Minimal Mirror Twin Higgs

In a Mirror Twin World with a maximally symmetric Higgs sector the little hierarchy of the Standard Model can be significantly mitigated, perhaps displacing the cutoff scale above the LHC reach. We show that consistency with observations requires that the Z2 parity exchanging the Standard Model with its mirror be broken in the Yukawa couplings. A minimal such effective field theory, with this sole Z2 breaking, can generate the Z2 breaking in the Higgs sector necessary for the Twin Higgs mechanism. The theory has constrained and correlated signals in Higgs decays, direct Dark Matter Detection and Dark Radiation, all within reach of foreseen experiments, over a region of parameter space where the fine-tuning for the electroweak scale is 10-50%. For dark matter, both mirror neutrons and a variety of self-interacting mirror atoms are considered. Neutrino mass signals and the effects of a possible additional Z2 breaking from the vacuum expectation values of B-L breaking fields are also discussed.


I. INTRODUCTION
An intriguing idea, that has persisted over several decades, is that of the Mirror World: the Standard Model, with quarks and leptons (q, l) and gauge interactions SU (3) × SU (2) × U (1), is supplemented by an identical sector where mirror quarks and leptons (q , l ) interact via mirror gauge interactions SU (3) × SU (2) × U (1) . There are two prime motivations for this idea. The discrete symmetry that interchanges the ordinary and mirror worlds can be interpreted as spacetime parity, P , allowing a neat restoration of parity [1,2]. Secondly, mirror baryons are expected to be produced in the early universe and to be sufficiently stable to yield dark matter, and this may lead to an understanding of why the cosmological energy densities of baryons and dark matter are comparable.
A third, more recent motivation for the Mirror World arises from the absence so far of new physics at colliders to explain the origin of the weak scale. If the Higgs doublets of the two sectors (H, H ) possess a potential with maximal SO(8) symmetry at leading order, the observed Higgs boson becomes a pseudo-Nambu-Goldstone boson with a mass that is insensitive to the usual Standard Model (SM) quadratic divergences; this is the Twin Higgs idea [3]. Furthermore, if the symmetric quartic coupling of this potential, λ, is large relative to the SM quartic coupling, λ SM , the Mirror World reduces the amount of fine-tuning by a factor of 2λ/λ SM to reach any particular UV cutoff of the effective theory. Since we now know that λ SM = 0.13 is small, this improvement can be very significant, allowing a Little Hierarchy between the weak scale and the UV cutoff, which may be beyond the LHC reach.
In this paper we formulate a minimal, experimentally viable, low energy effective theory for this idea, Minimal Mirror Twin Higgs, and study its signals. This is a pressing issue: mirror baryon dark matter, the Twin Higgs mechanism and consistency with cosmological limits on the amount of dark radiation all require a breaking of parity, P [4]. How is this to be accomplished? We do not attempt a UV completion, whether supersymmetric [5,6] or with composite Higgs [7][8][9][10], but note that both account for the approximate SO (8) symmetry of the Higgs potential.
The SO(8) invariant quartic interaction contains an interaction H † HH † H thermally coupling the two sectors at cosmological temperatures above a few GeV, so that the bound on dark radiation provides a very severe constraint on Mirror Twin Higgs. The Twin Higgs mechanism requires a parity breaking contribution to the Higgs mass terms in the potential, ∆m 2 H . We find this term by itself to be insufficient to solve the dark radiation problem, nomatter what other interactions connect the two sectors, at least for fine-tunings above the percent level. This then implies that the Yukawa couplings of the two sectors differ, y = y.
Hence we introduce an effective theory, Minimal Mirror Twin Higgs, where all P violation arises from a breaking of flavor symmetry, yielding different Yukawa couplings in the two sectors. This single source of P violation leads simultaneously to three key results • The ∆m 2 H term necessary for the Twin Higgs mechanism is generated via q loops.
• The mirror QCD phase transition temperature is raised above the decoupling temperature of the two sectors, solving the problem of excessive dark radiation.
• The mirror baryon mass is raised, allowing viable dark matter.
A striking signature would be the discovery at the LHC, or a future collider, of the mirror Higgs itself, decaying to W W or ZZ; see [4] and Fig. 10 of [11]. As the mirror Higgs mass depends on the SO(8) invariant quartic, λ, it could be beyond the LHC range, and here we focus on other signals. The size of P breaking in the Yukawa couplings to obtain the above three results leads to a preferred range of the lightest q mass of (2 − 20) GeV, leading us to compute signals for the following quantities • The signal strength, µ, and the invisible width, Γ(h → inv), of the Higgs boson.
• The amount of dark radiation, ∆N eff .
• The direct detection rate for mirror baryon dark matter from Higgs exchange.
• The effective sum of neutrino masses affecting large scale structure and the CMB.
These signals are tightly correlated as they all depend on the Higgs portal between the two sectors, the ratio of the weak scales, and on the masses of the light q . For dark matter, both mirror neutrons and a variety of self-interacting mirror atoms are considered.
After a brief review of the Twin Higgs mechanism in section II, we demonstrate that the breaking of P in the Yukawa couplings is necessary in section III. We define the Minimal Mirror Twin Higgs theory in section IV, and discuss the consequences of breaking P in the Yukawa couplings. We constrain the q Yukawa couplings from Γ(h → inv) and ∆m 2 H , and study how large the mirror QCD phase transition temperature T c can be. In section V we study the cosmological history of the two sectors when the only communication between them arises from the Higgs interaction H † HH † H and find that the decoupling temperature can be lower than T c , allowing ∆N eff to lie inside the observational limit. We predict the amount of dark radiation and the effective sum of neutrino masses. In section VI we examine an alternative cosmology when communication between the sectors is dominated by kinetic mixing of the hypercharge gauge bosons. We study a variety of candidates for mirror dark matter in section VII, and find that the H † HH † H interaction, together with the enhanced q Yukawa couplings, will allow direct detection at planned experiments over a large part of the mass range. In section VIII we briefly study ∆N eff and dark matter candidates when additional P breaking arises from the absence of Majorana masses for right-handed mirror neutrinos. In the Appendix we show that a PQ symmetry common to both sectors allows a solution to the strong CP problem, with the axion mass enhanced by the mirror sector by a factor of order 10 3 , leading to the possibility that f a is of order 10 TeV.

