The visible matter content in the present Universe is highly asymmetric, referred to as the baryon asymmetry of the Universe (BAU), a longstanding puzzle in particle physics and cosmology [1,2]. With the standard model (SM) of particle physics falling short of providing an explanation, baryogenesis [3,4] as well as leptogenesis [5] have been the most widely studied beyond standard model (BSM) frameworks explaining the observed BAU dynamically. In leptogenesis, a nonzero lepton asymmetry is generated first which later gets converted into baryon asymmetry by electroweak sphalerons [6]. One interesting aspect of leptogenesis is its natural realization within canonical seesaw mechanisms like type-I [7–11], type-II [11–15], and type-III [16] seesaws proposed to explain nonzero neutrino mass and mixing, another observed phenomena which the SM fails to address. Leptogenesis in such seesaw models typically involve the out-of-equilibrium CP-violating decay of a heavy particle like heavy right handed neutrino (RHN) in the type-I seesaw. In order to generate nonzero CP asymmetry as well as to explain neutrino data, at least two copies of such RHNs are required. Additionally, with the hierarchical RHN spectrum, there exists a lower bound on the scale of leptogenesis, known as the Davidson-Ibarra bound M1≳109  GeV [17]. While this keeps canonical leptogenesis models out of reach from terrestrial experiments, it is possible to bring the scale of leptogenesis to TeV scale via resonant enhancement with tiny mass splitting between RHNs, known as the resonant leptogenesis [18].

In addition to providing evidence for baryon asymmetry of the Universe, the cosmic microwave background (CMB) observations also suggest that we live in a Universe which is homogeneous and isotropic on large scales up to an impressive accuracy [2,19]. Such observations, however, lead to the so called horizon and flatness problems which remain unexplained in the description of standard cosmology. The theory of cosmic inflation which introduces a phase of rapid accelerated expansion in the very early Universe is the leading paradigm solving these problems in standard cosmology [20–22]. Inflation not only solves these problems, but also makes unique predictions which can be tested at present and future CMB observations.

In this work, we propose a novel scenario where leptogenesis and neutrino mass can be realized in a minimal setup with only one RHN and an additional Higgs doublet playing the role of inflaton. The same Higgs doublet also generates the desired lepton asymmetry via the Affleck-Dine (AD) mechanism [23]. With just two BSM fields, namely, a RHN and a second Higgs doublet, we show that the model can satisfy the requirements of generating the correct baryon asymmetry, neutrino mass, and mixing while being consistent with inflationary cosmology simultaneously.¹

A similar setup but with type-II seesaw was proposed recently by [24,25] with follow-up studies in [26]. Affleck-Dine Dirac leptogenesis was studied in [27] with two Higgs doublets and one massless right-handed neutrino without explaining the origin of neutrino mass.

This paper is organized as follows. In Sec. II, we briefly discuss the model followed by the details of inflationary dynamics in Sec. III. We then elaborate upon the origin of baryon asymmetry via Affleck-Dine mechanism in Sec. IV. Finally, in Sec. V, we summarize our results and conclude.

