Type-I Seesaw as the Common Origin of Neutrino Mass, Baryon Asymmetry, and the Electroweak Scale
Vedran Brdar, Alexander J. Helmboldt, Sho Iwamoto, Kai Schmitz
TType-I Seesaw as the Common Origin of NeutrinoMass, Baryon Asymmetry, and the Electroweak Scale
Vedran Brdar, a, ∗ Alexander J. Helmboldt, a, † Sho Iwamoto, b, c, ‡ and Kai Schmitz b, c, § a Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany b Universit`a degli Studi di Padova, Via Marzolo 8, 35131 Padua, Italy c INFN, Sezione di Padova, Via Marzolo 8, 35131 Padua, Italy
Abstract
The type-I seesaw represents one of the most popular extensions of the Standard Model. Previousstudies of this model have mostly focused on its ability to explain neutrino oscillations as well as onthe generation of the baryon asymmetry via leptogenesis. Recently, it has been pointed out thatthe type-I seesaw can also account for the origin of the electroweak scale due to heavy-neutrinothreshold corrections to the Higgs potential. In this paper, we show for the first time that all ofthese features of the type-I seesaw are compatible with each other. Integrating out a set of heavyMajorana neutrinos results in small masses for the Standard Model neutrinos; baryogenesis isaccomplished by resonant leptogenesis; and the Higgs mass is entirely induced by heavy-neutrinoone-loop diagrams, provided that the tree-level Higgs potential satisfies scale-invariant boundaryconditions in the ultraviolet. The viable parameter space is characterized by a heavy-neutrinomass scale roughly in the range 10 . ··· . GeV and a mass splitting among the nearly degenerateheavy-neutrino states up to a few TeV. Our findings have interesting implications for high-energyflavor models and low-energy neutrino observables. We conclude that the type-I seesaw sectormight be the root cause behind the masses and cosmological abundances of all known particles.This statement might even extend to dark matter in the presence of a keV-scale sterile neutrino. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] a r X i v : . [ h e p - ph ] O c t ontents The
Standard Model (SM) describes neutrinos in terms of massless left-handed (LH) Weyl fermions.The observation of neutrino flavor oscillations, however, points at nonvanishing neutrino masses,which provides direct experimental evidence for new physics (NP) beyond the Standard Model (BSM) [1]. One straightforward way of explaining nonzero neutrino masses is to supplement theStandard Model by massless right-handed (RH) neutrinos N R I that transform as complete singletsunder the SM gauge group. The presence of RH neutrinos (RHNs) in the theory then allows one towrite down a Yukawa term that couples LH and RH neutrinos to the SM Higgs doublet φ = ( φ + , φ ) T , L D N = i2 N R I /∂ N R I − y Iα N R I ˜ φ † L α + h.c. , I = 1 , , , α = e, µ, τ . (1)Here, y Iα is a matrix of complex Yukawa couplings, L α = (cid:0) ν L α , (cid:96) L α (cid:1) T represents the SM LH leptondoublet of flavor α , and ˜ φ = i σ φ ∗ = ( φ ∗ , − φ − ) T denotes the hypercharge-conjugated Higgs doublet.Equation (1) sets the stage for neutrino mass generation via the standard Higgs mechanism. Upon electroweak symmetry breaking (EWSB), the Higgs field acquires a nonzero vacuum expectationvalue (VEV), √ (cid:104) φ (cid:105) = v (cid:39)
246 GeV, such that LH and RH neutrinos combine into massive Diracfermions. This scenario is referred to as the Dirac-neutrino scenario. In this model, the electroweak (EW) scale v , which is induced by the tree-level Higgs mass parameter µ , can be identified as thefundamental energy scale that determines the masses of all SM particles, i.e. , the masses of the SMHiggs boson, EW gauge bosons, and all SM fermions. Another attractive feature of this minimalSM extension is that it provides a possibility to explain the origin of the baryon asymmetry of theUniverse (BAU) via the so-called neutrinogenesis mechanism [2]. This mechanism is based on theidea that, in the presence of RH neutrinos, the decay of heavy exotic degrees of freedom (DOFs) inthe early Universe can lead to a primordial asymmetry between LH and RH neutrinos. The leptonnumber carried by the LH neutrinos, L L , is then converted into a primordial baryon number B by1W sphaleron processes [3,4]. The lepton number carried by the RH neutrinos is of equal magnitudebut different sign, L R = − L L . It remains sequestered from the rest of the thermal bath, until LHand RH neutrinos eventually equilibrate at late times due the Yukawa interaction in Eq. (1).The Dirac-neutrino model manages to explain the small SM neutrino masses, offers a startingpoint for realistic models of baryogenesis (see, e.g. , Ref. [5]), and relates the masses of all knownelementary particles to a single energy scale, i.e. , the Higgs mass parameter µ . However, despite theseachievements, it also suffers from a number of shortcomings and calls for further model building:1. In order to relate tiny SM neutrino masses of O (0 .
1) eV to the Higgs VEV v ∼
100 GeV, theRHN Yukawa couplings y Iα need to be of O (10 − ). This aggravates the SM flavor puzzle.2. The particle content of the Dirac-neutrino model on its own is not sufficient to realize successfulbaryogenesis. In order to generate a primordial chiral neutrino asymmetry, it is necessary toextend the model by new DOFs whose masses may be as large as the energy scale of gaugecoupling unification, Λ GUT ∼ GeV, in grand unified theories (GUTs). The Dirac-neutrinoscenario features, in particular, no intrinsic connection between the generation of the baryonasymmetry at high energies and the phenomenology of neutrino oscillations at low energies.3. If the Higgs VEV is regarded as a fundamental energy scale, one would naively expect thatthe solutions to other SM problems, such as dark matter (DM) or the EW hierarchy problem,should also be related to new physics at or slightly above the EW scale. However, all experi-mental efforts thus far have failed to directly detect new particles beyond the Standard Model.This challenges the notion of the Higgs VEV as a fundamental scale and might be taken as anindication that the scale of new physics may, in fact, be vastly separated from the EW scale.4. Equation (1) is not the most general Lagrangian that is compatible with the field content ofthe Dirac-neutrino model. Indeed, without imposing any symmetry, one is allowed to writedown a Majorana mass term for the RH neutrinos, which explicitly breaks lepton number L .To forbid this term, one has to impose L as an exact global symmetry, or alternatively, B − L asa gauge symmetry. This represents a model-building constraint that needs to be accounted forwhen embedding the Dirac-neutrino model into a more comprehensive model at high energies. The shortcomings of the Dirac-neutrino model motivate the extension of Eq. (1) by a Majoranamass term for the RH neutrinos, which results in the Lagrangian of the type-I seesaw model [6–10], L M N = i2 N R I /∂ N R I − y Iα N R I ˜ φ † L α − N R I M IJ (cid:0) N R J (cid:1) C + h.c. (2)Here, M IJ is a symmetric matrix of L -violating Majorana masses, which are a priori unrelated to anyother SM mass scale. The matrix M IJ can always be chosen to be real and diagonal, M IJ = M I δ IJ ,without loss of generality. In the model defined by Eq. (2), the SM neutrinos turn into Majoranafermions upon EWSB, which is why this scenario is referred to as the Majorana-neutrino scenario.Typically, one assumes the RHN masses to be much larger than the EW scale, M I (cid:29) v . The SMneutrino masses then end up being suppressed not only by small RHN Yukawa couplings, but alsoby the large ratio of mass scales, v/M I (cid:28)
1. In the Majorana-neutrino scenario, it is therefore nolonger necessary to assume Yukawa couplings as small as y Iα ∼ − . Another advantage of this2irac-neutrino option Majorana-neutrino optionUnderlying symmetry Lepton number → M I = 0 Scale invariance → µ = 0Scale behind all SM particle masses Higgs mass parameter µ RHN Majorana masses M I Anticipated scale of new physics Λ NP (cid:38) µ Λ NP (cid:38) M I SM neutrino mass generation Higgs mechanism Type-I seesaw mechanismFermion type of SM neutrinos Dirac fermions Majorana fermionsGeneration of the baryon asymmetry Neutrinogenesis
Leptogenesis (this work)
Table 1: Properties of the Dirac-neutrino and Majorana-neutrino options. Note how the scales M I and µ exchange their roles in both scenarios due to the different underlying symmetries. In the Dirac-neutrino scenario, the RHN masses are absent and the masses of all known particle are set by thetree-level Higgs mass parameter. The Majorana-neutrino option is based on the reversed situation.As a consequence, the scale of new physics is expected to be much larger in the Majorana-neutrinoscenario than in the Dirac-neutrino scenario. In this paper, we show that the generation of neutrinomasses and the EW scale in the Majorana-neutrino scenario is compatible with leptogenesis.model is that it establishes a link between baryogenesis and low-energy neutrino phenomenology.In the type-I seesaw, the baryon asymmetry can be generated via the leptogenesis mechanism [11], i.e. , via out-of-equilibrium decays of heavy Majorana neutrinos in the early Universe. These decaysgenerate a primordial lepton asymmetry, which is again converted to a primordial baryon asymmetryby EW sphalerons. For a recent series of review papers on leptogenesis, see Refs. [12–16].However, also the Majorana-neutrino scenario comes with a number of challenges and drawbacks.One may, e.g. , complain that the type-I seesaw model requires the introduction of new mass param-eters that are unrelated to the EW scale. One is therefore no longer able to identify a common originof all particle masses, as it is possible in the Dirac-neutrino scenario. Furthermore, the large hierar-chy between the mass scales M I and v can lead to the destabilization of the EW scale because of largeradiative corrections to the Higgs mass from the RHN sector [17]. Consider, e.g. , standard thermalleptogenesis, which can be shown to require RHN masses as large as M I (cid:38) GeV [18–21]. In thiscase, the Higgs mass is necessarily fine-tuned, which may be regarded as a naturalness problem [22].A possible way out of these problems is to turn the issue of radiative corrections to the Higgsmass into a virtue. It has recently been pointed out that the Higgs mass parameter µ in the Higgspotential can be entirely induced by RHN threshold corrections, provided that µ = 0 at tree level.This scenario is consistent with the low-energy neutrino oscillation data and has been dubbed the“neutrino option” [23, 24] (see Ref. [25] for related earlier work). The main premise of the neutrinooption is that the classical SM Lagrangian L SM satisfies scale-invariant boundary conditions in the ultraviolet (UV), such that µ = 0 above the RHN mass threshold. Alongside other symmetry-based approaches, such as supersymmetry or new (strongly coupled) gauge dynamics, the conceptof classical scale invariance provides a well-motivated guiding principle for the construction of BSMmodels [26–33]. In recent years, it has been applied as a tool for BSM model building in a varietyof scenarios, ranging from neutrino physics over dark matter to inflation (see, e.g. , Refs. [34–50]).It is important to note that the RHN mass terms in Eq. (2) explicitly break scale invariance. Inthe context of the neutrino option, classical scale invariance should therefore be regarded as a workingassumption. One open question is why classical scale invariance should be a good symmetry of the3M Lagrangian L SM but not of the seesaw Lagrangian L M N . Another point is that one ultimatelyhas to explain how classical scale invariance at low energies can emerge as the remnant of a full-fledged quantum symmetry at high energies [51]. These aspects of the neutrino option require furtherinvestigation. A first step in this direction has been made in Ref. [52], which illustrates how the RHNmasses in Eq. (2) can be generated via the spontaneous breaking of scale invariance in a theory thatinitially preserves conformal symmetry at the level of the entire classical Lagrangian. Conformalsymmetry breaking in this model may be associated with a first-order phase transition in the earlyUniverse, which could give rise to an observable signal in gravitational waves [53]. In this paper, we will stick to the original formulation of the Majorana-neutrino option in Refs. [23,24]and not attempt to embed it into a BSM model that is fully scale-invariant at the classical level.In this sense, the ad hoc assumption of scale-invariant boundary conditions in the SM sector canbe seen as being on the same footing as the ad hoc assumption of lepton number conservation inthe Dirac-neutrino model. Both the Dirac-neutrino and Majorana-neutrino models offer no intrinsicexplanation for the symmetry principles that they are based on and eventually need to be extended.However, a fascinating consequence of replacing lepton number conservation by classical scale invari-ance as a guiding principle in the construction of the BSM Lagrangian is that the RHN Majoranamasses M I now supersede the Higgs mass parameter µ as the fundamental input scale that deter-mines the masses of all known particles. The Majorana-neutrino option amounts to the idea thatthe RHN Majorana masses first induce the EW scale, which then leads to the generation of all SMparticle masses via a combination of the Higgs and type-I seesaw mechanisms. This scenario for theorigin of the SM particle masses has several advantages over the standard Higgs mechanism:1. It sheds new light on the question at which energy scale one should expect to find new physics.Provided that the EW scale is an effective scale that only comes about because of radiativecorrections from the RHN sector, it is conceivable that the scale of new physics is actually to besought at energies above the RHN mass thresholds, Λ NP (cid:38) M I . This may explain the absenceof new physics in current experiments. In this case, one could speculate that the RH neutrinosactually play the role of messenger fields that communicate with both the Standard Model andthe BSM sector that is, e.g. , responsible for the spontaneous breaking of scale invariance.2. In the Standard Model, the Higgs mass term and its negative sign, which is crucial for EWSB,are introduced by hand. The Majorana-neutrino option provides, by contrast, a dynamicalorigin for the Higgs mass term and may explain its negative sign. The key observation is thatthe RHN one-loop diagrams that induce µ come with an overall minus sign because of theunderlying Fermi-Dirac statistics. For an appropriate choice of the renormalization scale (seeSec. 