Asymmetric Dark Matter and CP Violating Scatterings in a UV Complete Model
Iason Baldes, Nicole F. Bell, Alexander J. Millar, Raymond R. Volkas
PPrepared for submission to JCAP
Asymmetric Dark Matter and CPViolating Scatterings in a UVComplete Model
Iason Baldes, Nicole F. Bell, Alexander J. Millar and RaymondR. Volkas
ARC Centre of Excellence for Particle Physics at the Terascale,School of Physics, The University of Melbourne, Victoria 3010, AustraliaE-mail: [email protected], [email protected],[email protected], [email protected]
Abstract.
We explore possible asymmetric dark matter models using CP violating scatter-ings to generate an asymmetry. In particular, we introduce a new model, based on DMfields coupling to the SM Higgs and lepton doublets, a neutrino portal, and explore its UVcompletions. We study the CP violation and asymmetry formation of this model, to demon-strate that it is capable of producing the correct abundance of dark matter and the observedmatter-antimatter asymmetry. Crucial to achieving this is the introduction of interactionswhich violate CP with a T dependence. a r X i v : . [ h e p - ph ] S e p ontents A.1 Minimal Neutrino Portal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17A.2 Extended Neutrino Portal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
B Thermal Masses 20C Chemical Potentials 20D Extensions of the Neutron Portal 21
D.1 → scatterings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21D.2 What about the symmetric component? . . . . . . . . . . . . . . . . . . . . . 23 E Other → ADM Scattering Operators 24
Measurements of the CMB and BBN show that the baryon-to-entropy density ratio in ouruniverse [1, 2], Y B = (0 . ± . × − , (1.1)is much higher than the SM predicts [3, 4]. Further, observations show us that the StandardModel (SM) only describes 15% of the matter in the universe, the remainder being dark mat-ter (DM). In light of this, it is very suggestive that the mass densities of dark and visiblematter are so similar, with Ω DM (cid:39) V M where Ω DM ( Ω V M ) is the DM (visible matter)density expressed as a ratio to the critical density. In many standard DM scenarios this mustbe taken as a coincidence. On the other hand, in asymmetric dark matter (ADM) modelsthe DM density is due to an asymmetry which is linked to the baryon asymmetry in someway [5–7]. – 1 –o do this requires that the DM and visible sectors be related - possibly at very highenergies. Suppose there is a conserved global symmetry between dark and visible matter suchthat B − L − D is conserved, where B and L are the usual baryon and lepton numbers and D is the dark matter charge. For asymmetries to form, B − L + D must be broken. Then if Ω DM is due to a D asymmetry, the DM mass must lie in a narrow range to ensure the correctabundance [6], m DM (cid:39) Q D × (1 . − GeV , (1.2)where Q D is the charge of dark matter under D . The value of 1.7 GeV holds for totally asym-metric dark matter with Q D = 1 where baryogenesis occurs entirely before the electroweakphase transition (EWPT), as will be considered in this work.As in normal baryogenesis, a process must satisfy the Sakharov conditions (particle num-ber violation, C and CP violation and thermal non-equilibrium), to be capable, in principle,of forming a particle number asymmetry [8]. Decays of heavy particles, first order phasetransitions and the Affleck-Dine mechanism have all been proposed as mechanisms to createa baryon asymmetry and related DM asymmetry. A full list of references can be found inreviews [5–7]. In this paper we will instead study the creation of an asymmetry through CPviolating scatterings.The possibility of using CP violating scatterings or coannihilations to generate ADM wasfirst proposed in [9], the idea of using asymmetric freeze-out was introduced in [10] and baryo-genesis via annihilating particles was also studied in [11]. The details of such a mechanism inADM were further explored in a toy model context in [12]. A baryogenesis mechanism, usingthe neutron portal effective operator, in which CP violating scatterings typically dominateover decays, was discussed by the present authors in [13]. CP violating scatterings have alsobeen studied in leptogenesis, in which they are negligible compared to decays outside of theweak washout regime [14–16]. The aim of the current paper is to explore for the first timeviable and UV complete ADM models which use the mechanism of CP violating scatteringsduring freeze-out. The DM will be stabilised using a Z symmetry rather than couplings witha strong temperature dependence as in [10].CP violating ↔ processes are also crucial to the WIMPy baryogenesis scenarios, inwhich the baryon asymmetry is generated by the annihilations of weakly interacting massiveparticles (WIMPs) during freeze-out. However, the DM density is not due to an asymmetryin those scenarios [17–21]. CP violating scattering has also been studied in an ADM context,albeit for freeze-in scenarios [9, 22, 23].The structure of this paper is as follows. In Section 2 we introduce the neutrino portal.We couple LH to an exotic scalar and fermion, the lightest of which will form DM, anddiscuss possible UV completions. In Section 3 we analyse asymmetry production in theminimal inert two-Higgs-doublet UV completion of the neutrino portal. We find it cannotproduce an asymmetry of the observed size. Due to this we discuss a simple extension inSection 4, introducing an additional Higgs doublet, and show the observed asymmetry can beobtained due to the additional CP violation present. We then conclude. Alternative operatorsare discussed in Appendices D and E. – 2 – The Neutrino Portal
When trying to write down a model of ADM, it is simplest to start with the lowest massdimension gauge invariant operator involving lepton or baryon number, LH ( H ∼ (1 , , − ).This must be paired with at least two non-SM particles in order to stabilise dark matterwith a symmetry, leading to a neutrino portal [24, 25]. This can be achieved via the effectiveoperator gLY Hφ, (2.1)where g is a coupling with inverse dimension of mass, L ( H ) is the SM Lepton (Higgs) doubletand Y ( φ ) is an exotic fermion (scalar). The latter two fields will comprise our dark sector,in which the lightest particle’s stability will be ensured by a Z symmetry. Both φ and Y areodd under this symmetry, and all SM particles are even.It is preferable to have only one dark sector particle carry a baryon number equivalent D , because if both particles did, then they must both be stable. We can define a dark baryonnumber D = ( N φ + N Y ) / (where N X is the particle number of X ), so that B − L − D isconserved and B − L + D is broken, as desired for ADM. Unfortunately if both φ and Y carry D we have a U(1) symmetry in addition to B − L − D conservation. This is obtained byrephasing only the dark sector particles, giving ∆ φ − ∆ Y = 0 , (2.2)where ∆ X = N X − N X . If φ or Y decays, (2.2) requires that the asymmetry stored in theremaining particle vanishes. Then baryogenesis does not occur as B − L = 0 . This problemis avoided if only one dark sector particle carries D , e.g. D = N φ .To keep both φ and Y stable would require kinematically disallowing decays with a ratherimplausible mass difference smaller than the neutrino masses. This mass difference cannot beobtained as one particle species must go out-of-equilibruim above the EWPT (so sphaleronscan reprocess ∆ L into ∆ B ) and DM must be of order a few GeV. While it is possible thatadditional interactions could prevent Y or φ from carrying an extra dark conserved charge,for simplicity we will not consider that scenario. This leaves two possibilities: either φ is acomplex scalar and Y a Majorana fermion, or φ is a real scalar and Y a Dirac fermion. Wechoose the former case as it leads to more CP violating phases and permits a simpler methodto annihilate the symmetric component. Asymmetry production should be the same in bothcases, though they will have phenomenologically different dark matter (scalar versus fermion).We envision the Y s as the most massive particles (barring the mediator), going out-of-equilibrium before our dark matter candidate φ . As they do not store an asymmetry, theycan safely decay without removing the asymmetry generated by the scatterings. To satisfyunitarity constraints, which we will discuss in Section 3.1, there must be two generations of Y s and one of φ . Our ADM will be φ , which carries a non-zero baryon number equivalent D = N φ , preserving B − L − D . If there are additional dark sector particles carrying D , then an extension of this argument still holds: atleast two must be stable. As this is non-minimal, we do not consider that scenario in this work As a special case this can be avoided if the scatterings freeze-out before the EWPT but the particle decaysafter the EWPT, because B and L are no longer related by electroweak sphalerons. – 3 – igure 1 . The interaction LY Hφ together with its UV completions. We label them as case 1 (topright), case 2 (bottom left) and case 3 (bottom right).