We consider a seesaw model with one gauge singlet right-handed neutrino N which can give rise to one active neutrino mass via type-I seesaw [7–12,29]. A doublet scalar field η playing the role of inflaton can give rise to another active neutrino mass via radiative seesaw. The scalar potential has a Z2 symmetry preventing η from acquiring a vacuum expectation value (VEV), provided the condition mη2+12(λ3+λ4)v2>0is satisfied. We assume this to hold in our setup. The Yukawa Lagrangian, however, breaks the Z2 symmetry. The relevant part of the Yukawa Lagrangian is given by -L⊃YαL¯αη˜N+yαL¯αΦ˜N+12MNNc¯N+H.c.,(1)where L, Φ denote the lepton and SM Higgs doublet, respectively. The Z2 symmetry of the scalar potential can be retained in the Yukawa Lagrangian involving η by assigning both η and right-handed neutrino N to be Z2 odd, while all SM fields remain Z2 even. With this assignment, the Yukawa interaction YαL¯αη˜N is Z2 even and therefore does not break the symmetry. As a result, Z2-violating linear or trilinear η terms are not generated at one-loop from these couplings. However, the additional Yukawa interaction yαL¯αΦ˜N involving the SM-like Higgs doublet Φ and SM leptons introduces a mild Z2 breaking. This interaction can induce terms odd in η at one loop, but its magnitude is suppressed by the small type-I seesaw Yukawa couplings. Consequently, the induced η VEV after electroweak symmetry breaking remains negligibly small and does not affect the phenomenology discussed in the paper. Moreover, since the reheating temperature in our scenario is above the electroweak scale, the η field has already decayed before electroweak symmetry breaking, making the impact of any such loop-induced VEV-dependent decays insignificant. We also check the possibility of mη2 turning negative at some energy scales due to large Yukawa coupling with N. Using the one-loop renormalization-group evolution (RGE) of the scalar mass parameters we find that η mass-squared term becomes negative only in regimes where the right-handed neutrino mass is much larger than mη. Because of our consideration mη≳MN, the η mass squared term remains positive at all energy scales, in agreement with Ref. [30]. A complete treatment of such Z2-breaking effects via RGE corrections [31] is beyond the scope of this present work and is left for future studies. The U(1)L or global lepton number charges of the BSM fields η and N are assigned as -1 and 0, respectively. Note that the second term in the above Lagrangian violates lepton number, but it does not contribute to CP asymmetry from N decay. The Z2-symmetric scalar potential is V(η,Φ)=mη2|η|2+λ22|η|4+μϕ2|Φ|2+λ12|Φ|4+λ3(η†η)(Φ†Φ)+λ4(Φ†η)(η†Φ)+λ52[(Φ†η)2+H.c.],(2)where Φ is the SM Higgs doublet. The Z2-symmetric scalar potential keeps the two Higgs doublets in the alignment limit [32,33]. The scalar fields Φ and η can be parametrized as Φ=12(0ϕ+v),η=(η±(H0+iA0)2).(3)When the neutral component of Φ acquires a nonzero VEV v, one of the light neutrinos acquire mass via type-I seesaw (mνI)αβ=-yαyβv22MN.(4)Another light neutrino acquires mass at the one-loop level with η, N in the loop. The corresponding mass is given by (mνloop)αβ=YαYβMN32π2[Lk(mH02)-Lk(mA02)],(5)where the function Lk(m2) is defined as Lk(m2)=m2m2-Mk2lnm2Mk2.(6)Such a neutrino mass model with a combination of tree level and radiative contributions arising from one RHN and two Higgs doublets was discussed originally by Grimus and Neufeld [34]. Unlike the minimal type-I seesaw with two RHNs, this model can also explain the neutrino mass hierarchy naturally, without fine-tuning [35].

In order to incorporate the constraints from light neutrino masses, we use the Casas-Ibarra parametrization [36] for the type-I seesaw in combination with the one for the scotogenic model [37], as done for scoto-seesaw scenarios in [38,39]. This is given by Y=XM-1RmνdiagU†,(7)where R is an arbitrary complex orthogonal matrix satisfying RRT=1, and U is the usual Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix which diagonalizes the light neutrino mass matrix in a basis where charged lepton mass matrix is diagonal. The combined Dirac Yukawa coupling Y is Y=(Yα1×3yα1×3),(8)and XM is given by XM=(-X100X0),X1=MN32π2(Lk(mH02)-Lk(mA02)),X0=v22MN.(9)The R matrix for two heavy neutrino scenarios is given by [40] R=(0cosz1±sinz10-sinz1±cosz1),(10)where z1=a+ib is a complex angle. The diagonal light neutrino mass matrix, assuming normal hierarchy, is given by mνdiag=(0000Δmsol2000Δmatm2+Δmsol2).(11)For inverted hierarchy, mνdiag is mνdiag=(Δmatm2-Δmsol2000Δmatm20000).(12)The PMNS mixing parameters and mass squared differences are obtained from [1].

As mentioned earlier, we consider the Affleck-Dine (AD) mechanism [23] to generate the visible sector asymmetry. We adopt a simple nonsupersymmetric setup where baryogenesis occurs via leptogenesis. In a typical AD mechanism of this type, a lepton number (L) carrying field, to be referred to as the AD field hereafter, breaks L explicitly by virtue of its quadratic term. The cosmological evolution of the AD field then leads to the generation of lepton asymmetry. In the scalar potential given in Eq. (2), the λ5 term can be identified as the explicit L-breaking term relevant for generation of lepton asymmetry.

