepton flavor regime) charge asymmetry. The resulting network will of course-and unavoidably-involve the scalar doublet asymmetry, for which chemical equilibrium conditions have to be used in order to determine its dependence with the asymmetries that feed the heat bath (∆ ∆α and B − L), as it is done in the standard leptogenesis case [26]. In what follows we will derive in detail the appropriate set of equations suitable for tackling the problem. We will closely follow the notation of [27].
In the hot plasma, triplets are subject to reactions that either tend to washout the B − L asymmetry or to generate it. Depending on the interaction inducing the process one can distinguish-at tree-level-four kind of reactions: pure Yukawa, pure scalar, pure gauge and Yukawa-scalar reactions. Explicitly, for ∆ α , we have 4 : • Yukawa and scalar-induced decay and inverse decays, ∆ α ↔¯ ¯ and ∆ α ↔ φφ, described by the reaction densities: γ Dα ≡ i,j γ ∆α i j and γ φ Dα ≡ γ ∆α φφ . The total decay reaction density thus given by γ Dα = γ Dα + γ φ Dα . • Lepton flavor and lepton number (∆L = 2) violating Yukawa-scalar-induced and triplet-mediated s and t channel 2 ↔ 2 scatterings φφ ↔¯ i¯ j and φ j ↔φ¯ i , which are accounted for by the reaction densities γ φφ i j and γ φ j φ i . • Lepton-flavor-violating Yukawa-induced and triplet-mediated s and t channel 2 ↔ 2 scatterings: n m ↔ i j and j m ↔ i n , with reaction densities given by γ n m i j and γ j m i n . • Gauge-induced 2 ↔ 2 scatterings as follows: s-channel gauge-boson-mediated: ∆ α ∆ α ↔ F F (F standing for SM fermions), ∆ α ∆ α ↔ φφ and ∆ α ∆ α ↔ V V (V standing for SM gauge bosons); t and u channel triplet-mediated: ∆ α ∆ α ↔ V V and four-point vertex ∆ α ∆ α ↔ V V reactions. All together they are characterized by the reaction density γ Aα .
Note that if the flavor degrees of freedom were to be neglected, all the reactions-apart from those in the third item-would still be present in their unflavored form. The reactions in third item are therefore inherent to scalar flavored leptogenesis. All together, these reactions lead to the following network of flavored classical Boltzmann equations: Here we have adopted the following conventions (details can be found in Appx. A). A fraction of the asymmetry generated in i is transferred to RH charged leptons, e i , via SM Yukawa interactions, and so L i = 2 i + e i . We use the particle number density-to-entropy ratio defined as Y ∆ X = ∆n X /s = (n X − nX)/s, where n X (nX) is the number density of species X (X) and s is the entropy density. We have defined The derivative is denoted according toẎ ≡ sHz α dY /dz α , with H the expansion rate of the Universe, and as usual z α = m ∆α /T . Primed s-channel scattering reaction densities refer to the rates with resonant intermediate state subtracted: γ = γ − γ on-shell . Finally the matrices C ijk and C ijmnk are defined according to where the C and C φ matrices (asymmetry coupling matrices) relate the asymmetry in lepton and scalar doublets with the B/3−L k and triplet asymmetries-the "fundamental" asymmetries present in the plasma-according to In these relations the asymmetries Y ∆ k are given by the components of the asymmetry vector and the structure of the C and C φ asymmetry coupling matrices becomes determined by the constraints coming from the global symmetries of the effective Lagrangian and the chemical equilibrium conditions enforced by those SM reactions which in the relevant temperature regime (the regime at which the B − L asymmetry is generated) are faster than the Universe Hubble expansion rate. The final baryon asymmetry is then given by where the factor 3 accounts for the different SU (2) degrees of freedom of the scalar triplet. Before discussing chemical equilibration, we also write Boltzmann equations valid in the case where the top Yukawa-related reactions are either the only fast Yukawa processes (10 13 GeV T 10 15 GeV) or slow (T 10 15 GeV), or when quantum lepton flavor coherence is already broken but an alignment in lepton flavor space is fixed (i.e. ∆ couples to only one flavor combination). We will refer these cases as "the one lepton flavor approximation" (see Sec. 3.1 for further details). In addition to Eq. (24) which holds no matter the regime, one haṡ where in this case the asymmetry vector is reduced to , and so the relation between the lepton doublet asymmetry and Y ∆ reads like in (27), dropping the lepton flavor index. Note that the evolution equations derived in Ref. [15] match with (30)-(31) provided in the latter all SM Yukawa interactions effects are neglected, see Eq. (38) below.

Chemical equilibrium conditions
At very high temperatures (T 10 15 GeV) all SM reactions are frozen in the sense of item I. As the temperature drops, certain reactions (those driven by the largest couplings first) attain thermal equilibrium which demands kinetic as well as chemical equilibrium of the corresponding reactions, the latter in turn enforce constraints among the different chemical potentials of the intervening particles. Since for a relativistic species X the particle number density-to-entropy ratio is, at leading order in µ/T , related with the chemical potential according to [28]: (with g X the number of degrees of freedom) the chemical equilibrium constraints thus relate the different particle asymmetries of those species participating in fast reactions.
In principle, there is a chemical potential (an asymmetry) for each particle in the thermal bath, which implies that a priori there are as many chemical potentials as particles in the plasma: 61. This number, however, is largely reduced due to the constraints imposed by the set of chemical equilibrium conditions and the conservation laws of the early Universe effective Lagrangian. Depending on the temperature regime where the B − L asymmetry is generated, the possible constraints on the chemical potentials are: 1. Chemical potentials for gauge bosons vanish µ W i = µ B = µ g = 0, and so the components of the electroweak and color multiplets have the same chemical potentials [28]. This already reduces to 17 the number of independent asymmetries.
2. Regardless of the temperature regime, cosmological hypercharge neutrality must be obeyed, namely 3. Non-perturbative QCD instanton and electroweak sphaleron reactions-if in thermal equilibrium-enforce the following constraints: The temperature at which the QCD instanton reactions attain equilibrium has been estimated to be T ∼ 10 13 GeV [29,30] while for electroweak sphaleron processes, being controlled by α EW rather than α S , it has been found to be about a factor 20 smaller [30].

Finally Yukawa reactions when being in thermal equilibrium lead to the chemical equilibrium constraints:
Up-type quarks: Down-type quarks: Charged leptons: Top Yukawa-induced reactions are in thermodynamic equilibrium for T 10 15 GeV. Bottom and tau Yukawa-induced processes are in equilibrium at T 10 13 GeV while second and first generation Yukawa-induced processes at T 10 11 GeV and T 10 8 GeV respectively [31,32,33].
The exact number of non-vanishing chemical potentials as well as the number of chemical equilibrium conditions are fixed only when a specific temperature window is settled. Once this is done, the resulting system of equations is solved in terms of a single set of variables, which we take to be µ B/3−L i and µ ∆ . The solution thus provides the relations between the asymmetries of all the particles in the heat bath with the independent asymmetries {Y ∆ } = {Y ∆ ∆ , Y B/3−L i } appearing in the asymmetry vector in (28).