In this paper we consider UV completions of the effective interaction, eq. (2.1). Thesimplest tree level possibilities are depicted diagrammatically in fig. 1. The thermal historiesof all UV completions are essentially the same. At high temperature the Y s are kept inequilibrium by the rapid → interactions. These interactions eventually freeze-out andthe Y s depart from thermal equilibrium, satisfying the third Sakharov condition. As thesescatterings violate CP, a non-zero B − L is generated along with an associated D asymmetrystored in the φ , creating a baryon asymmetry via the sphalerons (as long as this occurs abovethe EWPT). Subsequently, the Y s decay into leptons and φ , potentially leading to furtherasymmetry formation. To see which process dominates, as well as the size of the asymmetryformed, it is necessary to solve the Boltzmann equations, which we will do in Section 3. Thethree simple UV completions are as follows.• The first case requires the mediator, f , to be a vector-like lepton. The Z symmetryprevents f from Yukawa coupling via the Higgs to the SM leptons, avoiding most ofthe limits on vector-like leptons [26, 27]. For CP violation to arise at one-loop it isnecessary to include two copies of f .• The second case has a Dirac SM singlet, f as intermediary. The Z symmetry precludesthe Y from acting as an intermediate particle and coupling to the leptons, which woulddestroy the global B − L − D . If Y is Dirac, rather than Majorana, this operator isvery similar to [9]. In [9], a Majorana fermion mediates an interaction between LH anda mirror copy, L (cid:48) H (cid:48) , using different initial temperatures to create out-of-equilibriumconditions. CP violation is the same as in the first completion.• The third case is an extension of inert two-Higgs-doublet models (IDM) [28]. Theintermediate particle, H , has the same quantum numbers as the SM Higgs but cannotplay the same role. While CP violation is possible with just one inert Higgs doublet,we will show that to get a sufficient asymmetry to form, two inert Higgs doublets arenecessary. While regular IDMs do have a dark matter candidate, they do not explainthe ratio of dark and visible matter; the mass ranges required for H to be ADM instead– 4 –f φ are excluded by a combination of collider searches, electroweak precision tests anddirect detection constraints [29, 30]. As the IDM completion is particularly illustrative we will study this case in more detail.
To see a specific model in action, we will study the IDM completion of the neutrino portal.As in IDM, we add a massive scalar SU (2) L doublet which will function as our heavy inter-mediary. We also introduce a new parity, with the Lagrangian symmetric under H → − H , Y → − Y , and φ → − φ . All SM particles are even under this parity. We call H inert as itdoes not acquire a non-zero vacuum expectation value (VEV), and does not play a role in thefermion mass generation. Where this work differs from traditional IDMs is that in our case H carries a non-zero B − L . The relevant additions to the SM Lagrangian are ∆ L = − λ ia H L i Y a − κH H † φ − λ | H | | H | − λ | H † H | − λ | H | | φ | − λ | φ | + H.c. (2.3)This induces (2.1) after integrating out the mediating H . After considering field rephasings,there are three physical CP violating phases, that we write as θ = Arg( λ ) , (2.4) θ = Arg( λ ) , (2.5) θ = Arg( λ ) , (2.6)with all other couplings real (without loss of generality). For our example solutions we choose θ = θ = θ = π/ .We consider the Y to have a mass higher than the scale of the EWPT, and H to beheavy enough to be considered always off shell, typically several orders of magnitude heavierthan the next lightest particle, Y . As the neutrino portal does not violate baryon numberdirectly, electroweak sphalerons are required to reprocess the lepton asymmetry into a baryonasymmetry. For the parameters considered in this paper, all relevant processes finish beforethe EWPT.We assume that there is an additional process that keeps the φ in equilibrium and anni-hilates the symmetric component, but remain agnostic as to the details of this process. For anexample, φ quartic coupling to a light real scalar field could be used for this purpose. In morebaroque extensions it is also possible for φ to decay into lighter particles, and for them to havenew interactions that annihilate their symmetric component. While for the purposes of thispaper the exact method is irrelevant, different mechanisms will provide different detectionprospects. For example, if φ annihilates into a light, stable state then there is a significantcontribution to dark radiation [6]. An exception to this can be found in [30], where an inert Higgs transfers a pre-existing asymmetry betweenthe dark and visible sectors. The main effect of this is on neutrino masses. In standard IDMs neutrinos gain Majorana masses throughradiative corrections [31], which requires a term λ ( H ∗ H ) . As this term breaks B − L − D , for simplicity weconsider instead neutrinos gaining a mass through the Higgs mechanism (with the addition of right handedneutrinos even under the Z ). This can be a Dirac mass or a Majorana mass if lepton number is softly broken. – 5 –riginally it was thought possible to use CP violating scatterings to simultaneouslycreate an asymmetry and annihilate the symmetric component [10]. Unfortunately, this isimpossible as it would require the freeze-out temperature to be of order m DM / . For op-erators like those of [10], decays prevent an asymmetry forming at arbitrarily low freeze-outtemperatures. If we make the coupling strength large enough to delay scattering freeze-outuntil T freeze-out ∼ m DM / , the lifetime becomes short enough that decays and inverse decayskeep the particle in equilibruim, so Sakharov’s conditions are never met. We find numeri-cally that the lifetime becomes shorter than t freeze-out well before T freeze-out ∼ m DM / . Asdiscussed in Appendix D.2, models that might have this feature are disfavoured by limits onnon-SM coloured particles. Due to the mass range of H we are considering most of the couplings related to asym-metry formation in our theory are unconstrained, but there are still some restrictions fromelectroweak considerations. While the population of H is negligible during the EWPT, it is still worth considering theeffects of the EWPT on H as it couples to φ . In addition, while φ is a SM singlet it canstill potentially influence the EWPT. During electroweak symmetry breaking (EWSB), H acquires a VEV v and acts as the SM Higgs; on the other hand H only experiences masssplitting. It is possible to parameterize H as H = (cid:18) H H − (cid:19) , (2.7)where H and H − are complex scalars. After EWSB we have m H = m H + ( λ + λ ) v , (2.8) m H − = m H + λ v . (2.9)There is also a contribution to the mass of φ when H acquires a VEV, as well as mass mixingbetween H and φ . This gives us a mass matrix in the ( φ, H ) basis (cid:32) m φ + λ v κvκv m H (cid:33) . (2.10)We expect this to diagonalise