Neutrinos, dark matter and Higgs vacua in parity solutions of the strong CP problem

The strong CP problem can be solved if the laws of nature are invariant under a space-time parity exchanging the Standard Model with its mirror copy. We review and extend different realizations of this idea with the aim of discussing Dark Matter, neutrino physics, leptogenesis and collider physics within the same context. In the minimal realization of ref. [1] the mirror world contains a massless dark photon, which leads to a rather interesting cosmology. Mirror electrons reproduce the dark matter abundance for masses between 500–1000 GeV with traces of strongly interacting dark matter. This scenario also predicts deviations from cold dark matter, sizable ∆Neff and colored states in the TeV range that will be tested in a variety of upcoming experiments. We also explore scenarios where the mirror photon is massive and the mirror particles are charged under ordinary electromagnetism with very different phenomenology. We also show that, for the measured values of the SM parameters, the Higgs effective potential can give rise to a second minimum at large field value as required to break spontaneously the parity symmetry.


Introduction
Discrete space-time symmetries, parity P, and time-reversal, T, are not fundamental symmetries of nature as we know it.Parity is maximally violated by the chiral electro-weak interactions, and the presence of a complex phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix breaks T perturbatively.In light of the CPT theorem, T violation is equivalent to CP violation and indeed all the observed CP violation observed in experiments is compatible with the one described by CKM matrix.This leads to the so called strong CP problem because one can add the topological term to the QCD lagrangian that breaks P and CP.Such term induces new CP violating effects in the Standard Model (SM), predicting in particular an electric dipole moment for the neutron d n ∼ 10 −15 θ e cm.The experimental constraint d n < 10 −26 ecm then implies that θ < 10 −10 , a value that appears inexplicable within the SM.Essentially all the solutions of the strong CP problem rely on the existence of new symmetries.In the axion solution (see [2] for a review) a U(1) global symmetry, known as Peccei-Quinn symmetry, is introduced that is anomalous under QCD but otherwise exact.Once the symmetry is spontaneously broken θ becomes a dynamical variable, the axion, whose potential is minimized at the CP preserving point, thus solving dynamically the strong CP problem.In the Nelson-Barr solution [3,4] CP is assumed to be an exact symmetry of nature.If the symmetry is spontaneously broken in a favourable way an order one CKM phase can be obtained with suppressed θ−term.
Similarly to Nelson-Barr models one might attempt to solve the strong CP problem imposing P as a fundamental symmetry that would immediately imply θ = 0.At first sight this possibility is excluded because the SM is a chiral gauge theory that maximally violates P. Nevertheless it was realized long ago that P could be a spontaneously broken symmetry in extensions of the SM.In [5] a solution of the strong CP problem was proposed extending the electro-weak gauge sector.In [6] it was further shown that a full mirror electro-weak sector with common color interactions could give sufficient suppression to the θ− term.The crucial observation was that the mirror sector should have mirror electro-weak interactions where SM fermions are also doubled but with opposite chirality as required by parity, see also [7][8][9] for recent works.
Recently an even simpler realization appeared in Ref. [1].Here P is used in connection with a Z 2 symmetry that exchanges the SM with a mirror copy that is parity related, Due to P the mirror sector SM has the same group structure of the SM and same matter content but representation are conjugated.In particular P enforces the condition A simple solution of the strong CP problem then emerges where SU(3) c of the strong interactions is identified with the diagonal subgroup of SU(3)× SU(3) so that the coefficient of the topological term is θ s .Crucially this solution is robust against spontaneous breaking of parity.The safest option that does not introduce new CP violating sources is to break P spontaneously.This can be done if the Higgs potential has a large field minimum so that in the mirror sector it has a large expectation value.
Remarkably we find that with the addition of the new colored states a consistent minimum below the Planck scale emerges precisely around the observed values of the top quark and Higgs boson masses, see [10] for related work.
In this work we study cosmology and particle physics implications of the parity solution of strong CP.An important ingredient turns out to be neutrinos.Neutrino masses can be generated through the see-saw mechanism that must be mirrored for consistency with P. The symmetry allows to mix SM with mirror neutrinos that thus act as sterile neutrinos from the point of view of the SM.The mirror neutrinos are often long lived and lead to entropy injection in the SM thermal bath.In the minimal scenario with massless mirror photon the mirror electron is stable an neutral under the SM and can be the Dark Matter (DM). 1 If the abundance is determined by thermal freeze-out with the mirror photon we predict the DM mass in the range 500-1000 GeV.In this scenario we also predict a small fraction of color dark matter and contribution to ∆N eff .Moreover DM is coupled to a long range force producing deviations from cold dark matter.We also explore scenarios where the mirror photon acquires mass.If the hypercharge is broken to the diagonal all the mirror states have identical electric charge as the SM partners.This forbids a large reheating temperature and can produce deviation from the SM parametrized by the Y parameter [15].
The paper is organized as follows.In section 2 we review the scenario of Ref. [1] and discuss generalizations to different symmetry breaking patterns.A study of the Higgs effective potential is presented showing that for the observed top quark mass consistent minima for the mirror world exist.We include right-handed neutrinos in section 3, discussing their lifetime and thermal leptogenesis.We study the cosmology with a massless dark photon in section 4 showing that the scenario is compatible with large reheating temperature and that DM can be reproduced by the mirror electron.Section 5 is devoted to the scenario with where hypercharge is broken to the diagonal.We summarize our findings in 6.
2 Solving the strong CP problem with P We start reviewing the main aspects of Ref. [1].The model has two exact copies of the SM related by the action of P. In particular, if ψ is a left-handed Weyl fermion in a representation r of SM, its P conjugate would be a (left-handed fermion) ψP Since our definition of P does not act on the representation of the gauge group, ψ is in the representation r of SM.
If P is a fundamental symmetry, the matter content of the model is completely fixed, as in table 1.In light of P the lagrangian of the mirror sector is determined by the one of the SM as, From the behavior of the Yukawa matrices it is manifest that also the physical θ angles of the two sectors are equal and opposite In absence of right-handed neutrinos (that we discuss in detail in section 3), the two sectors can communicate at the renormalizable level through the Higgs portal and kinetic mixing of the two hypercharges, these two terms are invariant under P. 2 As a boundary conditions at very high energies the 2 The presence of a kinetic mixing would have important implication for the scenario in [1] with a massless mirror photon.Upon diagonalizing the kinetic terms the low energy lagrangian can be cast in the form [16], As a consequence the mirror matter acquires a charge ϵ under ordinary electromagnetism.Very strong bounds apply on this scenario especially if states of the dark sector are the DM.We will mostly focus on ϵ ≈ 0 in what follows.
couplings of the two sectors must be equal, but running to low energy can be different when P is spontaneously broken, as we will discuss.
A necessary ingredient to solve the strong CP is to identify our QCD as the diagonal combination of the two SU(3)'s of the visible and mirror sector.This can be achieved introducing a scalar field Σ, transforming as a bi-fundamental (3,3) or (3,3) of the product group, where the covariant derivative acts on Σ as For a generic choice of the potential the minimum is realized for so that color groups are spontaneously broken to the diagonal that can be identified with the low energy QCD interactions.At tree level the couplings satisfy the boundary condition where g s (µ) is the low energy QCD coupling and where we have loosely defined M Σ as the energy scale where SU(3) × SU(3) is broken.While other mechanisms of symmetry breaking exist, see [1] for examples, many conclusions only depend on the symmetry breaking pattern.We will focus on the breaking with scalar bi-fundamental in the rest of the paper.The spontaneous breaking of SU(3) has two remarkable consequences.First, it solves the strong CP problem in the following way.The combination (g 3 G µ ± g3 Gµ )/ g 2 3 + g2 3 acquires mass m G = g 2 3 + g2 3 Σ 0 .Being an octet under SU(3) c the phenomenology reduces to the one of colorons [17,18].The orthogonal combination corresponding to g 3 G µ = ∓g 3 Gµ is massless and can be identified with the gluons at low energies.Due to P one finds Crucially the vanishing of the low energy θ angle holds even for g 3 ̸ = g3 so it holds even after the spontaneous breaking of P. Second, due to eq. ( 9), the mirror quarks are colored under ordinary QCD at low energy.This implies that P must be broken because otherwise the new colored states would have identical mass as the SM quarks.Experimentally the lightest mirror quark (the mirror up) must be heavier that 1.3 TeV [1], and given the observed value of the up mass, this requires ṽ/v ≳ 10 6 so that ṽ ≳ 2 × 10 8 GeV.Concretely this is obtained when the mirror Higgs H gets a VEV ṽ larger than v = 246GeV.All the mirror fermion and gauge boson masses will be then lifted as compared to the SM by a factor of ṽ/v.The mirror masses are thus The lightest states of the mirror world beside neutrinos are the dark electron and dark up quark with masses.According to lattice estimates the up quark mass is different from zero and in the M S scheme is found m MS u (2 GeV) ≈ 2.3 ± 0.1 MeV [19], four times the electron mass.Taking in account running in a world where the electro-weak VEV is 10 8 GeV the ratio changes roughly by factor 2 so that, m ũ ≈ 2m ẽ = 400 GeV ṽ 10 8 GeV .
We can distinguish two different scenarios, 1. Σ 0 > ṽ At energies E ≲ M Σ there are two copies of colored fermions of the SM.This scenario is closely related to the one of [6] where a single gauge group SU(3) was considered.At energies E > m Σ P demands g 3 = g3 .
2. Σ 0 < ṽ For Σ 0 ≲ 10 −5 ṽ the lightest color states are colorons.At energies greater than m Σ the symmetry is restored but the couplings of the two sectors run differently due to breaking fo P. For 10 −5 ṽ ≲ Σ 0 ≲ ṽ some fermions are lighter m Σ .P demands that g 3 = g3 at energies E > ṽ.

