Leptogenesis and Neutral Gauge Bosons

ULB-TH/16-15

Leptogenesis and neutral gauge bosons

Julian Heeck1, ∗ and Daniele Teresi1, † 1Service de Physique Théorique,UniversitéLibre de Bruxelles, Boulevard du Triomphe, CP225, 1050 Brussels, Belgium We consider low-scale leptogenesis via right-handed neutrinos N coupled to a Z0 boson, with gauged U(1)B−L as a simple realization. Keeping the neutrinos sufficiently out of equilibrium puts strong bounds on the Z0 coupling strength and mass, our focus being on light Z0 and N, testable in the near future by SHiP, HPS, Belle II, and at the LHC. We show that leptogenesis could be robustly falsified in a large region of parameter space by the double observation of Z0 and N, e.g. in the channel pp → Z0 → NN with displaced N-decay vertex, and by several experiments searching for light Z0, according to the mass of N.

I. INTRODUCTION II. FRAMEWORK

We consider the simplest SM extension that has a 0 Right-handed neutrinos NR are a popular minimal ex- Z boson coupled to (three) right-handed neutrinos NR, 1 tension of the Standard Model (SM) that can generate based on the anomaly-free gauge group U(1)B−L. To neutrino masses, the baryon asymmetry of the Universe, provide Majorana masses to NR we add one SM-singlet and even dark matter [1]. While neutrino masses via complex scalar Φ with B − L charge +2: the seesaw mechanism [2] and baryogenesis via leptogen- 2 2 esis [3] are typically assumed to arise from right-handed 2 w 2 8 L = iN RDN/ R + |DµΦ| − λS − |Φ| neutrino masses above 10 GeV [4, 5], low-scale realiza- 2 (1) tions such as resonant leptogenesis [6–8] exist and pro- c vide experimental testability. It was recently shown in − yLHNR + λN N RNRΦ + h.c. , Ref. [9] that an absolute lower bound MN > 2 GeV exists, assuming a thermal population of NR in the early suppressing ﬂavor indices. Without loss of generality we Universe, while a non-thermal population can lead to suc- can assume λN to be diagonal√ with positive entries. In cessful leptogenesis even for lower MN . unitary gauge, Φ√ = (w + S)/ 2, one physical scalar S with mass MS = 2λSw remains, while the vacuum ex- 0 0 pectation value (VEV) w induces a Z mass MZ0 = 2g w If the right-handed neutrinos NR are part of a more √ complete model, they are often endowed with addi- and a mass matrix MN = 2λN w for the sterile Majo- c tional interactions, potentially threatening the genera- rana neutrinos N = NR + NR. After electroweak sym- tion of a lepton asymmetry (for example Sakharov’s out- metry breaking the right-handed neutrinos mix with the of-equilibrium condition). A prime example here are left-handed ones and provide light neutrino masses in the 2 left–right symmetric models based on the gauge group standard way. Note that we further ignore mixing of S with the SM Brout–Englert–Higgs boson as well as ki- SU(2)L × SU(2)R × U(1)B−L [10, 11], which couple the − netic mixing of the Z0, which is a conservative choice. NR to a new charged gauge boson WR . Successful lep-

togenesis then requires MWR > 10 TeV; otherwise the gauge interactions with strength gR = gL = e/ sin θW heavily dilute the asymmetry [12–14]. Interactions via III. LEPTOGENESIS a neutral gauge boson Z0 are typically considered less dangerous because all such processes require two NR in- A. Boltzmann equations stead of just one [13, 15]. A well-motivated simple model for interactions of this sort can be obtained by promot- Assuming a Universe that started with total B − L ing the anomaly-free global symmetry U(1)B−L of the charge zero – possibly after an inﬂationary period – a SM to a gauge symmetry, which actually requires three non-zero baryon asymmetry can be dynamically gener- arXiv:1609.03594v2 [hep-ph] 29 Nov 2016 right-handed neutrinos for consistency. Previous work on ated at temperatures below the B − L thermal phase leptogenesis in a U(1)B−L context has been performed in transition, which typically occurs at a temperature of Refs. [13, 16–19].

In this article we (re)evaluate the constraints on the 1 0 µ U(1)B−L parameter space coming from successful lepto- Other realizations of the relevant coupling ZµNγ γ5N can be

