Symmetry origin of baryon asymmetry, dark matter, and neutrino mass

Symmetry origin of baryon asymmetry, dark matter, and neutrino mass Subhaditya Bhattacharya*and Arunansu Sil † Department of Physics, Indian Institute of Technology Guwahati, Assam-781039, India Rishav Roshan ‡ Department of Physics, Kyungpook National University, Daegu 41566, Korea Drona Vatsyayan § Departamento de Física Teórica and IFIC, Universidad de Valencia-CSIC, C/ Catedrático Jos´e Beltrán, 2—E-46980 Paterna, Spain (Received 25 May 2021; accepted 26 September 2022; published 11 October 2022) We propose a minimal model based on lepton number symmetry (and violation), to address a common origin of baryon asymmetry, dark matter and neutrino mass generation. The model consists of a vectorlike fermion to constitute the dark sector, three right-handed neutrinos (RHNs) to dictate leptogenesis and neutrino mass, while an additional complex scalar is assumed to be present in the early Universe the decay of which produces both dark matter and RHNs via lepton number violating and lepton number conserving interactions respectively. Interestingly, the presence of the same scalar helps in making the electroweak vacuum stable until the Planck scale. The unnatural largeness and smallness of the parameters required to describe correct experimental limits are attributed to lepton number violation. The allowed parameter space of the model is illustrated via a numerical scan. DOI: 10.1103/PhysRevD.106.075005 I. INTRODUCTION Standard Model (SM) has been extremely successful as a gauge field theory in describing the fundamental constituents of this Universe and their interactions via electromagnetic, weak and strong forces. After the discovery of a Higgs-like boson at the Large Hadron Collider (LHC) [1], SM also inherits a successful mass generation mechanism and can be deemed complete. However, many unanswered questions still persist. In particular, the quest for physics beyond the Standard Model (BSM) or new physics (NP) arises from observations like matter-antimatter asymmetry of the Universe [ηB¼nB−n¯ B nγ∼Oð10−10Þ][2], dark matter (DM) relic density [ΩDMh2∼Oð0.1Þ][3] and tiny but nonzero neutrino masses [mν≲Oð10−10ÞGeV] [4–6] amongst other theoretical/phenomenological motivations. All these observations have been well addressed in literature from theoretical as well as phenomenological points of view (albeit no experimental verification yet), but are most often considered one or at most two at a time. It is therefore tempting to consider a common framework that addresses all of them together. The type-I seesaw framework with three RHNs serves as the minimal framework that addresses leptogenesis, neutrino mass and DM together, where the lightest RHN plays the role of the DM candidate while the other two are responsible for explaining nonzero neutrino masses and baryogenesis via leptogenesis [7,8]. Apart from these minimal type-I setups, there also exist several extensions of type-I seesaw frameworks [9–17] which try to explain these three crucial issues under the same umbrella. For example, in Refs. [9–12], simultaneous decays of righthanded neutrinos to visible and dark sector particles are shown to account for the observed DM relic density as well as baryon asymmetry of the Universe. However, amongst existing efforts in this direction, a symmetry angle in tying the knot has mostly been ignored, which we focus on here. NP, although searched in different experiments, has not been confirmed yet, indicating either heavy NP scale or BSM fields feebly coupled to Standard Model (SM) or both. For example, heavy Majorana right-handed neutrinos (RHNs) are instrumental in realizing tiny neutrino mass via type-I seesaw [18–20], and can also explain matterantimatter asymmetry via leptogenesis [21,22]. Turning to *[email protected] †[email protected] ‡rishav[email protected] §drona.v[email protected].es Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 106, 075005 (2022) 2470-0010=2022=106(7)=075005(10) 075005-1 Published by the American Physical Society the genesis of DM, nonthermal freeze-in