Spontaneous breaking of parity
As we discussed P must be broken so that ṽ ≫ v.The breaking can either be spontaneous or soft compatibly with the solution of the strong CP problem.We find it remarkable that the SM has the necessary ingredients to generate spontaneous breaking of P because the Higgs potential develops a new minimum at large field values where the electro-weak symmetry is broken at a high scale.For the observed value of SM parameters however in the SM the second minimum is typically trans-Planckian so it is likely beyond the regime of validity of the SM effective field theory.In the present setup new colored states exist so this conclusion must be reconsidered.Indeed we will find consistent vacua below the Planck scale.Therefore the only assumption required for the parity solution to work is that the SM sits in the standard electro-weak minimum while the mirror world lives in the large field minimum.This also guarantees that the breaking does not introduce new potentially dangerous phases.
The second minimum is modified by two effects compared to the SM.First the new colored states modify the evolution of the couplings and the effective potential above Min[m ũ , m Σ ].Moreover the Higgs portal coupling λ 12 modifies the potential at tree level.We can always tune the electro-weak minimum to the observed value but then the second minimum will be predicted in terms of the SM couplings and it will be different from the one computed in the SM.Note that even if λ 12 is set to zero it will be generated by the running so that in general the potential is V (H, H). λ 12 is however induced only at high loop order so that λ 12 = 0 is in practise consistent.
The spontaneous breaking of a discrete symmetry leads to existence of topologically stable domain walls that would be disastrous if ever produced [20] (see also [21] for a recent discussion).For λ 12 = 0 the situation in our scenario is different.The domain walls interpolate between (v, ṽ) and (ṽ , v) that are exactly degenerate under exact P symmetry.As long as V = V (|H| 2 ) + V (| H| 2 ) the domain walls factorize into two independent profiles for H and H that are individually unstable so that no domain wall problem arises.The reason why this happens is that there exists a family of unstable solutions that interpolate between different minima.The presence of the Higgs portal is however expected to change this conclusion so that a single stable domain wall survives.The domain wall problem could be eliminated if the P symmetry is softly broken.Alternatively the domain wall problem is solved if P is broken during inflation, H I < ṽ.In this case any abundance of domain walls is inflated away. GeV . Parameter space of [1] with spontaneous P breaking.For the experimental value of the top quark mass, m t = 172.5 ± 0.5 GeV [22,23] a perturbative minimum below the Planck scale emerges for 10 6 GeV < M Σ < 10 10 GeV.We assume λ 12 = 0 and M Σ < m ũ.
2.2 Higgs effective potential at 2 loops: a second minimum before M Pl In this section we discuss under what circumstances the SM Higgs effective potential can develop a sub-Planckian minimum compatible with the P solution to the strong CP problem (see also [10] for a different realization).We work in the limit of vanishing quartic portal λ 12 .In this approximation and at the perturbative order of interest, the appearance of a mimimum at large Higgs vev for the mirror world can be studied by inspecting the SM effective potential, modified by the presence of new matter charged under the SM gauge symmetry as well as new dynamics.
The RGE running of the SM parameters is modified in several ways.First, the P symmetry enforces equality of couplings at and above ṽ.Second, importantly, above M Σ the strong coupling g s of our color SU(3) c is matched to the fundamental couplings of the two SU(3) factors as per eq.( 10), and this may happen before or after ṽ.For example if M Σ ≫ ṽ the model becomes the one of Ref. [6].While in the opposite regime above M Σ the running of g 3 is modified by the presence of new colored matter.
We here explore the case where M Σ < m ũ so that below the SU(3) × SU(3) breaking there is just the SM.The running of the SM parameter can be computed with great precision and we use [24] to run the MS parameters up to the scale μ = M Σ .Above this scale the running of g 3 is modified by the presence of the bifundamental complex scalar Σ.The modification of the β-function of g 3 at 2-loops reads 3 , Correlation between the large field minimum of the Higgs and the top quark mass in the scenario [1].
The matching condition (10) has the implication that the low-energy value of g 3 be limited to the range g s (M Σ ) ≤ g 3 (M Σ ) ≤ √ 2g s (M Σ ), since g3 will be inevitably larger due to the faster running in the infra-red.
We then study the SM effective potential in this context, by fixing M t , M h and α s (M Z ) to their standard values, and inspecting for which values of M Σ a new sub-Planckian minimum for h appears.The new physics acts as a stabilization for the Higgs quartic coupling: the matching condition will generate an effective larger g 3 coupling, that it is known to stabilize the SM potential, on top of the already smaller running due to the presence of Σ.Therefore, by fixing M t in the experimentally allowed range [22,23] , M Σ should be large enough to allow for g 3 (M Σ ) to be such that a minimum right before M Pl appear.Intuitively, very large values of g 3 (M Σ ) will make the second mimimum disappear.
The solution ṽ(M Σ ) will be then very precisely determined upon a judicious choice of g 3 (M Σ ) and g3 (M Σ ).This has to be determined iteratively satisfying both eq.( 10) and the P symmetric boundary conditions at ṽ, g 3 (ṽ) = g3 (ṽ).This is done by considering the running of g3 from ṽ to M Σ including all the thresholds of mirror quarks (below ṽ the mirror sector is simply SU(3)×U(1), we neglect the running of the mirror electromagnetism) up to three loops.We notice that the running of g3 is much faster and can result quite easily in confinement of SU(3), the running is faster since M Σ ≪ m ũ.This poses a tight constraint to identify solutions with g3 (M Σ ) in the perturbative realm.
The SM beta functions and effective potentials are taken from Refs [24,25].New physics only enters through the boundary conditions at M Σ and the new contributions to the β-functions.Here we compute the effective potential for the Higgs as In order to compute this we use the available β-functions and the expression of the Higgs anomalous dimension, constructing a set of differential equations for (g 2 i , y 2 i , λ, Γ).The effective potential then reads V eff (h) = 1 4 λ eff (h)h 4 .The position of the second SM minimum is found by solving By applying the algorithm discussed above we have found solutions with g3 (M Σ ) in the perturbative regime, satisfying to great accuracy (sub-per-mille level) the boundary conditions and having a minimum h < M Pl .Results are reported in figure 1 and 2. Notably consistent solutions are precisely found for the observed values of the top quark and Higgs mass.
In the scenario M Σ > ṽ below the SU(3) breaking scale there are two copies of colored fermions.The running of g s is thus modified compared to the SM and is larger at high scales.This in turn reduces y t compared to the SM and increases the instability scale until it disappears.In this case we have not found consistent minima with VEV smaller than the Planck scale and vanishing λ 12 .Many other possibilities exist where only a subset of mirror fermions are lighter than M Σ .A more detailed analysis will appear elsewhere.

