Dark matter, dark radiation and gravitational waves from mirror Higgs parity

An exact parity replicates the Standard Model giving a Mirror Standard Model, SM ↔ SM′. This “Higgs Parity” and the mirror electroweak symmetry are spontaneously broken by the mirror Higgs, 〈H′〉 = v′ ≫ 〈H〉, yielding the Standard Model Higgs as a Pseudo-Nambu-Goldstone Boson of an approximate SU (4) symmetry, with a quartic coupling λSM(v′) ∼ 10−3. Mirror electromagnetism is unbroken and dark matter is composed of e′ and e¯′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\overline{e}}^{\prime } $$\end{document}. Direct detection may be possible via the kinetic mixing portal, and in unified theories this rate is correlated with the proton decay rate. With a high reheat temperature after inflation, the et dark matter abundance is determined by freeze-out followed by dilution from decays of mirror neutrinos, ν′→ ℓH . Remarkably, this requires v′∼ (108–1010) GeV, predicting a Higgs mass of 123 ± 3 GeV at 1σ and a Standard Model neutrino mass of (10−2–10−1) eV, consistent with observed neutrino masses. The mirror QCD sector exhibits a first order phase transition producing gravitational waves that may be detected by future observations. Mirror glueballs decay to mirror photons giving dark radiation with ∆Neff∼ 0.03–0.4. With a low reheat temperature after inflation, the e′ dark matter abundance is determined by freeze-in from the SM sector by either the Higgs or kinetic mixing portal.


Introduction
At high energy colliders, precision measurements of the electroweak symmetry breaking sector of the Standard Model (SM) have been pursued for decades, but so far there has been no discovery of any physics that would lead to a natural explanation of the weak scale. If the SM Effective Field Theory is valid well above the weak scale, at what mass scale will it finally break down? A possible answer has been provided by the LHC: perhaps new physics enters at the scale where the SM Higgs quartic coupling passes through zero. For example, this new physics could be the breaking of PQ symmetry [1] or of supersymmetry [2][3][4][5].

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2 Vanishing Higgs quartic from a Z 2 symmetry In this section we review the framework of [6] that yields the near vanishing of the SM Higgs quartic coupling at a high energy scale. Consider a Z 2 symmetry that exchanges the SU (2) weak gauge interaction with a new SU(2) gauge interaction, and the Higgs field H(2, 1) with its partner H (1, 2), where the brackets show the (SU(2), SU(2) ) representation. We call the Z 2 symmetry as Higgs Parity. The scalar potential for H and H is We assume that the mass scale m is much larger than the electroweak scale. With m 2 positive, the Higgs parity is spontaneously broken and H acquires a large vacuum expectation value of H = v , with v 2 = m 2 /λ. After integrating out H at tree-level, the Low Energy potential in the effective theory for H is To obtain the hierarchy H = v v , it is necessary to tune λ to a very small value λ ∼ −v 2 /v 2 ; the quartic coupling of the Higgs H, λ SM , is then extremely small.
The vanishing quartic can be understood by an accidental SU(4) symmetry under which (H, H ) is in a fundamental representation. For |λ | 1, necessary for v v , the potential in eq. (2.1) becomes SU(4) symmetric. After H obtains a vacuum expectation value, the SM Higgs is understood as a Nambu-Goldstone boson with a vanishing potential. Note that in this limit of extremely small λ , the vacuum alignment in the SU(4) space is determined by the Coleman-Weinberg potential. The top contribution beats the gauge contribution so that the true vacuum is the asymmetric one, where the entire condensate lies in H (or in H, which is physically equivalent). (The SU(4) symmetry implies that the Higgs boson contribution to the Coleman-Weinberg potential does not affect the vacuum orientation.) Below the scale v , quantum corrections from SM particles renormalize the quartic coupling, and it becomes positive. From the perspective of running from low to high energies, the scale at which the SM Higgs quartic coupling vanishes is identified with v . The threshold correction to λ SM (v ) is calculated in the next section.
We note a connection of the mechanism with the Twin Higgs mechanism [18,19]. In the Twin Higgs mechanism, a global SU(4) symmetry involving the SM Higgs H and the mirror (twin) Higgs H is introduced to solve the little hierarchy problem. The global SU (4) symmetry results from the UV dynamics of the theory, such as the global symmetry of UV strong dynamics. In our mechanism, the approximate global SU(4) symmetry does not arise from the dynamics of the theory, but rather accidentally arises from the requirement v v . A further difference is that the Twin Higgs mechanism requires a discrete symmetry interchanging H and H that is explicitly broken, while Higgs Parity is exact.
Although the scale v is much smaller than the Planck scale and the typical unification scale, the theory is no more fine-tuned than the SM because of Higgs Parity. The required JHEP02(2020)078 fine-tuning of the theory is where the first factor in the left hand side is the fine-tuning to obtain the scale m much smaller than the cut off scale Λ, and the second one is the fine-tuning in λ to obtain the electroweak scale from m. The total tuning is the same as in the SM, v 2 /Λ 2 , and may be explained by environment requirements [20,21].
3 The mirror standard model The phenomenology of the theory crucially depends on the action of Higgs Parity on the SM gauge group. Refs. [6,11] considers the case where the SU(3) c × U(1)Y gauge group is not replicated. The theory solves the strong CP problem and can be embedded into SO (10) unification. Ref. [12] replicates the U(1)Y gauge group. The theory solves the strong CP problem and has an interesting dark matter candidate. In this paper we study a theory where the SM gauge group is entirely replicated by a Z 2 symmetry which maps where matter is described by 2-component, left-handed, Weyl fields. 1