2.3), this negative sign eventually leads to the correct sign of the Higgs mass term.3. In order to explain the two measured (solar and atmospheric) neutrino mass-squared differ-ences, the seesaw sector must contain at least two RH neutrinos (2RHNs) (see, e.g. , Refs. [54–58]). However, the number of RH neutrinos can easily be larger. In particular, it may appearappealing to extend the seesaw sector by an additional RH neutrino with a mass in the keVrange whose cosmological relic density accounts for the dark matter in the Universe [59–63].We will briefly discuss such a scenario towards the end of the paper in Sec. 3.3. In this case,4he type-I seesaw would not only set the masses of all known particles, but also be responsiblefor the mass and cosmological abundance of the DM particle.In Refs. [23, 24], it has been shown that the radiative generation of the Higgs mass parameterin the type-I seesaw typically requires RHN masses of O (10 ) GeV. In this case, the RHN Yukawacouplings that are necessary to explain the SM neutrino masses via the type-I seesaw mechanism aretoo small to allow for baryogenesis via standard thermal leptogenesis. This can also be expressed bysaying that RHN masses of O (10 ) GeV violate the Davidson-Ibarra (DI) bound, M I (cid:38) GeV, onthe RHN mass scale for standard thermal leptogenesis [18]. For this reason, it has been argued thatthe neutrino option is not compatible with the simplest (vanilla) version of thermal leptogenesis.In this paper, we are, however, going to show that the type-I seesaw does manage to simulta-neously generate SM neutrino masses, the EW scale, and the baryon asymmetry of the Universe,provided that baryogenesis proceeds via resonant leptogenesis [64–66]. In this leptogenesis scenario,the CP asymmetry in RHN decays is resonantly enhanced because of a nearly degenerate RHN massspectrum (see Ref. [14] for a review). The additional gain in CP asymmetry allows one to bypass theDI bound and lower the energy scale of leptogenesis down to values that are compatible with the neu-trino option. In our model, the small mass splitting among the RHN mass eigenstates correspondsto a second working assumption. When embedding the type-I seesaw into a UV completion, onewould have to show how this small mass splitting can be accounted for by a symmetry in the RHNsector (see, e.g. , Refs. [67–71]). In passing, we also mention that an alternative possibility to lowerthe energy scale of leptogenesis would be to rely on a concerted interplay of flavor effects [72–80].However, in this case, one would have to tune the tree-level SM neutrino masses against one-loopradiative corrections in order to realize successful leptogenesis [79, 80]. It is less clear to us how sucha parametric cancellation of different terms in perturbation theory may be achieved by imposing asymmetry at the Lagrangian level. For this reason, we will not consider this possibility in this paperand focus on resonant leptogenesis instead. In our analysis, resonant leptogenesis therefore actsas the counterpart of neutrinogenesis in the Dirac-neutrino scenario (see Tab. 1 for a comparisonbetween the Dirac-neutrino and Majorana-neutrino options).A remarkable outcome of our analysis is the realization that the type-I seesaw may not onlybe responsible for the masses of all SM particles, but also for the asymmetry between matter andantimatter in the Universe. In the context of the Majorana-neutrino option, it is therefore possibleto identify the type-I seesaw as the principle cause behind the masses and cosmological abundancesof all known particles. Moreover, if the seesaw sector also contains an additional keV-scale RHneutrino, this statement can be even extended to include dark matter. In this case, the type-Iseesaw would be the origin of the masses and cosmological abundances of visible and dark matter.The remainder of this paper is organized as follows: In the next section, we will review the type-Iseesaw and discuss how it manages to simultaneously generate SM neutrino masses (Sec. 2.1), thebaryon asymmetry of the Universe (Sec. 2.2), and the EW scale (Sec. 2.3). In Sec. 3, we will thenturn to the bulk of our analysis and show how the requirements of (i) successful baryogenesis and (ii)the neutrino option allow one to constrain the parameter space of the type-I seesaw model. We willpresent some analytical estimates (Sec. 3.1), the results of a comprehensive numerical parameter scan(Sec. 3.2), and discuss the implications of our analysis for high-energy flavor models and keV-scalesterile-neutrino dark matter (Sec. 3.3). Section 4 contains our conclusions and a brief outlook.5 Type-I seesaw
The type-I seesaw Lagrangian is given in Eq. (2). In this paper, we shall restrict ourselves tothe minimal type-I seesaw involving only two RH neutrinos, N R I ( I = 1 , y Iα in Eq. (2) is (in general) a rank-2 matrix. This is sufficient to explain the two knownnonzero mass-squared differences in the SM neutrino sector. At the same time, one of the three SMneutrino masses, m i ( i = 1 , , { m , m , m } = 0. Given the factthat the absolute neutrino mass scale, m tot = (cid:80) i m i , has not yet been measured, this is a perfectlyviable possibility at present. Similarly, resonant leptogenesis only demands two nearly degenerateRHN mass eigenstates; the presence of a third RH neutrino is not necessarily required. Working with only two RH neutrinos has two important advantages. First of all, from a physicalpoint of view, it is a relevant observation that already the minimal 2RHN seesaw succeeds in explain-ing neutrino masses, baryon asymmetry, and the EW scale. It is actually not necessary to considerthe standard scenario involving three RH neutrinos (3RHNs). This leaves room for adding a third,keV-scale RH neutrino N R3 that would not affect the low-energy neutrino observables, but whichcould act as a DM candidate (see Sec. 3.3). Such a scenario represents a highly attractive possibilitythat deserves further scrutiny in future work. A second advantage of working with only two RHneutrinos is that it leads to simplifications at the technical level. The 3RHN seesaw model features18 free parameters in the high-energy Lagrangian (3 RHN masses plus 9 complex Yukawa couplingsminus 3 unphysical charged-lepton phases), whereas the 2RHN seesaw model only contains 11 freeparameters at high energies (2 RHN masses plus 6 complex Yukawa couplings minus 3 unphysicalcharged-lepton phases). At the same time, the 3RHN seesaw gives rise to 9 observable quantities atlow energies (3 nonzero SM neutrino masses, 3 mixing angles, and 3 physical CP -violating phases),while the 2RHN seesaw only leads to 7 observables (2 nonzero SM neutrino masses, 3 mixing angles,and 2 physical CP -violating phases). Fixing the masses in the RHN spectrum for the purposes ofleptogenesis, this means that there is a mismatch of 6 real DOFs between high-energy and low-energyquantities in the 3RHN seesaw, but only a mismatch of 2 real DOFs in the 2RHN seesaw. In otherwords, the unconstrained theory space of possible flavor models has six real dimensions in the 3RHNseesaw, while it is only two-dimensional in the 2RHN seesaw. As a consequence, it is easier to scanover all possible flavor models in the 2RHN seesaw than in the 3RHN seesaw.Let us now review the mechanism of SM neutrino mass generation in the type-I seesaw model. Inthe course of EWSB, the SM Higgs doublet develops a nonvanishing VEV, √ (cid:104) φ (cid:105) = v (cid:39)
246 GeV,which generates a matrix of complex Dirac masses, ( m D ) Iα , for the LH and RH neutrinos in Eq. (2), L M N EWSB −→ i2 N R I /∂ N R I − [( m D ) Iα + y Iα φ ] N R I ν L α + y Iα φ + N R I (cid:96) L α − N R I M IJ ( N R J ) C + h.c. (3)Here, φ contains the real SM Higgs boson with a mass of m h (cid:39)
125 GeV after EWSB. The Diracmass matrix is directly proportional to the RHN Yukawa matrix, ( m D ) Iα = y Iα v/ √
2. After EWSB, Baryogenesis via heavy-particle decay always requires at least two particles that contribute to loop amplitudes [81]. L M N ⊃ − (cid:16) ( v L α ) C N R I (cid:17) (cid:32) αβ (cid:0) m TD (cid:1) αJ (cid:0) m D (cid:1) Iβ M IJ (cid:33) (cid:32) v L β ( N R J ) C (cid:33) + h.c. (4)The total neutrino mass matrix therefore corresponds to a complex symmetric 5 × M .Thus, there is a unitary matrix that diagonalizes M by means of a Autonne-Takagi factorization, M ( α,I )( β,J ) = (cid:32) αβ (cid:0) m TD (cid:1) αJ (cid:0) m D (cid:1) Iβ M IJ (cid:33) −→ D ( i,I )( j,J ) = (cid:32) D νij iJ Ij D NIJ (cid:33) , (5)where D ν and D N contain three light and two heavy Majorana mass eigenvalues, respectively, D νij = m i δ ij , D NIJ = M (cid:48) I δ IJ . (6)Regarding the heavy neutrinos, we are able to define 0 < M (cid:48) ≤ M (cid:48) , without loss of generality.However, as for the light neutrinos, we need to distinguish between the case of a normal hierarchy (NH), 0 = m < m < m , and the case of an inverted hierarchy (IH), 0 = m < m < m .The ordering among the light neutrino mass eigenstates then determines the sign of the largestpossible mass-squared difference in the light-neutrino mass spectrum, ∆ m l . For NH, we have∆ m l = ∆ m = m − m = m >
0, while for IH, we have ∆ m l = ∆ m = m − m = − m < m i and M (cid:48) I . However, in the following,we will restrict ourselves to the seesaw limit, in which the RHN masses in Eq. (2) are considerablylarger than the EW scale. In this case, the RH neutrinos decouple at high energies, such that thereis no appreciable mixing among the active and sterile neutrino states at low energies. In addition,we will also assume the RHN Majorana mass matrix to be diagonal from the outset. In the seesawlimit, we are able to perform an approximate block diagonalization of the total mass matrix M , M ( α,I )( β,J ) → M block( α,I )( β,J ) ≈ (cid:32) m αβ αJ Iβ M IJ (cid:33) , (7)where we carried out a perturbative expansion in ratios of the form ( m D ) Iα /M JK and only kept theleading terms. The mass matrix for the heavy Majorana neutrinos now coincides with the RHN massmatrix in Eq. (2). This means that, in the seesaw limit, the heavy neutrino mass eigenvalues coincidewith the RHN input masses at the Lagrangian level, M IJ = M I δ IJ = D NIJ = M (cid:48) I δ IJ . Meanwhile,we obtain the following expression for the Majorana mass matrix of the light SM neutrinos, m αβ = − ( m TD ) αI M − IJ ( m D ) Jβ , (8)which reflects the fact that, in the type-I seesaw model, the masses of the light SM neutrinos aresuppressed by the combination of small Yukawa couplings and large RHN masses.The Majorana mass matrix m αβ in Eq. (8) is again a complex symmetric matrix. Thus, there isagain a unitary matrix U that diagonalizes m αβ by means of a Autonne-Takagi factorization,( U T ) iα m αβ U βj = D νij = m i δ ij . (9)The matrix U relates the SM neutrino flavor eigenstates ν L α to the SM neutrino mass eigenstates ν i , ν L α = U αi ν i , ν i = ( U † ) iα ν L α = U ∗ αi ν L α . (10)7ormal hierarchy Inverted hierarchyBest-fit value 3 σ range Best-fit value 3 σ range∆ m / − eV .
39 6.79 – 8.01 7 .
39 6.79 – 8.01 | ∆ m l | / − eV .
525 2.431 – 2.622 2 .
512 2.413 – 2.606sin θ .
310 0.275 – 0.350 0 .
310 0.275 – 0.350sin θ .
582 0.428 – 0.624 0 .
582 0.433 – 0.623sin θ . . δ/ rad 3 .
79 2.36 – 6.39 4 .
