CTPU-PTC-21-21

Affleck-Dine Leptogenesis from Higgs Inflation

Neil D. Barrie,1, ∗ Chengcheng Han,2, † and Hitoshi Murayama3, 4, 5, ‡ 1Center for Theoretical Physics of the Universe, Institute for Basic Science (IBS), Daejeon, 34126, Korea 2School of Physics, Sun Yat-Sen University, Guangzhou 510275, China 3Department of Physics, University of California, Berkeley, CA 94720, USA 4Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa 277-8583, Japan 5Ernest Orlando Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA (Dated: June 8, 2021) We investigate the possibility of simultaneously explaining inflation, the neutrino masses and the baryon asymmetry through extending the Standard Model by a triplet Higgs. The neutrino masses are generated by the vacuum expectation value of the triplet Higgs, while a combination of the triplet and doublet Higgs’ plays the role of the inflaton. Additionally, the dynamics of the triplet, and its inherent lepton number violating interactions, lead to the generation of a lepton asymmetry during inflation. The resultant baryon asymmetry, inflationary predictions and neutrino masses are consistent with current observational and experimental results.

Introduction.—Despite the great successes of the [30–34]. However, to explain the existence of neutrino Standard Model (SM) at describing low energy scales, masses, only one triplet Higgs is required. We propose there remain many open problems that demand the ex- a mechanism by which successful Leptogenesis and istence of new physics. These issues include the mecha- neutrino mass generation can occur, with the addition nism for the epoch of rapid expansion in the early uni- of only a single triple Higgs. This triplet Higgs, in verse (inflation [1–5]), the origin of neutrino masses, and combination with the SM Higgs, will simultaneously give the source of the observed baryon asymmetry. Although rise to a Starobinsky-like inflationary epoch [35–41]. inflation is currently considered an integral component of standard cosmology, the exact nature of the inflation- Baryogenesis from a Complex Scalar— Following ary epoch remains unknown. The existence of non-zero [11], we consider a complex field φ with a global U(1) neutrino masses and the observed baryon asymmetry are charge Q and non-trivial potential V . Assuming it has a key pieces of evidence for physics beyond the SM. Each large initial field value in the early universe, φ will begin of these issues are tied to early universe physics, with any to oscillate once the Hubble parameter becomes smaller associated discoveries having significant implications for than its mass m. The dynamics of φ may give rise to both particle physics and cosmology. an asymmetry number density associated with its U(1) An exciting possibility to explore is whether each of charge, these unknowns could be explained within a unified † ˙ framework. There have been multiple attempts to do nφ = 2QIm[φ φ] , (1) so in the past, but it is difficult to provide solutions to all three problems simultaneously with a single addition which satisfies the equation of motion, to the SM. For example, the SM plus three right-handed neutrinos can explain the neutrino masses via the see-  ∂V  n˙ + 3Hn = Im φ . (2) saw mechanism [6–9] and generate the baryon asymme- φ φ ∂φ try through Leptogenesis [10], but not the inflationary arXiv:2106.03381v1 [hep-ph] 7 Jun 2021 sector. Inflationary Baryogenesis has been widely investi- To generate a non-vanishing U(1) asymmetry, the gated in the literature [11–22], but with few cases able to 1 potential V must contain an explicit U(1) breaking simultaneously explain each of the issues named above. term, which is generally assumed to be small. If the In this letter, we study the possibility that these three U(1) symmetry is composed of the global U(1) or problems can be solved through a simple extension to B U(1)L symmetries, a baryon number asymmetry can the SM by a single triplet Higgs. It has been known for a be generated prior to the Electroweak Phase Transition long time that with the addition of one triplet Higgs the (EWPT). In the following, we will consider a triplet baryon asymmetry cannot be generated through thermal Higgs which is doubly charged under the lepton number Leptogenesis, but rather requires the introduction of a symmetry U(1)L. There exist interaction terms between second triplet Higgs [25–29], or a right-handed neutrino the triplet and SM Higgs’ that explicitly violate lepton number, providing a possible path for Leptogenesis.

1 Inflation with Leptogenesisesis by the right-handed sneutrino has Model Framework— Consider the general La- been considered in [23, 24]. grangian describing the SM Higgs doublet H, and triplet 2

Higgs ∆, non-vanishing ∆0 VEV can be approximated in the limit M∆  vEW as, L 1 2 µν † √ = − MP R − f(H, ∆)R − g (DµH) (Dν H) 2 −g 2 0 µvEW h∆ i ' 2 , (8) µν † 2m −g (Dµ∆) (Dν ∆) − V (H, ∆) + LYukawa,(3) ∆ where the SM Higgs VEV is v = 246 GeV. Note that † † EW where f(H, ∆) = ξ H H + ξ ∆ ∆ + ..., M is the re- 0 0 H ∆ p the ∆ VEV is bounded by O(1) GeV > h∆ i & 0.05 duced Planck mass, and R is the Ricci scalar. The Higgs’ eV, in order to generate the observed neutrino masses, H and ∆ can be parameterized by, while ensuring yν is perturbative up to Mp. The upper √ 0  h+   ∆+/ 2 ∆++  bound on the ∆ VEV is derived from T-parameter H = , ∆ = √ , (4) h ∆0 −∆+/ 2 constraints determined by precision measurements [42]. where h and ∆0 are the neutral components of H and Trajectory of Inflation— The inflationary setting ∆ respectively. The LYukawa term contains not only will be induced by both Higgs’ through there non- the Yukawa interactions of the SM fermions, but also minimal couplings to gravity. These couplings act to a new interaction between the left-handed leptons and flatten the scalar potential for large field values. This the triplet Higgs ∆, form of inflationary mechanism has been utilized in stan- dard Higgs inflation, and results in a Starobinsky-like in- SM 1 ¯c LY ukawa = LYukawa − yijLi ∆Lj + h.c. (5) flationary epoch [35, 43–54]. We consider the following 2 non-minimal coupling, The neutrino mass matrix will be generated through 1 1 0 f(H, ∆) = ξ |h|2 + ξ |∆0|2 = ξ ρ2 + ξ ρ2 , (9) this interaction term, once ∆ obtains a non-zero vacuum H ∆ 2 H H 2 ∆ ∆ expectation value (VEV). Through this interaction we where we have utilized the polar coordinate parametriza- assign a lepton charge of QL = −2 to the triplet Higgs, tion h ≡ √1 ρ eiη, ∆0 ≡ √1 ρ eiθ. An inflation- thus opening the possibility that it may play a role in the 2 H 2 ∆ origin of the baryon asymmetry. ary framework consisting of two non-minimally coupled The potential for the Higgs’ V (H, ∆) is given by, scalars has been considered in the past, and found to ex- hibit a unique inflationary trajectory [52]. In the large 2 † † 2 2 † V (H, ∆) = −mH H H + λH (H H) + m∆Tr(∆ ∆) field limit the ratio of the two scalars is fixed by, +λ Tr(∆†∆)2 + λ (H†H)Tr(∆†∆) s 3 1 ρ 2λ ξ − λ ξ † 2 † † H ∆ H H∆ ∆ +λ2(Tr(∆ ∆)) + λ4H ∆∆ H ≡ tan α ' , (10) ρ∆ 2λH ξ∆ − λH∆ξH  λ + µ(HT iσ2∆†H) + 5 (HT iσ2∆†H)(H†H) M where the derivation of this trajectory is given in Section P IB. The inflaton can then be defined as ϕ which is related 0  λ5 T 2 † † + (H iσ ∆ H)(∆ ∆) + h.c. + ... , (6) to ρH and ρ∆ through, MP ρH = ϕ sin α, ρ∆ = ϕ cos α, where the terms given in the square brackets [...] violate ξ ≡ ξ sin2 α + ξ cos2 α . (11) lepton number, in which we have included dimension five H ∆ operators that are suppressed by MP . Even if they play We utilise the following Lagrangian, no role in low energy physics, they could be important L 1 1 1 √ = − M 2 R − ξϕ2R − gµν ∂ ϕ∂ ϕ during the early universe epochs of inflation and reheat- −g 2 P 2 2 µ ν ing. Note, the LHC has placed a lower limit on m∆ 1 of ∼ 800 GeV from searches for the associated doubly- − ϕ2 cos2 α gµν ∂ θ∂ θ − V (ϕ, θ) , (12) 2 µ ν charged Higgs. To simplify our analysis, we only focus on the neutral where components of ∆ and H, which have non-trivial VEVs. ˜ ! 1 2 2 λ 4 3 λ5 2 Thus, V can be simplified as follows, V (ϕ, θ) = m ϕ + ϕ + 2ϕ µ˜ + ϕ cos θ ,(13) 2 4 MP 0 2 2 2 0 2 4 0 4 V (h, ∆ )= −mH |h| + m∆|∆ | + λH |h| + λ∆|∆ | and  2 0 2 2 0∗ λ5 2 2 0∗ 2 2 2 2 2 +λH∆|h| |∆ | − µh ∆ + |h| h ∆ m = (m∆ cos α − mH sin α) , MP 4 2 2 4 0 λ = λH sin α + λH∆ sin α cos α + λ∆ cos α , λ ∗  + 5 |∆0|2h2∆0 + h.c. + ... , (7) 1 M µ˜ = − √ µ sin2 α cos α , P 2 2 where λ∆ = λ2 + λ3 , and λH∆ = λ1 + λ4 . All of the ˜ 1 4 0 2 3 λ5 = − √ (λ5 sin α cos α + λ sin α cos α) . (14) parameters must satisfy the stability condition, and the 4 2 5 3

