Leptogenesis Constraints on $B-L$ breaking Higgs Boson in TeV Scale Seesaw Models

In the type-I seesaw mechanism for neutrino masses, there exists a $B-L$ symmetry, whose breaking leads to the lepton number violating mass of the heavy Majorana neutrinos. This would imply the existence of a new neutral scalar associated with the $B-L$ symmetry breaking, analogous to the Higgs boson of the Standard Model. If in such models, the heavy neutrino decays are also responsible for the observed baryon asymmetry of the universe via the leptogenesis mechanism, the new seesaw scalar interactions with the heavy neutrinos will induce additional dilution terms for the heavy neutrino and lepton number densities. We make a detailed study of this dilution effect on the lepton asymmetry in three generic classes of seesaw models with TeV-scale $B-L$ symmetry breaking, namely, in an effective theory framework and in scenarios with global or local $U(1)_{B-L}$ symmetry. We find that requiring successful leptogenesis imposes stringent constraints on the mass and couplings of the new scalar in all three cases, especially when it is lighter than the heavy neutrinos. We also discuss the implications of these new constraints and prospects of testing leptogenesis in presence of seesaw scalars at colliders.


Introduction
The type-I seesaw mechanism [1][2][3][4][5] provides a simple way to understand the smallness of light neutrino masses in terms of the heavy right-handed neutrino (RHN) Majorana masses and their mixing with light neutrinos. Two pertinent questions arise here: (i) What is the seesaw (or the RHN mass) scale? (ii) Is there a symmetry (global or local) associated with the origin of the RHN masses that go into the seesaw formula? As far as the first question is concerned, in a phenomenological bottom-up approach, the seesaw scale could be anywhere between eV and 10 14 GeV [6,7]. However, due to observational bias (the 'streetlight effect'), we are particularly interested in the TeV scale seesaw models, which can be effectively probed in a plethora of current and future experiments at both energy [8,9] and intensity [10,11] frontiers. Low-scale seesaw can also be theoretically motivated from Higgs naturalness perspective [12][13][14]. As far as any associated symmetry behind the seesaw mechanism is concerned, B − L symmetry is an obvious and natural possibility [15,16], whose breaking leads to the heavy Majorana masses in the seesaw formula. Once this point of view is adopted, regardless of whether the symmetry is global or local, there should exist a Higgs field that breaks the symmetry and the real part of that complex Higgs field will be an important new particle intimately linked with the seesaw mechanism. Therefore, a detailed study of its properties is likely to provide another window to the physics behind the neutrino masses. The collider signatures of this B − L breaking scalar have been studied in some recent papers [17][18][19][20][21]. In this paper, we study some cosmological implications of this new scalar for the dynamical generation of matterantimatter asymmetry in the early universe.
An important consequence of the type-I seesaw is the possibility that it could provide a way to understand the matter-antimatter asymmetry of the universe via leptogenesis [22][23][24][25]. For TeV scale seesaw, it is necessary to use the resonant leptogenesis mechanism [26][27][28] (see Ref. [29] for a recent review) in order to avoid the Davidson-Ibarra bound [30]. An important aspect of leptogenesis is the dilution and wash-out effect induced by different particles and their interactions in the theory. In particular, the strong (exponential) dependence of the lepton asymmetry on the wash-out effect can be used to falsify high-scale leptogenesis at the LHC [31], as well as in low-energy experiments [32]. As for probing TeV scale leptogenesis, it depends on the details of the model under consideration [33]. In particular, the same dilution and wash-out effects can be used to put stringent constraints on the model parameters, if we want to have successful leptogenesis in these models. For example, in the left-right (LR) symmetric realization [34][35][36] of seesaw, dilution and washout of the lepton asymmetry due to the W R -mediated ∆L = 1 interactions [37][38][39][40] provides a lower bound of about 10 TeV on the mass of W R boson [39]. Similar (but weaker) lower bounds have also been derived on the mass of the extra Z boson that couples to RHNs as well as to the SM quarks and leptons in U (1) models [41][42][43][44][45].
In this paper, we consider generic seesaw models with B − L symmetry and derive leptogenesis constraints on the additional Higgs scalar H responsible for B − L breaking from its dilution and washout effect on the lepton asymmetry. 1 We consider three generic classes of theories: (i) where a scalar or pseudo-scalar has a generic coupling to the RHN in an effective field theory (EFT) framework; (ii) where B − L is a global U (1) symmetry (the so-called singlet Majoron model [47]); and (iii) where B − L is a local gauge symmetry. In all these cases, our constraints are independent of any details of leptogenesis, except for some of the key parameters such as the effective neutrino mass m and the CP asymmetry ε CP . We find that the allowed scalar mass regions are correlated with the mass of the RHNs for fixed B − L breaking scale (or the effective couplings to RHNs). In particular, the dilution of lepton asymmetry due to the seesaw scalar can be directly tested at the (high-luminosity) LHC and future hadron colliders. The RHNs can be pair produced in the Higgs portal through the mixing of seesaw scalar with the SM Higgs, and then decays into same-sign charged leptons plus jets as a result of the Majorana nature of RHN, i.e. pp → h/H → N N → ± ± + 4j. Therefore, if the heavy RHNs are discovered at the LHC or future higher energy/luminosity colliders [8,9], this will narrow the parameter region to search for the seesaw Higgs boson and its properties, which would provide a direct test of leptogenesis and existence of B − L symmetry in this scenario.