provides feebly interacting massive particles (FIMP) to acquire correct relic density with a tiny coupling [∼Oð10−10Þ] to the SM [23–28]. Such unusual small coupling or the heaviness of RHNs, however, demands justification. In this regard, we pose the following question: Is there any underlying (global) symmetry in nature whose presence and breaking are responsible for largeness like heavy RHN mass and smallness like tiny neutrino mass or the DM freeze-in couplings? We argue that the global lepton number symmetry (LNS) is one such possibility that can effectively justify the question above. While LNS is respected by the renormalizable Lagrangian in general, its breaking by certain terms in the setup can be attributed to the smallness and largeness of the respective coupling(s) and/or mass scale(s). The setup constitutes a dark sector and an extended SM sector comprising of three RHNs and a singlet scalar. While the absence of a Majorana mass term for RHNs is a consequence of the global LNS, their masses can be generated due to the Planck scale lepton number breaking (linked with gravity effect) in line with the recent finding [29].We further propose that DM production via freeze-in is also caused by lepton number violation (LNV), so that the associated tiny coupling can be justified. Together, our proposal explains baryogenesis via leptogenesis, nonthermal DM production via freeze-in and correct neutrino mass with minimal extension of SM where LNS and its violation, through interactions with a scalar mediator, play the central role in determining the strength of the associated interactions. In fact, the decay of this scalar to the dark sector (via LNV coupling) as well as RHN sector (via LN conserving interaction) bridges a connection between them in this model. Furthermore, the same scalar helps the electroweak (EW) vacuum to remain absolutely stable all the way to Planck scale. Our paper is organized as follows: we describe the model in Sec. II, production mechanism of RHN, DM and lepton asymmetry in Sec. III, Boltzmann equations and solutions to find allowed parameter space of the model in Sec. IV, vacuum stability in Sec. Vand summarize in Sec. VI. II. MODEL As already mentioned, the model aims to address leptogenesis, DM and neutrino mass generation in a correlated way and also aims to be minimal in construct. Concerning the field content of the model, apart from SM fields, the fermion sector consists of three RHNs Ni, responsible for neutrino mass generation via type-I seesaw as well as baryogenesis via leptogenesis, and a vectorlike singlet fermion χ, which serves as the DM component of the Universe. Apart, a scalar isosinglet ϕis ideated to connect DM and RHN sectors, which also aid to stabilize the EW vacuum [30], as elaborated later in Sec. V. To establish the connection via global LNS, we propose ϕto carry a lepton number Lof −2unit, RHNs carry the same L(þ1) as that of SM leptons, while χremains neutral. To obtain a stable DM, additional discrete Z2symmetry is imposed under which χtransforms as χ→−χ, while all other particles remain even. The charge assignments under Land Z2are shown in Table I. The lepton number conserving (LNC) renormalizable Lagrangian (LNP c), invariant under SM gauge symmetry and Z2, inherits the following interaction and mass terms: −LNP c⊂ðyνÞij¯ lLi ˜ HNjþYNiϕNc iNiϕþMχχχ þM2 ϕϕϕþλϕHH†HϕϕþH:c:; ð1Þ where fi; j ¼1;2;3gdenote family indices. His the SM Higgs isodoublet ( ˜ H¼iσ2H), which acquires a vacuum expectation value (VEV) υafter electroweak symmetry breaking (EWSB), parametrized by H¼1 ffiffi2 pð0;υþhÞT, where his the 125 GeV Higgs boson discovered at LHC [1].