Since the value of lepton asymmetry is dependent upon Lepton Number Density— Now that we have de- the motion of θ, we consider it to be a dynamical field. tailed the inflationary dynamics, consider the calculation In our model m  ϕ, meaning that the quartic poten- of the lepton number density, using Eq. (2), tial term dominates during the inflationary epoch. Thus, 2 ˙ 2 we require that the U(1) breaking is mainly sourced by nL = QLϕ θ cos α . (21) the dim-5 operator during inflation. If instead we con- From this relation, it is clear that we must determine the sidered the trilinear term to be the dominant source of dynamics of θ˙, and in particular its value at the end of lepton violation, it could begin to dominate the inflaton ˙ reheating. We shall assume that the initial θ0 will be zero, dynamics during the reheating epoch, breaking down the with a non-zero initial θ = θ0. It will become evident that efficient lepton asymmetry generating mechanism that the sign of the resultant asymmetry is dependent upon will be observed for the dim-5 operator. the choice of θ0. The scale µ will be chosen to be small We now translate the Lagrangian in Eq. (12) from the enough such that it does not contribute during inflation Jordan frame to the Einstein frame, utilizing the trans- and reheating, with the dim-5 term being the key source formations [55, 56], for lepton number violation. 2 2 2 2 Consider Eq. (20) at the beginning of inflation, g˜µν = Ω gµν , Ω = 1 + ξϕ /MP . (15) ˙ ! ˜ 5 To obtain a canonical kinetic term in the Einstein frame, ¨ f(χ) ˙ 2λ5 φ(χ) θ + 3H + θ + 4 sin θ = 0 , (22) we define the following field χ in terms of φ, f(χ) Mp f(χ)Ω (χ)

p p −1 p ˙ χ(ϕ) = 1/ ξ( 1 + 6ξ sinh ( ξ + 6ξ2ϕ) we have θ0 = 0 and θ0 6= 0. Under these initial condi- p p tions, θ will be dominated by the potential and acceler- −p6ξ sinh−1( 6ξ2ϕ/ 1 + ξϕ2) . (16) ation term, beginning to roll towards a potential mini- ˙ Subsequently, obtaining the final Einstein frame La- mum. Once a non-zero θ0 is generated a terminal veloc- grangian, ity is quickly reached, due to the balance of the Hubble friction and the slope of the potential; pushing the accel- L M 2 1 eration to zero. The resultant inflationary θ˙ is defined by √ = − P R − gµν ∂ χ∂ χ −g 2 2 µ ν the following equilibrium relation, 1 µν ˜ 2 q √ − f(χ)g ∂µθ∂ν θ − U(χ, θ) , (17) 2λ5Mp sin θ0 2 χ ˙ 3 Mp 2 3Hinfθinf ≈ 3 e − 1 . (23) ξ 2 cos2 α where During the inflationary epoch, this relation is only slowly ϕ(χ)2 cos2 α V (ϕ(χ), θ) f(χ) ≡ , and U(χ, θ) ≡ , (18) varying due to the slow roll of the χ field, and the Hubble Ω2(χ) Ω4(χ) µ rate remaining approximately constant, Hinf ∼ 2 . This field velocity is very small, and hence is consistent with with the χ potential replicating the Starobinsky form in our assumption of sub-dominance of the θ dynamics in the large field limit, the potential. Thus, θ also varies slowly over the course √ 2 3 2 2  2 χ  of inflation. Once inflation comes to an end (χ ∼ Mp), U (χ) = m M 1 − e 3 Mp , (19) inf 4 S p the oscillatory reheating epoch begins, and non-zero ac-

2 celeration is induced through the χ oscillations, λMp 13 where mS = 2 ' 3 · 10 GeV [40, 57]. From this 3ξ f˙(χ) Lagrangian we derive the following equations of motion, θ¨ + θ˙ ≈ 0 , (24) f(χ) 1 χ¨ − f 0(χ)θ˙2 + 3Hχ˙ + U = 0 , 2 ,χ resulting in the following approximate relation for θ˙, f 0(χ) 1 θ¨ + θ˙χ˙ + 3Hθ˙ + U = 0 . (20) 2λ˜ sin θ ϕ3 ,θ θ˙ ≈ 5 0 max , (25) f(χ) f(χ) reh 2 ξϕ2 3MpH cos α max 1 + M 2 We will assume that the model parameters are such that p the inflationary trajectory is negligibly effected by the where we have maintained the ϕ description due to the dynamics of θ, which will be justified below. Under this parameter dependence of nL. The value of ϕmax is assumption, the resultant inflationary observables are matched to the time dependent height of the ϕ poten- the same as the Starobinsky model, and are in excellent tial during the reheating, which decreases with each os- agreement with current observational constraints [57]. cillation due to Hubble friction. Note that, this relation See SectionI for details of the inflationary epoch and is approximately valid throughout both the inflationary observational predictions. and reheating epoch. Details regarding the derivation of the θ dynamics can be found in SectionII. 4