89 3.42 – 6.13 σ/ rad — 0 – π — 0 – π Table 2: Best-fit values and 3 σ confidence intervals for the low-energy neutrino observables accordingto the NuFIT 4.0 global-fit analysis [84, 85], including data on atmospheric neutrinos from Super-Kamiokande [86]. The largest mass-squared difference in the SM neutrino mass spectrum, ∆ m l , isdefined as m − m = m > m − m = − m < e , µ , and τ . This is possible becauseone is always able to perform unitary flavor transformations on the LH lepton doublet L α as wellas on the RH charged-lepton singlet (cid:96) R α prior to EWSB. In this case, there will be no contributionsto lepton mixing from the charged-lepton mass matrix, and the unitary matrix U can be identifiedwith the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) lepton mixing matrix [82, 83], U PMNS = U .In the 2RHN seesaw model, the PMNS matrix can be parametrized as follows, U = c c s c s e − iδ − s c − c s s e iδ c c − s s s e iδ s c s s − c s c e iδ − c s − s s c e iδ c c e iσ
00 0 1 . (11)Here, s ij and c ij are shorthand notations for sin θ ij and cos θ ij , respectively. θ , θ , θ ∈ [0 , π/ δ ∈ [0 , π ) is the CP -violating Dirac phase, and σ ∈ [0 , π ) rep-resents the only physical CP -violating Majorana phase in the 2RHN seesaw. The second Majoranaphase that is present in the 3RHN seesaw, τ , can always be rotated away by a phase transformationof the massless SM neutrino mass eigenstate. In Tab. 2, we list the current experimental constraintson the neutrino observables that are accessible in experiments according to the global-fit analysis inRefs. [84, 85]. Note that the CP -violating Majorana phase σ is at present unconstrained.The identities in Eqs. (8) and (9) can be used to write down the following matrix relation,( U T ) iα m αβ U βj = − (cid:104) M − / I ( m D ) Iα U αi (cid:105) T δ IJ (cid:104) M − / J ( m D ) Jβ U βj (cid:105) = m i δ ij , (12)which can be solved for the Dirac mass matrix ( m D ) Iα , or equivalently, for the Yukawa matrix y Iα , y Iα = ( m D ) Iα v/ √ v/ √ M / I R Ii m / i ( U † ) iα , (13)This is nothing but the famous Casas-Ibarra parametrization (CIP) of the RHN Yukawa matrix [87].The matrix R in Eq. (13) is a complex rotation matrix that satisfies RR T = × (but R T R (cid:54) = × ).It can be parametrized in terms of a complex rotation angle z ∈ C and a discrete parameter ζ = ± R + and a negative branch R − of possible R matrices.For NH and IH, we can respectively write the complex rotation matrix R as follows,NH: R ζ ( z ) = (cid:32) z ζ sin z − sin z ζ cos z (cid:33) , IH: R ζ ( z ) = (cid:32) + cos z ζ sin z − sin z ζ cos z (cid:33) . (14)The two real DOFs contained in z = z R + i z I reflect the mismatch between high-energy and low-energy parameters in the 2RHN seesaw. In our analysis in Sec. 3, we will use the CIP in Eq. (13)to scan over all possible flavor models that are consistent with the low-energy neutrino observableswithin their 3 σ ranges (see Tab. 2). In doing so, we will make use of several properties of theYukawa couplings y Iα as functions of z R and z I in the limit of a negligibly small RHN mass splitting, M − M →
0. First of all, we note that shifting the real part of the complex rotation angle by∆ z R = π/ y Iα .This leads to the following pattern when shifting z R by ∆ z R = π/ M = M , (cid:32) + y α + y α (cid:33) ∆ z R = π/ −→ (cid:32) + y α − y α (cid:33) ∆ z R = π −→ (cid:32) − y α − y α (cid:33) ∆ z R =3 π/ −→ (cid:32) − y α + y α (cid:33) ∆ z R =2 π −→ (cid:32) + y α + y α (cid:33) . (15)All relevant quantities in our parameter study in Sec. 3 will depend on products of at least twoYukawa couplings. For this reason, it will be enough if we restrict ourselves to scanning over theparameter range z R ∈ [0 , π ). Next to Eq. (15), the CIP also leads to two other useful relations, M = M , z R → π/ − z R , ( ζ, δ, σ ) → − ( ζ, δ, σ ) ⇒ (cid:32) + y α + y α (cid:33) → (cid:32) + y ∗ α + y ∗ α (cid:33) , (16) M = M , z R → π − z R , ( z I , ζ ) → − ( z I , ζ ) ⇒ (cid:32) + y α + y α (cid:33) → (cid:32) − y α + y α (cid:33) . These symmetry properties of the CIP will be directly visible in our numerical results in Sec. 3.Finally, we remark that the Yukawa couplings y Iα grow exponentially as functions of | z I | for | z I | (cid:29) | z I | (cid:29) ⇒ y α ≈ sign( z I ) i y α ∝ e | z I | . (17)By construction, the CIP still results in Yukawa couplings that are in accord with the small SMneutrino masses in this case. However, for | z I | (cid:29)
1, this can only be achieved at the cost of fine-tunedcancellations among the different entries in the RHN Yukawa matrix. In the context of the neutrinooption, there is no reason why such a situation should be realized. In our parameter study, we willtherefore restrict ourselves to the range z I ∈ [ − , +2] to avoid fine-tuned RHN Yukawa couplings. The baryon asymmetry of the Universe is typically quantified in terms of the baryon-to-photonratio in the present era, η B . The most precise value for η B follows from the observations of thetemperature anisotropies in the cosmic microwave background by the PLANCK satellite [88], η obs B = n B n γ (cid:39) . × − (cid:18) h Ω B . (cid:19) . (18) In the 3RHN seesaw, the matrix R is correspondingly parametrized in terms of three complex angles, z , z , and z , which correspond to the six real DOFs that cannot be constrained by the low-energy data in this model. n B and n γ are the number densities of baryons and photons, respectively, and h Ω B (cid:39) . quantum field theory (QFT) [89], which is beyond the scope of this work. Instead, we will carryout a semianalytical analysis similar to the one in Ref. [56], which also studies resonant leptogenesisin the 2RHN seesaw model. We begin by relating η B to the three LH-lepton-doublet asymmetries η lptg L α that are produced by the various L α -number-violating interactions during leptogenesis, η B = g ∗ ,s g ∗ ,s C sph (cid:88) α η lptg L α . (19)Here, g ∗ ,s and g ∗ ,s count the effective number of relativistic DOFs that contribute to the entropydensity of the thermal bath, s rad = 2 π / g ∗ ,s T , at the present time and at the time of leptogenesis,respectively. In the following, we will use the usual SM values for these two quantities, g ∗ ,s = 4311 , g ∗ ,s = 4274 ⇒ g ∗ ,s g ∗ ,s = 1724697 (cid:39) . . (20) C sph in Eq. (19) accounts for the conversion from the LH-lepton-doublet asymmetries η lptg L α to η B bymeans of EW sphalerons. Let us now compute C sph based on the analysis in Ref. [90]. In doing so, wewill neglect the effect of spectator processes [91–94] and make use of the fact that, at temperatures T ∼ GeV, all SM interactions eventually reach chemical equilibrium [95].First, we note that the total lepton asymmetry η L receives contributions from both LH leptondoublets L α and RH lepton singlets (cid:96) R α . In chemical equilibrium, one finds the following relation, η (cid:96) R α = 12 η L α − η L L ⇒ η L = η L L + η R L = 1714 η L L , η L L = (cid:88) α η L α , η R L = (cid:88) α η (cid:96) R α . (21)Second, it is important to notice that the transport equations that describe the evolution of the LH-lepton-doublet asymmetries η L α typically do not contain an explicit collision operator accounting forthe EW sphalerons processes, although EW sphalerons certainly do violate the L α number densities.In this case, one has to split the total lepton asymmetry η L into two contributions [90], η L = η lptg L + η sph L . (22)Here, η lptg L denotes the effective lepton asymmetry that is generated during leptogenesis and whoseevolution follows from solving a coupled set of transport equations. The asymmetry η sph L , on the otherhand, is induced by the interaction density of the EW sphalerons, γ sph . The effective leptogenesisasymmetry η lptg L can be related to the total baryon and lepton asymmetries, η L and η B , [90] η lptg L = η L − η B = η L − B = − η B − L . (23)This reflects the well-known fact that the lepton number generated during leptogenesis should actu-ally be regarded as a negative B − L charge whose B component happens to be zero. The total B , L , and B − L asymmetries are related to each other by standard factors, such that η B = − η L = 2879 η B − L = − η lptg L . (24)10q. (21) also applies at the level of the individual lepton asymmetries η lptg L and η sph L [90]. Combiningour results in Eqs. (21) and (24), we therefore obtain the following sphaleron conversion factor, η B = C sph (cid:88) α η lptg L α , C sph = − × − (cid:39) − . . (25)It is interesting to compare this results with other expressions for C sph that one frequently encountersin the literature. In our notation, we are also able to write down the following three relations, η B = − η L − B , η B = − η L , η B = − η L L . (26)None of these relations should be confused with the correct relation in Eq. (25). We anticipate the temperature scale of leptogenesis to be T ∼ · · · GeV in our scenario.In this case, the µ and τ Yukawa interactions have already reached chemical equilibrium duringleptogenesis, such that we do not need to worry about coherence / decoherence effects in flavorspace. Therefore, instead of solving matrix equations for quantum-mechanical density matrices, it issufficient to restrict oneself to semiclassical Boltzmann equations for number densities. In this paper,we will follow the analysis in Ref. [96] (see also Ref. [56]), which studies the evolution of the LH-lepton-doublet asymmetries η lptg L α during resonant leptogenesis based on a set of flavored Boltzmannequations. The final asymmetries at the end of leptogenesis can then be written as follows [56, 96], η lptg L α = 32 (cid:88) I ε Iα K eff α z α . (27)In writing down this expression, we assumed that leptogenesis always terminates before the EWsphalerons fall out of thermal equilibrium at temperatures around T sph (cid:39)
130 GeV [97]. Given thatwe are interested in leptogenesis at temperatures far above the EW scale, T (cid:29) T sph , this assumptionis always satisfied. Another assumption entering Eq. (27) is that one of the η lptg L α asymmetries shouldbe larger than the two others, such that leptogenesis is predominantly driven by the asymmetry inone specific lepton flavor. We performed an explicit numerical analysis of the Boltzmann equationsin Ref. [96] to confirm that this assumption is justified in our scenario to first approximation. Wealso confirmed that the semianalytical expression in Eq. (27) allows one to reproduce the fullynumerical result with a precision of O (10%). For these reasons, we will restrict ourselves to workingwith Eq. (27) in the following. For more details on the Boltzmann equations and the derivation ofEq. (27), we refer the reader to the detailed presentation in Ref. [96].Let us now spell out the meaning of ε Iα , K eff α , and z α in Eq. (27). The parameter z α quantifiesthe point in time when the lepton asymmetry η lptg L α ceases to evolve at the end of leptogenesis, z α (cid:39) .
25 ln (cid:16) K eff α (cid:17) . (28) K eff α is an effective measure for the strength of processes that wash out the asymmetry in the α -flavorchannel. Large values, K eff α (cid:29)
1, correspond to strong washout, while K eff α ∼ K eff α is defined in terms of the standard RHN decay parameters K I , K eff α = κ α (cid:88) I K I B Iα , K I = Γ I ζ (3) H ( T = M I ) , B Iα = | y Iα | ( yy † ) II . (29) The sphaleron factor in Eq. (25) can also be estimated by consistently neglecting the effect of the RH lepton singlets.In this case, one can simply write η B = − / η L L ≈ − / η L , such that η B ≈ − / η L − B = − / η lptg L ≈ − / (cid:80) α η lptg L α .This estimate results in C sph ≈ − / − .