ϕdoes not acquire a VEV and hence keeps L preserved. Note also the absence of a renormalizable DMSM interaction and Majorana mass term for RHNs in the limit of exact LNC. However, as any global symmetry is expected to be broken by gravity effects, it is possible to generate masses of RHNs by Planck scale (MP) LNV as in [29], which we adopt here. Furthermore, we propose additional LNV Yukawa interaction connecting DM χand the field ϕ. Together, we have −LNP v⊂ðYχϕ ¯ χχϕ þH:c:ÞþMiNc iNi;ð2Þ with M1<M 2≪M3ð≃MPÞ. Though both terms in Eq. (2) are of lepton number violating in nature, they can have different origins. In particular, the Yukawa interaction involving the DM and the ϕfield can be thought of as an explicit LNV operator as ϕcarries a lepton number of two units (negative) `alaϕNN interaction of Eq. (1). Hence, the corresponding dimensionless Yukawa coupling Yχϕ can be considered to be small enough which is technically natural in ‘t Hooft’s sense [31]. On the other hand, although the Majorana mass term for RHNs is also an LNV one, being super-renormalizable, it is assumed to be of gravitational origin in line with [29] and hence superheavy. The specific hierarchy of three heavy RHNs as M1<M 2≪M3ð≃MPÞcan be justified as follows. First, a democratic RHN mass matrix stems as a result of Planck scale LNV by gravity [29] which is flavor blind, given by TABLE I. Relevant particles and their charge assignments. Symmetries ϕNiχlL L−210þ1 Z2þ1þ1−1þ1 BHATTACHARYA, SIL, ROSHAN, and VATSYAYAN PHYS. REV. D 106, 075005 (2022) 075005-2 M¼MP0 B @ 111 111 111 1 C A;ð3Þ sparing a coefficient in front. A subsequent diagonalization provides eigenvalues as ð0;0;3MPÞ). Second, this exact democratic structure of the mass matrix is expected to be perturbed by topological fluctuations [32,33] resulting nonzero M1;2proportional to perturbations, so that M1;2≪ M3ð≃MPÞis realized at the Planck scale. It has been shown in [34] that M1;2also receive quantum corrections. However, it is quite possible that the quantum corrections remain subdominant compared to the tree level masses of M1;2introduced at the Planck scale itself, which we assume here.1We provide concrete numerical estimate of M1;2 shortly. Combining Eqs. (1) and (2), one finds that ϕcan simultaneously decay to visible sector (NiNi) via LNC interactions (proportional to coupling YNiϕof sizable magnitude) and to dark sector (χχ) via LNV interactions (with feeble coupling Yχϕ). The heavy RHNs (N1;2) can further decay to land H(and ¯ l¯ H) via the LNC channel. A schematic of the framework is shown in Fig. 1. Note also that the LNV term like ¯ lL ˜ HχRis prohibited by Z2symmetry and keeps DM stable.2We may also think of another LNV term: μϕHϕH†HþH:c:in Eq. (2), where μϕHis a dimension-full coupling. This would allow ϕto decay further to visible sector (hh) and help keeping ϕin thermal bath. Whether the term possesses gravitational origin or not, the effect of this term in the phenomenology concerning the DM production or leptogenesis is negligible as long as the related decay width remains subdominant compared to that to the RHNs and total decay width of ϕand not considered further. This translates into the condition μϕH<Y N1ϕMϕ. We will comment on the magnitude of such μϕHin the context of perturbativity and vacuum stability in Sec. V. Finally one should also note that due to the presence of BSM particles and their interactions, the present setup is also subject to different theoretical constraints. In order to make the electroweak vacuum stable, the scalar potential should be bounded from below which restricts the scalar quartic couplings of the model. On the other hand, the scalar quartic couplings (λi) along with all the Yukawa couplings (in general denoted by Yi) involved in the setup should remain perturbative provided jλij<4πand jYij<ffiffiffiffiffiffi 4π p:These constraints have been taken into account while doing the analysis. III. PRODUCTION OF RHNs, DM AND LEPTON ASYMMETRY Let us now turn to DM and RHN production processes in this setup. In Fig. 2, we show all possible production channels for DM (χ), and RHN N1(the lightest being the most relevant) from particles in thermal bath including ϕ(which thermalizes via sizable portal coupling λϕH). The production kinematics is dictated by the chosen hierarchy: FIG. 1. Schematic representation of the model: ϕcouples to the dark sector via LNV interaction while it interacts with the visible sector through the LNC one. Moreover, CP violating N→lHð¯ l¯ HÞdecays lead to lepton asymmetry. FIG. 2. Production of DM (χ) and RHN (N1). 