In what follows we briefly discuss the symmetries of the corresponding early Universe effective Lagrangian and the relevant chemical equilibrium conditions in 1-4 which enable us to calculate the rectangular matrices relating the lepton and scalar doublet asymmetries with {Y ∆ }, as given by Eqs. (27). In each regime, when applicable, we also discuss the one-flavor limit by taking flavor alignments as in Ref. [26] and deriving the corresponding C ,φ matrices needed in such approximations. We start by discussing the high temperature regime T > 10 15 GeV, proceeding then to the T ⊂ [10 13 , 10 15 ] GeV window and then fixing the remaining temperature windows as in Ref. [32].
• None SM reactions in thermal equilibrium, T 10 15 GeV: In this regime all SM reactions are slow in the sense of I, and so only triplet-related interactions are relevant. A proper treatment of the problem therefore should be done with the unflavored kinetic equations in (24), (30) and (31), bearing in mind that since in the heat bath the triplet is subject only to scalar-and Yukawa-induced interactions, in Eq. (24) the second term can be neglected. With all SM reactions frozen, only the triplet, lepton and scalar doublets develop chemical potentials: µ ∆ , µ , µ φ . These chemical potentials are subject only to the hypercharge neutrality constraint in 2, which leads to the following C and C φ matrices: The resulting Boltzmann equations match those derived in Refs. [11,15] since there, regardless of the temperature considered, no SM Yukawa interactions effects were taken into account, neither from the quarks nor from the charged leptons. Strictly speaking, this is an accurate procedure only in this regime, T 10 15 GeV, given that none of the SM Yukawa reactions are in thermal equilibrium (from now on, when necessary we will refer to this literature-reference-scheme as the "unflavored case"). It is worth stressing that through Eq. (27) the C φ matrix in Eq. (38) leads to the sum rule: This expression is nothing else but the sum rule employed in Ref. [15] (taking into account the fact that, in this reference, Y ∆ ∆ and Y ∆ φ involve a sum on the SU (2) degrees of freedom, so that one must substitute Note that above 10 15 GeV there is relatively little time for the reheating to occur before the temperature goes below the scalar triplet mass (assuming the reheating occurs below the Planck scale). So, unless the triplet Yukawa couplings are such that at T ∼ m ∆ the triplet still follows a thermal distribution (strong washout regime), the final baryon asymmetry produced will depend on the initial scalar triplet number density (initial condition), i.e. will further depend on the details of the reheating. We will not consider these possible effects here. Note also that above 10 15 GeV, for m ∆ ∼ 0.05 eV one gets non perturbative Yukawa couplings if B φ 8 · 10 −3 .
• Only top Yukawa-related reactions in thermal equilibrium, T ⊂ [10 13 , 10 15 ] GeV: Within this temperature regime, apart from top Yukawa-related interactions which are fast, all SM Yukawa-induced reactions fall in category I. Accordingly, the correct description of the problem is given by the one lepton flavor approximation equations in (24), (30) and (31).
The global symmetries of the effective Lagrangian are those of the SM kinetic terms broken only by the top Yukawa coupling, and so the group of global transformations is: The SU (3) factors combined with the exact U (1) B , U (1) PQ and the absence of Yukawa couplings for all SM particles, except the top quark, imply: Taking this constraints into account and the relevant chemical equilibrium conditions (33) and (35), the latter written only for the top quark, we obtain Including these effects will enhance the efficiency by about 20% with respect to the unflavored case, the precise value being of course dependent upon the parameter choice.
• QCD instantons, bottom and tau Yukawa-related reactions in thermal equilibrium, T ⊂ [10 12 , 10 13 ] GeV: In this temperature window the lepton doublets lose their quantum coherence due to the tau Yukawa-related interactions being in thermal equilibrium [32,33]. An appropriate study of the evolution of the B − L asymmetry should be then done by tracking the evolution of the flavored charge asymmetries B/3 − L i (i = a, τ , the state a being a coherent superposition of e and µ lepton flavors) with the network of Eqs. (23)- (25).
The QCD instantons reactions break the global U (1) PQ , the bottom and tau Yukawa couplings break the RH down-type quark and charged lepton SU (3) flavor multiplet and in addition the tau Yukawa coupling also breaks the global U (1) e . The Lagrangian is as expected "less symmetric", with the group of global transformations given by These global symmetries imply: while the complete set of chemical equilibrium conditions correspond to (33) for hypercharge neutrality (written so to include the now non-vanishing bottom and tau chemical potentials), (34) for QCD instantons and (35), (36) and (37) written for top, bottom and tau Yukawa interactions. When combined with the constraint µ B = 0 these conditions yield the following C ,φ matrices: As a final remark before we proceed with the following regime, we note that sticking to the one lepton flavor approximation means first choosing a direction in the τ − a flavor space. Taking either B aa = 1−B φ or B τ τ = 1−B φ , both cases are governed by the system of equations in (24), (30) and (31) with the structure of the C ,φ matrices determined by the corresponding choice. If B aa = 1−B φ , the asymmetry is entirely projected along the a flavor direction and so the one lepton flavor approximation C ,φ matrices are given by If instead the asymmetry is projected along the τ flavor the matrices read: • Electroweak sphalerons in thermal equilibrium, T ⊂ [10 11 , 10 12 ] GeV : In contrast to the previous regimes, in this case, with electroweak sphaleron reactions being in thermal equilibrium, baryon number is no longer conserved. The group of global transformations of the effective Lagrangian is therefore given by The constraints implied by the global symmetries imply Due to sphaleron reactions, lepton flavor is no longer conserved and so chemical potentials in three independent lepton doublets develop: τ , a and b . Conservation of the B/3 − L i charges however provide the constraint µ B/3−L b = 0, which when coupled with the corresponding chemical equilibrium conditions (those from previous regimes and constraint (34) for electroweak sphalerons) yields the following flavored C ,φ matrices: • Second generation quark and lepton Yukawa interactions in thermal equilibrium, T ⊂ [10 8 , 10 11 ] GeV: As pointed out in Ref. [32,33], in this temperature regime the lepton doublets completely lose their quantum coherence, implying that chemical potentials develop in each orthogonal lepton flavor doublet: τ , µ and e . Since the second generation Yukawa reactions are no longer of type I, the symmetries of the effective Lagrangian are reduced to U (1) factors: These constraints imply µ d = µ e = µ Q 1 = µ u = 0, and when combined with the corresponding chemical equilibrium conditions (the ones from previous item complemented with (35), (36) and (37) Since in this regime the flavor basis is completely defined, there are several flavor projections which render flavor alignment. The corresponding C ,φ matrices for the In this case as well one can define a one-flavor approximation by fixing an alignment in flavor space. For B ii = 1 − B φ (i = e, µ, τ ) the C ,φ matrices are given by

Formal integration of Boltzmann equations
Keeping only leading order terms in Eq. (25), i.e. dropping third and forth terms, an analytic formal integration of the equations responsible for the B − L asymmetry can be accomplished. For definitiveness we will focus on the two flavor regime, results for the three flavor regime can be readily derived following the same procedure we will outline. In the two flavor regime the asymmetry vector introduced in Sec. 3 (see Eq. (28)) is given by In terms of this vector, Eqs. (23) and (25) can be casted in matricial form, namely with and where the CP-asymmetry-vector ε is defined as while the flavor-triplet-coupling-matrix according to In the case M(z) = M, the system of equations in (59) can be decoupled via a rotation of the asymmetry vector Y ∆ , the matrix accounting for the rotation being determined by the similarity transformation which brings M to diagonal form. Strictly speaking M does depend on z, but it turns out that the z dependence of the rotation matrix P is quite moderate. As can be seen in Fig. 4, in the high as well as in the low temperature regime P(z) = P whereas within the window z ⊂ [0. 2,7] there is a dependence, which nevertheless is rather soft. Thus, taking a z independent change-of-basis-matrix P and rotating the asymmetry vector as we finally get a decoupled system of differential equations: where the rotated-CP-asymmetry-vector ε has been introduced: We have checked that this result is quite insensitive to changes in the flavor configuration. The vertical yellow stripe indicates the range where the matrix P slightly depends upon z.