40, which deviates from our result by less than 10 %. In the following, wewill, however, stick to our result C sph = − /
79. We thank Daniele Teresi for a helpful discussion on this point. B Iα denotes the branching ratio of N I decays into LH lepton doublets of flavor α at tree level,and H ( T = M I ) is the Hubble rate evaluated at a temperature equal to the RHN mass M I , H ( T = M I ) = (cid:18) π g ∗ ,ρ (cid:19) / M I M Pl , (30)where g ∗ ,ρ = 427 / ρ rad = π / g ∗ ,ρ T , at the time of leptogenesis, and where we employedthe reduced Planck mass M Pl = (8 πG ) − / (cid:39) . × GeV (with G being Newton’s constant).Γ I in Eq. (29) is the total N I tree-level decay rate at zero temperature,Γ I = Γ ( N I → L α + φ ) + Γ (cid:0) N I → L C α + φ ∗ (cid:1) = ( yy † ) II π M I . (31)These definitions illustrate that the RHN decay parameter K I characterizes how strongly the RHneutrino N I is coupled to the thermal bath. For K I (cid:29)
1, the RHN interactions have (nearly) reachedthermal equilibrium, for K I (cid:28)
1, they are far away from thermal equilibrium. Eq. (29) also showsthat (cid:80) I K I B Iα = K α can be regarded as the equivalent of K I in lepton flavor space. The washoutparameter K eff α = κ α K α , finally, is a rescaled version of K α , where the factor κ α accounts for theeffect of lepton-number-violating and lepton-flavor-violating two-to-two scattering processes, κ α = 2 (cid:88) I,J Re (cid:2) y Iα y ∗ Jα (cid:3) Re (cid:2)(cid:0) yy † (cid:1) IJ (cid:3) − Im (cid:2) y Iα y ∗ Jα (cid:3) Im (cid:2) y Iα y ∗ Jα (cid:3) ( y † y ) αα [( yy † ) II + ( yy † ) JJ ] (cid:18) − M I − M J Γ I + Γ J (cid:19) − (32)= 1 + 4 Re (cid:2) y α y ∗ α (cid:3) Re (cid:2)(cid:0) yy † (cid:1) (cid:3) − Im (cid:2) y α y ∗ α (cid:3) Im (cid:2) y α y ∗ α (cid:3) ( y † y ) αα [( yy † ) + ( yy † ) ] (Γ + Γ ) (Γ + Γ ) + 4 ( M − M ) . In the scattering operators leading to this expression for κ α , the contributions from on-shell RHneutrinos in the intermediate state have been subtracted. These contributions are already accountedfor by the decay and inverse-decay operators in the Boltzmann equations and must not be countedtwice. In our parameter scan in Sec. 3, we will mostly be interested in small RHN decay widths,Γ , (cid:28) M − M . In this case, the rescaling factor κ α typically obtains values close to one, κ α ≈ CP asymmetry parameter ε Iα , which represents the amountof CP asymmetry that can be generated per RHN decay. It is defined in terms of the partial decaywidths Γ (cid:96) ( N I → L α + φ ) and Γ (cid:96) (cid:0) N I → L C α + φ ∗ (cid:1) , which involve the RHN decay amplitudes attree level as well as the radiative RHN vertex (v) and self-energy (s) corrections at one loop [100], ε Iα = ε (v) Iα + ε (s) Iα = Γ (cid:96) ( N I → L α + φ ) − Γ (cid:96) (cid:0) N I → L C α + φ ∗ (cid:1) Γ I . (33)The vertex contribution ε (v) Iα (also referred to as the ε (cid:48) or direct CP asymmetry) reads, ε (v) Iα = (cid:88) J (cid:54) = I Im (cid:2) y Iα y ∗ Jα ( yy † ) IJ (cid:3) ( yy † ) II ( yy † ) JJ Γ J M I (cid:20) − (cid:18) M J M I (cid:19) ln (cid:18) M I M J (cid:19)(cid:21) , (34) In Ref. [96], the rescaling factor κ α is expressed in terms of resummed Yukawa couplings ¯ y Iα instead of the ordinarytree-level Yukawa couplings y Iα . The relation between these two sets of Yukawa couplings in the case of only two RHneutrinos can be found in Refs. [98, 99]. We checked that replacing y Iα by ¯ y Iα in our parameter study only leads tonumerically insignificant changes. For this reason, we decide to ignore this subtlety in the following. We also pointout that Eq. (32) agrees with the result in Ref. [96] after substituting ¯ y Iα → y Iα , whereas it does not agree with theexpression in Ref. [56], which involves a number of typos. We thank Bhupal Dev for a helpful discussion on this point. ε or indirect CP asymmetry), ε (s) Iα = (cid:88) J (cid:54) = I (cid:40) Im (cid:2) y Iα y ∗ Jα ( yy † ) IJ (cid:3) ( yy † ) II ( yy † ) JJ + M I M J Im (cid:2) y Iα y ∗ Jα ( yy † ) JI (cid:3) ( yy † ) II ( yy † ) JJ (cid:41) f IJ , (35)where the function f IJ ∝ / (cid:0) M I − M J (cid:1) originates from the N J propagator in the N I decay diagram.Resonant leptogenesis is based on the observation that ε (s) Iα can be resonantly enhanced in thecase of a small RHN mass splitting, ∆ M = M − M (cid:28) M , such that f IJ becomes exceptionallylarge. In fact, given the naive semiclassical estimate f IJ ∝ / (cid:0) M I − M J (cid:1) , one finds that ε (s) Iα appears to diverge for a vanishing mass splitting, ∆ M →
0. This, however, is an unphysical effectthat reflects the breakdown of the semiclassical approximation. A pair of exactly degenerate RHMajorana neutrinos represents, in fact, a single Dirac neutrino, such that lepton number remainsconserved and the CP asymmetry parameter identically vanishes, ε Iα = 0. This simple argumentindicates that the function f IJ needs to be regularized in quantum fields theory,1 M I − M J → M I − M J (cid:0) M I − M J (cid:1) + R IJ , (36)in order to avoid the singular behavior in the limit ∆ M →
0. The proper form of the regulator R IJ has been the subject of a long debate in the literature that has not yet been fully settled (see, e.g. , [98,99,101–105]). In this paper, we do not have anything new to add to this debate. Instead, wewill simply adopt the results in Refs. [98, 99, 104], which managed to reproduce the same form of theregulator both in an analysis based on quantum-mechanical density matrix equations [98, 99] and afull QFT analysis based on Kadanoff–Baym equations [104]. The main conclusion of Refs. [98,99,104]is that the function f IJ actually receives two contributions of similar magnitude, f IJ = f osc IJ + f mix IJ , f osc IJ = (cid:0) M I − M J (cid:1) M I Γ J (cid:0) M I − M J (cid:1) + R osc IJ , f mix IJ = (cid:0) M I − M J (cid:1) M I Γ J (cid:0) M I − M J (cid:1) + R mix IJ , (37)where f osc IJ and f mix IJ account for the contributions to the CP asymmetry parameter ε (s) Iα from RHNflavor oscillations and RHN mixing, respectively. If one decided to omit one these two contributions,the CP asymmetry parameter ε (s) Iα would roughly decrease by a factor 2. This should be regarded asan upper bound on the theoretical uncertainty of our expression for ε (s) Iα in Eq. (35). The regulators R osc IJ and R mix IJ in Eq. (37) are given by the following expressions, R osc IJ = ( M I Γ I + M J Γ J ) det (cid:2) Re( yy † ) (cid:3) ( yy † ) II ( yy † ) JJ , R mix IJ = M I Γ J . (38)This concludes our discussion of the different ingredients that are necessary to compute the finalbaryon asymmetry in our scenario. Combining all of the above results, we are now able to write η B (cid:39) C (cid:88) I,α ε Iα .
25 ln (25 K eff α ) K eff α , C = − (cid:39) − . × − , (39)which is the expression for η B that we will use to calculate the baryon asymmetry in Sec. 3. Let us now turn to the generation of the EW scale in the type-I seesaw. In the pure Standard Modelwithout RH neutrinos, the Higgs doublet φ possesses the following scalar potential, V SM = − µ | φ | + λ | φ | , | φ | = φ † φ = φ + φ − + φ ∗ φ . (40)13ere, µ denotes the Higgs mass parameter, and λ is the quartic Higgs self-coupling. The sign of themass term in Eq. (40) is chosen such that a real and positive mass parameter, µ >
0, results in EWSB.The Standard Model is based on the assumption that µ ∼
100 GeV > Q ∼
100 GeV; but it does not offer any intrinsic justification for this assumption. In the truevacuum after EWSB, the Higgs field has a nonzero VEV √ (cid:104) φ (cid:105) = v = µ/ √ λ , and the physicalHiggs boson h possesses a mass m h = √ µ = √ λ v . We emphasize that µ is the only explicitmass scale in the Standard Model. Once we set µ →
0, the SM Lagrangian becomes scale-invariant. Motivated by this observation, the Majorana-neutrino option is based on the assumption thatthe Higgs potential satisfies scale-invariant boundary conditions above the RHN mass thresholds,
Q > max { M I } ⇒ V UV = λ | φ | . (41)The Higgs mass parameter µ necessary for EWSB is then induced by RHN threshold correctionsto the Higgs potential, ∆ V ⊃ − µ | φ | , when matching the full theory (including dynamical RHneutrinos) at high energies onto the effective field theory (EFT) (without dynamical RH neutrinos)at low energies. In the case of two nearly degenerate RH neutrinos, this matching is done at Q = M (cid:39) M , which is the energy scale at which both RH neutrinos decouple [106]. The thresholdcorrections ∆ V are encoded in the one-loop effective Coleman–Weinberg (CW) potential [107],∆ V = V UV − V EFT = ( − π (cid:88) I M I ( φ ) ln M I ( φ ) Q . (42)The CW formula can be derived by computing one-loop vacuum (bubble) diagrams, using dimen-sional regularization to keep track of the infinities arising in the one-loop momentum integrals. Theseinfinities are canceled by an appropriate set of counterterms in the MS renormalization scheme. Inour notation, the MS renormalization scale ¯ Q is given by ¯ Q = e − / Q . In this sense, the scale Q canbe regarded as the renormalization scale in a different scheme that deviates from the MS scheme bya finite shift along the RG flow. The advantage of employing Q rather than ¯ Q in Eq. (42) is that, inthis scheme, the CW potential no longer contains nonlogarithmic terms that do not depend on therenormalization scale; in Eq. (42), all terms are proportional to logarithms of the form ln M I /Q .This allows one to minimize the absolute value of the CW potential by setting Q to the typical massscale of the theory. In our analysis, we will consequently fix the renormalization scale at the RHNdecoupling scale, Q = M , corresponding to ¯ Q = e − / M in the MS scheme. In this way, weintend to remove large logarithms from the CW potential, which we expect to improve the qualityof the perturbative series. This is in line with the discussion in Refs. [108, 109], which states thatthe choice Q = M amounts to a leading-log resummation to all orders in perturbation theory thatminimizes the Q dependence of the CW potential. In future work, it would be interesting to confirmthese statements by an explicit calculation of higher-order corrections in the type-I seesaw model.In the remainder of this paper, we will, however, simply stick to our choice Q = M , cautioningthat the scheme dependence of the EFT matching analysis requires further scrutiny.The ( −
2) prefactor in Eq. (42) follows from the Fermi-Dirac statistics of the RH neutrinos thatrun in the one-loop vacuum diagrams. In our renormalization scheme, the negative sign of thisfactor eventually leads to a negative sign of the Higgs mass term at the matching scale Q = M .This illustrates that the type-I seesaw is capable of explaining the tachyonic mass term in the The scale of quantum chromodynamics (QCD), Λ
QCD ∼
100 MeV, is not an explicit input scale, but generated viadimensional transmutation by the renormalization group (RG) running of the strong gauge coupling constant. φ -dependent masses M I ( φ ) inEq. (42) represent the two large mass eigenvalues that one finds when performing a Autonne-Takagifactorization of the total neutrino mass matrix M in Eq. (4) after replacing the Higgs VEV v bya classical and homogeneous Higgs field background φ . Again, similarly as in Sec. 2.1, the totalneutrino mass matrix can be approximately diagonalized by performing a perturbative expansionin powers of the Higgs field. While we were only interested in the leading-order result in Sec. 2.1[see Eq. (7)], we now have to compute the squared masses M I ( φ ) up to O ( φ ), in order to identifythe threshold corrections to the quadratic and quartic terms in the Higgs potential. This procedureallows us to write the threshold corrections ∆ V as a power series in the Higgs doublet φ ,∆ V = ∆ V − ∆ µ | φ | + ∆ λ | φ | + O (cid:0) | φ | (cid:1) , (43)where ∆ V denotes a constant contribution to the vacuum energy that we can ignore for our purposes.We obtain the following expressions for the threshold corrections ∆ µ and ∆ λ ,∆ µ = 18 π (cid:88) I (cid:20)(cid:16) yy † (cid:17) II M I (cid:18) ln M I Q + 12 (cid:19)(cid:21) , (44)∆ λ = 18 π (cid:88) I (cid:54) = J (cid:20)(cid:16) yy † (cid:17) II L (1) IJ −
12 Tr (cid:0) yy † y ∗ y T (cid:1) L (2) IJ + 12 Tr (cid:0) yy † yy † (cid:1) L (3) IJ (cid:21) , where the three loop functions L (1) IJ , L (2) IJ , and L (3) IJ are defined as follows, L (1) IJ = M J M J − M I ln M J M I − , L (2) IJ = M I M J M J − M I ln M J M I , L (3) IJ = M I M J − M I ln M I Q − . (45)In the limit of a small RHN mass splitting, M (cid:39) M , these expressions can be simplified to∆ µ (cid:39) Tr (cid:0) yy † (cid:1) π M (cid:18) ln M Q + 12 (cid:19) , ∆ λ (cid:39) − Tr (cid:0) yy † y ∗ y T (cid:1) π − Tr (cid:0) yy † yy † (cid:1) π (cid:18) ln M Q + 32 (cid:19) . (46)In particular, we obtain the following threshold corrections at the RHN decoupling scale Q = M ,∆ µ = ∆ µ (cid:12)(cid:12) Q (cid:39) Tr (cid:0) yy † (cid:1) π M , ∆ λ = ∆ λ (cid:12)(cid:12) Q (cid:39) − π (cid:20) Tr (cid:0) yy † y ∗ y T (cid:1) + 32 Tr (cid:0) yy † yy † (cid:1)(cid:21) . (47)These results conform with the naive expectations ∆ µ ∼ y / (cid:0) π (cid:1) M and ∆ λ ∼ y / (cid:0) π (cid:1) . Weobserve that the threshold correction to the Higgs mass parameter turns out to have a positive sign,∆ µ >
0, which is the sign needed to induce EWSB at low energies. The threshold correction to thequartic Higgs self-coupling, on the other hand, turns out to have a negative sign, ∆ λ <
0, whichmay raise the concern of a vacuum instability in the Higgs potential. However, at this point, it isimportant to remember that the Higgs potential already contains a quartic coupling at tree level [seeEq. (41)]. Compared to this tree-level coupling, ∆ λ is typically tiny, given that it is suppressed bythe combination of four Yukawa couplings and a loop factor. The effect of the threshold correction∆ λ can therefore be easily compensated for by a shift in the tree-level coupling. This allows us toneglect ∆ λ and focus on the matching of the Higgs mass parameter in the following.At the RHN decoupling scale Q = M , the type-I seesaw can be matched onto the SM effectivefield theory (SMEFT) (see Ref. [110] for a review). A complete one-loop matching of the two theoriesincluding operators up to dimension five on the SMEFT side has been performed in Ref. [24]. There,it has been shown that the dominant outcome of the one-loop matching consists of (i) the threshold15 Q = 173 . Q = 10 GeV g . . g . . g .