1Possibilities such as M1only or both M1and M2masses dominated by quantum corrections [34] are not suitable for our analysis due to their extreme hierarchical nature. 2It is easy to check that given the Z2charge assignments of the fields, discrete anomaly free conditions [35] are satisfied to have a gauge origin of the symmetry [36] to avoid Planck scale effects. SYMMETRY ORIGIN OF BARYON ASYMMETRY, DARK MATTER, …PHYS. REV. D 106, 075005 (2022) 075005-3 T>M ϕ>2M1>2Mχ;ð4Þ where Tdenotes maximum temperature available for the production of a species. We would like to also note here that we are agnostic about the inflationary scenario and do not identify ϕas inflaton, unlike [37,38]; this becomes apparent once we consider Mϕ<T , accessible to the reheat regime. N3 being superheavy is decoupled from the rest while N2is considered to be heavier than Mϕ=2for simplicity allowing ϕto decay only to the N1pair. Note that both the DM as well as RHN production is dominated by the decay or the inverse decay processes. For example, the scattering processes that produce DM are further subdued by feeble Yχϕ (LNV coupling) and/or large ϕmasses that appear in the propagator. The same is true for N1production via scattering processes as the heavy ϕmass either is present at both ends of the initial state or appears in the propagator in spite of sizable YN1ϕ(LNC coupling). Therefore, we consider ϕ→χ¯ χfor DM production and ϕ→N1N1and lh→N1for RHN production and neglect other processes safely. For number densities of DM and N1and their subsequent evolution, Boltzmann equations (BEQ) are used, which we elaborate shortly. At this moment, we note that the lightest RHN N1, once produced from ϕ, further decays (CP violating and out of equilibrium) to lHð¯ l¯ HÞfollowing the LNC Yukawa interaction to create lepton asymmetry as in the vanilla leptogenesis scenario (for a review, see Ref. [39]). The CP asymmetry produced in these decays as a result of the interference between tree level and one loop decay amplitudes is given by [40] ε1¼1 8πX j≠1 Im½ðˆ y† ν ˆ yνÞ2 1j ðˆ y† ν ˆ yνÞ11 FM2 j M2 1;ð5Þ where FðxÞ≃3=2ffiffiffi x pfor hierarchical RHNs and ˆ yνis the neutrino Yukawa coupling matrix in the mass diagonal basis of RHNs, the form of which can be obtained using Casas Ibarra (CI) parametrization [41]: ˆ yν¼ffiffiffi 2 p υU PMNS ffiffiffiffiffiffi md ν qRTffiffiffiffiffiffiffi MR p;ð6Þ where MRðmd νÞrepresents diagonal RHN (light neutrino) mass matrix, while UPMNS [42] is the unitary matrix (in charged lepton diagonal basis) required to diagonalize mν¼U PMNSmd νU† PMNS. Here, Ris a 3×3orthogonal matrix that can be chosen as [43] R¼0 B @ 0cos zRsin zR 0−sin zRcos zR 10 0 1 C A;ð7Þ where zR¼aþib is a complex angle. While M3is taken to be MP, the hierarchy between M1;2can be expressed as M2¼rM1. For example, with M1¼1011 GeV, r¼100 and zR¼0.016–0.105i, we get the following Yukawa matrix after CI parameterization: ˆ yν¼0 B @ 0.0029 −0.0004i−0.0127 −0.0196i0 0.0046 −0.0006i0.0790 þ0.0045i0 −0.0015 −0.0008i0.0989 −0.00193 0 1 C A:ð8Þ Such a choice of zRand M1;2is motivated by the fact that corresponding CP asymmetry turns out to be ε1¼ 1.4×10−6, which enters into BEQ and generates correct lepton asymmetry. Importantly, the model offers enough freedom to judicious choices of parameters like yν,zRto produce correct leptogenesis and neutrino mass, given M1, while the choice presented above is an example of its kind. IV. BOLTZMANN EQUATIONS AND EVOLUTION OF NUMBER DENSITY We now elaborate on the evolution of number densities via BEQs. ϕbeing the source of DM/RHN production, the yields of ϕ(Yϕ), N1(YN1), lepton asymmetry (YΔL) and DM (Yχ) are all coupled, dictated by the following set of BEQs: dYϕ dz ¼−s Hz½hσvϕϕ→SMiðY2 ϕ−ðYeq ϕÞ2Þ −1 sHz Yϕ Yeq ϕ½γðϕ→N1N1Þþγðϕ→χ¯ χÞ;ð9Þ dYN1 dz ¼1 sHz γðϕ→N1N1ÞYϕ Yeq ϕ −γN1YN1 Yeq N1 −1;ð10Þ dYΔL dz ¼1 sHz γN1ε1YN1 Yeq N1 −1−YΔL 2Yeq l;ð11Þ dYχ dz ¼1 sHz γðϕ→χ¯ χÞYϕ Yeq ϕ:ð12Þ Note that yield is defined by YðeqÞ¼nðeqÞ s,[nðeqÞis the (equilibrium) number density, s¼0.44gT3is the total entropy density]; and z¼Mϕ=T, where