Generalizations
The breaking of the two fundamental SU(3)'s to the diagonal is a structural part of the scenario.One might wonder if more general patterns of symmetry breaking can be considered.Focusing on two SM copies one can consider in general the pattern of symmetry breaking, The model of Ref. [1] corresponds to the case , where the electroweak interactions (visible and mirror) are unbroken by Σ.
Let us first consider the breaking SU(2) × SU(2) down to the diagonal.In the regime ṽ ≫ Σ 0 mirror SU( 2) is broken and one finds v 2 SM = v 2 + Σ 2 0 .The small VEV of Σ 0 implies the existence of light vector bosons charged under SU (2) with mass in the 100 GeV range or below that is excluded.The opposite regime ṽ ≪ Σ 0 is also not viable because v 2 SM = v 2 + ṽ2 so implying new light colored states.Therefore SU(2) × SU(2) cannot be broken to the diagonal.This also forbids the possibility of unified breaking pattern of the form SU(5) × SU(5) → SU(5) d .
A different conclusion holds for hypercharge that can be broken to the diagonal combination without introducing new states with SM charges.We discuss this possibility below.Alternatively U (1) could be broken while preserving U(1), namely H = SU(2) × U (1) × SU (2).In order for this to be consistent with P symmetry, one can for example add extra scalars charged under U(1)'s in both sectors or SU(2) triplets that spontaneously break only U (1).Eventually, upon P breaking, the only unbroken symmetries are the electroweak interactions of the SM.In this case the mirror states have the same quantum numbers of the model of Ref. [1] but without massless mirror photon.

