PHYSICAL REVIEW D 102, 055009 (2020)

New mechanism for matter- asymmetry and connection with

Arnab Dasgupta ,1 P. S. Bhupal Dev ,2 Sin Kyu Kang,1 and Yongchao Zhang 2,3 1School of Liberal Arts, Seoul-Tech, Seoul 139-743, Korea 2Department of Physics and McDonnell Center for the Space Sciences, Washington University, St. Louis, Missouri 63130, USA 3Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA

(Received 17 December 2019; revised 16 July 2020; accepted 31 August 2020; published 14 September 2020)

We propose a new mechanism for generating matter-antimatter asymmetry via the interference of tree- level diagrams only, where the imaginary part of the Breit-Wigner propagator for an unstable mediator plays a crucial role. We first derive a general result that a nonzero CP-asymmetry can be generated via at least two sets of interfering tree-level diagrams involving either 2 → 2 or 1 → n (with n ≥ 3) processes. We illustrate this point in a simple TeV-scale extension of the Standard Model with an inert Higgs doublet and right-handed neutrinos, along with an electroweak-triplet scalar field, where small Majorana neutrino masses are generated via a combination of radiative type-I and tree-level type-II seesaw mechanisms. The imaginary part needed for the required CP-asymmetry comes from the trilinear coupling of the inert doublet with the triplet scalar, along with the width of the triplet scalar mediator. The real part of the neutral component of the inert doublet serves as a candidate. The evolutions of the dark matter relic density and the baryon asymmetry are intimately related in this scenario.

DOI: 10.1103/PhysRevD.102.055009

I. INTRODUCTION Nanopoulos-Weinberg theorem [8] (see also Refs. [9,10]). Similar interference effects between tree and loop-level The observed asymmetry between the number densities diagrams have also been considered for generating the of baryonic matter and antimatter in the universe [1] cannot baryon asymmetry from 2 → 2 scattering [11–14] or be accounted for in the Standard Model (SM). Therefore, annihilation [15–20] processes. a viable baryogenesis mechanism is an essential ingredient In this paper, we argue that the interference between tree for the success of any beyond SM scenario. The dynamical and loop-level diagrams is not the only way to generate a generation of baryon asymmetry requires three necessary nonzero asymmetry from out-of-equilibrium heavy particle (but not sufficient) Sakharov conditions [2] to be satisfied: decays/annihilations. We propose a new mechanism where (i) baryon number (B) violation, (ii) C and CP violation, it suffices to consider two sets of interfering diagrams at and (iii) out-of-equilibrium dynamics. A well-known the tree-level only. This can be achieved through tree-level mechanism that satisfies these conditions involves the 2 → 2 scattering or 1 → n (with n ≥ 3) decay processes 1 → 2 decays of a heavy particle, such as in grand unified mediated by unstable particles. Then the CP-asymmetry theory (GUT) baryogenesis [3,4] or leptogenesis [5] (for can be generated from the complex couplings and the reviews, see e.g., Refs. [6,7]). To obtain a baryon/lepton propagator widths [see Eq. (3) below], which could even be asymmetry in these 1 → 2 decay scenarios, one must resonantly enhanced when the center-of-mass energy is consider the interference between tree- and loop-level close to the propagator mass. diagrams. Furthermore, some particles in the loop must To illustrate our new mechanism, we will consider a be able to go on-shell, and the interaction between the simple realistic model at TeV-scale, namely, combining the intermediate on-shell particles and the final-state particles scotogenic model [21] (with an inert Higgs doublet and should correspond to a net change in baryon/lepton number right-handed neutrinos) and the type-II seesaw framework for the net asymmetry to be nonzero; this is known as the – 2 [22 26] (with an SUð ÞL-triplet scalar) for small Majorana neutrino masses. For our parameter choice of the model, the CP-asymmetry originates from the complex trilinear cou- Published by the American Physical Society under the terms of pling of the inert Higgs doublet with the triplet scalar, along the Creative Commons Attribution 4.0 International license. with the imaginary part of the triplet scalar mediator width. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, Stabilized by a discrete Z2 symmetry, the neutral compo- and DOI. Funded by SCOAP3. nent of the inert doublet scalar plays the role of a TeV-scale

2470-0010=2020=102(5)=055009(12) 055009-1 Published by the American Physical Society DASGUPTA, DEV, KANG, and ZHANG PHYS. REV. D 102, 055009 (2020) weakly interacting dark matter (DM) candidate. Adopting three benchmark points (BPs), we illustrate that the gen- eration of baryon asymmetry via leptogenesis in this model is intimately correlated with the DM relic density. It is also found that successful leptogenesis sets limits on the triplet vacuum expectation value (VEV) vΔ and the trilinear scalar coupling jμηΔj (see Fig. 7)—a feature that could be directly (a) (b) tested at future high-energy colliders. The rest of this paper is organized as follows. Our general mechanism of attaining the CP asymmetry without having explicit loop diagrams is explained in Sec. II. The scotogenic plus type-II seesaw model is introduced in Sec. III. The lepton asymmetry generation in this model (c) (d) is detailed in Sec. IV. The collider signatures are touched upon in Sec. V. Our conclusions are given in Sec. VI. The FIG. 1. Generic topologies for tree-level 2 → 2 subprocesses scalar potential and scalar masses are collected in the that can give rise to a nonzero lepton or baryon asymmetry. Here Appendix A. The relevant thermal cross sections used in i1;2 and f1;2 are respectively the initial and final states, and m1;2 our analysis are given in Appendix B. The thermal cross are the masses of two different mediators. section relevant for the asymmetry in the narrow-width approximation is given in Appendix C.

