Mirror symmetry: from active and sterile neutrino masses to baryonic and dark matter asymmetries

We consider an SU(3)'_c\times SU(2)'_L\times U(1)'_Y mirror sector where the field content and dimensionless couplings are a copy of the SU(3)_c\times SU(2)_L\times U(1)_Y ordinary sector. Our model also contains three gauge-singlet fermions with heavy Majorana masses and an [SU(2)_L\times SU(2)'_L]-bidoublet Higgs scalar with seesaw-suppressed vacuum expectation value. The mirror sterile neutrino masses will have a form of canonical seesaw while the ordinary active neutrino masses will have a form of double and linear seesaw. In this canonical and double-linear seesaw scenario, we can expect one sterile neutrino at the eV scale and the other two above the MeV scale to fit the cosmological and short baseline neutrino oscillation data. Associated with the SU(2)_L and SU(2)'_L sphaleron processes, the decays of the fermion singlets can simultaneously generate a lepton asymmetry in the [SU(2)_L]-doublet leptons and an equal lepton asymmetry in the [SU(2)'_L]-doublet leptons to explain the existence of baryonic and dark matter. The lightest mirror baryon then should have a determined mass around 5 GeV to account for the dark matter relic density. The U(1) kinetic mixing can open a window for dark matter direct detection.


I. INTRODUCTION
Various neutrino oscillation experiments have firmly established the three active neutrino oscillation picture [1], however, some short baseline neutrino oscillation experiments [2] and recent re-evaluations of reactor antineutrino fluxes [3] hint the existence of additional sterile neutrinos with the eV-scale masses [4][5][6].The light active neutrinos can be elegantly understood in the seesaw [7][8][9] or other [10][11][12][13][14][15][16][17][18][19][20][21] extensions of the SU (3) c × SU (2) L × U (1) Y standard model (SM).If the light sterile neutrinos are eventually confirmed, we need explain not only the existence and small masses of the sterile neutrinos but also the mixing between the active and sterile neutrinos.Such sterile neutrinos can naturally appear in the mirror  universe models discussed in the literature [42][43][44][45][46][47][48][49][50][51].There are also other ideas for the light sterile neutrinos [52][53][54][55][56][57][58].On the other hand, precise cosmology has indicated that in the present universe the energy density of the dark matter is comparable with that of the ordinary matter [59].This raises an interesting possibility that the dark and ordinary matter may have a special relation although their properties are very different.For example, the dark matter relic density may be an asymmetry between the dark matter and antimatter  since the ordinary matter exists as a baryon asymmetry.In particular, the asymmetric dark matter can be well motivated in the mirror universe models .
In this paper we shall propose a novel mirror universe model to give the active and sterile neutrino masses as well as the baryonic and dark matter asymmetries.In addition to the SU (3) c × SU (2) L × U (1) Y ordinary sector and its SU (3) ′ c × SU (2) ′ L × U (1) ′ Y mirror partner, our model contains three gauge-singlet Majorana fermions and an [SU (2) L × SU (2) ′ L ]-bidoublet Higgs scalar.The mirror symmetry is softly broken to allow the symmetry breaking pattern in the mirror sector different from that in the ordinary sector.The mirror photon can become massive according to the mirror electromagnetic symmetry breaking.By integrating out the fermion singlets and the Higgs bidoublet, we can get the mirror sterile neutrino masses by canonical [7] seesaw, as well as the ordinary active neutrino masses by double [17] and linear [18] seesaw.In this canonical and double-linear seesaw scenario, two sterile neutrinos are above the MeV scale and the other one is at the eV scale, so that their existence can fulfill the cosmological and short baseline neutrino oscillation data [5].Our model can realize a leptogensis [94][95][96][97][98][99][100][101][102][103][104] as a common origin of the ordinary and dark matter.Specifically, the decays of the fermion singlets can simultaneously generate a desired lepton asymmetry in the [SU (2) L ]-doublet leptons and an equal lepton asymmetry in the [SU (2) ′ L ]-doublet leptons even if we do not resort to the fine tuned resonant effect [96].Through the sphaleron-induced lepton-to-baryon conversion [105], we can obtain an ordinary baryon asymmetry and an equal mirror baryon asymmetry to account for the number densities of ordinary and dark matter.The lightest mirror baryon then should have a determined mass around 5 GeV to explain the ratio between the ordinary and dark matter energy densities.In the presence of a U (1) kinetic mixing, the dark matter particle can be verified in the ongoing and future dark matter direct detection experiments.
Our model shares some ideas of Ref. [37], where the authors introduced two [SU (2)]-triplet Higgs scalars to generate the active neutrino masses by type-II [8] and inverse [19] seesaw.They also assumed all of the mirror neutrinos above the MeV scale.Furthermore, they res-onantly enhanced the CP asymmetries in the decays of the fermion singlets to produce the required lepton and baryon asymmetries.