Massive mirror photon Breaking
This can be done effectively promoting SU (3) → U (3) and considering the collective breaking Mirror states quantum numbers below the symmetry breaking scale m Σ in the model with massless dark photon (q = 0) and with hypercharge broken to the diagonal (q ̸ = 0).
We notice in fact that the scenario of [1] already contains the necessary ingredients and it is sufficient to to give Σ a U(1) charge (q , ±q).The presence of the additional U(1) constrains the potential of Σ of eq. ( 8) to be of the following form where we have absorbed the contribution from the Higgs vevs v and ṽ in the effective mass term µ 2 Σ ≥ 0. If both quartics are positive the only minimum is the one leaving a U(3) invariant, Σ ∝ I where Σ ij = µ Σ / 3λ Σ + λΣ δ ij . 4In the unitary gauge the model of eq. ( 8) produces and effective mass term Alternatively we could also give a Stueckelberg mass to the diagonal combination.Due to the breaking to the diagonal the mirror states are charged under the SM hypercharge and thus also carry electric charge.In particular the mirror states have Therefore all the mirror sector states have electric charge identical to the SM particles but with different masses controlled by ṽ.Note that instead the hyper-charge assignments of mirror matter (which is a good quantum number when ⟨H⟩ = 0) are different from the corresponding SM states.In this context the left-handed mirror neutrinos have vanishing hyper-charge (as well as, crucially, the VEV of the mirror Higgs field).For Σ 0 → ∞ the dark photon can be integrated out.In this limit where SU (2) L,R act on SM and mirror fermions respectively.The same scenario was also considered in [6].
More than one mirror The crucial ingredient of the strong CP solution is the convolution of space-time symmetry with mirror symmetry.This can be generalized to 2N copies of the SM where, Breaking color to the diagonal subgroup, gives rise at low energy QCD + a tower of states analogous to extra-dimensions.Indeed one could realize such scenario in extra-dimensions with replicas of the SM related by chirality.

Neutrinos from the mirror world
The generation of SM neutrino masses introduces new structure in the model that must be compatible with P invariance.In this section we consider a type-I see-saw with right-handed neutrinos.To give masses to all neutrinos one needs to add 3 copies (N i , Ñi ) of singlet Weyl fermions transforming under We can define the parity states, We consider the lagrangian invariant under P (see also [26] for a particular choice) 5 , where M + and M − are real and diagonal, while α and β are generic complex matrices in flavor space.
In the limit α = β and M + = M − two independent fermion number symmetries emerge for the visible and the mirror sector that guarantee that the lightest fermion of each sector is stable.
In a more symmetric notation, the above lagrangian can be recast as The Y, Y yukawa matrices are 3 × 6, and they are parametrized as Y = (α, β) and Ỹ = (α * , −β * ).In this notation P enforces the following relation This parametrization will be useful discussing leptogenesis.We now discuss the possible structures of the neutrino mass spectrum.Since we consider the limit where max[α, β]v ≪ min[M ± , max[α, β] ] the SM neutrinos are always majorana.We can distinguish two cases:

Mirror Majorana neutrinos
We start with the limit where M ± are the heaviest masses, so that N ± can be integrated out.They act as see-saw for both the visible and the mirror neutrinos, and they yield the following lagrangian where A ij is a symmetric matrix and B ij is an hermitian matrix in flavor space.In terms of the fundamental parameters they are given by A unitary rotation U and a diagonal rephasing P make the first two terms diagonal and real (and therefore equal).This amounts to the transformations L → U P L and L → U * P * L, that take the lagrangian into the form where mi is real 2 mi /ṽ The presence of six right-handed neutrinos guarantees that all the Majorana neutrinos (visible and mirror) are massive.
For vanishing ∆ the mass spectrum of mirror neutrinos is determined by the SM neutrinos, Therefore the mirror neutrinos are determined by the same neutrino mass ordering of the SM.
Importantly the masses are quadratic in the VEV while the masses of other particles grow linearly.As a consequence for m ν > 10 TeV the mirror neutrinos are not the lightest states in the mirror world.
For ∆ ̸ = 0 visible and mirror neutrino mix.After H gets a vev, the lagrangian can be written as and the mixing scales as Integrating out the mirror neutrinos we get the final Weinberg operator for the SM neutrinos For ∆ ij ≪ 1, the neutrino masses are controlled by the high-scale see-saw, and their masses are strongly correlated with the mirror neutrino masses as in eq.(32).Correlation is lost for ∆ ij ∼ O(1) when the mass of observable neutrinos becomes dominated by the see-saw with the mirror neutrinos.
Mirror neutrinos decay Due to the spontaneous breaking of parity the mirror neutrinos ν are unstable towards decay to SM even if they are the lightest mirror fermions.The decay rate of ν is important for the cosmological history.Upon breaking of the mirror weak interactions, L gets a majorana mass from the mirror Weinberg operator.In general, compatibly with P a mixing exists between SM and mirror neutrinos, see eq. (31).Such a term allows ν to decay even when electro-weak symmetry is unbroken.Clearly νi acts as three right-handed neutrinos for the SM neutrinos.The tree-level decay rate of the mirror neutrinos into LH + L * H * is given by Such a rate becomes faster than Hubble at a temperature Mirror neutrinos decay to SM final states from the LH ν term directly, or through off-shell W/Z and Higgs.The decay width depends on the available decay channels.We here consider the case where the lightest mirror neutrino is below the W mass, so that it decays to three fermion final states as well as lepton (charged or neutrino) plus meson final states below the QCD scale.The phenomenology here is totally analogous to a majorana 'heavy neutral lepton' (HNL), so that we can use results from the corresponding literature [26].We notice that the decay to the lightest SM leptons and neutrino are of the form Denoting with m ν,light.the mass of the lightest SM neutrino, the decay width is numerically given by where we assumed ∆ ij = ∆δ ij .To determine whether ν is long-lived we can compare it to Hubble at T ≈ m ν , their ratio reads From [26] we find C F ∼ 16 for m ν ∼ 10 GeV.For Γ/H < 1 the neutrinos are long lived.Their energy density for T < m ν redshifts nonrelativistically until they decay to SM neutrinos when Γ ∼ H.If the decay is sufficiently slow the mirror neutrinos temporarily dominate the energy density of the universe injecting entropy in the SM thermal bath.This process dilutes all the abundance in the dark sector as well as the baryonic asymmetry.In order to avoid constraints from BBN the reheating temperature should be somewhat larger than 5 MeV.This translates into Γ > 25/s.The regions where neutrinos are long lived leading to entropy injection and are shown in blue in Fig. 3 where the red region is excluded by BBN bounds.
Mirror neutrinos decouple after EWSB, at a temperature that can be computed by evaluating rates induced by Fermi operators with leptons (and light quarks).We call T ν,dec such a temperature, and mirror neutrinos decouples when relativistic.Their energy density at the time after they have become non-relativistic is where T m ≈ m is the temperature when they are non-relativistic.If Γ ≪ H(T m), mirror neutrinos are long lived and they reheat the SM, thanks to an entropy injection.
Most of the decay happens at a time around Γ ≈ H, which might be already in a phase of matter domination driven by the mirror neutrino.At that time the corresponding SM temperature is The maximal effect of entropy injection is when the neutrino decays while dominating the energy density of the universe.This gives the maximal amount of dilution, and correspond to the maximal reheating temperature The dilution is computed as the ratio of the SM entropy before and after the decay of the mirror neutrino.Using the simplified formulas in [27] the dilution factor reads, Note that the heaviest neutrinos whose mass should be larger than 10 GeV in light of the constraint on the mirror up quark does not produce significant dilution.The dilution is thus dominated by the lightest neutrino.BBN constraints require that the reheating temperature should be larger than 5 MeV or equivalently that Γ ≳ 25 s −1 .As a consequence only in the region around the physical values of neutrino masses the dilution is relevant.Dilution and BBN constrains are shown in 3 right for the reference value m µ = 0.008 eV.

