Asymmetric Twin Dark Matter

We study a natural implementation of Asymmetric Dark Matter in Twin Higgs models. The mirroring of the Standard Model strong sector suggests that a twin baryon with mass around 5 GeV is a natural Dark Matter candidate once a twin baryon number asymmetry comparable to the SM asymmetry is generated. We explore twin baryon Dark Matter in two different scenarios, one with minimal content in the twin sector and one with a complete copy of the SM, including a light twin photon. The essential requirements for successful thermal history are presented, and in doing so we address some of the cosmological issues common to many Twin Higgs models. The required interactions we introduce predict signatures at direct detection experiments and at the LHC.


Introduction
There is overwhelming evidence for the presence of Dark Matter in the Universe, however its nature is yet to be unveiled. One particularly suggestive observation is that Ω DM 5Ω B , raising the question of whether a common origin for the baryon and DM abundances is possible. One answer to this question is the Asymmetric Dark Matter (ADM) picture [1][2][3][4][5][6] (for recent reviews see [7][8][9]), which assumes the DM density to be determined by an asymmetry n DM in the same way the baryon number asymmetry n B sets the visible matter density. In general the abundances are related by where m DM is the DM mass and m N is the nucleon mass. If the two asymmetries are generated by the same mechanism or if one is responsible of the other then we expect n DM ∼ n B . It follows that m DM ∼ 5m N and the question to address is why this mass ratio is close to unity. The discovery of the Higgs boson and the lack of new physics signals at the LHC made Naturalness and the hierarchy problem a critical issue more than ever before. In search of solutions, renewed attention has been recently brought to Twin Higgs (TH) models [10][11][12][13][14] that solve the little hierarchy problem by introducing a copy of the SM. The mirroring of the SM Lagrangian is due to a Z 2 symmetry. The scalar potential has an accidental global SU (4) symmetry which protects the Higgs potential from quadratic corrections. The Higgs is indeed a pseudo-Nambu-Goldstone boson and it is naturally light even after symmetry breaking, which happens at loop level. For Naturalness reasons the order parameter f of the SU (4) breaking is expected to be f ∼ few v, with v = 246 GeV the SM Higgs vev.
Twin Higgs is a natural environment in which to implement the ADM idea. It is straightforward to draw a comparison between baryons and twin baryons and to consider the latter as DM candidates. Both are stable thanks to conservation of the SM (twin) baryon number B (B). Approximate Z 2 symmetry suggests n DM ∼ n B , for example by having only the combination B −B conserved above the scale f , and m DM ∼ m N , with the second easily perturbed by the breaking of Z 2 below f . Motivated by this tantalizing observation we study the viability of twin baryons as ADM.
Related work [15][16][17] on both thermal and asymmetric DM has been recently carried out in the context of Fraternal TH models [18], in which only the third generation is mirrored. Here instead we focus on the conventional TH scenario with a copy of all generations.

Twin sector properties
Our starting point is a Twin Higgs model with an effective cutoff Λ ∼ 4πf ∼ 5 − 10 TeV, where again f few v for Naturalness. We will suppose that there is a copy of each SM quark charged under twinSU (3) QCD and twinSU (2) weak gauge symmetries, whose couplings are expected to be close to the SM counterpartsg i (Λ) g i (Λ) (from now on all tilded symbols refer to twin sector quantities and fields). Hypercharge is not necessarily gauged, as recently shown in [18,19], and eventually the twin photon is either massless or very heavym γ Λ. The presence of leptons in the spectrum is not strictly necessary and we will comment on different possibilities in the next section.
Naturalness constrains the twin top and bottom Yukawas, withỹ t andỹ b expected to fall respectively within 1% and 10% of the SM values, so from now on we fixỹ t = y t . While in generalỹ i y i is still expected from Z 2 symmetry, naturalness does not impose any bound on the first two generations. Hence large deviations can be possible, but we will not consider this in the following. Similar considerations apply to the leptons when present; either they will be assumed to have massm f /v m , or to be integrated outm v. As for the interactions between the two sectors any TH model contains Higgs interactions where h is the SM-like Higgs and its interactions with twin particles are those responsible for cancellation of quadratic divergences. At low energy they induce an effective operator