167 0 . y t . . y b . . y τ . . λ . . µ .
54 GeV 101 . g i are the SM gauge couplings (where g = (cid:112) / g Y ), while y t , y b , and y τ denote the Yukawa couplings of the top quark, bottom quark,and tau lepton, respectively. The second column contains the input values for these parameters atthe top-quark mass scale, ¯ Q = 173 . Q = 10 GeV based on our numerical solution of the two-loop RG equations.corrections to the Higgs potential in Eq. (44) and (ii) nonzero Wilson coefficients for the dimension-five Weinberg operator [111], which results in the SM neutrino masses after EWSB. The one-loopWilson coefficients computed in Ref. [24] also capture radiative corrections to the tree-level neutrinomass matrix m αβ in Eq. (8). However, these corrections only become relevant in the case of large(and fine-tuned) Yukawa couplings, i.e. , for large absolute values of the parameter z I [see Eq. (17)].In our parameter study in Sec. 3, we will not be interested in this regime. This leaves the thresholdcorrection to the Higgs mass parameter, ∆ µ , as the only quantity that we need to explicitly accountfor, for the purposes of the matching analysis in this paper. On the SMEFT side, the RG runningof the Higgs mass parameter µ is controlled by the standard RG equations (RGEs) of the StandardModel. Our matching condition therefore amounts to the requirement that the running Higgs massparameter µ ( Q ) must equal the threshold correction ∆ µ at the RHN decoupling scale, µ = µ ( Q = M ) = Tr (cid:0) yy † (cid:1) π M = ∆ µ . (48)In order to determine the value of µ ( Q ) on the left-hand side of Eq. (48), we need to solvethe RG equations of the Standard Model, which we employ from Ref. [114]. The outcome of ouranalysis is shown in Fig. 1, alongside our results for the running quartic Higgs self-coupling. In afirst step, we solve the RG equations in the MS renormalization scheme at one-loop, two-loop, andthree-loop order [114]. As evident from Fig. 1, the result of the three-loop analysis for µ (cid:0) ¯ Q (cid:1) doesnot significantly improve over the result of the two-loop analysis. In the following, we will thereforerestrict ourselves to working with the results of the two-loop RGEs, in combination with the one-loop threshold corrections to the running SM parameters at the top-quark mass scale, ¯ Q = m t , inorder to fix the initial conditions of the two-loop RGEs. These initial conditions are given in Tab. 3,where we also show the corresponding values at the typical RHN scale of 10 GeV, which we obtainafter numerically solving the two-loop RGEs. At values of the renormalization scale relevant for the The RG running of the Wilson coefficients of the dimension-five Weinberg operator is numerically insignificant [24].This is the SMEFT version of the statement that the RG running of the type-I seesaw parameters can typically beneglected [56, 112, 113]. For this reason, we will not consider any RGEs in the BSM sector of our model. . . . . .
15 10 MS renormalization scale ¯ Q [GeV] λ ( ¯ Q ) µ ( ¯ Q ) [TeV] V = − µ | φ | + λ | φ | Figure 1: RG running of the Higgs mass parameter µ and quartic Higgs self-coupling constant λ asfunctions of the MS renormalization scale ¯ Q = e − / Q in the Standard Model. For the purposes ofthis paper, we are only interested in ¯ Q values of at most O (cid:0) (cid:1) GeV. The metastability / instabilityof the EW vacuum at higher energies needs to be addressed by new physics beyond the type-I seesaw.Majorana-neutrino option, ¯ Q ∼ · · · GeV, the Higgs mass parameter turns out to exhibit onlya very weak dependence on the renormalization scale (see Fig. 1). Over a broad range of ¯ Q values ofthis order of magnitude, we approximately find µ (cid:0) ¯ Q (cid:1) (cid:39) · · ·
102 GeV. As a consequence, Eq. (48)turns into the following constraint on the parameter space of the type-I seesaw model,Tr (cid:0) yy † (cid:1) M ∼ (1300 GeV) . (49)Once this relation is satisfied, the type-I seesaw induces a Higgs mass term of the right magnitudeand with the correct sign in the Lagrangian of the classically scale-invariant Standard Model. In Sec. 2, we have separately discussed the generation of SM neutrino masses, the baryon asymmetryof the Universe, and the EW scale. In the present section, we will now put together the pieces of thepuzzle and identify the viable parameter region where all three features of the type-I seesaw can berealized at the same time. To this end, we will make use of our results in Eqs. (13), (39), and (48), y Iα = i v/ √ M / I R Ii m / i ( U † ) iα , (cid:88) I,α
C ε Iα z α K eff α = η obs B , Tr (cid:0) yy † (cid:1) π M = µ . (50)That is, we will employ the CIP for the RHN Yukawa couplings to ensure that our analysis is inagreement with the low-energy neutrino data, and we will simultaneously impose the conditionsthat (i) leptogenesis results in the correct value of the baryon asymmetry and that (ii) the RHNthreshold corrections induce the correct Higgs mass parameter at the RHN decoupling scale. It17s interesting to note that, in the context of the 2RHN seesaw model, these conditions completelyremove the parametric freedom at low energies. Recall that, in the case of the 2RHN seesaw, thelow-energy EFT contains only one complex DOF, z = z R + i z I , that is not accessible in experiments[see Eq. (13)]. For a given RHN mass spectrum, this parameter can thus be eliminated by the twoconditions η B = η obs B and ∆ µ = µ . In the following, we will turn this argument around and solvethe conditions in Eq. (50) for the values of M and M that are required by leptogenesis and theneutrino option as functions of z R and z I . In this analysis, z R and z I can then be regarded as thecoordinates of the unconstrained theory space of all possible UV flavor models (see Sec. 2.1).We begin by examining the constraint in Eq. (48), which can also be written as follows, (cid:101) m = Tr (cid:2) m D m † D (cid:3) M = 16 π M v µ = (cid:101) m . (51)Here, we introduced the mass parameter (cid:101) m , which is defined in a similar way as the well-knowneffective neutrino mass parameters (cid:101) m I = (cid:0) m D m † D (cid:1) II /M I [21,115]. In this formulation, the neutrino-option constraint now states that the mass parameter (cid:101) m must obtain a particular value (cid:101) m , (cid:101) m (cid:39)
48 meV (cid:18) GeV M (cid:19) (cid:18) µ
100 GeV (cid:19) . (52)The mass (cid:101) m can be related to the total mass in the SM neutrino sector, m tot = (cid:80) i m i , as follows, (cid:101) mm tot = cosh (2 z I ) + δM z I ) + δm cos (2 z R )) , (53)where we introduced the dimensionless ratios δM and δm to parametrize the relative mass splittingsamong the heavy and light neutrino mass eigenstates, respectively, δM = ∆ MM = M − M M , δm = ∆ mm tot = ( m − m ) /m tot BFP (cid:39) . × − (NH)( m − m ) /m tot BFP (cid:39) . × − (IH) . (54)Here, the numerical values for δm correspond to the NH and IH best-fit points (BFPs) in Tab. 2.An important implication of Eq. (53) is that (cid:101) m turns out to be bounded from below by m tot , (cid:101) m > m tot , (55)which is reminiscent of the inequality (cid:101) m I > m min , where m min is the smallest nonzero SM neutrinomass eigenvalue (see the Appendix of Ref. [116]). Together with the neutrino-option constraint (cid:101) m = (cid:101) m in Eq. (51), this lower bound on (cid:101) m results in an upper bound on the RHN mass M , M = (cid:18) π v µ m tot (cid:19) / (cid:20) cosh (2 z I ) + δM z I ) + δm cos (2 z R )) (cid:21) − / < (cid:18) π v µ m tot (cid:19) / . (56)For both NH and IH, we thus find that M cannot obtain values larger than 10 GeV, m tot BFP (cid:39)
59 meV (NH)99 meV (IH) ⇒ M (cid:46) . × GeV (NH)7 . × GeV (IH) . (57) Note that (cid:101) m + (cid:101) m = m tot cosh (2 z I ), such that (cid:101) m → (cid:101) m + (cid:101) m for δM → values below this upper bound can always be realized at the cost of a larger value of | z I | . In thelimit of a small RHN mass splitting, δM (cid:28)
1, we find the following simple relation, δM (cid:28) ⇒ M ≈ (cid:18) π v µ m tot cosh (2 z I ) (cid:19) / . (58)Therefore, restricting the range of allowed z I values to z I ∈ [ − ,
2] (see Sec. 2.1), we recognize thata nonzero value of | z I | can lower M by roughly a factor of √ cosh 4 (cid:39) M values that are compatible with the successfulgeneration of the Higgs mass parameter without fine-tuned cancellations in the RHN Yukawa matrix, δM (cid:28) , z I ∈ [ − , ⇒ . GeV (cid:46) M (cid:46) . GeV . (59)Hence, all viable M values are of O (cid:0) (cid:1) GeV and spread across only half an order of magnitude.Next, let us turn to the baryon asymmetry η B in Eq. (39). In our analytical discussion, we shallrestrict ourselves to the resonantly enhanced part of the CP asymmetry parameter ε Iα only. Thatis, we will neglect the subdominant contribution ε ( v ) Iα in Eq. (34) for simplicity and only focus on thedominant contribution ε ( s ) Iα in Eq. (35). In our numerical analysis in Sec. 3.2, we will not make anysuch simplification, but work with the full expression for ε Iα instead. Furthermore, it is convenientto distinguish between two different regimes regarding the CP asymmetry parameter ε ( s ) Iα , dependingon whether the regulators R osc IJ and R mix IJ in Eq. (38) are numerically relevant or not. It is easy to seethat the transition between these two regimes is controlled by the size of the RHN mass splitting,∆ M = M − M , in relation to the RHN decay widths Γ I . For ∆ M (cid:29) Γ I , the function f IJ inEq. (35) does, in fact, not need to be regularized, whereas for ∆ M (cid:28) Γ I , regularization is crucial, f IJ ≈ sgn ( M I − M J ) × Γ J / (2 ∆ M ) ; Γ , (cid:28) ∆ M (cid:28) M , J M I M J ∆ M/R IJ ; ∆ M (cid:28) Γ , (cid:28) M , . (60)This behavior of f IJ determines the dependence of the final baryon asymmetry η B on the masssplitting ∆ M . For ∆ M (cid:29) Γ I and ∆ M (cid:28) Γ I , the baryon asymmetry will scale as η B ∝ / ∆ M and η B ∝ ∆ M , respectively. Let us now discuss these two cases in more detail one by one.In the first case, Γ I (cid:28) ∆ M , we can expand ε ( s ) Iα up to linear order in the RHN decay rates Γ I ,Γ I (cid:28) ∆ M ⇒ (cid:88) I ε ( s ) Iα ≈ A α M Γ + M Γ M − M , A α = 4 Re (cid:2)(cid:0) yy † (cid:1) (cid:3) Im (cid:2) y ∗ α y α (cid:3) ( yy † ) ( yy † ) , (61)where we dropped all higher-order terms in Γ I as well as all terms that are not resonantly enhanced.Making use of the CIP, the Yukawa prefactor A α in this expression can be written as follows, A α = 4 δm sin (2 z R ) ( B α sinh (2 z I ) − C α ζ cosh (2 z I )) δm cos (2 z R ) − cosh (2 z I ) , (62)where B α and C α capture the dependence on the PMNS matrix U and the SM neutrino masses m i , B α = (cid:88) i | U αi | m i m tot , C α = 2 √ m k m l m tot Im [ U αk U ∗ αl ] , ( k, l ) = (2 ,
3) (NH)(1 ,
2) (IH) . (63)These quantities also allow us to rewrite the effective washout parameter K eff α as a function of theCIP parameter z . In doing so, we can simply approximate M ≈ M , since K eff α does not experienceany resonant enhancement in the limit of a small RHN mass splitting. We thus obtain M ≈ M ⇒ K eff α ≈ κ α m tot ζ (3) m ∗ ( B α cosh (2 z I ) − C α ζ sinh (2 z I )) . (64)19ote that this expression is also valid in the ∆ M (cid:28) Γ I regime. m ∗ in Eq. (64) is a benchmark valuefor the SM neutrino masses that is sometimes referred to as the equilibrium mass [21]. It allows oneto express the RHN decay parameter K I in terms of the mass ratio (cid:101) m I /m ∗ , such that (cid:101) m I (cid:29) m ∗ and (cid:101) m I (cid:28) m ∗ are synonymous to the strong-washout and weak-washout scenarios, respectively, K I = (cid:101) m I ζ (3) m ∗ , m ∗ = (cid:18) π g ∗ ,ρ (cid:19) / πv M Pl (cid:39) . . (65)Combining Eqs. (61) and (64), we arrive at the following estimate for the final baryon asymmetry, η B ≈ C M Γ + M Γ M − M ζ (3) m ∗ m tot δm sin (2 z R ) δm cos (2 z R ) − cosh (2 z I ) (cid:88) α D α z α κ α , (66)where the dependence on the PMNS matrix and the SM neutrino masses is now encoded in D α , D α = B α sinh (2 z I ) − C α ζ cosh (2 z I ) B α cosh (2 z I ) − C α ζ sinh (2 z I ) . (67)This estimate is dominated by the leading 1 /δM term when expanding in powers of δM , δM (cid:28) ⇒ η B ≈ CδM ζ (3) m ∗ M πv δm sin (2 z R ) cosh (2 z I ) δm cos (2 z R ) − cosh (2 z I ) (cid:88) α D α z α κ α , (68)which is our final result for η B as a function of δM , M , z R , z I , etc. in the Γ I (cid:28) ∆ M regime.As expected, η B scales like one inverse power of the RHN mass splitting, η B ∝ /δM . On theother hand, it is linearly proportional to the mass splitting in the SM neutrino spectrum, η B ∝ δm ,to leading order in δm . This is similar to the situation in standard hierarchical leptogenesis, wherethe DI bound on the total CP asymmetry parameter also turns out to be proportional to the SMneutrino mass splitting [18]. An immediate consequence of this proportionality, η B ∝ δm/δM , isthat, given the numerical values of δm in Eq. (54), an inverted SM neutrino mass ordering alwaysrequires a RHN mass splitting that is roughly two orders of magnitude smaller than in the case of anormal SM neutrino mass ordering, δM IH ∼ − δM NH . Therefore, if we speculate that UV flavormodels may be biased towards larger RHN mass splittings, we are able to conclude that resonantleptogenesis tends to indicate a preference for the NH scenario rather than the IH scenario. This isan interesting result in light of the current low-energy neutrino data, which results in a χ differencebetween the NH and IH global fits of ∆ χ = χ − χ (cid:39) . z R = π/