Scattering of mirror neutrinos
The same couplings that control the decay of mirror neutrino are also relevant for scattering at energies above their mass.Roughly the rate is given by the decay rate replacing m ν with the temperature so that the rate goes as T 5 .As a consequence the mirror neutrinos decouple when the temperature drops below the mass.

Mirror Dirac neutrinos
Now we investigate the opposite regime where the scale of the right-handed neutrino is below or comparable to the P breaking scale, namely where ṽ ≫ M ± finding two different behaviors depending on the relative size of the two contributions.The discussion in this case will lead us to the vanilla see-saw for the SM neutrinos.In this limit, the mirror neutrinos gets a Dirac mass term, Li H(α * ij N j + − β * ij N j − ), that we can read out from eq. ( 26).Below the scale of the Dirac mirror neutrinos, the effective lagrangian involving SM neutrinos and other states is found along the solution α Neglecting the flavor structure one finds, Therefore, below the P breaking scale, we are left with the standard scenario for the neutrinos where the right-handed neutrino is a combination of N + and N − .SM neutrinos are set by the see-saw scale, and we lose the correlation with the mirror majorana case.In the case of 3 generations, the Yukawa couplings are complex allowing for CP violation in the neutrino sector as required by leptogenesis.

Leptogenesis
The structure of right-handed neutrinos N ± discussed above allows us to discuss baryogenesis via leptogenesis, exploiting the asymmetric out-of-equilibrium decay of Majorana neutrinos as in standard leptogenesis (see [28] and [29] for reviews).In our context, we identify two different scenarios, that here we briefly sketch leaving further developments for future work.

P-broken leptogenesis
Below the scale ṽ a combination of N ± acts as Majorana right-handed neutrino for the SM neutrinos (mirror neutrinos are instead Dirac).This falls into the category of standard leptogenesis.In this scenario the leptonic mirror sector plays no particular role since they are heavier than a Majorana combination of N ± .Therefore we can map this to the commonly discussed thermal leptogenesis.In particular, the interactions of the Majorana N can be matched to the following (schematic) effective lagrangian The branching fraction of N decaying to mirror states of is suppressed by factors of order

P-symmetric leptogenesis
In this case the scale M N is the highest in the system, higher than ṽ.This suggests that the decay of N respects P. In this case both SM and mirror neutrinos are inevitably Majorana.Since by symmetry, CP violation occurs equally in each sector, if the decay of N happens when the mirror weak interactions are unbroken (therefore P is not spontaneously broken) a net amount of lepton asymmetry in each sector will be produced.We refer to the lagrangian in eq. ( 26) in the phase where P is unbroken.The decays of N ± → LH, L * H * , L H, L * H * are such that the produced number of L is equal to the number of L * .In light of this it is convenient to define the total generalized lepton number, Given that P sends N L → −N L such number is odd under P, so that until P is exact, the decay of N ± will not generate any net L tot number.Essentially, the second Sakharov condition is not satisfied for L tot .This can be checked explicitly by computing the asymmetry in the decay of N in each sector, where N is the heaviest neutrino.In order to compute the above asymmetries it is convenient to use the symmetric notation of eq. ( 27).From an explicit computation we find (assuming N = F 1 ), where x ] (see [28] for a review).We notice that the loop function is symmetric under exchange of the two sectors, as it should.Moreover Y † Y = I(Y † Y ) * I. Notice also that the combination that enters the above expression is the square of (Y † Y ) ij .We then have ( , and also noticing that (Y † Y ) kk = ( Y † Y ) kk , yielding a relative sign between ε and ε upon taking the imaginary part we have explicitly ε = −ε, which supports the claim in eq. ( 49).
After N decays and before mirror electroweak symmetry breaking an equal and opposite amount of lepton asymmetries is present in each sector, which can be transferred to baryon number through the respective sphaleron transitions, leading to equal and opposite baryon and lepton asymmetries in each sector.We notice that at this epoch both B − L and B − L are individually conserved.
When the mirror Higgs develops a vev, mirror sphalerons shut down and B becomes conserved.However L violations become active again since the mirror neutrinos gets a majorana mass.Moreover, if ∆ is sizable this can recouple L and L. In turn, this can potentially lead to a washout of the SM baryon asymmetry produced thus far since both B + L (SM sphalerons) and L violating processes can now be active.This has the effect of (partially) washing out B generated before.We leave this interesting scenario open to future investigations.