The decoupled system of equations in (65) can then be formally integrated through their integrating factor. By doing so, and assuming vanishing primordial asymmetries, Y ∆ (z 0 ) = 0 with z 0 1, the solution reads In terms of the "new" asymmetries, and due to the diagonal structure of the matricial damping factor, one can define efficiency functions η i (z), which account for the evolution of the primed asymmetries and their corresponding values at freeze-out (z → ∞), namely where the efficiency functions can be directly read from (67) by taking into account that, as usual, they have been normalized to the scalar triplet equilibrium distribution evaluated at z 0 . The evolution of these asymmetries, however, does not describe the evolution of the actual B/3 − L i asymmetries and instead, as can be seen in (64), a superposition which involves the triplet asymmetry as well. A meaningful description requires switching to the non-primed variables, which yields 5 In the non-primed basis the matricial damping factor is no longer diagonal and therefore defining efficiency functions, as it was done in the primed basis, is no longer possible: both B/3 − L a and B/3 − L τ are a superposition of two terms weighted by the corresponding CP asymmetries a ∆α and τ ∆α . Let us discuss this in more detail. The i-th component of the asymmetry vector in (69) can be written as thus implying that in the primed basis the flavored asymmetries become with the flavored efficiency functions defined as: So, once lepton flavors are taken into account-in general-the efficiencies are no longer flavor diagonal. The presence of the flavor off-diagonal efficiencies is a manifestation of flavor coupling which, in contrast to the type-I seesaw-based leptogenesis case, persists even when C = I, due to the intricate structure of the flavor-triplet-coupling-matrix.
More precisely this occurs because, in contrast to the type-I seesaw leptogenesis case, an asymmetry in the state generating the B − L asymmetry develops (Y ∆ ∆ ), and so an additional kinetic equation accounting for this asymmetry turns out to be mandatory. Due to the presence of this equation the asymmetries in flavor a and τ are indirectly coupled, and such coupling becomes manifest in the exponential function in Eq. (72). In other words, unlike standard leptogenesis, flavor coupling effects are unavoidable in scalar triplet leptogenesis. A specific case where flavored efficiency functions, in the same sense of (68), can be properly defined corresponds to PFL scenarios. What actually happens in those cases is that due to the PFL condition i i ∆α = 0, which implies ∆α ≡ − a ∆α = τ ∆α , the off-diagonal efficiency functions can be hidden by suitable redefinitions: and so the total B − L asymmetry can be written as with the final value (the value at freeze-out) given by

Purely flavored triplet leptogenesis
For concreteness and in order to analyze as well as to demonstrate the viability of this scenario, we will fix the triplet mass spectrum to be hierarchical (m ∆α m ∆ β with α < β) and assume that the B − L asymmetry is entirely due to the dynamics of the lightest state ∆ (henceforth we drop the triplet generation index). We will also consider two-flavored regime situations where the B − L asymmetry is distributed along the τ and a lepton flavor directions (a being an admixture of µ and e flavors) 6 .
As previously argued (see Eq. (20) and the corresponding discussion), when the scalar triplet CP asymmetries arise from the presence of another scalar triplet, there exists an overall regime in which the purely flavored CP asymmetries are larger than the lepton number violating CP asymmetries, thus leading to a natural realization (to a very good approximation) of a PFL successful scenario. Strictly speaking PFL scenarios are defined by the condition i i = 0 [20], however in a more general fashion whenever the condition | i i ∆ | < | i ∆ | (for any given value of i) is satisfied a PFL scenario can be defined as well. This is actually the condition which is generically satisfied, as soon as Eq. (20) holds, i.e. if one or both scalar triplets couple substantially less to scalars than they do to leptons.
The viability of PFL scenarios demands leptogenesis to take place in the flavored regime, i.e. requires leptogenesis to occur at T ≤ 10 13 GeV, and furthermore it requires more than a dominance of the purely flavored CP asymmetries. Since the sum of the purely flavored CP asymmetries vanishes (total lepton number is conserved), if there were only source terms, a net non-vanishing B − L asymmetry would not develop due to an exact cancellation among the different B/3 − L i asymmetries. This cancellation has to be mandatorily avoided in order that a net non-vanishing total B − L asymmetry develops. In type-I seesaw, this is possible due to the lepton flavor dependence of the washout effect, which allows the B/3 − L i asymmetries to be washed-out in different amounts. In other words, the production of a net B − L asymmetry in the PFL type-I case, which involves L-conserving CP asymmetries as well, is closely related to the action of L-violating inverse decay rates larger than the Hubble Universe expansion rate (fast L-violating inverse decays), so that they reprocess the B/3 − L i asymmetries in different amounts, in such a way that these asymmetries do not compensate each other anymore.
In the type-II scenario a similar effect is also possible, provided decay/inverse decay to leptons and to scalars reach thermal equilibrium at some stage during the production of the B − L asymmetry, so that L-violating processes do induce a washout. Additionally, and this is a new effect which does not exist in the PFL type-I scenario, this is also possible even if the L-breaking processes present in the heat bath never reach thermal equilibrium.
Let us explain already at this point how does this new effect work. To this end we display in Fig. 5    10 11 GeV 7 ,m ∆ = 5 × 10 −2 eV and ∆ ≡ τ ∆ = − a ∆ = 1. As we will discuss further on in this section, this B ij flavor configuration maximizes the efficiency. Fig. 3 (left-hand side plot) clearly shows that for B φ = 10 −5 the inverse decays φφ → ∆ have always a rate slower than the Hubble expansion rate. The fact that for the type-II PFL case, we do get nevertheless a net non-vanishing B − L asymmetry can then at first sight appear to be counterintuitive. If for instance only the channel to leptons does get in thermal equilibrium, as it turns out to be the case for B φ = 10 −5 , the scalar triplets have effectively lepton number L = −2 and the only active (fast) inverse decays in the thermal bath, ∆ →¯ ¯ and∆ → , do not break total lepton number.