II. THE MODEL
There are two Higgs scalars in both of the ordinary and dark sectors, Here and thereafter the mirror fields are denoted by a prime on a symbol, and hence the parentheses following the ordinary fields are the quantum numbers under the Y gauge group, while the parentheses following the mirror fields are the quantum numbers under the ′ Y gauge group.Our model also contains three families of ordinary and mirror fermions, ) , with the family indices being omitted for simplicity.We further introduce three gauge-singlet fermions [42,43], and an [SU (2) L × SU (2) ′ L ]-bidoublet scalar [40,42], where the first and second parentheses are the quantum numbers under the G and G ′ gauge groups, respectively.Besides the gauge symmetries and the mirror discrete symmetry, we impose a U (1) G × U (1) ′ G global symmetry under which only the SU (2) doublets and the SU (2) L × SU (2) ′ L bidoublet are nontrivial: (1, 0) for the SU (2) L doublets, (0, 1) for the SU (2) ′ L doublets and (1, 1) for the SU (2) L × SU (2) ′ L doublet.

A. Interactions
For simplicity, we only write down the following terms of the full Lagrangian, where B µν and B ′ µν are the U (1) Y and U (1) ′ Y field strength tensors.Note the other gauge-invariant trilinear couplings have been forbidden by the As a result of the mirror symmetry, the Yukawa couplings of the [SU (2) L × SU (2) ′ L ]-bidoublet scalar to the [SU (2)]-doublet leptons should have a symmetric structure, Furthermore, the mass term of the gauge-singlet fermions and the trilinear coupling of the [SU (2) L × SU (2) ′ L ]bidoublet scalar to the [SU (2)]-doublet scalars softly break both of the ordinary and dark lepton numbers.Without loss of generality and for convenience, we will choose the base where the gauge-singlet fermions have a diagonal and real mass matrix, to define three Majorana fermions, Meanwhile, the [SU (2) L × SU (2) ′ L ]-bidoublet scalar can have a real cubic coupling to the [SU (2) L ]-doublet scalars by a proper phase rotation, i.e.

B. Vacuum expectation values
We allow the discrete mirror symmetry and the global G symmetry to be softly broken by the quadratic terms in the full scalar potential, which is not shown for simplicity.So, the mirror Higgs scalars can develop the vacuum expectation values (VEVs) different from those of the ordinary Higgs scalars.In particular, the charged components of the mirror Higgs scalars can have the nonzero VEVs [106,107].In this case, the symmetry breaking pattern should be Accordingly, the mirror photon can become massive although the ordinary photon keeps massless.The VEVs of the [SU (2) L ]-doublet Higgs scalars φ u and φ d should be fixed by [106] while the VEVs of the [SU (2) ′ L ]-doublet Higgs scalars φ ′ u and φ ′ d can be described by [106] φ The [SU (2) L ×SU (2) ′ L ]-bidoublet Higgs scalar Σ can pick up the VEV through its cubic coupling to the [SU (2)]doublet Higgs scalars φ u and φ ′ u , Clearly, the VEV Σ can be much smaller than the VEVs φ 0 u and φ ′0 u in the seesaw scenario, i.e.

C. Mirror photon
We can make a non-unitary transformation [108], to remove the U (1) Y × U (1) ′ Y kinetic mixing and then define the orthogonal fields, Here θ W with sin 2 θ W ≃ 0.231 is the Weinberg angle while W 3 µ and W ′3 µ are the SU (2) L and SU (2) ′ L gauge fields.In the above orthogonal base, the field A µ is exactly massless and is the physical mass-eigenstate field, the ordinary photon, according to the unbroken electromagnetic symmetry U (1) em in the ordinary sector, while the others Z µ , Z ′ µ and A ′ µ will mix together.The mirror electromagnetic symmetry U (1) ′ em is broken by the charged VEV φ + d given in Eq. (12).Consequently, the W ′± boson will also mix with the Z ′ boson and the mirror photon A ′ µ , which is massive now.The mirror photon can couple to the ordinary fermions besides the mirror fermions, In the case with φ ′+ d = O(100 MeV), the mirror photon can have a mass and its decay width will not be smaller than Here α = e 2 /(4π) ≃ 1/137 [109] is the fine structure constant.