The rest of this paper is organized as follows: In Sec. 2, we consider generic couplings of a scalar or a pseudo-scalar to the RHNs in an EFT framework and solve Boltzmann equations for the RHN number density and the lepton asymmetry in their presence to find viable regions allowed by the observed baryon asymmetry. The global U (1) B−L model is considered in Sec. 3, where the (almost) massless Majoron J plays an important role in the dilution process as it directly couples to the RHNs, in addition to the scalar interactions. The local U (1) B−L case is investigated in Sec. 4, where we have a heavy Z R boson in addition to the seesaw scalar, and the gauge interactions and self-trilinear interactions of the scalar are both involved in the Boltzmann equations. The collider implications of leptogenesis are presented in Sec. 5. We summarize in Sec. 6. The reduced cross sections and Z R boson couplings to SM fermions are given respectively in Appendix A and B.
2 Effective theory with a CP-even/odd scalar We begin the discussion by considering an effective "low energy" theory where the RHN N has a Yukawa coupling to a new (pseudo)scalar H (A). It also has a Dirac Yukawa coupling to the SM neutrinos ν, which is responsible for both type-I seesaw as well as the generation of lepton asymmetry via leptogenesis. The effective theory could be considered as the low energy simplified version of a more fundamental ultraviolet (UV)-completion at higher energy scale, such as the specific U (1) B−L models discussed in Sec. 3 and 4, yet some of the key features of dilution of lepton asymmetry could already be seen and easily understood in the EFT framework, c.f. the Feynman diagram in Fig. 1 and the plots in Figs. 2-6. We analyze in this section the generic dilution effect induced by these couplings and obtain bounds on the (pseudo)scalar mass and its coupling to RHNs.

Effective couplings
The lowest-order effective couplings of a CP-even (odd) scalar H (A) to the RHN can be parameterized as 2 where φ denotes the SM Higgs doublet, L the SM lepton doublet and we have suppressed the flavor indices for brevity. In most of the realistic seesaw models, there might be more than one physical scalar, or even multiplets, that couple to the RHN; see for instance the global U (1) B−L model in Sec. 3 and the local one in Sec. 4 (in the local case the CPodd component is eaten by the heavy Z R boson). Then the triple scalar couplings get 2 Here the Yukawa couplings fH and fA could be the same or different, depending on the specific model details. For example, in the global U (1) model in Sec. 3, they are equal in the original Lagrangian; however, the CP-even scalar H could mix with the SM Higgs but the CP-odd scalar does not, and as a result the couplings fH, A are not equal any more, with fH rescaled by the mixing angle cos θ. In addition, if the CP-odd scalar A mixes with the SM Higgs, as discussed in Sec. 5, it will no longer be a pure CP-odd state. However, the SM Higgs is well-determined to be CP-even like, with the mixing with a CP-odd scalar smaller than 0.2 at the 95% confidence level (CL) [48][49][50]. involved in the dilution of RHNs through the processes N N → S i → S j S k (i, j, k being scalar indices, see e.g. the diagrams in Fig. 7) and play an important role in leptogenesis, particularly when the Yukawa coupling f is comparatively smaller. To capture the most important consequence of the presence of a (light) scalar/pseudoscalar in the type-I seesaw leptogenesis, we neglect the model-dependent scalar interactions in this section, and consider only the t-channel process N N → HH/AA, mediated by the Yukawa coupling f in Eq. (2.1), 3 as shown in Fig. 1. It should be noted that even if the new scalar develops a non-vanishing vacuum expectation value (VEV), which is expected to be higher than the electroweak (EW) VEV at v EW 174 GeV, before the EW phase transition the heavy scalar does not mix with the SM Higgs in the early universe. Therefore, in addition to the RHN mass m N (and the leptogenesis-relevant quantities such as the effective neutrino mass m), there are only two free parameters in the effective theory, i.e. the scalar mass m H (m A ) and the effective Yukawa coupling f H(A) in Eq. (2.1).

Dilution of the lepton asymmetry
The reduced cross sectionσ(N N → HH/AA) can be found in Appendix A [cf. Eqs. (A.1) and (A.2)], which are dictated by the scalar masses m H, A and the Yukawa coupling f . The relevant Boltzmann equations, which govern the evolution of the RHN number density and the lepton asymmetry, are then given by where z ≡ m N /T is a dimensionless parameter, H N ≡ H(z = 1) 17m 2 N /M Pl is the Hubble expansion rate at temperature T = m N (with M Pl = 1.2 × 10 19 GeV being the Planck mass), n γ = 2T 3 ζ(3)/π 2 is the number density of photons, and η N ≡ n N /n γ is the normalized number density of RHN (similarly η ∆L = (n L − nL)/n γ for the lepton asymmetry). The γ's are the various thermalized interaction rates: γ D for the RHN decay 3 We consider the coupling of either H or A in Eq. (2.1) and not both of them simultaneously. So in what follows, we sometimes denote the coupling simply as f (without the subscript), whose meaning should be clear from the context. N → Lφ, and γ s = γ φs + γ V s and γ t = γ φt + γ V t the standard ∆L = 1 scattering processes as in Refs. [28,51] with the subscripts s, t denoting respectively the s and tchannel exchange of the SM Higgs doublet φ or the SM gauge bosons V = W i , B (with i = 1, 2, 3) before EW symmetry breaking. Here the integration over different momenta has already been performed, assuming implicitly kinetic equilibrium. The new scattering processes (cf. Fig. 1) in our model correspond to the scattering rates γ HH, AA , with the prefactor of 2 for the reduction of RHN by unit of two [45]. The thermal corrections to the SM particles are included in the calculation [51,52]. If the Yukawa coupling f is sizable and m N m H, A , or in other words, as long as the γ HH, AA term is comparable or larger than other terms on the right-hand-side (RHS) of Eq. (2.2), this extra process could significantly dilute the RHN number density before the sphaleron decoupling temperature T c 131.7 GeV [53], thus potentially making type-I seesaw freeze-out leptogenesis ineffective, as we show below. 4 In general, the dilution effect depends on the RHN mass m N , the scalar mass m H, A , the effective Yukawa coupling f H,A and other model parameters such as the effective neutrino mass m ≡ v 2 (Y † Y ) 11 /m N (or Dirac Yukawa coupling) and the CP asymmetry ε CP , as well as the CP property of the extra scalar. Since in this paper we are mostly concerned with the role of the new scalar in the lepton asymmetry generation in RHN decay N → Lφ, we fix m ∆m 2 atm 50 meV throughout, without any significant tuning or cancellation in the type-I seesaw formula for light neutrino masses: Also, for the sake of simplicity, we have assumed implicitly that the scalar H/A stays in equilibrium with the RHN through the Yukawa interaction and does not have any portals to talk (in)directly to the SM particles, otherwise we would have the extra process like N N → H/A → SM particles contributing to the dilution. No specific flavor structure is considered, as we are focusing on the role of scalars in leptogenesis. To be concrete, one can assume there are only two RHNs (say N e, µ ) above the EW scale with a universal coupling f , while the third one (N τ ) decouples from the "low-scale" leptogenesis. A large CP-asymmetry ε CP can then be generated by the resonant enhancement mechanism, if ∆m N Γ N /2 m N , where Γ N is the average RHN decay width [29].