0 Utilising Eq (25), we can now determine the generated where λ5 ∼ λ5, and θ0  1 have been assumed. lepton number density at the end of the reheating epoch, ˜ Parameter Constraints— In our analysis, we have reh 2λ5 sin θ0 5 nL ≈ QL ϕreh . (26) made simplifying assumptions regarding the model pa- 3MpHreh rameters. Here we consider the consistency with suc- An approximate value of ϕreh at the end of reheating cessful generation of Baryogenesis, inflation and neutrino λ 4 can be obtained from the total energy density, 4 ϕreh ' masses. To keep the dim-5 term from dominating the in- 2 2 3Mp Hreh. Hence, we obtain, flationary dynamics, we require, χ ˜ − √ 0 15λ5 sin θ0 3 ˜ −11 5/2 6Mp reh 2 λ5  6.25 · 10 ξ e , (31) nL ≈ QL 5 (MpHreh) . (27) λ 4 where χ is fixed by the number of inflationary e-folds This relation agrees with numerical simulations within 0 N. This constraint places an upper bound on the baryon an order of magnitude. asymmetry,

Baryon Asymmetry Parameter— After reheat- ηB 10 obs  5 · 10 sin θ0 , (32) ing, the generated non-zero nL will be present in the ηB form of neutrinos, which will be redistributed by equi- librium electroweak sphalerons prior to the EWPT, with where the minimum N = 50 has been chosen, with larger 28 e-foldings suppressing this limit. To ensure the µ term is nB ' − 79 nL [58–61]. The reheating temperature in this scenario is expected to be sufficiently large for this to oc- negligible during reheating, it is necessary for, 13 14 cur due to the SM Higgs couplings, Treh ∼ 10 − 10 ˜ χ0 µ˜ λ5 −11p − √ GeV, as in Higgs inflation [44]. Now, we can obtain the   6.25 · 10 ξe 6Mp . (33) M ξ2 predicted baryon asymmetry parameter from Eq. (27), p assuming no significant entropy production after reheat- The dependence on the mixing angle, α, in this con- 2π2 3 ing s = 45 g∗Treh [62], straint cancels out in the two extremal cases. Giving 0 µ λ5 µ λ5 0 ˜ ˜ M  ξ2 and M  ξ2 , for ∆ and h domination re- ηB 10 |λ5 sin θ0| 21 |λ5 sin θ0| p p obs ' 1.7 · 10 5 ' 7 · 10 5 ,(28) spectively. η λ 4 ξ 2 B The full inflationary and reheating trajectory of α is obs −11 where ηB ' 8.5 · 10 is the observed baryon asymme- defined by [52], 2 4 2 π Treh try parameter [63], and Hreh ' 90 g∗ M 2 . Interestingly, v p u ϕ2 the final asymmetry has a simple parameter dependence u 2λ∆ − λH∆ + M 2 (2λ∆ξH − ξ∆λH∆) tan α = u p . (34) and is independent of the reheating temperature. Choos- t ϕ2 2λH − λH∆ + M 2 (2λH ξ∆ − ξH λH∆) ing the following reference values ξ ∼ 300, λ ∼ 4.5 · 10−5 p θ ∼ 0.05, and λ˜ ∼ 5 · 10−15, we obtain the observed 0 5 To ensure α variation does not play a role, we assume baryon asymmetry parameter ratio, η ∼ ηobs . Given B B ξ ∼ ξ , fixing the field trajectory through both epochs. that the U(1) breaking term couplings are both very H ∆ L If we consider the reference values, λ ' 0.1, λ ' small, it is natural to consider that they originate from H ∆ 4.5 · 10−5, ξ ∼ ξ ' 300, we obtain α ∼ 0.022. The a spurion field which carries U(1) charge. The U(1) H ∆ L L observed baryon asymmetry can be obtained for θ ∼ 0.1 breaking terms are then generated by requiring that the 0 and λ0 ∼ 2.8·10−11. In this case, we obtain the following spurion field obtains a VEV of O(1) TeV. 5 upper limit of µ 1 GeV. The observed baryon asymmetry and required small . The regions of the parameter space with a larger µ couplings can be obtained for various parameter choices term lead to the overproduction of the baryon asymme- due to the dependence upon the mixing angle α. There try. This could be countered by inefficiency in the lepton are two extremal scenarios, namely, the Triplet Higgs ∆0 number production, wash-out or dilution processes, or the SM Higgs h dominating the inflationary epoch. allowing successful Leptogenesis. Both cases lead to different baryon asymmetries through determining which form of dim-5 term is most important. Wash-out Processes— Since the inflationary epoch If small α is chosen, Triplet Higgs domination occurs, involves the Higgs, we expect a large reheating tempera- 0 2 ηB 10 |λ5θ0|α ture as a result of its SM couplings. This means that the obs ∼ 1.7 · 10 5 , (29) ηB λ 4 triplet will be thermalized at the end of reheating, and ∆ we must consider possible wash-out processes. Firstly, we or for α ∼ π/2, we obtain SM Higgs domination, require that the processes LL ↔ HH are not effective, η |λ θ ||α − π/2| B 10 5 0 y2µ2 obs ∼ 1.7 · 10 5 , (30) η 4 Γ = nhσvi ≈ < H|T =m∆ , (35) B λH m∆ 5

FIG. 1. The allowed region of parameter space which is consistent with our model assumptions (Grey), and avoids detrimental wash-out processes (Blue). This includes constraints from non-perturbativity of the neutrino Yukawa couplings (Black).

q 2 2 where H = π g∗ T . Note that the triplet Higgs also vacuum expectation value is generically small in order to 90 Mp generates the neutrino mass, through the following rela- explain the neutrino masses, so this constraint does not tion, have any phenomenological implications. LHC searches apply lower bounds on the masses of the triplet Higgs µv2 components, for example the current limit on the mass mν ' y 2 , (36) 2m∆ of the doubly charged Higgs is ∼ 800 GeV. This limit may be improved at the upgraded high luminosity LHC where mν should be at least the order of the largest neu- trino mass 0.05 eV. Combining this with the above re- or at future proton colliders [64]. The lepton flavor viola- 12 tion processes induced by the doubly-charged Higgs may lation, we obtain the mass limit m∆ < 10 GeV for 0 be probed for masses of m∆ ∼ O(1) TeV for h∆ i less mν = 0.05 eV. The other processes it is necessary to consider are than 1 eV [65]. LL ↔ ∆ and HH ↔ ∆. They must not co-exist, oth- Finally, let us comment on the possibility of Q-ball erwise the lepton number will be rapidly washed out. [66–69] formation during the inflationary and reheating However, to maintain the lepton asymmetry, the process epochs. As discussed in SectionIC, it is unlikely that LL ↔ ∆ must be efficient while HH ↔ ∆ is out of stable Q-balls can form. However, even if they are able equilibrium. This leads to the following requirement, to be formed during reheating, it would decay much ear- lier than the EWPT due to the relatively large neutrino ΓID(HH ↔ ∆)|T =m∆ < H|T =m∆ . (37) Yukawa couplings we require. In Figure1, we depict the region of parameter space Concluding Remarks— We have shown, that the for which the generated lepton number density is safe introduction of a single triplet Higgs to the SM provides from significant wash-out effects. The black region is ex- a simple framework in which inflation, neutrino masses, cluded by requiring perturbative neutrino Yukawa y cou- and the baryon asymmetry can all be explained. The neutral component of the triplet Higgs and SM Higgs plings up to the Planck scale Mpl (y . 1). However, a small Yukawa coupling y < 0.1 is preferred for the size of induce an inflationary epoch with predictions in excellent −5 quartic coupling we consider, λ∆ ' 4.5 × 10 , to avoid agreement with observations. This inflationary setting fine-tuning at the high energy scale. The grey region is provides the venue for which successful Leptogenesis excluded by requiring that the µ-term does not dominate takes place, through the motion of the complex phase 0 the U(1)L breaking dynamics during reheating. The blue of ∆ . Simultaneously, the neutrino masses can be region describes the parameters that lead to significant explained by the VEV of ∆0 and the associated neutrino wash-out of the lepton asymmetry, through the equilib- Yukawa coupling. rium process HH ↔ ∆ in the thermal bath. Note that there is an additional limit from precision measurements Acknowledgment.—CH acknowledges support from on the vacuum expectation value h∆0i, namely, it must the Sun Yat-Sen University Science Foundation. NDB is be less than a few GeV. In our model, the triplet Higgs supported by IBS under the project code, IBS-R018-D1. 6