4, which maximizes (cid:12)(cid:12) η B (cid:12)(cid:12) as a function of z R , η B ≈ − C δm cosh (2 z I ) δM ζ (3) m ∗ M πv (cid:88) α D α z α κ α . (69)From this expression, it is evident that (cid:12)(cid:12) η B (cid:12)(cid:12) decreases exponentially as a function of | z I | for | z I | (cid:29) | z I | (cid:29) ⇒ η B ∝ e − | z I | . (70)This suppression can be traced back to the suppression of the CP asymmetry parameter in the caseof large (and fine-tuned) Yukawa couplings, ε Iα ∝ / (cid:0) yy † (cid:1) II ∝ e − | z I | for | z I | (cid:29) η B on cosh (2 z I ) in Eq. (69) allows us to finally combine our analysis ofresonant leptogenesis with our previous discussion of the neutrino option. According to Eq. (58), theneutrino option implies an approximate one-to-one relation between M and cosh (2 z I ), such that η B ≈ − C π δmδM ζ (3) m ∗ m tot M v µ (cid:88) α D α z α κ α , Γ , (cid:28) ∆ M (cid:28) M , . (71)This estimate for the baryon asymmetry is one of the main results in this paper. It illustrates inone compact expression how the type-I seesaw model succeeds in unifying the physics of neutrinomasses, leptogenesis, and EWSB. Roughly speaking, one may summarize the content of Eq. (71)as follows: The factors δm , m tot , and D α represent the masses and flavor oscillations of the lightSM neutrinos at low energies; the factors C , δM , ζ (3) m ∗ , M , z α , and κ α account for resonantleptogenesis in the early Universe; and the factors v and µ reflect the spontaneous breaking of EWsymmetry at the EW scale as well as the RHN threshold corrections to the Higgs mass parameter.Based on Eq. (71), we are now able to estimate the RHN mass splitting δM that is required forsuccessful baryogenesis. For illustration, let us set the RHN mass M to its maximally allowed value[see Eq. (57)] and maximize δM over the CP -violating phases δ and σ , while keeping the PMNSmixing angles fixed at their respective best-fit values given in Tab. 2. We thus find M = . × GeV (NH)7 . × GeV (IH) ⇒ δM BFP (cid:39) . × − (NH)1 . × − (IH) . (72)which translates into absolute mass splittings of ∆ M (cid:39) . × GeV (NH) and ∆ M (cid:39)
10 GeV (IH),respectively. Note that these values mark the upper boundaries of the viable M and δM ranges,meaning in particular that the largest possible RHN mass splitting that we find is of O (cid:0) (cid:1) GeV.If we allow the RHN mass M to vary by half an order of magnitude [see Eq. (59)], the strong depen-dence of η B on M in Eq. (71), η B ∝ M , implies a variation of δM by two orders of magnitude. TheNH case is therefore characterized by a relative RHN mass splitting in the range δM ∼ − · · · − ,while in the IH case, we expect a mass splitting in the range δM ∼ − · · · − .Next, let us consider the ∆ M (cid:28) Γ I regime, where the shape of the function f IJ in the CP asymmetry parameter ε (s) Iα is determined by the regulators R osc IJ and R mix IJ [see Eqs. (35) and (38)].We discuss this regime mostly for completeness. From a conceptual point of view, it is less appealingthan the Γ I (cid:28) ∆ M regime because of the strong dependence on the regularization procedure, whichintroduces a certain degree of theoretical uncertainty; and from a phenomenological point of view, itis less appealing because of the tiny RHN mass splitting that is required for successful baryogenesis.The only difference in the case of an extremely small mass splitting is that one should not expand ε (s) Iα in powers of Γ I [see Eq. (61)], but rather in powers of δM . To leading order, this results in∆ M (cid:28) Γ I ⇒ (cid:88) I ε ( s ) Iα ≈ A α δM πv m tot M (cid:18) X cosh (2 z I ) − δm + 1 X (cid:19) , (73)where X is a shorthand notation for the following function of z R and z I in the complex z plane, X = cosh (2 z I ) − δm cos (2 z R )2 cosh (2 z I ) . (74)Apart from this, all steps in the calculation remain the same. In analogy to Eq. (68), we thus obtain η B ≈ C δM π v ζ (3) m ∗ m M (cid:18) X cosh (2 z I ) − δm + 1 X (cid:19) δm sin (2 z R ) δm cos (2 z R ) − cosh (2 z I ) (cid:88) α D α z α κ α . (75)21 − − − − − − − − − − − − − η obs B (cid:39) . × − B a r y o n a s y mm e t r y η B Relative mass splitting δM = ∆ M/M NH (numerical)IH (numerical)analytical z = π/ . z = π/ . Figure 2: Final baryon asymmetry η B as a function of the relative RHN neutrino mass splitting δM = ∆ M/M = ( M − M ) /M at four representative points in the complex z plane. The RHNmass M is always chosen so as to satisfy the neutrino-option constraint in Eq. (48). All otherparameters are fixed at their best-fit values according to Tab. 2. The analytical estimates at smalland large δM values are based on Eq. (71) and Eq. (77), respectively. We find excellent agreementbetween the full numerical result and our analytical estimates, except for δM values close to theresonance peak, where the mass splitting is of the order of the RHN decay widths, ∆ M ∼ Γ , . Also,note how varying the CIP parameter z I affects the solutions of the leptogenesis condition η B = η obs B .In order to facilitate the comparison with Eq. (69), let us evaluate this expression at z R = π/ η B ≈ − C Y δm δM cosh (2 z I ) 16 π v ζ (3) m ∗ m M (cid:88) α D α z α κ α , Y = 5 + δm cosh (2 z I ) − δm . (76)Here, we introduced the function Y , which varies in the range Y (cid:39) · · · Y (cid:39)
5, in the IH scenario. Finally, let us impose the neutrino-optioncondition, which allows us to replace cosh (2 z I ) by an expression in terms of M [see Eq. (58)], η B ≈ − C Y π δm δM ζ (3) m ∗ m tot M v µ (cid:88) α D α z α κ α , ∆ M (cid:28) Γ , (cid:28) M , . (77)The expression in Eq. (77) is the ∆ M (cid:28) Γ I equivalent of our Γ I (cid:28) ∆ M result in Eq. (71). Themain difference between these two results is the different scaling with δM , M , and µ . While theresult in Eq. (71) scales as η B ∝ δM − µ − M , we now find η B ∝ δM µ − M . For ∆ M (cid:28) Γ I , thebaryon asymmetry is thus enhanced by a factor M /µ and suppressed by a factor δM compared tothe expression in the Γ I (cid:28) ∆ M regime. This leads to a situation where successful baryogenesis canonly be achieved for a RHN mass splitting that is significantly smaller than anything that we haveencountered thus far, δM (cid:28) − . In our numerical parameter scan in Sec. 3.2, we will thereforeignore the alternative solution in Eq. (77) and focus on our first solution in Eq. (71) instead. InFig. 2, we show a comparison of our analytical estimates in Eqs. (71) and (77) with the full numericalresult for the baryon asymmetry based on Eqs. (39). We find that, within their respective ranges ofapplicability, both of our analytical expressions are in excellent agreement with the exact result.22 .2 Numerical parameter scan Let us now cross-check and extend our analytical results by means of a full-fledged numerical analysis.As in the previous section, we are going to consider the conditions in Eq. (50); this time, however,we will refrain from applying any simplifying approximations and work with the full expressions thatwe derived in Sec. 2 instead. In a first step, we wish to perform a scan of our model in the complex z plane. We are specifically interested in the region z R ∈ [0 , π ) and z I ∈ [ − , +2] (see Sec. 2.1). Ateach point in this region, we would like to solve the leptogenesis and neutrino-option constraintsin Eq. (50) for the RHN mass M and the RHN mass splitting δM = ∆ M/M = ( M − M ) /M and determine the largest possible mass splitting δM max that is compatible with the current low-energy neutrino data. The fact that we decide to follow this strategy is motivated by our theoreticalprejudice that exceptionally small mass splitting may be hard to come by in concrete UV flavormodels. That is, if the size of the RHN mass splitting should, e.g. , be related to the quality of someapproximate global flavor symmetry in the ultraviolet, we would expect a bias towards larger masssplittings that do not require a strong suppression of symmetry-breaking effects at higher energies.In order to find the maximal mass splitting δM max as a function of z R and z I , we allow thelow-energy observables in Tab. 2 to vary within their 3 σ confidence ranges. As for the CP -violatingphase δ , which is currently only poorly constrained by the data, we consider the full range of possiblevalues, δ ∈ [0 , π ). Similarly, we allow the CP -violating phase σ , which is completely unconstrainedat the moment, to vary in the full range σ ∈ [0 , π ). In practice, we implement the variation of thelow-energy observables by drawing random numbers from the respective ranges of allowed values.Likewise, we let the discrete parameter ζ in Eq. (13) randomly flip between ζ = +1 and ζ = − z plane, we consider 10 different combinations of possible values forthe low-energy observables, and for each of these combinations, we solve the constraints in Eq. (50)for M and δM . In Fig. 3, we present our results for δM max for both the NH and IH scenarios; inFig. 4, we show our solutions for δM max and M in the NH case next to each other. In view of ournumerical results in Figs. 3 and 4, several comments are in order:1. All plots in Figs. 3 and 4 display the symmetry and reflection properties that we anticipatedin Eq. (16). Our results are thus invariant under the different operations on the RHN Yukawacouplings shown in Eq. (16). In particular, we observe that the dependence of δM max on z R is mostly controlled by the sin (2 z R ) term in Eq. (68), which stems from the fact that the CP asymmetry parameter ε (s) Iα in Eq. (61) is proportional to Re (cid:2)(cid:0) yy † (cid:1) (cid:3) ∝ sin (2 z R ). For2 z R /π ∈ Z , it is thus impossible to realize successful baryogenesis, which explains why we failto find solutions to the conditions in Eq. (50) on this hypersurface in parameter space.2. The dependence of δM max on z I is mostly controlled by the by the cosh (2 z I ) term in Eq. (69),which reflects the fact that ε (s) Iα is inversely proportional to (cid:0) yy † (cid:1) II ∝ cosh (2 z I ). Large values of | z I | therefore lead to large and fine-tuned Yukawa couplings that suppress the CP asymmetryparameter ε (s) Iα . This suppression can only be compensated for by a smaller RHN mass splitting.On top of that, the factor D α in Eq. (67) also contributes to the dependence of δM max on z I , D α = tanh (2 z I ) − E α − E α tanh (2 z I ) , E α = ζ C α B α . (78)The factor D α has only little impact on the overall magnitude of the final baryon asymmetry;its main effect is that it modulates the sign of the LH-lepton-doublet asymmetry η lptg L α . In the23 − z I z R /π − − − δM = ∆ M/M η B < η B < Normal hierarchy · − − − − e µ τ e µ τ e µ τ e µ τ z R /π − − − δM = ∆ M/M Inverted hierarchy − − . − − . e µ τ e µ τ e µ τ e µ τ Figure 3: Maximally allowed RHN mass splitting δM max as a function of z R and z I in the NHscenario (left panel) and in the IH scenario (right panel) . The solid contour lines represent thefull numerical result after varying the low-energy neutrino observables within their 3 σ confidenceintervals, while the dotted contour lines correspond to fixed low-energy input parameters ( δ = 3 π/ σ = 0, and all other observables set to their best-fit values in Tab. 2). The white triangles indicatethe locations of the texture-zero flavor models discussed in Sec. 3.3 at the same benchmark point.region 0 < z R < π/
2, we find that η lptg L α obtains positive values for z I > z α I = artanh ( E α ) / z I < z α I . In the NH case, this behavior leads to a sign flipof the total baryon asymmetry at z I values around z (0)I (cid:39) − .