The evolution of the RHN and lepton asymmetry number densities for a benchmark scenario are presented in Fig. 2, where we set the CP-asymmetry ε CP = 10 −3 and the Yukawa coupling f = 1. The scalar H (A) has a mass m H(A) = 500 GeV, lighter than the RHN mass m N = 1 TeV, such that the dilution process N N → HH (AA) is kinematically allowed in the parameter space of interest. It is clear from Fig. 2 that in presence of the new scalar interactions with the RHN, the lepton asymmetry decreases from its original value (lower dashed curve). We should also mention here that we solve the Boltzmann equations with the thermal initial conditions η N = η ∆L = 0, which explains why in the absence of new scalar interactions, the η N value (upper dashed curve) traces its equilibrium distribution (almost coincident with the upper solid curve) at a lower temperature (or higher z), as  can be seen from Eq. (2.2). Apart from these additional new features, Fig. 2 presents the standard picture of freeze-out leptogenesis, namely, the Boltzmann suppression of η N and thermal freeze-out of η ∆L for z 1. This is further illustrated in Fig. 3, where we show the dependence of the lepton asymmetry as a function of the scalar mass, while keeping the other parameter values same as in Fig. 2. We find that for m H, A m N , the RHN density and lepton asymmetry are strongly diluted, whereas for m H, A m N , the dilution effect becomes negligible, as expected from the kinematics.
In the strong washout regime, the RHN is typically close to the thermal equilibrium, i.e. |η N /η eq N − 1| 1, as can be seen from its number density in Fig. 2. In this case, the RHS of Eq. (2.2) can be simplified as The leptogenesis constraints on the new scalar can thus be qualitatively obtained by requiring that 4γ HH, AA < γ D + γ s + 2γ t . In most of the parameter space of interest, one can solve analytically the Boltzmann equations (2.2) and (2.3) by taking the equilibrium limit, which is a good approximation in presence of the additional Yukawa (and gauge) interactions. For instance, following Ref. [38], the final lepton asymmetry around z ∼ z c can be factorized as with the effective K-factor It is clear from Eq. (2.5) that the lepton asymmetry would be highly suppressed when the γ HH, AA term is large. Throughout this paper, we follow this analytic approximation to derive the bounds and switch to the numerical solutions whenever necessary. In Fig. 4, we show the required values of ε CP to produce the correct lepton asymmetry η ∆L obs 2.5 × 10 −8 [52], which reproduces the observed baryon asymmetry η obs Since ε CP cannot exceed one, a light H/A thus requires a lower bound on the RHN mass, as shown by the log 10 ε CP = 0 contours. Depending on the actual value of ε CP in a given model, a larger parameter space could be excluded, as illustrated by the other log 10 ε CP contours.
By comparing the left and right panels of Figs. 4 and 5 for different effective Yukawa couplings f = 1 and 0.1, we find that a larger f excludes larger regions of m N and m H, A , as expected, since the scattering cross section σ(N N → HH/AA) ∝ f 4 , and therefore, the dilution effect is stronger for larger values of f . This can be used to set limits on the coupling f , given the masses m N and m H, A . As a explicit example, we show the contours of ε CP in Fig. 6 as functions of f and m H, A for m N = 1 TeV. The shaded regions below these contours are excluded from successful leptogenesis requirement. The gray bands on the right side of the vertical dashed line in Fig. 6 show the non-perturbative regime f > √ 4π. We find that for a TeV scale RHN, the effective Yukawa coupling is required to be smaller than 0.2 for a CP-even scalar H below the 1 TeV scale if ε CP = 10 −2 and smaller than 0.03 for ε CP = 10 −5 , just to give some examples. Comparing the analytic cross sections in Eqs. (A.1) and (A.2), one finds that the wash-out effect of the CP-odd scalar A is somewhat weaker than that of the CP-even scalar H; therefore, in Fig . Within the shaded regions the lepton asymmetry is strongly washed out by the process N N → HH (cf. Fig. 1), the type-I seesaw leptogenesis is falsified.   If there are more than one physical scalars that couple to the RHN simultaneously, then roughly speaking the dilution effect will be strengthened by the extra degrees of freedom, unless there is a destructive interference due to some specific (fine-tuned) choice of the phases. For instance, assuming the interaction f i S i N N with roughly a universal coupling f i , the cross section σ(N N → S i S i ) ∝ N S σ(N N → SS) (SS could be either HH or AA in our case), with N S the numbers of scalars that are lighter than the RHN. Then more stringent limits could be imposed on the scalar masses and couplings f i . This gets more involved when the scalar self-interactions are also taken into account, e.g. in the case of the global U (1) B−L model in Sec. 3 (see below) where we have both a CP-even and odd scalars.