genesis and focus in particular on the case of light NR found for example in left–right models with MWR MZR or in and Z0, potentially testable and falsifiable in the future. models with gauged U(1)L and would not change our conclusions much. 2 Active–sterile neutrino mixing θ will lead to interaction terms Z0Nν that are linear in N and thus potentially more dangerous − for leptogenesis in direct analogy to the WR scenario. Since ∗Electronic address: [email protected] these couplings are however suppressed by g0θ they turn out to †Electronic address: [email protected] be subleading. 2 the order of the B − L breaking VEV w,3 and above the Notice that the interactions beyond the minimal see- 0 sphaleron decoupling temperature Tsph = 131.7 GeV [21]. saw model, i.e. the ones involving Z and S, do not An (unflavored) B − L asymmetry requires not only enter the equation for the asymmetry (4) at this order w 6= 0, but also y 6= 0 6= λN , CP-violating phases in y, as and therefore cannot wash out the asymmetry. However, well as out-of-equilibrium dynamics for N. While these they have a major role in keeping N close to equilibrium, conditions are most commonly satisfied by considering see (3), and therefore can heavily dilute the generated a slow decay N → LH compared to the expanding Uni- asymmetry. verse, they can also be satisfied via the decay H → LN [9] The Boltzmann equations (3) and (4) do not take into thanks to thermal effects, making it possible to consider account flavour effects in the charged-lepton sector. How- N masses as low as 2 GeV. Our analysis follows Ref. [9], ever, these are not expected to have a major impact but includes the new interactions involving Z0 and S, on the results [9], since here leptogenesis typically occurs in the strong-washout regimem ˜ ≡ v2(yy†)/M 0(∗) ¯ 0 0 0 N NN ↔ Z ↔ ff , NN ↔ Z Z ,Z S,SS, (2) 2 meV. Quantum coherences in the charged-lepton sec- which potentially keep N in thermal equilibrium with the tor can change significantly the lepton asymmetry for MN ∼ 200 GeV [14] if the Yukawa couplings are very SM and hence dilute the resulting lepton asymmetry. We −3 collect the cross sections in App. A. Since both Z0 and large, namely y ∼ 10 , which can be realized naturally S are in thermal equilibrium with the SM in the regime in models with additional symmetries. However, since considered in this work, we only have to consider the the asymmetry typically will be maximal for lower values of y, we may safely neglect this effect. We have also con- Boltzmann equations for the N number density nN and the total lepton asymmetry n , given respectively by servatively neglected the effect of tracking the evolution L of the different N’s, since this can only slightly increase n H dη ηN 2 the washout of the asymmetry [13], unless one is inter- γ N N = − − 1 2 γ z dz ηeq NN ested in very large values of y in models with additional N leptonic symmetries. Again, as discussed above, we have N η h i safely neglected this. Finally, quantum coherences in the − eq − 1 γD + 2(γHs + γAs) + 4(γHt + γAt) , ηN heavy-neutrino sector can give an additional O(1) contri- (3) bution to the asymmetry [14, 23], but this effect will not N have a significant impact on the results below and thus nγ HN dηL η 2 = γD eq − 1 εCP(z) − ηL it has been neglected, for simplicity, in Eqs. (3) and (4). z dz η 3 N For M in the GeV range, in addition to thermal lep- N N 4 η togenesis via N and H decay, as described by (3) and (4), − ηL 2(γHt + γAt) + eq (γHs + γAs) , (4) 3 ηN an asymmetry can be generated via the Akhmedov– Rubakov–Smirnov (ARS) mechanism [1, 24, 25]. How- where η ≡ n /n , z ≡ M /T , and H is the Hub- a a γ N N ever, this effect will be highly suppressed by the fact that ble rate at T = M . Here, γ is the reaction den- N D the evolution starts from a state of thermal equilibrium, sity for the decays N ↔ LH, H ↔ NL, including as discussed above. Moreover, since the bounds below thermal corrections, whose expression can be found in will be basically due to the equilibration of N due to Ref. [9]; γ denotes collectively the one for the pro- NN gauge interactions, which occurs also for the ARS mech- cesses NN ↔ ff¯ , Z0Z0 ,Z0S,SS, which can be calcu- anism for the same values of g0, the inclusion of this lated from the cross sections given in App. A, the factor mechanism (for which it is difficult to perform general of 2 in (3) being due to the fact that the two right-handed analyses; see e.g. Refs. [26–28]) would not significantly neutrinos are involved in the process; finally, the remain- change our results. ing reaction densities for ∆L = 1 scatterings can be found in Ref. [22]. We start the evolution from a zero lepton In Fig. 1 we show the evolution of the N number den- asymmetry and N at equilibrium, since for the values of sity as a function of the temperature, for a typical choice g0 considered in this work it is safe to assume that equi- of the model parameters, in the presence or absence of libration has occurred, possibly above the B − L phase gauge interactions. Although in low-scale leptogenesis the right-handed neutrino number density typically does transition. In Eq. (4), εCP(z) denotes the CP asymmetry, which we will assume to take a constant value ε (e.g. its not depart much from thermal equilibrium, the presence maximal value ε = 1), when either of the CP-violating of gauge interactions can suppress further the departure processes N ↔ LH, H ↔ NL is kinematically allowed, from equilibrium by many orders of magnitude (until after taking into account thermal effects [9], and zero their decoupling), thus diluting strongly the asymmetry, as explained above. otherwise. However, notice that for low MN in the GeV range, thermal effects imply that maximal values for the In Fig. 2 we show an example for the required CP asymmetry εCP ∼ 1 cannot be realized [9], and hence the asymmetry ε for a given set of parameters as a func- bounds given below should be seen as very conservative tion ofm ˜ . Clearly visible are the dips at the reso- in this regime. nances MN ∼ MZ0 /2 and MN ∼ MS/2, where the rates NN ↔ ff¯ and NN ↔ Z0Z0 become resonantly enhanced and make leptogenesis more difficult, especially in the natural-seesaw regimem ˜ ∼ 50 meV. For very 3 In the presence of additional scalar fields the critical temperature largem ˜ leptogenesis becomes difficult because of the can be much larger than the VEV; see e.g. Ref. [20]. strong washout of the generated asymmetry; for small 3

10-2

MN = 1 TeV g'=0 -3.5 10-4 /GeV) -1 Z' eq N 10-6 -4.0 /n N n /(M g' 10 10-8 g'=0.01 4.5

log -

10-10 10-1 100 101 102 -5.0 1000 2000 3000 4000 5000 6000 7000 z=MN/T MZ' [GeV]