6, which is consistent with thevalues that one typically finds for the three critical z I values z e I , z µ I , and z τ I . As a consequence,we are not able to construct viable solutions to the conditions in Eq. (50) for z I < z (0)I in theregion 0 < z R < π/ z I > − z (0)I in the region π/ < z R < π in the NH case. In the IHscenario, a similar effect occurs; however, in this case, the dependence of the excluded regionon z R and z I is more complicated. We will comment on this further below.3. The z dependence of M in Fig. 4 is dictated by the relation in Eq. (56). The dependence on z R in this relation is suppressed by the small value of δM , such that M effectively turns outto be a function of z I only, M ∝ cosh − / (2 z I ) [see Eq. (58)]. This dependence is a directconsequence of the neutrino-option constraint in Eq. (48), which can also be written as δM (cid:28) ⇒ π v µ = Tr (cid:2) m D m † D (cid:3) M ≈ cosh (2 z I ) m tot M . (79)Here, the cosh (2 z I ) factor is a consequence of the relation (cid:0) yy † (cid:1) II ∝ cosh (2 z I ), while the cubicpower of the RHN mass follows from the two powers of M in Eq. (48) and the single powerof M that relates Tr (cid:2) m D m † D (cid:3) to the effective mass parameter (cid:101) m [see Eqs. (51) and (53)].4. The dotted contour lines in Fig. 3 are the outcome of a restricted analysis based on δ = 3 π/ σ = 0 and all other observables kept fixed at their best-fit values in Tab. 2. While these24 − z I z R /π − − − δM = ∆ M/M η B < η B < Normal hierarchy · − − − − e e µ µ τ τ e µ τ z R /π M [PeV] η B < η B < Normal hierarchy
Figure 4: Maximally allowed RHN mass splitting δM max (left panel) and corresponding RHN mass M (right panel) as functions of z R and z I in the NH scenario after varying the low-energy neutrinoobservables within their 3 σ confidence intervals. The white contour lines in the plot on the left-handside indicate the orbits of the texture-zero flavor models discussed in Sec. 3.3.contour lines are also generated by our numerical code, we are able to confirm that they can bereproduced to excellent precision by the analytical expressions in Eqs. (58) and (68). In largeparts of the complex z plane, the dotted contour lines represent a good approximation of thesolid contour lines, which correspond to our full numerical results after varying all low-energyinput parameters. This indicates that the variation of the low-energy observables within their3 σ confidence ranges only has a small effect on the outcome of our analysis in a large partof parameter space. A notable exception to this statement occurs in the IH case in Fig. 3.For 0 < z R < π/ z I (cid:46) − π/ < z R < π and z I (cid:38)
1, we find a set ofsolutions that cannot be reproduced by simply assuming fixed input values for the low-energyobservables. We explicitly checked that this deviation is not a numerical artifact, but a genuineoutcome of our numerical algorithm. We thus conclude that, in this region of parameter space,the marginalization over the low-energy observables is essential.5. Overall, we find very good agreement between the results in Figs. 3 and 4 and our analyticalestimates in Sec. 3.1. As expected, the mass splitting varies in the range δM max ∼ − · · · − in the NH scenario and in the range δM max ∼ − · · · − in the IH scenario. Because ofthis relative suppression by a factor 10 − (and because of the slight experimental preferencefor a normal SM neutrino mass ordering), we will no longer consider the IH case from now.Regarding our numerical results for M , we find perfect agreement with Eqs. (57) and (58) byconstruction. All M values are of O (cid:0) (cid:1) GeV and spread across half an order of magnitude.6. As mentioned earlier, the CIP parameters z R and z I can be regarded as the coordinates of the25 .
45 0 .
50 0 .
55 0 . δ / π σ/π . . . . δM = ∆ M/M [10 − ]Normal hierarchy sin θ . . . . . δM = ∆ M/M [10 − ]Normal hierarchy Figure 5: Maximally allowed RHN mass splitting δM max as a function of σ and δ (left panel) andof sin θ and δ (right panel) for z I = 0, i.e. , for vanishing CP -violating phases at high energies,after varying all other low-energy neutrino observables within their 3 σ confidence intervals.unconstrained theory space of flavor models that might act as UV completions of the 2RHNseesaw. To illustrate this statement by means of a simple example, we show the position ofcertain (toy) flavor models in Figs. 3 and 4. We will elaborate on this in more detail in Sec. 3.3.Finally, we conclude this section by pointing out that our scenario also manages to account for thegeneration of the baryon asymmetry even if the CP -violating phases δ and σ in the PMNS matrix arethe only source of CP violation during leptogenesis. This observation allows us to conclude that ourscenario is compatible with the concept of leptogenesis from low-energy CP violation [80, 117–126].To show that this is the case all we have to do is to fix the CIP parameter z I at z I = 0 in ournumerical analysis. This renders the matrix R in Eq. (13) real, such that δ and σ are the onlynontrivial complex phases that enter the RHN Yukawa couplings y Iα . Once z I is set to z I = 0, weare able to perform a similar analysis as before. We scan the complex z plane along the real z R direction, solve the conditions in Eq. (50) for δM and M for random combinations of values of thelow-energy observables, and determine the maximally allowed mass splitting δM max .The outcome of this analysis is shown in Fig. 5, where we project our results for δM max into the σ – δ plane as well as into the sin θ – δ plane. Our motivation for picking these projections is thatsin θ , δ , and σ represent important observables that are expected to become better constrainedby experiments in the near future [1]. In both plots in Fig. 5, δM max varies only slightly aroundvalues of O (cid:0) − (cid:1) . This is in accord with our results in Fig. 3, where we also consistently find δM max ∼ − along the real z R axis, and demonstrates once more the comparatively small impactof varying the low-energy observables within their 3 σ confidence ranges. The main message from The overall factor i in Eq. (13) corresponds to a global phase that does not affect any of our results. δM and M across the entire rangesof sin θ , δ , and σ values. In particular, it is always possible to obtain the correct sign of the baryonasymmetry, independently of the values of δ and σ . The only exception to this statement are smallregions around the following six singular points, where the baryon asymmetry vanishes identically:( δ, σ ) ∈ { (0 , , ( π, , (2 π, , (0 , π ) , ( π, π ) , (2 π, π ) } . At these points, CP invariance is not violated,such that there is no source for the baryon asymmetry [127]. In their immediate vicinity, there is notenough CP violation to reproduce the observed value of the baryon asymmetry. Other than that,it is always straightforward to realize leptogenesis from low-energy CP violation in our scenario. Our scan of the complex z plane in Sec. 3.2 can be understood as a scan over all possible UV flavormodels that are consistent with the low-energy neutrino data. In the following, we will illustratethis point by means of a simple example — flavor models that are characterized by a single texturezero in RHN Yukawa matrix y Iα . The concept of texture zeros in the Yukawa sector is well knownfrom QCD, where it can be used to successfully predict the Cabbibo angle in the quark mixingmatrix [128]. In the SM neutrino sector, texture zeros in the Majorana mass matrix m αβ have beenextensively studied in the literature [129–133]. The same is true for texture zeros in the Yukawamatrix y Iα ; see, e.g. , Refs. [57, 58, 118, 134, 135] for studies of two-zero Yukawa textures in the 2RHNseesaw model. Typically, texture zeros in a fermion Yukawa matrix are assumed to be related to aflavor symmetry at high energies (see, e.g. , Ref. [136]), which demands that certain couplings areexactly zero or significantly suppressed compared to all other entries in the Yukawa matrix. Thisflavor symmetry could, e.g. , correspond to a Froggatt–Nielsen flavor symmetry [137] with a flavorcharge assignment such that some Yukawa couplings end up being vanishingly small.In this section, we shall simply consider one texture zero in the RHN Yukawa matrix y Iα . Twotexture zeros would only be possible, if we assumed an inverted SM neutrino mass hierarchy; morethan two texture zeros are always in conflict with the low-energy neutrino data in the 2RHN seesaw.The requirement that one element in the matrix y Iα must vanish can then be used to determine thecomplex rotation angle z . Let us denote by z ζ α and z ζ α those values of z that lead to vanishingvalues for the couplings y α and y α , respectively. Making use of the CIP in Eq. (13), we find z ζ α = arctan (cid:18) − ζ (cid:114) m k m l U ∗ αk U ∗ αl (cid:19) , z ζ α = arctan (cid:18) + ζ (cid:114) m l m k U ∗ αl U ∗ αk (cid:19) , ( k, l ) = (2 ,
3) (NH)(1 ,
2) (IH) . (80)These expressions depend on the PMNS matrix U as well as on the SM neutrino masses m i . However,in the following, we will fix the SM neutrino masses and the PMNS mixing angles at their best-fitvalues in Tab. 2, such that z ζ α and z ζ α turn into functions of the CP -violating phases δ and σ only.In Fig. 3, we evaluate these functions at δ = 3 / π and σ = 0 and indicate the locations of the twelvecomplex numbers z ζ α and z ζ α , where α = e, µ, τ and ζ = ±
1, in the complex z plane. Here, we useupwards and downwards pointing triangles to distinguish between the ζ = +1 and ζ = − z plane by varying δ and σ in the ranges δ ∈ [0 , π ) and σ ∈ [0 , π ). In this sense, bothfigures demonstrate in an illustrative manner how the complex z plane can be understood as a mapof the landscape of possible flavor models. Fig. 3 allows us, in particular, to conclude that it is notpossible to simultaneously demand the following four things in our model: (i) a successful solution27o the leptogenesis and neutrino-option constraints, (ii) δ = 3 / π and σ = 0, (iii) a normal SMneutrino mass ordering, and (iv) a vanishing RHN Yukawa coupling to the LH lepton doublet L e .Meanwhile, we can read off from Fig. 4 that a vanishing Yukawa coupling y Ie can be easily realizedas soon as we relax our assumptions on δ and σ . These statements are just simple examples of howour plots in Sec. 3.2 can help to constrain possible embeddings of the 2RHN seesaw into UV flavormodels. We expect that more complicated flavor models may give rise to a richer structure in thecomplex z plane that would allow for even more complex conclusions. We leave such an investigationof alternative flavor models for future work. In particular, it would be interesting to assess whetherthere are entire classes of models that are preferred or ruled out by our scenario.Finally, let us comment on dark matter. Throughout this work, we referred several times to thepossibility that, upon extending our RHN particle content by a keV-scale state, the type-I seesawmodel could in addition to neutrino masses, baryon asymmetry, and EW scale also account fordark matter. The simplest realization of keV-scale dark matter, compatible with the type-I seesawLagrangian employed in this work, was proposed by Dodelson and Widrow [59]. These authorsconcluded that the amount of dark matter consistent with observations can be produced by collisionprocesses, provided nonzero mixing between the active and sterile neutrino states. However, despiteits simplicity and minimality, this scenario is nowadays excluded by the combined constraints fromstructure formation [138], supernova 1987A data [139, 140], and X-ray searches [141]. The minimalscenario for keV-scale sterile-neutrino dark matter that is compatible with present limits is thereforeresonant production `a la Shi and Fuller [60]. In this scenario, the mixing between active and sterilestates can be enhanced because of lepton number asymmetries, which results in the well-known
Mikheev–Smirnov–Wolfenstein (MSW) effect [144–146]. The viable parameter space of the Shi–Fuller mechanism is currently probed by on-going astrophysical observations (see, e.g. , Ref. [147]).Resonant sterile-neutrino production is a basic ingredient of the so-called neutrino minimal Stan-dard Model ( ν MSM) [148,149], which successfully combines the physics of neutrino masses, dark mat-ter, and baryogenesis solely based on the type-I seesaw Lagrangian. In the ν MSM, the oscillationsof GeV-scale sterile neutrinos are responsible for the generation of primordial lepton asymmetriesvia the