Tis temperature. The reaction density γis given by γða→bcÞ¼neq K1ðzÞ K2ðzÞΓða→bcÞ;ð13Þ where K1;2are Bessel functions of first and second kind. The starting point to solve for the coupled BEQs above is T¼T(taken to be ∼10Mϕ), where we assume Yχ¼0, YN1¼0,YΔL¼0,Yϕ¼Yeq ϕ. The yield in each case is BHATTACHARYA, SIL, ROSHAN, and VATSYAYAN PHYS. REV. D 106, 075005 (2022) 075005-4 thereafter built by the dominant processes as mentioned in Eqs. (9)–(12). In Table II, we show a few benchmark points (BP1=2=3) that characterize the model with all relevant parameters (in agreement to LNS and violation) required to produce correct lepton asymmetry and observed DM relic density. We would like to point out that the benchmark values3 of M1;r and zRare chosen in a way so as to obtain the correct amount of baryon asymmetry via leptogenesis from the subsequent decay of N1with a fixed value of Mϕð≃M2Þ¼1013 GeV. The choices of these parameters also ensure correct neutrino mass generation via CI parametrization as described above. The value of LNC Yukawa coupling is kept at a moderate value, YN1ϕ¼0.01, so the RHN remains out of equilibrium in the early Universe and gradually thermalizes due to interactions with l-h. The parameters M1;r and zRhowever do not affect the dark sector significantly. DM relic density can be obtained to the desired value by choosing appropriate Yχϕ for a given DM mass (Mχ), so that DM is produced from the decay of ϕ with Mϕ>2Mχ. For example, with Mχ¼1500 GeV, the required Yχϕ ¼1.2×10−7. One can easily show that correct DM relic can also be obtained for other DM mass in the TeV ballpark by adjusting Yχϕ, with Mϕ¼1013 GeV as chosen for the benchmark points in Table II.Itis however intriguing to note: (i) the value of Mϕdictates a limit on the RHN mass M1so that it is produced from the ϕdecay as well as accounts for the correct leptogenesis and so is true for the DM mass (Mχ) and, (ii) the hierarchy between the Yukawa interactions Yχϕ=YN1ϕ∼ 10−9required for correct DM relic and leptogenesis can be attributed to that of LNV to LNC according to the model construct. Thus the model provides an interesting interplay of these two sectors connecting via LNS and its breaking. The numerical solution to BEQs for BP1, BP2 and BP3 is shown in Fig. 3. Concerning Yϕand its evolution [Eq. (9) and red thick lines in Fig. 3], interaction with SM via thermal average annihilation cross section hσvϕϕ→SMidue to sizable Higgs portal λϕH∼0.7(see, for example, [44]), helps ϕto keep up with the thermal equilibrium in the early TABLE II. Three characteristic benchmark values of M1and ratio r¼M2=M1are listed along with corresponding zRvalues those satisfy correct baryon asymmetry. Mϕð≃M2Þ¼1013 GeV and YN1ϕ¼0.01 are kept constant. The dark sector is mostly independent of the neutrino sector; for example, with DM mass Mχ¼1500 GeV, the LNV coupling is required to be Yχϕ ¼1.2×10−7which provides correct relic density. Benchmarks M1(GeV) r¼M2=M1zR BP1 5×1010 1030.195 −0.295i BP2 1011 1020.016 −0.105i BP3 1012 10 0.032 −0.025i FIG. 3. Solutions to BEQs [Eqs. (9)–(12)] for BP1, BP2 and BP3 for evolution of Yϕ(red), YN1(black), lepton asymmetry YΔL (orange) and DM Yχ(purple). The black dotted line represents Yeq N1 and the orange dashed line indicates the correct YΔLrequired to produce the observed baryon asymmetry of the Universe. The red dotted lines represent Yϕin absence of ϕ→N1N1;χχ. We assume λϕH∼0.7, while other parameters can be seen from Table II. 3Following Ref. [34], the quantum corrections to the tree level values of M1(also for M2) turn out to be approximately 0.1, 1, 10% of their tree level masses for BP1=2=3respectively. SYMMETRY ORIGIN OF BARYON ASYMMETRY, DARK MATTER, …PHYS. REV. D 106, 075005 (2022) 075005-5 Universe. It decouples from the thermal bath due to the depletion to SM final states as well as via decays to N1 pairs; in absence of ϕ→N1N1decay, ϕfreezes out as shown by the red dotted line. For Yϕ, we also neglect inverse decays N1N1→ϕ;χ¯ χ→ϕ, since the initial abundances of N1and χare vanishingly small, and neglect the decay contribution to DM (ϕ→χχ) as it is much much smaller due to Yχϕ ≪YN1ϕ;λϕH. We may note here that if we keep λϕHlarger or smaller within the same ballpark, there is no significant effect on YN1,Yχand YΔL, except that ϕfreezes out later (earlier). Turning to YN1[Eq. (10) and black thick line in Fig. 3], processes that contribute significantly apart from ϕ→N1N1are N1→lH,N1→¯ l¯ H,lH →N1,¯ l¯ H→N1. They eventually bring YN1into equilibrium (black dotted line). We see that in Fig. 3, the abundance of N1gradually increases with production from ϕand inverse decays of lh and reaches equilibrium. It then tracks the equilibrium distribution owing to its sizable Yukawa coupling with the SM leptons and Higgs. Decay of N1to lHð¯ l¯ HÞis responsible for generating lepton asymmetry YΔL¼Yl−Y¯ ldescribed via Eq. (11) and shown by the orange thick line in Fig. 3.YΔLbeing proportional to ε1[first term in Eq. (11)], is responsible for the rise in asymmetry, which gradually fades due to washout by inverse decays lHð¯ l¯ HÞ→N1denoted by the second term in Eq. (11). As temperature falls below M1, the washout processes get suppressed and once N1decays are complete, the asymmetry saturates (gray dashed line). The asymptotic yield Y∞ ΔLis eventually transferred to baryons (YB) (via electroweak sphalerons above T∼100 GeV) following, YB¼cY∞ ΔL, with c¼28=51 [39] to produce YB¼ð8.75 0.23Þ×10−11. Finally, BEQ for DM (χ) is shown in Eq. (12) and via the purple thick lines in Fig. 3, owing to the only nonthermal production from ϕ. It shows a typical DM freeze-in pattern for Yχto accumulate correct relic (Ωχh2¼0.120 0.001) [3], which follows a well-known relation with the asymptotic yield as Ωχh2¼2.755 ×108Mχ GeVYχðz∞Þ:ð14Þ Spot the absence of the late decay contribution of ϕto DM freeze-in, due to its tiny branching to DM, compared to RHN, thanks to LNS and its violation. Interestingly, the late decay contribution to N1yield is also not visible due to the presence of l;H interactions with N1, which dominates over the ϕN1N1interaction. At this point, it might seem obscure the importance of LNC interaction ϕNN in the framework as in absence of it, N1can still be produced from inverse decays and lepton asymmetry can also be generated. However, note that the allocation of lepton number to ϕis made via this interaction only and as a result, we could label the other interaction of ϕ(with DM χ) as a LNV one so as to attribute the smallness of the corresponding coupling to it. To be more specific, both the interactions of ϕ(with RHNs and DM) are relevant enough from the LNS symmetry point of view and its violation. DM relic density allowed parameter space in the Yχϕ −Mχplane for different Mϕin agreement to the benchmark point choices is shown in Fig. 4. The fall in the Yukawa coupling with the increasing dark matter mass can be easily understood by the expression of the relic density as in Eq. (14), proportional to both the dark matter mass and the dark matter yield; now, if the DM mass increases the DM yield has to decrease which can only be achieved with lower values of the Yukawa coupling, Yχϕ. V. EW VACUUM STABILITY We discuss here the fate of the EW vacuum in this model. This would be particularly interesting due to the presence of the additional scalar ϕand RHNs in our setup. It is well known that within the SM itself, the Higgs quartic coupling (λH) turns negative at a scale around ΛSM ∼ 1010 GeV [45–49] with top quark mass mt∼173.2GeV leading to a possible instability of the EW vacuum. The conclusion depends crucially on the precise value of the top mass though. The situation may worsen (i.e., λHcan be negative at a scale before ΛSM) in the presence of the RHNs [30,50] having sizable Yukawa coupling. On the contrary, the presence of the additional scalar ϕin the spectrum can potentially influence the running of the Higgs quartic coupling in a positive way, thanks to its Higgs portal interaction. Here comes the significance of ϕassumed in the model. While on one hand, ϕbridges the connection between the RHN and DM sector, an interesting interplay between the neutrino Yukawa coupling (yν) and