Cosmology with a massless mirror photon
The mirror sector contains colored states and a massless dark photon that seem at odds with standard cosmology, if the mirror sector is not empty.As we will show, quite surprisingly the presence of a massless dark photon does not exclude the scenario and moreover the mirror electron elegantly produces the DM abundance from thermal freeze-out from mirror photons.The region of parameters selected by DM in particular is close to the experimental bounds.
The only link between mirror world and the SM is provided by colored states and neutrinos 6 .For large reheating temperature the two sectors are in thermal equilibrium.Specifically if T R > ṽ the whole dark sector including neutrinos thermalizes with the SM at a common temperature.Neglecting for the moment the role of right-handed neutrinos, the first important event is the decoupling of mirror neutrinos, when electro-weak interactions of the type ν ẽ ↔ ν ẽ go out of equilibrium.Mirror neutrinos, in absence of other dynamics, decouple relativistically at a temperature If they are stable, they would overclose the universe given the masses (33).The only viable option in this case would be that the reheating temperature is below the mass lightest colored state, m ũ or m Σ , so that the dark sector never reaches thermal equilibrium with the SM.A much more interesting interesting possibility is that the mirror neutrinos decay to the SM, ∆ ̸ = 0.In this case the reheating temperature can be large and the two sectors are in thermal equilibrium initially.Even if T R < T ν so that the neutrinos are not in equilibrium due to the mirror weak interactions, the neutrinos portal will thermalize them.After the mirror neutrinos decay the dark sector contains massless mirror photon that contributes the dark radiation and massive stable states, mirror electron and hadrons made of mirror up quark contributing to DM.

Relativistic degrees of freedom
SM and mirror sector maintain equilibrium until T * ≈ Min[m ũ , m Σ ]/25 through ordinary color interactions.Below T * the mirror sector contains mirror electrons and mirror photons, while at the same temperature the SM has its full thermal degrees of freedom.(After the mirror electrons decouple the mirror photon temperature gets reheated compared to the SM).
Assuming that the two sectors decouple at temperature T * the evolution of the temperature is given in general by, g * (T ) g * ( T (T )) Assuming the mirror up quark to be the lightest colored state we estimate T * ∼ m ũ/25.At that temperature the only relativistic degrees of freedom in the mirror sector is the photon while mirror neutrinos typically remain into thermal contact with the SM.We find, where is the effective number of degrees of freedom of the SM plus mirror neutrinos.This allows us to compute ∆N eff from the presence of the light mirror photon, as  On the left the abundance assuming that the relevant entropy injection is associated to the SM neutrino with mass m ν = 0.008 eV (normal hierarchy).On the right m ν = 0.05 eV (inverted hierarchy).Light red region is excluded by BBN constraints.

Thermal Dark Matter: mirror electrons (and mirror up quark)
Mirror worlds contain automatically stable particles as a consequence of their accidental symmetries.By construction mirror baryon number and electric charge are accidental symmetries of the mirror sector.It follows that the lightest state carrying baryon number and the mirror electron are stable 7 .
For the latter, since the U(1) symmetry is gauged and unbroken, stability is exact while for baryon number conservation holds up to dimension 6 operators as in the SM.As we will show the mirror electron is a good DM candidate being neutral under the SM.Its mass is predicted by the DM abundance and it turns out to be borderline with current constraints.The lightest state with baryon number is the mirror up quark.This state is charged under ordinary QCD at low energy so naively it is not a viable DM candidate.The thermal abundance turns out to be suppressed compared to the mirror electron.Moreover as shown in [30] heavy colored states mostly form deep Coulombian bound states that are SM singlets and are thus good DM components.The residual small fraction of mirror up quark forms QCD-size exotic hadrons binding with ordinary matter.These states feature large hadronic cross-sections and they are very constrained experimentally, their fraction should be smaller than O(10 −4 ).Remarkably due to the double suppression of the abundance of hybrid QCD states the scenario appear to be viable.At the same time deviations from CDM are predicted that might be observable in future experiments.