II. REVIEW OF THE TWIN HIGGS MECHANISM
In this section, we review the Twin Higgs mechanism using a linear sigma model. The Let us take a closer look at the Higgs potential. A global symmetry preserving potential is given by We have neglected possible higher order terms which are expected in composite Twin Higgs models, as they are irrelevant for the following discussion. Since the global symmetry is explicitly broken by Yukawa couplings and the electroweak gauge interaction, we expect a breaking of the global symmetry in the Higgs potential, at least by quantum corrections. The quantum correction to the mass term is the most dangerous, and to suppress it we assume a Z 2 symmetry H ↔ H and call H the mirror Higgs. We also introduce appropriate mirrors of other SM particles. In the following, we use " " to denote mirror objects. The global symmetry of the Higgs potential is now SO (8). A Z 2 symmetric mass term is accidentally SO(8) symmetric, and an SO(8) breaking potential is given by 1 As we will see later, we need small Z 2 breaking terms to obtain a correct electroweak symmetry breaking scale. We assume Z 2 breaking in the mass term, The origin of the Z 2 breaking mass term is explained in the next section.
Let us derive the VEVs of the Higgs fields in the small SO(8) breaking limit. Assuming In the unitary gauge of SU (2) L × U (1) Y , the pseudo-Nambu Goldstone boson, namely the Standard Model like Higgs h, is given by Minimizing the potential of h given by Eqs. (2) and (3), we obtain The mass of h is given by whereas the mass of the mirror Higgs boson, h , is From Eqs. (6) and (7), the required Z 2 breaking is given by Let us comment on the fine-tuning in the Twin Higgs model. In order to obtain the hierarchy between v and v, ∆m 2 H must be tuned against ∆λ v 2 . The standard fine-tuning measure is given by The mixing in the physical higgs bosons, h and h , imply an overall reduction of the couplings of the Standard Model higgs to any Standard Model particles by the relative amount (1 − . The precision measurements of these couplings [13] require that v /v > 2 at 95% C.L. (see section IV B) 2 . Hence we need a tuning of at least 50%: this is the unavoidable, minimal fine-tuning in Twin Higgs models.
In general we expect a fine tuning also in the mass of the mirror Higgs boson. Assuming that the dominant contribution to this fine tuning comes from the top loop suitably cutoff at a scale Λ TH 3 , it is If ∆ m h > 1, the overall fine tuning in Twin Higgs is given by that can be compared with the fine tuning in the SM where λ SM is the SM quartic coupling and Λ SM is the cut off scale of the SM.
Thus the fine tuning in TH relative to the SM is As λ SM 0.13, this improvement is significant: for a moderate tuning Λ TH can be above the scales directly explorable at the LHC. 2 The electroweak precision measurement as well sets an indirect bound on v /v, which depends on the mass of the mirror Higgs. For m h = 1 TeV, considering the IR contribution only, one obtains v /v > 3 at 90% C.L. 3 For λ > 1 this requires a suppression of the Higgs loop contribution in the UV completion of the Twin Higgs model considered here.
Let us comment on the required quality of the Z 2 symmetry [9,12]. The top Yukawa coupling y t gives a one-loop quantum correction, where Λ is the cut off of the Higgs sector. In composite Twin Higgs models, Λ is the scale of higher resonances, which is expected be as large as Λ ∼ g * v , where g * is the coupling strength of hadrons. The naive-dimensional analysis [14,15] and the large N counting [16] suggest The strong interaction gives a two loop quantum correction, leading to the requirement Finally, the SU (2) L interaction gives a one-loop quantum correction, requiring A natural explanation for the quality of the Z 2 symmetry shown in Eqs. (16), (18) and (20) is that the Lagrangian is precisely Z 2 symmetric, with a complete copy of all Standard Model particles -this is nothing but the Mirror World. A key question then becomes the form and origin of the Z 2 breaking necessary to construct a fully realistic theory.