scalarHiggs portal coupling (λϕH) decides the fate of EW vacuum. FIG. 4. Yχϕ as a function of DM mass Mχto accumulate correct relic density (Ωχh2¼0.120 0.001) for Mϕ¼1013 GeV. BHATTACHARYA, SIL, ROSHAN, and VATSYAYAN PHYS. REV. D 106, 075005 (2022) 075005-6 The scalar potential involving the Higgs and ϕfield [part of which is already present in Eq. (1)] is given by VðH;ϕÞ¼−μ2 HjHj2þλHjHj4þM2 ϕϕϕþλϕðϕϕÞ2 þλϕHjHj2jðϕϕÞþμϕHðϕþϕÞH†H; ð15Þ where we retain the trilinear term proportional to μϕHand rescale it in terms of Mϕas μϕH¼αϕHMϕand wish to estimate its role in EW vacuum stability. Note that by integrating out the heavy scalar ϕ, the effective scalar potential below the scale mϕcan be written as Veff ¼−μ2 HjHj2þðλH−α2 ϕh=2ÞjHj4:ð16Þ The second term should coincide with the SM Higgs quartic term and, hence, the matching condition at scale mϕ, λSM H¼ðλH−α2 ϕh=2Þ;ð17Þ results. In order for the above scalar potential of Eq. (15) to be bounded from below, the conditions are λH;λϕ≥0; λϕHþ2ffiffiffiffiffiffiffiffiffiffi λHλϕ p≥0. Furthermore, the scalar quartic couplings should be less than 4πat any scale4below the Planck one. Using this perturbativity limit on λHand the value of the SM Higgs quartic coupling at Mϕ¼1013 GeV (due to running), we find αϕH≲1which in turn indicates μϕH≲Mϕ. Such a finding supports our previous consideration of ignoring the contribution of this trilinear term in the DM phenomenology. In the same line, we ignore its contribution in the following discussion on vacuum stability as well by considering its magnitude to be vanishingly small. A complete list of βfunctions of all the couplings involving the RHN and a singlet scalar can be found in existing literature including [51]. Among them, the contributions of RHN and the scalar singlet in the βfunction of Higgs quartic coupling can be written as βλH¼βSM λHþβRHN λHþβϕ λH;ð18Þ where βRHN λH¼4λHTr½ˆ y† ν ˆ yν−2Tr½ðˆ y† ν ˆ yνÞ2;βϕ λH¼2λ2 ϕH;ð19Þ in one loop. Note that the ˆ yνused above is defined in Eq. (6). The requirement of λH>0at high scale guarantees absolute stability of the EW vacuum. On the other hand, if it happens to be negative at some scale, a second deeper minimum may exist. In this case, if the tunneling probability PTof EW vacuum to the second minimum is longer than the age of the Universe (TU), then metastability of the EW vacuum can be ensured. The tunneling probability is given by [45,52] PT¼T4 Uμ4 Be−8π2 3jλHðμBÞj;ð20Þ where μBis the scale at which the tunneling probability is maximized and is determined from the condition βλHðμBÞ¼0. Metastability then requires λHðμÞ>−0.065 1−lnðv=μBÞ:ð21Þ Here the running of all the SM and BSM couplings of the present setup is performed in two loops (using SARAH [53])5in three steps: (i) μ¼mtto M1, (ii) μ¼M1to Mϕ, and (iii) μ¼Mϕð∼M2Þto MP. The initial conditions of all relevant SM couplings such as top-quark Yukawa yt, gauge couplings gi(i¼1, 2, 3) and Higgs quartic coupling λHare provided in Table III at μ¼mt[45], where we consider mh¼125.09 GeV, mt¼173.2GeV and αSðmZÞ¼ 0.1184. In Fig. 5, we show the running of the Higgs quartic coupling with the energy scale μfor the parameters associated with BP2 of Table II. The running of λHin the SM is shown in red while the effect of scalar-Higgs portal coupling is observed in the green portion for λϕH¼0.7and in the orange part for λϕH¼0.3between μ¼Mϕð∼M2Þto MP. The blue shaded line shows the evolution of λHbetween μ¼M1and μ¼Mϕwhich essentially overlaps with the SM running. This is because Tr½ˆ y† ν ˆ yν¼0.017 being relatively small (compared to 0.5 as observed in [30,50] in order to observe any significant deviation), we do not expect much influence of RHNs on the renormalization group evolution of λH.However,we note that a sizable Higgs portal coupling of ϕ∼0.7is capable of keeping the EW vacuum absolutely stable until the Planck scale. This being a salient feature of the presence of the ϕfield in the setup, we can also recollect that the same portal coupling was helpful in keeping the ϕin thermal bath and give birth to both DM and RHNs. TABLE III. Values of the relevant SM