The Lagrangian
The most general gauge and Higgs Parity invariant Lagrangian up to dimension 5 is where L SM is the SM Lagrangian up to dimension 4 and L SM its Z 2 mirror. The next two terms of (3.2) link the SM and mirror sectors: λ = λ + λ describes mixing between the ordinary and mirror Higgs doublets and kinetic mixing between ordinary and mirror hypercharge. The dimension 5 operators in the second line of (3.2) describe the neutrino sector. M M,D are large mass scales and η and ξ are 3 × 3 dimensionless flavor matrices.

The mirror spectrum
The charged mirror fermions acquire a mass m f = y f v from the vacuum expectation value of the mirror Higgs, v . The Z 2 symmetry sets y f = y f at the scale µ = v , so that mirror fermion masses are larger than their SM counterparts by a factor of approximately v /v, as shown in figure 1. Mirror electrons and up quarks are the lightest fermions charged under U(1) EM and SU(3) , respectively, and thus stable and viable DM candidates. We explore e and u DM in section 6.

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Unlike mirror quarks, mirror glueballs, S , acquire mass chiefly from SU(3) nonperturbative effects, with mass [22,23] The mirror QCD confinement scale, Λ QCD , is not a free parameter, but is determined by running α S (m Z ) .1181 up to the Z 2 restoration scale v , equating α S (v ) = α S (v ), and then running α S down to lower scales until it diverges at the scale Λ QCD . In the MS scheme the dynamical scale is given by (3.4) Mirror glueballs are unstable and dominantly decay to γ γ , and if heavy enough, subdominantly to HH † . The latter are visible decays which may occur during BBN if S is long-lived. We investigate such constraints in section 5. Standard and mirror neutrinos obtain mass from the dimension 5 operators on the second line of (3.2). We will be interested in small mixing between ν and ν with M D M M so that m ν /m ν (v /v) 2 , giving energy density of the universe and release significant entropy into the SM thermal bath upon decaying. We investigate the effect of such entropy dilution on freeze-out e and u DM in section 6.1.

Prediction for v
Between the electroweak scale and the scale v , the running of the Higgs quartic coupling λ SM is exactly the same as in the SM. We follow the computation in [24] and show the running in the left panel of figure 2 for a range of top quark mass m t = (173.0 ± 0.4) GeV, QCD coupling constant at the Z boson mass α S (m Z ) = (0.1181 ± 0.0011), and Higgs mass m h = (125.18 ± 0.16) GeV. The value of the SM quartic coupling at the scale v is not exactly zero because of the threshold correction [12], where the MS scheme is assumed. The prediction for the scale v is shown in the right panel of figure 2. For each top quark mass and QCD coupling constant, the range of the prediction corresponds to the 1-sigma uncertainty in the measured Higgs mass, m h = (125.18 ± 0.16) GeV. Within the uncertainties, v as small as few 10 8 GeV is possible. Future measurements can pin down the scale v with an accuracy of few tens percent [12].

Kinetic mixing
Even though quantum corrections to the kinetic mixing are small, 2 no symmetry forbids a tree-level from being order unity in the effective Lagrangian (3.2). However, as shown JHEP02(2020)078  For example, consider a theory where the SM gauge group and the mirror gauge group separately unify to G × G at scale v G , shown qualitatively in figure 4. Above v G the operators that induce kinetic mixing between the standard and mirror sectors are: where F, F are the gauge field strengths and Σ, Σ the Higgs fields. The first term is absent if Σ is not an adjoint representation of G or charged under some symmetry. When Σ and Σ acquire a vacuum expectation value v G , 3 the higher dimensional operators in (3.7) induce a kinetic mixing It is possible to freeze-in e as DM via the induced kinetic mixing of (3.8). As shown in figure 4, the correct DM abundance can be produced for a kinetic mixing parameter 4 × 10 −11 , essentially independent of DM mass. If the dim-6 coefficient c 6 is non-zero, the JHEP02(2020)078  4 Direct detection and the correlation with proton decay

Direct detection by nuclear recoils
Kinetic mixing induced from higher dimensional operators allows e dark matter to scatter electromagnetically with a nucleus. The Rutherford cross section for scattering between e and a nucleus of mass m N and atomic number Z, with relative velocity v rel is given by where q is the momentum transfer and F (q) is the nuclear form factor. The number of expected events in a direct detection experiment with an energy threshold E th , a total target mass M tar , an exposure time T , and atomic weight A is where we assume a local DM density of 0.3 GeV/cm 3 , as well as a velocity distribution of Here f (E th ) takes into account the suppression of the scattering by the form factor, Assuming the Helm form factor [26,27], we find f (E th ) 0.3.