III. SEESAW FOR ACTIVE AND STERILE NEUTRINO MASSES
From Eq. ( 5), the ordinary active neutrinos, the mirror sterile neutrinos and the gauge-singlet fermions will have the mass matrix as below, after the ordinary and mirror electroweak symmetry breaking.

A. Active and sterile neutrino masses and mixing
As long as the gauge-singlet fermions are heavy enough, i.e.
we can make use of the seesaw mechanism to get the mass matrix of the active and sterile neutrinos, Here the mass eigenvalues have been introduced, If the entries in the mass matrix (22a) have the following hierarchy, the sterile neutrino masses should have a form of the canonical seesaw, while the active neutrino masses should have a form of the double and linear seesaw, Under the seesaw condition (24), the active mixing matrix U ν and the sterile mixing matrix U ν ′ can approximate to the Pontecorvo-Maki-Nakagawa-Sakata [110] (PMNS) matrices, while the active-sterile mixing matrices U νν ′ and U ν ′ ν can be simplified by

B. Sterile neutrino decays
Due to their mixing with the ordinary neutrinos, the sterile neutrinos can decay into the ordinary fermions [111], if the kinematics is allowed.Here G F = 1.16637 × 10 −5 GeV −2 is the Fermi constant, s 2 W = sin 2 θ W is the Weinberg angle, while V ud ≃ 0.97419 is an element of the Cabibbo-Kobayashi-Maskawa matrix [109].

IV. LEPTOGENESIS FOR ORDINARY AND MIRROR BARYON ASYMMETRIES
If CP is not conserved, the decays of the heavy Majorana fermions N i can simultaneously generate two types of lepton asymmetries: one is stored in the [SU (2) L ]doublet leptons l L , i.e.
and the other is stored in the mirror leptons l ′ L , i.e.
Here n l L and n l ′ L are the number densities while s is the entropy density.The relevant diagrams are shown in Fig. 1.
The SU (2) L sphaleron processes [105] then will partially transfer the ordinary lepton asymmetry to an ordinary baryon asymmetry, Similarly, the mirror lepton asymmetry will be partially converted to a mirror baryon asymmetry, through the SU (2) ′ L sphaleron processes [105].Due to the Yukawa couplings in Eq. ( 5), the ordinary lepton asymmetry and then the ordinary baryon asymmetry should equal to the mirror ones, i.e.
Here ε N i is the CP asymmetry in the decays of the heavy Majorana fermions N i .

A. CP violation in decays
The total decay width in the decays of the heavy Majorana fermions N i can be calculated at tree level, We then can compute the CP asymmetry ε N i appeared in Eq. ( 34) at one-loop level, with S(x) being the self-energy correction, V (x) and Ṽ (x, y) being the vertex corrections, In the limit M 2 , M 2 Σ , the CP asymmetry ε N i can be simplified as As we will show later the dark matter relic density and the BNN constraint enforce with m max ν ′ being the maximal mass eigenvalue of the mirror neutrinos.So, the simplified CP asymmetry (38) can have an upper bound, which is similar to the Davidson-Ibarra bound [98] in the canonical seesaw scenario.

B. Scattering processes
The Majorana fermions N i and the Higgs bidoublet Σ can mediate some lepton-number-violating scattering processes such as L φ u and so on.These scattering processes will not be kept in equilibrium below the temperature T D given by [112] [ Here Γ S is the interaction rate of the scattering processes, while is the Hubble constant with M Pl ≃ 1.22×10 19 GeV being the Planck mass and g * = 2 × (106.75 + 2) = 217.5 being the relativistic degrees of freedom.Below the masses of the mediators N i and Σ, the interaction rate can be given by [113] Γ S = 2 π 3 which tends to for the constraint (39).Alternatively, the scattering processes can decouple at a temperature above or near the mediator's mass if the interactions are weak enough to satisfy [112] K

C. Final baryon asymmetries
In the case the lightest Majorana fermion N has a mass smaller than the decouple temperature of the scattering processes mediated by the other Majorana fermions N 2,3 and the Higgs bidoublet Σ, i.e.
the final baryon asymmetries can be approximately solved by [112] V. IMPLICATIONS AND CONSTRAINTS Before giving the concrete parameter choice, we shall demonstrate some general implications and constraints on the model.