FIG. 1: The deviation of the N number density from equi- 0 −1 librium in the presence (continuous line) or absence (dashed FIG. 3: Upper limit on g /MZ0 = (2w) vs. MZ0 for decou- line) of gauge interactions. For illustrative purposes, we have pled S, MN = 1 TeV, and ε = 1. chosen MN = 10 TeV, MZ0 = 5 TeV, MS = 30 TeV, and m˜ = 1 eV. < < for ε = 1 in the region 200 GeV ∼ MN ∼ 1 TeV, and 0 stronger for lower MN . If the Z is light enough (roughly m˜ < 10−4 eV the generation of the asymmetry is sup- < ¯ ∼ 2MN < MZ0 ∼ 3.5MN ) the rate NN ↔ ff is resonantly pressed by the slow rate of the CP-violating decays. In enhanced (see Fig. 3), which will severely increase the the following we will always maximize the final asymme- limit on w. For MZ0 < 2MN , this s-channel resonance try with respect tom ˜ in order to get conservative limits. < 0 0 disappears, and for MZ0 ∼ MN the channel NN ↔ Z Z opens up.4 Since the rate NN ↔ Z0Z0 grows for small 0 4 M 0 like (g /M 0 ) due to the dominant emission of the B. Decoupled S Z Z would-be Goldstone boson,5 limits on the B − L parameter space for MZ0 MN depend again on the VEV, Let us consider the decoupled-scalar limit MS → ∞ first: for the extreme case MZ0 MN , only the rate w/5 TeV > 10−MN /1.5 TeV , (6) NN ↔ ff¯ is relevant, which is proportional to w−4. Limits are then cast on the VEV w as if we assume a maximal CP asymmetry ε = 1, up to the w/12 TeV > 10−MN /1.5 TeV , (5) perturbative-unitarity bound MN < 1.6 TeV (see discussion below). Let us remind the reader that it is difficult to actually obtain CP asymmetries ε of order 1, since this requires two of the Majorana neutrinos to be highly degenerate -1 in order to resonantly enhance the asymmetry [6], 5

MNi − MNj ∼ Γi,j/2 MNi,j , (7) -2 0 where Γj denotes the decay width of Nj. In light of this, let us discuss the eﬀect of smaller ε. As can be 4 seen from Fig. 4, lower values of ε result in stronger lim- /GeV N its on w but are diﬃcult to give in analytic form. In

M -3 addition, we see the increasing lower bound on M [9]. 10 N Notice that, as discussed above, for MZ0 MN the rele- log 0 -3 vant rate depends on the combination g /M 0 ; therefore, -2 Z 3 the results shown in Fig. 4, although explicitly given for 0 MZ0 = 1 GeV, are of general validity, as long as Z is much lighter than N. 0

2 -4 -2 0 2 4 4 0 0 The threshold for NN → Z Z is technically not MN but the m˜ > log10 /eV relevant temperature T ∼ 2MZ0 and analogously for the NN → SS process. 5 0 In the limit g → 0 the U(1)B−L becomes a global symmetry and FIG. 2: Contours of log10 ε needed for successful leptogenesis 0 0 the longitudinal Z component Im(Φ) becomes the (massless) for M 0 = 5 TeV, M = 30 TeV, and g = 0.2. Perturbative Z S Majoron, with qualitatively similar impact on leptogenesis [20, unitarity excludes the region MN > 44 TeV. 29, 30]. 4

-2 -2.5 λN> 2π λS>2π

-3 ϵ=10-2 ϵ=1 -3.0 /GeV) /GeV) Z' Z' -4 ϵ=10-4 /(M g' /(M g' -3.5 MS = 3 TeV 10 10 log -5 log ϵ=10-6

-4.0 λN> 2π -6 0 1000 2000 3000 103 104 105 MN GeV [ ] MN [GeV]

0 -2.0 FIG. 4: Upper limit on g /MZ0 vs. MN for MZ0 = 1 GeV (valid for any M 0 M ) and decoupled S for various CP λN > 2π Z N -2.5 asymmetries. In the shaded region perturbative unitarity would be violated. -3.0 MS = 10 GeV /GeV) -3.5 MS = 100 GeV Z'

C. The eﬀect of S MS = 1 TeV

g'/(M -4.0 M

10 S = 2 TeV

The previous results are fairly simple because we ig- log -4.5 MS = 3 TeV nored/decoupled the additional new particle, the scalar MS = 4 TeV -5.0 S. This is of course a much too simplifying assumption, S decoupled 0 0 as S is required to unitarize e.g. the NN → Z Z cross -5.5 section. Diﬀerently stated, since the quartic interaction 0 500 1000 1500 2000 2500 3000 4 2 2 M S has a coupling constant λS = MS/2w , a hierarchy N [GeV] MS w leads to a non-perturbative coupling, outside 0 of our region of calculability. A similar argument holds FIG. 5: Upper limit on g /MZ0 vs. MN for MZ0 = 1 GeV (valid for any M 0 M ), ε = 1, and various S masses. for the Yukawa coupling λN . We adopt the perturbative- Z N unitarity limits from Refs. [31, 32], which can be cast in The shaded regions (top and bottom) and the dashed parts the form of the lines (bottom) correspond to the perturbative-unitarity √ √ bounds (8). MN,S ≤ 4πw or λN ≤ 2π, λS ≤ 2π . (8)