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XENON1T searches for a recoil between DM and Xenon with a threshold energy around 10 keV [28]. The bound obtained there can be interpreted as an upper bound of 16 on the expected number of the events. Currently, the strongest bound on for m e > 10 2 GeV comes from XENON1T [25], requiring < 1 × 10 −10 m e 10 2 GeV 1/2 (4.5) as shown in figure 3. If is close to this bound, future experiments may detect e dark matter.

Correlation between proton decay and direct detection
Let us consider a case where the SM gauge group is embedded into a unified gauge group with heavy gauge bosons mediating proton decay. The proton decay rate is where |W 0 | = 0.103 ± 0.041 GeV 2 encodes the relevant hadronic matrix element extracted from a lattice computation [29]. We also assume that below the heavy gauge boson mass scale the gauge group contains a U(1) factor which eventually joins the U(1)Y gauge group.
(This case excludes, for example, the Pati-Salam gauge group breaking at an intermediate scale.) The kinetic mixing is given by eq. (3.8) and we assume c 6 = 0. The direct detection rate N event /M tar t of (4.2) and the proton decay rate are correlated with each other, as shown in figure 5. The blue region shows that if XENON1T were to detect a nuclear recoil signal, the proton lifetime would generally be longer than Hyper-Kamiokande could detect, for c 8 = 1. The orange region shows the analgous signal region for LZ. For v ≤ 10 9 GeV, Hyper-Kamiokande and LZ both can detect correlating proton decay and nuclear recoil signals, respectively. If c 8 > 1, the kinetic mixing parameter is stronger for fixed v G so that nuclear recoil experiments and proton detect experiments may find correlating signals for v 10 9 GeV. For example, the dashed blue and orange contours of figure 5 show the reach of XENON1T and LZ, respectively, for c 8 = 10.

High and low reheat scenarios; BBN and dark radiation
Since all the parameters of the SM have been determined, the only free parameters that affect the cosmology of the Mirror Higgs Parity theory are the reheat temperature after inflation and the portal parameters that connect the SM and mirror sectors. A key question is whether the two sectors were brought into thermal equilibrium after inflation.
At sufficiently high temperatures, the SM and mirror sectors are kept in thermal equilibrium by the Higgs portal; the sectors then decouple at a temperature  Proton Lifetime Hyper-Kamiokande Figure 5. Correlation between the proton decay rate and the DM-nuclear scattering rate as a function of v . The rates are related as they both depend on the unification scale v G via higherdimensional operators.
Our two cosmological scenarios correspond to whether the reheat temperature after inflation, T RH is above or below T dec , and lead to very different mechanisms for the abundance of e and u dark matter. For T RH > T dec , the u and e abundances are given by freezeout as the temperature drops below their masses, followed by dilution from ν decay; for T RH < T dec we assume that only the SM sector is reheated, so that DM arises from freeze-in. These two schemes for DM production are discussed in the next section.
In both high and low reheating cosmologies, long-lived mirror glueballs are produced whose decay products may yield substantial dark radiation or alter the relic abundances of light elements. In this section we study the general constraints on the maximum production of mirror glueballs. These results will be used in the next section to place limits on the high T RH scheme and identify regions of parameter space that give signals of dark radiation and perturbed light element abundances.
The mirror QCD confinement transition occurs when the mirror thermal bath cools to a temperature T c = 1.26 Λ QCD [22]. At this point, the mirror bath contains only γ and g so that the ratio of entropies of the two sectors at T c is about r = (16/106.75)(T c /T c ) 3 . If the reheat temperature after inflation is greater than T dec , the two sectors were initially in thermal equilibrium and r = (8/9)(g * (T dec )/106.75). On the other hand, if the reheat temperature after inflation is below T dec , the two sectors were never in thermal equilibrium and ratio of temperatures T /T is generally much less than one.
After the mirror QCD transition, g confine to form mirror glueballs S , whose energy density normalized by the entropy is given by The factor A takes into account the non-trivial dynamics before and after the phase transition and is estimated in appendix B. A = 1 corresponds to the limit where a massless ideal JHEP02(2020)078 gas of mirror gluons suddenly becomes pressureless mirror glueballs at T c and the mirror glueball number density is conserved afterward.
Mirror glueballs are typically long-lived. The lifetime of the mirror glueball is dominantly set by its decay rate to mirror photons, described by the dimension-8 operator F F G G , generated by a loop of mirror quarks of mass m q and charge Q as shown in the left panel of figure 6. After confinement this becomes a dimension-5 operator connecting S to γ γ [30] Since the amplitude is dominated by the smallest m q , we take q = u giving Q = 2/3, so that the mirror glueball decay rate to mirror photons is (5.4) The mirror glueball can also decay to the SM sector via the Higgs portal as shown by the right panel of figure 6. The decay rate to HH † is given by , exceeds about 1 s, S decays during BBN. If this occurs, S may inject substantial energy density, ρ vis , into the SM hadronic sector altering the neutron to proton ratio before nucleosynthesis or disassociating light elements immediately after, leading to the constraint [32] Here, D is a generic dilution factor which may arise if there exists a particle which comes to dominate the energy density of the universe and decays before BBN, thereby injecting entropy into the SM thermal bath.
In the cosmology with T RH > T dec , mirror neutrinos are a natural candidate to provide such dilution since they are abundantly produced, decouple from the mirror bath while relativistic, and are long-lived. In this scenario, D = T MD,ν /T RH,ν , where T MD,ν is the JHEP02(2020)078  Figure 7. ∆N eff contours (purple) and BBN constraints (orange) from S → γ γ , HH † . In the left (right) panel the two sectors were (were not) initially thermally coupled so that DM is thermally produced via freeze-out and dilution (freeze-in). The temperature ratio of the two sectors, T /T , is evaluated at the mirror confinement temperature. For clarity, we take A = 1.
temperature of the SM bath when ν induced matter-domination begins and T RH,ν when it ends. If T RH < T dec , there is no particle in the mirror standard model to provide such dilution and D = 1. We show the BBN constraints as a function of v in figure 7 in orange using the precise energy yield constraints calculated in [32]. When T RH > T dec , r is known so D is constrained, as shown in the left panel of figure 7. When T RH < T dec , D is known so r is constrained, as shown in the right panel of figure 7.
In addition, the energy deposited by S into mirror photons is constrained, even if S does not decay during BBN. The mirror photons behave as dark radiation, whose energy density is conventionally expressed as an excess in the effective number of neutrinos ∆N eff . For the high T RH cosmology, with ν decay leading to a dilution factor D, ∆N eff depends on whether S decays before, during, or after the ν matter-dominated era For the low T RH cosmology few ν are produced, so they do not give a matter dominated era and D = 1. Contours of the dark radiation abundance produced from S → γ γ are shown in figure 7.