Mirror electrons as DM
The lightest charged states in the mirror world are electrons ẽ that are coupled to massless dark photons.The system at energies below m ũ is just dark QED, Mirror electrons and up quark follow an equilibrium distribution due to annihilation into mirror photons.Allowing for a different temperature between visible and mirror sector at decoupling one finds [27], Ωh 2 0.12 = ξ 106.75 g * (T * ) g * (T * ) + g * (T * )ξ 4 106.75 1 ⟨σ eff v⟩(23TeV) 2  (58) where g * is the effective number of degrees of freedom of visible and mirror sector at freeze-out and ξ the ratio of temperatures.
The tree level annihilation cross-sections of electrons is given by, This should be corrected by two related effects: and Sommerfeld enhancement and bound state formation that adds an annihilation channel [31].For α EM ∼ 10 −2 however the modification to the tree level cross-section is very small and be neglected.At freeze-out ξ ≈ 1 the temperature is between 10 and 50 GeV so that the pre-factor in eq. ( 58) can be neglected in what follows.Computing the abundance with the tree level cross-section with α EM = 1/128 one finds that mirror electrons reproduce the DM abundance for m ẽ ≈ 300 GeV.Using the fact that m ũ ∼ 2m ẽ we would thus predict m ũ ∼ 600 GeV that is excluded experimentally.This estimate however neglects the slow decay of mirror neutrinos.If this happens for T ≲ m ẽ/25 (the temperature where mirror electrons decouple from the thermal bath) it leads to entropy injection in the SM thermal bath that the depletes the DM abundance according to the factor (45) 8 .This effect is generic in most regions of parameter space.We show the DM abundance accounting for the entropy dilution in 4. The critical abundance is reproduce for, allowing to evade possible collider bounds on colored states.A precise determination of the DM depends on the details of the neutrino sector and requires the solution of the coupled Boltzmann equations that we leave to future work.
There are two potential constraints on DM charged under a long-range interaction: i) the delay of kinetic decoupling (suppressing power on small scales); ii) the modification to DM properties in clusters and galaxies.The first problem is not very relevant for heavy DM.The kinetic decoupling can be estimated to happen T k ≈ MeV(M/TeV) 3/2 /ξ 2 × (0.01/α) [32], safely before BBN for heavy masses, while the second constraint extends to even larger DM masses [33].The strongest constraints arise from the structure of galaxy halos as for the observed ellipticity of the NGC720 halo [34].Demanding that the interaction are sufficiently weak not to deform the DM velocity distribution over the lifespan of a galaxy one can estimate the bound [33] so that mirror electrons in the TeV range can be good DM candidates.Another constraint on this scenario arise from underground direct detection experiments.As discussed in [11,12] even if absent at tree level a kinetic mixing between hyper-charges arises at loop level.For M Σ > m ũ running effects at 4-loop order generate ϵ ∼ 10 −8 that is already in tension with Xenon 1T.This conclusion assumes that the local DM density is not modified by the supernova shock waves that tend to expel it from the galaxy [35], see however [36].The effect is suppressed if M Σ ∼ m ũ as in that case the running only arises at higher loop level.This would however require a coincidence of scales.In any case direct detection experiments appear very relevant for this scenario and might already exclude this scenario.

Colored DM fraction
The abundance of color triplets with no visible electric charge leads to a component made of colored DM [37].Naively DM made of strongly interacting particles is not viable because for example it would form exotic nuclei where one of the ordinary quarks is replaced by the exotic quark.These nuclei have QCD like cross-sections and lead to very strong constraint on their abundance that can be at best a small fraction of DM.However for heavy DM, colored particles can form deep Coulombian bound states with binding energy E B ∼ α 2 3 M Q much larger than the confinement energy.These bound states are small color singlets that behave as collision-less DM for practical purposes.The fraction of DM that ends up into exotic nuclei with QCD cross-sections and into Coulombian bound states is a dynamical question that requires to solve the coupled Boltzmann equations.As it turns out most of the DM actually forms deep bound states that are energetically favoured leaving just a small fraction of strongly interacting heavy quarks that hadronize with ordinary matter.
This general mechanism was explored in details in [37] for a scenario with a Dirac color octet.The QCD-like stable bound states have in this case zero electric charge leading to weaker constraints.It was in particular shown that the abundance can be reproduced for a mass M Q = 12.5 TeV.In the current scenario the heavy quark is a color triplet with zero charge leading to exotic nuclei with fractional electric charges.The experimental constraints are in this case stronger and it is unclear whether the scenario would be allowed if this made 100% of DM.In our case however only a fraction of DM is in the form of colored particles leading to weaker constraints.Extremely strong bounds were however derived in [35] re-interpreting searches of magnetic monopoles assuming that they are accelerated by supernova shock-waves and arrive to the detector with sufficient energy.
The abundance from freeze-out can be simply estimated using the where S is the Sommerfeld enhancement (computed in the massless limit) and we neglected the subleading decay to mirror photons.Using v rel ∼ 0.3 at decoupling and α 3 (M u ) = 0.1 one finds an abundance of colored states Ω uū /Ω eē ∼ 0.1.We note that this estimate is conservative.As discussed in [37] bound state formation enhances the effective annihilation cross-section reducing the abundance.Moreover after QCD confinement about half of the particles bind into unstable bound states leaving half of colored particles in states with net baryonic charge.Therefore the fraction of strongly interacting DM is very suppressed.
In summary the scenario of [1] leads to sharp predictions for DM made of mirror electrons and up quarks with mass in the TeV range.The scenario is currently close to experimental limit on DM charged under a long range force predicting new colored states just above the collider constraints that give rise to a small component of strongly interacting DM.Future experiments can thus test this scenario.

Phenomenology of massive mirror photon
Let us now turn to the scenario where hypercharge as color is broken to the diagonal combination.In this case a combination of U(1) gauge bosons has mass and the mirror states acquire SM quantum numbers as in the Table 2 right.In particular the states have identical electric charge as the SM partners. 9o understand the phenomenology of the mirror photon in this context, it is useful to focus on the effective lagrangian of the neutral gauge bosons, where we neglect the difference of mirror couplings due to different running below ṽ and we also added a kinetic mixing for hypercharge as allowed by the symmetries.Given that for phenomenological reasons ṽ ≫ v, it is a good approximation to work in the limit ṽ → ∞.To leading order this corresponds to replace W = g 1 /g 2 × B in the above lagrangian so that, The 4 derivatives term gives a contribution to the precision electro-weak parameter Y parameter [41] Current bound require |Y | < 0.2 × 10 −3 [42] so that m A > 10 TeV that is stronger than the one from direct searches.The bound is also expected to improve with future LHC data [39,43,44].