We now extend the 'effective theory' results of Sec. 2 to two realistic models, which are based on the U (1) B−L symmetry. In this section we deal with a global B − L symmetry (the local B − L symmetry case will be considered in the Sec. 4). This is a well-motivated simplified model to generate the RHN masses and implement the seesaw mechanism, and could originate from a more fundamental UV-theory at high scale [47,61]. In this class of models, we have both the CP-even and odd scalars as well as new interaction in the scalar sector beyond the diagram in Fig. 1, that could lead to resonant enhancement of the dilution effect. See also Ref. [62] for related discussion.
The physical scalars in this scenario arise from a complex scalar singlet ∆ R that carries two units of lepton number and couples to the RHNs through the Yukawa Lagrangian After symmetry breaking with the VEV ∆ R = v R , we obtain the RHN Majorana mass m N = 2f v R . The scalar potential of ∆ R reads (omitting its possible interactions with the SM Higgs boson) Expanding the ∆ R field about its VEV, we can obtain mass m 2 H 3 = 4ρv 2 R for the physical CP-even scalar H 3 , 5 while the accompanying Goldstone component is the massless Majoron particle J. In principle, the Majoron could have a non-vanishing mass through the gravitational interactions [63,64] or by interacting with other particles beyond the type-I seesaw scheme. Even if it is the case, as long as its mass m J m N , that does not essentially make any difference to our following discussion. For a light Majoron, e.g. with mass GeV, in the type-I seesaw its couplings to the active neutrinos are proportional to the heavy-light neutrino mixing angle V 2 νN . For a TeV scale N , the mixing V νN ∼ 10 −6 , and we are then safe from the Majoron limits from supernova observations [65] and other astrophysical and terrestrial constraints [66,67]. Here we assume this to be the case. For simplicity, we assume that the Majoron decouples from the SM particles much above the QCD phase transition temperature (∼ 200 MeV) as is the case for the singlet Majoron model [47], so that its effect on N eff is very small due to entropy dilution.
From the potential (3.2) we could get the triple scalar couplings, which can be parameterized as functions of the CP-even scalar mass m H 3 and the VEV v R [68] with 6 and 2 the symmetric factors for identical particles. These couplings induce the extra diagrams in Fig. 7, that contributes to the processes N N → H 3 H 3 , H 3 J, JJ and interfere with those from the pure Yukawa interactions in Fig. 1. The partial decay widths of H 3 into heavy RHNs and massless Majorons are respectively where we have traded the Yukawa and scalar coupling into the particle masses via f = m N /2v R and λ H 3 JJ = −m 2 [62,69]). It should be noted that in the third diagram in Fig. 7, i.e. N N → H ( * ) 3 → JJ, if the H 3 mediator is on-shell, this corresponds to the inverse decay process N N → H 3 with the scalar H 3 decays further into two massless Majorons H 3 → JJ. Compared to the two-to-two process N N → JJ (combining both the s and t channels in Fig. 1 and 7, with the H 3 propagator off-shell), the inverse decay could enhance largely the dilution effect, as a result of the resonance structure.
An example is presented in Fig. 8 where one can see clearly that the resonance structure occurs at m H 3 2m N . The corresponding Boltzmann equation in the global U (1) B−L model is easily obtained by replacing the 2γ HH, AA term in Eq. (2.2) by summation of all the terms which reduce the RHN number by unit of two, i.e. from RHN decay, with the indicated values of ε CP and m = 50 meV. The gap structure corresponds to the resonance N N → H 3 → JJ [cf. Fig. 8]. Note that for fixed value of v R in both panels, the Yukawa coupling changes as f = m N /2v R . Together with m 2 H 3 = 4ρv 2 R , we set the perturbative boundaries f < √ 4π and ρ < 4π in Fig. 9. 6 By comparing the two panels, one can see that when the VEV v R is larger, the Yukawa coupling f gets smaller, and more regions are allowed, as shown by the broader colorful regions in the right panel of Fig. 9.

2(γ H
In analogy to Fig. 6 we can set limits on the v R scale and scalar mass m H 3 , which are shown in Fig. 10 with m N = 1 TeV. In this plot one can see clearly the leptogenesis limit on the v R scale, and the dependence on the physical scalar mass m H 3 . For a 1 TeV RHN, the v R scale is required to be larger than 2.1 TeV for ε CP = 10 −2 and 19 TeV if ε CP = 10 −5 , in the large m H 3 limit. The limits become more stringent when H 3 is lighter. At the resonance m H 3 2m N , the constraints on v R could even be an order of magnitude higher, which is determined largely by the H 3 width. One should note that in the large m H 3 limit the dilution is dominated by the (almost) massless final state N N → H * 3 → JJ, which however does not decouple as 1/m 4 H 3 , as naively expected, as the triple scalar coupling λ H 3 JJ in Eq. (3.3) scales like m 2 H 3 which cancels out the scalar mass 1/m 2 H 3 in the propagator. Thus the limits on v R approaches to be a constant in Fig. 10 in the large m H 3 limit, as long as the quartic coupling ρ in Eq. (3.2) is perturbative.
before symmetry breaking, which can be viewed in some sense as the "effective" theory of LR model [34][35][36] at the TeV scale with the SU (2) R breaking scale and the mass of the heavy W R bosons much higher. 7 The SM fermion doublets Q, L and singlets u R , like N L → W * R → QuR and the three-body decay N → LW * R → Lqq if mW R > mN , which would dilute the RHN number density before the temperature Tc, thus imposing very stringent limits on the WR boson mass [37,39]. If mW R < mN , then the situation becomes worse, as the two-body decay N → LWR is open, which does not generate any lepton asymmetry, as it is dictated by the gauge interaction, but only contributes to dilution of the asymmetry.