The work of HM was supported by the Director, Office Phys. Rev. D 101, 043014 (2020), arXiv:1909.12300 [hep- of Science, Office of High Energy Physics of the U.S. De- ph]. partment of Energy under the Contract No. DE-AC02- [18] N. D. Barrie, A. Sugamoto, T. Takeuchi, and K. Ya- 05CH11231, by the NSF grant PHY-1915314, by the mashita, Higgs Inflation, Vacuum Stability, and Lepto- genesis, JHEP 08, 072, arXiv:2001.07032 [hep-ph]. JSPS Grant-in-Aid for Scientific Research JP20K03942, [19] C.-M. Lin and K. Kohri, Inflaton as the Affleck-Dine MEXT Grant-in-Aid for Transformative Research Areas Baryogenesis Field in Hilltop Supernatural Inflation, (A) JP20H05850, JP20A203, by WPI, MEXT, Japan, Phys. Rev. D 102, 043511 (2020), arXiv:2003.13963 [hep- and Hamamatsu Photonics, K.K. ph]. [20] M. Kawasaki and S. Ueda, Affleck-Dine inflation in su- pergravity, JCAP 04, 049, arXiv:2011.10397 [hep-ph]. [21] A. Kusenko, L. Pearce, and L. Yang, Postinflationary Higgs relaxation and the origin of matter-antimatter ∗ [email protected] asymmetry, Phys. Rev. Lett. 114, 061302 (2015), † [email protected] arXiv:1410.0722 [hep-ph]. ‡ [email protected], [email protected], [22] Y.-P. Wu, L. Yang, and A. Kusenko, Leptogenesis from Hamamatsu Professor spontaneous symmetry breaking during inflation, JHEP [1] R. Brout, F. Englert, and E. Gunzig, The Creation of the 12, 088, arXiv:1905.10537 [hep-ph]. Universe as a Quantum Phenomenon, Annals Phys. 115, [23] H. Murayama, H. Suzuki, T. Yanagida, and J. Yokoyama, 78 (1978). Chaotic inflation and baryogenesis by right-handed sneu- [2] K. Sato, First Order Phase Transition of a Vacuum and trinos, Phys. Rev. Lett. 70, 1912 (1993). Expansion of the Universe, Mon. Not. Roy. Astron. Soc. [24] H. Murayama, H. Suzuki, T. Yanagida, and J. Yokoyama, 195, 467 (1981). Chaotic inflation and baryogenesis in supergravity, Phys. [3] A. H. Guth, The Inflationary Universe: A Possible Solu- Rev. D 50, R2356 (1994), arXiv:hep-ph/9311326. tion to the Horizon and Flatness Problems, Phys. Rev. [25] E. Ma and U. Sarkar, Neutrino masses and leptogene- D 23, 347 (1981). sis with heavy Higgs triplets, Phys. Rev. Lett. 80, 5716 [4] A. D. Linde, A New Inflationary Universe Scenario: A (1998), arXiv:hep-ph/9802445. Possible Solution of the Horizon, Flatness, Homogeneity, [26] T. Hambye, E. Ma, and U. Sarkar, Supersymmetric Isotropy and Primordial Monopole Problems, Phys. Lett. triplet Higgs model of neutrino masses and leptogenesis, B 108, 389 (1982). Nucl. Phys. B 602, 23 (2001), arXiv:hep-ph/0011192. [5] A. Albrecht, P. J. Steinhardt, M. S. Turner, and [27] G. D’Ambrosio, T. Hambye, A. Hektor, M. Raidal, and F. Wilczek, Reheating an Inflationary Universe, Phys. A. Rossi, Leptogenesis in the minimal supersymmetric Rev. Lett. 48, 1437 (1982). triplet seesaw model, Phys. Lett. B 604, 199 (2004), [6] P. Minkowski, µ → eγ at a Rate of One Out of 109 Muon arXiv:hep-ph/0407312. Decays?, Phys. Lett. B 67, 421 (1977). [28] E. J. Chun and S. Scopel, Soft leptogenesis in Higgs [7] T. Yanagida, Horizontal gauge symmetry and masses of triplet model, Phys. Lett. B 636, 278 (2006), arXiv:hep- neutrinos, Conf. Proc. C 7902131, 95 (1979). ph/0510170. [8] S. L. Glashow, The Future of Elementary Particle [29] E. J. Chun and S. Scopel, Analysis of Leptogenesis in Physics, NATO Sci. Ser. B 61, 687 (1980). Supersymmetric Triplet Seesaw Model, Phys. Rev. D 75, [9] M. Gell-Mann, P. Ramond, and R. Slansky, Complex 023508 (2007), arXiv:hep-ph/0609259. Spinors and Unified Theories, Conf. Proc. C 790927, [30] T. Hambye and G. Senjanovic, Consequences of triplet 315 (1979), arXiv:1306.4669 [hep-th]. seesaw for leptogenesis, Phys. Lett. B 582, 73 (2004), [10] M. Fukugita and T. Yanagida, Baryogenesis Without arXiv:hep-ph/0307237. Grand Unification, Phys. Lett. B 174, 45 (1986). [31] P. J. O’Donnell and U. Sarkar, Baryogenesis via lepton [11] I. Affleck and M. Dine, A New Mechanism for Baryoge- number violating scalar interactions, Phys. Rev. D 49, nesis, Nucl. Phys. B 249, 361 (1985). 2118 (1994), arXiv:hep-ph/9307279. [12] M. P. Hertzberg and J. Karouby, Generating the Ob- [32] W.-l. Guo, Neutrino mixing and leptogenesis in type served Baryon Asymmetry from the Inflaton Field, Phys. II seesaw mechanism, Phys. Rev. D 70, 053009 (2004), Rev. D 89, 063523 (2014), arXiv:1309.0010 [hep-ph]. arXiv:hep-ph/0406268. [13] K. D. Lozanov and M. A. Amin, End of inflation, os- [33] S. Antusch and S. F. King, Type II Leptogenesis and cillons, and matter-antimatter asymmetry, Phys. Rev. D the neutrino mass scale, Phys. Lett. B 597, 199 (2004), 90, 083528 (2014), arXiv:1408.1811 [hep-ph]. arXiv:hep-ph/0405093. [14] M. Yamada, Affleck-Dine baryogenesis just after infla- [34] P.-H. Gu, H. Zhang, and S. Zhou, A Minimal Type II See- tion, Phys. Rev. D 93, 083516 (2016), arXiv:1511.05974 saw Model, Phys. Rev. D 74, 076002 (2006), arXiv:hep- [hep-ph]. ph/0606302. [15] K. Bamba, N. D. Barrie, A. Sugamoto, T. Takeuchi, and [35] A. A. Starobinsky, A New Type of Isotropic Cosmological K. Yamashita, Ratchet baryogenesis and an analogy with Models Without Singularity, Phys. Lett. 91B, 99 (1980), the forced pendulum, Mod. Phys. Lett. A33, 1850097 [,771(1980)]. (2018), arXiv:1610.03268 [hep-ph]. [36] B. Whitt, Fourth Order Gravity as General Relativity [16] K. Bamba, N. D. Barrie, A. Sugamoto, T. Takeuchi, Plus Matter, Phys. Lett. 145B, 176 (1984). and K. Yamashita, Pendulum Leptogenesis, Phys. Lett. [37] A. Jakubiec and J. Kijowski, On Theories of Gravita- B785, 184 (2018), arXiv:1805.04826 [hep-ph]. tion With Nonlinear Lagrangians, Phys. Rev. D37, 1406 [17] J. M. Cline, M. Puel, and T. Toma, Affleck-Dine inflation, (1988). [38] K.-i. Maeda, Towards the Einstein-Hilbert Action via 7