Lowering MS to perturbative values will typically IV. PHENOMENOLOGY strengthen the limits on the B − L parameter space, be- 0 0 cause the destructive interference in the NN → Z Z A. Light Z0 rate is less relevant than the s-channel S resonance that becomes readily accessible in the thermal bath. Lowering Having derived limits on the B − L parameter space M below 2M turns this resonance off, but opens the S N from successful leptogenesis, we can compare to exist- channel NN → SZ0 if the Z0 is sufficiently light. Finally, ing bounds and regions to be probed in the near fu- for MS < MN , the channel NN → SS opens up (see also ∼ ture, focusing for now on 10 MeV ≤ M 0 ≤ 10 GeV. Ref. [19]).4 All of this is illustrated in Fig. 5. We see that Z 0 We follow Ref. [33] to translate limits from beam-dump the bounds on M 0 /g can increase by more than an or- Z experiments [34], BaBar [35], and ν –e− scattering der of magnitude compared to the decoupled S. Only for e,µ data [36, 37] (see Fig. 6). Also shown is the potential M < 20 GeV do the bounds become weaker than those S ∼ reach of the proposed SHiP experiment [38], adopted of the decoupled S case, and only for the similarly small from Refs. [39, 40]; of Belle II (for an integrated lumi- M . N nosity of 50 ab−1) [41, 42]; and of the Heavy Photon From Figs. 3, 4 and 5 we see that the weakest, most Search (HPS) experiment [43], naively converted from conservative bound on the B − L parameter space com- limits on hidden photons. Finally, solar neutrino scat- patible with perturbative unitarity arises for the hierar- tering with electrons in future second-generation xenon- chy M 0 < M M and reads Z ∼ N S based dark matter direct detection experiments such as 0 > LZ can further probe the parameter space with limits up MZ0 /g = 2w ∼ 1 TeV , (9) to w ' 1 TeV (LZ line in Fig. 6) [44]. Prospects for LHCb achievable at MN ∼ 1.6 TeV. For more realistically ob- sensitivity require a more careful adaption from the hid- tainable CP asymmetries ε = 10−2 (10−4), the most con- den photon case [45, 46] and are beyond the scope of this > servative bound can be read off of Fig. 4 as w ∼ 800 GeV article. 0 (8 TeV). We stress that these limits increase rather dra- Notice that the observation of a U(1)B−L Z with < 0 < < matically if the hierarchies MZ0 ∼ MN MS are invalid. MZ0 /g ∼ 7 TeV (much stronger for MZ0 ∼ 5 GeV [33]) 5

10-3 BaBar

beam -4 TeV 10 dump =1.6 , M N =1 Ν scattering ¶ TeV =1 TeV ' N =1 -1 g , M GeV N Belle II, 50 ab 1 -3 M GeV HPS ¶= 500 , = 500 M N GeV =10 = 1, ¶ N ¶= =10 -3 , M M N -5 1, =10 GeV 10 ¶= ¶ =10 LZ N -3 , M 10 ¶=

SHiP

10-6 10-2 10-1 1 10

MZ' GeV

FIG. 6: Parameter space of a gauge boson Z0 coupled to B − L@ . TheD shaded areas are excluded at 90% C.L., and future reach is shown in dashed lines; see text for details. The diagonal black and blue lines show the upper bound for successful −3 leptogenesis with CP asymmetries ε = 1 and 10 , respectively, for various N masses. The black line ε = 1, MN = 1.6 TeV corresponds to the most conservative limit from leptogenesis within the perturbative region. would already establish that neutrinos cannot be Dirac different experiments. Successful leptogenesis can only > particles, because the νR would be thermalized by the be obtained for MN ∼ 1.5 TeV (Fig. 4), so the obser- Z0 interactions and disturb Big Bang nucleosynthesis by vation of a sub-TeV right-handed neutrino would falsify contributing to Neff . If neutrinos are Majorana particles, leptogenesis. Since N masses up to O(100 GeV) can be the same argument requires the sterile Majorana partners probed at FCC-ee, ILC and CEPC [49–51] – depending N to be typically heavier than the MeV scale. As shown on the active–sterile mixing angle – there indeed exists here, successful leptogenesis in the presence of such a Z0 the possibility to falsify leptogenesis in models where the severely increases this lower bound on MN . right-handed neutrino has additional couplings to a neu- As shown above, the most conservative leptogenesis tral gauge boson. Here we are of course glossing over the < intricacies of establishing the true identities of Z0 and N, constraint (9) (obtained for ε = 1, MZ0 ∼ MN MS) 0 > i.e. all their relevant couplings. already implies MZ0 /g ∼ 1 TeV, and is hence stronger than the neutrino scattering constraints below MZ0 = 0.1 GeV (Fig. 6). For MZ0 > 0.1 GeV, neutrino scattering limits, among others, are superior to the most con- B. Heavy Z0 servative leptogenesis constraint. This is no longer true for realistically achievable CP asymmetries ε 1, which Let us discuss a second region of interest, with MZ0 > can easily provide limits even beyond future neutrino- 10 GeV, which in particular opens up the leptogene- scattering reach. Since the largest realistic value for ε is 0 sis region around the Z resonance MZ0 ∼ 2MN , since however impossible to quantify (or to measure), this is the lower bound on MN is around a few GeV [9]. at best a qualitative constraint on the parameter space. On the experimental side, there are no B-factory lim- Nevertheless, even for ε = 1 one can obtain strong limits its for M 0 > 10 GeV, so the strongest constraints < Z if the hierarchy MZ0 ∼ MN MS is invalid, which is (and future prospects [44]) come from neutrino scatter- 0 experimentally testable by observing both Z and N, the ing, specifically CHARM-II [37, 52], together with LHC scalar S being practically impossible to observe unless searches [53] (see also Refs. [54, 55] for dilepton bounds); we turn on scalar mixing. In particular, the requirement see Fig. 7. As has been emphasized in Ref. [56] (see of successful leptogenesis becomes competitive to con- also Refs. [18, 57]), the additional gauge interactions of straints from direct searches for MN TeV, and it is N can severely improve the discovery prospects of both precisely this region where one could hope to find N. N and Z0, provided the decay of N is slow enough to Since leptogenesis is infamously hard to verify as a lead to displaced vertices. This is naturally the case if mechanism, a more relevant alternative question is thus the active–sterile mixing angle θ fulfills the naive see- 2 whether it is falsifiable [13, 47, 48]. As an optimistic ex- saw relation θ ∼ Mν /MN , which impliesm ˜ ∼ 50 meV. ample, let us assume that a Z0 is found right around The second ingredient for the displaced-vertex observa- the corner of existing limits at MZ0 ' 20 MeV with tion of N is the hierarchy MZ0 > 2MN , resulting in a g0 ∼ 2 × 10−5, which implies a VEV w ∼ 500 GeV and very efficient on-shell production of Z0 at the LHC (or offers ample opportunity to study the new particle in SHiP), followed by Z0 → NN. Due to the very low back- 6