to a heavy state and a light state. We will label the light state φ (cid:48) as the dark matter admixture is mostly φ , with mixing angle κv/m H . We have a lightenough dark matter candidate for small mixing angles, vκ/m H (cid:46) ( m φ + λ v ) / . (2.11) While directly searching the full possible mass range of H is impossible at the LHC, it is possible to ruleout high scale baryogenesis in general. If low scale lepton number violation (LNV) is observed, either throughdirect observation of LNV at the LHC or through a combination of neutrinoless double beta decay and leptonflavour violation, then the washout induced by this LNV makes any high scale baryogenesis irrelevant [32, 33]. – 6 –his requirement for small mixing is the only real constraint relevant to asymmetry forma-tion. For κ (cid:39) m Y and m φ (cid:39) , we have m φ (cid:48) (cid:39) λ v . As we envision φ (cid:48) to be ADM, we musthave λ (cid:46) × − to get the correct relic density ( m φ (cid:48) (cid:39) . GeV as in (1.2)). Introducing φ can potentially alter the EWPT, but only via its quartic coupling to H . One might worry that if the EWPT is made first order, there would be two competingmethods for baryogenesis - electroweak baryogenesis and scatterings. From the analysis in[34] it can be shown that the EWPT is first order if λ π − λ H ( λ + λ ) > (cid:20) λ H (cid:18) λ y t (cid:19) − λ π (cid:21) (cid:16) m φ v (cid:17) , (2.12)where λ H is the quartic self coupling of H and y t is the Yukawa coupling of H to the topquark. As λ is small, the EWPT remains second order. We make the approximation thatthe electroweak phase transition occurs the same way as in the SM. As H still couples to the electroweak gauge bosons, electroweak precision tests are, in prin-ciple, sensitive to H . More precisely the electroweak precision tests are only sensitive to themass splitting of H [28]. Standard electroweak precisions tests can be cast in terms of the"oblique" or Peskin-Takeuchi parameters S, T and U which parametrize the radiative correc-tions due to new physics [35]. From [28] the main contributions to electroweak precision tests(at one loop order) can be written as ∆ T (cid:39) λ π α (cid:39) . λ , (2.13) ∆ S (cid:39) λ v π m H − (cid:39) × − × λ TeV m H − . (2.14)The experimental values are ∆ T U =0 = 0 . ± . and ∆ S U =0 = 0 . ± . , so λ (cid:46) . isrequired to satisfy electroweak precision tests [36]. Fortunately, λ does not affect asymmetryproduction, so this constraint does not seriously affect the model. ∆ S is negligible for allsensible parameter choices. While the self couplings of φ do not affect asymmetry production directly, they will contributeto the thermal mass of φ . The most stringent limits on dark matter self interactions come fromthe Bullet Cluster and similar colliding clusters. From this there is the constraint, [37–39] λ (cid:46) × (cid:18) m φ (cid:48) GeV (cid:19) . (2.15)This leaves λ essentially unconstrained. As the thermal mass of φ will be determined by λ ,this freedom is useful for the kinematics. We use the thermal masses described in Appendix B In this case λ is much too small for φ to be detected at the LHC via invisible Higgs decays. However, ifthere are significant cancelations between m φ and λ v it is possible for λ to be large enough for the invisibledecays of the SM Higgs into φ to be detectable at the LHC, though this is not required for our scenario towork [40]. – 7 –hroughout this paper.Fortuitously only one of these constraints affect parameters required for asymmetryformation, so we are free to choose parameters to maximise the asymmetry formed. The onlyreal constraint is (2.11). Now that we have invented some interactions satisfying the Sakharov conditions, we need toknow the magnitude and nature of the asymmetry they generate. Our main technique forcalculating the evolution of an asymmetry will be Boltzmann equations. We adopt the same W ( ψ, a . . . → i, j . . . ) ≡ n eqψ n eqa ... (cid:104) vσ ( ψ, a . . . → i, j . . . ) (cid:105) notation as [13], where (cid:104) vσ ( ψ, a . . . → i, j . . . ) (cid:105) denotes a thermally averaged cross section, and parameterize the CP violation as (cid:15) ≡ W ( ψ, a . . . → i, j . . . ) − W ( i, j . . . → ψ, a . . . ) W ( ψ, a . . . → i, j . . . ) + W ( i, j . . . → ψ, a . . . ) . We define the CP symmetric reaction rate density as: W sym = 12 (cid:104) W ( i, j . . . → ψ, a . . . ) + W ( ψ, a . . . → i, j . . . ) (cid:105) . (3.1)In general both (cid:15) ψ,a... → i,j... and W sym will be temperature dependent. As we will be usingMaxwell-Boltzmann statistics as a (good) approximation throughout this paper, we can factorout the chemical potential from the phase space f ψ = e ( µ ψ − E ψ ) /T = e µ ψ /T f eqψ , (3.2)where f eq refers to the equilibrium values when the chemical potential is zero, and define: r ψ ≡ n ψ n eqψ = e µ ψ /T . (3.3)Putting this all together, for the process ψ + a + b + . . . −→ i + j + . . . , evolving in theFriedmann-Robertson-Walker metric, the evolution of n ψ is given by [41]: dndt + 3 Hn = C ( ψ ) , (3.4)where H is the Hubble expansion rate and the collision term C ( ψ ) is: C ( ψ ) = W sym (cid:104) r i r j . . . (1 − (cid:15) ψ,a... → i,j... ) − r ψ r a . . . (1 + (cid:15) ψ,a... → i,j... ) (cid:105) . (3.5)We compute (cid:104) vσ ( ψ + a → i + j ) (cid:105) from the cross sections by using the result from [42], (cid:104) vσ ( ψ + a → i + j ) (cid:105) = g ψ g a T π n eqψ n eqa ˆ ∞ ( m ψ + m a ) p ψa E ψ E a v rel σK (cid:18) √ sT (cid:19) ds, (3.6)where s is the square of the centre-of-mass energy, p ψa is the centre-of-mass momentum of ψ and a and K i ( x ) is a modified Bessel function of the second kind of order i . As a note– 8 –f caution, we will not use the Einstein summation convention when discussing Boltzmannequations.The requirement that CPT and unitarity must hold imposes an important restrictionon our CP violating terms. In our language the unitarity constraint is given by: [43–45] (cid:88) j W ( i → j ) = (cid:88) j W ( j → i ) = (cid:88) j W ( i → j ) = (cid:88) j W ( i → j ) , (3.7)where i is the CP transform of i . In general (3.6) must be solved numerically, which can allowsmall artificial violations of unitarity to arise. To deal with these small errors, we enforce theunitarity relations (3.7).To calculate the CP violation we use Cutkosky rules [46]. It should be noted that theseare the cutting rules for zero temperature quantum field theory; when thermal effects dom-inate it is necessary to use the rules contained in [47, 48]. As we will only be interested inthe dominant contributions to CP violation near freeze-out, we will use the zero temperaturecutting rules.With these techniques, it is possible to analyse the asymmetry production. We now catalogue the relevant interactions. Our CP violating scatterings are given by: W ( Y a L → H φ ) CPT = W ( φ ∗ H ∗ → LY a ) = (1 + (cid:15) a ) W a , (3.8) W ( LY a → φ ∗ H ∗ ) CPT = W ( H φ → Y a L ) = (1 − (cid:15) a ) W a , (3.9)where we have included an implicit sum