However, although the scalar doublet channel never reaches thermal equilibrium, still a portion of the scalar triplets in the heat bath undergoes decays to scalar doublets (∆ → φφ), and these processes do break L. If the processes ∆ → φφ and∆ →φφ take place at different rates, the thermal bath gets a fraction of total lepton number each time these reactions occur. Quantitatively this means that we can define an effective B − L yield, Y eff ∆ B−L , determined by the counting of how many scalar triplets decay times their branching ratio into scalar doublets, namely where the factor 2 comes from the fact that the decay to scalar doublets violates lepton number by 2 units. This effective quantity holds for the total B − L asymmetry available if one assigns to ∆ (∆) a lepton number equal to −2 (2), as we have previously pointed out. It is related to the usual B − L yield (where triplets have vanishing lepton number) according to Since ultimately all triplets decay (their density vanishes), the final B − L asymmetry simply reads In order to prove that this formula reproduces the correct B−L asymmetry yield at freezeout, we have inserted in Eq. (75) the Y ∆ ∆ asymmetry obtained by solving numerically the set of Boltzmann equations. The result is shown in Fig. 5 (left-hand side plot) by the dashed gray curve. It clearly shows that Eq. (77) reproduces very well the numerical result (red dashed curve) for the B − L asymmetry yield at freeze-out, up to a small deviation of order 30%. This deviation can be fully traced back to the effect of the inverse decay processes, φφ → ∆ andφφ →∆, i.e. of the term ∝ B α φ in Eq. (23). These scalar inverse decays are not as numerous as scalar decays but not negligible either.
The generation of a baryon asymmetry, through decays rather than through inverse decay washout effects, is thus closely related to the possibility of creating a scalar triplet asymmetry (something obviously not possible for a right-handed neutrino due to its Majorana nature). The role of flavor effects is in fact to generate such a triplet asymmetry. To see that, it is useful to write down the relevant terms in Eq. (23), This expression clearly shows that a triplet asymmetry can be generated by two kinds of flavor effects: -The first possibility arises if the C ij have a flavor structure. For instance, if the τ Yukawa is in equilibrium, once a lepton doublet τ is produced, it has the time to interact through the Yukawa coupling and a fraction of the τ flavor is transferred from τ lepton doublets to e τ lepton singlets, while this is not the case for flavor a. These transferred fractions are just given by the C matrices which are dictated by the chemical potential equilibrium equations, see Eq. (26). This means that there are less τ than a lepton doublets available for inverse decays to scalar triplets. So, even if there is no flavor structure in the branching ratios (i.e. B a = B τ ) and even if, at the onset, Y ∆ Lτ = −Y ∆ La , the number of ∆ produced is different from the number of∆ produced because their production rate is proportional to Y τ + Y a and Y τ + Y a respectively, which are unequal 8 . 8 It is worth noting that we have the same reprocessing concerning the φ asymmetry created from the slow ∆ decays. This latter asymmetry is partly reprocessed through L-conserving SM Yukawa interactions into chiral asymmetries for charged leptons, which modifies back the ∆ asymmetry, hence the number of ∆ decaying into SM scalars, hence the B − L asymmetry. This effect is nevertheless mild.
-The second possibility arises from the flavor structure of scalar triplet decays, i.e. the B i . If B a = B τ , a triplet asymmetry can be produced even if the C coefficients do not distinguish the a and τ flavors. In this case, even if at the onset, Y ∆ Lτ = −Y ∆ La , with for example B τ B a and Y Lτ > 0, inverse decays involving the τ flavor are much more frequent than those involving the a flavor and inverse decays ∆ inverse decays, resulting in the generation of a Y ∆ ∆ asymmetry (of negative sign in this case).
In other words, in the PFL case there is no L-violating CP asymmetry. The fact that a final B − L asymmetry can be generated in this case, even without L-violating processes attaining thermal equilibrium, i.e. B φ B , can be understood as a three step process, summarized in Fig. 6. Firstly, an asymmetry Y ∆ Lτ = |Y ∆ La | = 0 is created from the source term in Eq. (25). Secondly, thanks to flavor effects, this asymmetry induces a triplet asymmetry via Eq. (78), due to the flavor structure encoded in C ij and/or due to the flavor structure encoded in the B i . And finally, once a scalar triplet asymmetry is created, a B − L asymmetry develops in turn because each time a triplet (anti-triplet) decays to scalars, a pair less of anti-leptons (leptons) is produced back from the decay of a triplet (anti-triplet). The more Y ∆ ∆ asymmetry is produced, the bigger the efficiency. This PFL production mechanism, based on the chain of processes i j ↔∆ →φφ and i¯ j ↔ ∆ → φφ, is therefore very different from the PFL type-I scenario. It stems from the fact that in the type-II scenario, a seesaw state asymmetry develops, and in its last step this asymmetry generates a final B − L asymmetry from a production mechanism which is due to out-of-thermal equilibrium decays, i.e. from the ∆ → φφ and∆ →φφ L-violating processes (B φ B ) 9 . Let us emphasize once again that this B φ 1 case is the situation which leads naturally to PFL, since this condition leads to a natural dominance of the purely flavored CP asymmetries. It must be noted that PFL could nevertheless work for larger values of B φ too, in a way more similar to the more involved PFL scenarios in the type-I context, see Sec. 4.1.3 below.
In the following section we will analyze, along these lines, the efficiency dependence upon the relevant parameters. We will discuss in particular the flavor configurations which minimize, or maximize, the production of Y ∆ ∆ . We will then discuss the flavored CP asymmetry parameter dependence and show how the configurations that maximize the efficiency minimize the flavored CP asymmetry. The production of the B − L asymmetry, which is given by the product of the flavored CP asymmetry and the efficiency, results therefore from the balance of both effects.

Initial stage
Final stage Figure 6: Sketch of the type-II PFL mechanism. See text for further details.

PFL scenario efficiency
The problem of quantifying the efficiency is-in principle-an eight parameters problem: τ,a ∆ , m ∆ ,m ∆ , B φ , B aa , B τ τ and B aτ , which reduces to six parameters due to the constraints B + B φ = 1 and ∆ ≡ τ ∆ = − a ∆ . Since the efficiency does not depend on ∆ -see Eq. (74)-we will analyze the dependence of the efficiency upon the 5 remaining parameters: m ∆ ,m ∆ , B φ , B aa and B τ τ .
We start by analyzing the dependence upon B ij for fixed B φ , m ∆ andm ∆ . We will see that different flavor configurations (B ij configurations) will produce a minimal or maximal efficiency. However, as we will latter show in Sec. 4.2, the configurations that maximize the efficiency do not necessarily maximize the final B − L asymmetry. We then proceed by analyzing the dependence of the efficiency withm ∆ for fixed B φ , m ∆ and B ij , and finally the dependence of the efficiency with B φ for fixedm ∆ , m ∆ and B ij . This will allow us to understand and distinguish the main features of the type-II seesaw PFL scenario.