Conformal Transformation, Phys. Rev. D39, 3159 [59] V. A. Kuzmin, V. A. Rubakov, and M. E. Shaposhnikov, (1989). On the Anomalous Electroweak Baryon Number Non- [39] J. D. Barrow and S. Cotsakis, Inflation and the Confor- conservation in the Early Universe, Phys. Lett. 155B, mal Structure of Higher Order Gravity Theories, Phys. 36 (1985). Lett. B214, 515 (1988). [60] M. Trodden, Electroweak baryogenesis, Rev. Mod. Phys. [40] T. Faulkner, M. Tegmark, E. F. Bunn, and Y. Mao, Con- 71, 1463 (1999), arXiv:hep-ph/9803479 [hep-ph]. straining f(R) Gravity as a Scalar Tensor Theory, Phys. [61] A. Sugamoto, The neutrino mass and the monopole– Rev. D76, 063505 (2007), arXiv:astro-ph/0612569 [astro- Anti-monopole dumb-bell system in the SO(10) grand ph]. unified model, Phys. Lett. 127B, 75 (1983). [41] F. L. Bezrukov and D. S. Gorbunov, Distinguishing be- [62] L. Husdal, On Effective Degrees of Freedom in the Early tween R2-inflation and Higgs-inflation, Phys. Lett. B713, Universe, Galaxies 4, 78 (2016), arXiv:1609.04979 [astro- 365 (2012), arXiv:1111.4397 [hep-ph]. ph.CO]. [42] S. Kanemura and K. Yagyu, Radiative corrections to elec- [63] N. Aghanim et al. (Planck), Planck 2018 results. VI. Cos- troweak parameters in the Higgs triplet model and impli- mological parameters, (2018), arXiv:1807.06209 [astro- cation with the recent Higgs boson searches, Phys. Rev. ph.CO]. D 85, 115009 (2012), arXiv:1201.6287 [hep-ph]. [64] Y. Du, A. Dunbrack, M. J. Ramsey-Musolf, and J.-H. [43] F. L. Bezrukov and M. Shaposhnikov, The Standard Yu, Type-II Seesaw Scalar Triplet Model at a 100 TeV Model Higgs boson as the inflaton, Phys. Lett. B659, pp Collider: Discovery and Higgs Portal Coupling Deter- 703 (2008), arXiv:0710.3755 [hep-th]. mination, JHEP 01, 101, arXiv:1810.09450 [hep-ph]. [44] F. Bezrukov, D. Gorbunov, and M. Shaposhnikov, On [65] C. Han, D. Huang, J. Tang, and Y. Zhang, Probing the initial conditions for the Hot Big Bang, JCAP 0906, 029, doubly charged Higgs boson with a muonium to antimuo- arXiv:0812.3622 [hep-ph]. nium conversion experiment, Phys. Rev. D 103, 055023 [45] J. Garcia-Bellido, D. G. Figueroa, and J. Rubio, Pre- (2021), arXiv:2102.00758 [hep-ph]. heating in the Standard Model with the Higgs-Inflaton [66] S. R. Coleman, Q Balls, Nucl. Phys. B 262, 263 (1985), coupled to gravity, Phys. Rev. D 79, 063531 (2009), [Erratum: Nucl.Phys.B 269, 744 (1986)]. arXiv:0812.4624 [hep-ph]. [67] T. D. Lee and Y. Pang, Nontopological solitons, Phys. [46] J. L. F. Barbon and J. R. Espinosa, On the Natural- Rept. 221, 251 (1992). ness of Higgs Inflation, Phys. Rev. D79, 081302 (2009), [68] A. Kusenko, Solitons in the supersymmetric extensions arXiv:0903.0355 [hep-ph]. of the standard model, Phys. Lett. B 405, 108 (1997), [47] A. O. Barvinsky, A. Yu. Kamenshchik, C. Kiefer, A. A. arXiv:hep-ph/9704273. Starobinsky, and C. Steinwachs, Asymptotic freedom in [69] A. Kusenko, Small Q balls, Phys. Lett. B 404, 285 (1997), inflationary cosmology with a non-minimally coupled arXiv:hep-th/9704073. Higgs field, JCAP 0912, 003, arXiv:0904.1698 [hep-ph]. [48] F. Bezrukov and M. Shaposhnikov, Standard Model Higgs boson mass from inflation: Two loop analysis, JHEP 07, 089, arXiv:0904.1537 [hep-ph]. [49] G. F. Giudice and H. M. Lee, Unitarizing Higgs Inflation, Phys. Lett. B694, 294 (2011), arXiv:1010.1417 [hep-ph]. [50] F. Bezrukov, A. Magnin, M. Shaposhnikov, and S. Sibiryakov, Higgs inflation: consistency and generali- sations, JHEP 01, 016, arXiv:1008.5157 [hep-ph]. [51] C. P. Burgess, H. M. Lee, and M. Trott, Comment on Higgs Inflation and Naturalness, JHEP 07, 007, arXiv:1002.2730 [hep-ph]. [52] O. Lebedev and H. M. Lee, Higgs Portal Inflation, Eur. Phys. J. C71, 1821 (2011), arXiv:1105.2284 [hep-ph]. [53] H. M. Lee, Light inflaton completing Higgs inflation, Phys. Rev. D98, 015020 (2018), arXiv:1802.06174 [hep- ph]. [54] S.-M. Choi, Y.-J. Kang, H. M. Lee, and K. Yamashita, Unitary inflaton as decaying dark matter, JHEP 05, 060, arXiv:1902.03781 [hep-ph]. [55] R. M. Wald, General Relativity (Chicago Univ. Pr., Chicago, USA, 1984). [56] V. Faraoni, E. Gunzig, and P. Nardone, Conformal trans- formations in classical gravitational theories and in cos- mology, Fund. Cosmic Phys. 20, 121 (1999), arXiv:gr- qc/9811047 [gr-qc]. [57] Y. Akrami et al. (Planck), Planck 2018 results. X. Con- straints on inflation, (2018), arXiv:1807.06211 [astro- ph.CO]. [58] F. R. Klinkhamer and N. S. Manton, A Saddle Point So- lution in the Weinberg-Salam Theory, Phys. Rev. D30, 2212 (1984). 1

Affleck-Dine Leptogenesis from Higgs Inflation Supplemental Material

This Supplemental Material is organized as follows.