over the three families of leptons. Because all theleptons have essentially the same mass and chemical potential, it is not really necessary toconsider them as separate species except when summing the couplings for a process. Theseannihilations will be the main generators of the asymmetry. The relevant unitarity constraint,derived from (3.7), is (cid:15) W = − (cid:15) W . (3.10)The only CP violation for the process Y a L → H φ comes from interference betweengraphs in fig. 2, involving a Majorana mass insertion. No other one loop graphs lead to acomplex combination of couplings. Taking generic couplings λ ∼ λ ia for the Lagrangian in(2.3), the tree level cross section scales as σv ∼ λ κ M H . (3.11)Denoting σ as the CP conjugate cross section, the CP violation at the cross section levelscales as ( σ − σ ) v ∼ λ κ m Y m Y M H . (3.12)– 9 – i Y a H H φ L i Y a H H φH L j Y b Figure 2 . Graphs contributing to (cid:15) a . The one loop graph (right) has a Majorana mass insertion. Inserting this into (3.6), noting the momenta and energies scale as √ s and the Bessel function,which falls as K ( z ) ∼ (cid:112) π z e − z for z (cid:38) , imposes an effective cut-off on the integral at √ s ∼ T , one finds W ∝ λ κ T M H (3.13)and (cid:15) W ∝ λ κ m Y m Y T M H . (3.14)Hence, from this dimensional analysis, it is clear there is no T dependence in the CP violation, (cid:15) for T (cid:38) M Y . We also have CP conserving scatterings, which we label W ( Y a H → Lφ ∗ ) = T a , (3.15) W ( Y a φ → H ∗ L i ) = U a , (3.16) W ( Y a L → Y b L ) = S ab , (3.17) W ( LL → Y a Y b ) = P ab . (3.18)While LL → Y a Y b is not technically CP conserving, CP violation in this term only leads to aflavour asymmetry. As Y is Majorana, and not ultra-relativistic, there is no method to storethis asymmetry or transfer it to a non-zero B − L so these terms can be safely neglected. Forthe t-channel graphs, no loop graph with an absorptive part is kinematically allowed and sothere is no contribution to the CP violation. The expressions for the cross sections of theabove processes are given in Appendix A. We consider the following decays of Y , Y and H , though we are only interested in the CPviolation of the Y decays. We will label these decays by: Γ( H → Y a Lφ ∗ ) = Γ H → Y a (3.19) Γ( Y → Y LL ) = Γ A (3.20) Γ( Y → LH φ ) = 12 (1 + (cid:15) D )Γ B (3.21) Γ( Y → LH ∗ φ ∗ ) = 12 (1 − (cid:15) D )Γ B (3.22) Γ( Y → LH φ ) = Γ( Y → LH ∗ φ ∗ ) = 12 Γ , (3.23) This is, of course, ignoring the implicit temperature dependence of the thermal masses, but the overallpoint still stands. – 10 – igure 3 . Graphs contributing to (cid:15) D . The one loop graph (right) has a Majorana mass insertion. where we have parameterized the CP violation in decays by (cid:15) D . Note that (3.19) is kinemati-cally allowed at high temperatures due to the thermal mass of H . Contributions to (cid:15) D comefrom the graphs in fig. 3. We can see from this figure that Y decays are CP conserving: CPviolation would require an on shell Y in the loop, which is kinematically forbidden. Thereis a further contribution to the CP violation from the → scattering LH φ → LH ∗ φ ∗ ,with the real intermediate Y s subtracted [14]. While the CP symmetric component of thisscattering is negligible, CP violation in this process serves to cancel the CP violation in thedecays at thermal equilibrium [13]. This can be seen by applying the unitarity relation (3.7)to the state LH φ , to derive (cid:15) LH φ → LH ∗ φ ∗ W sym ( LH φ → LH ∗ φ ∗ ) = 12 (cid:15) D n eqY Γ B , (3.24)which we include in our Boltzmann equations. The chemical potentials of the SM fields and φ depend only on the B − L asymmetry (seeAppendix C), so we have only three coupled differential equations to solve. These Boltzmannequations are similar to those in [13]; in both cases we have two heavy Majorana particleswhich experience CP violation through decays and → scatterings. For Y and Y , wehave: dn Y dt + 3 Hn Y = Γ n eqY (cid:104) ( r l r H r φ + r l r H r φ ) / − r Y (cid:105) − Γ A n eqY (cid:104) r l r l r Y − r Y (cid:105) + Γ H → Y n eqH (cid:104) r H + r H − r l r φ r Y − r l r φ r Y (cid:105) + W (cid:104) r H r φ + r H r φ − r Y ( r l + r l ) (cid:105) − (cid:15) W (cid:104) r H r φ − r H r φ + r Y ( r l − r l ) (cid:105) + T (cid:104) r φ r l + r φ r l − r Y ( r H + r H ) (cid:105) + U (cid:104) r H r l + r H r l − r Y ( r φ + r φ ) (cid:105) − S (cid:104) ( r l + r l )( r Y − r Y ) (cid:105) + 2 P (cid:104) r l r l − r Y (cid:105) + P (cid:104) r l r l − r Y r Y (cid:105) . (3.25)– 11 –nd dn Y dt + 3 Hn Y = Γ B n eqY (cid:104) ( r l r H r φ + r l r H r φ ) / − r Y (cid:105) + Γ A n eqY (cid:104) r l r l r Y − r Y (cid:105) + Γ H → Y n eqH (cid:104) r H + r H − r l r φ r Y − r l r φ r Y (cid:105) − (cid:15) D Γ B n eqY (cid:104) r l r H r φ − r l r H r φ (cid:105) + W (cid:104) r H r φ + r H r φ − r Y ( r l + r l ) (cid:105) + (cid:15) W (cid:104) r H r φ − r H r φ + r Y ( r l − r l ) (cid:105) + T (cid:104) r φ r l + r φ r l − r Y ( r H + r H ) (cid:105) + U (cid:104) r H r l + r H r l − r Y ( r φ + r φ ) (cid:105) + S (cid:104) ( r l + r l )( r Y − r Y ) (cid:105) + 2 P (cid:104) r l r l − r Y (cid:105) + P (cid:104) r l r l − r Y r Y (cid:105) (3.26)The Boltzmann equation for B − L is: dn B − L dt + 3 Hn B − L = Γ n eqY (cid:104) r l r H r φ − r l r H r φ (cid:105) + Γ B n eqY (cid:104) r l r H r φ − r l r H r φ (cid:105) + Γ H → Y n eqH (cid:104) r H − r H + r l r φ r Y − r l r φ r Y + r l r φ r Y (cid:105) + Γ H → Y n eqH (cid:104) r H − r H + r l r φ r Y − r l r φ r Y + r l r φ r Y (cid:105) + (cid:15) W (cid:104) ( r l + r l )( r Y − r Y ) (cid:105) − (cid:15) D Γ B n eqY (cid:104) r Y − ( r l r H r φ + r l r H r φ ) (cid:105) + (cid:88) a =1 , (cid:16) W a (cid:104) r H r φ − r H r φ + r l r Y a − r l r Y a (cid:105) + T a (cid:104) r H r Y a − r H r Y a + r φ r l − r φ r l (cid:105) + U a (cid:104) r H r l − r H r l + r φ r Y a − r φ r Y a (cid:105)(cid:17) = dn D dt + 3 Hn D . (3.27)The terms with (cid:15) and (cid:15) D are source terms for the asymmetry, while all other terms tendto wash out the asymmetry. We have used (3.10) to write the Boltzmann equations solelyin terms of these two CP violating terms. The decay rates which appear here are thermallyaveraged (see Appendix A).We can now numerically solve these to see the evolution of an asymmetry. Our numerical solutions to the Boltzmann equations are stable under changes to the preci-sion, starting temperature (for initial temperatures m H (cid:29) T (cid:29) m Y ) and initial conditions.We start our solutions at high temperature, where → scatterings can wash out any pre-existing asymmetry, and then track the evolution of B − L down to the EWPT. To goodapproximation, at the EWPT sphalerons simply switch off, freezing the value of B . In gen-eral, if Y or Y is sufficiently long lived they may come to dominate the energy density atsome time, causing the universe to become matter dominated (instead of radiation dominatedas we assume). This would lead to a dilution factor due to reheating when the Y decay [49].For the interesting regions of parameter space (those which lead to large asymmetries) thiscondition is not satisfied but for completeness we do include the dilution factor in our code.– 12 – igure 4 . Asymmetry formed as a function of κ and m H . There is a ridge of values where theasymmetry formed is significant, corresponding to a freeze-out temperature of order M Y / . Thistemperature is due to the interplay between decays and scatterings: increasing the couplings leads to τ Y (cid:46) t freeze-out , which forces the Y to remain in equilibrium, and decreasing the couplings decreasesCP violation. Maximal asymmetry corresponds to κ ∼ m Y and m H ∼ × m Y . In this example, m Y = 90 TeV and m Y = 100 TeV.