In order to generate asymmetry through the Affleck-Dine mechanism, the U(1)L breaking λ5 term in the potential given by Eq. (2) should be effective during the inflationary and reheating dynamics. Inflation in such a case can be realized by a combination of the inert doublet η and the SM Higgs Φ [27,41], with nonminimal couplings to gravity of the form [42] f(|Φ|,|η|)=ξ1|Φ|2+ξ2|η|2.(13)Here, ξ1,2 are dimensionless couplings of the respective scalar doublets to gravity. Earlier works on a common origin of inflation and baryogenesis or leptogenesis via the AD mechanism can be found in [24,43–57]. Considering Φ≡(012ρ1eiθ1) and η≡(012ρ2eiθ2), in order to realize the inflationary trajectory in the large field limit, following the analysis of Refs. [25,27,41], we have ρ1ρ2≡tanα=λ2ξ1-(λ3+λ4)ξ2λ1ξ2-(λ3+λ4)ξ1≃λ2ξ1λ1ξ2,(14)where, from now on, we consider λ3, λ4 to be small enough for simplicity, without any loss of generality. The inflation can now be described in terms of a single field φ, such that ρ1=φsinα,ρ2=φcosα,ξ=ξ1sin2α+ξ2cos2α.The potential can then be written as²

While radiative corrections to the potential can be important, we can always tune the scalar couplings at low energy scale appropriately to get the desired values at the scale of inflation after incorporating one-loop corrections.

Going to the Einstein frame through the conformal transformation, g˜μν=Ω2gμν,Ω2=1+ξφ2/MP2,(16)results in the following Einstein frame Lagrangian: L-g=-MP22R-12gμν∂μχ∂νχ-12f(χ)gμν∂μθ∂νθ-U(χ,θ),(17)where f(χ)≡φ(χ)2cos2αΩ2(χ),U(χ,θ)≡V(φ(χ),θ)Ω4(χ),(18)and R represents the Ricci scalar. The canonical scalar field χ is obtained through the field redefinition given by dχdφ=6ξ2φ2/MP2+Ω2Ω2.(19)In the large field limit, the potential reduces to that of the well-known Starobinsky inflation, given by Uinf(χ)=34Λ2MP2(1-e-23χMP)2,(20)where Λ=λMP23ξ2 and MP is the reduced Planck mass.

In this section, we discuss the generation of baryon asymmetry via Affleck-Dine leptogenesis. From the Lagrangian in Einstein frame given by Eq. (17), the equations of motion for χ and θ are derived to be [27] χ¨-12f′(χ)θ˙2+3Hχ˙+U,χ=0,θ¨+f′(χ)f(χ)θ˙χ˙+3Hθ˙+1f(χ)U,θ=0.(40)Here,  ′ in f denotes derivative with respect to χ. During inflation, θ is in the approximate slow-roll phase, during which we have θ˙≃-MPU,θf(χ)3U.(41)Because of the presence of the U(1)L breaking term, asymmetry number density nL∝φ2θ˙ is generated, which at the end of inflation is given by nL|end=QLφend2(χ)θ˙endcos2α(42)≃QLMPU,θΩ23Uend≃8QLλ˜5sin(2θend)3λξMP3,(43)where we used Eq. (18) and the slow-roll condition for θ˙ given by Eq. (41). In the last step, we have taken the quartic term to be dominant at the end of inflation. In Eq. (43), QL denotes the U(1)L charge of the AD field η.

The final asymmetry depends on the details of the reheating dynamics. To understand this clearly, let us look into Eq. (40) for θ, which can be rewritten as ddt(a3f(χ)θ˙)=ddt(nLa3QLΩ2)=-a3U,θ,(44)where we have substituted θ˙ in terms of nL and used Eq. (18) for f(χ). In addition to the form of the χ potential during reheating, the asymmetry also depends on the form of the U(1)L breaking term, which is determined by U,θ.

Now, if U,θ redshifts faster than a-3 (matterlike), then the right-hand side of Eq. (44) becomes diminishing over time, and the quantity nLa3QLΩ2 remains conserved. After the end of inflation, the potential during reheating is initially dominated by the quartic term after a few e-folds as discussed above. Hence, a3U,θ∝a3χ4∝a-1 becomes subdominant with time. When the magnitude of φ=χ reduces to 2λm, the potential energy density behaves like matter. In this case, we again have a3U,θ∝a3χ4∝a-3, which remains subdominant. Thus, the quantity nLa3QLΩ2 remains approximately conserved, after the start of radiation domination with the quartic potential.