II. THE GENERAL MECHANISM We propose that a net lepton or baryon asymmetry can be generated from the interference effect of two sets of tree- level decay or scattering diagrams with the same initial and (a) (b) final states, as long as the following two conditions are satisfied: FIG. 2. Generic topologies for tree-level 1 → 3 subprocesses (i) There is a net nonzero lepton or baryon number that can give rise to a nonzero lepton or baryon asymmetry. Here i 1 between the initial and final states. and f1;2;3 are respectively the initial and final states, and m1;2 are (ii) At least one set of decay or scattering amplitudes is the masses of two different mediators. complex such that the squared amplitudes for particles and antiparticles are different, giving rise stand for the rest of the subamplitudes. The corresponding ¯ ¯ ¯ ¯ 2 to a net CP-asymmetry. amplitude for the conjugate process i1i2 → f1f2 is The simplest way to achieve this is through 2 → 2 scatterings (see Fig. 1)or1 → 3 decays (see Fig. 2) ¯ M ¼ðC M1 þ C M2ÞW : ð2Þ involving two different intermediate state particles, with 1 2 the outgoing particles (or decay products) carrying a net Comparing the modular squares of the amplitudes, we nonzero baryon or lepton number. Without loss of general- obtain the CP-asymmetry factor ity, we focus here on the simplest 2 → 2 scattering case with the initial states i1, i2 and with only two subprocesses 2 ¯ 2 δ ≡ jMj − jMj for the final states f1, f2 (here i1;2 and f1;2 generically stand 2 for bosons and/or fermions), mediated by intermediate- ¼ −4Im½C1C2Im½M1M2jWj ; ð3Þ state particles of mass m1 and m2, respectively. The total i1i2 → f1f2 amplitude for the process can be written as where Im½C1C2 is the imaginary part coming from the couplings, which is required to be nonzero for CP M ¼ðC1M1 þ C2M2ÞW; ð1Þ violation, and Im½M1M2 incorporates the imaginary part from the subamplitudes M1;2, which is reminiscent where Ci contain only the couplings, W contains the wave M of the imaginary part coming from the interference of tree functions for the incoming and outgoing particles and i and loop-level diagrams in the 1 → 2 decay scenario. Equation (3) is a general result applicable to 2 → 2 1In principle, this condition can be somewhat relaxed if we scatterings, as well as 1 → n decays (for n ≥ 3). consider flavor-dependent asymmetries, with zero net lepton or baryon number in the final state, as in flavored leptogenesis (for recent reviews, see e.g., Refs. [27,28]). For simplicity, here we 2Note that the CPT theorem only guarantees the equivalence ¯ ¯ ¯ ¯ will not consider such flavor-dependent effects. of rates for i1i2 → f1f2 and f1f2 → i1i2.

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In the tree-level 2 → 2 processes shown in Fig. 1,we largely enhanced at the s-channel resonance, 2 have only one source for the complex subamplitudes, i.e., s − m1 ≃ m1Γ1. namely, the finite widths of the mediators. One may argue (iii) If both subprocesses are in the t-oru-channel that the finite width is also a loop-induced effect for [cf. Fig. 1(c)+(d)], then the width terms in the unstable mediators, since it is related to the imaginary part denominator of Eq. (5) can be neglected, i.e., of self-energy [29,30]. However, the crux of our new mechanism is that we only require a nonzero width, Im½M1M2 whereas the 1 → 2 decay case needs both nonzero width 2 2 A1A2½ðx1 − m Þm2Γ2 − ðx2 − m Þm1Γ1 and interference between tree and loop (self-energy and/or ≃ 1 2 : − 2 2 − 2 2 ð7Þ vertex correction) diagrams. In general, the subamplitudes ðx1 m1Þ ðx2 m2Þ M1 2 for the processes in Fig. 1 can be written as ; In this case, the CP-asymmetry is suppressed by the ratio m Γ =ðx − m2Þ with i, j ¼ 1,2. A i i j j M j ; In the next section, we will consider an explicit example j ¼ − 2 Γ ð4Þ xj mj þ imj j that realizes the possibility (ii) discussed above. with j ¼ 1,2,xj ¼ s, t, u the Mandelstam variables, mj III. AN EXAMPLE and Γ respectively the mediator masses and widths, and A j j To illustrate our new mechanism in a minimal realistic some arbitrary real parameters. The imaginary component extension of SM, we consider an amalgamation of the of the product of subamplitudes appearing in Eq. (3) can scotogenic model [21] and type-II seesaw [22–26] mech- then be written as anisms at TeV-scale. For the purpose of scotogenic mecha- nism, an inert SUð2Þ -doublet scalar η ≡ ðηþ; η0Þ and three − 2 Γ − − 2 Γ L A1A2½ðx1 m1Þm2 2 ðx2 m2Þm1 1 right-handed neutrinos (RHNs) N (with i ¼ 1, 2, 3) are Im½M1M ¼ ; i 2 − 2 2 2Γ2 − 2 2 2Γ2 2 ½ðx1 m1Þ þ m1 1½ðx2 m2Þ þ m2 2 introduced. To implement type-II seesaw, an SUð ÞL-triplet Δ ≡ Δþþ Δþ Δ0 η ð5Þ scalar ð ; ; Þ is added. The inert doublet and the three RHNs Ni are odd under a discrete Z2 symmetry, 2 while all other particles are even. In this model, we assume which is nonvanishing as long as x1 − m m2Γ2 ≠ x2 − ð 1Þ ð the RHNs, as well as the triplet scalar components, are 2 Γ m2Þm1 1 in the numerator. With the imaginary part of the heavier than the η scalars, so any asymmetry generated by C C ≠ 0 couplings Im½ 1 2 , we can then produce a nonzero the conventional decays of N and/or Δ is not relevant at the asymmetry [cf. Eq. (3)]. This general argument holds, temperature scale of interest. An added advantage of our irrespective of the specific subprocesses or the model mechanism is that the lightest neutral component η0 plays details. the role of DM [21], with its relic density intimately 2 → 2 For the tree-level case in Fig. 1, we can have three connected to the lepton asymmetry. A nonminimal cou- distinct possibilities for the two subprocesses to realize pling of the inert doublet to gravity can also successfully M M ≠ 0 Im½ 1 2 in Eq. (5): accommodate inflation [33,34]. (i) If both subprocesses are in the s-channel [cf. Fig. 1 The relevant Yukawa couplings are given by the (a)+(b)], one just needs to replace x1;2 by s in Eq. (5). Lagrangian In this case, the CP-asymmetry factor δ in Eq. (3) can be largely enhanced in the vicinity of resonance N † Δ C −L ¼ Y η˜ LαN þ Y Lα ΔLβ þ H:c:; ð8Þ − 2 ≃ Γ 1 Y iα i αβ (s), with s mi mi i (with i ¼ , 2), similar to the enhancement effect in resonant leptogenesis [31,32]. with L ≡ ðν; lÞ being the SM lepton doublet, C the charge (ii) If one of the subamplitudes is in the s-channel and conjugation operator, η˜ ¼ iσ2η (σ2 being the second Pauli the other one in the t-oru-channel [cf. Fig. 1(a)+(d) matrix), α; β ¼ e, μ, τ the lepton flavor indices, and i ¼ 1, or (b)+(c)], one can safely neglect the imaginary part 2, 3 the RHN mass indices. For simplicity, we assume there for the t-oru-channel propagator. For concreteness, is no mixing nor CP violation in the RHN sector. The mass we take M1 as the s-channel and M2 as the parameter μηΔ in the scalar potential x-channel (x ¼ t or u) amplitude. In this case, Eq. (5) can be simplified to † † V ⊃ μηΔη Δ η˜ þ H:c: ð9Þ