III. NECESSITY OF Z 2 SYMMETRY BREAKING IN YUKAWA COUPLINGS
As reviewed in the previous section, the Twin Higgs mechanism requires a Z 2 symmetry under which the Standard Model and mirror particles are exchanged. The Z 2 symmetry must be broken eventually to obtain the correct electroweak symmetry breaking scale. In this section, we show that it is mandatory to break the Z 2 symmetry in Yukawa couplings to suppress the abundance of dark radiation, no matter what the origin of Z 2 breaking is and no matter what interactions might be added to the theory to couple the two sectors.
Thus the minimal phenomenologically viable way to break the Z 2 symmetry is via y f = y f .

The Standard Model and mirror particles interact with each other by the interaction in
Eq. (1) so that in the early universe at temperatures of order the weak scale the two sectors are kept in equilibrium via Higgs exchange. At lower temperatures the mirror particles eventually annihilate/decay into mirror photons (and mirror neutrinos), giving an extra component of relativistic particles, which is often referred to as dark radiation. The amount of dark radiation depends on the decoupling temperature between the two sectors, T d , below which the interaction rate between Standard Model and mirror particles is smaller than the Hubble expansion rate. Without introducing extra interactions, T d is determined by Higgs exchange which depends on the Yukawa couplings [4,5]. In this section, to be general, we allow additional interactions coupling the sectors and treat T d as a free parameter.
Assuming the Yukawa couplings are Z 2 symmetric, the mirror charged lepton masses are determined solely by the ratio v /v. (The masses of mirror hadrons could be also affected by the Z 2 symmetry breaking in the SU (3) c gauge coupling.) As their masses are relatively small, they remain in the thermal bath until low temperature, and contribute too much dark radiation. In Fig. 1, we show the prediction for the abundance of dark radiation, which by convention is expressed as an effective extra number of neutrinos, ∆N eff , which is given by where g(T ) and g The d.o.f of the Standard Model particles g(T ) is extracted from [17]. We assume that only the mirror electron, muon, tau and photon are light and contribute to the dark radiation.
Light mirror neutrinos, as considered in sections IV to VII, make the claim in this section even stronger. Then g r = 2 and The Planck collaboration puts a bound on the effective number of neutrinos, N eff = 3.2 ± 0.5 [21], which leads to the upper bound of ∆N eff < 0.65 (2σ) and 0.40 (1σ), indicated by dashed lines in Fig. 1. The 2σ bound can be only marginally satisfied. The 1σ bound can be satisfied when v /v 40, which requires a fine-tuning of more than 0.1% to obtain the electroweak symmetry breaking scale. Note the importance to reach this conclusion of the deviation of the actual g (T d ) from a naive stepwise function. We conclude that it is necessary to break the Z 2 symmetry in the Yukawa couplings to further raise the mirror charged lepton masses.
The Yukawa couplings can naturally acquire Z 2 symmetry breaking if they arise from VEVs of fields, as in the Froggatt-Nielsen (FN) mechanism [22]. Once these fields take asymmetric VEVs, the Z 2 symmetry of the Yukawa couplings is spontaneously broken. As long as the top Yukawa coupling does not depend on these field values, the Twin Higgs mechanism is maintained. In this paper we do not specify the model of the FN mechanism, but study the physical consequences of a low energy effective field theory for Twin Higgs with y f = y f .

IV. MINIMAL MIRROR TWIN HIGGS
The effective field theory below Λ TH for Minimal Mirror Twin Higgs is The Twin Higgs mechanism is imposed by boundary conditions at Λ TH : As discussed in footnote 1, there could also be a boundary condition giving non-zero ∆λ.
Z 2 breaking is the minimal consistent with the requirement of the previous section, namely This breaking of Z 2 is hard, meaning that the boundary conditions of (24) are not exact and are broken by loop corrections at Λ TH . We restrict the size of y f , for f = t, so that these corrections maintain the Twin Higgs mechanism and satisfy Eqs. (16,18,20 Such Dirac masses from a seesaw are an interesting possible consequence of parity restoration in the mirror scheme. Next we constrain the Z 2 breaking in y f by requiring that quantum corrections to the Higgs masses yield non-zero ∆m 2 H as required for the Twin Higgs mechanism in Eq. (9). We then place experimental bounds on y f from the invisible decay width of the SM-like Higgs.
Finally, we study mirror fermion spectra, consistent with these constraints, that maximize the QCD scale.
A. Constraint on Yukawa couplings from ∆m 2

H
We derive a constraint on Z 2 symmetry breaking in the Yukawa couplings by requiring that quantum correction to the soft Higgs masses yields the Twin Higgs mechanism. We assume the top Yukawa couplings of the Standard Model and mirror sectors are identical but allow other Yukawa couplings to differ, inducing a Z 2 breaking Higgs mass term where N f is the multiplicity of the mirror fermion f ; 3 for mirror quarks and one for mirror leptons. It should be remarked that the sign of the mass term is the required one. For this correction to explain ∆m 2 H of Eq. (9), the sum of the square of the mirror Yukawa couplings, and hence the mirror fermions masses, are determined Here, δ f ,µ ≡ y f (m f )/y f (µ) encodes the effect of the renormalization between a scale µ and m f . δ q ,Λ is about 1.3 − 1.5 for m q = (100 − 10) GeV, and δ l ,Λ 1. In the following we take δ q ,Λ = 1.4. Note that the estimation of ∆m 2 H | Yuk is UV sensitive and hence involves O(1) uncertainty, which we formally treat by varying the value of Λ in Eq. (29). The Z 2 breaking correction to the Higgs quartic couplings is proportional to y 4 f and is negligible.