Anomaly freedom necessitates the introduction of three RHNs N i : (1, 1/2, −1). In the minimal scalar sector, we have the SM Higgs doublet φ(2, −1/2, 0) and a singlet ∆ R (1, −1, 2). The singlet VEV ∆ R = v R breaks the gauge symmetry down to the SM gauge group SU (2) L × U (1) Y which is further broken by the doublet VEV φ 0 = v EW to U (1) em , leading to the type I seesaw formula for neutrino masses. In this model, H 3 = Re(∆ R ) is a physical CP-even scalar that couples directly to the RHNs as in Eq. (3.1), and the Goldstone mode is eaten by the heavy Z R boson, which acquires a mass where g R and g BL are the gauge couplings associated with the U (1) I 3R and U (1) B−L gauge groups. The Z R couplings to the chiral fermions f L,R with electric charge Q f and thirdcomponent of isospin I 3,f are given by with tan φ ≡ g BL /g R the right-handed gauge mixing angle, θ w the weak mixing angle and e the electric charge. One can find more details on how to obtain these gauge couplings in Appendix B.
In presence of the scalar H 3 and the Z R boson, we can have the processes N N → Z ( * ) R → ff (f being the SM fermions) as wall as N N → H 3 H 3 , H 3 Z R , Z R Z R , which all reduce the RHN number by two units [45]. Our model in this section is however different from Ref. [45] in the following aspects: (i) The gauge structure in our model is different from that in Ref. [45], although both are claimed to be U (1) B−L models. The U (1) B−L in [45] does not contribute to electric charge and their Z is a pure B − L boson, while in our case the Z R is a mixture of the I 3R and B − L; (ii) Ref. [45] concentrates on the gauge boson, while in this paper we focus on the impact of B −L breaking scalars on leptogenesis, which are largely complementary to each other.
Before proceeding to the leptogenesis constraints on the scalar and gauge bosons, we would like to mention that the Z R mass is tightly constrained by the LHC dilepton data pp → Z R → + − with = e, µ [71][72][73][74]. The current limits can be cast onto our model by rescaling the couplings and production cross sections of the sequential Z boson adopted in the data analysis [71,72]. In our model, the gauge coupling g R is a free parameter, which is related to the To keep g BL real, there is a lower limit on g R , i.e. g R > g L tan θ w 0.55g L [68]. 8 For simplicity, we take g R = g L for which the dilepton limit is m Z R > 3.72 TeV [75].
All the reduced cross sections are collected in Appendix A, with the widths of H 3 and Z R bosons properly included, respectively, in the channels N N → H 3 → Z R Z R and N N → Z R → ff . As in the global U (1) B−L model where we have N N → H 3 → JJ, the  processes with on-shell s-channel H 3 /Z R bosons correspond to the inverse decay N N → H 3 /Z R which significantly enhances the dilution cross sections, as shown in Fig. 11. The leptogenesis predictions in two benchmark scenarios are shown in Fig. 12, where for the sake of comparison, two different values of the v R scale are adopted, i.e. v R = 4 TeV and 10 TeV, which are both above the current dilepton limits on Z R mass. As in Fig. 9, only within the colorful regions, the observed lepton asymmetry could be generated for the indicated ε CP , with m = 50 meV. It is transparent in both panels that when m N m Z R , there is almost no limit on the scalar mass m H 3 , as in this case the dilution is dominated by the Z R mediated process N N → ff , benefiting from the (almost) massless fermions in the final states and the large number of degrees of freedom. When the RHN is relatively light, the Yukawa coupling is to some extent suppressed via f = m N /2v R ; when H 3 is light, the triple scalar coupling could also be suppressed by m 2 H 3 , as shown in Eq. (3.3), thus the process N N → H 3 H 3 could not compete against N N → ff .
When the RHNs are sufficiently heavy, e.g. heavier than the current dilepton limits on Z R boson mass of around 3.7 TeV for g R = g L , such that they could annihilate into two Z R bosons, the scalar H 3 returns to play an important role, in particular when it is close to the resonance m H 3 2m N 2m Z R . One can see the clear resonance structure around 4 TeV in the left panel of Fig. 12. 9 When the v R scale is higher, as exemplified in the right panel of Fig. 12, the Z R gets heavier, and the leptogenesis limits on the scalar mass become less stringent, and making it more challenging for the LHC and future collider tests (see Sec. 5 below). Analogous to Figs. 6 and 10, we present in Fig. 13 the leptogenesis limits on m H 3 , as functions of the scale v R , with fixed m N = 5 TeV, m = 50 meV and the indicated values of ε CP . All the shaded regions below (or on the left of) these curves are excluded, which correspond to the lighter scalar mass. The peak structure at around m H 3 10 TeV corresponds to the resonance m H 3 2m N , which helps to exclude large regions in the m N − m H 3 plane. Note that here we do not consider any flavor structure; if the couplings of the Z R boson is flavor-dependent, e.g. dominantly to the tau-flavor, then it might be lighter, because the limits from Z R → τ + τ − searches at the LHC [76, 77] are not so stringent as the dilepton limits (for = e, µ). With lighter Z R and RHN, the leptogenesis limits on H 3 for a given g R might change accordingly.

Collider prospects
In this section, we focus on collider prospects for the seesaw scalar. Since we are considering the role of beyond SM scalar in leptogenesis via the processes like N N → HH, we will assume the scalar to be lighter than the RHNs involved, e.g. scalar mass m H, A (or m H 3 ) 9 There exists another resonance effect that occurs at mN mZ R /2 2 TeV where the process N N → ZR → ff is largely enhanced, and thus we have the gap at around 2 TeV in the left panel of Fig. 12. down from m N to about 1 GeV. 10 Since the scalar primarily couples to the RHNs, it is largely free of existing collider constraints. We will be interested in two mass ranges: (a) masses heavier than 5-10 GeV and (b) masses below 5 GeV and try to delineate possible ways to test leptogenesis in both cases.