A1A2m1Γ1 M M ≃− is chosen to be complex, which is crucial for the CP- Im½ 1 2 2 2 2 2 2 ; ð6Þ ½ðs − m1Þ þ m1Γ1ðx − m2Þ asymmetry [cf. Eq. (3)]. The full scalar potential and the resultant physical scalar masses are collected in Appendix A. which is proportional to the s-channel mediator In this setup, the neutrino mass is generated from both width Γ1. Here also the CP-asymmetry could be loop-level scotogenic and tree-level type-II seesaw

055009-3 DASGUPTA, DEV, KANG, and ZHANG PHYS. REV. D 102, 055009 (2020) mechanisms, which are induced respectively by the Yukawa couplings YN and YΔ given in Eq. (8):

N T N Δ mν ¼ðY Þ ΛY þ Y vΔ; ð10Þ where Λ is an effective loop-suppressed RHN mass scale, given by [21,35] FIG. 3. Feynman diagrams for the 2 → 2 scattering processes 2 2 ηη → LαLβ in our example model. m mη m Λ Ni R ln Ni ii ¼ 16π2 2 − 2 2 X mN mη mη i R R δ 4 μ N Δ N T 2 2 ¼ Im½ ηΔfY Y ðY Þ gii mη m − I Ni : i 2 − 2 ln 2 ð11Þ mN mη mη sm mΔΓΔ 1 1 i I I Ni ; × − 2 2 2 Γ2 − 2 þ − 2 ð15Þ ðs mΔÞ þ mΔ Δ t mN u mN We have assumed that the RHNs do not mix with each i i other, therefore Λ is a diagonal matrix. The Yukawa Γ couplings in Eq. (10) are related to the where Δ is the triplet scalar width. With the width in the 0 numerator of Eq. (15), the asymmetry in this simple model data, Λ and the triplet VEV hΔ i¼vΔ as follows: can be viewed as the interference effect of RHN-mediated 1=2 −1 2 1=2 † tree-level diagram in the right panel of Fig. 3 and the one- YN ¼ F ðΛ = Omˆ ν U Þ ; ð12Þ iα I PMNS iα loop correction to the triplet propagator in the s-channel, where the s-channel process corresponds to the following Δ −1 ˆ † Yαβ ¼ FIIvΔ ðUPMNSmνUPMNSÞαβ; ð13Þ subprocesses where mˆ ν ¼fmν ;mν ;mν g the diagonal neutrino mass 1 2 3 ηη → ΔðÞ; ΔðÞ → LL ð16Þ eigenvalues, and UPMNS the PMNS lepton mixing matrix. In Eq. (12) we have used the Casas-Ibarra parametrization [36] for the coupling YN, with O an arbitrary orthogonal which incorporate the conventional one-loop decay of (on- shell) triplet scalars into a lepton pair. matrix. FI and FII are the fractions of contributions to neutrino mass matrix from the radiative scotogenic and From an effective field theory (EFT) perspective, if we mΔ m tree-level type-II seesaw mechanisms respectively, with integrate out the heavy mediator masses and Ni , the 1 width effect has to be consistently incorporated into the FI þ FII ¼ . effective coupling. This can be understood by inserting a self-energy diagram in the Δ-propagator in Fig. 3 and then IV. GENERATION OF LEPTON ASYMMETRY integrating out the resulting two Δ propagators to obtain a AND DARK MATTER RELIC DENSITY loop-level effective coupling that includes the width of Δ, As stated above, the matter asymmetry is generated from giving rise to a non-vanishing asymmetry. the interference effect between two tree-level diagrams, In Eq. (15), the imaginary part of the product of the Δ which are shown in Fig. 3 for our scotogenic type-II seesaw Yukawa couplings YN, Y [cf. Eq. (8)] and the trilinear 3 μ model with mΔ ≳ 2mη. In particular, we analyze the 2 → 2 coupling ηΔ [cf. Eq. (9)] can be parametrized using ΔL ¼ 2 scattering processes Eqs. (12) and (13) as follows:

ηη → LαLβ; ð14Þ μ N Δ N T Im½ ηΔfY Y ðY Þ gii −1 μ Λ−1=2O ˆ 2OTΛ−1=2 0 ¼ FIFIIvΔ Im½ ηΔf mν g : ð17Þ which include η η → lα lβ , η η → lα νβ and ii 0 0 η η → νανβ. These processes can be mediated by an s- In general, the orthogonal O-matrix might be complex, thus channel triplet scalar Δ, and also by RHNs Ni in the t- and u-channels, as shown in Fig. 3. The effective CP-asym- potentially contributing to the imaginary part in Eq. (17). 2 → 2 metry factor [cf. Eq. (6)] is given by It is interesting that part of the same processes in Eq. (14) containing η0 contributes also to the (co)annihi-

3 lation of DM particles. In this sense, the time evolutions of In the opposite regime where mΔ < 2mη, an asymmetry can be generated via the interference between tree-level Δ → LL DM relic density and the lepton asymmetry are correlated, decay mediated by the Yukawa coupling YΔ and the vertex as we will see below. The freeze-out mechanism for DM is correction to this decay induced by two η’s and a N mediated by identical to the standard inert doublet case [37,38], where N the couplings μηΔ and Y respectively. DM annihilates into the SM particles.

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A. Boltzmann equations however, the dominant process for DM freeze-out is ηη → þ − 2 The cogenesis of DM relic density and leptonic asym- W W via SUð ÞL gauge interaction, whereas the dominant ηη → metry is governed by the coupled Boltzmann equations washout process is LL via the Yukawa couplings; therefore, we can satisfy the freeze-out condition for suitable choice of the Yukawa couplings without requiring any of the dYη −s 2 eq 2 ¼ ½ðYη − ðYη Þ Þhσviðηη → SMSMÞ; ð18Þ dz HðzÞz final states to be massive.

dYΔL s 2 eq 2 B. Numerical results ¼ ½ðYη − ðYη Þ Þhσviδðηη → LLÞ dz HðzÞz The three BPs used in our numerical analysis of the − 2 eq 2 σ ηη → Ω 2 YΔLYl rηh vitotð LLÞ baryon asymmetry YΔB and DM relic density DMh eq ¯ [cf. Fig. 5] are collected in Table I. These are obtained − 2YΔ Yη hσviðηL → ηLÞ; ð19Þ L by implementing our model in SARAH 4 [42] and after ðeqÞ ðeqÞ checking consistency with all lepton flavor violating where z ¼ mη=T, Y ≡ n =s are the normalized num- i i constraints using SPheno 4.0.4 [43]. The observed value of ber densities (in equilibrium) for the particles i (s being the DM relic density is obtained in each case by fixing the − eq eq entropy density), YΔL ¼ YL YL¯ , rη ¼ Yη =Yl , and λ −λ0 Higgs-DM quartic couplings Hη ¼ Hη in Eq. (A1) for a rffiffiffiffiffiffiffiffiffiffiffiffi given mass scale μη as shown in Table I. This assumption is 8π3 2 g mη taken in order to ensure the mass of the charged scalar is HðzÞ¼ 90 2 ð20Þ z MPl always higher than the neutral scalar masses (i.e., ηR and ηI). All the quartic couplings in Eq. (A1) not listed in this with MPl the Planck scale and g the number of relativistic table are set to be zero. degrees of freedom at temperature T. Here the hσvi’s We solve the Boltzmann equations (18) and (19) numeri- are the thermally averaged annihilation/scattering rates: cally for the three representative BPs in Table I. We assume σ ηη → h við SM SMÞ is the DM annihilation rate, and F ¼ F ¼ 1=2 in Eqs. (12) and (13), i.e., equal contribu- σ ηη → I II h vitot;δð LLÞ are respectively given by tions from scotogenic and type-II seesaw to neutrino masses. This choice maximizes the CP-asymmetry in Eq. (17), σ ηη → ≡ σ ηη → h vitotð LLÞ h við LLÞ subject to keeping other factors the same. In addition, the O O ˆ 2OT ˆ 2 þhσviðηη → L¯ L¯ Þ; ð21Þ matrix is taken to be identity, so that mν ¼ mν, and the mass parameter μηΔ is assumed to be purely imaginary in