B. Standard Model like Higgs decays
The Standard Model like Higgs h is an admixture of the two doublets H and H , h = In turn this leads to a universal reduction by a factor c γ of the Higgs couplings to all pairs of SM particles as well as to a coupling of the same Higgs to the mirror fermions where the QCD running factor for a mirror quark f = q , δ q ,m h , is about 1.3 for m q around 10 GeV, and the precise value depends on the details of the mirror quark spectrum 5 .
Higgs decays to mirror fermions leads to an invisible branching ratio where phase space has been neglected. Fig. 2 shows the correlations between v/v , Br inv , and the universal deviation from unity of the Higgs signal-strengths at the LHC into any SM final state, versus the relevant combination of mirror fermion masses We adopt the bound µ > 0.75 (95% C.L.) for the gluon fusion channel [13], as it has the smallest uncertainty. The bound is so strong that mirror fermions with m f < m h /2 give subdominant contributions to ∆m 2 H (see Eq. (29)): there must be at least one mirror fermion heavier than m h /2 other than t . The high-luminosity LHC can probe µ < 0.93 [23], which is shown by a dashed line in Fig. 2. Irrespective of the mirror fermions masses, v /v < 4 can be probed. The ILC is expected to measure the Higgs signal-strength with an accuracy of 1% [24], which probes v /v < 10.

C. Mirror QCD phase transition temperature
The larger Yukawa couplings of the mirror quarks leads to a larger mass of the mirror quarks, and hence to a larger mirror QCD phase transition temperature, T c . Since the number of degrees of freedom of the mirror sector changes rapidly near the phase transition, increasing T c above the decoupling temperature of the two sectors is critical to obtaining ∆N eff below the current bound, as specifically illustrated in section V C. However, there is a limit to how much T c can be increased, and here we derive an upper bound on T c .
First, we take the masses of N + mirror quarks (m + ) above m h /2 and degenerate, and determine their mass according to Eq. (29). Next, we take the masses of the remaining 5 − N + mirror quarks to be the same (m q ), and constrained by the bound µ > 0.75. With these masses, we solve the two-loop renormalization group equation of the mirror QCD  Note that the sum of the square of Yukawa couplings is bounded as Eq. (29) and µ > 0.75, while the dynamical scale of mirror QCD depends on the product of the mirror quark masses.
Thus, for a given N + , the universal m + and m − saturating these bounds maximize T c : the maximal T c can be read off from the right hand end of each line.