(2.2)
Higher dimensional operators will also be introduced in the following, with an expected effective scale M ∼ 4πf . In particular we have in mind recent realization of TH in the composite holographic Higgs framework [19][20][21], in which effective operators are generated by the strong sector and by integrating out heavy resonances. Finally the twin sector respects the same accidental global symmetries as the SM: twin baryon number, lepton number and charge. We will neglect possible small breakings generated by higher dimensional operators and consider stable each of the lightest particles carrying global charges.

Twin Dark Matter
Due to conservation of twin baryon numberB, as already explained, twin protonp and eventuallyñ are stable on cosmological time scales. If in the early Universe comparable asymmetries are generated then n B nB and How likely is it to happen? The running of α s plays a crucial role. First the different masses of the twin quarks change the thresholds in the β function. It is also possible to directly introduce a small difference δα s =α s (Λ) − α s (Λ), for instance this can be due to threshold effects in composite TH realizations, in which heavy resonances are expected to have different representations under the twin and the SM gauge groups. We compute the confinement scale of twin QCD by runningα s down from 4πf , with the β function computed for 6 flavours of massm q = f / √ 2ỹ q . This leaves δα s , f and rescalings of all Yukawas (apart theỹ t ) as variables. The results are presented in Fig. 1 which shows how Λ QCD /Λ QCD 5 is a typical result once a O(10%) splitting of the UV coupling constants is introduced. The effect can be explained by a simple approximate equation obtained by fixing . (2.6) The first term accounts for the difference in the quark mass thresholds and its evident suppression explains the negligible sensitivity to the choice of f andỹ. On the other hand the initial condition is exponentially amplified by the running. We also show the fine tuning (FT) contours given by two loop contribution to the Higgs mass [18] which is of the same order of the standard F T ∼ (f /v) 2 . Then f /v 5 is expected to avoid FT larger than few %.
Having successfully accounted for the DM to baryon mass ratio, how can their number density be related? Following the ADM picture [6] some mechanism connecting the baryon asymmetries in the two sectors should be efficient in the early Universe. First consider an effective operator connecting SM and twin singlets, Z 2 invariant and preserving only B −B where again M ∼ 4πf and all fields are right handed. The operator flavour structure, embedded in the coupling g lmn ijk , has been introduced to evade constraints on DM lifetime. A structure of the form g lmn ijk ∝ y i y j ..., as in partial compositeness, would be enough to avoid any bound and to ensure that the operator involving third generation quarks is in equilibrium at temperatures f T 4πf . Indeed for g ∼ 1 the operator is efficient above temperatures T ∼ M and, because it conserves B −B, it will convert any excess in B intoB and viceversa so that it enforcesñ B ∼ n B .
A different possibility is the sharing of asymmetries through non-perturbative effects, for example leptogenesis followed by generation of both B andB. This solution depends crucially on the UV completion of the model and it is beyond the scope of the present work.
We conclude, also anticipating the results in Section 3, that our DM candidate is a twin neutronñ with massm n 5 GeV. Theñ is typically stable and it is the main DM component due to charge neutrality of the Universe if theB asymmetry is the only asymmetry generated in the twin sector. We will study its experimental signatures in Section 4. Let us remark that DM self interaction scales like (Λ QCD /Λ QCD ) 3 resulting in a cross section σñ/m n 0.25 cm 2 g −1 which is below the current bound σñ/m n 0.5 cm 2 g −1 [22,23].