hσviδðηη → LLÞ ≡ hσviðηη → LLÞ Eq. (17). In doing so, the contribution for the asymmetry − hσviðηη → L¯ L¯ Þ: ð22Þ TABLE I. Three BPs for the numerical analysis. All the quartic The expressions for all the thermal cross sections appearing couplings in Eq. (A1) not listed in this table are set to be zero. Here Δmη0 ¼ mη − mη is the mass splitting between the two in Eqs. (18) and (19) are collected in Appendices B and C. R I scalars η and η . Evaluating the Boltzmann equations above, one can R I obtain the lepton asymmetry YΔLðzÞ, which is then con- BP1 BP2 BP3 verted to baryon asymmetry YΔB ¼ −ð28=51ÞYΔL [39] via vΔ 1 keV 1 keV 1 keV the standard electroweak sphaleron processes [40] at the μη 600 GeV 1 TeV 1.5 TeV sphaleron transition temperature T ¼ð131.72.3Þ GeV sph μHΔ 33.6 keV 93.5 keV 210 keV [41]. In an analogous way, one can also calculate the μηΔ 15i GeV 7.1i GeV 6i GeV evolution of the DM density Yη from Eq. (18) and get the mN1 6 TeV 10 TeV 15 TeV Ω 2 2 755 108 final relic abundance DMh ¼ . × Yηðmη=GeVÞ mN2 6.6 TeV 11 TeV 16.5 TeV ≃ 20 m at DM freeze-out temperature Tf mη= . N3 7.2 TeV 12 TeV 18 TeV We note here that our mechanism for generating the lepton mη0 600 GeV 1 TeV 1.5 TeV Δ 0 asymmetry and DM relic density simultaneously is similar to mη 506 keV 300 keV 200 keV the WIMPy baryogenesis mechanism [17]. A crucial cri- mη 606 GeV 1 TeV 1.5 TeV terion for achieving successful asymmetry in both cases is mΔ0 1.2 TeV 2 TeV 3 TeV that the washout of the asymmetry processes must freeze-out mΔ 1.2 TeV 2 TeV 3 TeV mΔ 1.2 TeV 2 TeV 3 TeV before the freeze-out of the DM annihilation processes, i.e., λ σ ηη → σ ηη → H 0.253 0.253 0.253 h vitotð LLÞ < h við SMSMÞ.InRef.[17],both λHη 0.19 0.56 0.91 washout and DM freeze-out are governed by the same final λ0 −0 19 −0 56 −0 91 Hη . . . states; therefore, one of the final states is required to be 00 −5 −5 −5 λ η 1 × 10 1 × 10 1 × 10 massive to satisfy the above freeze-out condition. In our case, H

055009-5 DASGUPTA, DEV, KANG, and ZHANG PHYS. REV. D 102, 055009 (2020) coming from the standard decay of N’s will not come into play as it requires nontrivial orthogonal matrix O. The RHNs are taken to be much heavier than the η particles to avoid the wash-out of lepton asymmetry from the inverse decay processes Lαη → Ni. For the BPs we take, the mass splitting − η η0 mηI mηR (with I the imaginary part from ) is larger than 100 keV scale, such that the direct detection constraints for inelastic scattering of DM with nucleons [44–46] can be evaded. Ω 2 The evolutions of the DM relic density DMh and the baryon asymmetry YΔB are evaluated using micrOMEGAs 5.0 [47] and the results are presented in Fig. 4. The time evolutions of DM relic density for BP1, BP2, and BP3 are denoted, respectively, by the red solid, dashed and dot- dashed curves. For each choice of the DM mass, the maximal contribution to baryon asymmetry comes in the vicinity of the s-channel resonance in Fig. 3, i.e., when Δ 2mη → mΔ. In Fig. 4 we have fixed the Δ-mediator mass at FIG. 5. Net baryon number density YΔB as function of the - μ the resonance point and have satisfied the required baryon mediator mass, for three different values of j ηΔj (with the argument of π=2) for each of the three BPs in Table I. The solid asymmetry by fixing the trilinear coupling μηΔ as shown in horizontal black line indicates the central value of the observed Table I. obs −11 baryon number density YΔ ¼ð8.718 0.004Þ × 10 . The dependence of baryon asymmetry on the absolute B μ value of the trilinear coupling j ηΔj and the triplet scalar effect in Eq. (15) when the center-of-mass energy is close to mass mΔ is shown in Fig. 5. For each of the three BPs given the triplet scalar mass mΔ. The enhancement of baryon in Table I, we show the variation of YΔ as function of the B asymmetry at the resonance mΔ ≃ 2mη can be clearly seen μ mediator mass for different values of ηΔ, as shown by the in Fig. 5. Note that in the narrow-width approximation, solid, dashed and dot-dashed lines with green, blue and red, the thermal-averaged cross section hσviδ asymptotically which correspond respectively to BP1, BP2, and BP3. In reaches a finite value [cf. Eq. (C3)] which determines the the numerical calculations we have included the resonance height of the peak in Fig. 5, whereas the sharp drop right after the resonance is due to the Boltzmann suppression in Eq. (C3). As expected in Eq. (17), increasing the magnitude of μηΔ results in a larger baryon asymmetry. We have fixed jμηΔj for each BP in Table I to be the minimum value for which the observed baryon asymmetry can be obtained at the resonance. Note that for larger trilinear couplings, one can also achieve the observed asymmetry away from the resonance point. This plateau region is due to a mutual cancellation between the s- and t-channel contributions in Fig. 3, which in turn lowers the wash-out rate, and as a result, slightly enhances the baryon asymmetry. The dependence of baryon asymmetry on the triplet VEV vΔ can be seen in the plots in Fig. 6. Here the solid lines are for the evolution of YΔB as function of T, and the dotted, dashed and dot-dashed lines denote the evolu- tion of the thermally averaged cross sections hσvi, respec- ηη → ηη → tively for the processes ð LLÞδ, ð LLÞtot and ðηη → SMSMÞ. The red, blue, green and magenta lines are FIG. 4. Net baryon number density YΔ , DM density Y and B DM respectively for the BPs with jμηΔj¼1 GeV, 10 GeV, neq σv =H as functions of temperature T, for the three BPs in DMh iδ 100 GeV and 1 TeV, and the left and right panels are Table I. The solid horizontal black line indicates the observed −11 respectively for the VEVs of vΔ ¼ 0.1 eV and vΔ ¼ Yobs 8 718 0 004 10 baryon number density ΔB ¼ð . . Þ × , and the 100 dashed horizontal black line indicates the central value of eV. Other parameter are set to be the same as for the observed DM relic density Ωobs h2 ¼ 0.120 0.001 [1]. BP1. When the VEV vΔ gets larger, the Yukawa coupling DM Δ ∝ −1 The vertical solid line represents the central value of the Y vΔ will be smaller, so the wash-out effect will be 131 7 2 3 sphaleron freeze-out temperature Tsph ¼ð . . Þ GeV. suppressed and the resultant baryon asymmetry YΔB will be