For regimes where the two Y are relatively close in mass, scatterings dominate the asym-metry, as also found in [13]. As there are two mass scales in this problem (once we choosea mass for Y and Y ), we scan over κ and m H , keeping in mind (2.11). The asymmetry ismaximised when κ ∼ m Y and m H ∼ × m Y . Unfortunately, this model does not generatethe full asymmetry required for baryogenesis; from fig. 4 it can be seen that the maximumasymmetry is of order Y B ∼ − .This is quite puzzling: with such similar Boltzmann equations and asymmetry produc-tion methods, how can this model fail where [13] succeeded in obtaining Y B ∼ − ? The an-swer lies in the CP violation. Whereas in the neutron portal case there were graphs that gavethe CP violation in scatterings the property (cid:15) ( Xu → dd ) W ( Xu → dd ) ∝ T W ( Xu → dd ) ,there is no temperature dependence in our CP violating terms. As was suggested in [13],which we will show explicitly, in the neutron portal case this temperature dependence madethe scatterings relevant at a higher temperature, enhancing the asymmetry production. Toobtain similar temperature dependence, we must have CP violating graphs without Majoranamass insertions. To this end we must include a second copy of the mediating scalar. In fact,this conclusion also holds for [12, 13]: to get the full asymmetry found in the EFTs studied itis necessary to have two heavy intermediate scalars regardless of the number of CP violatingphases. – 13 – igure 5 . New one loop graphs contributing to CP violation. The graph on the left (right) contributesto (cid:15) a ( (cid:15) D ). Neither graph has a Majorana mass insertion. A simple extension of the neutrino portal, which allows the generation of the observed Y B ,is the addition of another inert Higgs. Three-Higgs models have been explored in both aninert and general context [50, 51]. We will not write down the full potential, as most of theterms are irrelevant for asymmetry formation, but simply note that to satisfy electroweakprecision tests it is necessary to avoid significant mixing between the two inert scalars [52].The Lagrangian is now ∆ L = − m H p | H p | − λ iap H p L i Y a − κ p H H † p φ + H.c, (4.1)where p = 2 , . With 14 relevant couplings there are now nine CP violating phases which, forthe sake of the example, we will choose to be Arg( λ ) = 0 , Arg( λ ) = π , Arg( λ ) = π , Arg( λ ) = π , Arg( λ ) = π λ ) = 2 π , Arg( λ ) = π , Arg( λ ) = 2 π , Arg( κ ) = π . (4.2)We have chosen phases that avoid cancellations between the various CP violating graphs.There are now additional graphs which lead to complex couplings, involving a closed fermionloop (fig. 5). As there are no Majorana mass insertions, these graphs exhibit the desiredtemperature dependence, (cid:15) W ∝ T W (see fig. 6). This can be seen by looking at thesecond term in (A.12), which corresponds to this new graph; relative to the first term, whichis due to the original CP violation with Majorana mass insertions (fig. 2), there are twoextra factors of energy and momentum, E and p . When velocity averaged with (3.6), athigh temperatures E ∼ p ∼ T . At low temperatures the CP violation becomes constant,though due to the different combination of phases involved it is not necessarily equal to theCP violation in fig. 2. There are similar contributions to the CP violation in decays, which wealso include. Armed with this new CP violation, we can again solve the Boltzmann equationsfor the evolution of B − L . With the temperature dependent CP violation, we see a significant increase in the asymmetry.As long as there are no significant cancellations between the various contributions to the (cid:15) a the full baryon asymmetry of the universe can be generated (see fig. 7). Fittingly for ADM,cancellations are minimised when the λ iap are not symmetric: it is preferable for the leptonsto couple more strongly to one of the generations of the Y and for not all generations of– 14 – igure 6 . CP violation in the extended neutrino portal with m Y = 90 TeV and m Y = 100 TeV.At high temperature the CP violation from the graphs in fig. 5 grows as T , and the CP violationfrom the graphs in fig. 2 is constant. At low temperatures both contributions become constant,and approximately equal in value. We have plotted both sources of CP violation with phases andcouplings chosen to maximise their CP violation, though no parameters maximise both simultaneously.At T ∼ × GeV, where maximal asymmetry production occurs, the CP violation from the graphsin fig. 5 dominates. leptons to couple with the same strength. This asymmetry in the couplings does not needto be more than an order of magnitude for significant enhancement, as in fig. 8. Similarly, itis preferable for κ and κ to be an order of magnitude different, and m H and m H to bewithin an order of magnitude of each other. Comparing this to the asymmetry formed whenonly the mass insertion graphs are included (fig. 7) we see that the asymmetry starts formingearlier, culminating in a significantly higher asymmetry.Due to the similarity of the asymmetry production, Boltzmann equations and CP viola-tion between this model and [13], it is clear that main asymmetry production in the neutronportal EFT studied in [13] was also due to this temperature dependent CP violation (ratherthan the mass insertion diagrams which are also there in the neutron portal case). This pro-vides compelling evidence that for scatterings to dominate over decays a heavy intermediateparticle is necessary (to provide the dimensionality for temperature dependence). As manycosmological models are EFTs, this will often be satisfied. Further, multiple intermediateparticles (leading to bubble graphs) seem to be a generic feature of scattering models, beingnecessary in not only this work but also [12, 13, 17]. We have shown that it is possible to create a realistic UV complete model of ADM that usesscatterings to generate the asymmetry. Further, we have demonstrated explicitly the impor-tance of temperature dependence in the CP violating terms.To do this, we introduced a new model, using an neutrino portal. By examining the
