Higgs Parity, Strong CP, and Dark Matter

An exact spacetime parity replicates the $SU(2) \times U(1)$ electroweak interaction, the Higgs boson $H$, and the matter of the Standard Model. This"Higgs Parity"and the mirror electroweak symmetry are spontaneously broken at scale $v' = \left\langle{H'} \right\rangle \gg \left\langle{H}\right\rangle$, yielding the Standard Model below $v'$ with a quartic coupling that essentially vanishes at $v'$: $\lambda_{SM}(v') \sim 10^{-3}$. The strong CP problem is solved as Higgs parity forces the masses of mirror quarks and ordinary quarks to have opposite phases. Dark matter is composed of mirror electrons, $e'$, stabilized by unbroken mirror electromagnetism. These interact with Standard Model particles via kinetic mixing between the photon and the mirror photon, which arises at four-loop level and is a firm prediction of the theory. Physics below $v'$, including the mass and interaction of $e'$ dark matter, is described by $\textit{one fewer parameter}$ than in the Standard Model. The allowed range of $m_{e'}$ is determined by uncertainties in $(\alpha_s, m_t, m_h)$, so that future precision measurements of these will be correlated with the direct detection rate of $e'$ dark matter, which, together with the neutron electric dipole moment, will probe the entire parameter space.


I. INTRODUCTION
For decades, a natural weak scale has been the key guide to constructing theories of physics beyond the Standard Model (SM), leading to new physics at or below the TeV scale. However, so far LHC data points to an alternative view where the SM, with a highly perturbative Higgs boson, is the effective theory to extremely high energies. In this case, the Higgs quartic coupling, λ SM , exhibits extraordinary behavior taking an absolute value of 10 −2 or less at energies above about 10 9 GeV. Indeed, at 2σ where µ c (10 9 − 3 × 10 12 ) GeV [1] (see [2][3][4][5][6][7][8][9][10] for earlier works).
In a recent paper [11], two of us introduced a new framework, "Higgs Parity", to understand this behavior of the SM quartic. A Z 2 symmetry replicates the SU (2) and µ c is close to v .
The strong CP problem [12] can be addressed by introducing spacetime parity [13,14], and a viable theory was first constructed by Babu and Mohapatra [15]. Higgs Parity provides a solution to the strong CP problem if it is promoted to a spacetime parity, P , and does not replicate QCD [11].
Thus, simple theories with Higgs Parity can simultaneously solve the strong CP problem and account for the extraordinary behavior of the SM quartic, making them a significant competitor to axion theories [16,17]. However, without a Weakly Interacting Massive Particle or an axion, the nature of dark matter (DM) in these theories becomes pressing. In this paper we show that such theories have a built-in DM candidate provided P replicates the entire electroweak gauge group as well as the quarks and leptons. DM is composed of mirror electrons and positrons, stabilized by an unbroken U (1) QED . The mirror baryon made of three mirror up quarks is also stable. However, a strong upper bound on the abundance of exotic hadrons, made both of mirror up quarks and SM quarks, requires that only a very small fraction of DM can arise from such mirror baryons. The suppression of the mirror up quark abundance requires that e is produced non-thermally as we will discuss.
In this paper we study a theory that has the same number of parameters as the SM.
Remarkably, these parameters allow us to compute the DM mass, its self interactions, and its interactions with SM particles.
At energies above µ c , the gauge group is SU Thus mirror quark and charged lepton masses are larger than those of the SM by v /v (and calculable renormalization factors) and the strong CP parameterθ = 0. Since the gauge and Yukawa couplings in our theory are the same as in the SM, the change in parameter space may be described by The last stage signifies that the mirror Higgs mass has no effect on any experimental observable. Particle physics and dark matter physics are described by one parameter less than in the SM; however, additional physics is required to understand the DM abundance. There could be extra parameters in the mirror neutrino masses, but ν are very heavy and play no role in this paper.  (2), SU (2) ) representation. The scalar potential for H and H is given by We assume that the mass scale m is much larger than the electroweak scale. With m 2 positive, the Z 2 symmetry 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 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, µ c of (1), is identified with v , v µ c . The threshold correction to λ SM (v ) is calculated in the next section.
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 Standard Model because of the Z 2 symmetry.
The required fine-tuning 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 Standard Model, v 2 /Λ 2 , and may be explained by environment requirements [18,19].
In [11] it was shown that the strong CP problem [12] is solved if the Z 2 symmetry includes space-time parity and leaves the QCD interaction invariant. In this paper we choose to have Z 2 replicate the full electroweak interaction, so that there is an unbroken mirror QED symmetry that stabilizes light mirror matter [20] allowing it to be DM [21].