Efficiency: B ij dependence
In order to proceed, we first solve numerically the system of kinetic equations in (23)- (25) for different flavor configurations. We then provide some physical arguments supporting the special flavor configurations that maximize/minimize the efficiency. For concreteness, we fix three out of the five relevant parameters as follows: Once these parameters are fixed, the efficiency is entirely dictated by the flavor configurations determined by the values of the B ij parameters. It turns out that the flavor dependence is well described by the quantity: The green (red) dots indicate the allowed range for the efficiency, as required by neutrino data (3σ level [1]) for the inverted (normal) hierarchical light neutrino mass spectrum. In addition to the constraints on R, we also took into account the constraints imposed by data on the different B ij elements (see Fig. 8). We stress that these constraints apply only if the neutrino mass matrix is entirely dominated by the lightest scalar triplet contribution.
which represents the ratio of triplet decay branching ratios to different lepton-flavor final states. The importance of this quantity can be understood from Eq. (78), where we see that it is precisely through the B i that a triplet asymmetry is generated. We plot in Fig. 7 the efficiency as a function of this parameter R for the parameters fixed according to Eq. (79). A viable scalar triplet leptogenesis setup requires-of course-consistency with neutrino data [1,3,2]. If the most relevant contribution to the neutrino mass matrix in Eq. (6) is given by the lightest triplet, which can be regarded as a quite reasonable possibility (assumption), the determination of the available flavor configurations can be done directly via neutrino oscillation data. We present in Fig. 8 the constraints on R (left panel) and on the ratios of branching ratios B ii /B ij (right panel) as a function of the lightest neutrino mass, for the normal (green) and inverted (red) light neutrino mass spectrum. We fixed the neutrino oscillation parameters according to their upper and lower 3σ limits [1]. It can be seen that the R configuration leading to a vanishing B − L asymmetry, although showing up at the 3σ level in the normal spectrum case, can be readily evaded, thus showing the viability of the PFL scenario even in its most constrained form. Fig. 7 clearly shows that the efficiency exhibits four special configurations, namely (i, ii) two global maxima at R 1 and R 1, (iii) one local maximum and (iv) one global minimum near R ∼ 1. We now aim to understand the physical reasons behind these special configurations. Inverted hierarchy Normal hierarchy Figure 8: Allowed ranges for R (left plot) and the ratios of branching ratios B ii /B ij (right plots) as a function of the lightest neutrino mass for both normal (green) and inverted mass spectrum (red). The results have been derived by varying the neutrino oscillation parameters in the 3σ range according to [1].
Configurations (i) and (ii): following Eq. (80), these global maxima correspond to the flavor alignments B a B τ and B a B τ . The effect seems entirely driven by the B i , so we will not consider the possible effects of the C ij and C φ k elements in the analysis. More precisely, these maxima are reached whenever the inverse decays involving the a or τ flavor never enter in thermal equilibrium, i.e. for where we used in the last equality the parameter values given in Eq. (79). This value is in good agreement with the numerical results shown in Fig. 7, where the two maxima are reached for R 10 −4 and R 10 4 . For these configurations, only the asymmetry produced in one flavor is transferred through inverse decays i i →∆ to a triplet asymmetry, which is therefore maximal since the two flavor asymmetries have opposite signs. As a consequence, one asymmetry is depleted through the chain i i ↔∆ →φφ, while the other flavor asymmetry remains unaffected, clearly leading to a maximal efficiency.
Configuration (iii): this local maximum is in fact reached for R ≈ 1 when B ii B aτ . In this configuration, only the inverse decays a τ →∆ reach thermal equilibrium, and one expects no production of a triplet asymmetry, and therefore no production of a final B − L asymmetry, since the flavor asymmetries are depleted by the same amount. However, this is not the case because the C ij elements have a flavor structure, which plays a crucial role. The point is that when inverse decays are in thermal equilibrium, the combination of processes a τ ↔∆ →φφ and¯ a¯ τ ↔ ∆ → φφ tends to equilibrate the flavor asymmetries in lepton doublets Y ∆ τ ≈ −Y ∆ a , while in the meantime decreasing the separated asymmetries by a small amount 10 . But due to the chemical equilibrium conditions, the total lepton flavors asymmetries Y ∆ B/3−L i are in general different. Indeed, using Eq. (27), the total B − L asymmetry at freeze-out is related to the lepton flavor doublet asymmetries through: In the PFL regime, in the case where the final lepton doublet asymmetries are equal and opposite, Y ∆ τ ≈ −Y ∆ a (as for the case B ii = 0), a final B − L asymmetry can be produced only if the C ij elements have a flavor structure. This B − L asymmetry can be quite large because the flavor asymmetries in lepton doublets Y ∆ i decrease only slightly for this special configuration. Any significant deviation from this special configuration, e.g. B ii > B Eq ii , would not only tend to equilibrate the flavor asymmetries in the lepton doublets, but also the Y ∆ i separately through the chain¯ i¯ i ↔ ∆ → φφ. All in all, the efficiency has in consequence a local maximum for B aτ ≈ (1 − B φ )/2.
Configuration (iv): shifted to the left of the maximum defining configuration (iii), a minimal efficiency (almost vanishing efficiency) can be seen, it lies at about R ≈ 3/4. In order to understand the reason for this configuration to show up, we can look in a first step if analytically the efficiency may vanish for some value of the flavor parameters B ij . Using Eq. (74), we see that a vanishing efficiency is obtained whenever η τ (z) = −η a (z) for all z, which means through Eq. (72): which is satisfied as long as all the coefficients of the exponential power series expansion vanish, i.e. i,k=2,3 (84) We have found this turns out to be the case if the flavor-triplet-coupling-matrix entries satisfy the following two conditions where the corresponding elements must not depend on z, which is indeed our case-see Eq. (62). This result in turn can be understood using Eqs. (23) and (25). In the two flavor PFL scenario, since the source terms for both flavors are equal and opposite, a vanishing efficiency will be generated if the washouts of the two flavors are also equal and opposite, which is nothing but the conditions in Eq. (85). More precisely, for this to be achieved, we need that Y ∆τ = −Y ∆a remains valid at any time. As Eq. (25) shows, this requires: (a) Y ∆ ∆ = 0 and (b) i,j,k C ijk B ij Y ∆ k = 0 at any time. These two relations hold simultaneously if both conditions in Eq. (85) are fulfilled. Indeed, if relation (a) holds, (b) can be rewritten as the second condition in Eq. (85). On the other hand, if relation (b) holds, (a) can be rewritten using Eq. (25) as which is nothing but the first condition in Eq. (85). Using Eq. (62), these conditions can be simultaneously fulfilled only in the limit B φ → 0, in which case the triplet flavor configuration must satisfy the simple relation: Strictly speaking, since B φ = 0, the efficiency is not vanishing for any value of B ij . However, for small B φ , the efficiency does not vanish exactly anymore but shows now a minimum for R ≈ 3/4, which is in good agreement with the numerical results shown in Fig. 7.