V. THERMAL HISTORY WITH HIGGS EXCHANGE
In this section, we discuss the thermal history of the Minimal Mirror Twin Higgs theory of (23) in the limit of < 10 −5 , so that the sectors are coupled only via Higgs exchange, and focus on the amount of dark radiation. Note that radiative corrections to in the effective theory below Λ TH vanish at 3 loops 6 , and any 4-loop contribution would be of order 10 −10 .
In the early universe, at temperatures larger than several GeV, the Standard Model and mirror particles interact with each other and have the same temperature. Below some temperature, T d , the interaction between the sectors becomes inefficient and they evolve independently. Heavy mirror particles eventually decay or annihilate into mirror photons and neutrinos, which are observed as dark radiation estimated in Eq. (21). To satisfy observational constraints requires g (T d ) g(T d ) so that at T d the colored states u, d, s, g contribute to g(T d ) but QCD states contribute very little to g (T d ); roughly speaking, decoupling must occur between QCD and QCD phase transitions, and we explore this further below.
A. QCD and constraints on the q spectrum If some q are light compared to the QCD scale Λ , then T c is lower than computed in the previous section resulting in large QCD contributions to g (T d ) that are excluded by bounds on ∆N eff . Hence the QCD phase transition is purely gluonic with zero q flavors. The temperature dependence of this zero flavor QCD, g QCD (T ) has been accurately computed on the lattice [26]. As the temperature decreases from high values, g QCD (T ) drops from its large perturbative value of 16 only very gradually, and then sharply drops very close to the critical temperature T c ; immediately after the phase transition at T c the glueball contribution to g (T c ) is 0.6. If T d > 1.1 T c , much of the entropy of the mirror gluon plasma is distributed solely to the mirror particles, which leads to too large ∆N eff . Hence, in the next sub-section we seek regions of parameter space where T d < T c , which is at most (1-2.8) GeV, as shown in Fig. 3.
Below T c , mirror glueballs S decay to γ γ via generated by integrating out the lightest mirror quark, of mass m q , giving a decay rate Λ is the scale at which QCD becomes strongly coupled, and is comparable to T c . In the case with all q heavier than m h /2, labelled N + = 5 in Fig. 3, the mirror glueball is not that the sub-diagram involving a photon and two Higgs vanishes due to the Bose statistic of the Higgs. Intuitively speaking, the symmetrized two Higgs have vanishing angular momentum, and vanishing correlators with a photon.
in thermal equilibrium just above T c where its contribution to g is large. In this case, the mirror glueballs decay late, well after T d computed below, and hence is excluded by the limit on ∆N eff . Fig. 3 shows that, in all the remaining cases of N + = 1-4, there must be one q lighter than 22 GeV. Over much of the parameter space of the q spectrum, the decay rate of (35) is fast enough to ensure that the glueballs at T c are indeed in thermal equilibrium with γ , and contribute 0.6 to g (T c ). Cases with Γ(S → γ γ ) less than the Hubble expansion rate at T c must be discarded as they give too much dark radiation.
Mirror glueballs also decay to ν ν via giving a decay rate which is negligible in comparison with Γ(S → γ γ ). maintained, even if f is heavy and its number density is much smaller than that of relativistic particles. We discuss the thermal equilibrium of mirror neutrinos later.
Let us first estimate the interaction rate assuming that the dynamics of f in the thermal bath is described by that of free fermions. This is certainly correct for the mirror leptons.
We comment on the case with mirror quarks later. The scattering cross section between f and f by Higgs exchange is given by where we assume a non-relativistic limit. Here p cm is the momentum of the fermion in the center of mass frame. In the thermal bath, it has a typical size The annihilation cross section of a pair of f into a pair of f is given by Here p f is the momentum of f in the center of mass frame. In the thermal bath, it is as large as p 2 f 3m f T /2. N f is the multiplicity of the Dirac fermion f ; for one lepton (quark), The energy density of mirror particles, ρ , is transferred into Standard Model particles at a rate Here 3.0 Λ=3 TeV  region is excluded by the measurement of the Higgs-signal strength. The high-luminosity LHC can probe µ < 0.93, whose corresponding invisible branching ratio is shown by a dashed line in the right panel. It is also sensitive to the invisible branching ratio of 10% [23]. For v /v ∼ 3, the deviation of the Higgs signal-strength from unity as well as non-zero invisible branching ration may be detected in the high-luminosity running of the LHC. The ILC is sensitive to an invisible branching fraction of sub-percent level [24]. The region with T d < T c can be probed by the Higgs-signal strength as well as the invisible decay of the Higgs at the high-luminosity LHC and the ILC.
In the above analysis we calculated T d assuming that the dynamics of mirror fermions is described by that of free fermions. This is not correct for mirror quarks when the temperature is smaller than the binding energy B D of the mirror QCD interaction. As the temperature drops below the binding energy, some mirror quarks form bound states, namely mirror quarkonia. This effect is expected to enhance the energy transfer rate by the annihilation of mirror fermions. The annihilation rate of mirror fermions inside quarkonia is where the second factor is the inverse volume of a quarkonium. The energy transfer rate by annihilation is given by where N q is the number of mirror quarks with a mass m q , and n B (m, T ) is the number density of a real scalar with a mass m in the thermal bath with a temperature T . The ratio of the energy transfer rate by the annihilation of free fermions to that by the annihilation inside quarkonia is Here we have used the non-relativistic approximation for n F,B . In the parameter space we have discussed, the binding energy E B is comparable to the temperature, and hence the formation of the bound state does not change the result in Fig. 4 significantly. But we note that it is possible that T d < T c is achieved for a wider parameter region.
Here we show that mirror neutrinos can be in thermal equilibrium down to T d . Chemical equilibrium of mirror neutrinos is maintained by the annihilation process f f ↔ ν ν , with cross section The number of mirror neutrinos produced/annihilated per unit volume and time is given by Comparing this rate with H × n(ν ), we find that chemical equilibrium as well as kinetic equilibrium are maintained down to a temperature of about m f /10. Kinetic equilibrium alone is maintained by the scattering f ν → f ν , with a cross section and is effective down to a temperature of about m f /20. Comparing m f /10 and T d in Fig. 4, thermal equilibrium of mirror neutrinos is also maintained down to temperature T d .

C. Dark radiation
As we have shown, mirror photons and mirror neutrinos can be in thermal equilibrium with Standard Model particles down to temperature T d < T c . Taking account mirror photons and neutrinos, the prediction for ∆N eff is which is consistent with the upper bound, ∆N eff < 0.65.
To keep T d smaller than T c , some mirror fermions must have masses not far above T d and so they also contribute to dark radiation, giving where g f is the effective d.o.f. of mirror fermions f . In Fig. 6, we show the prediction for ∆N eff as a function of m f . Here we have neglected the contribution from the mirror gluon plasma, which is correct for m f that give T d < T c . Regions that give T d > T c are depicted by dotted lines. We also assume that g s,f is well approximated by that of the ideal gas of f . This is correct for mirror leptons; for mirror quarks, the actual g s,f is smaller. Lines in the left panel are terminated if the Higgs coupling-strength falls below 0.75. The predicted amount of the dark radiation can be consistent with the experimental bound.
To summarize: for the amount of dark radiation to be below the experimental bound, there must be a mirror fermion with mass in the range (4 − 28) GeV.  particles, the ν number density is For the case of Majorana neutrinos, the effective total mass of light neutrinos, constrained by data on structure formation, is Here we have used the prediction of Fig. 6, ∆N eff > 0.3, and the experimental constraint v /v > 3, to obtain the last inequality. Although the current cosmological data are more constraining on ∆N eff than on ( m ν ) eff , both parameters may play a comparably important role in observations in the near future. We note that ( m ν ) eff can be larger or smaller if the Z 2 symmetry is also broken in the Dirac mass term of neutrinos.
For the case of Dirac neutrinos, the effective total mass of neutrinos is so that m ν have a small effect on structure formation and the CMB spectrum.