III. THE MIRROR ELECTROWEAK THEORY
In this paper we study a theory where the electroweak gauge group, SU (2) × U (1), is replicated by a parity symmetry, while the QCD interaction is invariant; thus the gauge and Higgs are neutral under SU (2) × U (1) , and the action of parity is where matter is described by 2-component Weyl fields.

A. Renormalizable interactions
The most general gauge and parity invariant Lagrangian up to dimension 4 is given by where L KE contains canonical kinetic energies for all fields, B describes kinetic mixing between ordinary and mirror hypercharge and the QCD θ parameter is absent due to parity.
V (H, H ) is the Higgs potential of (4), and Yukawa couplings are described by L Y =(q y uū )H † + (q y dd )H + ( y eē )H + (q y * uū )H † + (q y * dd )H + ( y * eē )H + h.c. (9) where y u,d,e are the SM 3 × 3 Yukawa coupling matrices and parity implies that the mirror Yukawa matrices are the complex conjugate of the SM ones.
As V (H, H ) has three parameters, this theory possesses a single extra parameter compared to the SM. The analysis of the previous section applies: without loss of generality, in the limit of small λ , the vacuum has H = v H = v , and λ SM (v ) = 0 at tree level.
In this theory the observed values of G F and the Higgs mass determine v and v , and the third parameter of the Higgs potential determines the mirror Higgs mass and is irrelevant for physics below the scale v .

B. Strong CP problem
The 6 × 6 mass matrices for the (u, d) quarks of the two sectors are Mirror and standard quarks give equal and opposite phases to the determinant of their mass matrices, so thatθ = 0 at tree level. Loop corrections give rise toθ ∼ O(10 −16 ) as in the Standard Model [22], corresponding to a neutron electric dipole moment of order 10 −31 e cm, so that the strong CP problem is solved. This method of using parity to solve the strong CP problem was invented by Barr, Chang and Senjanovic [20]. The vanishing Dirac mass limit of the model by Babu and Mohapatra [15] reduces to this method.
The effective field theory contains the Higgs Parity even, dimension 6 operator where G is the field strength of SU (3) c , M P l = 2.4 × 10 18 GeV is the reduced Planck mass, and C is a dimensionless coupling. Condensation of H yields the strong CP phase We will find that DM places a lower bound on v , giving a result for θ close to the experimental constraint, θ < 10 −10 [23][24][25], that could be discovered in on-going searches for the neutron electric dipole moment [26][27][28].
The strong CP problem can be also solved by a CP symmetry, which forbids the theta term. Since CP symmetry also requires Yukawa couplings to be real, the CKM phase is obtained by spontaneous breaking of CP. A one-loop quantum correction to the strong CP phase can be suppressed by sophisticated setups [29][30][31][32]. In the parity solution, parity does not require Yukawa couplings to be real and the CKM matrix is easily reproduced.