Efficiency:m ∆ dependence
By fixing B φ = 10 −5 as in the previous section, and taking as an example B aa = B aτ = 0, and B τ τ = 1 − B φ , we display in Fig. 9 the dependence of the efficiency withm ∆ , for the three benchmark triplet masses (m ∆ = 10 11 , 10 12 , 10 13 GeV). It can be seen that irrespective of the triplet mass, the smallerm ∆ the smaller the resulting efficiency. The reason for this behavior follows directly from the relative strength of gauge and Yukawa induced reactions: the largerm ∆ the most likely triplets will decay rather than scatter, thus implying a larger efficiency. On the other hand, we see that the efficiency decreases with m ∆ . This is also due to gauge reactions: the smaller m ∆ the most likely the triplet will scatter rather than decay, thus implying a smaller efficiency. More precisely, as for the unflavored case, when gauge scatterings are faster than decays they suppress Y Σ − Y Eq Σ in Eq. (24) by a factor γ D /γ A , which implies an equal suppression of the source term in Eq. (25).
In the unflavored case one can distinguish two regimes [15], the gauge and Yukawa regimes, depending on the values of m ∆ andm ∆ . While in the unflavored case a maximum efficiency is obtained at the transition between both regimes, this is in general not anymore the case in the flavored leptogenesis scenario. Depending on the flavor configuration, a maximum efficiency can be obtained far in the Yukawa regime because of flavor effects, see Sec. 5 for a more detailed explanation.

Efficiency: B φ dependence
We present in Fig. 10  For B φ ≤ 10 −1 , which is the interesting region for this PFL scenario, we can distinguish two distinct regimes. They are separated by B Eq φ , the value at which the inverse decays φφ → ∆ become active, determined by the condition where we used in the last equality the parameter valuem ∆ = 0.05 eV. The way the efficiency scales with B φ depends on the flavor configurations. For B φ B Eq φ the efficiency always increases with B φ as a result of the fact that the larger B φ the faster the decay to SM scalars, as can be seen in Eq. (77), but the exact scaling actually also depends on the interplay of the Y ∆ ∆ and Y ∆ B/3−L i asymmetries. Now, as soon as B φ B Eq φ , inverse decays φφ → ∆ become efficient, implying that lepton number is broken by processes in thermal equilibrium (fast processes). This brings a new B φ suppression in the efficiency, resulting in an efficiency increasing less with B φ or even decreasing, depending on the flavor configuration, see Fig. 10.
To conclude, we see that for the flavor configuration that maximizes the efficiency, the value of B φ which gives the maximal efficiency is obtained for B φ ∼ B Eq φ , that is to say for the value of B φ at which the φφ → ∆ inverse decays are about to be active. In this case, the efficiency can be as large as unity for values of m ∆ 10 12 GeV, or less for smaller values of m ∆ (due to the gauge scattering thermalization effect). For other configurations that lead to smaller efficiencies, the maximum efficiency is obtained for much larger values of B φ ∼ 1.

Minimal and maximal B − L asymmetry
As stressed above, a PFL scenario is naturally favored as soon as i ( F ) ∆α dominates the CP asymmetry, which naturally holds if Yukawa couplings are larger than scalar couplings, i.e. when Eq. (20) holds. This equation can also be recasted in terms of the triplet branching ratios to scalar and lepton final states where the last inequality comes from the Cauchy-Schwarz inequality: As an example, taking a smooth triplet mass hierarchy m ∆α /m ∆ β ∼ 10 −1 (10 −2 ) and assuming the upper bound Tr[M ν α M ν † β ] ≈m ∆αm∆ β , a PFL scenario will be naturally dominant as soon as B α,β φ 10 −1 (10 −2 ). We have seen in the previous sections that the efficiency strongly depends on the flavor parameters B ij . Explicitly, we have shown that the efficiency has a minimum at R ≈ 3/4, global maxima at B aa = B aτ ≈ 0 and B τ τ = B aτ ≈ 0, and a local maximum at B aa = B τ τ ≈ 0. However, a maximal efficiency does not imply a maximal B − L asymmetry. Indeed, using Eq. (15), we can actually compute a general upper bound for the purely flavored CP asymmetry: where we assumed perturbative Yukawa couplings for the second triplet generation, i.e. |Y β | ≤ 1. This expression shows clearly that the three configurations that maximize the efficiency give vanishing CP asymmetries! This can be understood easily from the fact these configurations involve a Yukawa coupling only for one flavor. We see also that the upper bound on the CP asymmetry is directly related to the hierarchy between the different triplet masses, which is compatible with the requirement in Eq. (88), i.e. a smooth triplet mass hierarchy favors PFL scenario and allows for a large CP asymmetry. We plot in Fig. 11 the resulting maximal B − L final asymmetry that can be achieved, as a function of the flavor parameter R, for m ∆α /m ∆ β = 10 −1 . To this end we have considered the same parameter configuration used in Fig. 7. It can be seen that the maximal B − L asymmetry that can be achieved can account for the observed baryon asymmetry of the Universe for a large range of R values, except at R ≈ 3/4. We also point out that, if the neutrino mass matrix is dominated by the light scalar triplet, the constraints coming from neutrino data are compatible with successful PFL scenario. One realizes as well that two of the B −L asymmetry global maxima are shifted with respect to the efficiency maxima, and are now located around the points at which i i →∆ inverse decay rates are of the order of the Universe Hubble expansion rate, where B ii = B Eq ii , see Eq. (81). As a final remark, it is worth noting that the local maximum at R ≈ 1 has gone away.
This result has to be compared with the unflavored case, where the CP asymmetry is very suppressed for B φ B or B φ B , since the CP asymmetry is proportional to B φ B -see Eqs. (14) and (19). This is no more the case in PFL leptogenesis, since the lepton number conserving and flavor violating CP asymmetries depend only on Yukawa couplings.

General triplet flavored leptogenesis
Having discussed the viability of the PFL scenario in pure type-II seesaw models, we are now in a position to analyze the impact that flavor effects may have in general triplet flavored leptogenesis models. Here, as already defined in the introduction, by "general models" we refer to models where the lepton number violating CP asymmetries are relevant or even dominate over the lepton number conserving CP asymmetries which drive PFL. Accordingly, if the extra degrees of freedom enabling a non-vanishing CP asymmetry are additional triplets, a general model will be defined by Eq. (13), while if the extra degrees of freedom are RH neutrinos-as will be the case in models featuring interplay between type-I and type-II seesaws-the CP asymmetry in Eq. (21), being lepton number violating, will always define a "general model". In what follows we will assume the asymmetry is entirely generated via the decays of the lightest triplet, something that can be achieved by taking a heavy mass spectrum obeying the following hierarchy: m ∆ M ∆α,Nα . In "general" scenarios, since the CP asymmetries are lepton number breaking, successful leptogenesis is possible in the absence of lepton flavor effects, in contrast to PFL where flavor effects are mandatory. In what follows we will quantify the enhancement that the inclusion of flavor effects may have in the final B − L asymmetry, and in order to do that and to put the discussion in context we will start by briefly reviewing some general well known results of the unflavored regime, which have been derived from kinetic equations in which none of the SM reactions were taken into account (see Sec. 3.1) [11].