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Sector Decoupling   6 Cosmological abundance of mirror dark matter 6.1 Freeze-out and dilution from ν decay In this section, we take the reheat temperature of the universe larger than the temperature at which the two sectors decouple, T RH > T dec . In this case, the relic abundances of mirror e and u dark matter are set by freeze-out followed by dilution from the late decays of ν . 4 As the temperature of the universe drops, unstable mirror particles decay, while stable e and u annihilate and freeze-out. Although heavier mirror charged leptons and quarks are unstable, their decay widths are much smaller than their masses because of the large mirror electroweak scale. Figure 8 shows the temperatures around which each particle freezes-out (solid lines) and decays (dashed lines). Here we ignore the effects caused by late decays of JHEP02(2020)078 mirror neutrinos, and include them momentarily. For v in the range of (10 8 − 10 11 ) GeV, the e and u abundances are determined by the following processes in chronological order: 1. b freezes-out.
2. c , µ and s freeze-out. During these annihilations, b and c decay producing c , µ and s . The annihilations also produce e , u and d , but they thermalize quickly.
3. d , u and e freeze-out. During these annihilations, s and µ partially decay producing e , u and d . We note that τ is short-lived and does not affect the above processes. A set of Boltzmann equations describing the freeze-out dynamics is shown in appendix A. We elaborate on the fourth process. After the mirror QCD phase transition, mirror quarks are tied with each other by strings and form bound states. For v < 10 10 GeV, the Coulomb binding energy of mirror hadrons containing a u or d is comparable to T c [35], and so an O(1) fraction of these mirror quarks form loosely bound states with large radii ∼ Λ QCD . With such a large cross-section, these mirror hadrons scatter among themselves efficiently, rearranging their quark constituent until they contain a q q pair, and subsequently annihilate into γ [35,36]. For v > 10 10 GeV the Coulomb binding energy of mirror hadrons is larger than T c , and so most of the mirror quarks initially form tightly bound states with a smaller radius ∼ (m q α S ) −1 [37]. Nevertheless, these tightly bound states still have a relatively large radius and scatter and annihilate relatively efficiently. The mirror baryon containing only mirror strange quarks, s s s , generally forms a tightly bound state for all v . Still, s annihilates efficiently so that its beta decay contributions to e are small.
The thermal abundances of e and u are shown in figure 9. The solid lines conservatively assume that the annihilation cross-section of mirror hadrons is π/(m q α S ) 2 . The abundance of e does not change even if the cross-section is as large as Λ −2 QCD . For comparison, the dashed line assumes mirror hadrons completely cease annihilating after confinement. Even though the annihilation cross-section of e does not change in either case, the relic abundance of e drops when annihilations of mirror hadrons continue after the QCD' phase transition since any beta decays from s or d that produce e below T c are effectively absent (see figure 8). To the left of the vertical dotted line, the QCD' phase transition occurs before u freezes-out, which is why its abundance dramatically increases if hadronic annihilations are assumed to cease below T c .
We see from the solid lines of figure 9 that e is the dominant component of DM. On the other hand, efficient annihilations after the QCD phase transition make u a small component of DM, which exists today in the form of mirror hadrons like u u u . For all v > 6×10 7 GeV, the thermal abundance of e is too large to be DM. This is problematic as such a low v requires m t and α S (m Z ) to lie beyond their current 3σ experimental values. Ωh 2 e' and u' Freeze-Out Abundances Figure 9. The cosmological abundance of mirror electrons and up quarks from freeze-out and from decays of heavier charged mirror fermions. Dilution from mirror neutrino decays is not included.
Nevertheless, in the above discussion, we have ignored mirror neutrinos which are cosmologically stable if m ν < m e + m u + m d and M D of (3.2) is sufficiently large. The former prevents decays to the mirror sector, due to mirror fermion number and mirror electromagnetic charge conservation, and the latter suppresses decays to the SM sector. However, as M D is reduced, mirror neutrinos can decay well after they becoming nonrelativistic to SM particles, thereby diluting e and u . Consequently, the v required to produce e DM shifts to higher scales.
Shortly after the two sectors decouple at T dec , ν decouple from the mirror thermal bath as the mirror weak interaction rate drops below the Hubble expansion rate, as shown in figure 8. Since T dec = T dec m ν , ν decouple while relativistic with an initial yield Y ν n ν (T dec )/s(T dec ) = 0.004. With this initial abundance, if ν are sufficiently long-lived they dominate the energy density of the universe prior to decaying.