In the latter case (b), the tree-level scalar quartic coupling ρ = m 2 H 3 /4v 2 R is required to be very small, in order to generate small or vanishing scalar masses at the tree level in both the global and local U (1) B−L models, as in Refs. [19,20]. The small scalar mass can be assumed to be generated at 1-loop level via the Coleman-Weinberg mechanism, inspired by a conformal theory set-up [78]. The searches for displaced vertices and long-lived particles at the high energy frontier, i.e. the LHC and proposed dedicated experiments such as MATHUSLA [79,80], and the high intensity experiments like SHiP [11] and DUNE [81] provide excellent motivations to study the light scalars.
In case (a) on the other hand, the scalar mass has an upper limit m H 3 < √ 16πv R , from the perturbativity of ρ < 4π, since it breaks the B − L symmetry and is responsible for neutrino mass generation via the seesaw mechanism.
As we have seen above, successful leptogenesis forbids certain ranges for the seesaw scalar mass depending on the RHN masses, and therefore, any evidence for scalar masses in these forbidden mass regions would rule out leptogenesis as a mechanism for origin of matter.

Constraints
In both regimes (a) and (b), constraints and predictions for signals of H, A come from the fact that a beyond SM scalar could in principle mix with the SM Higgs after EW symmetry breaking. This scalar mixing however does not play any role in freeze-out leptogenesis since the latter takes place prior to EW symmetry breaking. First of all, the scalar mixing would universally rescale the couplings of SM Higgs and contribute to the oblique parameters. It could also induce flavor-changing rare meson decays to the extra scalar when the latter is light, leading to stringent constraints in that mass range [19,20]. There are also direct searches of these extra scalars at LEP and LHC, e.g. when they decay into two SM fermions [82], gauge bosons [83][84][85][86][87][88][89], or di-Higgs [90][91][92]. All these data could be used to set limits on generic beyond SM scalars, i.e. on its mass and mixing angle sin θ with the SM Higgs, as shown in Fig. 14. The purpose of this section is to connect the rich phenomenology of heavy or light scalars to leptogenesis: if a (light) scalar could be found at the LHC or in the future higher energy colliders, that would have strong implications on the type-I seesaw leptogenesis and the heavy RHNs, thus providing a complementary probe of the seesaw mechanism.
For a CP-even scalar, the constraints from SM Higgs data and EW precision tests in the effective theory framework of Sec. 2 can be found e.g. in Ref. [93], depicted as the gray and brown regions in Fig. 14, respectively. The direct searches have been performed at the LHC in the final states of H → W W/ZZ [83][84][85][86], induced by mixing with the  Figure 14. Limits on the CP-even scalar mass m H and its mixing with the SM Higgs sin θ in the effective theory framework. The limits from precision test (brown) and SM Higgs coupling measurements (gray) are from Ref. [93]. The limits from direct searches at LEP (magenta), LHC (blue), and the flavor limits from B → Kχχ (purple) and B d −B d mixing (yellow) in the low mass regime are from Ref. [20]. Also shown are the LLP prospects at LHC [20]. The future sensitivity from invisible decay of the SM Higgs (red dashed) apply to the CP-even scalar in the global U (1) B−L model with v R = 1 TeV. See text for more details. SM Higgs. All these data are combined, following the procedure in Ref. [94], and are shown as the blue shaded region in Fig. 14. There are also searches for heavy CP-even or odd scalar resonances in the diphoton spectra [87][88][89]. However, in these searches, the interference between the continuum background gg → γγ and the signal gg → H/A → γγ is in general very important, even dominating over the pure resonance contribution [94]. The interference effect is not taken into account properly in the data analysis, thus these exclusion data can not be naïvely interpreted as limits on the effective mixing angle sin θ in our model. When kinematically allowed (i.e. for m H > 2m h ), the heavy scalar could decay also into the SM Higgs pair, H → hh; however, the limits from Refs. [90][91][92] can not set any limits on the mixing angle sin θ, suppressed by the small branching ratio. 11 The limits from Higgs searches at LEP are shown as the magenta region in Fig. 14 [95], excluding the scalar mass from ∼ 12 GeV up to 120 GeV and setting an upper limit on the mixing angle of order O(0.1).
For a light scalar H, the most stringent limits are from the flavor sector, as H could have loop-level flavor-changing couplings to quarks, such as Hsb, thus contributing significantly to neutral meson oscillations like B d −B d and rare meson decays like B → K + − . A detailed discussion can be found, e.g. in Ref. [20]. In Fig. 14 we only show the most stringent limits in the low mass regime, i.e. those from B → Kχχ (with χ SM charge leptons) combining the data from Refs. [96][97][98] and from B d −B d oscillation [20]. Enhanced by the small scalar mass m H , the limits on the mixing could go even up to the order of 10 −3 . For such small mixing values, when H is light enough, the decay length at the LHC 11 Here for simplicity we have assumed the cubic scalar coupling stems only from the term λ SM hhh in the SM via the mixing sin θ. In principle, one could also have contributions from the beyond SM cubic terms like λHHH . could be sizable compared to the detector radius, thus motivating a long lived particle (LLP) search. The region below the orange curve in Fig. 14 can be probed at LHC with a decay length 1 cm < bτ 0 < 1.5 m for an integrated luminosity of 3000 fb −1 at √ s = 14 TeV [20].