In the unflavored case, an efficiency function accounting for the z (temperature) evolution of the unflavored B − L asymmetry can be defined 11 : where at freeze out η ≡ η(z → ∞). As in fermion triplet leptogenesis, in this case one can also define a gauge and a Yukawa regime, which boundaries in them ∆ − m ∆ parameter space plane are determined by the values of B φ , as displayed in Fig. 12 (upper left-hand side plot). While in the gauge regime triplet dynamics is dominated by gauge-mediated triplet annihilation, in the Yukawa regime the dynamics is driven by Yukawa-induced

Gauge Regime
Yukawa Regime reactions, and so it is in the latter where flavor effects can have striking implications. For a fixed triplet mass, the transition between both regimes becomes determined by a "critical"m ∆ , that we denote bym ∆ 12 . The behavior of the efficiency (i.e. of the B − L asymmetry) is to a large extent determined by the regime where leptogenesis takes place (gauge or Yukawa), or in other words by the location of the boundary in them ∆ − m ∆ plane, determined in turn by the value of B φ . For the parameter space points shown in Fig. 12 (upper left-hand side plot), the behavior of the B − L asymmetry goes along the following lines: • The B φ = 1/2 case: In the gauge (Yukawa) regime the asymmetry increases (decreases) withm ∆ . In the gauge regime this is due to the fact that there is no substantial production of the asymmetry until z approaches the value z = z A where γ A /γ D goes below 12 In practice, for a given value of m ∆ ,m ∆ is defined as the value ofm ∆ above (below) which the inverse decays are (not) in thermal equilibrium once the gauge scatterings cease to dominate the whole process (i.e. it leads to γ D /n Eq ∆ H = 1 when γ A goes below γ D at a temperature z = z A ).
unity, z A ∼ 3 in the right-hand side plot in Fig. 3. The generation of the B − L asymmetry can then be understood as proceeding in two stages determined by two z (temperature) windows: z < z A and z > z A . For z < z A the B − L asymmetry is suppressed by a factor γ D /γ A ∝ m ∆m∆ /g 4 , so for a fixed scalar triplet mass the smallerm ∆ , the smaller the ratio γ D /γ A , and so the asymmetry produced. In this range most of the production occurs when z z A . Before, γ D /γ A is exponentially suppressed. For z > z A instead, γ A becomes irrelevant and, since in the gauge regime there is no suppression effect from inverse decays, the asymmetry produced within this z range is simply equal to the number of triplets left times the CP asymmetry, . The relevance of this contribution is determined bym ∆ : large values of this parameter imply small values for z A , which then in turn imply a less Boltzmann suppressed Y Eq ∆ (z A ). Thus, the B − L asymmetry generated consist of two contributions, one generated at z < z A and a second produced at z > z A , namely In the Yukawa regime instead, the efficiency decreases withm ∆ because in this case still there is no substantial asymmetry produced until z approaches z A , and because the asymmetry produced afterwards is further washed-out by the inverse decay whose magnitude increases withm ∆ .
• The small B φ case (B φ = 10 −3 or B φ = 10 −5 ): As can be seen in the upper right-hand and lower left-hand plots in Fig. 12 the efficiency goes on to increase withm ∆ well inside the Yukawa regime. This can be understood from the fact that in this case, even if the total decay rate is well in thermal equilibrium and faster than the gauge scattering rate, the decay to a pair of scalars remains out-of-equilibrium (see Fig. 3, left-hand side plot), which implies that lepton number is not broken by inverse decay, resulting in no washout from these processes. The efficiency is suppressed only by gauge-mediated triplet scatterings which, as pointed out in the previous item, precludes any substantial production of B − L asymmetry until z reaches ∼ z A . All in all, despite standing in the Yukawa regime, the B − L asymmetry is only suppressed by gauge scatterings in the way stressed in the previous item, and so the total B − L asymmetry is again given by Eq. (92). For large values ofm ∆ , the gauge suppression is nevertheless faint because z A is not much larger than unity. This results in very large efficiency for B φ 1/2. Only whenm ∆ m ∆ is lepton number effectively broken by scalar doublet-triplet inverse decays and the efficiency decreases withm ∆ . Note that, even if large efficiencies can be obtained in this way for B φ 1/2, since lepton number is unbroken in the B φ → 0 limit, these efficiency enhancements are accompanied by a suppression of the CP asymmetry, so that still the maximum B − L asymmetry is obtained for values of B φ not far from its maximum value 1/2. The picture described in the items above is expected to change as soon as one hits the flavor regime, if the parameters are such that triplet dynamics takes place in the Yukawa regime. In order to discuss the impact that flavor effects may have, it is convenient to introduce an efficiency-like parameter. Let us discuss this in some more detail. Flavor coupling does not allow a conventional definition of an efficiency, however a parameter resembling the efficiency of the unflavored case can be defined: withη given byη where the flavored efficiency functions have been defined in Eq. (72) and with Note that the definition ofη is such that when taking the limit a ∆ → τ ∆ , one recovers the usual definition of the efficiency. This parameter proves to be useful in particular when comparing the results obtained in the flavored regime with those arising from the unflavored limit. Instead, the parameter ∆ , introduced in the definition ofη, has a twofold utility: first of all it "measures" the deviation from the PFL (¯ ∆ 1) and the general scenarios (¯ ∆ 1); secondly, it "measures" the flavor misalignment of the source terms in the evolution equations of the B/3 − L i charges.
In order to quantify the impact that flavor effects have on the B − L asymmetry, it is useful to consider first a case where both CP flavored asymmetries are equal, i.e. ∆ = 0, that is to say in a way the extreme opposite to the PFL case. This will allow to discuss flavor effects that are different from the ones we discussed in the previous section for the PFL case. For this case, we show in Fig. 12 the efficiency-like parameterη as a function of m ∆ for different values of (m ∆ , B φ ), overlapped with the results we got for the unflavored case.
Some comments are in order regarding these results. Either in the gauge or in the Yukawa regime (form ∆ ∼m ∆ ), gauge scatterings preclude any substantial creation of a B − L asymmetry as long as γ A /γ D 1, that is to say as long as z is below ∼ z A . The B/3 − L i asymmetry production is anyway suppressed by a γ D /γ A factor as in Eq. (92). Gauge scatterings, being flavor "blind", are insensitive to lepton flavor effects and so the suppressions they induce cannot be overcome. This means that, as long as we consider values of parameters which in the unflavored case gives Eq. (92) i.e. the maximum efficiency allowed by gauge scattering (in the gauge regime, or in the Yukawa regime for B φ < 1/2 and not too large values ofm), flavor effects cannot further enhance the efficiency. However, in the Yukawa regime, for large values ofm ∆ , since inverse decay washouts are flavor sensitive, flavor effects allow to largely avoid this effect, so that the efficiency goes on to increase also there, as Fig. 12 shows. As a result in this case too, To further emphasize the effects of flavor in the¯ ∆ = 0 case, we have calculated the efficiency-like parameterη as a function of B φ . The calculation has been done for two flavor configurations (a) and (b)-those already used in Fig. 10-and for fixed parameters m ∆ andm ∆ . The results are displayed in Fig. 13 (left-hand side plot), where the flavored and unflavored (as e.g. in Refs. [11,15]) outputs are compared. It can be seen that considering only the effects of the SM interactions (i.e. configuration (b)), one can get an enhancement of order 2 with respect to the unflavored case, whereas for the flavor configuration (a) one can get a further one-order of magnitude enhancement, as can be seen in particular for B φ = B = 1/2.