M M M M M M 10-1 Z'/ N = 0.1 Z'/ N =1 Z'/ N =2 MZ'/MN =3 MZ'/MN = 10 MZ'/MN = 100

LHC8 0 10-2 CMS7 -1 Ν scattering LZ SHiP ' -3 g 10 BaBar -2

HL-LHC g' 10 -3 -4 10 leptogenesis log -4 Belle II, 50 ab-1 10-5 -5 1 10 102 103 -6 MZ' = 3 MN GeV 100 101 102 103 104 M XM 0 Z' = N [GeV] FIG. 7: Upper limit on g vs. MZ0 @= 3MD N for decoupled S and ε = 1 (thick black line). Existing limits (shaded) and fu- 0 FIG. 8: Upper limit on g vs. MZ0 = XMN for X ∈ ture prospects (dashed) under the assumption M 0 /M = 3 Z N {0.1, 1, 2, 3, 10, 100}, decoupled S and ε = 1. The dashed and the seesaw relation for the active–sterile mixing angle are part of the line corresponds to the perturbative-unitarity taken from Ref. [56]. In particular, the dark red dot-dashed bound (8). For M lighter than a few GeV the leptogene- (blue dashed) lines show the projections for the HL-LHC N sis bounds reported should be considered very conservative; (3 ab−1) in the channel pp → Z0 → `+`− (pp → Z0 → NN see the discussion in Sec. III A. with displaced vertices), while the dashed green line shows the SHiP reach for displaced vertices. For MN lighter than a few GeV the leptogenesis bound reported should be considered very conservative; see the discussion in Sec. III A. V. CONCLUSION

The matter–antimatter asymmetry of our Universe is ground, these displaced-vertex searches could in the fu- a longstanding mystery given our confidence in the (in- ture be even more sensitive to a Z0 than the standard flationary) Big Bang theory. Leptogenesis is championed dilepton channel pp → Z0 → `` [56]. as a simple explanation that also resolves the neutrino For leptogenesis, the hierarchy MZ0 > 2MN implies mass problem of the SM. that the limits on w become stronger than before be- In this paper we have performed a comprehensive anal- cause the thermal distribution of N energies in the early ysis of leptogenesis in the presence of a neutral gauge Universe makes it very easy to hit the s-channel Z0 res- boson Z0 interacting with the right-handed neutrinos N, 0 onance, if the mass of Z is below the TeV range. Tak- taking the gauged U(1)B−L model as a benchmark sce- ing as a benchmark the fixed ratio MZ0 /MN = 3 (as nario. The gauge interactions put N in thermal equi- in Ref. [56]), we can give the conservative leptogenesis librium at high temperatures; the additional annihila- constraints for ε = 1 (decoupled S); see Fig. 7. (Our tion channel NN → ff¯ via heavy s-channel Z0 or results agree to a good accuracy with the simplified anal- NN → Z0Z0 for light Z0 will dilute the lepton asym- ysis of Ref. [18].) Some comments are in order: the metry, allowing us to provide a very conservative lower 0 0 > future prospects for observing Z and N in Fig. 7 as- limit MZ0 /g ∼ 1 TeV for successful leptogenesis. This sume the production of one pair of right-handed neu- limit increases drastically for realistically achievable CP < trinos with decay length fixed by the naive seesaw re- asymmetries or if the hierarchy MZ0 ∼ MN is violated. 2 lation θ ∼ Mν /MN for the mixing angle. The limits The latter case, with MZ0 > 2MN , is of particular inter- and prospects should thus slightly increase for our case est because it allows for an efficient production of N via with (at least two) degenerate N. Furthermore, fixing on-shell Z0 → NN, potentially followed by a displaced- θ implies a fixedm ˜ , a parameter we varied to maximize vertex decay of N. This is a promising detection channel ε in our leptogenesis limits, so our limits will also in- for both Z0 and N at SHiP or the (HL-)LHC. As it can be crease (slightly). Increasing the ratio MZ0 /MN far above seen from Fig. 7, leptogenesis provides extremely strong 3 will not change much the current or projected lim- constraints in this region of parameter space, so that any its of the dilepton channel, which rescale slightly with observation of Z0 and N via displaced vertices would ef- BR(Z0 → ``), but will reduce the efficiency for recon- fectively rule out leptogenesis. structing the displaced vertex of the boosted N. The lep- We have also considered for the first time the opposite togenesis constraints on the other hand become stronger hierarchy, MZ0 < 2MN , and shown that it still poses 0 still for MZ0 /MN 3 (Fig. 8), making it possible to ex- strong limits on the Z parameter space (Fig. 6). Even clude leptogenesis robustly through a double observation here it is possible to falsify leptogenesis by detecting both 0 > 0 of Z and N with MZ0 /MN ∼ 3. Let us emphasize again Z and N in a large region of parameter space. In this 0 that the ε = 1 limit given in Figs. 7 and 8 for MN lighter case of a light Z , whose discovery prospects will increase than a few GeV is extremely conservative (see Sec. III A) significantly in the near future, it has been important to and will realistically fall below even the Belle-II reach in consider NN → Z0Z0 processes, in addition to NN → Fig. 7. ff¯, which is in turn dominant for heavy Z0. A careful 7