VI. THERMAL HISTORY INCLUDING KINETIC MIXING
In the last section, we discussed the thermal history of the Minimal Mirror Twin Higgs theory with < 10 −5 , so that Higgs exchange is the unique interaction coupling the two sectors, and found that light mirror fermion must be in the range of about (4-20) GeV.
In this section we allow larger values of , from the UV completion above Λ TH , so that the Standard Model and mirror sectors also interact by kinetic mixing between U (1) Y and U (1) Y gauge bosons, described by the B µν B µν term of Eq. (23). As we will see, the allowed range of mirror fermions masses are wider than the case without the kinetic mixing.

A. Decoupling temperature
Here we list the decoupling temperatures of various processes that maintain thermal equilibrium of mirror photons and/or neutrinos. We take a field basis such that the mirror photon is shifted to eliminate kinetic mixing, A → A + A. In this basis, Standard Model charged particles interact only with photons, while mirror particles interact with both photons and mirror photons.
Mirror photons are in thermal equilibrium with mirror charged fermions f , which also interact with photons and through mixing, maintaining thermal equilibrium between Standard Model particles and mirror photons. The cross section for f γ ↔ f γ is given by, where we assume T m f , and q f is the electromagnetic charge of f . The scattering rate is smaller than the expansion rate of the universe below a temperature T d,γ , U (1) kinetic mixing also mixes the mirror Z boson and the Standard Model photon, allowing mirror neutrinos to interact with Standard Model fermions with a cross section, This scattering becomes ineffective below the temperature T d,ν ,  Here we estimate ∆N eff for a representative point of parameter space, illustrating the importance of a variety of reactions between mirror and QCD phase transitions. Suppose that the lightest mirror charged fermion is a quark of mass 20 GeV, so that mirror glueballs decay into mirror photons just below T c . In Fig. 7, we show the decoupling temperatures of various processes as a function of . We also show the mirror QCD phase transition temperature, which we assume to be the maximal one we estimated in section IV C for v /v = 3 and Λ = 3 TeV. In region A, kinetic mixing is insufficient to transfer the energy of the mirror The last factor in the second term accounts for the conservation of the comoving number density of mirror neutrinos. However, we find this factor is ≥ 0.94, so that ∆N eff is well approximated by Eq. (48) with T d = T d,γ . In region D, ∆N eff is given by Eq. (48) with We find that in region B, C, and D, ∆N eff is about 0.3. In region E, ∆N eff is larger than 0.3, and can saturate the bound on ∆N eff for 10 −2 .
If the lightest mirror quark is heavier than 20 GeV, mirror gluons do not decay immediately below T c , but decay later. This is excluded if kinetic mixing is absent, because T c and T d are close to each other. With kinetic mixing, the energy of mirror photons can be transferred into SM particles well below T c , if a charged mirror lepton is light enough. As long as mirror glueballs decay into mirror photons before the QCD phase transition, sufficiently light mirror charged leptons can suppress ∆N eff to be within the allowed range.

C. Milli-charged particle
With U (1) kinetic mixing, mirror fermions of mirror charge q f can be understood to carry SM electric charge q f . In the range m f = O(1-10) GeV, the most stringent constraint comes from collider experiments [28,29], which is much weaker than the bound from ∆N eff , 10 −2 . A proposed search at the LHC can search down to mixings of = 10 −2 − 10 −3 [30].

VII. MIRROR BARYON DARK MATTER
The mirror sector, like the SM, possesses accidental baryon and lepton symmetries. Mirror baryons and leptons may account for dark matter in the universe [31] (see [32,33]   with an obvious notation. The meson M ud is captured by mirror baryons, e.g.
As the sum of the masses of M ud and B ddd is larger than the mass of B udd by 2m d , the meson M ud disappear from the thermal bath by this capture process. Finally, B scatter with each other and almost all become the mirror neutron B udd . For example, the scattering process eliminates B uud and B ddd from the thermal bath. Note that the sum of the masses of B uud and B ddd is larger than twice of the mass of B udd as the mirror baryon B ddd is spin-3/2 and has a contribution to its mass from a spin-spin interaction.

Without any IR effects, the scattering cross section between mirror neutrons is
and does not affect structure formation. The cross section can be enhanced up to the unitarity limit by some IR effects, e.g. the Sommerfeld effect [34,35] or the existence of resonance states [36]. In our case, mirror pions are also heavy due to large mirror quark masses, and those effects are expected to be suppressed.