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eq σ ηη → ηη → ηη → FIG. 6. Net baryon number density YΔB and n h vi=H for the processes ð LLÞδ, ð LLÞtot and ð SMSMÞ as functions of temperature T, for the values of jμηΔj¼1 GeV, 10 GeV, 100 GeV and 1 TeV for the BP1 in Table I, with vΔ ¼ 0.1 eV (left) and vΔ ¼ 100 eV (right). The solid horizontal black line indicates the central value of the observed baryon number density obs 8 718 0 004 10−11 YΔB ¼ð . . Þ × , and the dashed horizontal black line indicates the central value of the observed DM relic density Ωobs 2 0 120 0 001 DMh ¼ . . . The vertical solid line represents the central value of the sphaleron freeze-out temperature 131 7 2 3 Tsph ¼ð . . Þ GeV.

Δ −1 larger, as can be seen by comparing the left and right panels when vΔ is very small, the Yukawa coupling Y ∝ vΔ is so of Fig. 6. large that the washout effect from LL → ηη is too strong. On μ Δ Varying vΔ and j ηΔj and fixing all other relevant the other hand, when vΔ is very large the coupling Y is too parameters as in Table I, we obtain the allowed regions of small to produce sufficient baryon asymmetry. Although one vΔ and jμηΔj which simultaneously satisfy the observed would expect that the suppression of YΔ can be compensated obs 8 718 0 004 10−11 baryon number density YΔB ¼ð . . Þ × and Ωobs 2 0 120 0 001 the observed DM relic density DMh ¼ . . ,as shown by the shaded regions in Fig. 7 for the three BPs.4 From this parameter scan, we find both lower and upper bounds on the triplet VEV and the trilinear coupling for each BP:

BP1∶ 40 eV≲vΔ ≲1.5 MeV; 0.3 GeV≲jμηΔj≲80 GeV;

BP2∶ 20 eV≲vΔ ≲1.2 MeV; 0.3 GeV≲jμηΔj≲380 GeV;

BP3∶ 10 eV≲vΔ ≲20 MeV; 0.3 GeV≲jμηΔj≲1.2 TeV: ð23Þ

The lower limit on jμηΔj is set by the YΔB requirement, while the upper limit is set by the relic density requirement, which is governed by the ηη → WW process that is independent of vΔ for most part of the parameter space. As for the limits on vΔ,

4 To find the maximum allowed parameter space for YΔB,we take the maximal CP phase for μηΔ and keep all the points with ≥ obs μ μ YΔB YΔB. Since the CP-asymmetry depends on Im( ηΔ), FIG. 7. Allowed parameter space of vΔ and j ηΔj for the three cf. Eq. (15), we can always adjust the CP phase accordingly BPs simultaneously satisfying the observed relic density and μ obs μ for a fixed j ηΔj to get YΔB ¼ YΔB. Therefore, we only show j ηΔj baryon asymmetry, with all other relevant parameters set to be the and not its phase in Fig. 7. same as in Table I.