The lepton number density at the end of reheating can be obtained by integrating Eq. (44), which gives nL(tre)≃Ω2(tre)Ω2(tend)(a(tend)a(tre))3[nL|end-Ω2(tend)∫tendtreU,θa3a(tend)3dt]≃nL|endΩ2(tre)Ω2(tend)exp(-3C)(χ(teq)χ(trd))3(teqtre)2[1+c2].(45)The constant c2=-∫tendtre3UendMPa3a(tend)3dt indicates the effect of the source term during reheating and is determined numerically. Here, we used Eq. (43) to write U,θ in terms of nL|end. We find c2≃55.5. Recall that after the beginning of radiationlike behavior of the potential, its effect becomes subdominant, as explained above. teq denotes the time when the quadratic term in the potential dominates and can be written as teq=trd(χ(trd)χ(teq))2.(46)Using the above in Eq. (45), we get nL(tre)∼1.4exp(-3C)g*λ˜5sin2θξλ1/4Ω2(tre)Ω2(tend)(TRHm)×(TRHρrd1/4)3MP3(1+c2),(47)where we used Eq. (43). Recall that C and ρrd are obtained numerically as explained below Eq. (30). The final baryon number yield after the decoupling of electroweak sphalerons is obtained as YB∼exp(-3C)λ˜5sin2θendξλ1/4Ω2(tre)Ω2(tend)(TRHm)MP3ρrd3/4(1+c2).(48)

Lepton asymmetry can also be washed out due to L-violating scattering processes with interactions of the type ηΦ†↔η†Φ or LΦ(η)↔L¯Φ†(η†). The rates for such washout processes can be calculated as Γ1=Γ(ηΦ†↔η†Φ)≈nηeqλ52T2,(49)Γ2=Γ(LΦ↔L¯Φ†)≈|yα†yα|28πMN2nLeq,(50)Γ3=Γ(Lη↔L¯η†)≈|Yα†Yα|28πMN2nηeq.(51)Another source of washout comes from the inverse-decay process: NL→η. The corresponding washout rate in comparison to the Hubble rate can be approximated as WID=14ΓηH(T=mη)(mηT)3K1(mηT),(52)where K1 is the modified Bessel function of the second kind. We constrain the parameter space in such a way to keep all these washout processes out of equilibrium.

Finally, we make a rough estimate of the isocurvature perturbations, which can be induced by the quantum fluctuations of θ [56,70]. The variance of these fluctuations is given by [71] ⟨δθ2⟩≃(Hk2π)21χ*2.(53)The isocurvature perturbations can be related to the temperature fluctuations observed in CMB as [72,73] (δTT)iso≃-25ΩBΩmδnBnB.(54)Here, ΩBΩm≃0.16 [2] is the ratio of the baryon to matter energy densities. nB is the baryon number density, which in our case can be found using Eq. (47) as δnBnB≃2cot(2θi)δθ.(55)On the other hand, the adiabatic perturbations can be written as [72,73] ⟨(δTT)2⟩adi≃120Hk28π2MP2εV,(56)where εV is the slow-roll parameter defined in Eq. (21). The ratio of these two perturbations is calculated to be αnon-adi=⟨(δTT)2⟩iso⟨(δTT)2⟩adi≃128MP2εV5ΩB2Ωm2cot2(2θi)χ*2.(57)Using the upper bound αnon-adi≲2×10-2 [74], we arrive at a lower bound on θi, given by θi≳0.153(1-ns)2+92(1-ns)3,(58)where we used Eqs. (25) and (26) for our inflationary predictions in terms of ns. For ns in the range 0.95 to 0.98, the above bound on θi lies in the range 0.013 to 0.005. As we see, we have sufficient allowed parameter space for these values of θi, which could also potentially be investigated by future CMB experiments. We leave a detailed numerical calculation of these isocurvature perturbations to future works.

In Fig. 2, we show the evolution of φ (left panel) and the asymmetry density nL∝φ2θ˙ (right panel) for different choices of relevant parameters. The top panel shows the variation with respect to m parametrizing the mass of the inflaton. The middle panel shows the variation with respect to the L-violating coupling λ5 coupling, whereas the bottom panel indicates the effect of varying the initial value of θ field, namely, θi. As we can see, ϕ trajectories naturally overlap as the evolutions are very similar for changes in parameters λ5 and θi, and visible differences appear only when the mass is varied. In the top right panel, we observe that the initial asymmetry after the end of inflation scales like radiation for all masses. For a larger mass m, the scaling later transitions to matterlike behavior, since the field reaches equality (teq) sooner, and the field χ starts oscillating as matter. In contrast, smaller m continues to scale like radiation over the same period. Using Eq. (48), we see that the asymmetry produced at the end of inflation is directly proportional to λ5 and θend (or θi). However, the subsequent scaling of the asymmetry with time is independent of λ5 and θend.