s treatment of the scalar S, responsible for the breaking of 0 2 ! 2 0 α MZ0 2Mq 4Mq B − L, has shown that its eﬀect can make the bounds Γ(Z → qq¯ ) = 1 + 2 1 − 2 , (A2) 9 M 0 M 0 on successful leptogenesis even more than an order of Z Z magnitude stronger. 0 2 3/2 0 α MZ0 4MN Γ(Z → NN) = 1 − 2 , (A3) 6 MZ0 0 α M 0 Acknowledgements Γ(Z0 → νν) = Z , (A4) 6 3/2 JH thanks Sebastian Ohmer, Felix Kahlhoefer, and M 2 4M 2 Γ(S → NN) = α0M N 1 − N , (A5) Camilo Garcia-Cely for discussions and Brian Shuve and S 2 2 MZ0 MS Christopher Hearty for kindly sharing their data shown 2 s 2 in Fig. 7. We thank Jean-Marie Fr`ereand Thomas Ham- 0 0 0 MS 4MZ0 Γ(S → Z Z ) = α MS 2 1 − 2 bye for comments on the manuscript. DT is supported by 2M 0 M Z S (A6) the Belgian Federal Science Policy through the Interuni- 2 4 4MZ0 12MZ0 versity Attraction Pole P7/37. JH is a postdoctoral re- × 1 − 2 + 4 . searcher of the F.R.S.-FNRS. We acknowledge the use of MS MS Package-X [58]. The spin-averaged total cross section for NN → ff¯ via 0 s-channel Z depends on the B − L charge QB−L and Appendix A: Formulas color multiplicity Nc of the fermion f:

2 04 For the convenience of the reader, we collect the rel- Nc(f)QB−L(f) g 0 σ(NN → ff¯ ) = evant decay widths of the new particles Z and S into h 2 2 2 2 i 12πs (M 0 − s) + M 0 Γ 0 leptons ` ∈ {e, µ, τ}; quarks q ∈ {u, d, c, s, t, b}; light Z Z Z (A7) q neutrinos ν ∈ {νe, νµ, ντ }; and heavy neutrinos N ∈ 2 q 2 2 0 02 × s + 2Mf s − 4Mf s − 4MN . {N1,N2,N3}, using α ≡ g /4π: s 0 2 2 The spin-averaged cross section for NN → SS can be 0 ¯ α MZ0 2M` 4M` Γ(Z → ``) = 1 + 2 1 − 2 , (A1) written as 3 MZ0 MZ0

4 " p g0 M 2 s − 2M 2 (s − 4M 2 )(s − 4M 2) σ = N S N S 4 2 2 2 2 M 4 + M 2 (s − 4M 2) 4πMZ0 (MS − s) (2MS − s) s (4MN − s) S N S n 2 o × −32M 6 M 2 − s + 9M 8s + 3M 2 M 4 s2 − 6M 4 − 16M 2s + 4M 4 20M 6 + 4M 4s + 4M 2s2 − s3 N S S N S S S N S S S (A8) 2 2 6 4 2 4 2 2 3 2 2 4 2 + 2MN MS − s 18MS + 32MN MS − s + 10MSs − 11MSs + s + 16MN s − 5MS + MSs " p 2 2 2 ## (s − 4MN )(s − 4MS) s − 2MS ×arctanh 2 2 4 2 2 , 2MN (s − 4MS) − 2MS + 4MSs − s while the spin-averaged cross section for NN → Z0Z0 (including s-channel S exchange) is more complicated:

04 " 4 2 p 2 2 g M 0 2M 0 − s (s − 4M )(s − 4M 0 ) σ = Z Z N Z h 2 i 4 2 2 8 2 2 2 2 2 2 (MZ0 + MN (s − 4MZ0 )) 4πMZ0 (MS − s) + MSΓS (2MZ0 − s) s (s − 4MN ) 4 2 2 4 2 4 2 2 4 2 × −2MZ0 MZ0 − 4MN 2MSMN + MS − 8MSMN + 48MN MZ0 2 4 2 2 2 2 2 3 2 2 2 4 4 + MZ0 MSMN −16MN + MZ0 + 4MS −2MN MZ0 + MZ0 + 8MN MZ0 −28MN + MZ0 s 4 4 4 2 2 2 2 2 2 4 2 6 8 2 + 2 M M + 4M M + 8M M 0 + M M + 8M M 0 + 2M M 0 − M 0 s S N N S N Z N S N Z N Z Z (A9) 2 4 2 2 4 3 1 4 2 2 2 6 2 2 −MN 8MN + 8MN MZ0 + MZ0 s + 4 MZ0 MS − s −4 MS − 8MN MZ0 4MN − MZ0 2 4 2 2 4 6 2 4 2 2 − 4MZ0 32MN MZ0 − 8MN MZ0 + MZ0 + MS −4MN + 3MN MZ0 s 2 4 2 2 2 2 2 2 4 2 + −4MSMN + 4MN MS + 8MN MZ0 + MS + 12MN MZ0 s  2 2 p 2 2 ! 2 2 2 3o 2MZ0 − s + (s − 4MN )(s − 4MZ0 ) − 2M + M 0 s log . N Z  2 p 2 2  2MZ0 − s − (s − 4MN )(s − 4MZ0 ) The expression for NN → SZ0 is even more involved and will not be shown here. In the non-relativistic limit, all four annihilation cross sections NN → ff¯ , SS, Z0Z0, SZ0 match those of Ref. [32]. Notice that to calculate the relevant 8 thermally averaged rates entering the Boltzmann equations, one needs the spin-summed reduced cross sections [59] which can be readily inferred from the formulas above.