C. Kinetic Mixing at 4 loops
Kinetic mixing between the standard and mirror sectors is induced at four loops by the shared color charge of standard and mirror quarks, as shown in Fig. 1. We may directly compute the kinetic mixing between the SM photon and the mirror photon by projecting the external gauge field into the massless combination. The renormalization group equation of the kinetic mixing parameter can be read off from the four-loop beta function of QCD [33], Here i runs over all the quark charges, q i , while j is summed only over mirror quarks with mass below the scale µ. The prediction for is shown in Fig. 2  The result, = O(10 −8 ), is important for placing a limit on the mass of e from DM direct detection, and the large numerical factor of (13) plays a crucial role.

D. Neutrino Masses
Standard and mirror neutrinos obtain mass from operators of dimension 5 where M M,D are large mass scales and η and ξ are 3×3 dimensionless flavor matrices. Taking (15) The prediction for the kinetic mixing parameter as a function of v , for three values of the cutoff of the theory. If DM is e , the shaded region is excluded by the direct detection limit of XENON1T. For various values of the coupling C, defined in (11), the present limit on the neutron electric dipole moment excludes the region to the right of the vertical lines.

E. Threshold correction to λ(v )
We start from the one-loop Coleman-Weinberg potential of the theory above the mirror electroweak scale, where M is an arbitrary scale. A change of M can be absorbed by a change of λ. We take M to be the vev of H , which is given by After integrating out H , the potential of H, to the leading order in c and λ , is given by To obtain the electroweak scale much smaller than v , λ c/2 is required. Then the Higgs potential is given by We match this potential to the one-loop Coleman-Weinberg potential of the SM, where we take the MS scheme. By matching A numerical evaluation shows that λ SM (v ) is negative and O(10 −3 ).
In Table I

A. Hadronization of u
After the QCD phase transition, u quarks form bound states by combining with other colored particles, namely, they hadronize. Hadronization of massive colored particles and their subsequent evolution was investigated in [35].
Since the ordinary SM quarks, which we collectively denote as q, are much more abundant than u , the u first form bound states u qq and u q, having B number of 1/3. These hadrons have a large radius ∼ Λ −1 QCD and, if sufficiently dense, can efficiently scatter with each other to rearrange constituents [35]. In particular, states with B = 2/3 are formed by u qq + u qq → (u u q) * + qqq, u qq + u q → (u u q) * , u q + u q → (u u q) * +qqq (23) and similarly there is a processing of B = −1/3 hadrons to those with B = −2/3 by the corresponding antiparticle reactions. The rearrangements may involve emission of pions, which we omit here and hereafter. In addition, rearrangements can form B = 0 mesons containing u ū u qq +ūqq → (u ū ) * , u qq +ū q → (u ū ) * + qqq, u q +ū q → (u ū ) * .
The The cross section of the rearrangement and the subsequent falling in the ground states is suppressed by the destruction of the excited states before falling. Taking this effect into account, the production cross section of the ground states is [36] σ The production cross section of u u u is of this order. This is also effectively the annihilation cross section of h as u ū annihilate into SM particles.
The abundances of (u u u ) and h is estimated as follows. If the cross section times the number density of u is larger than the Hubble expansion rate around the QCD phase transition, the abundance of (u u u ) is comparable to the initial abundance of u . The abundance of h is given by the freeze-out abundance determined by the cross section in Eq. (26). If the cross section is small, the abundance of (u u u ) is given by the freeze-in abundance, while that of h is close to the initial abundance of u . The abundance of (u u u ) and h are given by

B. The ICRR Limit on u Dark Matter
The abundance of h is strongly constrained. Stringent constraints come from monopole searches of the 1980's, which are sensitive to ionization from fractionally charged h . The bound from the ICRR experiment [39] is derived in [40] taking into account the acceleration by supernova remnants. For m u = 10 6−7 GeV the bound is Y h < 10 −25 . 3 This is much smaller than Y crit of (29), so that the bound on the u abundance before the QCD phase transition is the same, Y u < 10 −25 . The abundance of u u u is even smaller and almost all of DM is composed of e . Possible cosmological scenarios leading to the hierarchy of the abundances of e and u are discussed in Section V.