Figure 3 shows the summary of our results in the λ5-mη plane with Δm/mη≡(mη-MN)/mη being shown in the color bar. The shaded regions are disfavored by P-ACT 2025 and PLANCK 2018 data, respectively, whereas the meshed region in green violates perturbative upper limit on Yukawa coupling. The contours of different shapes indicate leptogenesis and neutrino mass satisfying points for different choices of Yukawa coupling Yα and θend, while keeping the reheating temperature fixed. We have used θend and θi interchangeably, as θi is approximately equal to θend, especially for smaller values of λ5 (10-4 and below). In this regime, the value of θ remains the same up to two significant digits. In the evolution plots, we determined the asymmetry numerically by setting the initial condition θi, whereas in the summary plots, we used the analytical expression from Eq. (48) and therefore substituted θend instead. From Eq. (39), it is clear that Δm/mη should decrease with mη for fixed Yα and TRH. This pattern is visible from both the plots shown in Fig. 3. Due to the choice of a low TRH, the corresponding relative mass difference Δm/mη needs to be fine-tuned as the corresponding Yukawa coupling cannot be made arbitrarily small due to the constraints from CMB isocurvature as well as light neutrino mass. Such a small mass splitting is not technically natural in our minimal setup, and additional symmetries and nonminimal particle content may be required to explain such fine-tuning, which we do not pursue further in this work. Since the reheat temperature is chosen to the small and Yukawa coupling Yα≳O(0.1), one requires phase-space suppression in the decay width of η as can be noticed from the color bar in Fig. 3. The yellow shaded portion of these contours indicate the region where washout effects are dominant, and hence, lepton asymmetry is likely to get washed out. Due to large Yukawa coupling, the washout Γ3 in Eq. (51) is the most dominant one, and the yellow shaded region in Fig. 3 corresponds to the parameter space where Γ3H|T=TRH>1.The red shaded region is disfavored by the limits on lepton flavor-violating (LFV) decay BR(μ→eγ) from MEG-II [75] while the cyan shaded region is within future sensitivity [76], details of which can be found in the Appendix. While there exist other LFV processes like μ→3e [77] and μ→e conversion [78], the corresponding constraints on the parameter space remain weaker compared to the one for μ→eγ. However, future sensitivity of PRISM/PRIME to μ→e conversion [79] remains promising, as indicated by the pink shaded band. Additionally, the CMB isocurvature constraint given by Eq. (58) disfavors regions with too small θend∼θi, which is indicated by the meshed purple region. Clearly, P-ACT 2025 allows only a small region of parameter space consistent with successful leptogenesis, neutrino mass, and mixing. On the other hand, a large part of the parameter space remains consistent with the PLANCK 2018 data.

To conclude, we have proposed a novel and very minimal framework for explaining the baryon asymmetry of the Universe together with neutrino oscillation data by extending the standard model of particle physics with only two new fields: one right-handed neutrino and a second Higgs doublet with the latter also playing the role of cosmic inflaton. Light neutrino masses arise from a combination of type-I and radiative seesaws with the lightest neutrino mass being zero. While canonical leptogenesis from CP-violating decay is not feasible due to the chosen minimal field content, we implement the Affleck-Dine mechanism by identifying the second Higgs doublet as the AD inflaton. Future data from cosmology and terrestrial experiments will be able to shed more light into the currently allowed parameter space of the model. The model also predicts the vanishingly lightest active neutrino mass which keeps the effective neutrino mass much out of reach from ongoing tritium beta decay experiments like KATRIN [80], while future data should be able to confirm or refute it. Additionally, near-future observations of neutrinoless double beta decay can also falsify our scenario, particularly for normal ordering of light neutrinos. While the minimal setup proposed here does not have any particle dark matter candidate, one could explore the possibility of forming primordial black hole dark matter due to Higgs fluctuations during inflation [81]. We leave exploration of such interesting possibilities to future works.