U V completions, we separately studied temperature independent and temperature dependent CP– 15 – igure 7 . Example solution with m Y = 90 TeV, m Y = 100 TeV. The asymmetry generated is Y B = 1 . × − . We use (cid:15) ∝ T and (cid:15) ∝ T to refer to the CP violation in the scatterings fromfigures 2 and 5, respectively. The asymmetry generated by the decays is negligible. Figure 8 . Asymmetry formed vs m H , with m H = 2 . m H . For asymmetric couplings the asym-metry formed is as much as two orders of magnitude higher than when all the couplings are equal.Asymmetric couplings can avoid cancellations between the various CP violating graphs. The asym-metric couplings have been chosen for this example to maximise asymmetry production. Coupling Y and Y to one generation of leptons, with the couplings differing by an order of magnitude, is sufficientfor a large asymmetry to form. When there is little or no discrimination between the couplings, theasymmetry formed is essentially the same as the unextended neutrino portal. If the two inert Higgshave the same masses and couplings then the temperature dependent CP violation cancels exactly. violation, with the latter proving to be far more significant. For future ADM model buildersto see a significant asymmetry caused by → scatterings, quadratic temperature depen-dence in the CP violation will be a key feature.– 16 –here are many other potential operators if one is willing to allow a slightly more compli-cated dark sector, including variations on the WIMPy baryogenesis operators. An interestingavenue for future research would be the annihilation of the symmetric component of the φ number density; while the same scatterings cannot simultaneously create an asymmetry andannihilate the symmetric component of ADM in existing models, other minimal options canstill be explored. Acknowledgments
IB and AM were supported by the Commonwealth of Australia. NFB and RRV were sup-ported in part by the Australian Research Council.
AppendixA Cross Sections of the Neutrino Portal
For the sake of completeness, here we catalogue the cross sections of the case study in Sections2.1 and 4. E , E , E and E refer to the energies of the particles in the order listed. For the → scatterings, we will denote the initial momentum p i and the final momentum p f . Allcross sections are written in the centre of mass frame, except when otherwise stated. A.1 Minimal Neutrino PortalA.1.1 Cross sections (CP conserving component)
We include some of the factors from (3.6) to highlight the symmetry under interchange ofparticles. Y a L i → H φ p i E E σv = | κ λ ia | π √ sm H p i p f ( E E + p i ) . (A.1) L i φ → Y a H p i E E σv = | κ λ ia | π √ sm H p i p f E E . (A.2) L i H → Y a φ p i E E σv = | κ λ ia | π √ sm H p i p f E E . (A.3) Y a L i → Y b L j p i E E σv = | λ jb λ ia | π √ sm H p i p f (cid:2) ( E E + p i )( E E + p f ) + E E E E + 1 / p i p f (cid:3) . (A.4) L i L j → Y a Y b p i E E σv = | λ jb λ ia | + | λ ja λ ib | π √ sm H p i p f (cid:2) E E E E + 1 / p i p f (cid:3) . (A.5)– 17 – .1.2 Cross sections (CP violating component) Y L i → H φp i E E ( σ − σ ) v = − (cid:88) j m Y m Y Im( λ i λ j λ ∗ i λ ∗ j ) κ π sm H p i p f p loop E E L j , (A.6)where E j is the energy of L j in the loop and p loop is the momentum of the (on shell) particlesin the loop. The factor of m Y m Y comes from projecting out the Majorana masses. CPviolation in Y a L i → Y b L j is similar but only leads to flavour violation and is so neglected. A.1.3 Decays (CP conserving component)
To calculate the three body decays, we used the usual trick of decomposing N-body phasespace into a series of 2-body phase spaces. All integrals are numerically integrated. The decayrates appearing in section 3.3 are in fact thermally averaged [14], Γ thermal = K ( m/T ) K ( m/T ) Γ . (A.7)Since the relative masses of the particles changes with temperature, at different times the Y , H and φ can all decay depending on the thermal masses (in particular the choice of λ ). Wewill only write down the Y decays as the others are similar and not particularly important. Inour example solutions we chose λ so that at high temperatures H → L i Y a φ occurs. Keepingthis in mind we now catalogue the decays. Y a → L i H ∗ φ ∗ Γ = ( m Ya − m L ) ˆ ( m φ + m H ) ds | κ λ ia | (2 π ) √ sm H m Y a E L ( s ) (cid:18)(cid:113) s + p L ( s ) + (cid:113) m L + p L ( s ) (cid:19) p L ( s ) p φ ( s ) , (A.8)where E L ( s ) is the energy of L i , p L ( s ) [ p φ ( s ) ] is the momentum of L i [ φ ] in the centre-of-momentum frame [rest frame of the mediating particle H ] and s is p µφ p H µ . When we saythe rest frame of H , we mean a fictitious on-shell particle with mass √ s in place of H . Y → L i Y L j Γ = ( m Y − m Li ) ˆ ( m Lj + m Y ) ds | λ λ i j | + | λ j λ i | )(2 π ) √ sm H m Y E L i ( s ) (cid:16)(cid:113) s + p L i ( s ) + (cid:113) m L i + p L i ( s ) (cid:17) (A.9) × p L i ( s ) p L j ( s ) (cid:16) E L j ( s ) E Y ( s ) + p L j ( s ) (cid:17) , where E L i ( j ) ( s ) and p L i ( j ) ( s ) are the energy and momentum of L i ( j ) , respectively, and s is p µY p L j µ . Note that p L j ( s ) is in the rest frame of H and all others are in the rest frame of H . – 18 – .1.4 Decays (CP violating component) Y → L i H ∗ φ ∗ Γ − Γ = ( m Y − m L ) ˆ Low ds (cid:88) j m Y Im( λ i λ j λ ∗ i λ ∗ j ) κ (2 π ) s / m H m Y E L J ( s ) (cid:0) m Y − s − m L i (cid:1) p L i ( s ) p L j ( s ) p φ ( s ) , (A.10)where s is p µL i p H µ and Low ≡ Max[( m φ + m H ) , ( m Y + m L j ) ] . The kinematics of L i werewritten in the centre-of-momentum frame, and the other particles kinematics are in the restframe of H . A.2 Extended Neutrino Portal
When a second inert Higgs doublet is added, H , we get many more diagrams contributingto the various processes. For most processes, these diagrams are simply allowing the inter-mediate particle to be H or H , with some interference terms. Due to the triviality of theextension, we do not write these down. But there are important new contributions to the CPviolation in both the decays and scatterings. The new cross section is: Y L i → H φp i E E ( σ − σ ) v = − (cid:88) jprs m Y m Y Im( λ i r λ j s λ ∗ i p λ ∗ j r κ ∗ p κ s )64 π sm H p m H r m H s p i p f p loop E E L j (A.11) + (cid:88) jprs Im( λ ∗ j r λ j s λ ∗ i p λ i r κ ∗ p κ s )32 π sm H p m H r m H s p i p f p loop (cid:0) E E + p i (cid:1) (cid:0) E L j E Y + p loop (cid:1) , where p, r, s ∈ , and E Y is the energy of Y in the loop. The first term is essentially thesame as (A.6), and is due to an extended analogue of fig. 2. The second term is due to fig. 5:as there are no Majorana mass insertions we have a more complicated function of energy andmomenta in place of the Majorana masses. We note that if all the masses and couplings arethe same for H and H the second term cancels, and CP violation is reduced to the minimalneutrino portal. The contributions to the decays are given by: Y → L i H ∗ φ ∗ Γ − Γ = − ( m Y − m L ) ˆ Low ds (cid:88) jprs m Y Im( λ i r λ j s λ ∗ i p λ ∗ j r κ ∗ p κ s )(2 π ) s / m H p m H r m H s m Y E L J ( s ) (cid:0) m Y − s − m L i (cid:1) × p L i ( s ) p L j ( s ) p φ ( s )+ ( m Y − m L ) ˆ Low ds (cid:88) jprs λ ∗ j r λ j s λ ∗ i p λ i r κ ∗ p κ s )(2 π ) s / m H p m H r m H s m Y E L J ( s ) (cid:16)(cid:113) s + p L i ( s ) + (cid:113) m L i + p L i ( s ) (cid:17) × (cid:0) E L j E Y + p loop (cid:1) p L i ( s ) p L j ( s ) p φ ( s ) . (A.12)Similar to the CP violation in the scatterings, in the limit of degenerate couplings and massesthe second term disappears. This completes the cataloguing of the cross sections used to solvethe Boltzmann equations in section 3.3. – 19 – Thermal Masses