C. Bulk Matter Constraints on u Dark Matter
Additional constraints on h come from searches for fractionally charged particles in bulk matter, implemented via Millikan drop experiments or ferromagnetic levitometers [41].
While such experimental constraints are strong (no more than one h per ∼ 10 21 nucleons) and mass-independent, the results should be interpreted carefully, taking into account the distribution of h on Earth from billions of years of geologic churning, the potential contamination of the sample during the refinement process pre-experiment [42], and the uncertainty to what materials h may bind to due to the exotic chemistry of fractionally charged particles [42,43]. We (very) roughly estimate the relative number of h compared to nuclei in the crust as well as in meteorites and find that the flux constraints f h ≡ Ω h /Ω DM The volume of each non-refined terrestrial sample tested for fractionally charged particles is 10 −3 cm 3 [41], so that (30) suggests fewer than one h resides in a given sample. It is thus highly plausible that h has escaped detection in such samples.
Bulk matter searches for fractionally charged particles have also been tested on meteorites which have the advantage of lacking the uncertainty associated with geological weathering.
Moreover, iron meteorites are naturally ferromagnetic and hence can be minimally processed in principle before testing on ferromagnetic levitometers.
Meteorites are made of heavy elements which are synthesized in stars. As is argued in [36], For low momentum, the Fermi-accelerated differential spectrum of h , d(nv)/dp = Φ/p, [40], so that the number density of h a distance X below the meteorite surface is approximately where t CR is the exposure time of the meteorite to cosmic rays before falling to Earth. We set the surface depth equal to the typical atmospheric ablation for meteorites like the Hoba sample, approximately 0.5 m.
The ablation length as well as the exposure time can be inferred by measuring the abundance of isotopes and the tracks of cosmic rays in a meteorite [45]. For example, the Hoba meteorite experienced 40 cm of ablation and about 2 × 10 8 years of exposure to cosmic rays [46]. Since 10 −4 cm 3 by volume of Hoba has been tested with null results [47], there is a good chance that no h are detected for f h = 10 −8 . Besides Hoba, only three other meteorites have been tested, totaling less than 10 −3 cm −3 by volume [41,47,48]. The exposure time to cosmic rays for each of these meteorites is far less than Hoba [45,49], and thus give weaker constraints.

D. The XENON1T Limit on e Dark Matter
Mirror electrons interact with SM particles via kinetic mixing. The cross section of the 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, XENON1T searches for a recoil between DM and Xenon with a threshold energy around This result is translated to a bound in the (v , ) plane in Fig. 2. Together with the prediction for , this requires that the mirror electroweak scale is above (3 × 10 11 − 10 12 ) GeV, for a UV cutoff ranging from v to M P l . The LZ experiment [53] is expected to provide about 10 times better sensitivity and probe v values an order of magnitude larger. An experiment whose sensitivity is saturated by the neutrino-floor will have about 100 times better sensitivity [54] and probe v values two orders of magnitude larger. Note that larger values of v are expected to yield larger values of θ via the dimension 6 operator of (11), as shown by vertical lines in The uncertainty from the Higgs mass is sub-dominant, as seen in Fig. (3).
In Fig. 5, we show the prediction for the expected number of events, in experiments with Xenon targets, as a function of the top quark mass for a given strong coupling constant.
We take a UV boundary condition for the kinetic mixing parameter of (Λ) = 0 with The strong coupling constant can be measured with an accuracy of 0.1% by improving lattice computation as well as the conversion of the coupling at the lattice scale to that of higher energy scales [55]. Further measurements at the Z-pole at lepton colliders can achieve similar accuracy [56]. The uncertainty in the prediction of the event rate from the last term of (37) is then very small compared with that from the cutoff Λ. The top quark mass can be measured with an accuracy of 0.2 GeV at high-luminosity running at the LHC [57], below which the uncertainty is saturated by the theoretical ambiguity associated with the definition of the pole mass and its conversion to MS [58][59][60]. The Higgs mass can be determined with an accuracy of few 10 MeV at high luminosity running of the LHC [61]. At this stage the direct detection rate is predicted within a factor of about 6, where the uncertainty from the top quark mass dominates. Further improvement is possible by determining the MS top quark mass directly by the measurement of the top quark production cross section which is free from the ambiguity. Lepton colliders can determine the top quark mass with an accuracy of few 10 MeV [62][63][64][65], allowing for the prediction of the direct detection rate within few ten percents. With this accuracy, uncertainties from the local DM density, the velocity dispersion [66,67], the cutoff Λ, and the theoretical uncertainty in the determination of v become important.