Akhmedov–Rubakov–Smirnov (ARS) mechanism [150] (see Ref. [151] for very recent work onthis topic), which then set the stage for the production of keV-scale sterile-neutrino dark matter.Given the similarity between the ν MSM and our setup, it would be interesting to investigate thepossibility of an extended type-I seesaw sector featuring a split RHN spectrum. That is, if the RHNspectrum should contain states with masses in the PeV, GeV, and keV range, one could attemptto simultaneously realize (i) the generation of the EW scale via RHN threshold corrections, (ii)baryogenesis via a combination of resonant and ARS leptogenesis, and (iii) the production of keV-scale sterile-neutrino dark matter via the Shi–Fuller mechanism. Alternatively, one should study inmore detail whether the scenario of resonant leptogenesis explored in this paper might not suffice togenerate the lepton number asymmetries needed for DM production. In this case, one would be ableto bypass the ARS mechanism at lower energies and would not need to introduce GeV-scale statesin the RHN spectrum. Both of these scenarios go beyond the scope of this paper and deserve moreattention in future work. In passing, we note that keV-scale sterile-neutrino dark matter was alsoexplored in gauge [152, 153] and scalar extensions [154–158]. The latter are particularly appealingbecause the realization of scale-invariant boundary conditions calls for an extended scalar sector. In view of the recently discovered unidentified X-ray line at around 3.55 keV [142, 143], X-ray searches have led toa strong interest in this type of dark matter, which could radiatively decay into pairs of photons and active neutrinos. Conclusions and outlook
RH neutrinos are key players in the field of BSM model building, which allow one to address severalshortcomings of the Standard Model at the same time. In this paper, we have considered two viableRHN scenarios that manage to simultaneously explain (i) the SM neutrino oscillations, (ii) the baryonasymmetry of the Universe, and (iii) the origin of all SM particle masses. The first scenario, whichwe dubbed the Dirac-neutrino option, is based on the assumption that the RHN sector preserveslepton number. In this case, neutrinos turn into massive Dirac fermions in consequence of theHiggs mechanism, baryogenesis might proceed via neutrinogenesis, and the EW scale plays therole of a universal mass scale that determines the masses of all SM particles. The Dirac-neutrinooption, however, suffers from (i) exceptionally small RHN Yukawa couplings, (ii) a missing connectionbetween baryogenesis at high energies and the phenomenology of neutrino oscillations at low energies,and (iii) the fact that the EW scale should likely not be considered a fundamental scale, given theabsence of signals for new physics in current experiments. Motivated by these observations, wetherefore turned to a second scenario based on an alternative underlying symmetry principle. Thissecond scenario, which we dubbed the Majorana-neutrino option, postulates that the SM Lagrangiansatisfies scale-invariant boundary conditions in the ultraviolet, whereas the RHN sector is allowedto feature Majorana masses for the RH neutrinos that explicitly break classical scale invariance andlepton number. In this scenario, the Higgs mass parameter in the Higgs potential is forbidden at treelevel and only induced via RHN one-loop threshold corrections. As a consequence, the RHN massscale replaces the EW scale as the input scale that translates into the masses of all SM particles.This observation may be taken as a sign that the scale of new physics should, in fact, not be soughtclose to the EW scale but rather at energies above RHN thresholds. The RH neutrinos should thenbe regarded as messengers between the Standard Model and the BSM sector in this framework.According to the Majorana-neutrino option, the SM neutrinos obtain small masses via the type-Iseesaw mechanism. Moreover, it can be shown that the generation of the Higgs mass term (with thecorrect magnitude and the correct sign) requires RHN masses of O (cid:0) (cid:1) GeV. Our main contribu-tion in this paper was to show that these two features of the type-I seesaw model are compatible withbaryogenesis via resonant leptogenesis. We focused on the minimal type-I seesaw model involvingonly two RH neutrinos and studied resonant leptogenesis both from an analytical and a numericalperspective. We found excellent agreement between our two approaches and concluded that resonantleptogenesis succeeds in explaining the baryon asymmetry of the Universe for an absolute RHN masssplitting as large as ∆ M ∼ GeV, or equivalently, for a relative RHN mass splitting as large as δM ∼ − . These values apply in the case of a normally ordered SM neutrino mass spectrum. Inthe case of an inversely ordered SM neutrino mass spectrum, we found an additional suppression bya factor of O (cid:0) (cid:1) . In addition, it is interesting to note that the success of resonant leptogenesisis not contingent on the presence of additional sources of CP violation at high energies. We couldshow that the leptonic CP violation encoded in the PMNS matrix is enough to explain the observedbaryon asymmetry. In light of these results, we arrive at the conclusion that the type-I seesaw cannot only explain the masses of all known particles but also the cosmological relic density of matter.In this sense, the type-I seesaw may be regarded as the origin of all mass and matter in the Universe.This statement might even extend to dark matter if the type-I seesaw sector should also contain aRHN state with a mass at the keV scale whose relic density accounts for dark matter.29he production of keV-scale sterile-neutrino dark matter in our scenario should be investigatedmore carefully in future work. Similarly, it would be interesting to extend our discussion of flavormodels in Sec. 3 and assess which classes of flavor models turn out to be favored or disfavored by ourscenario. In this paper, we merely restricted ourselves to a class of simple texture-zero models thatallowed us to illustrate the physical meaning of our parameter plots in Figs. 3 and 4. In particular, itwould be worthwhile to seek an embedding of the type-I seesaw in a flavor model that automaticallyexplains the small splitting among the RHN mass eigenvalues. Besides that, there are several furtherdirections in which our analysis could be extended: (i) The renormalization scheme dependence ofthe RHN threshold corrections should be cross-checked by explicit higher-order computations in thetype-I seesaw model. (ii) One should repeat our analysis in the 3RHN seesaw model and study towhich extent this model enables one to loosen the parameter constraints that we derived in thiswork. (iii) Finally, one should embed the type-I seesaw in a fully scale-invariant UV completion thatexplains how the RHN masses originate from the spontaneous breaking of classical scale invariance.Such a UV-complete model will necessarily feature an extended scalar sector, and it would beinteresting to study the consequences of this extended scalar sector for leptogenesis. All of thesequestions are, however, beyond the scope of this work. We conclude by emphasizing that the type-Iseesaw model is a truly intriguing extension of the Standard Model. Despite the fact that it hasbeen around for some 40 years, it still calls for further exploration and promises further surprises. Note added
During the final stages of our project, we became aware of work by Ilaria Brivio, Kristian Moffat,Silvia Pascoli, Serguey T. Petcov, and Jessica Turner [159] that is closely related to ours. Here,we comment on the relation between our results and those obtained in Ref. [159]. The authorsof Ref. [159] arrive at the conclusion that resonant leptogenesis in the context of the Majorana-neutrino option requires a relative mass splitting of δM ∼ − . In addition, they derive a lowerbound M min1 on the RHN mass M of M min1 (cid:39) . M min1 (cid:39) . M are realized at z maxI (cid:39) . z maxI (cid:39) . i.e. , at values of the CIP parameter z I larger than those that we considered in this paper.The lower bound on M is associated with the fact that the baryon asymmetry becomes expo-nentially suppressed at large values of | z I | [see Eq. (70)]. This suppression can only be compensatedfor by resorting to smaller mass splitting δM up to the point where one reaches the resonance peakin Fig. 2 [see also Eq. (60)]. Beyond that point, it is no longer possible to realize successful lepto-genesis because of a strongly suppressed CP asymmetry parameter. By extending our numericalparameter scan to larger values of | z I | , we are able to locate this boundary of the viable parameterregion in the complex z plane. In the NH scenario, we find that M and δM can become as smallas M min1 (cid:39) . δM min (cid:39) . × − around z maxI (cid:39) .
7, while in the IH scenario, we obtain M min1 (cid:39) . δM min (cid:39) . × − around z maxI (cid:39) .
5. Our bounds on M and z I are thusmore or less consistent with those in Ref. [159]. They just appear to be slightly weaker; this may berelated to details such as the sphaleron conversion factor [see Eq. (25)] or the fact that the analysisin Ref. [159] accounts for the running of the SM neutrino masses. Besides that, we observe that themass splitting quoted in Ref. [159], δM ∼ − , actually corresponds to the smallest possible valuethat is consistent with resonant leptogenesis and the neutrino option. As we have demonstrated inour paper, the RHN mass splitting can, in fact, become as large as δM ∼ − (see Fig. 3).30he upper bounds on z I can also be estimated analytically based on the expressions in Sec. 3.1.It is easy to show that, at large | z I | , the CP asymmetry parameter ε (s) Iα attains a maximal resonantenhancement for ∆ M/ Γ (cid:39) ∆ M/ Γ (cid:39) . z R → π/
4, we aretherefore able to write the final baryon asymmetry at the resonance peak as follows, η B ≈ − CF ζ (3) m ∗ m tot δm cosh (2 z I ) (cid:88) α D α z α κ α . (81)Here, the numerical factor F corresponds to the difference f − f [see Eq. (37)] that appears inthe CP asymmetry parameter, evaluated at the resonance peak and at large | z I | , such that Γ (cid:39) Γ , F = (cid:20) x x + 4 x x + O (cid:0) y (cid:1)(cid:21) x (cid:39) . (cid:39) . , x = ∆ M Γ , y = Γ M . (82)Making use of Eq. (81), one can then solve the leptogenesis constraint η B = η obs B for z I . For instance,for the benchmark point discussed in Sec. 3 ( i.e. , for δ = 3 π/ σ = 0, and all other observables setto their best-fit values in Tab. 2), we obtain z maxI (cid:39) . z maxI (cid:39) . M ,which is more or less consistent with our findings. However, regarding the required RHN masssplitting in the case of close-to-maximal M values, the authors of Ref. [159] refrain from performinga quantitative analysis. Instead, they restrict themselves to the qualitative statement that δM doesnot necessarily need to be as small as δM ∼ − for larger M values. In this sense, the masssplitting stated in the abstract of Ref. [159], δM ∼ − , must be understood as a lower bound onthe mass splitting that is necessary for successful leptogenesis. As we have shown in our analysis, δM can, in fact, be as large as δM ∼ − and δM ∼ − for NH and IH, respectively. Acknowledgements
We are grateful to Bhupal Dev and Daniele Teresi for helpful discussions and comments on resonantleptogenesis as well as to Simon J. D. King for collaboration at the early stages of this project.We acknowledge the friendly and constructive communication with Ilaria Brivio, Kristian Moffat,Silvia Pascoli, Serguey T. Petcov, and Jessica Turner towards the completion of this project. S. I.is supported by the MIUR-PRIN project 2015P5SBHT 003 “Search for the Fundamental Laws andConstituents”. K. S. wishes to thank the Joˇzef Stefan Institute (IJS) in Ljubljana, Slovenia for itshospitality during an extended stay in May 2019, when parts of this manuscript were written. Thestay of K. S. at IJS was supported by the Slovenian-Italian bilateral cooperation project “The flavorof the invisible universe”, grant numbers BI-IT-18-20-002 / SI18MO07. This project has receivedfunding / support from the European Union’s Horizon 2020 research and innovation programmeunder the Marie Sk(cid:32)lodowska-Curie grant agreement No. 690575 (K. S.).31 eferences [1]
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