[1] L. Canetti, M. Drewes, T. Frossard, and Leptogenesis in the Minimal B − L Extended Standard M. Shaposhnikov, “Dark Matter, Baryogenesis and Model at TeV,” Phys. Rev. D83 (2011) 093011, Neutrino Oscillations from Right Handed Neutrinos,” arXiv:1011.4769. Phys. Rev. D87 (2013) 093006, arXiv:1208.4607. [20] A. Pilaftsis, “Electroweak Resonant Leptogenesis in the [2] P. Minkowski, “µ → eγ at a Rate of One Out of 109 Singlet Majoron Model,” Phys. Rev. D78 (2008) Muon Decays?,” Phys. Lett. B67 (1977) 421–428. 013008, arXiv:0805.1677. [3] M. Fukugita and T. Yanagida, “Baryogenesis Without [21] M. D’Onofrio, K. Rummukainen, and A. Tranberg, Grand Unification,” Phys. Lett. B174 (1986) 45–47. “Sphaleron Rate in the Minimal Standard Model,” Phys. [4] S. Davidson and A. Ibarra, “A Lower bound on the Rev. Lett. 113 (2014) 141602, arXiv:1404.3565. right-handed neutrino mass from leptogenesis,” Phys. [22] G. F. Giudice, A. Notari, M. Raidal, A. Riotto, and Lett. B535 (2002) 25–32, arXiv:hep-ph/0202239. A. Strumia, “Towards a complete theory of thermal [5] T. Hambye, Y. Lin, A. Notari, M. Papucci, and leptogenesis in the SM and MSSM,” Nucl. Phys. B685 A. Strumia, “Constraints on neutrino masses from (2004) 89–149, arXiv:hep-ph/0310123. leptogenesis models,” Nucl. Phys. B695 (2004) 169–191, [23] P. S. Bhupal Dev, P. Millington, A. Pilaftsis, and arXiv:hep-ph/0312203. D. Teresi, “KadanoffBaym approach to flavour mixing [6] A. Pilaftsis, “CP violation and baryogenesis due to and oscillations in resonant leptogenesis,” Nucl. Phys. heavy Majorana neutrinos,” Phys. Rev. D56 (1997) B891 (2015) 128–158, arXiv:1410.6434. 5431–5451, arXiv:hep-ph/9707235. [24] E. K. Akhmedov, V. A. Rubakov, and A. Yu. Smirnov, [7] A. Pilaftsis and T. E. J. Underwood, “Resonant “Baryogenesis via neutrino oscillations,” Phys. Rev. leptogenesis,” Nucl. Phys. B692 (2004) 303–345, Lett. 81 (1998) 1359–1362, arXiv:hep-ph/9803255. arXiv:hep-ph/0309342. [25] T. Asaka and M. Shaposhnikov, “The nuMSM, dark [8] P. S. Bhupal Dev, P. Millington, A. Pilaftsis, and matter and baryon asymmetry of the universe,” Phys. D. Teresi, “Flavour Covariant Transport Equations: an Lett. B620 (2005) 17–26, arXiv:hep-ph/0505013. Application to Resonant Leptogenesis,” Nucl. Phys. [26] P. Hernández,M. Kekic, J. López-Pavón,J. Racker, B886 (2014) 569–664, arXiv:1404.1003. and N. Rius, “Leptogenesis in GeV scale seesaw [9] T. Hambye and D. Teresi, “Higgs doublet decay as the models,” JHEP 10 (2015) 067, arXiv:1508.03676. origin of the baryon asymmetry,” Phys. Rev. Lett. 117 [27] M. Drewes, B. Garbrecht, D. Gueter, and J. Klaric, (2016) 091801, arXiv:1606.00017. “Leptogenesis from Oscillations of Heavy Neutrinos with [10] R. N. Mohapatra and G. Senjanovic, “Neutrino Mass Large Mixing Angles,” arXiv:1606.06690. and Spontaneous Parity Violation,” Phys. Rev. Lett. 44 [28] P. Hernández,M. Kekic, J. López-Pavón,J. Racker, (1980) 912. and J. Salvado, “Testable Baryogenesis in Seesaw [11] R. N. Mohapatra and G. Senjanovic, “Neutrino Masses Models,” JHEP 08 (2016) 157, arXiv:1606.06719. and Mixings in Gauge Models with Spontaneous Parity [29] P.-H. Gu and U. Sarkar, “Leptogenesis Bound on Violation,” Phys. Rev. D23 (1981) 165. Spontaneous Symmetry Breaking of Global Lepton [12] S. Carlier, J. M. Frère,and F. S. Ling, “Gauge dilution Number,” Eur. Phys. J. C71 (2011) 1560, and leptogenesis,” Phys. Rev. D60 (1999) 096003, arXiv:0909.5468. arXiv:hep-ph/9903300. [30] D. Aristizabal Sierra, M. Tortola, J. W. F. Valle, and [13] J.-M. Frère,T. Hambye, and G. Vertongen, “Is A. Vicente, “Leptogenesis with a dynamical seesaw leptogenesis falsifiable at LHC?,” JHEP 01 (2009) 051, scale,” JCAP 1407 (2014) 052, arXiv:1405.4706. arXiv:0806.0841. [31] F. Kahlhoefer, K. Schmidt-Hoberg, T. Schwetz, and [14] P. S. Bhupal Dev, C.-H. Lee, and R. N. Mohapatra, S. Vogl, “Implications of unitarity and gauge invariance “TeV Scale Lepton Number Violation and for simplified dark matter models,” JHEP 02 (2016) Baryogenesis,” J. Phys. Conf. Ser. 631 (2015) 012007, 016, arXiv:1510.02110. arXiv:1503.04970. [32] M. Duerr, F. Kahlhoefer, K. Schmidt-Hoberg, [15] N. Cosme, “Leptogenesis, neutrino masses and gauge T. Schwetz, and S. Vogl, “How to save the WIMP: global unification,” JHEP 08 (2004) 027, analysis of a dark matter model with two s-channel arXiv:hep-ph/0403209. mediators,” JHEP 09 (2016) 042, arXiv:1606.07609. [16] M. Plümacher, “Baryogenesis and lepton number [33] J. Heeck, “Unbroken B − L symmetry,” Phys. Lett. violation,” Z. Phys. C74 (1997) 549–559, B739 (2014) 256–262, arXiv:1408.6845. arXiv:hep-ph/9604229. [34] S. Andreas, C. Niebuhr, and A. Ringwald, “New Limits [17] J. Racker and E. Roulet, “Leptogenesis, Z0 bosons, and on Hidden Photons from Past Electron Beam Dumps,” the reheating temperature of the Universe,” JHEP 03 Phys. Rev. D86 (2012) 095019, arXiv:1209.6083. (2009) 065, arXiv:0812.4285. [35] BaBar, J. P. Lees et al.,“Search for a Dark Photon in [18] S. Blanchet, Z. Chacko, S. S. Granor, and R. N. e+e− Collisions at BaBar,” Phys. Rev. Lett. 113 Mohapatra, “Probing Resonant Leptogenesis at the (2014) 201801, arXiv:1406.2980. LHC,” Phys. Rev. D82 (2010) 076008, [36] R. Harnik, J. Kopp, and P. A. N. Machado, “Exploring arXiv:0904.2174. nu Signals in Dark Matter Detectors,” JCAP 1207 [19] S. Iso, N. Okada, and Y. Orikasa, “Resonant (2012) 026, arXiv:1202.6073. 9