For the CP-odd scalar A, if it mixes with the SM Higgs, then the two physical scalars from h − A mixing are no longer CP eigenstates but rather admixtures of the CP-even and odd components, with one at 125 GeV and the other one heavier or lighter depending on the mass m A . A mixing angle of 0.2 is still compatible with current LHC data [48][49][50] at 95% CL. The limits on the scalar H presented in Fig. 14 apply also to the CP-odd A.
The limits on the physical scalar H 3 in the global and local U (1) B−L model (cf. Sec. 3 and 4 respectively) are roughly similar to the CP-even scalar H in the EFT framework (cf. Through the mixing with H 3 , the SM Higgs could also decay into two (almost) massless Majorons, with the partial width The Majorons are expected to escape the detectors without leaving any tracks even if they can decay into active neutrinos when they are massive, thus contributing to the invisible branching ratio of the SM Higgs BR(h → inv.). At the √ s = 14 TeV LHC, with an integrated luminosity of 3000 fb −1 , the Higgs invisible BR can be constrained to be smaller than 9% at the 95% C.L [99], while at √ s = 1 TeV ILC with a luminosity of 1000 fb −1 , the BR limit can reach up to 0.26% [100]. These prospects can be used to set future limits on the scalar mass m H 3 and mixing angle sin θ for a given v R scale. A representative example is shown in Fig. 14 as the red dashed lines (the region above which can be excluded), where we have set v R = 1 TeV. This is largely complementary to the direct search of heavy scalars, in particular when the mixing angle is relatively small and the scalar mass relatively large, until it hits the perturbativity limit (roughly at 3 TeV for v R = 1 TeV in Fig. 14).
In the global U (1) B−L model, the Majorons can be pair produced from its couplings to the SM Higgs h and H 3 in Eq. (3.3) and the h − H 3 scalar mixing, via the gluon fusion processes gg → h/H 3 . If one gluon is emitted from one of the initial-state gluons, as shown in Fig. 15, this resembles the mono-jet process at the LHC, with the two Majorons acting like (long-lived) dark matter particles leaving only missing transverse energy at hadron colliders. 12 Thus the production cross section for gg → gJJ in Fig. 15 is constrained by the monojet searches at the LHC [102][103][104]. The parton level production cross section is estimated by implementing the simulations gg → gh and gg → gH 3 in CalcHEP [105] and summing up the cross sections from both the two portals via where the BRs can be evaluated using Eqs. (3.5) and (5.1). Then the monojet limits on the H 3 mass and mixing angle sin θ can be obtained by comparing with the experimental data, after imposing appropriate cuts on the gluon jet. It turns out that when the scalar mass m H 3 is small, H 3 decays mostly into the (almost massless) Majorons, while the SM Higgs decay h → JJ is suppressed by both the mixing angle sin θ and the small coupling ; then the production is dominated by the H 3 portal, as shown in Fig. 16 where we have set explicitly sin θ = 0.1 and v R = 1 TeV. The low m H 3 5 GeV range is however excluded by the B meson data for a sizable sin θ (cf. the B → Kχχ decay and B d −B d oscillation limits in Fig. 14). When the scalar H 3 gets heavier, the production rate gg → gH 3 decreases largely, which diminishes the contribution from the H 3 portal; at the same time, the triple scalar coupling m 2 H 3 /v R becomes larger and larger which enhance the branching ratio of h → JJ, and eventually the SM Higgs takes over as the dominant channel, at m H 3 100 GeV, as shown in Fig. 16. For m H 3 230 GeV, the BR(h → JJ) exceeds the current limit of 28% for the invisible decay of the SM Higgs [106], and hence, excluded as shown in Fig. 16. With both the limits from B meson and the invisible decay of the SM Higgs taking into consideration, the scalar mass is restricted to be within a range of 5 GeV m H 3 230 GeV; within this range, the current monojet data [102][103][104] does not provide any limits on the mass m H 3 and the mixing angle sin θ, if the scale v R = 1 TeV.

Testing leptogenesis at the LHC
In the EFT framework, the RHNs have only the scalar portal to couple to the SM particles, 13 i.e. through the h − H scalar mixing sin θ; then the RHNs are produced predominantly from the gluon fusion process via the SM quark loops: gg → h * /H ( * ) → N N . After production, due to their Majorana nature, the RHNs could decay into both positively and negatively charged leptons, emitting a SM W boson: N → ± W ∓ . One expects the signals with same-sign charged leptons in the final state, with roughly a branching fraction 12 With a non-zero but small mass, the Majoron could decay into the active neutrinos through the heavylight neutrino mixing [47] and other SM fermions at loop level [101], which renders it a good long-lived decaying dark matter candidate. 13 The RHNs could also be produced through their Dirac Yukawa coupling [107][108][109][110][111][112] to the SM Higgs [cf. Eq. (2.1)]; however, in the TeV-scale type-I seesaw with Majorana RHNs, the Yukawa coupling is 10 −6 , thereby suppressing this production channel. where is the effective scalar coupling to the gluons through the SM quarks (dominated by the top quark loop), τ X = m 2 X /4m 2 q (m q being the SM quark masses), and the loop function [113] The relative minus sign in Eq. (5.5) comes from the orthogonal scalar mixing matrix The resultant production amplitude is √ŝ the center-of-mass energy at the parton level. When H is lighter than the SM Higgs, it dominates the RHN production, while when H is heavier, the SM Higgs takes over to be the dominant term. When the two scalars are almost degenerate, which is mildly disfavored by the precision Higgs data [106] (barring fine-tuned cancellations), the two terms in Eq. (5.10) almost cancel out with each other, diminishing largely the production cross section of RHNs. 14 The production cross sections for four different benchmark values of m N = 200 GeV, 300 GeV, 500 GeV and 1 TeV are shown in the left panel of Fig. 17, respectively, as the solid blue, green, orange and red curves, where the Yukawa coupling f = 1, the scalar mixing angle sin θ = 0.1 and the cross sections are obtained by rescaling the next-to-nextto-leading order production of a SM Higgs in the gluon fusion process [114]   The dependence of production cross section on the RHN mass m N and the scalar mass m H, A is more clearly seen in the right panel of Fig. 17, where we show the contours for some benchmark values of 1, 0.1, 0.01 and 10 −3 fb. With highly suppressed SM background of same-sign charged lepton events, it is promising to test leptogenesis at the LHC for a large range of m H,A 30 GeV and 200 GeV with m N TeV. Recall from Sec. 2 that when the RHN mass m N TeV, even if the scalar H, A is heavier than N , there is large parameter space of inefficient leptogenesis, see e.g. Fig. 4 and 5. At higher energy colliders such as FCC-hh [115][116][117] and SPPC [118], with larger production cross section σ(pp → N N ), a broader parameter region of leptogenesis could be tested.