Cosmological history
Naively since the dark photon is massive one might expect that this scenario would be less constrained.
In fact the opposite is true.Due to the spontaneous breaking of hypercharge to the diagonal all the mirror states have the same electric charge as the SM states.Since the mirror electron and up quark are stable in order to escape from bounds their abundance should be negligible.As we have shown if the mirror sector is initially in thermal equilibrium with the SM the abundance of electrons saturates the DM for masses below TeV.The only possibility is thus that the mirror sector is never in thermal equilibrium with the SM.Given that the bridge between the two sectors is given by colored states we estimate, When this condition is satisfied the abundance of mirror is negligible.Since the reheating temperature is low the mechanisms thermal leptogenesis cannot be realized in this framework.The only states that can be populated are thus neutrinos that are produced through the neutrino portal.
Mirror neutrinos as Dark Matter?An interesting possibility for DM is offered by the mirror neutrinos that are the only neutral states when hypercharge is broken to the diagonal, see Table 2. Mirror neutrinos with masses in the KeV range can give rise to sterile neutrino DM, see [45,46] for reviews.Due to the mixing with the SM neutrinos they can be produced via freeze-in from the SM thermal bath.The abundance is roughly reproduced for, Note that if the mirror neutrinos are Majorana the mixing is given by (35) so that it is difficult to obtain the critical abundance from freeze-in given the bound m ũ > 1.5 TeV.Other mechanisms could however be considered.Moreover in the regime where mirror neutrinos are Majorana the mirror neutrinos have masses given by eq. ( 32).This implies that the lightest neutrino must be lighter than 10 −9 eV if the lightest mirror neutrino is in the KeV range.

Conclusions
All known solutions of the strong CP problem rely on the existence of a new symmetry.In the case of the QCD axion the SM should be augmented with a U (1) PQ global symmetry only broken by the QCD anomaly.The existence of an almost exact (apart from anomalies) global symmetry raises the question of axion quality problem [47], especially since in quantum gravity no global symmetries are known to exist.In the Nelson-Barr solution CP must be assumed to be an exact symmetry that is spontaneously broken and a non-generic structure of couplings of new fermions must also be enforced.
From this point of view we find it fascinating that the strong CP problem might be solved by a space-time symmetry such as a generalized parity, P.This idea was originally advocated in the context of left-right models [5] and further generalized in [6] but has not received as much attention as the QCD axion or even the Nelson-Barr mechanism.Here we have built upon Ref. [1] where an even simpler scenario was considered.The SM is extended with a mirror copy related by spacetime parity.The breaking of SU(3) × SU(3) → SU(3) c then implies a boundary condition θ s = θ + θ = 0 solving the strong CP problem.
In this work we extended Ref. [1] in several ways.First we have identified different patterns of symmetry breaking that allow to solve the strong CP problem but with different phenomenological implications.The existence of other minima in the SM Higgs effective potential suggests the possibility to break P spontaneously.We have shown that in the simplest scenario where the breaking of color is realized by a scalar bi-fundamental consistent high scale vacua exist precisely for the measured value of the top quark and Higgs masses.We have then included right-handed neutrinos necessary to give masses to visible and mirror neutrinos.This leads to an interesting structure of mirror neutrinos whose masses scale quadratically with the mirror Higgs VEV.The neutrino dynamics is particularly important for the cosmological history of the model as they are typically long lived.Moreover if the reheating temperature is large thermal leptogenesis can be realized in the SM without producing an asymmetry in the mirror world.
The phenomenology depends crucially on the fate of the mirror photon.If the mirror hypercharge remains unbroken as in [1] all the mirror states have no electric charge under the SM.This leads to a very predictive scenario that is determined by the VEV of the mirror Higgs.Assuming a large reheating temperature the dark photon contributes to the effective number of relativistic degrees of freedom, predicting sizable ∆N eff .The mirror electrons reproduce the DM abundance for mass in the range 500-1000 GeV, that is marginally allowed by astrophysical and collider constraints and would produce signals in direct detection experiments.A small fraction of strongly interacting DM is also predicted in the form of bound states of the mirror up quark that however leads to severe experimental constraints.The detailed predictions depend on the mirror neutrinos whose decay leads entropy injection in the SM plasma.Overall the DM scenario appears somewhat in tension with experimental constraints motivating extensions of the minimal setup.
If the pattern of symmetry breaking also includes hypercharge so that U (1) × U (1) → U (1) Y the mirror states have the same electric charge as the SM states.This implies that the mirror electron and up quarks that are stable have electric charge.As a consequence this scenario is only viable for a reheating temperature lower than the mirror electron mass.The only viable DM candidate is the lightest mirror neutrino that could realize sterile neutrino DM depending on the mass of the lightest neutrino.Direct and indirect collider constraints require that the dark photon mass is larger than 5-10 TeV.
All in all solving the strong CP problem through P appears as an attractive possibility that should be considered seriously.In this work we have just scratched the surface of the phenomenology of mirror P world.A more detailed analysis will be surely needed to study the details of dark matter, leptogenesis and collider physics in this scenario.Many questions lie ahead that we are hoping to answer in the future.For example the spontaneous breaking of parity gives a logic to the existence of a second minimum in the SM effective potential.It would be interesting to find a cosmological mechanism to populate the biverse.

5 Figure 3 .
Figure 3. On the left, lifetime of mirror neutrinos for the reference choice of SM neutrino m ν = 0.008 eV.On the right, isolines of dilution (45) are shown and bounds from BBN and mass mirror up quark.

Ω
is currently consistent with experimental bounds it can be tested in future experiments .This result is modified if neutrinos are long lived.The entropy injection (45) modifies the previous results as, ∆N eff ≈ 0.06η DM for m ν =0.05eV

Figure 4 .
Figure 4. Mirror electron DM mass as function of mixing ∆ accounting for entropy injection (solid black line).On the left the abundance assuming that the relevant entropy injection is associated to the SM neutrino with mass m ν = 0.008 eV (normal hierarchy).On the right m ν = 0.05 eV (inverted hierarchy).Light red region is excluded by BBN constraints.

Figure 5 .
Figure 5. Prediction for the SM lightest neutrino mass in connection with the DM relic abundance, for different values of ∆.Light red region is excluded by BBN.

Table 1 .
Quantum number of left-handed Weyl fermions of SM and mirror SM.