[37] S. Bilmis, I. Turan, T. M. Aliev, M. Deniz, L. Singh, [49] FCC-ee study Team, A. Blondel, E. Graverini, and H. T. Wong, “Constraints on Dark Photon from N. Serra, and M. Shaposhnikov, “Search for Heavy Neutrino-Electron Scattering Experiments,” Phys. Rev. Right Handed Neutrinos at the FCC-ee,” in Proceedings, D92 (2015) 033009, arXiv:1502.07763. 37th International Conference on High Energy Physics [38] S. Alekhin et al.,“A facility to Search for Hidden (ICHEP 2014): Valencia, Spain, July 2-9, 2014. 2016. Particles at the CERN SPS: the SHiP physics case,” arXiv:1411.5230. Rept. Prog. Phys. 79 (2016) no. 12, 124201, [50] S. Antusch and O. Fischer, “Testing sterile neutrino arXiv:1504.04855. extensions of the Standard Model at future lepton [39] D. Gorbunov, A. Makarov, and I. Timiryasov, colliders,” JHEP 05 (2015) 053, arXiv:1502.05915. “Decaying light particles in the SHiP experiment: Signal [51] S. Antusch, E. Cazzato, and O. Fischer, “Displaced rate estimates for hidden photons,” Phys. Rev. D91 vertex searches for sterile neutrinos at future lepton (2015) 035027, arXiv:1411.4007. colliders,” arXiv:1604.02420. [40] K. Kaneta, Z. Kang, and H.-S. Lee, “Right-handed [52] CHARM-II, P. Vilain et al.,“Precision measurement neutrino dark matter under the B − L gauge of electroweak parameters from the scattering of interaction,” arXiv:1606.09317. muon-neutrinos on electrons,” Phys. Lett. B335 (1994) [41] T. Ferber, “Towards First Physics at Belle II,” Acta 246–252. Phys. Polon. B46 (2015) no. 11, 2285. [53] I. Hoenig, G. Samach, and D. Tucker-Smith, “Searching [42] Christopher Hearty, private communication. for dilepton resonances below the Z mass at the LHC,” [43] HPS, A. Celentano, “The Heavy Photon Search Phys. Rev. D90 (2014) 075016, arXiv:1408.1075. experiment at Jefferson Laboratory,” J. Phys. Conf. Ser. [54] A. Alves, A. Berlin, S. Profumo, and F. S. Queiroz, 556 (2014) 012064, arXiv:1505.02025. “Dirac-fermionic dark matter in U(1)X models,” JHEP [44] D. G. Cerdeño,M. Fairbairn, T. Jubb, P. A. N. 10 (2015) 076, arXiv:1506.06767. Machado, A. C. Vincent, and C. Bœhm, “Physics from [55] M. Klasen, F. Lyonnet, and F. S. Queiroz, “NLO+NLL solar neutrinos in dark matter direct detection Collider Bounds, Dirac Fermion and Scalar Dark experiments,” JHEP 05 (2016) 118, arXiv:1604.01025. Matter in the B − L Model,” arXiv:1607.06468. [45] P. Ilten, J. Thaler, M. Williams, and W. Xue, “Dark [56] B. Batell, M. Pospelov, and B. Shuve, “Shedding Light photons from charm mesons at LHCb,” Phys. Rev. D92 on Neutrino Masses with Dark Forces,” JHEP 08 (2015) 115017, arXiv:1509.06765. (2016) 052, arXiv:1604.06099. [46] P. Ilten, Y. Soreq, J. Thaler, M. Williams, and W. Xue, [57] K. Huitu, S. Khalil, H. Okada, and S. K. Rai, “Proposed Inclusive Dark Photon Search at LHCb,” “Signatures for right-handed neutrinos at the Large Phys. Rev. Lett. 116 (2016) 251803, arXiv:1603.08926. Hadron Collider,” Phys. Rev. Lett. 101 (2008) 181802, [47] F. F. Deppisch, J. Harz, and M. Hirsch, “Falsifying arXiv:0803.2799. High-Scale Leptogenesis at the LHC,” Phys. Rev. Lett. [58] H. H. Patel, “Package-X: A Mathematica package for 112 (2014) 221601, arXiv:1312.4447. the analytic calculation of one-loop integrals,” Comput. [48] F. F. Deppisch, J. Harz, M. Hirsch, W.-C. Huang, and Phys. Commun. 197 (2015) 276–290, H. Päs,“Falsifying High-Scale Baryogenesis with arXiv:1503.01469. Neutrinoless Double Beta Decay and Lepton Flavor [59] M. A. Luty, “Baryogenesis via leptogenesis,” Phys. Rev. Violation,” Phys. Rev. D92 (2015) 036005, D45 (1992) 455–465. arXiv:1503.04825.