Thermal History
A successful thermal history must address two main points: the absence of other, possibly overproduced, relics and the influence on BBN and CMB of additional relativistic degrees of freedom.
Notice that we cannot rely on low reheating temperatures. The Higgs portal of Eq. (2.2) will always keep the two sectors in thermal equilibrium above temperature T few GeV, while the DM (and baryon asymmetry) are expected to be generated at T v. As such we discard reheating as a solution and from now on we always consider the two sectors to be in equilibrium up to at least T few GeV. In the following we will present the thermal history, including a possible twin nucleosynthesis phase, for two scenarios differing in content of the twin sector: • Scenario A: a minimal scenario with only twin quarks.
• Scenario B: a complete copy of the SM content, including a massless twin photon.

Scenario A
We start with a minimal scenario with only the necessary degrees of freedom, which are the twin quarks. We consider either the twin photon heavy or the twin hypercharge not gauged.
In the latter case there is no gauge anomaly and no leptonic field is required 1 , in the former all charged leptons and neutrinos are taken to be heavy enough to be integrated out. We will comment on this choice at the end of the section.
This scenario does not include any (nearly) massless particle and has no impact on the CMB. Only twin pions as possible symmetric relics are left to study. Their mass is given bỹ and similarly their decay constant scales asf π Λ QCD /Λ QCD f π 450 MeV. Theπ ± are stable as they are the lightest charged particles. On the other handπ 0 is a singlet and can decay to SM states, with its widthΓ being the crucial parameter. We assume it to be induced by higher dimensional operators.
To study the cosmological history of the twin-pions we write the corresponding Boltzmann equations. We assume that operators responsible forπ 0 decay keep the twin pions in thermal equilibrium via the processesπ SM →π SM. Neglecting the twin baryons, the Boltzmann equations arė where n 0 and n ± are the number densities ofπ 0 andπ ± respectively, andn represents the equilibrium distribution. Here, angle brackets represents thermal averages and σ is the ππ scattering cross section Upon inspection of Eq. (3.2) it is clear that ifΓ 2H(T =m π ) the decay is efficient in keepingπ in equilibrium and no relic density will be produced. ForΓ 2H(T =m π ), equilibrium is not maintained and theπ will become overabundant, with each degree of freedom having a freeze-out densitỹ (3.4) After freeze-out the term proportional ton in Eq. (3.2) drops to zero. The first term on the right hand side describes the decay ofπ 0 , depleting their number, while the second term replenishes it as long as σñ Γ . Above a certain temperatureT the scattering will be efficient, n ± will track n 0 and the total number density will decrease as On the other hand belowT theπ ± number density will freeze-out while allπ 0 will eventually decay. A rough estimate ofT can be computed by where s(T ) is the entropy density. The final density ofπ ± is then given by An upper bound on theπ width can be estimated by plugging the solution of Eq. (3.5) in Eq. (3.6). Imposing a conservative limit Ω π h 2 < 1/10 Ω obs DM h 2 0.012, we obtaiñ Moreover, if theπ 0 lifetime is too long its decays could inject entropy during BBN. Thus we require 1/Γ 1 s, corresponding tõ Γ 6.6 × 10 −25 GeV . (3.8) The bound of Eq. (3.8) automatically satisfies Eq. (3.7) and suggests that no significant relic density ofπ ± is left and DM can be considered as solely composed of twin baryons. It is interesting to notice that such a late decay could be linked, and possibly solve, the cosmological lithium problem [25].
Finally it is possible that for sufficiently long lived pions the Universe enters an early period of matter domination with subsequent reheating once the pions decay. This would dilute both baryon and DM asymmetries. While irrelevant for present discussion, it is an important point to address in any UV completion.
If twin hypercharge is gauged theπ 0 decays in SM via anomaly and γ-γ mixing [26] and the width is 2Γ (3.9) wherem γ ≈ TeV is the twin photon mass. However the conditionΓ 2H is never satisfied when compared with the experimental constraint of 10 −1 -10 −2 [27,28]. Alternatively, and in the case of hypercharge not gauged theπ 0 decay can be obtained by introducing higher dimensional operators. In particular, an axial current term (1/M 2 )(qγ µ γ 5 q)(qγ µ γ 5q ) induces a decay through mixing with the SM mesons. Notice that such operators involve isospin singlets and so we include isospin breaking effects in the form of theπ-η mixing angle sin θ ∼ 10 −2 . The width roughly scales as where the first factor accounts for the mixing between the pion and an X meson, while the mass ratio accounts for the phase space difference. To avoid additional isospin breaking suppressions from the SM sector we consider mixing with the η meson and so Eq.  It is clear that form π m η = 958 MeV the required scale would be too small for any reasonable model. The price to pay for heavyπ is in fine tuning asm π ∝ √ f and thus we expect f /v 10 (or drastic changes in the Yukawas).
Another possibility is a 4-fermion interaction between twin quarks and SM leptons. The width isΓ where m appears because of helicity suppression and decay to muons is the dominant channel. The bound from Eq. (3.8) gives which is less stringent than Eq. (3.11), but the presence of such operators is less easily justifiable.
What if twin leptons are added? Apparently theπ ± stability problem would be eliminated ifπ ± →l ±ν decay is allowed. However˜ annihilate only through weak interactions toν, and a modified version of the Lee-Weinberg bound [29] then requiresm > few GeV to avoid overproduction. It would clearly reintroduce the stableπ ± problem. On the other hand, it is an interesting observation that a heavy˜ could be a WIMP candidate, a possibility recently explored in [15,16]. The presence of light neutrinos does not significantly change theπ 0 decay width, cause of the mass suppression from chirality flip, and bounds from CMB could be possibly constraining. We do not pursue this direction any further.
Finally let us briefly comment on the possibility of twin nucleosynthesis, which could involvep andñ once the temperature of the Universe is around the twin deuteronD binding energy. Notice thatñ is stable thanks to phase space. The fusion processes can proceed only with a pion in the final state withpñ,pp,ññ →Dπ 0,± . At temperatures much above the binding energy the ratio of nucleons to nuclei is set by the high entropy of the Universe and the abundance of nuclei is negligible. In particular, this is valid at T m π when the fusion is not efficient thanks to Boltzmann suppression. We conclude that DM is entirely composed ofñ and possiblyp (if hypercharge is not gauged).