couplings [top-quark Yukawa yt, gauge couplings gi(i¼1, 2, 3) and Higgs quartic coupling λH] at energy scale μ¼mt¼173.2GeV with mh¼ 125.09 GeV and αSðmZÞ¼0.1184. Scale λHytg1g2g3 μ¼mt0.125932 0.93610 0.357606 0.648216 1.16655 4The unitarity constraints are found to be less stringent compared to this. 5We neglect running of Yϕχ due to its tiny value ∼10−7. SYMMETRY ORIGIN OF BARYON ASYMMETRY, DARK MATTER, …PHYS. REV. D 106, 075005 (2022) 075005-7 VI. SUMMARY The paper outlines the possibility of addressing neutrino mass generation, the plethora of matter over antimatter in the Universe and FIMP-type dark matter to provide the correct relic density together via lepton number symmetry (and violation) naturally justifying the heaviness and smallness of NP parameters and null observation in current experiments. Here, the SM particle spectrum is extended minimally with a heavy complex scalar ϕ, three right-handed neutrinos (Ni), and a vectorlike fermion (χ) all singlet under the SM gauge symmetry. An additional unbroken Z2symmetry, under which the newly introduced vectorlike fermion is nontrivially charged while all the other particles remain even, makes χa stable DM candidate. In addition, we also assign −2unit of lepton charge to the complex scalar so that it can couple directly to the RHNs with a LNC Yukawa interaction (YN1ϕ). On the contrary to this, the complex scalar interacts with the DM with a LNV Yukawa interaction (Yχϕ). Therefore, the singlet scalar ϕ, which is assumed to be in a thermal bath in the earlier Universe through sizable Higgs-portal coupling, simultaneously decays to produce the RHN and the DM via LNC and LNV interactions respectively. This naturally addresses the out-of-equilibrium nonthermal DM production via tiny LNV DM-SM coupling, hitherto unexplained in literature. Considering the fact that one of the RHNs, N3, gets mass at Planck scale due to the breaking of lepton number by gravity effects, the other two RHNs acquire masses at a much lower scale by quantum effects and address neutrino mass generation by type-I seesaw. Once the lightest RHN is produced from the decay of the ϕfield, its further decay to the SM lepton and Higgs to generate the asymmetry in the visible sector by judicious choice of the lepton-Higgs Yukawa coupling. It is worth reiterating that the model not only connects the DM genesis and leptogenesis via symmetry (and breaking) arguments, but also explains the motivation for the connection via ϕto achieve vacuum stability of EW potential all the way up to Planck scale. The feasibility of the model and possible choices of the parameters are demonstrated by a few representative benchmark points to successfully address all three phenomena together. The numerical solution to the coupled BEQs responsible for generating lepton asymmetry and DM relic are explicitly demonstrated, yielding a correlation between the LNC and LNV Yukawa interactions. While the benchmark points chosen are not exhaustive, they indicate the range of masses and interaction strengths that validate the model. The “common link”ϕbetween the DM and the RHN sector plays a crucial role in guiding the parameters. For example, absent the decay of ϕto RHN, the mass of ϕ and Yχϕ can be of completely different strengths unlike the ones in Fig. 4. The model naturally consists of either very heavy or feebly coupled NP, and does not promise an early detection in next-generation experiments. However, a further extrapolation of the setup with the identification of the ϕas the inflaton may open up new directions. ACKNOWLEDGMENTS S. B. acknowledges Grant No. CRG/2019/004078 from SERB, Government of India. A. S. acknowledges the support from Grants No. CRG/2021/005080 and No. MTR/2021/ 000774 from SERB, Government of India. R. R. was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (No. NRF2020R1C1C1012452) [1] S. Chatrchyan et al. (CMS Collaboration), Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Phys. Lett. B 716, 30 (2012). [2] N. Aghanim et al. (Planck Collaboration), Planck 2018 results. VI. 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