Light d , e : mirror atom
If u is sufficiently heavy, it decays to d , e and ν and disappears from the thermal bath.
The mirror electron e is now stable, as its decay is kinematically forbidden. After the QCD phase transition, d combines into B ddd . To preserve charge neutrality, there are the same number ofē . The mirror baryons B ddd and positronsē eventually "recombine" into mirror atoms.
The self scattering cross section of mirror atoms is given by [37] σ/m D 100 α 2 where R = max(m B ddd /m e , m e /m B ddd ) and m D is the mass of the dark atom. Here we assume that R ∼ 1 and the kinetic energy of the dark atom is much smaller than the binding energy. The cross section is minimized for R = 1, which we assume in the following.
The cored dark matter halo profiles could be explained by m D ∼ 10 GeV. The upper bound on the cross section in dwarf galaxies, σ/m D < 10 cm 2 /g [38] requires that m D > 9 GeV.
The constraint from galaxy clusters is weak, since the velocity dispersion of dark matter is so large that its kinetic energy is comparable to the binding energy, and the self interaction cross section is suppressed.
The recombination of mirror baryons and electrons is incomplete. From the numerical estimation in Ref. [39] we obtain the ionized fraction, The ionized components interact with each other and with atomic dark matter strongly, and may affect halo shape [39] and the scattering of clusters. The estimation of these and other constraints on the ionized fraction is beyond the scope of this paper.
With the sizable kinetic mixing that we considered in section VI, the ionized components participate in acoustic oscillations and affect the CMB spectrum [40,41] 7 . The upper bound on the ionized fraction is Ω ionized /Ω atom < 0.01 [42] which, together with the limit from self scattering in dwarfs, disfavors atomic dark matter for such kinetic mixing 8 .

Light u , e : mirror atom
If d is sufficiently heavy, B uuu and e are stable and eventually form a mirror atom, (B uuu + 2e ) with a binding energy larger than that of (B ddd +ē ). The constraints from self interactions and the ionized components of the atomic dark matter are weakened. The precise determination of the constraints is beyond the scope of this paper.
where s(T ) is the entropy density, and ρ DM /s 3.6 × 10 −9 GeV is the energy density of dark matter divided by the entropy density. The first factor in the left-hand side is the cross section of the W mediated process, and the second factor is a rough estimate on the number density of mirror protons and electrons. The freeze-out temperature is then given by If the freeze-out temperature is much smaller than m p + m e − m n , mirror protons and mirror electrons disappear from the thermal bath. This is the case for m 10 GeV. For m 10 GeV, a non-negligible amount of mirror electrons and protons remain. Some of them later recombine into mirror atomic states. Unlike the cases of sections VII A 2 and VII A 3, these states decay into mirror neutrons and mirror neutrinos through mirror electron capture.
Thus there is no constraint from the self-interaction of atomic dark matter. However, there is a constraint on the ionized component.

B. Direct detection of dark matter
The above dark matter candidates, N , interact with Standard Model nucleons, N , through the exchange of the Standard Model Higgs, and may be observed in direct detection experiments [44][45][46]. The interaction of the Standard Model Higgs relevant for the direct detection is given by where we have taken into accout the one-loop QCD correction to the coupling with gluons [47]. The relevant matrix elements of N is given by Here we assume that the mass of N is mainly given by the masses of mirror fermions. Using the trace anomaly formula for the Standard Model nucleon [48], and matrix elements derived by a lattice calculation [49] 9 , q=u,d,s the scattering cross section between N and N through Higgs exchange is given by If kinetic mixing is absent, m N is bounded from below. Let us consider the mirror neutron, which is free of constraints from self interactions as well as from the efficiency of mirror recombination. It is made of one u and two d and the lines in Fig. 6 for N f = 6 show that the constraint on ∆N eff requires the mirror neutron to be heavier than 19 GeV  In Fig. 8, we show the prediction on σ N N as a function of m N . The regions depicted by thin (dotted) lines are disfavored by ∆N eff at the 1(2)σ level, if the mirror neutron is dark matter and kinetic mixing is absent. We show the constraints from the LUX experiment [52] and the Panda-II experiment [53] by solid lines. The higher mass region, m N > 40 (110) GeV is excluded for v /v = 3 (5). We also show the sensitivity of the XENON1T experiment [54], the LZ experiment [55], and the neutrino floor [56] by dashed lines. We conclude that dark matter can be detected by near future experiments in the region consistent with the upper bound on ∆N eff without kinetic mixing.
Once kinetic mixing between hypercharge gauge bosons is introduced, m N may be smaller and hence direct detection via Higgs exchange may be difficult to observe. In this case, however, direct detection via the kinetix mixing is possible. The mirror neutron dark matter interacts through its magnetic dipole moment. The magnetic moment of mirror neutron is expected to be of order Translating the bound derived in [57], we obtain the bound 10 −3 -10 −4 for m N = O(1-10) GeV. Projected low-threshold experiments such as CRESST III [58] and SuperCDMS SNO-LAB [59] will probe smaller kinetic mixing.
Mirror atoms have a large radius and interact through screened charges of mirror baryons and electrons. The scattering cross section between atomic dark matter and a nucleus (per nucleon) is given by [60] σ 4πα 2 2 α 4 bind m 2 N 10 −34 cm 2 α α bind where α bind is the fine-structure constant of the binding force that forms the atom. Here we assume R 1. A kinetic mixing that allows T d < T c ( > 2 + g f (T d ) 2 ∆N eff can be as small as 0.08.
Next we consider the case with a massless B − L gauge field. As we will see in the next sub-section, the massless B − L gauge field is beneficial in identifying mirror baryons with dark matter. The prediction for the amount of the dark radiation are given by ∆N eff can be as small as 0.17.