055009-7 DASGUPTA, DEV, KANG, and ZHANG PHYS. REV. D 102, 055009 (2020) by increasing μηΔ [cf. Fig. 3 (left)], this in turn increases the background for ll þ W þ MET is expected to be higher N mass splitting between ηR and ηI, which decreases Y than that without MET, and a detailed simulation is needed to [cf. Fig. 3 (right)] to maintain the neutrino mass. Due to estimate the prospects of RHN signals in this model at future this reason there is a sharp upper bound on vΔ. colliders. The leptogenesis mechanism in this model can thus be directly tested at future lepton colliders by measuring the VI. CONCLUSION Yukawa couplings of doubly charged scalars [48], which is intimately related to vΔ and the active neutrino masses and We have proposed a new technique to generate the mixings via Eq. (13). This model can also be tested at future observed baryon asymmetry of the universe only via the hadron and lepton colliders by searches of the beyond SM tree-level interference of 2 → 2 scatterings or 1 → 3 particles in the scotogenic and type-II seesaw models, as decays. The CP-violating asymmetry comes from the detailed in Sec. V. absorptive part of the propagators. We have illustrated this mechanism explicitly in a simple scotogenic model with V. COLLIDER SIGNALS type-II seesaw, in which the asymmetry is generated in the ΔL ¼ 2 processes ηη → LL mediated by s-channel triplet The BPs chosen for our model to simultaneously explain scalars and t or u-channel RHNs. The neutrino masses baryogenesis, DM and neutrino masses involve TeV-scale receive contributions from both scotogenic and type-II beyond SM scalars and heavy RHNs which can be directly seesaw mechanisms. The real part of the neutral component tested at current and future high-energy colliders. For of the inert doublet η serves as a DM candidate. As shown instance, the neutral and charged triplet scalars can be in Fig. 4 the baryon asymmetry and DM relic density are directly searched for at the Large Hadron Collider (LHC) correlated and both can be matched to their observed values [49,50], as well as in future High-Luminosity LHC (HL- – for (sub-)TeV inert doublet and triplet masses. The baryo- LHC) [51 55], 100 TeV hadron colliders [54,56] and genesis requirements impose both lower and upper bounds lepton colliders [57], or indirectly probed in the high- on the triplet VEV, as shown in Fig. 7, with testable precision low-energy experiments like MOLLER [58].Itis consequences at future colliders. important to note here that the allowed range of vΔ in Fig. 3 correspond to the Yukawa couplings 5 × 10−9 ≲ YΔ ≲ 3 × ACKNOWLEDGMENTS 10−3 which give rise to prompt dilepton signals in the Δ decays for the triplet masses considered in Table I. We thank Pei-Hong Gu for comments on the draft. A. D. The charged η scalars can be produced in association thanks the Particle Theory Group at Washington University with the neutral DM particle η0 through the W boson, i.e., in St. Louis for warm hospitality, where this work was pp → W → ηη0 → η0η0WðÞ [59]. The inert doublet initiated. Y. Z. thanks Kaladi Babu for useful discussions, scalars can also be produced from their couplings to the and the High Energy Theory Group at Oklahoma State SM Z boson via pp → ηRηIj or the SM Higgs through University for warm hospitality, where this work was pp → η0η0j (with j being an energetic jet) [60]. The inert completed. The work of P. S. B. D. and Y. Z. was supported doublet sector can then be constrained by the mono-W in part by the U.S. Department of Energy under Grant [61,62] and monojet [63,64] searches at the LHC. Our No. DE-SC0017987 and in part by the MCSS. This work model can in principle be distinguished from the pure was also supported in part by the Neutrino Theory Network scotogenic or pure type-II seesaw model at colliders using Program under Grant No. DE-AC02-07CH11359. The both inert doublet and triplet scalar signatures. work of S. K. K. and A. D. was supported by the NRF In the scotogenic model, the RHNs do not mix directly of Korea Grants No. 2017K1A3A7A09016430, and with the light active neutrinos. For our chosen BPs, the heavy No. 2017R1A2B4006338. RHNs are heavier than the inert doublet and can only be produced at high-energy colliders from the off-shell decay APPENDIX A: SCALAR POTENTIAL η → l → lη∓ðÞ → lη0 ∓ðÞ α Ni, followed by Ni α α W . AND SCALAR MASSES Due to the Majorana of the heavy RHNs, we can get same-sign dileptons, as in the Keung-Senjanović process The most general scalar potential for the SM Higgs [65], but now with significant missing transverse energy doublet H ≡ ðHþ;H0Þ, inert doublet η ≡ ðηþ; η0Þ and (MET) due to the presence of η0 in the final state. The SM triplet Δ ≡ ðΔþþ; Δþ; Δ0Þ is given by

−μ2 † μ2 η†η − μ2 Δ†Δ μ ˜ †Δ μ η†Δ†η˜ V ¼ HðH HÞþ ηð Þ ΔTr½ þð HΔH H þ ηΔ þ H:c:Þ λ † 2 λ η†η 2 λ Δ†Δ 2 λ0 Δ†ΔΔ†Δ λ †η 2 λ0 † η†η þ HðH HÞ þ ηð Þ þ ΔfTr½ g þ ΔTr½ þ HηjH j þ HηðH HÞð Þ λ00 †η 2 λ † Δ†Δ λ0 †ΔΔ† λ η†η Δ†Δ λ0 η†ΔΔ†η þ HηððH Þ þ H:c:Þþ HΔðH HÞTr½ þ HΔTr½H Hþ ηΔð ÞTr½ þ ηΔTr½ ; ðA1Þ