One generation of long-lived ν
For our first example, we assume that two flavors of ν decay rapidly and study e dilution from decays of the single long-lived flavor. The long-lived ν decays to H via the neutrino portal operator of (3.2) 5 with a decay rate The mass of the mirror neutrino is given by eq. (3.5), and for sufficiently large v , the mirror neutrino is massive enough that it can beta decay into e , u andd , with a decay rate Γ ν →e u d =  When ν dominantly decay into the SM sector, the decay products heat up the SM thermal bath, thereby diluting the frozen-out abundance of e and u relative to n γ by a factor is the lifetime of the mirror neutrino. The numerical factor of 1.2 is taken from [38]. We solve the Boltzmann equation for the abundance of mirror fermions in appendix A, including freeze-out, the change of the expansion rate during the mirror neutrino matter-dominated era, and dilution from ν decays. An approximation for the resulting e yield from freeze-out and dilution is where D is the dilution factor provided by mirror neutrino decays (6.3). For a given (v , m ν ), the parameter M D is determined to yield the correct e DM abundance. Furthermore, the resulting values of M D are large enough that m ν can be mapped to m ν by the scaling Further constraints on this scenario are shown in the (v , m ν ) plane in figure 10.

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In the allowed white region, we find M D must lie within the range (10 18 -10 23 ) GeV. In the red-shaded region, the e abundance is smaller than the dark matter abundance without dilution. For too small a neutrino mass, the required T RH,ν ≈ √ Γ ν M Pl to reproduce the dark matter abundance is below the MeV scale and affects BBN as well as the effective number of neutrinos [39,40]. We adopt the bound T RH,ν > 4 MeV [41], excluding the pinkshaded region. In the blue-shaded region, the mirror beta decay ν → e u d is kinematically allowed, creating too much e and u abundance. In the orange-shaded region the sum of the SM neutrino masses are above 0.3 eV, which is disfavored by the observations of the Cosmic Microwave Background (CMB) [42]. The gray-shaded region is excluded at the 3σ level from measurement of α S and the Higgs and top masses. If the long-lived species is the lightest ν then beta decay to ν e ē cannot occur. However, if the long-lived ν is one of the heavier states, then the lightly green-shaded region of figure 10 is also excluded since the long-lived ν creates e . The corresponding SM neutrino mass should be above ∆m 2 31(23) , excluding the lightly yellow-shaded region. The allowed region is not large: m t should be above its present central value and, remarkably, the neutrino mass must be within a factor of 10 of its present upper bound of 0.1 eV.
In the resulting allowed region of parameter space for e dark matter, the purple contours show our prediction for ∆N eff from decays of mirror glueballs, produced at the QCD confining transition, to mirror photons. Throughout the entire region ∆N eff is in the range 0.03-0.4, allowed by Planck [42] and within range of the sensitivities of CMB Stage IV experiments [43].

Universal coupling strength of neutrino portal
As a second illustration of e freeze-out and dilution from ν decays, we take the strength of the neutrino portal coupling to be independent of generation. Thus, in a neutrino mass basis, we take ν → lH decays to be given by (6.1) for all three generations of ν . To avoid overproducing e , all three ν must be light enough that beta decay is forbidden. Thus the total decay rate of each mirror neutrino is given by (6.1) and is proportional to m ν . Consequently, the dilution (6.3) is dominated by the heaviest mirror neutrino. For a normal hierarchy (m ν 1 m ν 2 < m ν 3 ) of SM neutrinos, the mirror neutrino responsible for dilution is ν 3 ; for an inverted hierarchy (m ν 3 m ν 1 < m ν 2 ), ν 2,1 give comparable dilutions; and for a quasi-degenerate spectrum ν 3,2,1 all give comparable dilutions.
The bounds from BBN, too much dark matter from ν → e u d decay, and too little dark matter from freeze-out are approximately as in figure 10, with the vertical axis interpreted as the heaviest neutrino, which is constrained by oscillation data to be at or above 0.05 eV. Thus the larger values of v and ∆N eff are excluded in this case. The upper bound on the heaviest neutrino from the cosmological limit on the sum of the neutrino masses is 0.1 eV.
In addition to these bounds, there is a constraint from the decay ν 3 → ν 1,2 e ē for a normal hierarchy or ν 2,1 → ν 3 e ē for an inverted hierarchy. In either case, too much e is produced. Regardless of whether the SM neutrinos obey a normal or inverted hierarchy, JHEP02(2020)078  Figure 11. Constraints on (v , m ν ) when e dark matter arises from freeze-out and dilution from ν with universal neutrino portal couplings. Here m ν is the mass of the lightest neutrino. Purple contours show ∆N eff resulting from decays of S to γ . Vertical gray contours show v when m t and α S (m Z ) deviate from their central values by 0 to 3σ. In the allowed white region, ∆N eff is always greater than 0.03, which will be probed by CMB Stage IV [43].