B. Additional dark matter candidates
The mirror neutrino masses now arise from Dirac Yukawa couplings and the lightest, ν , could have a mass of order the lighter states of u , d and e , or could be heavier than these light states. Also, a massless B − L gauge field may alter dark matter phenomenology.
When B − L is exact, so that B − L charge is conserved, the B and L asymmetries must be produced after the mirror sphaleron process freezes out to avoid washout of B and L asymmetries. On the other hand, if the B − L symmetry is only approximate, a B − L asymmetry may be produced before the mirror sphaleron process freezes out. Reactions mediated by W are absent, so that u , d and e numbers are separately conserved.
The following mirror particles are stable, If B − L symmetry is broken at a sufficiently high scale, a non-zero B − L asymmetry may be generated above the mirror electroweak phase transition temperature. After the mirror sphaleron process freezes out, the B and L asymmetries are given by Here we assume that the matter content of the mirror sector is identical to the standard model plus three right-handed neutrinos just before the mirror sphaleron process freezes out.
The resultant number densities of mirror protons, neutrons and electrons are given by Dark matter is composed of comparable numbers of mirror atoms and mirror neutrons. The constraints from self interactions and ionized fraction of atomic dark matter are relaxed.

IX. CONCLUSIONS
The Twin Higgs mechanism significantly relaxes fine-tuning of the electroweak scale, and allows for a larger cut off scale. The cut off of the Standard Model is while that of the Twin Higgs theory is Λ TH = 5.7 TeV × ∆ TH 10 The minimal theory has no new colored states to be produced at the LHC. It does offer the possibility of discovery modes at the LHC, such as production of the mirror Higgs via Higgs mixing; but the larger cut off may raise the masses of new particles above the LHC reach.
Mirror Twin Higgs models, however, predict the existence of extremely light particles, mirror photons and mirror neutrinos, that contribute to the dark radiation of the universe, leading to constraints on a realistic theory. We have found that, independent of the interactions that couple the two sectors, it is necessary to break the mirror Z 2 symmetry in the Yukawa couplings.

Minimal Mirror Twin Higgs
We have constructed a completely realistic effective field theory of Twin Higgs below the cut off Λ TH . It contains a complete mirror sector, so that a UV completion, which we did not study, can restore spacetime parity symmetry. In  As illustrated in Fig. 4, it is non-trivial that Higgs mixing can lead to a decoupling temperature less than the QCD phase transition temperature, necessary for a solution of the dark radiation problem. Fig. 4 shows that, with Higgs mixing alone, there must be light with a sizable kinetic mixing such that the thermal history is affected is excluded.
In the Minimal Mirror Twin Higgs the seesaw mechanism yields light neutrino masses for both sectors. These neutrinos can be Majorana with those in the mirror sector heavier by a factor (v /v) 2 than the observed neutrinos, leading to important effects in both CMB and LSS. The effective sum on neutrino masses relevant for cosmological data is at least 2.3 times greater than in the Standard Model. Small mixing between these Majorana standard and mirror neutrinos could lead to mirror neutrinos being observed as massive sterile neutrinos.
Alternatively the seesaw could lead to Dirac neutrinos, with the mirror states as right-handed neutrinos degenerate with the observed left-handed states, leading to only very small effects on CMB and LSS. The predictions of Eqs. (26,27) for the masses of mirror neutrinos, however, rely on the assumption that Z 2 breaking does not substantially affect either the neutrino Yukawa couplings or right-handed neutrino masses, and therefore our predictions for the effective sum of neutrino masses, ( m ν ) eff , are less robust than the predictions for ∆N eff , Γ(h → inv) and the mirror baryon dark matter direct detection cross section.
As we have shown in this paper, the Z 2 symmetry breaking in the Yukawa couplings is mandatory. This may lead toθ =θ, ruining the Peccei-Quinn solution to the strong CP problem [74]. Here we show that if the Z 2 symmetry is broken by VEVs of FN symmetry breaking fields we regainθ =θ. We note that this mechanism is not peculiar to Twin Higgs models, but is generically applicable to Mirror World scenarios.
Let us denote the FN symmetry breaking field as φ. The determinants of the mass matrix of SU (3) c and SU (3) c charged fermions are proportional to where A is the anomaly coefficient of (FN symmetry)-SU (3) c -SU (3) c , and f is a function.
Note that the phases of the mass matrices do not depend on the absolute value of the VEVs of φ and φ . The difference of the FN symmetry breaking scales itself does not ruin the axion solution to the strong CP problem.
If the FN symmetry has no color anomaly (i.e. A = 0), the determinants of the mass matrices have the same phases for any phases of the φ and φ VEVs, givingθ =θ. If the FN symmetry has a color anomaly, the theta angles may or may not be identical, depending on how the phases of the VEVs of φ and φ are determined. For example, suppose that the phase directions of φ and φ are determined by explicit breaking of the continuous FN symmetry to a discrete Z N subgroup, so that vacua are given by φ = | φ | × exp(2πi k N ), φ = | φ | × exp(2πi k N ) (k, k = 0, 1, · · · , N − 1). (A2) If A/N is an integer, the theta angles remain identical in any vacua. If not, the theta angles remain identical in specific vacua.
Assuming that mirror quark masses are larger than the dynamical scale of mirror QCD, the mass of the QCD axion is given by