I. INFLATIONARY OBSERVABLES

In transforming from the Jordan to Einstein frame, the field ϕ no longer has a canonically normalised kinetic term. To rectify this, we make the following field redefinition,

q 2 2 2 2 dχ (6ξ ϕ /Mp ) + Ω = , (S1) dϕ Ω2

 ϕ 2 where Ω2 = 1 + ξ . There are three distinct regimes that can be seen in the evolution of the relation between Mp χ and ϕ, namely,  ϕ ϕ 1  for  (after reheating)  M M ξ  p p  r3  ϕ 2 1 ϕ 1 χ  ξ for   √ (reheating) ≈ 2 M ξ M ξ (S2) Mp  p " # p  r3 r3  ϕ 2 1 ϕ  2  ln Ω = ln 1 + ξ for √  (inflation)  2 2 Mp ξ Mp

Translating this to the effects on the scalar potential, we can see how the non-minimal coupling leads to a flattening of the potential in the large field limit. The scalar potential in each regime is signified by,  1 χ 1  λχ4 for  (after reheating)  4 M ξ  p  1 2 2 1 χ U(χ) ≈ mSχ for   1 (reheating) (S3) 2 ξ Mp  √ 2  3  2  χ  2 2 − 3 (χ/Mp)  mSMp 1 − e for 1  (inflation) 4 Mp Considering the inflationary potential parametrised by χ above, we see that this is exactly the Starobinsky infla- tionary potential. This scenario has the following characteristic slow roll parameters, 3 1  ' , and η ' , (S4) 4N 2 N where N is the number of e-foldings of expansion during the inflationary epoch. Thus, the spectral index and tensor to scalar ratio are given by 2 12 n ' 1 − , and r ' , (S5) s N N 2 respectively. The value of N is related to the field displacement of the inflaton, and approximately given by,

Z χf √ U 3 2 (χ /M ) N ≈ − dχ ≈ e 3 0 p , (S6) χ0 U,χ 4 where χ0 and χf are the initial and final inflaton field values respectively. For N > 50, it is required that χ0 & 5.25Mp. We can now compare the inflationary predictions to the current Planck results,

ns = 0.9649 ± 0042 (68%C.L.) , (S7)

r0.002 < 0.056 (95%C.L.) , (S8) for the Λ−CDM+r model. Comparing this with the Starobinsky inflation predictions, we see that they are extremely well fitted for the required number of inflationary e-folds 50 < N < 60 . 2

In addition, agreement with the scalar perturbations observed in the CMB places a requirement on the Starobinsky mass scale mS. The following parameter relation is necessitated,

λ ' 5 · 10−10 , (S9) ξ2

13 leading to the Starobinsky mass scale mS ' 3 · 10 GeV.

A. Isocurvature Fluctuations

Note that our baryon asymmetry depends on the initial θ0. The quantum fluctuations of θ during the inflationary epoch would induce iso-curvature fluctuations, which can be constrained by observation. The perturbations size is,

2 2 1 Hi hδθ i ≈ 2 2 , (S10) 4π χ0 and the isocurvature perturbations are defined by,

δT  6 Ω δn = − b b , (S11) T iso 15 Ωm nb where,

δnb = cot(θ0)δθ , (S12) nb so the predicted isocurvature perturbations are,

* 2+ 2 2 δT 9 Ωb Hi 2 ≈ 2 2 2 cot (θ0) . (S13) T 225π Ωm χ iso 0

The adiabatic fluctuations are defined as,

* 2+ 2 δT 1 Hi ≈ 2 2 , (S14) T 20 8π Mp  adi

Now we can calculate the ratio of these perturbations, * + * + δT 2 δT 2 α = / , (S15) non-adi T T iso adi 2 2 2 24 cot (θ0) Ωb Mp = 2 2 2 . (S16) 5N Ωm χ0 which has the following observational constraint,

−2 αnon-adi < 1.9 × 10 , (S17) from Planck data. From this we can derive a constraint on the initial θ,

2 θ0 > 4N
(S18) N ln 3

Ωb where we have taken ≈ 0.16 and assumed θ0 is small. This means that we cannot choose arbitrarily small θ0. Ωm 3

B. Derivation of Inflationary Trajectory in Higgs Portal Inflation

The inflationary context we utilise exhibits a flat direction fixed by the ratio of the two fields h and ∆0 . Following the derivation in Ref [52], we show the presence of this trajectory. Firstly, consider the field redefinitions of h and ∆0,

r  2 0 2  3 ξH h ξ∆(∆ ) h χ = Mp log 1 + 2 + 2 and κ = . (S19) 2 Mp Mp s

The kinetic terms of the Lagrangian in Eq. (3) are then given by,

1 1 κ2 + 1  M (ξ − ξ )κ 2 √p ∆ H µ Lkin = 1 + 2 (∂µχ) + 2 2 (∂µχ)(∂ κ) 2 6 ξH κ + ξ∆ 6 (ξH κ + ξ∆) 2 2 2 2 Mp ξH κ + ξ∆ 2 + 2 3 (∂µκ) . (S20) 2 (ξH κ + ξ∆)

We require large non-minimal couplings in our analysis, ξ ≡ ξH + ξ∆  1, so at leading order in 1/ξ the kinetic terms become,

2 2 2 2 1 2 Mp ξH κ + ξ∆ 2 Lkin = (∂µχ) + 2 3 (∂µκ) . (S21) 2 2 (ξH κ + ξ∆)

µ The large ξ limit suppresses the mixing term (∂µχ)(∂ κ) giving a canonically normalised χ, while also suppressing the kinetic term of κ. There are three key regimes for κ each with corresponding canonically normalized variable κ0,

0 κ ξ∆  ξH or κ → 0 , κ = √ , ξ∆ 0 1 ξH  ξ∆ or κ → ∞ , κ = √ , ξH κ 0 1 ξH = ξ∆ , κ = √ arctan κ . (S22) ξH Now consider the potential in terms of κ,

4 2 λH κ + λh∆κ + λ∆ 4 U = 2 2 Mp , (S23) 4(ξH κ + ξ∆) for large χ. This potential has the following minima, dependent upon the chosen coupling relations, s 2λ∆ξH − λh∆ξ∆ (1) 2λH ξ∆ − λh∆ξH > 0 , 2λ∆ξH − λh∆ξ∆ > 0 , κ = , 2λH ξ∆ − λh∆ξH

(2) 2λH ξ∆ − λh∆ξH > 0 , 2λ∆ξH − λh∆ξ∆ < 0 , κ = 0 ,

(3) 2λH ξ∆ − λh∆ξH < 0 , 2λ∆ξH − λh∆ξ∆ > 0 , κ = ∞ ,

(4) 2λH ξ∆ − λh∆ξH < 0 , 2λ∆ξH − λh∆ξ∆ < 0 , κ = 0, ∞ . (S24)