Insufficient Dark Matter
this constraint can be translated to a bound on the lightest SM neutrino: (6.6) ∆m 2 31 ≡ |m 2 3 − m 2 1 | (0.05 eV) 2 is the atmospheric neutrino mass difference squared and m e is the electron mass. We have made the good approximation that ∆m 2 31 is also the mass squared difference between the lightest and heaviest SM neutrino in an inverted hierarchy. This bound is shown in the yellow hatched region of figure 10.
The constraints on this scheme for e dark matter are shown in figure 11, where the vertical axis is the lightest SM neutrino mass. The bound of (6.6) appears in green. If v turns out to be larger than 4 × 10 9 GeV, the lightest neutrino mass is predicted to be in a narrow range. The lightest mirror neutrino is longer-lived than the heaviest mirror neutrino for a universal M D , but decays before the onset of the BBN for m ν > 10 −3 eV.
The sum of the masses of the three neutrinos can be constrained through its imprint on the structure of the universe. Future measurements of the CMB, BAO, and 21 cm emission are expected to determine the sum of the masses with an uncertainty of 10 meV [44][45][46]. One can check the consistency of the the measurements and the bounds we have obtained. During the matter dominated era by ν , cosmic perturbations of massive components can grow. Since e tightly couples to mirror photons, the perturbation of e does not grow by itself. The perturbation of mirror glueballs grows, decays into mirror photons, which scatter with e and grow the perturbation of e , like the growth of a weakly interacting massive particle during a matter dominated era [47]. We will discuss the implication of the growth to the future searches for ultra compact mini halo elsewhere.

Freeze-in from Higgs portal and kinetic mixing
In this section, we consider the relic abundances of mirror e when the reheat temperature of the universe is below T dec and only the SM sector is reheated. Since the SM and mirror sectors are weakly coupled below T dec , mirror DM is produced via freeze-in through the Higgs portal, as shown in figure 12. Although the mirror fermion and gauge boson production rates are UV-dominated, the entropy production during reheating negates far-UV production so that the dominant production occurs around T RH . Reheat temperatures below the mirror electron mass yield insufficient e to be DM since the small e freeze-in abundance is further diluted by (m e /T RH ) 6 as production almost ceases below T ≈ m e . 6 Consequently, we focus on T RH m e . A set of Boltzmann equations describing the freezein dynamics is shown in appendix A. The thermal evolution of the mirror electrons is as follows.
At T RH , the mirror electrons carry a typical energy T RH and a freeze-in number density 7 n(T RH ) = 4 9 n H is the SM Higgs thermal number density, H is Hubble, and σv is the freeze-in crosssection given by (6.8) 6 Some e production still occurs for TRH < T < m e by scatterings involving highly energetic particles produced by inflatons [48,49], which we find is not efficient enough to reproduce the DM abundance. 7 For low v and high TRH, e and γ may thermalize during reheating, altering (6.7). Thermalization cools the mirror bath so that mirror particles freeze-out instantly but are then replenished by the Higgs portal. Since freeze-in production is maximized at TRH, any pre-thermalized contribution is typically small. Even so, we consider this effect in appendix A.
JHEP02(2020)078  Figure 13. Constraints on the mirror electroweak scale v and the reheat temperature T RH of the universe. In the blue region, e is overproduced via freeze-in from the Higgs portal. In the red region, the required to freeze-in e as DM via the kinetic mixing portal (shown by the dotted counters) is large enough to produce nuclear recoil signals in XENON1T. In the orange region, the reheat temperature is high enough that the two sectors were originally thermally coupled and the freeze-in regime reduces to the freeze-out regime (see section 6.1).
For all v , the frozen-in abundance of e at T RH exceeds that of dark matter for T RH m e . For v 4 × 10 8 GeV, annihilations of e are ineffective during subsequent freeze-out. The freeze-in yield of e from the Higgs portal is T RH M Pl (Higgs Portal) (6.9) In this regime, a reheat temperature approximately equal to the mirror electron mass reproduces the correct DM abundance, as shown in figure 13.
For v 4 × 10 8 GeV, annihilations of e are effective during subsequent freeze-out and the allowed T RH rises, as shown in figure 13. However, as T RH increases, mirror fermions heavier than e are produced at T RH , which transfer much of their abundance to γ and e as they annihilate and thermalize via 2 → 2 and 2 → 3 processes as discussed in appendix A.
For T RH ≥ T dec , the two sectors were once in thermal equilibrium and the situation reverts to traditional freeze-out discussed in section 6.1. ∆N eff and BBN constraints from frozen-in mirror glueball decays are not shown in figure 13 as they are much weaker than the bound on overproduction of e DM.
Finally, as mentioned in section 3.4, e DM can also be frozen-in via kinetic mixing induced from higher dimensional operators (3.8). On one hand, the freeze-in abundance JHEP02(2020)078 of e through the Higgs portal is dominantly set by its yukawa coupling, which is fixed and whose smallness prevents sufficient e to be produced as DM for T RH < m e . On the other hand, the freeze-in abundance of e through kinetic mixing is set by , which is a free parameter (indirectly set by the unification scale v G ), and whose value can be chosen to sufficiently produce e DM for reheat temperatures as low as ∼ m e /25.
For T RH < m e , the freeze-in yield of e from kinetic mixing is The black dotted contours in the region T RH < m e of figure 13 show the necessary for e to be frozen-in as DM. The shaded red region is excluded if e is the DM since the required to freeze-in e DM via kinetic mixing is large enough to already produce recoil signals at XENON1T. 8 A similar calculation for the proposed LZ experiment, which can probe an order of magnitude smaller, produces the green contour 'LZ'. For low v , LZ has the potential to probe nearly all reheat temperatures capable of freezing-in e .