Regarding the global U (1) B−L model, in the simplest scenario without any CP violation in the scalar sector (i.e. assuming all the quartic couplings in the scalar potential to be real and no spontaneous CP violations in the VEV v R ), the SM Higgs mixes with only the CP-even scalar H 3 . Without the CP violating mixing h − J, the collider prospects for the scalar H 3 is the same as that for H, with the effective Yukawa coupling of f eff = m N / √ 2v R . In other words, for a fixed value of v R , the solid contours in Fig. 17 have to be just rescaled by the factor of (m N / √ 2v R ) 2 . So for lighter N , the production cross section is relatively suppressed by the RHN mass m N . An explicit example with v R = 1 TeV and sin θ = 0.1 is shown in Fig. 18. For larger mixing angles which is still allowed by current Higgs precision data and direct searches at LHC, as well as for different v R values, the same-sign dilepton cross section in Fig. 18 should be rescaled by a factor of (sin θ/0.1) 2 × (v R /1 TeV) −2 . It is clear that for a small scalar mixing sin θ = 0.1, if the RHNs is pair produced through an on-shell heavy seesaw scalar, i.e. m H 3 > 2m N , then the production cross section σ(pp → H 3 → N N ) > 0.01 fb in a large region of parameter space of the global U (1) B−L model. So it seems promising that leptogenesis in this model could be directly tested at the high luminosity LHC, if both the seesaw scalar and RHNs are in the few hundred GeV range.
In the gauged U (1) B−L model, the RHNs could also be pair produced through the gauge portal [119][120][121][122][123], i.e. through both the Z and Z R bosons, as the SM Z boson mixes with the Z R boson [68,75] while Z R couples directly to the SM quarks and leptons [cf. Eqs. (4.3) and (4.4)]. However, in the heavy Z R limits, both the Z and Z R mediators are effectively suppressed by the large mass m 2 Z R (note that the Z − Z R mixing angle is suppressed by ∼ m 2 Z /m 2 Z R ). As in the global model, the collider prospects of the RHNs can be easily obtained by rescaling the solid contours in Fig. 18 by the factor of (sin θ/0.1) 2 × (v R /1 TeV) −2 . However, in light of the stringent dilepton limits on the Z R mass [71,72], the v R scale in this case has to be larger than about 3.4 TeV [cf. Eq. (4.2)], which implies that the production cross sections will be suppressed for both gauge and scalar portals. 15

Conclusion
In this paper, we have studied the impact of the neutral scalar field that breaks B − L symmetry in a TeV scale seesaw on leptogenesis. The Yukawa coupling of the new CPeven (odd) scalar H (A) to RHNs induces dilution of the RHN number density through 2 ↔ 2 scattering processes like N N ↔ HH(AA), which renders leptogenesis ineffective as a mechanism for understanding matter-antimatter asymmetry of the universe for a wide range of the scalar mass and coupling parameters. We have reported the leptogenesis constraints in the context of three different scenarios: in an effective low-energy theory symmetry. We also comment on ways to probe these mass ranges for the seesaw scalar in collider settings and point out how this could provide a new window to probe the origin of neutrino masses and the baryon asymmetry.

A The reduced cross sections
In this appendix, we give the explicit analytic formulas for the reduced cross sections for various 2 ↔ 2 scatterings involving the RHNs used in our leptogenesis calculations in Sec. 2, 3 and 4.

A.1 Effective theory
In the EFT framework of Sec. 2, the reduced cross sections read where x = s/m 2 N , r X = m 2 X /m 2 N (with X = H, A), and the functions Note the difference in the two cross sections (A.1) and (A.2) for the CP-even and odd scalar, respectively, which is due to the γ 5 structure of the CP-odd scalar coupling to the Majorana neutrinos in Eq. (2.1).

A.2 Global
The A SS terms are defined as At the H 3 resonance, the propagator 1/|x − r| should be modified accordingly to include the H 3 width. the effective couplings of Z R boson to the fermions, and c N = sin θw sin 2φ to the RHNs (with the RH gauge mixing tan φ = g BL /g R ). In Eq. (A.11) f runs over all the flavors of quarks, charged leptons and active neutrinos, with N f C the color degrees of freedom (3 for quarks and 1 otherwise) and the symmetry factor S f = 1 for the charged fermions and 1/2 for neutrinos. The A V V and A V S terms are with β Z R = β X (X = w = m 2 Z R /m 2 N ) and β 3 = β XY (X = r, Y = w).

B Couplings of the Z R boson to fermions
In the gauged SU (2) L × U (1) I 3R × U (1) B−L model of Sec. 4, neglecting the mixing with the SM Z boson, the Z R boson is given by the linear combination [68] Z R, µ = cos φ U I 3R , µ − sin φ U BL, µ . The scalar H 3 could decay into two RHNs with the width given in Eq. (3.4). If kinematically allowed, it also decays into the heavy gauge bosons, with the partial width (B.10)