Scenario B
Now we consider a complete copy of the SM and again assumeỹ i y i . Various experimental constraints [27,28] require a small photon-twin photon kinetic mixing 10 −9 , which could be possible if mixing arises at 4-loop or higher order. A second constraint comes from the number of effective neutrino species, the recent Planck measurements [30] give ∆N ef f = 0.11 ± 0.23. The number of relativistic degrees of freedom is defined where g i accounts for the internal degrees of freedom. In general g * in the twin sector is comparable to the one in the SM. Above the decoupling temperature the only difference is given by the larger mass of twin particles, which become non relativistic at a higher temperature. After decoupling the different temperature of the two sectors must be taken into account. If the twin neutrinos are light, the contribution ofν andγ is equal to the SM one, g * = 3.36, rescaled to take into account the different temperatures of the two sectors where the normalization factor corresponds to g * = 0.45 of one SM neutrino species. By conservation of entropy the temperatures ratio is with T d the temperature of decoupling of the two sectors. If the decoupling happens at the freeze-out of Higgs portal interactions, theng * (T d ) ∼ g * (T d ) and ∆N ef f ∼ 7, which is ruled out. The best possibility is to have the sectors decouple between the two QCD phase transitions when g * (T d ) ∼ 61.75 whileg * (T d ) ∼ 10.75. Thus we obtain ∆N ef f ∼ 0.7, which is however still in tension with Planck measurements.
The most natural solution is to decouple theν by lifting their masses, for instance by a suitable choice of the twin seesaw scale. The effect is two-fold: increasing the entropy ratio at decoupling,g * (T d ) ∼ 5.5, and decreasing the total number of relativistic species as can be readily seen by a modified version of Eq. which is compatible with present constraints. We conclude that the two sectors must decouple between the two phase transitions with 0.15 GeV T d 1 GeV. How can this be achieved? Additional interactions between the two sectors must be introduced andẽ should be involved. For a 4-fermion operator suppressed by a scale M the decoupling temperature is roughly given by the condition Then the required scale is 7 TeV M 30 TeV. We stress that the above results are independent of f . The estimate only involvesm e , which is few order of magnitudes lower than Λ QCD forỹ e y e , and thus our conclusions are not sensitive to f .
If the neutrinos are integrated out, then each leptonic species is stable due to individual lepton number conservation and could in principle be a thermal relic. However, the˜ ˜ →γγ annihilation channel is efficient enough to eliminate any symmetric component [31].
In this scenarioπ ± as well asñ are stable due to phase space, whileπ 0 will decay tõ γγ through the twin chiral anomaly. The relic density ofπ ± is negligible, as shown in the previous section.
What about nucleosynthesis? First consider the case in which the only asymmetry generated in the twin sector is inB. Then charge neutrality of the Universe imposes that onlyñ are generated after the twin QCD phase transition. Twin neutrinos are heavy and so weak processes of the formẽ +ñ ↔νp are suppressed. We are left with fusion processes nñ →Dπ − and as in the previous section we conclude that DM is completely composed of n.
On the other hand, suppose aL asymmetry is generated in the early Universe as well. Nowp andẽ are also present, and by charge neutralityñ p =ñ e . So the fusion process pñ →Dγ is open and twin atoms will be formed. The ratio of different nuclei is set by the initial leptonic asymmetry and we leave a detailed study of this case for future work.