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˜ where H ¼ iσ2H , η˜ ¼ iσ2η and the mass parameters 1 2 2 2μ2 λ 2 λ λ0 2 μ 0 Δ m ¼ ð η þ ηv þð ηΔ þ ηΔÞvΔÞ: ðA9Þ H;η;Δ > so that both H and obtain nonvanishing η 2 H 0 0 VEVs, i.e., hH i¼v ≃ 246 GeV and hΔ i¼vΔ. To gen- erate the baryon asymmetry from the tree-level interference effect, the coupling μηΔ is assumed to be complex, and all APPENDIX B: THERMAL CROSS SECTIONS other parameters in Eq. (A1) are assumed to be real. The physical masses for the neutral and charged scalars The general expression for the thermally averaged cross can be obtained from minimization of the scalar potential in section for the processes in Eqs. (18) and (19) is [47] Eq. (A1). Note that here the doublet η is odd under the Z2 symmetry, which is essential to provide a DM candidate, 1 σv i1i2 → f1f2 and does not mix the SM Higgs and the triplet. In particular, h ið Þ¼2 2 2 Tmi1 K2ðmi1 =TÞmi2 K2ðmi2 =TÞ when we take the first-order derivative of the potential with Z Z ∞ 1 1 jMj2 respect to the VEVs v and vΔ, the solutions of the tadpole ffiffiffi × p pi1i2 pf1f2 2 2 −1 32π equations for fμ ; μ g are given by sin s H Δ pffiffiffi × K1ð s=TÞds d cos θ; ðB1Þ 1 pffiffiffi μ2 λ 2 − 2 2μ λ 2 H ¼ 2 ð Hv HΔvΔ þ HΔvΔÞ; ðA2Þ where T is the temperature, Ki the modified Bessel functions of order i, M is the amplitude for the process 1 ffiffiffi μ 2 2 p HΔ 2 2 i1i2 → f1f2, and μΔ ¼ λΔvΔ − 2 2 v þ λHΔv : ðA3Þ 2 vΔ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ≡ λðs; m2;m2Þ=s; ðB2Þ μ2 μ2 ij 2 i j After replacing f H; Δg in the scalar potential, the mass matrix for the real scalars reads s ≡ max½ðm þ m Þ2; ðm þ m Þ2; ðB3Þ ffiffiffi in i1 i2 f1 f2 2 p λ v − 2μ Δv 2 2 2 0 ffiffiffiH H λðx; y; zÞ ≡ x þ y þ z þ 2xy þ 2xz þ 2yz: ðB4Þ M ¼ p μ 2 ; ðA4Þ − 2μ pHffiffiΔv HΔv 2 vΔ σ ηη → σ η ¯ → η In Eq. (19), h vitotð LLÞ and h við L LÞ are from which we can get two mass eigenvalues for the real respectively for the amplitudes: component from H0 and the scalar Δ0 which is the real part R ˆ 2 2 μ 2 Δ0 μ ∼ O 100 2 mν FI j ηΔj s of . In the case of HΔ ð Þ keV, the two CP-even jM ðηη → LLÞj ¼ tot v2 ðs− m2 Þ2 þ m2 Γ2 scalar masses turn out to be Δ Δ Δ Δ X 2 2 2 mˆ ν 2 1 1 2 þ F 2 m s 2 þ 2 2 2 2 μHΔv II Λ Ni − − ≃ λ ≃ ffiffiffi ii t mN u mN mh Hv ;mΔ0 p ; ðA5Þ i i i R 2 vΔ 2 X 2 F F jμηΔjðs− mΔÞ mˆ ν þ I II m s − 2 2 2 Γ2 Λ Ni with the first one (h) identified as the SM-like Higgs boson. ðs mΔÞ þ mΔ Δ iivΔ i The masses of the pseudoscalar and the charged scalars 1 1 from the triplet are respectively × þ ; ðB5Þ t − m2 u −m2 Ni Ni 2 μHΔ 2 2 mΔ ¼ pffiffiffi ðv þ 4vΔÞ; ðA6Þ 2 2 2 2 I mˆ ν F jμηΔj ðmη − tÞ 2vΔ jMðηL¯ → ηLÞj2 ¼ I v2 ðt − m2 Þ2 Δ Δ μ 1 X 2 2 2 2 2 HΔ 2 2 mˆ ν F m s m ¼ m ¼ pffiffiffi þ λ0 ðv þ 2v Þ: ðA7Þ II Ni Δ Δ 4 HΔ Δ þ 2 2 2 2 2 2vΔ Λ ðs − m Þ þ m Γ i ii Ni Ni Ni μ 2 − X ˆ 2 Finally the masses for real scalar η , pseudoscalar η and FIFIIj ηΔjðmη tÞ mν R I þ 2 2 mN s − Λ i the charged scalars η from the Z2-odd doublet η are ðt mΔÞ i iivΔ respectively s − m2 Ni × − 2 2 2 Γ2 : ðB6Þ 1 ðs mN Þ þ mN N 2 2 0 00 2 i i i mη ¼ ½2μη þðλ η þ λ η λ ηÞv R;I 2 H H H pffiffiffi The cross section hσviðηη → SMSMÞ in Eq. (18) can be λ ∓ 2 2 μ þð ηΔvΔ j ηΔjÞvΔ; ðA8Þ found in [37,38], with “SM SM” referring to the all the

055009-9 DASGUPTA, DEV, KANG, and ZHANG PHYS. REV. D 102, 055009 (2020) possible channels involving the quarks, leptons, scalar and hσviδðηη → LLÞ Z ffiffiffi gauge bosons in the SM. ∞ p z pηηpLL sz ds ffiffiffi K1 δ ¼ 128π2 5 2 p APPENDIX C: MAXIMUM ASYMMETRY mηK2ðzÞ sin s mη 4π r4 In Eq. (19) hσviδðηη → LLÞ is for the amplitude given in ≃ P Δ Γ Γ 2 μ˜ 2 2 Δ→ηη Δ→LL Eq. (15) which produces the lepton asymmetry. In the mη ηΔ α;βjYαβj narrow-width approximation, the asymmetry in Eq. (15) at X K1 rΔz μ N Δ N T ð Þ the resonance point simplifies to × Im½ ηΔfY Y ðY Þ gii 2 ; ðC3Þ r K2ðzÞ X i Ni δ ≃ 4 μ N Δ N T Im½ ηΔfY Y ðY Þ gii μ˜ ηΔ μηΔ Δ η i where ¼ =m , ri ¼ mi=m , pij and sin are defined in Eqs. (B2) and (B3) respectively, and 2 1 1 × sm δðs − mΔÞ þ ; ðC1Þ Ni t − m2 u − m2 1 Ni Ni Γ μ˜ 2 Δ→ηη ¼ 16π ηΔpηη; ðC4Þ where we have used 1 2 Γ Γ →0 ΓΔ→ ¼ jYαβj p : ðC5Þ mΔ Δ ⟶Δ=mΔ πδ − 2 LL 16π LL 2 2 2 2 ðs mΔÞ: ðC2Þ ðs − mΔÞ þ mΔΓΔ ItcanbeseenfromEq.(C3) that the thermally averaged Plugging the amplitude (C1) back into Eq. (B1) and asymmetry asymptotically reaches a finite value at the integrating over s, we get resonance, which determines the height of the peak in Fig. 5.

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