Gravitational waves from the mirror QCD phase transition
In the range of v consistent with the observed top quark mass, mirror quark masses are much larger than the mirror QCD scale. The mirror QCD phase transition is then first order [50,51]. The phase transition proceeds by nucleation of bubbles, which collide with each other and produce gravitational waves [52].
We consider the case where the e dark matter abundance is set by freeze-out followed by dilution from late ν decays. The abundance of gravitational waves Ω GW,col h 2 directly produced by the bubble collisions as a function of a frequency f is given by f p is close to the frequency at the peak of the distribution and T c 1.3 Λ QCD is the temperature of the mirror QCD phase transition. Here we use the results of ref. [53], assuming that the velocity of the bubble wall is the speed of light, and take into account the dilution D from ν decay. The ratio (β/H) parametrizes the duration of the phase transition β −1 in comparison with the Hubble time scale H −1 . ρ tot is the total energy density, ρ g is the energy density of the mirror gluon bath, ρ lat is the latent heat of the JHEP02(2020)078 10 -8 10 -8 phase transition, ρ kin is the kinetic energy of the bubble wall and ρ SM is the energy density of the SM bath, all of which are evaluated at the phase transition. Lattice calculations show ρ lat /ρ g is O(1), and so we approximate this ratio as unity [54]. Likewise, we expect ρ kin is comparable to ρ g as is the case for a weakly coupled, relativistic transition, and so we take ρ kin /ρ lat = 1 [55]. g dec and g dec are the degrees of freedom of the SM and the mirror sector at the decoupling of the two sectors, respectively. b parametrizes the energy density of the mirror gluons just before the phase transition, ρ g = b T QCD 4 . The ratio ρ SM /ρ g is estimated in appendix B. Gravitational waves are also produced by the turbulent motion of fluids induced by the bubbles [56]. The abundance of such gravitational waves Ω GW,tub h 2 is Here we use the results of refs. [55,57] assuming that the bubble walls expand at the speed of light. 9 Numerically, this contribution is smaller than the one from the bubble collision.

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The prediction (7.1), (7.3) for the gravity wave spectrum depends on v via T c and especially D. With v determined by the top quark mass, we show in figure 14 the prediction for the spectrum of the gravitational waves for various m t , taking β/H of (10,100) in the (left, right) panel. The dashed and dotted lines show the contribution from the bubble collision and the turbulent motion respectively, and the solid lines show the sum of them. In the blue shaded region, the freeze-out followed by the dilution from ν fails as is shown in figure 10. The ratio (β/H) is likely to be O(100) [58]. If the top quark mass is large enough, gravitational waves can be detected by future experiments such as LISA, DECIGO and BBO [59]. We note that prediction for the gravitational wave spectrum assumes that the phase transition occurs before the ν matter-dominated era. This condition is satisifed in the region where future experiments may detect the gravitational wave spectrum, that is, at the 2 − 3 σ level for m t and α S (m Z ).
We also note that many aspects of the phase transition in QCD-like theories, such as (β/H) and ρ kin /ρ lat , are not well-understood because of the non-perturbative nature. Once the phase transition is well-understood, it will become possible to check the consistency of future measurements of the top quark mass and the gravitational wave spectrum. For a survey of gauge theories exhibiting first order phase transitions, see [60,61].

Conclusions and discussions
We have introduced the Mirror Higgs Parity theory, described by (3.2). The entire SM Lagrangian, including dimension 5 operators for neutrino masses, is replicated by Higgs parity and the only unknown parameters are those of the kinetic mixing, Higgs and neutrino portals that connect the two sectors. The spectrum of the mirror sector is a scaled up version of the SM spectrum, as shown for the light mirror particles in figure 1. The scaling depends only on the Higgs Parity breaking scale v , which sets the scale at which the SM Higgs quartic vanishes and will become better determined by precision measurements of (m t , α S ).
There are several interesting theories containing the Higgs Parity mechanism for the vanishing of the Higgs quartic at high energies. Mirror Higgs Parity is the simplest theory where the Higgs Parity partner of the electron, e , is dark matter, with an abundance set by thermal mechanisms. Direct detection of e dark matter can occur via kinetic mixing and leads to a recoil spectrum characteristic of photon exchange. The present bound from XENON1T and the future reach of LZ on the kinetic mixing parameter are shown in figure 3.
If the SM gauge group is unified at scale v G into a group such as SU (5), the proton decay rate scales as Γ p ∝ 1/v 4 G . Furthermore, since kinetic mixing vanishes in the unified theory, it may arise from a higher dimensional operators, such as in eq. (3.7), leading to ∝ v n G , where n is a model-dependent, positive integer. Thus proton decay excludes small v G and direct detection excludes large v G . The correlation of these two rates for n = 4 is shown in figure 5. A large fraction of the allowed parameter space of the theory will be probed by a combination of Hyper-Kamiokande and LZ. turbulent magnetic fields similar to that from turbulent motion of fluids, and hence we simply use the fitting provided in ref. [57].