A. Dark matter mass
From the Yukawa couplings in Eq. ( 5), we can easily read the relation between the ordinary and mirror charged fermion masses, As a result of the mirror symmetry, the ordinary and mirror strong coupling constants should become equal at sufficiently high scales.Therefore, the beta functions of the ordinary and mirror QCD can govern the dependence of the mirror hadronic scale on the ordinary one [37], where we have defined In the ordinary sector, the current quark masses m u and m d are much smaller than the hadronic scale Λ QCD so that they can only have a negligible contribution to the nucleon masses, In addition, the ∆ baryons and the neutron has a mass split from the hyperfine interaction among the constituent quarks [37], with m q ≃ 300 MeV being the constituent quark mass.In the mirror sector, the quark masses m u ′ and m d ′ may be larger than the hadronic scale Λ QCD ′ .The mirror proton and neutron masses then can approximately equal to the sum of the mirror quark masses, which implies In the case (54b), the mirror ∆ ′− baryon can be lighter than the mirror neutron if the mirror hyperfine interaction doesn't compensate the mass difference m u ′ − m d ′ [37], As the lightest mirror baryon is expected to serve as the dark matter particle, its mass should be to explain the cosmological observations,

B. Dark matter direct detection
In the presence of the U (1) Y × U (1) ′ Y kinetic mixing, the mirror photon can mediate a scattering of the dark matter particle off the ordinary nucleons.For example, the mirror proton p ′ or the mirror ∆ ′− baryon has a spinindependent cross section, which can be close to the XENON10 limit [114].Here X denotes the mirror proton p ′ or the mirror ∆ ′− baryon, Z and A − Z are the numbers of proton and neutron within the target nucleus, while is the reduced mass.Alternatively, the mirror neutron n ′ can serve as the dark matter particle if it is the lightest mirror baryon.The mirror neutron as the dark matter particle can have an energy-dependent cross section.The detailed studies can be found in [37].

C. Constraints on sterile neutrinos and mirror photon
The Big-Bang Nucleosynthesis (BBN) stringently restricts the existence of the new relativistic degrees of freedom.The constraint on the new degrees of freedom is conventionally quoted as the effective number of additional light neutrinos.The latest Planck 2013 results show N ef f = 3.30 ± 0.27 [115].So, one light sterile neutrino can be allowed at 3 σ level.A very recent analysis [5,6] on the neutrino oscillation data also hint at the existence of an additional neutrino with an eV-scale mass.This means the other two sterile neutrinos should have the masses heavier than a few MeV and have a lifetime shorter than 1 second.From Eqs. ( 28) and ( 29), we hence can put The mirror photon should also satisfy the BBN constraint.From Eq. ( 19), it is easy to see Clearly, the mirror photon A ′ with a mass m A ′ = 100 MeV can have a lifetime shorter than 1 second if we take ǫ > 4 × 10 −11 .Currently, the measurement on the muon magnetic moment constrains ǫ 2 cos 2 θ W cos 2 θ ′ W < 2 × 10 −5 for m A ′ = 100 MeV [116].
Furthermore, the active-sterile neutrino mass matrix (22b) should be constrained by the neutrinoless double beta decay experiments.In the regime of m max ν ′ 100 MeV, we can perform [117]

D. Leptogenesis scale
From Eqs. ( 40) and ( 47), the CP asymmetry |ε N 1 | should be bigger than to explain the observed baryon asymmetry.Accordingly, we can have a low limit on the leptogensis scale, which is expected to be below the critical temperature (44),

VI. AN EXAMPLE OF PARAMETER CHOICE
As an example, let us set to give the mirror charged fermion masses [109], We hence can determine the mirror hadronic scale (49), and then the mirror baryon masses ( 53) and ( 55 The Yukawa couplings in the sterile neutrino masses (25) can be parameterized by [118] with Ω being an arbitrary orthogonal matrix.By taking the masses of the fermion singlets, So, the final baryon asymmetry (47) can explain the observation [59], Y mirror partner.In our model, the mirror sterile neutrino masses can have a form of the canonical seesaw, while the ordinary active neutrino masses can have a form of the double and linear seesaw.The mixing between the active and sterile neutrinos can also be seesaw-suppressed.Two sterile neutrinos can be above the MeV scale to avoid the BBN constraint, while the third sterile neutrino can be at the eV scale to fit the short baseline neutrino oscillation data.An ordinary lepton asymmetry and an equal mirror lepton asymmetry can be simultaneously produced from the decays of the fermion singlets.The baryonic and dark matter asymmetries then can equal each other since the ordinary and mirror sphaleron processes have a same efficiency of lepton-to-baryon conversion.Consequently, the lightest mirror baryon should have a mass around 5 GeV to serve as the dark matter particle.The U (1) Y and U (1) ′ Y kinetic mixing can mediate a testable dark matter scattering.