While throughout this paper most thermal field theoretic effects are neglected, there is onethat must be included. At high temperatures particles can gain an effective mass throughinteractions with the plasma. For H , L and φ , these thermal masses are kinematicallysignificant. H is too massive to appear in the plasma at the temperatures we consider andso does not acquire a thermal mass, nor does it contribute to thermal masses. Similarly,as the Y only interact via H they also do not acquire a thermal mass. The main effectof thermal masses is to change the kinematics; it has been shown that effects such as theapparent breaking of chiral symmetry can be neglected to good enough approximation forour purposes [53–55]. The thermal masses are given by: [55, 56] m H = (cid:18) g + 116 g Y + 14 y t + 12 λ H (cid:19) T , (B.1) m L = (cid:18) g + 132 g Y (cid:19) T , (B.2) m φ = λ T , (B.3)where g is the coupling constant of SU (2) L and g Y is the coupling constant of U (1) Y inthe SM. We have neglected all SM Yukawa couplings except to the top quark, as well ascontributions from λ . In our solutions we use m H = 0 . T , m φ = 0 . T and m L = 0 . T .In general, the asymmetry formed increases for large thermal masses.Interestingly, as the Y do not acquire a thermal mass, at temperatures above . m Y a the SM Higgs can decay into Y rather than the other way round (depending on the thermalmass of φ ). While there could be concerns about the CP violation of these decays, these areineffective as H is in equilibrium. This is in agreement both with ansatz calculations of theCP violation and the results from [54]. C Chemical Potentials
Since φ , as well as the leptons, will be kept close to equilibrium above the EWPT it isunnecessary to have a Boltzmann equation for each particle species. Rather, we can solveour Boltzmann equations with the chemical potentials of these species, using (3.2) to obtaintheir number densities. This is where the advantage of using Maxwell-Boltzmann statisticslies: we can separate out the chemical potentials. Here we are not considering processes (orthe relevant chemical potentials) below the EWPT. Our task is made simpler by the fact thatall our chemical potentials can be written in terms of µ φ . As our model is similar to standardtreatments of B and L violation, such as leptogenesis, we can borrow the chemical potentialsfrom [57], and just note that the segregation of B − L into φ is the only source of B − L violation. At the temperatures we will be considering, the population of H is negligible andso does not affect the chemical potentials. By making the replacement B − L = µ φ , (C.1)– 20 –e obtain the chemical potentials: µ L = − µ φ (C.2) µ H = 479 µ φ . (C.3)These can be related to the asymmetry generated in φ by using n φ − n φ = T µ φ . (C.4)To write this all in terms of the baryon asymmetry, we use B = 2879 µ φ . (C.5) D Extensions of the Neutron Portal
D.1 → scatterings We have considered the simplest gauge invariant combination of B − L carrying SM particles, LH , but what about the second simplest? The next lowest dimensional gauge singlet B − L combination of SM particles is udd , the neutron portal. The neutron portal’s main issueis stability - the dark matter tends to decay. There are two ways to stabilise the neutronportal: temperature dependent couplings, as suggested in [10], or a symmetry. Both ofthese approaches require additional particle degrees of freedom. If we wish to impose a newsymmetry, we must move beyond → scatterings. The essential neutron portal operator is Xu R d cR d R , (D.1)where X is Dirac. It also possible to use left handed quarks, or for X to be Lorentz contractedwith down type quarks, but there are no differences relevant to this discussion.Following the method of Section 2, the simplest way to make a stable version of theneutron portal is to introduce the effective operator gXu R d cR d R σ, (D.2)where X is a Majorana fermion and σ is a complex SM singlet scalar. As before, to satisfy(3.7) we require two generations of X . We will only consider one generation of quarks: asthey remain in equilibrium any CP violation from having multiple generations will not changeour discussion below.There are multiple ways to open up this effective operator, shown in fig. 9, but theonly simple UV completion capable of generating an asymmetry is the last. In all others,there are rapid → flavour changing scatterings (fig. 10), which are problematic. As the → scatterings are mediated by only one heavy intermediate scalar, they dominate over the → scatterings. As CP violating effects involving the → scatterings must be as largeas possible during freeze-out the delay of thermal non-equilibrium due to → scatteringsresults in a serious suppression of the final asymmetry.– 21 – igure 9 . Tree level UV completions of the stable neutron portal operator (D.2). Figure 10 . Flavour changing scatterings.
If the collision terms for u and the X are given by W ( u + X → d + d + σ ) = (1 + (cid:15) ) W ,W ( u + X → d + d + σ ) = (1 + (cid:15) ) W , (D.3) W ( u + X → u + X ) = (1 + (cid:15) ) W , then the relevant unitarity constraints are given by (cid:15) W + (cid:15) W = 0 ,(cid:15) W + (cid:15) W = 0 , (D.4) (cid:15) W − (cid:15) W = 0 . From this, it is clear that for any of these process to violate CP, flavour changing scatteringsmust exist. Further, to get significant CP violation without delaying the departure fromthermal equilibrium these scatterings must be comparable in size to the → scatterings.This argument holds for any of the particles X could scatter with; at least one flavour chang-ing process must exist. As in this last UV completion there are no tree level scatterings toaccomplish this, another process must be added. An alternative ADM scenario, using the neutron portal and CP violating decays, can be found in [58]. Mediators for this process are contained in the other UV completions. – 22 –t seems that the only way to obtain a working ADM neutron portal using → scatter-ings is to have correlated couplings, which is not obviously an improvement over temperaturedependent couplings. Any full theory of → scatterings must explain why two seeminglyunrelated processes conspire to allow an asymmetry to be formed. While the neutron portal isthe second simplest SM gauge singlet operator involving baryon or lepton number, it actuallyrequires a complex description. We must look elsewhere for a generic method for ADM viascatterings. D.2 What about the symmetric component?