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For large values of the reheat temperature after inflation, T RH , the SM and mirror sectors reach thermal equilibrium via the Higgs portal interaction. The e relic abundance arises first from freeze-out and is then diluted by ν decay to H. Fixing the neutrino portal parameters to obtain the observed abundance, the remaining relevant parameters are v , which determines m e , and m ν which determine m ν . The constraints on this scheme for dark matter are shown in the (v , m ν ) plane in figure 10, for the case that dilution is dominated by a single ν . Remarkably, the corresponding neutrino is required to have a mass larger than 0.01 eV, in the range of masses determined from oscillation data. Furthermore, v must be in the range of (10 8 -10 10 ) GeV, predicting a Higgs mass of 123 ± 6 GeV at 3σ in (m t , α S ), and 123 ± 3 GeV at 1σ, by requiring the Higgs quartic to vanish at v . Fixing (m h , α S ) at their measured central values, we predict m t between (173.2-175.5) GeV.
Within this allowed unshaded region of figure 10, we predict the contribution to dark radiation arising from decays of mirror glueballs to mirror photons. The resulting ∆N eff , shown by purple contours, varies from about 0.04 to 0.4, and is highly correlated with v and therefore with m t .
Since all the mirror quarks are much heavier than the mirror confining scale, the mirror QCD phase transition, which occurs at T ∼ (40-1000) GeV for v = (10 8 -10 12 ) GeV, is first order and produces gravitational waves from bubble dynamics and turbulent fluid motion at the transition. The spectral energy density today, normalized to the critical energy density, is then obtained by including the ν decay dilution factor, and is shown in figure 14. Part of the allowed region of the theory can be probed by LISA, DECIGO and BBO, and a gravity wave signal in these experiments would be correlated with m t and ∆N eff .
For low values of the reheat temperature after inflation, T RH , e DM can arise via freezein production. The observed DM abundance may be obtained anywhere in the unshaded region of figure 13. On the edge of the blue shaded region this occurs via the Higgs portal, which is UV dominated around T RH . In the rest of the unshaded region this occurs via kinetic mixing, dominated at temperatures near m e , for a suitable value of .
Mirror Higgs Parity exchanges SU(3) with SU(3) and hence does not solve the strong CP problem. One possible solution is to introduce a QCD axion [62][63][64][65]. If Higgs Parity transforms the QCD axion into a mirror QCD axion, the mirror QCD axion is an axionlike-particle with a mass where the topological susceptibility is taken from [66]. The mass is correlated with v and hence with the top quark mass. Both axions may contribute to the dark matter density. Alternatively, if the QCD axion is neutral under Higgs Parity it couples to QCD and mirror QCD with the same decay constant. Since Higgs Parity ensures the equality of the theta angles in the two sectors, the strong CP problem is still solved [67][68][69][70]. The mass is given by eq. (8.1). An advantage of such a heavy axion is that it is easier to understand the PQ symmetry as an accidental symmetry [70]. In this case, it is even possible to have a small decay constant 10 9 GeV, since the large mass prevents the production of axions in JHEP02(2020)078 stellar objects and meson decays. We will discuss the phenomenology of axion dark matter in Mirror Higgs Parity in future works.

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f is the mirror fermion with the largest mass below T RH and subscript H is the SM Higgs. The production cross sections from the SM Higgs are [72,73] σ HH † →ff v rel 1 8π where the summation on f and q is taken for mirror fermions and quarks with masses greater than T . Initially possessing a typical energy ∼ T , the thermalization cross-section among mirror charged fermions is given by while the soft, number-changing (ff → ff γ, ff → ff g, gg → ggg) bremsstrahlung crosssections are given by is the Hubble scale during the reheating matter-dominated era. Here, α i equals α EM or α S (T ) and n i equals n e or n f depending on whether the exchange involves mirror photons or gluons. Soft-scattering keeps the mirror bath in kinetic equilibrium (but not necessarily chemical equilibrium), establishing an effective temperature where ρ tot (T ) is the total energy density of the mirror sector frozen in via the Higgs portal when the universe is at a temperature T , and n tot is the total number density of the mirror sector determined from the Boltzmann equations. For mirror photons, γ, and gluons, g, the equilbrium number densities are n eq m T , µ = g mT 2π  Figure 15. The QCD modification factor A as a function of a f /a c . A is defined as the energy density ratio of the actual glueball gas to that derived by a non-interacting ideal gas approximation and glueball number conservation.
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