V. COSMOLOGICAL PRODUCTION OF e DARK MATTER
In this section we describe how the relic DM abundance is set in the early universe. As we have seen in the previous section, the abundance of e must be much larger than that of u . We first show that thermal production mechanisms do not work. The hierarchy of the abundances can be achieved by non-thermal production from the decay of the inflaton, or generically from a particle that dominates the energy density of the universe.

A. Freeze-Out and Dilution
For a large enough reheat temperature (T RH m u ), both the SM and mirror sectors are in thermal equilibrium. As a result, the abundance of e is set by thermal freeze-out and is given by Ω e ≈ Ω DM (v /10 8 GeV) 2 5 . To produce the observed DM abundance, v is so low that it is already ruled out by kinetic mixing (v 10 12 GeV), besides predicting an unrealistically large top quark mass.
One way to increase v while maintaining Ω e = Ω DM is to dilute the frozen-out e by entropy produced from the decays of a massive particle that subsequently dominates the hierarchy between e and u abundances from developing.

B. Freeze-In
Another potential thermal mechanism for producing e DM is through freeze-in from the SM plasma via electromagnetic interactions and kinetic-mixing, with ∼ 10 −8 . Taking the reheat temperature after inflation below the e mass, T RH m e leads to a freeze-in abundance with an exponential Boltzmann suppression, ∼ exp(−2m e /T RH ). The hope is that when this is chosen to give the observed DM abundance in e , the heavier u will be even more Boltzmann suppressed so that its relic abundance is sufficiently small. However, e has the observed DM abundance if T RH ≈ m e /10 and, at this value of T RH , the freeze-in abundance of u is larger than for e : the closeness of m e and m u means that the additional Boltzmann suppression of u production is more than compensated by the much stronger coupling of u to the SM via gluons. For the reasons discussed in Sec IV A, u must be highly sub-dominant relative to e , hence the freeze-in origin for DM fails.
C. Non-Thermal Production from Decays of φ We have seen that u is overproduced by many orders of magnitude in both freeze-out and freeze-in production of e DM. Nevertheless, non-thermal production of e DM from the decay of an inflaton φ, (or any field which dominates the energy density of the universe), can produce e DM with a sufficiently small and innocuous abundance of u (Ω u /Ω DM 10 −8 ) if certain constraints on the inflaton reheat temperature and the e and u branching ratios are imposed. 6 These general constraints are as follows: First, the reheat temperature must be sufficiently low so that the thermally produced freeze-in abundance of u from the SM bath is 10 −8 Ω DM , implying 7 Next, decays of the inflaton must directly produce the observed DM abundance, requiring a branching ratio into e of Last, the inflaton branching ratio into u must be sufficiently small that Ω u /Ω DM 10 −8 , This small branching fraction requires m φ to be in a narrow range, as it is challenging to obtain B u B e except by a kinematic suppression.
This seems to require a coincidence among the mass scales, which may be understood by an anthropic argument. Let us consider a landscape of vacua, scanning over the scale v while fixing other parameters of the theory. Suppose that the structure of the theory is such that u is abundantly produced where kinematically allowed so that matter-radiation equality occurs much earlier than in our universe. A few examples are provided below. There are two possible obstacles for the formation of a habitable environment in such a DM-rich universe [72]. First, the collapse of halos occurs much earlier, and hence galaxies are much denser than in our universe. A planet then has more frequent close encounters with stars, disturbing the habitable orbit around its own star. Second, the mass fraction of baryons is much smaller than ours. The baryons inside a disk are no longer self-gravitating and are stable against further collapse to form stars. Both obstacles require that the DM abundance should not exceed O(10 − 100) times the DM abundance in our universe, so that universes with copious u production do not contain observers. 8 On the other hand, universes with e production kinematically forbidden have no DM. Almost no galaxies are formed before domination by dark energy, after which structure formation is prevented.
The requirements on T RH , B e , and B u described above can be satisfied, for example, in a model where the inflaton directly couples to quarks and gluons but not to leptons. To satisfy (40), the upper bound on the inflaton mass is m φ < 2m u . e DM is produced through decays φ →ē e γ via an off-shelf loop of mirror quarks and a virtual γ . The inflaton coupling is determined so decays to quarks and gluons give T RH appropriately small to satisfy (39) and ensure that the freeze-in abundance of e is negligible.
Another model, which we will explore in detail in the future, can incorporate baryogenesis.
The inflaton directly couples to heavy right-handed neutrinos N, N , that are integrated out to yield dimension 5 operators of (14), leading to masses for the neutrinos ν and ν . The inflaton decays to ν via the mixing between the right-handed neutrinos and ν . The beta decay of ν into e ē and a lighter ν , which is suppressed by the large mirror electroweak scale, produces e DM with a small branching ratio. The decay into u is forbidden by imposing m ν < m u + m d + m e . The anthropic argument is applicable if the beta decay of ν into e ē 8 Note that we fix the magnitude of the primordial cosmic perturbation as well as the baryon density. The first and the second obstacles are avoided by decreasing the cosmic perturbation and increasing the baryon density, respectively. and a lighter ν involves a small mirror MNS angle. ν also decays into the Standard Model left-handed leptons and the Higgs, and leptogenesis [73] occurs non-thermally [74,75].