Case 2 and 3 concern inflationary scenarios dominated by ∆0 and h, respectively, while the inflaton in scenario 1 is characterised by a mixture of the two scalars. In scenario 1, the potential has the following minimum,

1 4λ λ − λ2 U = ∆ H h∆ M 4 , (S25) 2 2 p min (1) 16 λ∆ξH + λH ξ∆ − λh∆ξ∆ξH

3 2 2 which will be related to the Starobinsky mass scale through ∼ 4 msMp . In cases 2 and 3 we derive the usual single 2 4 2 4 field non minimally coupled inflationary potential, λ∆/(4ξ∆)Mp and λH /(4ξH )Mp , respectively. We require that the numerator of Eq. (S25) is positive to ensure that we do not have a negative vacuum energy at large field values. In 0 √ each case, the canonical field κ obtains a large mass of order Mp/ ξ. This mass is always greater than the Hubble rate during inflation, so the κ can be integrated out. 4

C. Possibility of Q-ball Formation

In Baryogenesis scenarios consisting of scalar fields carrying global charges, there exists the possibility of forming Q-balls. It is important to investigate their stability and regime of formation as they can have interesting phenomeno- logical implications. If the Q-balls are absolutely stable, they can be a component of the dark matter relic density, and potentially prevent successful Baryogenesis through sequestering the generated asymmetry from the thermal plasma. In our scenario, any Q-balls that are produced will have decay pathways into fermions through the neutrino Yukawa coupling. This means that as long as the decay time of these processes is such that the Q-balls decay before the EWPT, their should be no phenomenological implications of Q-ball formation during the inflationary and reheating epochs. Firstly, it must be determined whether Q-ball formation is possible in our model. To do this we investigate the 1 2 2 effective potential Ueff(χ, ω) = U(χ) − 2 ω χ , and obtain the range of ω frequencies for which bounce solutions exist. The upper and lower bounds on the frequency, ω+ and ω− are defined as follows,

2 00 2 2U(χ) ω+ = U (0) , and ω− = 2 , (S26) χ min where U(χ) is given in Eq. (18). From these relations we obtain the following requirement for Q-ball formation to occur,

00 2U(χ) U (0) > 2 . (S27) χ min

p 00 where we have m∆ ' U (0). The ω− appears when the potential just obtains two degenerate minima, one is at the origin and the other is at a large χ field value. Since the potential in Eq. (18) becomes flat for χ  Mp, all ω− > 0 will allow Q-ball formation due for sufficiently large χ. However, it would be expected that higher dimensional operators in φ should be present in the potential. In terms of the canonically normalised Einstein frame field χ, these higher order terms are exponentially χ √n 6 6 Mp φ enhanced by e , for a n + 4 dimensional term. A dim-6 operator of the form λ6 2 , would increase faster with Mp χ than the ω− dependent term, with the lower bound on ω dependent upon the choice of λ6 . Note that λ6 should be sufficiently small as to not distort the inflationary dynamics. An example is depicted in Figure S1, for which −6 −8 ω− ' 1.27 × 10 Mp is the lower bound for the choice λ6 = 10 . This value of ω− is much greater than the range of masses m∆ that we consider; see Figure.1. An approximate relation for when Q-Ball formation occurs can be found between the λ6 coupling and the frequency ω−, as follows,

m ω− − S λ ' 430 e ω− (S28) 6 Mp

From this relation, we see the exponential suppression that is required to allow Q-ball formation for small frequencies. −13 −3×1010 If we consider ω− ∼ m∆ ∼ 1 TeV, then the necessary λ6 must be incredibly tiny, less than ∼ 2 · 10 e . Thus, we can conclude that the formation of stable Q-balls does not occur in this model.

2.× 10 -10

1.5× 10 -10

1.× 10 -10

5.× 10 -11

0 0 5 10 15 20

FIG. S1. Potential for bounce solutions. All the parameters are in Planck unit. 5

II. DYNAMICS OF θ √ Consider the equation of motion for θ given in Eq. (20). In the inflationary setting, which assumes ϕ  Mp/ ξ and equivalently χ  Mp, the f(χ) components become, √ 2 χ 2 Mp − 2 f(χ) ' cos α (1 − e 3 Mp ) , (S29) ξ which comes from the transformation properties described in Eq. (S2), √ √ χ 2  χ  2 2 2 Mp 2 Ω ' e 3 Mp and ϕ ' e 3 Mp − 1 . (S30) ξ

In what follows, α will be assumed to be constant for simplicity, which requires ξH ∼ ξ∆ during the reheating epoch. Thus, the θ equation of motion during inflation becomes, ¨ ˙ θ + 3Hθ + Cinf sin θ ' 0 , (S31) where slow rolling of χ has been assumed, and, ! χ  √ 1/2 2 q √  √  2Mp − √ − 2 χ c˜5 Mp 2 χ − 2 χ C = µe˜ 6Mp 1 − e 3 Mp + e 3 Mp − 1 1 − e 3 Mp , (S32) inf 1/2 2 ξ cos α Mp ξ which upon simplifying, 2 ˜ q √ 2Mp λ5 2 χ C ' e 3 Mp − 1 , (S33) inf ξ3/2 cos2 α We will assume µ is sufficiently small that it is unimportant during the reheating epoch. Consequently, the µ term will also not contribute during inflation due to the exponential suppression relative to the dim-5 term.

A. Beginning of Inflation

˙ At the beginning of inflation, we assume that θ0 = 0, such that we have potential dominated evolution, ¨ θ ≈ Cinf sin θ , (S34) if we additionally assume that θ0 is relatively small such that we can make the approximation sin θ0 ≈ θ0, then we can find a simple solution for this equation of motion, √ √ Cinft p ˙ p Cinft θ ' θ0(e − Cinft) and θ ' θ0 Cinf(e − 1) . (S35) This explains why we see the rapid increase in θ˙ at the beginning of inflation. The rise is approximately exponential, and as such this behavior is cut-off very quickly as the friction term becomes important. The transition from this exponential growth to the next regime is embodied in the plateau structure seen in Figure S2.

B. Inflationary Epoch

As inflation proceeds from this early stage, the motion of θ can be described by the equilibrium between the friction and potential terms, ˙ 3Hθ ≈ Cinf sin θ . (S36) Given that this equilibrium is reached rapidly, and the change in θ over that period was small, despite the exponential growth, we have, C θ θ˙ ≈ inf 0 . (S37) 3H

This equilibrium is maintained throughout the inflationary epoch, but it should be noted that both Cinf and H each have χ dependence and decrease as inflation proceeds. Thus, the relation is maintained, but decreases slowly over the √ q 2 χ period of inflation due to the e 3 Mp − 1 term. The change from the beginning to the end of inflation is dependent on the initial χ0, but the final equilibrium value is independent of the length of the inflationary epoch. 6

1× 10 8

 5× 10 7 θ

1× 10 7

5× 10 6

1.× 10 -15 5.× 10 -15 1.× 10 -14 5.× 10 -14 t

FIG. S2. The numerically solved θ˙ (Blue) vs the analytical solution (yellow) in Eq. (S35), where t is coordinate time for which the scale factor is given by a = eHt.