As discussed in Section 2.1, no existing models can use the same process to annihilate thesymmetric component and generate an asymmetry but is it possible to design one that does?Any model satisfying this condition must have four properties:1. The particles going out-of-equilibrium must be stable.2. The process must freeze-out late enough for the symmetric component to be annihilated, T freeze-out (cid:46) m DM / .3. Dark matter must couple to quarks, as a lepton asymmetry cannot be reprocessed below T EW P T and ADM requires that m DM is below the EW scale.4. CP violation must be significant at freeze-out, at least − (a conservative lower boundby analogy with [18], which used kinematic suppression of the washout to enhance theasymmetry formed).As there are strict bounds on light non-SM coloured particles [59], DM must couple to a gaugesinglet combination of quarks carrying baryon number, so some extension of the neutron por-tal is necessary. At least one of the mediators for any interaction involving the neutron portalmust be coloured, making it potentially detectable at the LHC. We can derive an upperbound on m mediators from properties 2 and 4. As dark matter is very light (order GeV), this isin tension with collider searches, which require that new coloured particles are heavier thanabout a TeV (for gluino-like particles) and 440 GeV (for single light flavoured squark-likeparticles) [59]. Models with small mass splittings can weaken these bounds, however as weare considering ADM the dark matter will always be significantly lighter than the colouredmediators.The simplest model which satisfies these conditions is (D.2), so we can get the mostforgiving bounds from it. More complicated extensions of the neutron portal will in generalhave tighter bounds, as additional mediators suppress the interaction. To simplify matterswe will consider all mediators to have the same mass; light SM gauge singlet mediators tendto lead to rapid D conserving scatterings amongst the dark sector particles. We estimatethe freeze-out temperature by comparing the interaction rate Γ to the Hubble time: freeze-out occurs when Γ ∼ H . These results agree with the numerical calculations of the similarneutrino portal model. Requiring that freeze-out occurs late enough (point 2 above) gives m mediator (cid:46) × λ GeV , (D.5) If we make X Dirac, rather than Majorana, and kinematically disallow decays dark matter will be anadmixture of X and σ , similar to [58]. – 23 –here λ is the couplings strength, assuming all couplings are the same. The CP violation atlow temperatures is bounded by a loop factor m DM πm mediator . From 4 we get m mediator (cid:46) × λ GeV . (D.6)While limits on coloured particles are model dependent, coloured particles of order GeVwould almost certainly have been seen at the LHC. The squark limits can be avoided for λ (cid:38) , but this raises the issue of perturbativity and low energy Landau poles. So strong aYukawa interaction between quarks and the mediating particles means that there would bea very strong detection possibility via, for example, monojet searches. This is an interestingavenue for future work.Unfortunately it seems that no operator allows one to use a single scattering process toannihilate the symmetric component and generate an asymmetry through asymmetric freeze-out without using couplings greater than one. Further, this requires at least → scatterings,so for the same process to annihilate the symmetric component and generate an asymmetrysimultaneously, it must rely on a conspiracy with flavour violating scatterings. E Other → ADM Scattering Operators
As was shown in Appendix D, → scatterings have significant advantages over → scatterings. Other than the neutrino portal, what other → scatterings are possible? Ifthe SM particles involved are not required to form a gauge singlet then there are dozens ofoperators that can be used. Due to the sheer number of operators (many being only triviallydifferent) an exhaustive list would be exhausting both to the reader and the authors. Instead,we will discuss the possibilities using an example.For the example, consider WIMPy baryogenesis [17]. WIMPy baryogenesis also usesCP violating scatterings to generate a baryon asymmetry, albeit using the "WIMP miracle"instead of an asymmetry to set the relic abundance of dark matter. Despite this, it is possibleto tweak WIMPy baryogenesis so that it becomes a theory of ADM. Fortuitously, the differentways of modifying WIMPy baryogenesis reveal all the salient model building concerns of → ADM scattering operators. For stock WIMPy baryogenesis, the relevant additions tothe Lagrangian are [17] ∆ L = gLψX c X + λHψf + H.c, (E.1)where X and f are singlet Dirac fermions, the dark matter, and ψ carries the same quantumnumbers as L . In the original model, the X freeze-out, not carrying any baryon number,behave as WIMPs. CP violation in these scatterings stores lepton number in the ψ . The ψ s (which are heavier than X ) decay via ψ → H ∗ f . Dangerous decays giving lepton num-ber back to the visible sector are forbidden by a Z symmetry. Of course, there are otherWIMPy baryogenesis models with, for example, X coupling to quarks but these give a similarstory. There are two different approaches to turning the WIMPy baryogenesis operator into amodel of ADM, corresponding to the two broad categories of → ADM scattering operators.– 24 –he first approach corresponds to a class of theories similar to the toy model of [12]. In this first class of theories, particles that store an asymmetry go out-of-equilibrium and thendecay into lighter particles, transferring the asymmetry. This decay is required to allow theannihilation of the symmetric component. To accomplish this, and to get a tighter relationshipbetween dark and visible matter, we can do away with X . There is already a conserved U(1)symmetry between L and ψ , so the process LL → ψψ (E.2)could create an asymmetry through asymmetric freeze-out, with ψ going out of equilibrium.The asymmetry stored in ψ is transferred to f as in WIMPy baryogenesis, but now dark mat-ter consists solely of f . In place of X , a mechanism to annihilate the symmetric componentof f is necessary but for essentially the same degrees of freedom we now have Ω DM (cid:39) V M naturally.In the second approach, which is similar to the neutrino portal model and [13], particlesthat do not carry B − L − D depart from equilibrium. In the WIMPy baryogenesis scenario,we can replace one of the X s in this operator with a Majorana fermion Y, giving ∆ L = gLψX c Y + λHψf + H.c. (E.3)Now X is free to be light, and to remain in thermal equilibrium. As in the neutrino portalmodel, the Majorana fermion Y departs from equilibrium, and subsequently decays. UnlikeWIMPy baryogenesis, we insist that X carry a baryon number equivalent. This leads totwo component dark matter, consisting of X and f . In order to ensure the stability of f weintroduce a Z symmetry, with Y , ψ and f being the negative parity states. If X is lighterthen f , N X − ( N f + N ψ ) conservation ensures the stability of X . To annihilate the symmetriccomponent, it is possible to gauge U (1) N X − ( N f + N ψ ) . This should produce a similar result tothe neutrino portal as the Boltzmann equations and types of CP violation are quite similar,though it is non-minimal as it only involves one SM particle.All other operators using this mechanism of asymmetric freeze-out should fall broadlyinto one of these two categories. While the neutrino portal is the minimal case, addingcomplexity to the dark sector can lead to some potentially interesting phenomenology, suchas dark forces. Bibliography [1]
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