Experimental signatures
The experimental signatures predicted by the model fall in two main categories: DM direct detection and possible collider signals. Both of them are crucially dependent on the higher dimensional operators that we have introduced in Section 3. In general the presence of 4fermion operators induce a DM-nucleon scattering cross section. For simplicity we consider an operator of the form b qbq M 2 (qγ µ q)(qγ µq ) . (4.1) As previously explained we assume that DM is in the form of twin neutron with massm n 5 GeV. The spin independent cross section on a nucleon N is where µ is the reduced mass while b p = 2b u + b d , b n = b u + 2b d and similarly for twin baryons. The parameter space with M 3 TeV is already probed by present experiments [32][33][34] as shown is shown in Fig. 2 for flavour universal couplings b N =b n = 3.
Notice that the lowest values of M are demanded forπ 0 decay in Scenario A, and only axial couplings are strictly necessary. In that case only spin dependent are induced with cross section of magnitude comparable to Eq. (4.2) which is below present bounds.
On the other hand, Higgs portal interactions will always contribute with wheref n f N 0.3 is the nucleon matrix element. The predicted cross section is below the neutrino floor. The presence of effective operators will give additional collider signatures on top of the usual TH signatures [13]. Twin quarks are pair produced and any twin particle will typically escape the detector. Mono-jet [35,36] and mono-photon searches [37] can be safely evaded for M 1 TeV. For a more tuned scenario in which theπ are heavy enough to decay inside the detector it could be possible to have signatures similar to emerging jets [38].

Conclusions
We have shown that Twin Higgs models provide a natural framework for Asymmetric Dark Matter. They automatically contain the required global symmetries and particles, in particular considering the parallel between baryons and twin baryons.
Indeed, twin baryons with mass ∼ 5 GeV can be naturally obtained by small Z 2 breaking effects which induce a higher twin QCD confining scale. It is also natural to expect the same order of magnitude for the baryon asymmetries in the two sectors. We have studied two scenarios differing in spectrum and interactions, and shown that twin neutrons are typical DM candidates. Regardless of the twin particle content, higher dimensional operators with effective scale M ∼ O(1 − 10) TeV are expected and often necessary for a successful thermal history. Their introduction provides experimental signatures within current or near future reach, both at Dark Matter direct detection experiments and at the LHC.