1× 10 8

8× 10 7

 6× 10 7 θ 4× 10 7

2× 10 7

0 0 5.× 10 -12 1.× 10 -11 1.5× 10 -11 t

FIG. S3. Evolution of θ˙ during inflation, where t is coordinate time for which the scale factor is given by a = eHt.

C. End of Inflation and Reheating Epoch

As inflation comes to an end, the large χ assumption begins to fail and the slow roll parameters are violated, meaning we can no longer ignore the contribution from the χ dependent friction term, ! f˙(χ) θ¨ + 3H + θ˙ + C(χ) sin θ = 0 . (S38) f(χ)

We will now consider the equation of motion in terms of the ϕ parametrisation.   2ϕ ˙ 2λ˜ ϕ3 θ¨ + 3H + θ˙ + 5 sin θ = 0 . (S39)   2  2 ξϕ2 ξϕ Mp cos α 1 + ϕ 1 + 2 M 2 Mp p

This can be rewritten in terms of the nL,  2ξϕ  2λ˜ ϕ5 n˙ + 3H − ϕ˙ n + Q 5 sin θ = 0 . (S40) L 2 L L ξϕ2 1 + ξ(ϕ/MP ) Mp 1 + 2 Mp where we can ignore the second term in bracket when ϕ becomes small enough. Now consider the behaviour of ϕ and ϕ˙ during each oscillation. There are two key points, when it reaches the maximum in its potential ϕmax andϕ ˙ = 0, and when it is at the minimum ϕ = 0 andϕ ˙ =ϕ ˙ max. Take the first of these points, ˜ 3 ˙ 2λ5 ϕmax 3Hθ∗ + 2 2 sin θ∗ ≈ 0 . (S41) M cos α ξϕmax p 1 + 2 Mp ˙ ˙ This fixing of θ at the peak of ϕ with each oscillation leads to the progressive decrease of θ and subsequently nL throughout reheating. This behaviour is demonstrated in Figure S4. 7

10-8 10-7

10-11 10-11 10-14 10-15 10-17 10-19 10-20 10-23 0 100 200 300 400 500 350 355 360 365 370 375 380 τ τ

FIG. S4. Evolution of 3Hθ˙ (yellow) and the potential term (blue), where τ = mS t/2 .

The second of these equations is most important for following the motion of θ˙ through each oscillation, having the following solution,

ϕ 2 θ˙ ≈ max θ˙ , (S42) ϕ ∗ which translates to a constant nL, with the initial equation of motion equivalent ton ˙ L ≈ 0. This almost constant evolution of nL during this epoch is depicted in Figure S5 for τ & 350.

10-5

nL 10-8

10-11 0 200 400 600 800 1000 τ

2 FIG. S5. Numerical solved lepton number density, where τ = mS t/2 . Note, nL is in Planck units and has been scaled by mS .

From this we can write down the resulting approximation for nL, ˜ 5 4λ5 sin θ0 ϕmax nL ≈ 2 , (S43) 3M H ξϕmax p 1 + 2 Mp where we have identified θ∗ = θ0, which is approximately the initial θ at the onset of reheating. A comparison of this approximation to the numerical solution is given in Figure S6. Note, that this solution is approximately valid throughout the whole of the inflationary and reheating epoch. 8

0.01 1.4 1.2

10-5 1.0 R 0.8 nL L 10-8 0.6 0.4

10-11 0.2 0 200 400 600 800 1000 0 200 400 600 800 1000 τ τ

FIG. S6. In (a) we compare the numerical (yellow) and analytical solutions (blue), and provide the ratio RL in (b), where 2 τ = mS t/2 . Note, nL is in Planck units and has been scaled by . mS

III. BEHAVIOUR OF µ TERM

In the above analysis, we have assumed that the cubic lepton number violating term is subdominant throughout the inflationary and reheating epochs. This is ensured by the requirement that,

˜ χ0 µ˜ λ5 − √ −11p 6Mp  2  6.25 · 10 ξe . (S44) Mp ξ This assumption is made to allow analytical and numerical analysis of the θ dynamics and resultant lepton number density. If the parameter µ is increased above this limit, the lepton number density enters a highly oscillatory regime during the reheating epoch, once it dominates over the ϕ quartic interaction. The dependence of the oscillation amplitude in nL on µ can be observed in Figure S7. As µ increases, the amplitudes are enhanced until they begin to cross into negative values. This means that for µ values that violate the above constraint we begin to obtain results that are highly oscillatory around zero for the lepton number density. In such cases, it becomes difficult to determine the predicted lepton number density, as it will be highly dependent upon the end point of reheating. Note that the maximum of the oscillations still retains a similar behaviour to that in the small µ case. During the inflationary stage, the cubic interaction is suppressed relative to the quartic and Dim-5 term due to the non-minimal interaction. The µ term can be chosen such that it would effect the inflationary trajectory despite this suppression. This behaviour is shown in Figure S8, where increasing µ leads to significantly different behaviour during the inflationary stage. For even larger µ values the inflationary trajectory will no longer be described by a Starobinsky-like cosmology. 9

(a) (b) 1.×10 -8 1.×10 -8 5.×10 -9 5.×10 -9

|nL| |nL| 1.×10 -9 1.×10 -9 5.×10 -10 5.×10 -10

320 340 360 380 320 340 360 380 τ τ

(c) (d) 1.×10 -8 1.×10 -8 5.×10 -9 5.×10 -9

|nL| |nL| -9 1.×10 -9 1.×10 -10 5.×10 -10 5.×10

320 340 360 380 320 340 360 380 τ τ

-7 (e) 10 10-8 (f) 10-8 10-9 -9 10-10 10 |nL| |nL| -10 10-11 10 10-12 10-11 -13 10 10-12 10-14 320 340 360 380 320 340 360 380 τ τ 10-6 (g) 10-5 (h) 10-7 10-6 10-8 10-7 n n | L| 10-9 | L| 10-8 10-10 10-9 10-11 10-10 320 340 360 380 320 340 360 380 τ τ

FIG. S7. The lepton number density during reheating for varying cubic coupling term µ, where τ = mS t/2 . The cubic µ −6 −5 −4 −4 −3 ˜ −10 couplings chosen are ˜ = (a) 0, (b) 5 · 10 , (c) 5 · 10 , (d) 10 , (e) 5 · 10 , (f) 5 · 10 , (g) 0.05, (h) 0.5 where λ5 = 10 λ5 and ξ = 300 . The blue and yellow curves denote positive and negative lepton number densities respectively. Note, nL is in Planck units and has been scaled by 2 . mS 10

0.001 (b) 10-4 10-5 n -7 L |nL| 10 10-7

-10 10-9 10

0 100 200 300 400 500 0 100 200 300 400 500 τ τ

0.01 0.01 (c) (d)

-5 10 10-5

|nL| |nL| 10-8 10-8

10-11 10-11 0 100 200 300 400 500 0 100 200 300 400 500 τ τ

(e) 0.01

10-4 |nL| 10-6

10-8

0 100 200 300 400 500 τ

FIG. S8. The lepton number density during inflation and reheating for varying cubic coupling term µ, where τ = mS t/2 . µ ˜ −10 The cubic couplings chosen are ˜ = (a) 0, (b) 0.05, (c) 0.5, (d) 1, (e) 5 , where λ5 = 10 and ξ = 300 . The blue and yellow λ5 curves denote positive and negative lepton number densities respectively. Note, nL is in Planck units and has been scaled by 2 . mS