Finally, let us discuss what happens very qualitatively in cases other than the pure PFL case, τ ∆ = − a ∆ and the "opposite" case, τ ∆ = a ∆ . In these "intermediate" cases the "efficiency" as defined in Eq. (93) cannot be considered as an efficiency anymore, because it can be larger than one. For instance in the pure PFL case it is infinity since ∆ = 0. As a result it is difficult to span the range of possibilities in simple terms for these cases. To get a reliable idea of the behavior of the efficiency-like parameter, and thus of the B − L asymmetry, in a specific case, the most efficient procedure is probably to integrate first the full set of Boltzmann equations in a "blind" way and see what the result looks like before trying to understand it by simple means. But the basic picture qualitatively remains clear. As long as z < z A any flavor asymmetry production is suppressed by a factor of γ D /γ A , and afterwards the B − L asymmetry that can be produced can anyway not be larger than the number of triplets remaining at z ∼ z A times the sum of the absolute values of the flavor asymmetries. The important flavor effects stressed above, from the L-violating inverse decays as well as from the L-violating decays, will be operative in a way which may depend non trivially on basically all parameters, the flavor CP asymmetries, the C ,φ constants, the total decay rate and the various branching ratios.
As an illustration of the efficiency dependence on the mismatch between the flavored CP asymmetries, parameterized by¯ ∆ , on the right-hand side plot in Fig. 13 we show the dependence ofη with¯ ∆ for B φ = 1/2 for the flavor configuration (a). In the region where ∆ 1 is small ( a ∆ ∼ τ ∆ ), as previously stressed, any possible mismatch between the asymmetries in flavor a and τ can only be due to the flavor dependence of the washout terms. As ∆ increases, the source terms start having a flavor dependence as well, and so an imbalance between production in flavor a and τ appears. The flavor dependence of both production and washout at large ∆ , yields larger values forη. In other words, flavor effects are diminished in those regions of parameter space where ∆ 1 and become more remarkable in regions where ∆ 1. Accordingly, in the various flavor regimes, enhancements of the efficiency-like parameterη with respect to the unflavored case are a consequence of combined effects: the mismatch between the different flavored CP asymmetries i ∆ , the SM interactions through the C and C φ matrices, and the flavor configurations encoded in B ij .

Conclusions
We have considered scalar triplet leptogenesis scenarios where the states enabling successful production of the cosmic baryon asymmetry are either extra triplets or RH neutrinos. We have derived for the first time the complete set of flavored classical Boltzmann equations governing the evolution of the different relevant asymmetries, including the effects of those SM reactions which in the leptogenesis era may be fast: charged lepton and quark Yukawa reactions as well as QCD and electroweak sphaleron processes. The resulting network of kinetic equations combined with the different asymmetry coupling matrices, which follow from the chemical equilibrium conditions enforced by the fast SM processes, provide the tools for studying triplet scalar leptogenesis in full generality.
In scenarios involving an additional triplet (purely type-II seesaw scenarios), we have identified a novel class of models where the flavored CP asymmetries, consisting of lepton number violating and lepton number conserving contributions, become dominated by the lepton number conserving piece. Such a dominance naturally shows up as soon as the couplings of at least one triplet (i.e. not necessarily of all seesaw states as for PFL type-I seesaw scenarios) approximately conserve L, in practice simply that it couples more to leptons than to scalars. The purely flavored CP asymmetries have no reasons to be suppressed by the smallness of the light neutrino masses since, in contrast to the leptonnumber-violating CP asymmetries, they only involve L-conserving couplings.
With the aid of the derived flavored Boltzmann equations and asymmetry coupling matrices, we have carried out a throughout study of the PFL scenario in the two flavor regime, for definitiveness. The way this PFL scenario works is totally novel (for small values of B φ which gives natural dominance of the purely flavored CP asymmetries): in this case there is no L-violating process in thermal equilibrium at any epoch but yet flavor effects do allow the creation of a B − L asymmetry from the L-violating slow decay of the triplet to SM scalars. We have proved its viability by calculating the B − L yield, finding that, for utterly reasonable and wide ranges of parameter values, a baryon asymmetry consistent with observation can always be achieved. By exploring the B − L asymmetry parameter space dependence, we have determined the lepton flavor configuration that maximizes the efficiency, finding that the same structure renders the flavored CP asymmetry minimal. Our findings show that maximal B − L yield is achieved for intermediate lepton flavor configurations.
Finally, we discussed general scenarios, which we have defined by the condition of the CP asymmetry involving lepton number violation. These scenarios can arise either in models with extra triplets or with RH neutrinos (models exhibiting interplay between type-I and type-II seesaw). We discussed the impact that lepton flavor effects may have in the final B − L asymmetry, showing that relevant flavor effects can only be achieved in the Yukawa regime, being more striking as deeper one moves into that regime, and depending on the parameter flavor configuration. Our results show that for certain flavor structures-once lying in the Yukawa regime-the asymmetry may be enhanced by several orders of magnitude. In both regimes, the B − L asymmetry production is suppressed as long as the gauge scattering rate is faster than the decay rate (i.e. for z z A ), and the asymmetry produced afterwards is proportional to the number of triplet remaining afterwards. The latter being Boltzmann suppressed if z A > 1, the suppression is more pronounced for smaller values of m ∆ , given that as m ∆ decreases z A increases. Deep in the Yukawa regime, however, the decay rate becomes faster than the gauge scattering rate at very early epochs, z A 1 and this Boltzmann suppression goes away. In this way, deep in the Yukawa regime, one can basically avoid all efficiency suppressions, from gauge scattering as well as, through flavor effects, from L-violating inverse decays. +3 r 2 − 1 2C 1 + C 2 r 2 − 1 ln where the following notation has been adopted: r = 1 − 4/x and C 1 = 12g 4 + 3g 4 Y + 12g 2 g 2 Y and C 2 = 6g 4 + 3g 4 Y + 12g 2 g 2 Y (with g and g Y the SU (2) and U (1) SM gauge coupling constants).
The reaction densities with a resonant intermediate state subtracted can be calculated from Eqs. (98), (99), (100) and (101) as follows: Rates for the different SM reactions are approximately given by [29,30,35,36]: Yukawa reactions: where h f i denotes the Yukawa coupling of fermion f i .