VI. CONCLUSIONS AND DISCUSSIONS
The Standard Model is remarkable: it correctly describes a wide wealth of data, while giving a highly incomplete understanding of particle physics. At its inception, there was an immediate realization that one must seek a deeper theory beyond. A particularly elegant idea is to unify the three gauge forces [76,77], despite their manifest differences. Furthermore, if there is a desert above the weak scale, v, the unification of couplings at a very large energy scale M G leads to a prediction for the proton decay rate, Γ p In the intervening decades, despite a succession of ever more powerful experimental tests, the Standard Model, with three generations, neutrino masses and a single Higgs doublet, has shown ever wider applicability. We are motivated to pursue an alternative completion far in the UV because the observed value of the Higgs mass implies that the SM possesses another scale, µ c , where the Higgs quartic coupling vanishes and we take the view that this is the next symmetry breaking scale of nature. Which deeper symmetries of nature should be introduced and broken at µ c ? Motivated by the strong CP problem we introduce a Higgs Parity that includes spacetime parity but does not replicate QCD, and motivated by DM we introduce mirror electroweak gauge symmetry.
We have constructed the minimal theory with gauge group SU ( We comment on the comparison between grand unification (41)  Here one is greatly aided by two features: N event falls only linearly with v , and there is a second observable, the neutron electric dipole moment, that grows as v 2 . Figure 5 shows that, no matter how the values of {α s , m t , m h } evolve as uncertainties are reduced, the entire parameter space of the theory will be tested. As in grand unification, adding particles in the desert could destroy the prediction; however, extra particles added at the scale v do not easily affect our prediction. There is an uncertainty coming from the UV completion scale for the calculation of the kinetic mixing parameter, but this is a logarithmic effect that leads at most to an uncertainty of 2.5 around the central prediction. Unlike minimal grand unification, our theory implies that the gauge structure gets more complicated before any ultimate simple unification.